THREE ESSAYS ON UNBALANCED PANEL DATA
MODELS

By
Do Won Kwak

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY

Economics
2011

Abstract
THREE ESSAYS ON UNBALANCED PANEL DATA MODELS
By
Do Won Kwak

This dissertation consists of three chapters on panel data models. The ﬁrst chapter examines unconditional logit and conditional logit, and pooled correlated random eﬀect (CRE)
logit approaches in binary panel data models with highly dependent data. Simulation results
show that, ﬁrst, the conditional logit method is not robust to violation of the conditional
independence (CI) assumption. The magnitude of bias for the model coeﬃcient is greater
for the data with smaller time dimensions and higher serial correlations. Second, we ﬁnd
no signiﬁcant ﬁnite sample biases for average partial eﬀects (APEs) and associated rejection
frequency of CRE logit in the presence of high serial correlation under correct speciﬁcation
of unobserved heterogeneity. Finally, we quantify two sources of bias into the part due to
unobserved heterogeneity and the part due to serial correlation by using unconditional logit
and conditional logit methods. Finite sample biases by both sources are important in binary panel models with highly persistent outcome and correlated individual ﬁxed eﬀects.
As an empirical example, we apply conditional logit and pooled CRE logit to the Survey
of Income and Program Participation (SIPP) data where welfare participation exhibits high
persistence. The results imply that it is important to account for both state dependence and
unobserved heterogeneity simultaneously.

The second chapter introduces two formal tests of the missing completely at random
(MCAR) assumption in unbalanced panel data. In a Monte Carlo (MC) simulation, we provide the evidence of the substantial powers of tests. Inverse probability weighted (IPW) estimator and multiple imputation-based (MI) estimator under the missing at random (MAR)
assumption are studied as methods for missing data when the MCAR assumption is violated. We suggest combining MI and IPW methods such that the MI method is applied to
non-monotone missing data and the IPW method is applied to monotone missing data sequentially. Proposed tests and the MI-IPW method for missing data are applied to estimate
the eﬀect of class size reduction (CSR) on student scores for grades in K-3 using Project
STAR. The result of empirical application shows the violation of the MCAR assumption and
the MI-IPW estimates for the eﬀects of CSR on student achievement scores in grades K-3
are about 4.5 to 6 percent while unweighted estimates are about 6 to 7.5 percent.
In the last chapter, we extend the robust inference method with heterogenous variance and
autocorrelation in Hansen (2007) to unbalanced panel data with large time dimension. With
the homogeneity assumption and random attrition, we derive the robust inference result
using weighted least squares (WLS) for unbalanced panel data. We show that, our inference
method with WLS in unbalanced panel data provides conservative t-test results in Bakirov
and Szekely (2006) in the presence of attrition or heteroskedastic variance. We study the size
and the power of a proposed inference method in unbalanced panel data using the Monte
Carlo simulation. A MC simulation reveals that a proposed WLS method reduces the size
distortion and improves power over the methods of Ibragimov and Muller (2009) and Hansen
(2007).

ACKNOWLEDGMENTS
I would never been able to ﬁnish my dissertation without guidance, suggestions and support from several people. First, my deepest appreciation goes to my advisor, Jeﬀrey M.
Wooldridge. He was readily available at every stage of the dissertation process and provided me his perspective and helpful comments all the time. I could never have reached the
depths of this dissertation without his mentorship and insightful advice. Besides my advisor, I would like to sincerely thank the faculty members in Michigan State University, Todd
Elder, Peter Schmidt, Timothy Vogelsang, and Tapabrata Maiti. Whenever I needed advice
about my research, they helped make this dissertation a better work with their insights and
suggestions. I also have been very fortunate to have conducted some part of my dissertation
research at StataCorp as an intern. Simulations in my dissertation would not have been
developed as it is now without computing and data expertise I had learnt at StataCorp. I
would like to extend a special thanks to David Drukker and Rafal Raciborski.
I would like to thanks to my fellow graduate students. Jeﬀ Brown, Cheol-Keun Cho,
Dooyeon Cho, Junghwan Hyun, Myoung-Jin Keay, Jongduk Kim, Jinyoung Lee, Cuicui Lu,
Suhyeon Nam, Iraj Rahmani, Seunghwa Rho, Shengwu Shang, Wei-Siang Wang, and Yali
Wang had numerous discussions with me and I appreciate for their thoughtful comments
and enjoyable conversation.
I would like to express my gratitude to my parents and younger brother for their unwavering love and support. Last but not least, I would like to thank my wife who was always
there cheering me up and stood by me through the good and bad times.

iv

Contents

List of Tables

viii

List of Figures

xii

1

The robustness of conditional logit estimator in a binary panel data model
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Ideal conditions for conditional logit estimator . . . . . . . . . . . .
1.3 DGP for correlated logistic errors . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Correlated errors from multivariate logistic distribution . . . . . . . .
1.4 Monte Carlo experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 The procedures of Data Generation Process: a continuous covariate .
1.4.2 The bias and size distortion for coeﬃcient: a continuous covariate . .
1.4.3 The bias for APEs and rejection frequency: a continuous covariate . .
1.4.4 DGP: A model with both binary and continuous covariates . . . . . .
1.4.5 The bias and size distortion for δ: binary covariate . . . . . . . . . .
1.4.6 The bias for APEs and rejection frequency: binary covariate . . . . .
1.5 Panel data models for married women’s welfare participation . . . . . . . .
1.5.1 Static panel data model for married women’s welfare participation . .
1.5.2 Dynamic Unobserved eﬀects logit models under strict exogeneity . .
1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
5
6
7
8
11
12
16
21
27
28
30
32
34
39
44

2

The MI-IPW method in an unbalanced panel data model: An empirical
application on the eﬀect of class size reduction on SAT scores for students
in grades K-3
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Linear panel data model with missing and noncompliance . . . . . . . . . . .
2.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Tests of missing completely at random . . . . . . . . . . . . . . . .
2.2.3 Implications from the rejection of tests . . . . . . . . . . . . . . . .
2.3 Formal tests of MCAR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Hausman-type test I - Comparison of θpls and θF E using E(ci ,sit xit )=
0 as null . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45
45
52
52
55
57
57

v

58

2.3.2
2.3.3

2.4

2.5

2.6

2.7
2.8
3

Monte Carlo experiment I . . . . . . . . . . . . . . . . . . . . . . . .
Hausman-type test II - Comparison of θF E and θF D , Null: strict
exogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Monte Carlo experiment: Test of MCAR using strict exogeneity assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.5 Variable addition test of strict exogeneity . . . . . . . . . . . . . . .
Estimations with MAR assumption: IPW and MI-IPW methods . . . . . .
2.4.1 IPW method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 AIPW estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.3 Multiple imputation . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 MI-IPW method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Monte Carlo experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 DGP I: Correct speciﬁcation of both outcome and probability of selection models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.2 DGP II: The correct probability of selection model with endogenous
treatment eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.3 The robustness of IPW estimator to misspeciﬁcation of the probability
of selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.4
The robustness of ﬁrst-diﬀerencing estimator with small withinvariation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Empirical application: The return to class size reduction (CSR) on SAT score
2.6.1 Background information on Project STAR experiment of CSR . . . .
2.6.2 Evidence of experimental violation . . . . . . . . . . . . . . . . . . .
2.6.3 The impact of CSR on SAT score for grades in K-3 . . . . . . . . . .
2.6.4 Test of strict exogeneity assumption . . . . . . . . . . . . . . . . . .
2.6.5 Estimation with inverse probability weighted(IPW) and multiple imputation(MI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.6 IPW estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.6.7 MI-IPW method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Direction (sign) of the bias for unweighted estimators . . . . . . . . . . . . .
2.7.1 Bound for the eﬀects of small class . . . . . . . . . . . . . . . . . . .
conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Cluster robust inference in unbalanced panel data when T is large
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Robust inference with t-test in unbalanced panel data . . . . . . . .
3.2.2 WLS with heterogenous covariance matrix in unbalanced panel data
3.3 Monte Carlo experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 The t-test using t∗ -statistic with scaled factor under homegeniety
wls
and attrition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

60
66
67
69
74
75
79
81
83
84
84
92
97
108
115
116
117
127
134
137
140
151
155
155
159
160
160
164
166
171
172
173

The t-test using t∗ -statistic with scaled factor under heterogenous
wls
variance and attrition . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.3.3 Trade-oﬀ calculation: Dropping observation to make balanced panel
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

3.3.2

3.4

APPENDICES

189

A Gaussian copula density

189

B Proof for the consistency of conditional logit estimator in chapter 1

191

C Monte Carlo simulation results: Coeﬃcient β for continuous covariate

193

D Monte Carlo simulation results: APEs for continuous covariate

204

E Monte Carlo simulation results: Coeﬃcient δ for binary covariate

217

F Monte Carlo simulation results: APEs for binary covariate

222

G Proof of Hausman-type test in chapter 2

228

H Proof of Corollary 3 in Chapter 3: Robust inference in unbalanced panel
data
232
I

Monte Carlo simulation results in chapter 3: Size-adjusted power for t-test
in Ibragimov and Muller [2009]
236

vii

List of Tables
1.1

Basic Descriptive Statistics for the Generated Errors . . . . . . . . . . . . .

14

1.2

Pairwise correlation of the generated errors for ρ = 0.1 . . . . . . . . . . . .

14

1.3

Pairwise correlation of the generated errors for ρ = 0.5 . . . . . . . . . . . .

15

1.4

Pairwise correlation of the generated errors for ρ = 0.9 . . . . . . . . . . . .

15

1.5

Decomposition of bias for β for n = 100

. . . . . . . . . . . . . . . . . . . .

21

1.6

Decomposition of bias for β for n = 200

. . . . . . . . . . . . . . . . . . . .

22

1.7

The bias for β due to serial correlation using conditional logit for n = 200

.

22

1.8

Decomposition of bias into two parts using unconditional logit for n = 200 .

26

1.9

The bias of unconditional logit n = 200

. . . . . . . . . . . . . . . . . . . .

26

1.10 Decomposition of bias into two parts n = 100 . . . . . . . . . . . . . . . . .

29

1.11 Decomposition of bias using unconditional logit for n = 100

. . . . . . . . .

32

1.12 Basic statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

1.13 Logit estimation for married women’s welfare participation . . . . . . . . . .

37

1.14 Probit estimation for married women’s welfare participation . . . . . . . . .

38

1.15 Probit estimation of dynamic model for married women’s welfare participation 43
2.1

Hausman-type test I: Null and Alternative (cov(sit dit , ci ) = 0)

. . . . . . .

62

2.2

Hausman-type test I: Null and Alternative (cov(sit dit , ci ) = 0)

. . . . . . .

62

2.3

Pooled LS and FE estimates for α and p-value for Wa

. . . . . . . . . . . .

64

2.4

RE and FE estimates for α and p-value for Wa

. . . . . . . . . . . . . . . .

65

2.5

FD and FE estimates for α and p-value for Wb

. . . . . . . . . . . . . . . .

70

2.6

Test of strict exogeneity variable addition test and Hausman test for Wb

. .

73

2.7

Speciﬁcation of models for missing variables . . . . . . . . . . . . . . . . . .

87

2.8

DGP I: MTE estimates for unweighted estimators . . . . . . . . . . . . . . .

89

2.9

DGP I: MTE estimates for IPW estimators

90

viii

. . . . . . . . . . . . . . . . . .

2.10 DGP I: MTE for estimators with multiple imputations . . . . . . . . . . . .

91

2.11 DGP II: MTE for unweighted estimators . . . . . . . . . . . . . . . . . . . .

94

2.12 DGP II: MTE estimates for IPW estimators . . . . . . . . . . . . . . . . . .

95

2.13 DGP II: MTE estimates for estimators with multiple imputations . . . . . .

96

2.14 Selection model misspeciﬁcation: MTE for unweighted estimators, δ=0 . . . 100
2.15 Selection model misspeciﬁcation: MTE for IPW estimators, δ=0

. . . . . . 101

2.16 Selection model misspeciﬁcation: MTE for unweighted estimators, δ=.2
2.17 Selection model misspeciﬁcation: MTE for IPW estimators, δ=.2

. . . . . . 103

2.18 Selection model misspeciﬁcation: MTE for unweighted estimators, δ=.5
2.19 Selection model misspeciﬁcation: MTE for IPW estimators, δ=.5

. . 102

. . 104

. . . . . . 105

2.20 Selection model misspeciﬁcation: MTE for unweighted estimators, δ=1.0 . . 106
2.21 Selection model misspeciﬁcation: MTE for IPW estimators, δ=1.0

. . . . . 107

2.22 Sensitivity analysis to small within-variation: MTE with full within-variation 112
2.23 Sensitivity analysis: MTE with within-variation of 50% sample only

. . . . 113

2.24 Sensitivity analysis: MTE with within-variation of 15% sample only

. . . . 114

2.25 Students in each class types for grades in K-3 . . . . . . . . . . . . . . . . . 118
2.26 Attrition and reassignments of class types (noncompliance)

. . . . . . . . . 121

2.27 SAT percentile scores at t − 1 for Leavers and stayers . . . . . . . . . . . . . 122
2.28 Data availability for covariates . . . . . . . . . . . . . . . . . . . . . . . . . . 124
2.29 Data availability by selection indicator . . . . . . . . . . . . . . . . . . . . . 125
2.30 Basic statistics: The mean and standard deviation

. . . . . . . . . . . . . . 126

2.31 Cross-section unweighted LS and reduced-form estimates . . . . . . . . . . . 130
2.32 Unweighted pooled LS and reduced-form . . . . . . . . . . . . . . . . . . . . 132
2.33 Unweighted FD estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.34 Hausman-type test II: strict exogeneity . . . . . . . . . . . . . . . . . . . . . 135
2.35 Variable addition test: strict exogeneity . . . . . . . . . . . . . . . . . . . . . 136
2.36 Cross-section IPW LS and IPW reduced-form estimation . . . . . . . . . . . 145

ix

2.37 IPW pooled LS, pooled IV

. . . . . . . . . . . . . . . . . . . . . . . . . . . 149

2.38 Weighted FD estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
2.39 Covariates and distributional assumption used for MICE . . . . . . . . . . . 152
2.40 MI-IPW pooled LS, pooled IV

. . . . . . . . . . . . . . . . . . . . . . . . . 154

2.41 MI-IPW FD estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
2.42 corr(vit , sit+j ) ∀ j≥ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
2.43 Upper bound estimates for cross-section IV estimator . . . . . . . . . . . . . 158
3.1

selection process

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.2

The size of test, 1(p < 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

3.3

The size adjusted power of test . . . . . . . . . . . . . . . . . . . . . . . . . 179

3.4

The size adjusted power of test, 1(p < 0.05), T = 100 . . . . . . . . . . . . . 180

3.5

Conﬁguration of heteroskedastic variance . . . . . . . . . . . . . . . . . . . . 181

3.6

The size of test, 1(p < 0.05) . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

3.7

The size adjusted power of test, 1(p < 0.05), T = 100 . . . . . . . . . . . . . 184

3.8

The size of test, 1(p < 0.05) where AR(1) error with correlation coeﬃcient 0.75 186

3.9

The size adjusted power of test, 1(p < 0.05), T = 50

. . . . . . . . . . . . . 187

C.1 Unconditional logit estimates for β with no serial correlation . . . . . . . . . 194
C.2 Conditional logit estimates for β with no serial correlation . . . . . . . . . . 195
C.3 Unconditional logit estimates for β with serial correlation . . . . . . . . . . . 196
C.4 Conditional logit estimates for β with serial correlation . . . . . . . . . . . . 197
C.5 Unconditional logit estimates for β with serial correlation . . . . . . . . . . . 198
C.6 Conditional logit estimates for β with serial correlation . . . . . . . . . . . . 199
C.7 Unconditional logit estimates for β with serial correlation . . . . . . . . . . . 200
C.8 Conditional logit estimates for β with serial correlation . . . . . . . . . . . . 201
C.9 Unconditional logit estimates for β with serial correlation . . . . . . . . . . . 202
C.10 Conditional logit estimates for β with serial correlation . . . . . . . . . . . . 203

x

D.1 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 205

D.2 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 206

D.3 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 207

D.4 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 208

D.5 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 209

D.6 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 210

D.7 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 211

D.8 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 212

D.9 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 213

D.10 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 214

D.11 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 215

D.12 LPM, unconditional logit, CRE logit APE estimates for β

. . . . . . . . . . 216

E.1 Unconditional logit for δ with low serial correlation for n = 100 . . . . . . . 218
E.2 Unconditional logit for δ with high serial correlation for n = 100 . . . . . . . 219
E.3 Conditional logit for δ with low serial correlation for n = 100 . . . . . . . . . 220
E.4 Conditional logit for δ with high serial correlation for n = 100 . . . . . . . . 221
F.1 LPM, unconditional logit, CRE logit APE estimates for δ . . . . . . . . . . . 223
F.2 LPM, unconditional logit, CRE logit APE estimates for δ . . . . . . . . . . . 224
F.3 LPM, unconditional logit, CRE logit APE estimates for δ . . . . . . . . . . . 225
F.4 LPM, unconditional logit, CRE logit APE estimates for δ . . . . . . . . . . . 226
F.5 LPM, unconditional logit, CRE logit APE estimates for δ . . . . . . . . . . . 227
I.1

The size adjusted power of test, 1(p < 0.05), T = 100 . . . . . . . . . . . . . 237

I.2

The size adjusted power of test, 1(p < 0.05), T = 300 . . . . . . . . . . . . . 238

I.3

The size adjusted power of test, 1(p < 0.05) , T = 1, 000 . . . . . . . . . . . 239

xi

List of Figures

1.1

Bias for β with continuous covariate . . . . . . . . . . . . . . . . . . . . . . .

17

1.2

Bias of unconditional and conditional logit for β with continuous covariate .

18

1.3

Bias of unconditional logit and CRE-logit estimators for APEs . . . . . . . .

24

1.4

Bias of unconditional and conditional logit estimators for β with binary covariate 28

1.5

Unconditional logit and CRE-logit estimates of APEs for binary covariate . .

2.1

Bias of unweighted and IPW LS estimator . . . . . . . . . . . . . . . . . . . 109

2.2

Bias of unweighted and IPW FD estimator to small within variation . . . . . 115

2.3

Estimated πit for t = 1, 2, 3

2.4

Estimated πit conditional on sit = 0 or sit = 1 for t = 1, 2, 3 . . . . . . . . . 144

2.5

Estimated πit and pit for t = 1, 2, 3 with attrition sample . . . . . . . . . . . 147

31

. . . . . . . . . . . . . . . . . . . . . . . . . . . 143

xii

Chapter 1
The robustness of conditional logit
estimator in a binary panel data
model

1.1

Introduction

Standard maximum likelihood estimators are usually inconsistent in binary panel data models with individual ﬁxed eﬀects when individual ﬁxed eﬀects are unspeciﬁed with a large
number in the cross-section (N) and a ﬁxed number of time (T). The problem arises since
there are so many nuisance parameters (individual ﬁxed eﬀects) to estimate while the time
dimension is ﬁxed.1 Fixed-T identiﬁcation diﬃculty for micro panel can be overcome by using
1 This

situation is called incidental parameters problem. The references on the incidental
parameters problem include Neyman and Scott [1948], Lancaster [2000] and Arellano and
Hahn [2007]. Individual ﬁxed eﬀects is also called as unobserved heterogeneity hereafter.

1

the conditional logit approach if the logistic distributional assumption is true.2 Chamberlain
[2010] showed that conditional logit delivers

√
n consistent estimation of the parameter of

interest if individual-and-period speciﬁc errors are independent over time and logistic, and
covariates are unbounded. However, these conditions for

√

n consistency of conditional logit

can be very restrictive in micro panel data applications. In particular, the independence
assumption is restrictive in the applications with highly persistent response variables such
as married women’s labor force participation (Chay and Hyslop [2001]), married women’s
welfare participation (Chay and Hyslop [2001]), the incidence of external debt crises in developing countries (Hajivassiliou and McFadden [1998]) and habitual smoking behavior (Collado
and Browning [2007]). Moreover, as ﬁxed-T consistency results for model coeﬃcients can not
be extended to APEs, it becomes a serious limitation in the applications where researchers
are interested in the eﬀect of a policy on outcome. As we do not make any distributional
assumption on heterogeneity (i.e. individual ﬁxed eﬀects), it is diﬃcult to ﬁgure out what to
plug in for unobserved heterogeneity when calculating the APEs. However, the conditional
logit approach is useful for the estimations of log odd ratio and relative eﬀects between
covariates.
In this paper, we study the robustness of ﬁxed-T consistency results for the conditional
logit estimator to the violation of conditional independence assumption using a Monte Carlo
(MC) experiment. Speciﬁcally, we introduce data generating processes (DGPs) which are
2

These types of conditional methods are not available in general non-linear models and
can not be extended to APEs estimation since they do not make any distributional assumption on unobserved heterogeneity. For general non-linear models, a recent literature focuses
rather on approximately unbiased estimator than on the estimators with no bias at all. Arellano and Hahn [2007], Hahn and Newey [2004], Arellano [2003], Fernandez-Val [2009] and
others provide bias correction methods for non-linear ﬁxed eﬀects panel data model. An
extensive review is provided in Arellano and Hahn [2007].

2

designed to satisfy ideal conditions for the conditional logit estimator except the conditional
independence (CI) assumption to focus on the eﬀects by the violation of CI assumption.
For simplicity, the violation of CI in DGPs is induced by correlated logistic errors of AR(1)
using a copula. We introduce a copula in simulation since it allows the separation of the
dependence structure over time from the marginal distribution for each time period. Thus,
by specifying marginal distribution of outcome for each time period and using a copula to
specify the dependence structure of marginal distributions of outcomes over time, we specify
joint non-normal (logistic) distribution for outcomes over time. In a MC experiment, using
DGPs with correlated logistic errors, we examine the performance of conditional logit, CRE
logit with pooled MLE, and unconditional logit methods.3 Finally, we quantify ﬁnite sample
bias due to unobserved heterogeneity and due to true state dependence using unconditional
logit and conditional logit estimators. Simulation with highly persistent binary panel data is
important for the conditional logit estimator as well as for approximately unbiased estimators
with bias correction since these methods require the conditional independence assumption
but the eﬀect of violation has not been studied much.
When interpreting the simulations results, we focus on ﬁnite sample bias for coeﬃcients β
and APEs in (1.1) while in DGPs, the focus is on the generation of correlated errors uit T
t=1
which are drawn from T-multivariate logistic distribution with serial correlation of AR(1).
3 Correlated

random eﬀect logit approach is based on parametric speciﬁcation of the distribution for unobserved heterogeneity as in Chamberlain [1980]. In the simulation and application, we use parametric speciﬁcation for unobserved heterogeneity as follows. D(ci |xi ) ∼
2
logistic(ψ + xi ξ, σa ). Unconditional logit approach is based on logit model with MLE where
it estimate unobserved heterogeneity(ci ) as parameters. For the properties of unconditional
logit see Arellano and Hahn [2007] and Andersen [1973]. For the case of T =2, Abrevaya
[1997] shows the bias of unconditional MLE is 100 %.

3

yit = 1[xit β + ci + uit ≥ 0], i = 1, 2, ..., n, and t = 1, 2, ..., T

(1.1)

where n is large, T is ﬁxed and T ≥2, xit is continuous or binary, ci is correlated unobserved
heterogeneity (with xit ) and uit is idiosyncratic error.
In simulation, the conditional logit estimator for β in (1.1) shows ﬁnite sample bias when
the CI assumption is violated by serial correlation among errors. Finite sample bias is
greater for the high ρ(serial correlation coeﬃcient) and small T (the size of time dimension).
The size distortion of inference (the bias of rejection frequency) with conditional logit is
very severe for the data with ρ ≥.4 and it increases with n (the size of the cross-section).
CRE logit with pooled MLE for APEs and the associated rejection frequency shows no
ﬁnite sample bias regardless of the magnitude of ρ. Unconditional logit exhibits the bias
due to both (inconsistent estimation of) unobserved heterogeneity and serial correlation.
For unconditional logit estimator β with DGPs of ρ ≤.4, more bias is due to unobserved
heterogeneity than due to serial correlation. For the unconditional logit estimator for APEs,
more bias is due to unobserved heterogeneity than to serial correlation for DGPs with ρ ≤.6.
We analyze married women’s welfare participation using the conditional logit model with
MLE and CRE logit model with pooled MLE for β and APEs with the data of 8 waves
for 1,934 married women, taken from Chay and Hyslop [2001]. Married women’s welfare
participation shows substantial serial persistence. This persistence could be due either to
certain individual characteristics or due to true state dependence.4 The estimation results
4 In

our example of welfare participation, true state dependence implies ”narcotic” eﬀect
in Chay and Hyslop [2001]. Once a person starts to depend on welfare, participation itself
change the person’s mental attitude and behavior so that leads the person to rely more on
welfare.

4

and a comparison of the estimates imply that much of the serial persistence of welfare
participation is explained by unobserved heterogeneity (54%) but true state dependence
(46%) is also an important source for serial persistence. Therefore, the empirical example
shows that proper control of true state dependence as well as unobserved heterogeneity is
very important in the model speciﬁcations of married women’s welfare participation.
The rest of the chapter is organized as follows. In section 2, we explicitly explain the
model and ideal assumptions for the conditional logit method and section 3 describes the
procedure of DGPs using copula to satisfy ideal conditions for the conditional logit model
except CI. Section 4 shows ﬁnite sample simulation results for conditional logit, CRE logit,
and unconditional logit methods. Section 5 provides an empirical application pertaining to
married women’s welfare participation and section 6 contains a brief summary of the results
and the conclusion.

1.2

Model

The conditional logit estimator allows to estimate β in (1.1) without making any assumption
on unobserved heterogeneity since the logit transformation allows to separate the estimation
of β from unobserved heterogeneity. (Andersen [1973], Chamberlain [1984] and Wooldridge
[2010]) However, this comes with imposing CI assumption, which is not necessary for other
competing estimators such as CRE logit with pooled MLE. In this section, we introduce
conditional logit model and conditions for its ideal performance.

5

1.2.1

Ideal conditions for conditional logit estimator

Assumption 1 Our baseline model has latent variable representation.

yit = 1[xit β + ci + uit ≥ 0]

∗
yit = xit β + ci + uit

where i = 1, 2, ...n, t = 1, 2, ..., T
ci ∼ logistic (ψ + xi ξ,

π2
)
6

2

xit is continuous and uit ∼ logistic(0, π )
6
Assumption 2 Conditional strict exogeneity (SE)

E(yit |xi , ci ) = E(yit |xit , ci ) for each t
P (yit = 1|xi , ci ) = P (yit = 1|xit , ci ) for each t
Assumption 3 (Functional form marginal distribution) Conditional mean of binary response yit is speciﬁed as

E(yit |xit , ci ) = P (yit = 1|xit , ci ) = Λ(xit β + ci ) for each t = 1, 2, 3, ...T
where Λ(xit β + ci ) =

exp(xit β + ci )
1 + exp(xit β + ci )

6

(i.e.) conditional marginal distribution of each yit is logistic distribution.
Assumption 4 Conditional Independence (CI)

T

D(yi1, yi2 , ......, yiT |xi , ci ) =

D(yit |xit , ci )
t=1

Assumption 5 Other regularity conditions for the consistency of β such as convex support
of β, and continuity and diﬀerentiability of objective function. We also assume random
sampling over cross-section.
Theorem We assume, under assumptions 1-5, conditional logit estimator for β is a consistent estimator.
Proof See appendix.
The proof in appendix shows that the transformation using the sum of response variables
as suﬃcient statistics eliminates unobserved heterogeneity. Obtained βc−logit estimator is
√
a consistent and n-asymptotically normal. Chamberlain [2010] also shows that the only
binary panel model that provides ﬁxed-T

√

n consistent estimators for β is logit model.

Remark 1 Proof explicitly shows that CI assumption is required to apply conditional MLE
for β. CI can be easily violated when response variables show state dependence.

1.3

DGP for correlated logistic errors

This section introduces the details of data generating processes(DGPs) in simulation for
studying the robustness of conditional logit to the violation of CI assumption. The key is to
7

generate variables that satisfy ideal conditions for conditional logit except CI.

1.3.1

Correlated errors from multivariate logistic distribution

We introduce dependence among errors using a copula and logistic distribution. In general,
specifying variables from multivariate logistic distribution can be very complicated as we have
to deﬁne joint distribution. However, using copula, we can draw variables from multivariate
logistic distribution by only specifying marginal distribution for each variable and copula
which specify intended dependence among marginal distribution of each variable separately.

Copula
Copula is a function which links univariate marginal CDFs to their multivariate CDF with
intended dependence among marginal distributions. Copula is useful for our purpose since
any T -dimensional joint distribution function can be decomposed into its T marginal distributions and a copula function which completely describes the dependence structure among
marginal distributions. For instance, in DGP, we use a copula to generate errors from a
multivariate logistic distribution for which each marginal error has standard logistic distribution with intended dependence among marginal distribution of errors. We specify only
conditional marginal distribution of zit for each time and use a Gaussian copula to specify
dependence among CDFs of zi1 , zi2 , ..., ziT . We choose a Gaussian copula since most statistical packages allow users to randomly draw a set of variables from joint normal distribution
with user-speciﬁed pair-wise covariance. Therefore, we specify the marginal distribution for
each zit and Gaussian copula for dependence among CDFs of zi1 , zi2 , ..., ziT as in (1.2).

8

Φ(z1 , z2 , ....., zT ) = C(Φ1 (z1 ), ..., Φt (zt ), ..., ΦT (zT ); ρ)

(1.2)

where Φ(·) is joint normal, Φt (·) is univariate normal of marginal distribution of each t, and
C(·, ρ) is copula function where ρ governs dependence among marginal distributions.
Then, we transform normal marginal to logistic marginal at each period as in (1.3) and
the dependence of multivariate normal by a Gaussian copula is inherited to the dependence
among logistic marginal CDFs since copula is invariant to increasing transformation. Thus,
after transforming of marginal distribution, we obtain (1.4).

→

Φt (zt )

Ft (ut )

(1.3)

transformation of marginal CDF

F (u1 , u2 , ....., uT ) = C(F1 (u1 ), ..., Ft (ut ), ..., FT (uT ); ρ)

(1.4)

where F (·) is joint logistic distribution and Ft (·) is univariate logistic marginal distribution.
Joint logistic density is derived using multivariate gaussian copula in the appendix.
Because of this invariance property of copula, dependence structure on zit T is maintained
t=1
in ui T after increasing transformation. Generated ui T follows joint logistic distribution
t=1
t=1
with correlation parameters ρ as in (1.2). Joint logistic density of f (u1 , u2 , ....., uT ) and its
derivation is provided in appendix.

Extension: Generate correlated variables from non-normal distribution Dependent variables generation using Gaussian copula can be extended to other non-normal multivariate distribution. For the method to work, the only requirement is to obtain the intended

9

marginal distribution from increasing transformation of the normal distribution. For instance, in our DGPs, each normal marginal CDF is transformed to logistic marginal CDF by
increasing transformation and the Gaussian copula speciﬁes the dependence structure among
the marginal distributions. Dependence structure is maintained after increasing transforming
by invariance property of copula.

Generation of a random vector from T -joint logistic distribution with dependence structure of AR(1) correlation
Consider a random vector z = (z1 , z2 , ......, zT ) which is from T -variate multivariate normal
distribution with AR(1) covariance matrix Cov(z) of (1.5) and zt ∼ N (0, 1) ∀ t. 5


ρ
 1


 ρ
1



.
.
Cov(z) =  .
.
.
 .


 ρT −2 ρT −3



ρT −1 ρT −2

· · · · · · ρT −1 

. 
. 
··· ···
. 


..
. · · · ρ2 



··· 1
ρ 



··· ···
1

(1.5)

Using gaussian copula in (1.6), we can express T -variate joint normal distribution as
follows. Let Φ be a joint normal CDF for z = (z1 , z2 , ......, zT ) with marginal CDF Φt for
each zt , then, by Sklar’s theorem in Sklar [1959], there exists unique copula C that connect
each marginal CDF Φ1 (z1 ), Φ2 (z2 ), ...., ΦT (zT ) ∈ RT with dependence structure of (1.5)
5 We

choose pairwise correlation of in (1.5) for simplicity. We have corr(Z)=cov(Z) since
zt is standard normal ∀t. The main advantage of using copula is the ﬂexibility of model
marginal distribution and dependence among marginal distribution separately. Moreover,
the speciﬁcation of dependence relationship among marginal CDFs is not limited to 2nd
moment or linear relationship but is allowed to nonlinear and tail dependence. However, in
the simulation, our focus is serial correlation among errors so that we use copula with only
linear relationship speciﬁcation for simplicity.

10

such that equation (1.6) is satisﬁed.

Φ(z1 , z2 , ...., zT ) = C(Φ1 (z1 ), ...., ΦT (zT ); Cov(z))

(1.6)

where Φt is normal CDF ∀t = 1, ..., T .
The transformation from a normal marginal distribution to a logistic marginal distribuΦ(z )

t
tion is performed by ut = Ft−1 (Φ(zt )) = ln( 1−Φ(z ) ) where Ft (·) is logistic CDF. We can
t

rewrite (1.6) as the equation (1.7) for a copula C and u = (u1 , u2 , ...., uT ) ∈ RT , such that
F (u1 , u2 , ...., uT ) = C(Ft (u1 ), ...., Ft (uT ); ρ), where F is joint logistic CDF.

(1.7)

Because of the invariance dependence property of copula for increasing transformation of
marginal distribution, {uit }, t = 1, 2, ..., T , the dependent structure of z in (1.6) is maintained
for u in (1.7) with the same C. 6 Lee [1979] also showed similar relationship between normal
and logistic distribution for bivariate case.

1.4

Monte Carlo experiment

We generate data based on two model speciﬁcations. First one is where the model has only
one covariate with unbounded support and individual ﬁxed eﬀects are correlated with the
covariate. Second model has two covariates of unbounded continuous covariate and binary
covariate and has individual ﬁxed eﬀects which are correlated with the binary covariate. In
both models, we generated dependent errors from joint logistic distribution using Gaussian
6

Linear dependence, (for instance, pairwise correlation), that we use here can not be
maintained after transformation of marginal distribution from normal to logistic but general
dependence measures such as Kendal’s rank correlation are maintained under monotonic
transformation such as from normal to logistic. ( Schweizer and Sklar [1983] and Trivedi and
Zimmer [2007])
11

copula. We perform simulation with binary covariate since conditional logit models with
MLE requires covariates to have unbounded supports for

√
n consistency result in Chamber-

lain [2010]. Thus, simulation with binary covariate provides information about how bounded
support of covariate aﬀects the property of conditional logit estimator.

1.4.1

The procedures of Data Generation Process: a continuous
covariate

We generate data {yit , xit , ci , uit } for t = 1, 2, .., T and i = 1, 2, ..., n based on model (1.1).
We follow Heckman [1981], Green [2004] and Fernandez-Val [2009] for the design of the DGPs
in this section. 7 We draw four items with 2,000 replications. The correlation across timedimension is induced through error components u using a copula as illustrated in previous
section 1.3.1. We generate four items as follow:
1. xit ∼ iid N (0, 1). We generate an univariate covariate from identical and independent
standard normal distribution.
2. Unobserved heterogeneity:
x + xi2 + ..... + xik
a
π2
T
√
+ √i where ai ∼ logistic(0, ) and k = [ ].(1.8)
D(ci |xi ) ∼ i1
3
2
2
2k
3. We generate each uit such that it has standard logistic marginal distribution (i.e.
2
uit ∼ log istic(0, π )), Cov(uit , uis ) = ρ|t−s| for t = s, and (u1 , u2 , ...., uT ) is drawn
3

from joint logistic distribution by using the procedure in section 1.3.1.
7 Devroye

[1986] and Johnson [1987] contain extensive methods of generating multivariate
random variables and vectors from non-normal distribution.

12

4. We generate yit as follow.

yit = 1[xit β + ci + uit ≥ 0] where β = 1 and using xit , ci , uit as in1, 2, 3.

(1.9)

Generation of {xit }T and ci is straightforward and generation of {yit }T is also straightt=1
t=1
forward once we have {uit }T . We use a copula in generating errors, (ui1 , ui2 , ...., uiT ) from
t=1
T -variate logistic distribution while each uit has a univariate logistic distribution and dependence structure among (ui1 , ui2 , ...., uiT ) is inherited from dependence structure of joint
normal distribution for (z1 , z2 , ......, zT ).

Generation of correlated logistic errors, u
Let z = (z1 , z2 , ......, zT ) is variables drawn from T -variate multivariate normal distribution
with dependent structure of AR(1) serial correlation. We obtain (u1 , u2 , ...., uT ) by the
increasing transformation of marginal distributions, ut = Ft−1 (Φt (zt )) where Ft is logistic
CDF and Φt is standard normal CDF.
u = (u1 , u2 , ...., uT ) is generated from the following algorithm:
(Step 1) Draw z1 , z2 , ...., zT from T -variate normal distribution with dependence structure
of cov(z) in (1.5).
(Step 2) Obtain each marginal CDF et = Φt (zt ) where Φt (·) is standard normal CDF, and
∀t = 1, ..., T .
(Step 3) Transform zt to ut based on following formula. ut = Ft−1 (et ) = Ft−1 (Φt (zt ))
2

e

Φ (z )

t
t t
where Ft (·) ∼ logistic(0, π ), (i.e. ut = ln( 1−e ) = ln( 1−Φ (z ) ))
3
t
t t

(Step 4) Repeat steps of (1), (2), and (3) for all i = 1, 2, ..., n.
13

Table 1.1. Basic Descriptive Statistics for the Generated Errors
u1
ρ = 0.1
ρ = 0.25
ρ = 0.5
ρ = 0.75
ρ = 0.9

u2

u3

u4

u5

u6

u7

u8

-.07(1.9)
-.03(1.8)
-.03(1.8)
-.02(1.8)
-.02(1.8)

.07(1.7)
.03(1.8)
.01(1.8)
-.01(1.8)
-.01(1.8)

.12(1.8)
.08(1.9)
.06(1.9)
.02(1.9)
.02(1.9)

.02(1.9)
.03(1.9)
.04(1.9)
.02(1.8)
.02(1.8)

-.04(1.8)
-.02(1.8)
.01(1.9)
.01(1.9)
.01(1.9)

-0.05(1.9)
-0.03(1.8)
-0.02(1.8)
-0.01(1.9)
-0.01(1.9)

.02(1.9)
.06(1.9)
.04(1.9)
.03(1.9)
.03(1.9)

.08(1.9)
.11(1.8)
.11(1.9)
.07(1.9)
.07(1.9)

Table 1.2. Pairwise correlation of the generated errors for ρ = 0.1
u1
u1
u2
u3
u4
u5
u6
u7
u8

u2

u3

u4

u5

u6

u7

u8

1
0.09
0.06
0.02
0.03
0.03
0.02
0.02

1
0.13
0.01
0.01
0.06
-0.05
-0.002

1
0.11
-0.02
0.01
0.01
0.02

1
0.11
0.02
0.05
0.07

1
0.11
0.01
0.04

1
0.09
0.02

1
0.08

1

Basic Statistics for correlated logistic errors

Table 1.1 reports the MC point es-

timates of 2,000 replications for the mean and the standard deviation of generated errors
from multivariate logistic distribution with n=100 and T =8. The estimates for standard
deviations are reported in parenthesis. Table 1.2, 1.3, and 1.4 report pairwise correlations
among generated errors for n=100, T =8 and ρ for 0.1, 0.5 and 0.9, respectively. Tables of
basic statistics verify that pairwise correlations of z are maintained to the pairwise correlations among u. Statistics for uit such as mean and standard deviation is equal to mean and
standard deviation of a variable from standard logistic distribution.

14

Table 1.3. Pairwise correlation of the generated errors for ρ = 0.5
u1
u1
u2
u3
u4
u5
u6
u7
u8

u2

u3

u4

u5

u6

u7

u8

1
0.51
0.30
0.14
0.07
-0.01
0.02
0.05

1
0.54
0.28
0.09
0.04
0.03
0.06

1
0.52
0.26
0.13
0.08
0.12

1
0.48
0.26
0.14
0.10

1
0.50
0.23
0.12

1
0.52
0.22

1
0.50

1

Table 1.4. Pairwise correlation of the generated errors for ρ = 0.9
u1
u1
u2
u3
u4
u5
u6
u7
u8

u2

u3

u4

u5

u6

u7

u8

1
0.90
0.80
0.72
0.67
0.59
0.53
0.48

1
0.90
0.80
0.74
0.66
0.59
0.54

1
0.90
0.81
0.73
0.65
0.59

1
0.90
0.82
0.73
0.67

1
0.91
0.81
0.74

1
0.89
0.81

1
0.91

1

15

1.4.2

The bias and size distortion for coeﬃcient: a continuous
covariate

All the results presented in this section are based on 2,000 replications and the robust
standard error of sandwich form that allows weakly dependent serial correlation. We vary
three components T , n, and ρ in the DGPs since it is important to examine how conditional
logit estimator for β behaves in ﬁnite sample as T , n, and ρ change. We are particularly
interested in typical micro panel such as small T , large n with high persistence(ρ > 0.4).
Throughout the tables in this section, we report the mean of the MC point estimate for β
(Mean), the standard deviation of the MC point estimate for β (SD), the mean of the MC
point estimate of the robust standard error for β (SE), the rejection frequency (1(pval <
0.05)), and 95-percent coverage rate for the rejection frequency.

Conditional logit for β
Figure 1.2 summarizes the estimation results for unconditional logit and conditional logit
estimators for coeﬃcient β. As the second panel of ﬁgure 1.2 shows, for ρ=0, conditional logit
estimator for β shows no bias and rejection frequency in hypothesis test with conditional
logit estimator for β shows no size distortion regardless of n and T . For ρ=0.2, the bias
of conditional logit estimator of β is less than 10% for T ≥4. The rejection frequency
reveals that a simulation with ρ=0.2 leads to over-rejection by 100% for n = 400. The
size distortion increases with n. For ρ=0.4, the bias of conditional logit estimator for β
is greater than 10% for ∀T ≤ 8. The rejection frequency reveals that a simulation with

16

Figure 1.1. Bias for β with continuous covariate

(Note) x-axis: time dimension, y-axis: estimated β. True β is 1. Each color of solid line
represents serial correlation coeﬃcient, ρ - blue (0), brown (.2), green (.4), purple (.6), and
teal-blue (.8). We use n = 200 as benchmark since the bias does not change much as we
increases n over 200. The left panel of ﬁrst row shows the estimates for unconditional logit
estimator and the right panel of ﬁrst row shows the estimates for conditional logit
estimator. The left panel of second row shows the estimates from both of conditional
(dashed line) and unconditional (solid line) estimators together. More results on simulation
with various n, T, ρ are reported in Appendix. “For interpretation of the references to color
in this and all other ﬁgures, the reader is referred to the electronic version of this
dissertation.”

17

Figure 1.2. Bias of unconditional and conditional logit for β with continuous covariate

(Note) x-axis: time dimension, y-axis: estimated β and legend represent number of
cross-section units - true (teal), blue (100), red (200), green (400) and purple (600) - by
solid line for unconditional logit estimates and by dashed line for conditional logit
estimates. ρ represents serial correlation coeﬃcient. Each panel of ﬁgure represents
estimated β for each ρ. Figures in the ﬁrst row report estimated β for ρ =0 and ρ=.2.
Figures in the second row report estimated β for ρ =.4 and ρ=.6.

18

ρ=0.4 leads to over-rejection by 700% for n = 400. For ρ=0.6, the bias of conditional logit
estimator of β is greater than 20% for ∀T ≤ 8. The rejection frequency with nominal level
0.05 approaches 0.95 and beyond as n increases over 400 with ∀T ≥ 4. Finally, for ρ=0.8,
the bias of conditional logit estimator for β is greater than 50% for ∀T ≤ 8. The rejection
frequency with nominal level 0.05 approaches 1 as n increases over 200 with ∀T ≥ 3. Table
1.7 shows the magnitude of bias in % as ρ changes using unconditional logit estimates in
MC experiment.
Figure 1.1 shows that the increase in T leads to the smaller bias for coeﬃcient estimates
but the increase in n does not change the magnitude of ﬁnite sample bias for coeﬃcient β
especially for T ≥ 3. Overall, the magnitude of ﬁnite sample bias for coeﬃcient β is greater
with the higher value of ρ and the smaller T but is invariant to n. The size distortion becomes
more severe as n increases and ρ increases while the change in T (∀T ≥ 3) does not change
the rejection frequency much. Regardless of the lengths of T and n, the size distortion
in inference is quite severe for ρ >0.2. In sum, simulation reveals that conditional logit
estimator with correlated logistic errors provides biased coeﬃcient estimates and rejection
frequency in inference and, therefore, is not robust to the violation of CI assumption by
serial correlation especially for ρ >0.2.

Unconditional logit for β: Decomposition of Bias
Left panel in the ﬁrst row of ﬁgure 1.1 shows ﬁnite sample bias of unconditional logit for
β. The bias for β is greater for smaller T and higher value of ρ while ﬁnite sample bias
does not change much as n changes in ﬁgure 1.2. Furthermore, in table 1.5 and 1.6, using

19

unconditional logit estimator for β, simulation shows the decomposition of bias into the part
due to unobserved heterogeneity(ci ) and the part due to serial correlation(ρ) for n = 100
and n = 200, respectively.

Bias for β

Finite sample bias for unconditional logit is greater with smaller T and larger

ρ. For instance, when there is no serial correlation, ρ = 0, the magnitude of bias for
unconditional logit is greater than 100 % with T =2 and ∀n ≤ 600 while the magnitude of
bias for unconditional logit is about 20 % with T =8 and ∀n ≤ 600. Even for T as large as 8,
ﬁnite sample bias of unconditional logit estimator is about 17 % with no serial correlation
and ﬁnite sample bias for unconditional logit increases up to about 20 %, 30 %, 45 % and
90 % as ρ increases up to .2, .4, .6, and .8 respectively.

Decomposition of bias For ρ ≤0.4, ﬁnite sample bias due to ci (unobserved heterogeneity)
dominates the bias due to ρ (serial correlation among errors) ∀n, T . For T ≥5, the bias and
the decomposition of bias does not change much as n changes. For instance, in table 1.6,
for the case of n=200 and T =8, ﬁnite sample bias due to unobserved heterogeneity is about
18 % while ﬁnite sample bias due to serial correlation is about 4 %, 13 %, 29 % and 73
% for ρ=.2, .4, .6 and .8 respectively. As ρ increases ﬁnite sample biases by both sources
are substantial so that it is important to control both unobserved heterogeneity and serial
correlation properly. Sometimes, we can control only one source of bias and, in this case,
the choice of allowing serial dependence on response variable(D(yi |xi , ci )) or of allowing no
restriction on D(ci |xi ) should be depending on whether researchers have more information

20

Table 1.5. Decomposition of bias for β for n = 100
ci ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8
t=2 146
42
113
286
624
t=3 75
19
51
112
478
t=4 48
12
33
73
186
t=5 36
9
24
54
136
t=6 28
7
19
43
108
t=7 23
5
15
35
90
t=8 20
5
13
29
76
(Note) % of bias are reported in the cells. ci represents the proportion of bias due to
unobserved heterogeneity. ρ represents serial correlation coeﬃcient for errors. Abrevaya
[1997] shows that unconditional logit MLE for β has probability limit 2β. Thus, 46 % of
bias is due to ﬁnite sample bias which has nothing to do with unobserved heterogeneity.
ˆ
ˆ
With no serial correlation of error, we obtain βc−logit =1.23 and βunc−logit =2.46.

on D(yi |xi , ci ) or D(ci |xi ).8 For instance, for the case where there exists high persistence
of response variables even after conditioning on covariates, pooled methods that allowing
serial dependence with parametric assumption on D(ci |xi ) probably more appropriate than
those methods that impose no restriction on D(ci |xi ) and require conditional independence
assumption on outcome D(yi |xi , ci ).

1.4.3

The bias for APEs and rejection frequency: a continuous
covariate

Since conditional logit approach avoids specifying individual ﬁxed eﬀects, APEs cannot be
obtained for conditional logit estimator. In the estimation of APE for β in (1.1), we study the
APEs for CRE logit with pooled MLE under correct speciﬁcation for D(ci |xi ) and we also
compare these estimates with APEs for ﬁxed eﬀects(FE), LPM, and unconditional logit with
8 It

is like determining which functional forms assumption either D(yi |xi , ci ) or D(ci |xi )
is causing less distortion in statistical analysis.

21

Table 1.6. Decomposition of bias for β for n = 200
ci ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8
t=2 119
31
80
175
493
t=3 67
17
45
99
238
t=4 45
11
30
68
168
t=5 33
8
23
51
128
t=6 26
6
18
41
100
t=7 22
5
14
33
84
t=8 18
4
13
29
73
(Note) % of bias are reported in the cells. ci represents the proportion of bias due to
unobserved heterogeneity. ρ represents serial correlation coeﬃcient for errors. 19 % of bias
is due to ﬁnite sample bias which has nothing to do with unobserved heterogeneity. With
ˆ
ˆ
no serial correlation of error, we obtain βc−logit =1.095 and βunc−logit =2.19.

Table 1.7. The bias for β due to serial correlation using conditional logit for n = 200
ci ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8
t=2 0
25
50
97
269
t=3 0
14
28
57
129
t=4 0
9
21
44
101
t=5 0
8
18
36
84
t=6 0
6
15
31
72
t=7 0
6
13
27
64
t=8 0
5
11
24
57
(Note) % of bias are reported in the cells. ci represents the proportion of bias due to ﬁnite
sample estimation error for unobserved heterogeneity. ρ represents serial correlation
coeﬃcient for errors.

22

MLE.9 The APEs of CRE pooled-MLE and LPM with FE does not impose independence over
time while unconditional logit does not impose any restriction on unobserved heterogeneity.
Thus, unconditional logit is inconsistent estimator due to both unobserved heterogeneity and
serial correlation. Figure 1.3 summarizes MC estimates of APEs for CRE logit with pooled
MLE, unconditional logit with MLE and FE LPM. With correct speciﬁcation of unobserved
heterogeneity for CRE logit, only unconditional logit for APEs shows substantial bias. Finite
sample bias for APEs of unconditional logit is positively correlated with the magnitude of
serial correlation.

CRE logit with pooled MLE: APEs
Left panel of the second row in ﬁgure 1.3 summarizes the APEs for CRE logit with pooled
MLE. The results with n=200 are reported as benchmark.10LPM and CRE logit with pooled
MLE show no ﬁnite sample bias for APEs and associated rejection frequency. The correct
coverage rate of rejection frequency of hypothesis test on APEs of β is obtained regardless
of serial correlation (ρ) and the size of time dimension(T ).11Finite sample bias is less than
6%(= .165−.156 ) ∀T and ∀ρ and is less than 2% for most T and ρ. In short, CRE logit
.165
with pooled MLE provides reliable estimates for APEs of β and shows no size distortion in
inference under correct speciﬁcation of unobserved heterogeneity.12
9 Unconditional

logit is logit model where individual ﬁxed eﬀects are estimated as parameters. CRE logit with pooled MLE is logit model with parametric assumption for unobserved
heterogeneity as in DGPs of previous section. Pooled estimation is based on the assumption
of wrong model.
10 More results with various n and T are reported in appendix.
11 The results do not change much as n changes. See appendix for details.
12 The correct speciﬁcation of unobserved heterogeneity is implicitly assumed in DGPs
since ci is generated to satisfy CRE logit model.

23

Figure 1.3. Bias of unconditional logit and CRE-logit estimators for APEs

(Note) x-axis: time dimension, y-axis: estimated β. In ﬁgures of ﬁrst row, solid lines report
APEs for unconditional logit and dashed lines report APEs for conditional logit. Color
represent estimates with diﬀerent number of cross-section units from n=100 to n=600.
Figures in ﬁrst row, left panel reports APEs with ρ=0 and right panel reports APEs with
ρ=.6. For ﬁgures in second row, left panel report APEs for CRE logit with n=200 and
various ρ - blue (true), red (0), greed (.2), purple (.4), teal (.6), and orange (.8) and right
panel report APEs for unconditional logit with n=200 and various ρ - blue (0), red (.2),
greed (.4), purple (.6), and teal (.8).

24

Unconditional logit with pooled MLE: APEs
Unconditional logit with MLE shows substantial ﬁnite sample bias for APEs and shows
substantial size distortion in inference. Table 1.9 reports the estimates of β for unconditional
logit. Finite sample bias for APEs is greater for smaller T and greater value of ρ. Even for
ρ=0, ﬁnite sample bias is greater than 15 % ∀T . The bias for APEs is greater than 17 %, 42
%, 96 % and 204 % with ρ=.2, .4, .6 and .8, respectively, ∀T . The bias in rejection frequency
is greater for smaller T and greater value of ρ but does not depend on n.13

The bias decomposition for APEs using unconditional logit Table 1.8 reveals that
the decomposition of the bias for APEs into the part due to unobserved heterogeneity (ci )
and the part due to serial correlation (ρ) for n=200. The bias by ci dominates the bias by
serial correlation with ρ ≤0.6. This implies that, compared to the bias for β, the relative
inﬂuence to the bias from ci is greater for APEs. % of bias due to serial correlation decreases
as T increases for T ≥ 4.
In sum, ﬁrst, serial correlation does not aﬀect APEs for LPM and CRE logit with pooled
MLE with correct speciﬁcation of individual eﬀects. Second, both estimates of LPM and
CRE logit with pooled MLE under correct speciﬁcation of individual eﬀects show no significant ﬁnite sample bias for APEs and rejection frequency regardless of n, T and ρ. Third,
the estimates of APEs for unconditional logit show substantial bias and its magnitude is
positively correlated to serial correlation and negatively correlated to the magnitude of time
dimension (T ).
13 See

appendix for the results of the bias for rejection frequency.

25

Table 1.8. Decomposition of bias into two parts using unconditional logit for n = 200
ci , ρ = 0 ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8
t=2
114
9
17
24
17
t=3
67
10
21
37
52
t=4
44
6
16
32
55
t=5
30
5
13
25
48
t=6
22
4
10
21
41
t=7
17
3
7
16
34
t=8
15
2
6
14
30
(Note) % of bias of unconditional logit APEs are reported in the cells. ci represents the
proportion of bias due to ﬁnite sample estimation error for unobserved heterogeneity. ρ
represents serial correlation coeﬃcient for errors. n = 200 case was reported as benchmark
since very small diﬀerence of bias decomposition occurs as we change n. % of bias due to ci
is obtained from

AP Eunc,ρ=0 −AP ET rue
and % of bias due to serial
AP ET rue
AP Eunc,ρ>0 −AP Eunc,ρ=0
.
obtained from
AP Eunc,ρ>0

correlation (ρ) is

Table 1.9. The bias of unconditional logit n = 200
true ci , ρ = 0 ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8
t=2 .163
.349
.363
.378
.388
.277
t=3 .163
.273
.289
.308
.334
.258
t=4 .164
.236
.246
.262
.288
.326
t=5 .165
.214
.222
.235
.255
.294
t=6 .165
.201
.208
.218
.235
.269
t=7 .166
.194
.199
.206
.220
.251
t=8 .164
.188
.192
.198
.211
.237
(Note) The bias are reported in the cells. n = 200 case was reported as benchmark since
very small diﬀerence of bias decomposition occurs as we change n.

26

1.4.4

DGP: A model with both binary and continuous covariates

We generate data {yit , xit , Dit , ci , uit } for t = 1, 2, .., T and i = 1, 2, ..., n based on
model (1.10) with 1,000 replications.

yit = 1[xit β + Dit δ + ci + uit ≥ 0]

(1.10)

We draw the following ﬁve items as below. The correlation across time is induced through
error components u as in section 1.3.1.
1. xit ∼ iid N (0, 1). We generate an univariate covariate from identical and independent
standard normal distribution.
2. Dit = 1(dit − .5 ≥ 0) where dit ∼ iidU nif (0, 1)
3. Unobserved heterogeneity:
D(ci |xi ) ∼

a
Di1 + Di2 + ..... + Dik
√
+ i
2k
π2

(1.11)

3

where ai ∼ logistic(0,

π2
T
) and [k = ].
3
2

4. We generate each uit such that it has standard logistic marginal distribution (i.e.
2

uit ∼ log istic(0, π )), Cov(uit , uis ) = ρ|t−s| for t = s, and joint distribution of
3
(u1 , u2 , ...., uT ) is logistic.
5. We generate yit

yit = 1[xit β + Dit δ + ci + uit ≥ 0]
27

(1.12)

Figure 1.4. Bias of unconditional and conditional logit estimators for β with binary covariate

(Note) x-axis: time dimension, y-axis: The estimates for β are reported for n = 100. The
color of line represents serial correlation coeﬃcient - orange (true), blue (0), red (.2), green
(.4), purple (.6), teal-blue (.8). The left panel of ﬁrst row shows the estimates for
unconditional logit estimator and the right panel of ﬁrst row shows the estimates for
conditional logit estimator. The left panel of second row shows the estimates for both
conditional (dashed line) and unconditional (solid line) estimators together.

using xit , Dit , ci , uit as in 1,2,3,4, β=.2 and δ =1.

1.4.5

The bias and size distortion for δ: binary covariate

Conditional logit for δ
As ﬁgure 1.4 shows, in the MC experiment with binary covariate, we reach the same conclusion as in the MC experiment with a continuous covariate. Smaller T and greater value of ρ

28

Table 1.10. Decomposition of bias into two parts n = 100
ci ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8
t=2 157
48
212
486
14343
t=3 65
18
47
104
450
t=4 41
10
28
63
162
t=5 29
8
21
47
116
t=6 26
6
15
36
92
t=7 20
6
15
32
80
t=8 18
4
11
25
68
(Note) % of bias are reported in the cells.

leads to greater magnitude of ﬁnite sample bias for coeﬃcient estimate, δ, but the change in
n does not aﬀect the bias much. The size distortion becomes more severe as n and ρ increase
while the change in ∀T ≥ 3 does not have much eﬀect on the rejection frequency. Regardless
of the values of T and n, the size distortion is quite severe for ρ >0.2. Conditional logit
estimator for δ and rejection frequency are biased and, therefore, are not robust to violation
of CI assumption for ρ >0.2, ∀n, T .

Unconditional logit for δ: Decomposition of Bias
We obtain the same conclusion as DGP with a continuous covariate although the magnitude
of bias is greater with a binary covariate especially for T ≤ 4. The bias of unconditional
logit for δ is greater for data with smaller T and higher value of ρ while the bias does not
change much as n changes as shown in ﬁgure 1.4. Table 1.10 shows that, for ρ ≤0.4, the bias
due to ci (unobserved heterogeneity) dominates the bias due to ρ (serial correlation among
errors) ∀T .

29

1.4.6

The bias for APEs and rejection frequency: binary covariate

In the estimation of APEs for β in (1.12), we study the APEs for CRE logit with pooled MLE
and compare these estimates with APEs for LPM and unconditional logit with MLE. Figure
1.5 summarizes MC estimates for APEs of CRE logit with pooled MLE, unconditional logit
with MLE and LPM.
As the left panel in the second row of ﬁgure 1.5 shows, LPM and CRE logit with pooled
MLE exhibit no substantial ﬁnite sample bias of APEs for δ regardless of serial correlation
(the magnitude of ρ) and size of T . The rejection frequency also shows no signiﬁcant size
distortion regardless of the magnitude of serial correlation (ρ) and size of T .14In short, CRE
logit with pooled MLE provides reliable ﬁnite sample estimates for APEs of δ and a valid
inference under correct speciﬁcation of unobserved heterogeneity.

The bias decomposition for APE: binary covariate
Unconditional logit with MLE shows substantial bias for APEs and rejection frequency. The
right panel in the second row of ﬁgure 1.5 shows the bias of APE of δ for unconditional logit.
The bias for APE is greater for smaller T and larger value of ρ. Even for ρ=0, the magnitude
of bias for APE of δ is greater than 24 % ∀T . The bias increases with the magnitude of ρ
for all T . The bias for rejection frequency is greater for smaller T and larger value of ρ.15

14 The results are reported
15 See appendix for further

in appendix.
results of the bias for rejection frequency.
30

Figure 1.5. Unconditional logit and CRE-logit estimates of APEs for binary covariate

(Note) x-axis: time dimension, y-axis: APEs for β with n = 100 are reported. ρ represents
serial correlation coeﬃcient. Figures in ﬁrst row report APEs with ρ=0 (left) and ρ=.6
(right) respectively. The colors of solid line represent various APE estimators; blue (true),
red (LPM), green (unconditional logit), and purple (CRE logit). As we see, purple
completely cover blue. The left panel of second row shows the estimates for CRE logit
estimator and the right panel of second row shows the estimates for unconditional logit
estimators with various serial correlation coeﬃcient; blue (0), red (.2), green (.4), purple
(.6) and teal-blue (.8).

31

Table 1.11. Decomposition of bias using unconditional logit for n = 100
ci ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8
t=2 164
27
65
140
255
t=3 87
14
37
73
197
t=4 55
7
23
48
104
t=5 41
7
18
38
82
t=6 33
4
13
28
65
t=7 27
5
12
26
59
t=8 25
4
10
22
51
(Note) % of bias of unconditional logit APEs are reported in the cells. ci represents the
proportion of bias due to ﬁnite sample estimation error for unobserved heterogeneity. ρ
represents serial correlation coeﬃcient for errors. n = 100 case was reported as benchmark
since very small diﬀerence of bias decomposition occurs as we change n. % of bias due to ci
is obtained from

1.5

AP Eunc,ρ=0 −AP ET rue
and % of bias due to serial
AP ET rue
AP Eunc,ρ>0 −AP Eunc,ρ=0
obtained from
.
AP Eunc,ρ>0

correlation (ρ) is

Panel data models for married women’s welfare
participation

In this section, we illustrate the application of conditional logit for β and CRE logit with
pooled MLE for APEs using the Survey of Income and Program Participation (SIPP) data
on the welfare participation of married women from Chay and Hyslop [2001].16 As welfare
participation of married women exhibits substantial serial persistence over time, we apply
CRE logit with pooled MLE to obtain APEs of marital status and the number of children
on welfare participation.
Welfare participation data are from 1990 panel of the SIPP which contains eight waves
at four month intervals covering a 32-month period. The data set contains 1,934 married
16 Pooled

MLE with CRE logit model does not assume that the full conditional density
of y give x, c is correctly speciﬁed but it is very useful since full conditional density of y give
x, c is likely to be very complicated and pooled MLE does not restrict on serial dependence.
32

women over 8 periods. The sample contains women who are 18-65 years old and either
receive AFDC payments during or before the sample period or whose average total family
income during the sample period is below the family-speciﬁc average poverty level.17Welfare
participation status shows clear serial correlation over time. For instance, out of 1,934
married women, 1,076 married women never participate in welfare program and 364 married
women participate in welfare program for all sample periods. Basic statistics for variables
which is used in estimation are provided in table 1.12.
We start estimation with static model of (1.13) for simplicity. We control unobserved
heterogeneity with chamberlain device while we do not control for true state dependence
explicitly.

participationit = 1[zit β + ft + ci + uit ≥ 0], i = 1, 2, ..., n, and t = 1, 2, ..., T

(1.13)

where zit includes black race dummy, years of education, family poverty level status, age/10,
age2 /100, marital status and number of children age less than 18. And ft is period ﬁxed
eﬀects.

17In

summer 1993, the Nation had 36 million mothers 15 to 44 years old; 3.8 million
of them (10 percent) were receiving AFDC (Aid to Families with Dependent Children)
payments to help with the rearing of 9.7 million children. An additional 0.5 million women
over 45 years old and 0.3 million fathers living with their dependent children also received
AFDC.
33

Table 1.12. Basic statistics
Obs
AFDC
15,472
married
15,472
kids
15,472
Black
15,472
Educ
15,472
Poverty
15,472
age/10
15,472
2 /100 15,472
age
Note: There is

1.5.1

Mean
.302
.291
1.849
.310
11.260
1026.426
3.405
12.917
no missing

S.D
Min
Max
.459
0
1
.454
0
1
1.479
0
10
.463
0
1
2.700
0
18
352.849 519.755 2377.113
1.149
1.6
6.3
8.864
2.56
39.69
data. 15,472 = 8×1,934

Static panel data model for married women’s welfare participation

Table 1.13 shows that allowing unobserved heterogeneity to be correlated with marital status
and number of children has important eﬀects on estimated APEs. LPM provides estimated
coeﬃcients of -0.27 and 0.05 on marital status and number of children, respectively, with
both coeﬃcients being statistically signiﬁcant. Each child is estimated to increase the welfare
participation probability by about 0.05 while being married is estimated to reduce the welfare
participation probability by about 0.27. If we use logit and assume unobserved heterogeneity
is independent of explanatory variables, the estimated APEs are a little bit higher for both
marital status and number of children. When we use CRE logit model with pooled MLE
which controls for unobserved heterogeneity, we obtain APEs that are very similar to the
estimates by LPM. Moreover, the coeﬃcients on the time averages of time-varying covariates
are statistically signiﬁcant and practically large. All these results imply that accounting for
unobserved heterogeneity correctly is very important. CRE logit model with full MLE reveals
the same conclusion as with partial MLE. Using GEE with logit link and correlated random

34

eﬀect provides very similar estimates to the estimates for CRE logit model with full MLE.
Finally, column (6) contains the estimates from conditional logit. Unfortunately, the
coeﬃcient magnitudes are diﬃcult to interpret because of scaling factor. The ratio of coeﬃcients using conditional logit is − 2.80 = (−3.69), which is quite diﬀerent from the one,
0.76
− 1.76 = (−5.18), from CRE logit model with Pooled MLE. Moreover, if we do not con0.34
trol for unobserved heterogeneity, the ratio of coeﬃcient for RE logit with Pooled MLE
2.28
is − 0.54 = (−4.22). The discrepancy for the ratio of coeﬃcients between CRE logit and

RE logit which does not control unobserved heterogeneity is possibly caused by either misspeciﬁcation for the distribution of unobserved heterogeneity or uncontrolled unobserved
heterogeneity. Moreover, the discrepancy for the ratio of coeﬃcients between CRE logit and
conditional logit is possibly caused by either misspeciﬁcation for the distribution of unobserved heterogeneity or serial correlation. Therefore, for instance, if we assume distribution
of unobserved heterogeneity is correct with Chamberlain’s device, then we can infer that the
discrepancy is originated from the bias by serial correlation for conditional logit and the bias
by unobserved heterogeneity for RE logit. Moreover, in this case, because CRE logit with
pooled MLE in column (3) allows arbitrary weak dependence, it seems sensible to rely on
the estimates of CRE logit with pooled MLE under the assumptions that marital status and
number of children variables are strictly exogenous conditional on unobserved heterogeneity
and the assumption which unobserved heterogeneity can be speciﬁed by the time averages
of time-varying covariates. However, without correct speciﬁcation on D(ci |xi ), no procedure
strictly dominates the other. In particular, the choice among these procedures should be
based on whether we are more conﬁdent for the correct speciﬁcation about D(yi |xi , ci ) or
D(ci |xi ).
35

Table 1.14 shows results for probit model using the same estimation methods as in table
1.13. Although standard errors are smaller for probit model, we can draw the identical
conclusion as using logit model.

36

Table 1.13. Logit estimation for married women’s welfare participation

Model
Estimation
married
kids
married
kids

(1)
Linear
FE
coef
-.2707
(.0278)
.0524
(.0098)

(2)
logit
pooled
coef
-2.276
(.152)
.539
(.056)

MLE
APE
-.3704
(.0211)
.0877
(.0086)

(3)
CRE logit
pooled
coef
-1.757
(.196)
.343
(.060)
-.613
(.231)
.231
(.062)

(4)
CRE logit
MLE
GEE
APE
coef
-.2851
-1.591
(.0308)
(.185)
.0557
.294
(.0096)
(.054)
-.723
(.230)
.242
(.061)

√1

1+ˆ 2
σ

(5)
CRE logit
Full
APE
coef
-.2613
-3.305
(.0298)
(.248)
.0483
.724
(.0088)
(.091)
-1.762
(.349)
.547
(.111)
.242

(6)
FE logit
MLE
MLE
APE
coef
-.2293
-2.802
(.0122)
(.242)
.0671
.761
(.0084)
(.104)

log likelihood
-7571.91
-7552.25
-4028.83
-1412.62
number of women
1934
1934
1934
1934
1934
494
*Note: Cluster robust standard errors are reported in parentheses below all coeﬃcients and APEs. All model include a full set
of period dummy variables, time constant variables years of education, black race dummy variable, age/10, and age2 /100
which are not reported.

37

Table 1.14. Probit estimation for married women’s welfare participation

Model
Estimation
married
kids
married
kids

(1)
Linear
FE
coef
-.2707
(.0278)
.0524
(.0098)

(2)
probit
pooled
coef
-1.251
(.078)
.309
(.033)

MLE
APE
-.3490
(.0189)
.0861
(.0086)

(3)
CRE probit
pooled
MLE
coef
APE
-.951
-.2644
(.103)
(.0281)
.198
.0550
(.035)
(.0097)
-.354
(.126)
.132
(.036)

√1

1+ˆ 2
σ

(4)
CRE probit
GEE
coef
APE
-.906
-.2525
(.105)
(.0289)
.177
.0492
(.032)
(.0091)
-1.304
(.134)
.404
(.037)

(5)
CRE probit
Full
coef
-1.870
(.134)
.400
(.052)
-.726
(.217)
.290
(.071)
.349

(6)
FE logit
MLE
MLE
APE
coef
-.2078
-2.802
(.0124)
(.242)
.0590
.761
(.0078)
(.104)

log likelihood
-7600.63
-7580.68
-3976.44
-1412.62
sample
1934
1934
1934
1934
1934
494
*Note: Cluster robust standard errors are reported in parentheses below all coeﬃcients and APEs. All model include a full set
of period dummy variables, time constant variables years of education, black race dummy variable, age/10, and age2 /100
which are not reported.

38

1.5.2

Dynamic Unobserved eﬀects logit models under strict exogeneity

Instead of controlling serial correlation by the models with pooled MLE methods, we probably control serial dependence among yi by using a dynamic model with full MLE. Chay
and Hyslop [2001] emphasizes the importance of introducing dynamic models to seriously
consider both unobserved heterogeneity and true state dependence in welfare analysis. Explicitly distinguishing spurious sate dependence by individual permanent characteristics and
true state dependence is possible only with dynamic models. Our treatment for dynamic
model follows Wooldridge [2005] and Wooldridge [2010].
Suppose we date our observations starting at t = 0, so that yi0 is the ﬁrst observation on
outcome, y. For t = 1,...,T, we are interested in dynamic model of (1.14).

E(yit |yi,t−1 , ..., yi0 , zi , ci ) = G[ρyit−1 + zit β + ci + uit ≥ 0]

(1.14)

where zit is a vector of contemporaneous explanatory variables, and G can be the probit or
logit function. There are several important points about this model. First, zit is assumed to
satisfy a strict exogeneity assumption (conditional on ci ). Second, the probability of success
at time t is allowed to depend on the outcome in t − 1 as well as unobserved heterogeneity,
ci .
Following Chay and Hyslop [2001], we focus the speciﬁcation of dynamic model to investigate following two points. First, we are interested in hypothesis test with the null,
H0 : ρ=0. ρ =0 implies the existence of state dependence after controlling for the unobserved heterogeneity, ci . This provides information about the decomposition of welfare

39

participation persistence to state true dependence and unobserved heterogeneity. Second,
the estimation for ρ and the marginal eﬀects of marital status and number of children on
welfare participation. We start with the speciﬁcation of dynamic model of (1.15).

T

f (y1 , y2 , ....yT |y0 , z, c, θ) =

f (yt |yt−1 , ...., y0 , zt , c, θ)

(1.15)

t=1

We condition on yi0 as well to solve the initial conditions problem. Furthermore, we
propose a D(ci |y0 , zi ) as in (1.17) to obtain f (y1 , y2 , ...., yT |y0 , zi ) using (1.16). Given a
density h(c|y0 , z, δ), we have

∞

f (y1 , y2 , ....yT |y0 , z, c, θ) =

∞

f (y1 , y2 , ....yT |y0 , z, c, θ)h(c|y0 , z, δ)dc

(1.16)

where G is standard normal CDF for probit model and

2
h(c|y0 , z, δ) is Normal(ψ + ς0 yi0 + zi ξ, σa )

(1.17)

where z include AF DCt−1 , number of children, marital status, and family poverty level.
We can rewrite our estimation equation as in (1.18) using chamberlain device.

yit = 1[ψ + ρyit−1 + zit β + ς0 yi0 + zi ξ + ai + uit ≥ 0]

(1.18)

Pooled MLE does not consistently estimate the scaled coeﬃcients or the APEs in estimating dynamic model. This is because while P (yit = 1|zi , yit−1 , ...., yi0 , ai )=Φ(ψ + ρyit−1 +
zit β + ς0 yi0 + zi ξ + ai ) is true, P (yit = 1|zi , yit−1 , ...., yi0 )=Φ(ψa + ρa yit−1 + zit βa + ς0a yi0 +
zi ξa ) does not hold true unless ai is identically zero. Therefore, only random eﬀect probit consistently estimate the scaled coeﬃcients and the APEs. For comparison purpose we
40

also report CRE probit with pooled MLE and RE probit with MLE which does not control
unobserved heterogeneity in table 1.15.
The column (4) in table 1.15 for dynamic model provides that CRE probit with MLE for
ρ which is 1.290(s.e=.072) and is statistically diﬀerent from zero even after controlling for
ˆ
unobserved heterogeneity. This is strong evidence for the existence of true state dependence.
ˆ ˆ
ˆ ˆ
ˆ
The computation of APE for AF DCt−1 is based on averaging Φ(ψa +ρa +zit βa +ς0a yi0 +zi ξa )
ˆ
ˆ
ˆ
- Φ(ψa + zit βa + ς0a yi0 + zi ξa ) across all i and t.18 CRE probit with full MLE for the
ˆ
APE of AF DCt−1 is provided in the model (4) of table 1.15 and is about .175. In other
words, averaged across all women and all time periods, the probability of participating in
welfare at time t is about .175 higher if the women was in welfare at time t − 1 than if she
was not. This estimate controls for unobserved heterogeneity, number of children, marital
status, family poverty level and women’s education, race, age, age2 . The column (2) of
table 1.15 reports the APEs for random eﬀect probit with MLE which does not control
unobserved heterogeneity. Estimated APE for state dependence is .384 which is higher than
random eﬀects estimates for which heterogeneity is controlled for. This implies that much
of observed serial dependence in welfare participation could be attributed to unobserved
heterogeneity. ( .384−.175 = .544) About 54 % of observed serial dependence of welfare
.384
participation is due to heterogeneity while about 46 % of observed serial dependence is due to
state dependence. In other words, simple dynamic RE probit model that ignore unobserved
heterogeneity exaggerate true state dependence by 54 %. In sum, the analysis with dynamic
model implies that both sources of persistent welfare participation are important since the
18 Original

coeﬃcients are scaled by √ 1

1+ˆ 2
σ

of all APEs.

41

=1, where σ 2 =.00000045 in the calculation
ˆ

diﬀerence between the estimates clearly shows that welfare participation is depending on
individual permanent characteristics as well as ”narcotic” eﬀects of welfare program.

42

Table 1.15. Probit estimation of dynamic model for married women’s welfare participation

Model
Estimation Method
AF DC1
married
kids
√1

1+ˆ 2
σ

(1)
Linear
FE
coef
.3703
(.0163)
-.1922
(.0230)
.0350
(.0075)

(2)
RE probit
Full
coef
2.028
(.073)
-.988
(.094)
.211
(.037)
.734

MLE
APE
.3840
(.0058)
-.1076
(.0057)
.0220
(.0024)

(3)
CRE probit
pooled
MLE
coef
APE
1.290
.1343
(.063)
(.0085)
-1.595
-.1545
(.143)
(.0115)
.280
.0180
(.059)
(.0034)

(4)
CRE probit
Full
MLE
coef
APE
1.290
.1747
(.062)
*
-1.595
-.1545
(.136)
*
.280
.0178
(.051)
*
.999

(5)
FE logit
MLE
coef
1.643
(.097)
-2.506
(.303)
.670
(.129)

unobs heterogeneity controlled not controlled
controlled
controlled
controlled
log likelihood
-2624.13
-1658.99
-1658.99
-936.45
sample
1934
1934
1934
1934
441
Note: Cluster robust standard errors are reported in parentheses below all coeﬃcients and APEs. All models include a full set
of period dummy variables, time constant variables years of education, black race dummy variable, age/10, age2 /100, initial
period value for AFDC and married and kids variables for all periods which are not reported.

43

1.6

Conclusion

This paper examines the robustness of the conditional logit estimator to the violation of the
CI assumption using correlated logistic errors in a binary panel data model with correlated
individual ﬁxed eﬀects. Simulation results show that the conditional logit method is not
robust to violation of the CI assumption. The magnitude of bias for coeﬃcient is greater for
the data with a smaller time dimension (T ) and a higher serial correlation coeﬃcient. (ρ)
Under correct speciﬁcation of unobserved heterogeneity, CRE logit with pooled MLE provides reliable estimates for APEs with no signiﬁcant ﬁnite sample bias. Simulation for the
rejection frequency of the hypothesis test shows evidence that CRE logit provides a reliable
test in the presence of high serial correlation. Finite sample bias due to both unobserved
heterogeneity and serial correlation is substantial. An empirical example of welfare participation which exhibits high persistence shows that both sources of bias are important. The
decomposition shows that bias due to uncontrolled unobserved heterogeneity is about 54%
and that due to serial correlation is about 46%. This implies the importance of accounting
for both state dependence and unobserved heterogeneity.

44

Chapter 2
The MI-IPW method in an
unbalanced panel data model: An
empirical application on the eﬀect of
class size reduction on SAT scores for
students in grades K-3

2.1

Introduction

Randomized experiments have become more popular in evaluating the impact of social programs recently. Angrist and Pischke [2009] reﬂects this trend by arguing that the causal
inference for the eﬀect of a program can be best answered by a randomized experiment.

45

However, perfect implementation of a randomized trial is extremely rare. In most experiments, experimental violations occur typically through noncompliance and attrition and can
invalidate any statistical conclusions drawn from them. The possible bias from noncompliance can be addressed by the IV method using initial treatments as IV for actual treatments.
However, standard methods for missing data, which include complete-case analysis, simple
imputation and last-observation-carried-forward analysis, are not appropriate for a valid inference without assuming missing completely at random (MCAR). 1 Unfortunately, in most
empirical applications, the MCAR assumption is unrealistic and the violation inevitably
leads to biased estimates, incorrect standard errors and invalid inferences for unweighted
complete-case analysis.2
In this chapter, we focus on the marginal treatment eﬀect (MTE) of a program in a
random experiment with ideal design at the beginning but subsequent missing data at the
implementation stage. The purpose of this article is twofold. First, we provide robust
Hausman tests of the missing completely at random (MCAR) assumption in an unbalanced
linear panel data model for least-square (LS), ﬁxed-eﬀects (FE) and ﬁrst-diﬀerencing (FD)
estimators for which the rejections of tests imply invalid statistical analysis. Proposed Wald
statistics for robust Hausman tests are based on comparing two consistent estimators under
the null and it is robust in the sense that estimators used in the construction of the Wald
statistics do not assume conditions for the estimators to be eﬃcient.3 The idea of the test
1

MCAR and missing at random (MAR) assumption is in the sense deﬁned by Rubin
[1976] and Little and Rubin [2002]. The MCAR implies that the probability of missing is
independent of both observed and unobserved variables. MAR implies that, conditional on
observed data, the probability of missing does not depend on the unobserved data.
2 Data is complete in the panel data for a particular period if a cross-section unit has
observations for outcome and all covariates at that period.
3 Original Hausman tests(Hausman [1978]) which compare random eﬀect estimator and

46

is that, under the null, the diﬀerence in the estimates of two consistent estimators should
be entirely due to sampling errors. This is an extension of a robust Hausman test of a
balanced panel in [Wooldridge, 2010, p.321-334] to unbalanced panel data. We also conduct
a Monte Carlo experiment of regression based variable addition tests for ﬁxed-eﬀects and
ﬁrst-diﬀerencing estimators to test the MCAR assumption.
Second, we introduce practical methods with valid statistical analyses for handling missing
data under the missing at random (MAR) assumption which include the inverse probability weighted (IPW) method (Horvitz and Thompson [1952] and Robins et al. [1994]), the
likelihood-based maximum likelihood (ML) method (Rubin [1987] and Little and Rubin
[2002]) and the multiple imputation (MI) method ( Buuren et al. [1999] and Schafer and
Graham [2002]). Many microeconomic applications, that typically use the linear models involve various types of high-dimensional covariates which include continuous and categorical
variables of various types (binary, count, fraction and so forth). Moreover, data can be missing for all covariates and outcome. Therefore, the likelihood based ML and the MI methods
are not appropriate for unbalanced panel data with these types of missing variables since
the ML and the MI methods require modeling the joint distribution of missing variables
and the correct speciﬁcation of a model for joint distribution would be extremely diﬃcult,
if not impossible. In many applications, although joint normality is assumed, we have very
little information on the joint distribution of missing variables. On the other hand, the IPW
method with missing outcome and covariates does not need to specify a fully parametric
model for joint distribution of missing variables since probability weights can be modeled by
a univariate normal (probit) or a logistic (logit) distribution.
ﬁxed eﬀect estimator assume eﬃciency for random eﬀect estimator.

47

Meanwhile, the IPW method under MAR is criticized mainly due to the following three
concerns. First, the IPW method is generally less eﬃcient than the MI method with correct speciﬁcation for missing variables since the IPW method only uses complete-case data
while the MI method uses information in covariates and outcome even from incomplete case
samples. (Clayton et al. [1999]) Therefore, for instance, if a linear regression model has highdimensional covariates and only one variable in the covariates has missing data, then the loss
of eﬃciency for the IPW method should be signiﬁcant. However, for monotone missing data
(i.e. attrition), there is no loss of information due to this factor since data missing occurs
in outcome and all covariates simultaneously for a cross-section unit.4 The second concern
is that IPW estimates can be very unstable if the estimated probability weights are close to
zero for certain subpopulation units(Little and Rubin [2002]). However, we can verify the relevance of this criticism by probing the estimates with actual data in the application. Finally,
the IPW estimates can be sensitive to the model speciﬁcation for the probability of selection
(or missing). We provide an example of using an economic theory in the speciﬁcation of
the selection process and the MAR assumption. Particularly, in the empirical example, we
extend Becker [1993]’s model of parent’s school choice between public and private schools
to specify the selection process that satisﬁes the MAR assumption. For simplicity, we use
4

Recently, Robins and Rotnitzky [1995] and Scharfstein et al. [1999] propose an AIPW
(i.e. doubly robust) estimator which allows to construct a theoretically eﬃcient version of
IPW so that the extension of IPW to AIPW estimator can overcome ineﬃciency problem
at least asymptotically. An estimator is called doubly robust if it is consistent when either
a model for conditional expectation(imputation model for missing variables) or a model
for probability of selection is correctly speciﬁed. An AIPW estimator can attain eﬃciency
in the sense that its variance can reach semi-parametric eﬃciency bound among the class
T
of estimators that use n
i=1
t=1 sit · momentit (Observedit , θ) = 0. More discussion for
eﬃciency properties of AIPW estimators is provided in Bickel et al. [1998], Tsiatis [2006],
Robins et al. [1994], and section in 2.4.1.

48

the logit or the probit models for the distribution of probability of selection.5 Unfortunately,
it is very hard to test how inaccurate the approximation of the speciﬁcation for predictors
and the logit or the probit models for probability of selection are in practice so we conduct
a Monte Carlo simulation to provide evidence on the sensitivity of the IPW estimator to
selection model misspeciﬁcation.
We explicitly categorize unbalanced panel data into four types based on missing patterns.
The four data types are the following: (i) complete case data, (ii) individual units leave
the sample but re-enter the sample later (type-I missing data), (iii) individual units are in
the sample but some variables are missing (type-II missing data), and (iv) attrition (typeIII missing data). For type-I missing data, we drop the observations and do not use in the
estimations while we assume these types of data missing are completely random or negligibly
small. For type-II missing data, we apply multiple imputations based on the missing variable
types. Finally, we propose a method which applies MI and IPW sequentially. Speciﬁcally,
after multiple imputations of type-II missing variables, we treat type-II missing data as if
they are complete case data. We then combine imputed type-II missing data with complete
case data and hen apply the IPW method.
Tests of MCAR and the estimations with unweighted, IPW and MI-IPW methods under
the MAR are conducted to gauge the eﬀect of class size reduction (CSR) on SAT scores
for students in K-3 grades using Tennessee’s Project STAR. We employ Tennessee’s Project
STAR to evaluate the MI-IPW method in the application since it is a randomized experi5

Nonparametric estimation of the probability of selection is also available from Hirano
et al. [2003] but it suﬀers from the curse of dimensionality in practical applications.(Hall and
Presnell [1999]).

49

ment with signiﬁcant numbers of noncompliance and attrition.6 For instance, 13% of 11,601
students switched from treatment groups to the control group or vice versa (i.e. 13% of the
sample are deﬁers) and about 45% of the students either entered the program after kindergarten or left before the program ended and about 5% of the participating students had
missing variables during in the STAR program. Moreover, potential diﬀerential attrition
across treatment and control groups appears in basic statistics which show that a larger
fraction of students left the sample in regular and regular-with-aide classes than from small
classes for all grades from K to 3. As reported in Table 2.27, the diﬀerence in the average
scores between students who stayed and drop-out students is greater in small classes than
the diﬀerence in regular and regular-with-aide classes for both ﬁrst and second grades although the diﬀerences are small. This implies that a relatively greater proportion of better
performing students left the program from regular and regular-with-aide classes than from
small classes. This is informal evidence of diﬀerential drop-out rates of students across different types of classes and suggests possible overestimation in the eﬀect of CSR on SAT
scores. Therefore, the MCAR assumption is possibly violated and a standard estimation
with complete-case is probably inappropriate.7
Our application of proposed tests and estimations with Project STAR data proceed as
follows. First, we apply formal tests of diﬀerential attrition across class types which lead
to violation of the MCAR assumption. We test the MCAR assumption using Hausmantype tests and regression based variable addition tests using FD and FE estimators. In
6 Tennessee’s

Project STAR is a longitudinal data with a single cohort of 11,601 students
for grades in K-3. Missing data information for STAR is reported in table 2.26.
7 Further details of ﬁndings with unweighted estimation with complete-case are provided
in Word et al. [1990] and Krueger [1999].

50

the application with Project STAR, both Hausman-type tests and regression-based variable
addition tests reject the MCAR and IPW and MI-IPW methods under MAR are proposed to
overcome the endogeneity problem by violation of MCAR. Meanwhile, we cannot directly test
with the null of weak exogeneity for a pooled LS estimator. This is because, for sit = 0 (see
(2.2) for deﬁnition), we do not have observations for either outcome or for some covariates
so we cannot estimate E(uit |sit , xit ) directly. (see (2.1) for the deﬁnitions of variables).
Second, we use LS, reduced-form(IV)8 and ﬁrst-diﬀerencing estimators with IPW and MIIPW methods in a linear unbalanced panel data model to estimate the eﬀect of CSR on SAT
score for grades K-3 using Tennessee’s project STAR data. Speciﬁcally, we apply the IPW
method using the logit (or probit) model for which all observed past covariates and outcome
are adopted as predictors for probability of selection which is based on Becker [1993]’s school
choice model. In the application, the estimated probability weights are strictly greater than
0.2 so that we can avoid a criticism of instability for the IPW method from the probability
weights being too close to zero. Multiple imputations of chained equation estimates are also
obtained and compared to IPW estimates. The estimates for the eﬀect of CSR on SAT
scores are about 4 to 6.5 percent for IPW and MI-IPW while unweighted estimates for the
return of CSR are 5 to 7.5 percent. Unweighted estimators of complete-case overestimate
the eﬀect by about 1 to 2 percentage points. Direction of bias analysis for pooled estimators
also suggests that unweighted estimators with complete-case reveal upward bias under an
assumption of positive correlation between E(sit |xit ) and E(sit+1 |xit ).
The rest of chapter is organized as follows. In section 2, we provide pooled LS, FE and FD
8 Reduced-form

estimation (IV hearafter) implies LS estimation where initial assignment
of treatments is used as covariates instead of actual treatments as in Krueger [1999].

51

estimators in an unbalanced linear panel data model. Section 3 introduces robust Hausmantype tests and variable addition tests to test the MCAR assumption and to show the power
of the tests. In section 4, we describe IPW and MI-IPW methods under MAR to deal with
missing data and to perform robust analysis for the IPW method to the misspeciﬁcation of
selection model using Monte Carlo(MC) simulation. Section 5 illustrates the tests of the
MCAR and the estimations of IPW and MI-IPW methods under MAR using project STAR
in the analysis of the eﬀect of CSR on SAT score for students in grades K-3. Section 6
provides a method with which to calculate the direction of bias when the MAR assumption
is violated. Section 7 contains concluding remarks.

2.2

Linear panel data model with missing and noncompliance

We are mainly interested in the estimation of marginal treatment eﬀect (MTE hearafter) of
a program in linear unbalanced panel data model. In particular, we focus on three popular
estimators, in linear panel data models, which include pooled LS, FE and FD estimators.

2.2.1

Model

We introduce selection indictor (2.2) to express explicitly the conditions of consistency and
MCAR for LS, FE and FD estimators.
yit = dit · β + x1it δ + x2i γ + ft + ci + uit for , i = 1, 2, .., N and t = 1, 2, .., T

52

(2.1)







 0


sit =



 1 if all of y , x are observed

it it





otherwise

(2.2)

where yit is an outcome, dit is a vector of indicators for treatments, x1it is a vector of
time varying covariates, x2i is a vector of time constant covariates, ft is time eﬀect, ci is
unobserved heterogeneity and uit is idiosyncratic error. Let xit = (dit , x1it , x2i , ft ), θ =
(β , δ , γ , 1) and vit = ci + uit . Then we can rewrite (2.1) as (2.3).
yit = xit · θ + vit for , i = 1, 2, .., n and t = 1, 2, .., T

(2.3)

We can express pooled LS, FE and FD estimators with selection indicator as (2.4), (2.5)
and (2.6) respectively and explicitly state conditions for MCAR.
N

T

N

sit xit xit

θpls = (
i=1 t=1
N

T

θF E = (
..

where xit = xit − xi , xi =

i=1 t=1
T
1
sit xit
Ti
t=1
N

T

)−1

sit xit yit
i=1 t=1

.. ..
sit xit xit )−1

and Ti =

N

T

.. ..

sit xit y it

(2.5)

sF D ∆xit ∆yit
it

(2.6)

i=1 t=1
T
sit .
t=1

T

N

T

sF D ∆xit ∆xit )−1
it

θF D = (

(2.4)

i=1 t=2

i=1 t=2

where ∆xit = xit − xit−1 for t = 2, 3, .., T and sF D = min{sit , sit−1 }.
it

Conditions for consistency with missing data
Pooled Least Squares For a pooled least squares(PLS) estimator, we essentially need
rank condition (2.7) and exogeneity condition (2.8) to obtain a consistent estimator.
rankE(sit xit xit ) = dim(θ) , t = 1, 2, ..., T

(2.7)

E(sit xit vit ) = 0 , t = 1, 2, ..., T

(2.8)

53

Fixed eﬀects Rank condition and exogeneity condition for ﬁxed-eﬀects estimator are (2.9)
and (2.10), respectively.
.. ..

rankE(sit xit xit ) = dim(θ) , t = 1, 2, ..., T
.. ..

E(sit xit uit ) = 0 , t = 1, 2, ..., T

(2.9)

(2.10)

First diﬀerencing Rank condition and exogeneity condition for ﬁrst diﬀerencing estimator
are (2.11) and (2.12), respectively.
rankE(sF D ∆xit ∆xit ) = dim(θ) , t = 2, ..., T
it

(2.11)

E(sF D ∆xit ∆uit ) = 0 , t = 2, ..., T
it

(2.12)

Exogeneity assumption Strict exogeneity of (2.13) is suﬃcient for the exogeneity conditions of FE and FD estimators in (2.10) and (2.12) while weak exogeneity of (2.15) is
suﬃcient for pooled estimator.
E(uit |si , xi ) = 0, for t = 1, 2, ..., T

(2.13)

where si = (si1 , si2 , ...., siT ) and xi = (xi1 , xi2 , ...., xiT )

(2.14)

E(uit |sit , xit ) = 0 and E(ci |sit , xit ) = 0, for t = 1, 2, ..., T

(2.15)

Pooled estimator does not need strict exogeneity while it requires no correlation between
selection and unobserved heterogeneity.

54

2.2.2

Tests of missing completely at random

If missing completely at random (MCAR) is satisﬁed, all three estimators should be consistent with random experiment.9 We suggest a Hausman-type test using null hypothesis of
strict exogeneity and E(ci |sit , xit ) = 0 in section 2.2.1.

MCAR MCAR in random experiment implies both weak and strict exogeneity while violation of either weak or strict exogeneity leads to violation of MCAR. Moreover, with proper
rank conditions, MCAR assumption is suﬃcient for all of pooled LS, FE and FD estimators
to be consistent while violation of either weak or strict exogeneity leads unweighted pooled
LS or FD and FE estimators to be inconsistent respectively.

Null I: strict exogeneity Consider a case in which the violation of strict exogeneity is
the only source of bias. Under strict exogeneity, both FE and FD estimators are consistent
and the diﬀerence of FE and FD estimates should be entirely due to sampling error. We
extend a robust Hausman test which compares FE and FD estimators in balanced panel data
as in [Wooldridge, 2010, p.324-325] to unbalanced panel data model by constructing a Wald
statistic based on

√
n(βF E − βF D ). The rejection of null hypothesis implies the violation of

strict exogeneity (Cov(uit ,si di ) = 0) and both FE and FD estimators are inconsistent. In
short, MCAR is violated and wnweighted FE and FD estimators are inconsistent.

Singularity Since we can not obtain the estimates for FE and FD coeﬃcient of time
constant covariates, the set of parameters that we can use to construct a Wald statistic
9 See

Wooldridge [2009] for more discussion of equivalence of three estimators under the

null.

55

should exclude the coeﬃcients from time constant variables. Thus, in typical applications,
the indicators for the treatments of a program should vary over time to apply our method
of comparing two consistent estimators under the null.

Null II: no correlation between selection and unobserved heterogeneity We propose a subsequent test for the case when we can not reject the null of strict exogeneity. We
assume strict exogeneity hold both under the null and the alternative. Therefore, under
the null, all conditions for the consistency of pooled, FE and FD estimators are satisﬁed
while, under the alternative, E(ci |sit , xit ) = 0 is not satisﬁed. Still both ﬁxed eﬀect and
ﬁrst diﬀerencing estimators provide consistent estimators both under the null and the alternative while pooled estimator is consistent only under the null.10Thus, we can test the null
hypothesis of MCAR by using diﬀerential correlation between selection and unobserved heterogeneity across treatments and control as alternative. We construct a Wald statistic based
on

√

n(βpls − βF E ) to perform a test of MCAR. The rejection of the null implies the violation

of E(ci |sit , xit ) = 0 (conditional on xit Cov(ci ,sit xit )= 0) and this leads pooled estimator to
be inconsistent because of Cov(ci ,sit xit )= 0. Under ideal conditions and E(ci |sit , xit ) = 0,
MCAR is violated and unweighted pooled estimator is inconsistent while, with FE and FD
transformed linear models, MCAR assumption holds and unweighted FE and FD estimators
are consistent.
10 Strict

exogeneity assumption is maintained both under the null and under the alternative. We also assume that rank conditions are satisﬁed for all estimators we consider in this
section.

56

2.2.3

Implications from the rejection of tests

The rejection of strict exogeneity implies that both unweighted FE and FD estimators with
complete-case are inconsistent. The failure of rejection of strict exogeneity and the success
of rejection for Cov(ci ,sit |xit )= 0 leads unweighted pooled estimator with complete-case to
be inconsistent while we can not reject the null that unweighted FE and FD estimators with
complete case are consistent.11
The rejection of both tests implies missing indicator is probably correlated with both timevarying and time-constant unobserved variables even conditional on observed variables(ci
and ui are correlated with sit xit ) so that MCAR is violated and unweighted FE and FD
estimators with complete case are invalid. Therefore, we need to control for missing data
to produce a valid statistical analysis. Either FD or FE transformations are not enough to
obtain a valid inference since the correlation between ui and si xi is not zero.

2.3

Formal tests of MCAR

In this section, we introduce a test statistic used in performing robust Hausman tests for
MCAR assumption and investigate the power of the test via Monte Carlo experiment.
11 Under

the null of these two Hausman-type tests, all other conditions for consistency
including rank condition and other regularity conditions are assumed to be satisﬁed.

57

2.3.1

Hausman-type test I - Comparison of θpls and θF E using
E(ci ,sit xit )= 0 as null

A test statistic is constructed to perform a Hausman-type test for the null of no correlation
between unobserved heterogeneity and missing data. First, we stack two estimators θpls and
θF E . We denote stacked estimator as θa = (θpols , θF E ) and denote restriction matrix as R
of k × 2k.

√

n

−1

T

1
0
sit xit xit
 n
 i=1 t=1
nθa = 
n T

.. ..
1
0
sit xit xit
n








1
sit xit yit
 √n

i=1 t=1

 1 n T
.. ..
√
sit xit y it
n

i=1 t=1

2k×2k

n



T

i=1 t=1





2k×1

and R = [Ik | − Ik ].
We write the null hypothesis as (2.16) and this implies that under the null both pooled
LS and FE estimators are consistent.

H0 : Rθa = 0





 1 0 · · · · · · 0 −1 0 · · · · · · 0


 0 1 0 · · · 0 0 −1 0 · · · 0



. .
.
..
..
. .
.
Rθa =  .
.
.
.
. .
.
 .

 .
.
.. . .
..
 .
. . .
. .
. .
.
 .


0 0 ··· 0 1 0
0 · · · 0 −1
















k×2k























θ1,ls 


.

.
.



θk,ls 



θ1,F E 



.

.
.



θk,F E

2k×1

58

(2.16)





 θ1,ls − θ1,F E


 θ
 2,ls − θ2,F E


.

.
.


= 
.

.
.



.

.
.



θk,ls − θk,F E





















=0

k×1

Using the null H0 : Rθa = 0 12and consistency conditions in (2.2.1), we construct a Wald
statistic which converges asymptotically to a chi-squared distribution with degree of freedom
equal to number of restrictions, k in (2.16).

Assumptions
1. Conditional strict exogeneity : E(uit |xi , si ) = 0, ∀t
2. no correlation between unobserved heterogeneity and missing data: E(ci |xit , sit ) = 0.
3. Rank conditions: E(

T
F E .. ..
t=1 sit xit xit )

1
p lim
n
p lim

1
n

n

and E(

T
t=1 sit xit xit )

T

T

sit xit xit = E(
i=1 t=1
n T

have full rank

.. ..

sit xit xit )
t=1
T

sit xit xit = E(
i=1 t=1

.. ..

sit xit xit )
t=1
n

T

1
sit xit uit and
4. Conditions for applying CLT to following two objects, √n
i=1 t=1
1
√
n

n

T

i=1 t=1

12 This

.. ..

sit xit uit .

is equivalent to H0 : θj,P LS = θj,F E ∀j = 1, 2, ...k

59

A Wald Statistic for a robust Hausman test
Proposition 2 Under assumption 1,2,3,and 4, for the test with null hypothesis of H0 :
Rθa = 0, we can deﬁne a Wald type statistic Wa which converges to χ2 distribution.
k
√
√
√
Wa = [R n(θa )] [Rvar( nθa )R ]−1 R n(θa ) ∼ χ2
k
where






Rθa

=

















1 0 · · · · · · 0 −1
0 1
.
.
.

..

··· 0

0

0

.
.
.

.
.
.

.
.
.

.
.
.

1

0

.

.
.
.

..

0 0 ···

0

.

0

··· ···

0 


−1 0 · · · 0 


. 
..
. 
.
. 

. 
..
. . 
. 


0 · · · 0 −1

k×2k























θ1,P LS 


.

.
.



θk,P LS 



θ1,F E 



.

.
.



θk,F E

=

2k×1

























θ1,P LS − θ1,F E 


θ2,P LS − θ2,F E 



.

.
.



.

.
.



.

.
.



θk,P LS − θk,F E

k×1

Proof. See appendix.

2.3.2

Monte Carlo experiment I

We investigate the power of the test using Monte Carlo simulation. We generate a selection
process which reduces full sample to either 75 or 50 percent of full sample. Generated

60

selection process induces a diﬀerential correlation between unobserved heterogeneity and
selection across treatment and control. Thus, this particular selection process preserves
asymptotic consistent results for FE and FD estimators. We focus on how the power of test
changes as the fraction of complete-case data to full data changes to 75 and 50 percents.

Data generating process
Data generation process is based on following outcome and selection process.

The model for DGP
yit = dit · α + xit · β + ci + uit , for i = 1, 2, .., n and t = 1, 2, 3, 4
where dit a binary variable and xit is continuous variable.
w
e
si1 = 1; sit = 1(dit · γ + √i + it > a), where γ = 1 for i = 1, 2, .., n and t = 2, 3, 4
2
2
Variables generation
h +v
√
1. dit = Φ( it i ) where Φ(·) is standard normal CDF. dit = 1 if dit > 0.7. α = 1,
2

hit ∼ iidN (0, 1) and vi ∼ iidN (0, 1).
2. xit ∼ iidN (0, 1), β = 0.2
3. uit ∼ iidN (0, 1)
√

4. ci ∼
5. sit

3wi
2

=

a
+ 2i , wi ∼ iidN (0, 1) and ai ∼ iidN (0, 1)
w
eit
Φ(dit γ + √i + 2 )
2

where

γ=1 .

where a determines the proportion of missing data.
61

sit

=

1

if

sit

>

a,

Table 2.1. Hausman-type test I: Null and Alternative (cov(sit dit , ci ) = 0)

Pooled LS
FE estimator

Null
Alternative
Consistent Inconsistent
Consistent Consistent

Table 2.2. Hausman-type test I: Null and Alternative (cov(sit dit , ci ) = 0)

RE estimator
FE estimator

Null
Consistent and Eﬃcient
Consistent

Alternative
Inconsistent
Consistent

With this data generating process, the bias of pooled LS estimator for α is originated from
wi which causes a diﬀerential correlation between sit and ci across dit while, for FE estimator,
FE transformation eliminate ci and FE estimator for α is consistent. This particular selection
process leads to general missing.
The size and power of the Hausman-type I test is based on the estimators with characteristics in 2.1 under the null and alternative.
As DGP satisfy the conditions for Random eﬀects (RE) estimator to be ideal under the
null we can also carry out Hausman type-I test by stacking RE and FE estimators. For
this case, the size and power of the Hausman-type I test is based on the estimators with
characteristics in 2.2 under the null and alternative.

Simulation results
MC simulations are performed with 1,000 replications with T =4 and various n from 100 to
5,000 and cluster robust standard error is used in all simulations. We choose the size of T =4
to mimic the actual sample size of Tennessee’s Project STAR in the application.
In DGP, a determines the missing proportion of data. Therefore, a=0 implies no missing
62

data so it is a balanced panel while a=0.5 implies 50 percent of data is missing and only
25 percent of data is complete case. Third column and fourth column in tables 2.3 and 2.4
represent the means of pooled LS and FE estimates for α respectively. P-value column shows
that the mean of p-value for Wald statistic, Wa for Hausman-type test I. 1(P < .05) column
shows the mean of the rejection of null at the nominal level 0.05. Finally, 95-CI column
represents, the coverage for 1(P < .05).
Both in table 2.3 and table 2.4, the power of test increases with cross-sectional size, n,
for all a > 0. For example, for the data of same size as Project STAR data for which n=
6,800 in each grade and a=0.5(missing fraction of data), the power of test is 1 in both 2.3
and 2.4. Overall, if n is greater than 5,000, the power of test is 1 regardless of the value for
a(fraction of non-complete data to full data) in the simulation. In short, the power of robust
Hausman-type test using the diﬀerence between pooled LS and FE is quite reliable to test
MCAR for n > 5, 000. As we see in tables 2.3 and 2.4, under alternative DGPs, both bias
and standard error is smaller for RE estimator than those for pooled estimator.
In both 2.3 and 2.4, the panel with a=0 shows the results for balanced panel data and it
reports no size distortion for all n and the coverage rate of rejection frequency contains true
nominal level, 0.05, for all n.

63

Table 2.3. Pooled LS and FE estimates for α and p-value for Wa
a=0
True α Pooled FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
1.003 1.002 0.502
0.054
(0.040,0.068)
n = 500
1
1.003 1.002 0.510
0.062
(0.047,0.076)
n = 1, 000
1
0.996 0.998 0.496
0.047
(0.034,0.060)
n = 2, 000
1
0.999 1.001 0.490
0.047
(0.034,0.060)
n = 5, 000
1
0.998 1.000 0.510
0.048
(0.035,0.061)
a = 0.25 True α Pooled FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
0.810 1.000 0.374
0.155
(0.132,0.177)
n = 500
1
0.812 1.000 0.132
0.533
(0.502,0.564)
n = 1, 000
1
0.804 0.997 0.033
0.846
(0.823,0.868)
n = 2, 000
1
0.810 1.001 0.002
0.990
(0.984,0.996)
n = 5, 000
1
0.807 0.999 0.000
1
(1,1)
a = 0.5
True α Pooled FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
0.758 1.002 0.348
0.182
(0.158,0.206)
n = 500
1
0.754 1.000 0.089
0.664
(0.635,0.693)
n = 1, 000
1
0.747 0.999 0.014
0.936
(0.920,0.951)
n = 2, 000
1
0.751 0.999 0.001
0.995
(0.991,0.999)
n = 5, 000
1
0.750 1.000 0.000
1
(1,1)
*Note: 1(P<.05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; a is
the ratio of missing sample to full sample.; Cluster robust standard error has been used in
all estimations.; p-value is the mean of p-value for Wa , r is rejection rate, and CI-95 is
coverage for r.

64

Table 2.4. RE and FE estimates for α and p-value for Wa
a=0
True α Pooled FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
.999
.999
0.512
0.051
(0.037,0.065)
n = 500
1
1.000 1.000 0.495
0.055
(0.040,0.069)
n = 1, 000
1
1.000 1.001 0.497
0.050
(0.036,0.064)
n = 2, 000
1
1.001 1.002 0.507
0.054
(0.040,0.068)
n = 5, 000
1
1.000 1.000 0.496
0.045
(0.032,0.058)
a = 0.25 True α Pooled FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
0.960 1.000 0.397
0.141
(0.119,0.163)
n = 500
1
0.962 1.001 0.166
0.489
(0.458,0.520)
n = 1, 000
1
0.958 0.998 0.055
0.768
(0.742,0.794)
n = 2, 000
1
0.962 1.001 0.007
0.965
(0.954,0.976)
n = 5, 000
1
0.960 1.000 0.000
1
(1,1)
a = 0.5
True α Pooled FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
0.973 1.005 0.433
0.119
(0.099,0.139)
n = 500
1
0.968 1.000 0.225
0.325
(0.296,0.354)
n = 1, 000
1
0.967 1.001 0.105
0.616
(0.586,0.646)
n = 2, 000
1
0.965 0.999 0.026
0.879
(0.859,0.899)
n = 5, 000
1
0.967 1.001 0.000
1
(1,1)
*Note: 1(P<.05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; a is
the ratio of missing sample to full sample.; Cluster robust standard error has been used in
all estimations.; p-value is the mean of p-value for Wa , r is rejection rate, and CI-95 is
coverage for r.

65

2.3.3

Hausman-type test II - Comparison of θF E and θF D , Null:
strict exogeneity

A test statistic for a Hausman-type test of strict exogeneity is derived. We stack two
unbiased estimators θF E , θF D under strict exogeneity and denote stacked estimator as
θb = (θF E , θF D ) and restriction matrix as R of k × 2k.


√

nθb = 


1
n

n

T

i=1 t=1

−1 

.. ..
sit xit xit

0
1
n

0

n

T

i=1 t=2

sF D ∆xit ∆xit
it









 1
√

1
√
n

n

T

.. ..
sit xit y it

i=1 t=1
n T
sF D ∆xit ∆yit
it
n
i=1 t=2







Using the null H0 : Rθb = 0 and the assumptions for consistency in (2.2.1), we can
construct a Wald statistic which converges asymptotically to a chi-squared distribution with
following assumptions.

Assumptions
1. Conditional strict exogeneity : E(uit |xi , si ) = 0, ∀t
.. ..
T
t=1 sit xit xit )

2. Rank conditions: E(

1
p lim
n

1
p lim
n

n

n

T

and E(

.. ..
sit xit xit

T
FD
t=2 sit ∆xit ∆xit )
T

= E(

i=1 t=1

have full rank

.. ..

sit xit xit )
t=1

T

T

sF D ∆xit ∆xit
it
i=1 t=2

sF D ∆xit ∆xit )
it

= E(
t=2

n

T

.. ..

1
3. Conditions for applying CLT to following two objects, √n
sit xit uit and
i=1 t=1
1
√
n

n

T

i=1 t=2

sF D ∆xit ∆uit .
it

66

Proposition 3 Under assumption 1,2,and 3, for the test with null hypothesis of H0 : Rθb =
0, we can derive a Wald type statistic Wb that converges to χ2 distribution.
k
√
√
√
Wb = [R n(θb )] [Rvar( nθb )R ]−1 R n(θb ) ∼ χ2
k
Proof. See appendix. It can be obtained in similar manner as Hausman type I test.

Singularity A test using FE and FD estimators can only include the coeﬃcients for time
varying covariates.13

2.3.4

Monte Carlo experiment: Test of MCAR using strict exogeneity assumption

We investigate the power of test for DGPs with selection processes which reduce full sample
to either 75, 50 or 25 percent of full sample and induce a diﬀerential correlation between
selection and idiosyncratic errors across dit (treatment and control). Thus, these selection
processes violate strict exogeneity assumption. We focus on how the power of test changes
as we increase the missing proportion of data from 25 percent to 75 percent.

Data generating process
Data generation is based on following outcome and selection process.

The model for DGP
yit = dit · α + xit · β + ft + ci + uit
13 See

Wooldridge [2010] for more discussion.

67

(2.17)

where i = 1, 2, .., n and t = 1, 2, 3, 4
v + w + eit
si1 = 1; sit = 1 if Φ( it √i
> cutof f ) and sit = 0 otherwise
3

(2.18)

Variables generation
h +v
√
1. dit = Φ( it i ) where Φ(·) is standard normal CDF. dit = 1 if dit > 0.5. and
2

dit = 0 otherwise. α = 1, hit ∼ iidN (0, 1) and vi ∼ iidN (0, 1).
2.







 



 0   1 ρ 
 uit 
 ∼ BN   , 



  


0
ρ 1
eit
where BN is bivariate normal and ρ=0.7 in the simulation.
3. xit ∼ iidN (0, 1), β = −0.2
4. ft ∼ unif orm(−1, 1)
√

5. ci ∼

3wi
2

v
+ 2i , wi ∼ iidN (0, 1) and vi ∼ iidN (0, 1)

h +wi+e
6. sit = Φ( it √ it ). sit = 1 if sit > a and sit = 0 otherwise ,
3

where a determines proportion of missing data since sit is normal CDF.
Both FE and FD estimators for α are not valid by a diﬀerential correlation between sit and
uit across dit since sit dit is correlated with uit and the source of diﬀerential correlation hit
is time-varying. The diﬀerential correlation remains even after FE and FD transformations
so that both FE and FD estimators are not reliable for unbalanced panel data with a >0.

68

Simulation results
Table 2.5 reports the size and power of Hausman-type II tests. Simulation performed with
1,000 replications with data set size of n ranges from 100 to 5,000 and T =4.
First, from the a=0 panel, we see in column r ≡ 1(P<.05) that there is no size distortion
and the coverage rate of rejection frequency contains true nominal level 0.05 for all n from
100 to 5,000. Second, the power of test increases with the proportion of missing data, a,
and cross-sectional sample size, n. Third, for Project STAR data of a=0.5 and n=6,000,
the power of test is about 0.6. Therefore, it is quite possible that we may not be able to
reject strict exogeneity assumption even if selection and time-varying unobserved variables
are correlated since ﬁnite sample power is strictly less than 1.

2.3.5

Variable addition test of strict exogeneity

Wooldridge [2009] proposed a simple regression based test of strict exogeneity for FE and
FD estimators in an unbalanced panel data. This test simply adds selection indicator of
sit+1 as an additional regressor in the linear regression model for FD and FE estimations.
Then the t-statistics for the coeﬃcient, λ on sit+1 in (2.19) and (2.20) can be used in t-tests
of null hypothesis of H0 : λ = 0. This is because the strict exogeneity implies any function
of si = {si1 , si2 , ....siT } should not be correlated with ui = (ui1 , ui2 , ...uiT ) conditional on
xi . The rejection of test implies the violation of strict exogeneity and violation of strict
exogeneity is suﬃcient for violation of MCAR assumption so that both unweighted FE and
FD estimators are invalid.

69

Table 2.5. FD and FE estimates for α and p-value for Wb
a=0
True α FD
FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
0.994 1.002 0.492
0.061
(0.046,0.076)
n = 500
1
1.001 1.002 0.504
0.045
(0.032,0.058)
n = 1, 000
1
0.998 0.998 0.513
0.047
(0.034,0.060)
n = 2, 000
1
0.999 0.999 0.499
0.044
(0.031,0.057)
n = 5, 000
1
1.001 1.000 0.507
0.054
(0.040,0.068)
a = 0.25 True α FD
FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
0.831 0.849 0.487
0.078
(0.061,0.095)
n = 500
1
0.834 0.854 0.469
0.061
(0.046,0.076)
n = 1, 000
1
0.829 0.849 0.456
0.082
(0.065,0.099)
n = 2, 000
1
0.831 0.850 0.418
0.121
(0.101,0.141)
n = 5, 000
1
0.830 0.850 0.311
0.263
(0.236,0.290)
a = 0.5
True α FD
FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
0.749 0.802 0.457
0.087
(0.070,0.104)
n = 500
1
0.743 0.803 0.388
0.167
(0.144,0.190)
n = 1, 000
1
0.752 0.802 0.361
0.205
(0.180,0.230)
n = 2, 000
1
0.753 0.808 0.259
0.352
(0.322,0.382)
n = 5, 000
1
0.751 0.807 0.162
0.599
(0.569,0.629)
a = 0.75 True α FD
FE p-value r ≡ 1(P<.05)
95-CI
n = 100
1
0.725 0.879 0.409
0.126
(0.105,0.147)
n = 500
1
0.717 0.869 0.296
0.320
(0.291,0.349)
n = 1, 000
1
0.708 0.869 0.218
0.495
(0.464,0.526)
n = 2, 000
1
0.711 0.869 0.159
0.611
(0.581,0.641)
n = 5, 000
1
0.712 0.866 0.106
0.752
(0.725,0.779)
*Note: 1(P < .05) column shows the mean of the rejection of null at the nominal level
0.05.; a is the ratio of missing sample to full sample.; Cluster robust standard error has
been used in all estimations.; p-value is the mean of p-value for Wb . FD column and FE
column represents the mean of FD and FE estimates for α respectively. Finally, 95-CI
column represents, the coverage for 1(P < .05).

70

..

..

..

..

y it = λ · sit+1 + xit θ + ft + uit , for sit = 1

(2.19)

∆yit = λ · ∆sit+1 + ∆xit θ + ∆ft + ∆uit , for sF D = 1
it

(2.20)

Although we can not directly test weak exogeneity by variable addition test for pooled
estimator in (2.21) since we can not observed all xit , yit for individual i when sit =0, we can
obtain indirect evidence from variable addition test in the pooled LS based on the assumption
of non-zero serial correlation of selection indicator. If we further assume that E(sit |xit ) and
E(sit+1 |xit ) is correlated, we can t-statistic on λ in (2.21) to test weak exogeneity and
MCAR.

yit = λ · sit+1 + xit θ + ft + ci + uit , for sit = 1

(2.21)

Data generating process
We use the same DGP processes as in section 2.3.4 where we study the power of Hausmantype test for strict exogeneity. We only change the model of (2.17) from 2.3.4 to (2.19) and
(2.20) for FE and FD estimators. In these DGPs, the selection and unobserved heterogeneity
or time-varying idiosyncratic errors are diﬀerentially correlated across dit so that both FD
and FE transformations cannot eliminate the correlation and both FD and FE estimators
are not valid.

The power of variable addition test for strict exogeneity assumption
Table 2.6 reports the estimates for λ in (2.19) and (2.20) for FE and FD estimators. In all
simulations, the number of replication is 1,000 with the data set of T =4 and n from 100
71

to 5,000 and the cluster standard error are used in all estimations. Selection indicators are
generated to produce the fraction of missing data to full data to be ranged from a=0.1 to
a=0.75.
For pooled LS estimator, the power of the test for the null, H0 : λ = 0 in (2.21) is 1 for
all a >0 and n ≥500 but this is not a direct test of MCAR for the pooled LS estimator.
However, if we assume further non-zero correlation between E(sit |xit ) and E(sit+1 |xit ), this
test become valid test of weak exogeneity and MCAR.
For both FE and FD estimators, the power of test increases with n and a. For instance,
for the case of a=0.5, the power of test increases from 0.07 to 0.26 for FE estimation while
the power of test increases from 0.07 to 0.67 for FD estimation. Still, the power of test is
signiﬁcantly lower than 1 for the data with the actual size of the Project STAR. The power
of variable addition tests is greater for FD estimator than FE estimator with the same n
and a.
Compared to a robust Hausman-type test at the last column in table (2.6)which uses strict
exogeneity, the power of variable addition test is greater for when a ≤ 0.5 while the power
of variable addition test is lower when a=0.75.

72

Table 2.6. Test of strict exogeneity variable addition test and Hausman test for Wb
LS
a = 0.1
t-stat 1(P<.05)
n = 100
3.40
0.92
n = 500
7.30
1
n = 1, 000 10.35
1
n = 2, 000 14.54
1
n = 5, 000 23.04
1
a = 0.25 t-stat 1(P<.05)
n = 100
3.66
0.96
n = 500
8.14
1
n = 1, 000 11.58
1
n = 2, 000 16.35
1
n = 5, 000 25.91
1
a = 0.5
t-stat 1(P<.05)
n = 100
3.52
0.93
n = 500
7.94
1
n = 1, 000 11.21
1
n = 2, 000 15.97
1
n = 5, 000 25.19
1
a = 0.75 t-stat 1(P<.05)
n = 100
3.04
0.85
n = 500
6.84
1
n = 1, 000 9.62
1
n = 2, 000 13.59
1
n = 5, 000 21.43
1

95-CI
(0.90,0.94)
(1,1)
(1,1)
(1,1)
(1,1)
95-CI
(0.95,0.97)
(1,1)
(1,1)
(1,1)
(1,1)
95-CI
(0.92,0.95)
(1,1)
(1,1)
(1,1)
(1,1)
95-CI
(0.83,0.87)
(1,1)
(1,1)
(1,1)
(1,1)

t-stat
-0.13
-0.33
-0.48
-0.63
-1.06
t-stat
-0.23
-0.66
-0.84
-1.18
-1.85
t-stat
-0.32
-0.79
-1.05
-1.50
-2.37
t-stat
-0.31
-0.64
-0.87
-1.27
-2.04

FD
1(P<.05)
0.08
0.05
0.07
0.10
0.18
1(P<.05)
0.07
0.10
0.15
0.22
0.45
1(P<.05)
0.07
0.12
0.19
0.31
0.67
1(P<.05)
0.10
0.11
0.14
0.25
0.52

FE
95-CI
t-stat 1(P<.05)
95-CI
(0.06,0.09) -0.07
0.07
(0.05, 0.08)
(0.04,0.06) -0.18
0.05
(0.04,0.06)
(0.05,0.09) -0.23
0.06
(0.04,0.07)
(0.08,0.11) -0.39
0.07
(0.06, 0.09)
(0.16,0.21) -0.59
0.07
(0.06, 0.09)
95-CI
t-stat 1(P<.05)
95-CI
(0.05,0.08) -0.12
0.06
(0.04,0.07)
(0.08,0.12) -0.44
0.07
(0.05,0.09)
(0.13,0.17) -0.47
0.08
(0.07,0.10)
(0.20,0.25) -0.67
0.10
(0.08,0.11)
(0.42,0.48) -1.08
0.19
(0.16,0.21)
95-CI
t-stat 1(P<.05)
95-CI
(0.06,0.09) -0.20
0.07
(0.06,0.09)
(0.10,0.14) -0.46
0.07
(0.06,0.09)
(0.16,0.21) -0.60
0.09
(0.07,0.11)
(0.28,0.34) -0.86
0.15
(0.13,0.18)
(0.64,0.70) -1.31
0.26
(0.23,0.29)
95-CI
t-stat 1(P<.05)
95-CI
(0.08,0.12) -0.18
0.09
(0.07,0.11)
(0.09,0.12) -0.36
0.06
(0.04,0.07)
(0.12,0.16) -0.46
0.09
(0.07,0.11)
(0.23,0.28) -0.70
0.11
(0.09,0.13)
(0.49,0.55) -1.12
0.19
(0.16,0.21)

Hausman
1(P<.05)

1(P<.05)
0.08
0.06
0.08
0.12
0.26
1(P<.05)
0.09
0.17
0.21
0.35
0.60
1(P<.05)
0.13
0.32
0.50
0.61
0.75

*Note: 1(P<.05) is the rejection rate of the null of H0 :α=1. a is the ratio of missing sample to full sample.

73

2.4

Estimations with MAR assumption: IPW and MIIPW methods

We introduce the methods under MAR to deal with missing data when MCAR is violated.
MAR assumes that the variables that explain selection (or missing) process are either included in outcome model, as covariates or past outcomes, or they are observed as auxiliary
variables.

p(sit = 1|yit , zit ) = p(sit = 1|zit ) = p(zit ) ≡ Git

(2.22)

where t = 1, 2, 3 and zit is all observed variables at t.
Therefore, under MAR assumption of (2.22), individuals who have the same past outcome
and covariates have the same conditional probability of selection (or missing) status. In
particular, we focus on two of most popular methods to deal with missing data under MAR
assumption which are IPW and MI methods.14
In this section, we ﬁrst introduce the idea of the justiﬁcation for IPW and MI-IPW methods. Then, we examine ﬁnite sample properties for IPW and MI-IPW methods via Monte
Carlo simulation. We consider two DGPs: (i) outcome and selection models are correctly
speciﬁed in ﬁrst DGP and (ii) outcome model is correctly speciﬁed but selection model is
misspeciﬁed. Simulation with misspeciﬁed selection model provides information on the robustness property of IPW estimators for MTE to the violation of MAR where the violation
14 Unfortunately,

it is not possible to test MAR assumption since it is never possible to
know that all appropriate observed variables have been included as predictors in probability
of selection model. Therefore, we suggest a method based on an economic theory for the
speciﬁcation of selection process where the justiﬁcation of valid MAR is economic theory for
the selection process.

74

of MAR is induced by the selection model misspeciﬁcation.

2.4.1

IPW method

We ﬁrst consider IPW method (or eﬃcient version of AIPW method) in Scharfstein et al.
[1999], Wooldridge [2007] and Tsiatis [2006]. These methods are based on the idea that,
complete-case data which are weighted by valid probability of selection can create pseudodata that mimic a random sample from the population of interest. Therefore standard
estimation method applied to inverse probability weighted complete-case provides a consistent estimator. Correctly speciﬁed probability of selection model leads to the satisfaction of
MAR assumption so that IPW estimator is consistent.

Consistency for IPW estimator
We consider a linear model with selection process as in (2.23) and (2.24) and we are primarily
interested in MTE for dit , α.
yit = dit · α + x1it · β + f1t + c1i + uit

(2.23)

where for i = 1, 2, .., n, t = 1, 2, 3, 4, dit is treatment status indicator and x1it is time-varying
continuous covariates.
We exclusively focus on the missing pattern of monotone drop-out (i.e. attrition) in the
modeling of selection process for IPW estimator.15
pi1 = 1; pit = 1 if Φ(zit · γ + f2t + c2i + eit > a)
15

(2.24)

We consider both monotone and non-monotone missing in the modeling of selection
process for MI-IPW estimator.

75

and pit = 0 otherwise for i = 1, 2, .., n and t = 2, 3, 4. Then the objective function is provided
in (2.25) for the case of balanced panel data model.
n

4

mit (Wit , θ) = 0 where mit = xit vit

(2.25)

i=1 t=1

where vit = ci + uit , xit = (dit , x1it , ft ) and Wit = (yit , xit ).
As we introduce missing data, we can rewrite the objective function (2.25) for unbalanced
panel data with complete-case as the following estimating equation of (2.26).
n

4

sit · mit (Wit , θ) = 0

(2.26)

i=1 t=1

The estimating equation for IPW estimator with complete-case in unbalanced panel data
is provided in (2.27).
n

4

i=1 t=1

sit
· mit (Wit , θ) = 0
pit

(2.27)

We suppose pit is obtained and converges to E(sit |Wit ) in (2.28). Then, under MAR
assumption, we can extend the consistency result of unweighted pooled LS estimator in
balanced panel data to IPW-pooled IPW estimator of (2.28) in unbalanced panel data.

n

T

βP LS = (
i=1 t=1

sit xit xit −1
) (
p(zit )

n

T

i=1 t=1

sit xit yit
)
p(zit )

(2.28)

where pit is estimated with either probit or logit model in the application.
Recall ﬁrst that the crucial exogeneity condition for Pooled LS estimator to be consistent
in balanced panel is given in (2.29).
T

E(

xit (ci + uit )) = 0

(2.29)

t=1

We show the exogeneity condition of (2.29) for IPW-pooled LS estimator as below. We
start from (2.30).
76

T

sit xit (ci + uit ))

E(

(2.30)

t=1

Using law of iterated expectation and MAR, we can rewrite (2.30) as the following.
T

E(

T

t=1

t=1
T

T

P (sit = 1|yit , zit )xit (ci + uit ))

= E(

E(sit |yit , zit )xit (ci + uit ))

sit xit (ci + uit )) = E(
= E(

p(sit = 1|zit )xit (ci + uit ))
t=1

t=1
T

= E(

p(zit )xit (ci + uit ))
t=1

Thus, provided that p(zit ) is the consistent estimate for the correctly speciﬁed p(zit ),
we obtain consistent IPW-Pooled LS estimator if conditional strict exogeneity of (2.31) is
satisﬁed.
T

E(
t=1

sit xit (ci + uit )
) = E(
p(zit )

T
t=1
T

= E(

x (ci + uit )
E(sit |yit , zit ) it
) = E(
p(zit )

T
t=1

x (ci + uit )
p(zit ) it
)
p(zit )

xit (ci + uit )) = 0

(2.31)

t=1

As shown in (2.31), MAR assumption for selection process is critical for the validity of
IPW method.

An example of using MAR for IPW Estimator
We consider the estimation of MTE for CSR on student achievement with project STAR
data in illustrating the use of MAR assumption for the probability of selection.
We consider IPW-Pooled LS estimator for missing data.
n

T

βP LS = (
i=1 t=1

sit xit xit −1
) (
p(zit )

77

n

T

i=1 t=1

sit xit yit
)
p(zit )

Speciﬁcation for the probability of selection We estimate p(sit = 1|zit ) using binary
logit and probit with maximum likelihood model for Git (zit , γ0 ).

Git (zit , γ) ≡ p(sit = 1|zit ) = p(zit ), t = 2, 3, 4
zit = (xit−1, yit−1 ), or zit = (xit−1,..., x1 , yit−1 , .., yi1 )
where zit is observed at t
(i) xit−1 : class types (initial small, initial aide, cumulative years in small class, cumulative years in small class) , student characteristics (gender, race, free lunch status), teacher
characteristics (race, experiences, highest degree dummy), class characteristics (fraction of
classmates with kindergarten attendance, fraction of classmates with free lunch, fraction of
female classmates , fraction of minority classmates) (ii) yit−1 : previous year SAT percentile
score
We use logit binary response models for probability of selection
πit = p(sit = 1|zit , sit−1 = 1) = P (zit γ + it > 0) = Λ(zit γ), sit−1 = 1.
where Λ(zit γ) =

exp(zit γ)
1 + exp(zit γ)

For the case of using logit model for selection probability, we can use conditional expectation
argument sequentially to estimate selection probability as follows.
Gi1 = 1, Gi2 = πi2 , Gi3 = πi3 · πi2 , Gi4 = πi4 · πi3 · πi2

Gi1 = 1, Gi2 = Λ(si1 · zi2 γ), Gi3 = Λ(si1 · zi2 γ) · Λ(si2 · zi3 γ)

Gi4 = Λ(si1 · zi2 γ) · Λ(si2 · zi3 γ) · Λ(si3 · zi4 γ)
78

Once we obtain γ from logit estimation, we can construct weights for inverse probability
weighted (IPW) pooled LS estimator as in (2.32).
n

T

n

T

sit xit xit −1
sit xit yit
βP LS = (
) (
)
Git
Git
i=1 t=1
i=1 t=1

(2.32)

Idea behind the consistency of IPW estimator
Under MAR, those cross-section units that have the same predictors values (past outcome,
covariates and auxiliary variables) should have the same probability of selection (missing).
Thus, remaining cross-section units with predictors of low(high) probability of selection get
high weights and play the role of as many cross-section units who leave sample and had the
same predictors values. Using valid probability weights eliminates bias due to diﬀerential
attrition across treatment and control. For instance, in the study of the MTE for CSR on
SAT score, if those students who left sample and might have perform well if they remained
in sample without treatment were exclusively in regular class, ignoring missing distorts MTE
of small class since students in regular class misrepresent population. IPW method recovers
random sample property of balanced panel since remaining students with the same predictors
in probability of selection model as students who left sample play the role of themselves and
missing students by receiving more weights.

2.4.2

AIPW estimator

The objective function of (2.33) provide an AIPW(Augmented IPW) estimator which is
consistent and optimally eﬃcient if data are MAR and conditional expectation is correctly

79

speciﬁed.
n

4

i=1 t=1

s
s
[ it · mit (Wit , θ) + (1 − it )E(mit |It−1 )] = 0
pit
pit

(2.33)

where It−1 is all information available at t − 1.
IPW estimator is consistent if probability of selection is correctly speciﬁed but is generally
ineﬃcient. While an AIPW estimator can be eﬃcient and is consistent if either a model
speciﬁcation for probability of selection or a model speciﬁcation for conditional expectation.
(i.e. joint distribution of missing variable) An eﬃcient AIPW estimator can be constructed
with appropriate model for conditional expectation in (2.33).16
Unfortunately, we do not have much information for parametric speciﬁcation of the model
on conditional expectation, E[sit · mit (Wit , θ)] = 0, to use in a practical application since
it basically requires a correct speciﬁcation for joint distribution of missing variables and,
therefore, in the implementation, we focus on either the consistency of IPW estimator which
only requires correct speciﬁcation for probability of selection or MI-IPW estimator for which
MI is based on MICE for E[sit · mit (Wit , θ)] = 0. In the application of MI, imputation is
16 AIPW

is doubly robust and an estimator is called doubly robust if it is consistent when
either a model for conditional expectation or a model for selection is correctly speciﬁed.
Moreover, AIPW estimator is eﬃcient as it delivers semi-parametric eﬃciency bound among
4
the class of estimators that use n
i=1
t=1 sit · mit (Wit , θ) = 0. For more discussion on
AIPW estimators is provided in Tsiatis [2006] and Robins et al. [1994]. Recently, we
observed theoretical development of AIPW estimator(doubly robust(DR) estimator) which
combines IPW method with imputation method as in equation (2.33). For instance, the
ﬁrst-term in objective function (2.33) is the same as IPW use for identiﬁcation of α with
complete-case and second-term in (2.33) is the same objective function as in imputation
model.17 Theoretical attractiveness of DR estimator such as robustness and eﬃciency has
been studied in great details in Carpenter et al. [2006], Tsiatis [2006], Robins and Rotnitzky
[1995], and Wooldridge [2007]. However, up to our knowledge, there is no simulation study to
evaluate the usefulness of theoretical properties of IPW or DR estimators for the regression
model with high-dimensional missing covariates of various forms including binary, count,
ordinal and fraction. Most of simulation studies has only one missing continuous variable in
the model and uses joint multivariate normal distribution for conditional expectation term in
the simulations.(Bang and Robins [2005], Carpenter et al. [2006] and Graham et al. [2008]).

80

applied only to cross-section units who are remained in the sample with missing variables
but not to cross-section units who leave sample once but come back to sample later.

2.4.3

Multiple imputation

We consider imputation based method to deal with missing data as in Rubin [1987], Buuren et al. [1999] and Ragnuthan et al. [2001]. Although multiple imputation has become
increasingly popular in epidemiology, psychology, statistics and other social sciences, up to
our knowledge, there has been no MI method appropriately designed for multiple treatments and missing occurs for outcome and all of covariates with various types. However,
high-dimensional covariates with general missing or attrition is quite common in economic
applications and the necessity of correct speciﬁcation for the joint distribution of all missing
covariates and outcome makes MI method unattractive in practice. For instance, in the
application with Project STAR data, missing occurs in outcome and covariates including
class size, free lunch status dummy, teacher’s years of experience, teacher’s master degree
dummy, teacher’s race dummy, fraction of female classmates, fraction of minority classmates
and so forth. Therefore, we restrict the use of imputation based method to within sample
individuals who have missing variables so that we do not apply imputation for the units who
are not in the sample so outcome and all covariates are missing. In most application, for the
cross-section units in the sample with missing variables, it is very rare that the number of
missing variables is more than three so the burden of approximation is reduced signiﬁcantly
compared to imputing all variables for cross-section units who leave sample. In implementation, we apply a MI method using speciﬁcation of a logistic distribution for binary variables

81

and normal distribution for all other continuous variables.
Among many MI methods used in practice, we implement fully conditional approach of
multiple imputation in Royston [2004] and Royston [2005], using Stata ice command. 18

Set-up for MICE Estimator
Conditional predictive distribution is estimated from the observed data under MAR. Missing
variable are multiply imputed for within sample units. In implementation, we adopt a
univariate conditional models in a Gibbs sampler type approach in Buuren et al. [1999].
We model each variable separately conditional on others observed variables. Sequences of
missing variables imputation follow from most sample to least sample in terms of number of
sample.

yit = (yO , yM ), yO : observed , yM : missing

(2.34)

xit = (xO , xM ), xO : observed , xM : missing

(2.35)

We need to model all variables with missing data as in (2.36) and (2.37).
f (yM,it |WO,it−1 ) , ∀t

(2.36)

f (xM,it |WO,it−1 ) , ∀t

(2.37)

where WO,it−1 is all observed variables at t − 1.
18

There are currently two widely used methods of model-based imputation: multiple
imputation based on multivariate normal distribution for missing variables and multiple
imputation by chained equation(MICE) in Buuren et al. [1999]. We implement a MI method
as comparison purpose to IPW method since it is readily available to most of statistical
packages such as Stata, SAS, and R.
82

We conduct multiple imputation by chained equations method in Stata implementation
of Royston [2005] and type in below command.
ice y d x ‘auxiliary’ using temp.dta, m(10) cmd(d:logit)
where m(10) implies that ice produces 10 imputed data sets.

2.4.4

MI-IPW method

In unbalanced panel data, we suggest MI-IPW method for which implementation is based
on the following. First, we sort data for each cross-section unit at each time into four
category: complete case, monotone(attrition), non-monotone-I, and non-monotone-II. Nonmonotone-I missing is for cross-section unit which is in sample but has missing variables
while non-monotone-II missing is cross-section unit who leave sample but re-enter sample
later. We assume MCAR for cross-section unit with missing patterns of non-monotone-II
so that we ignore these types of data and still obtain valid statistical analysis. Second,
we impute missing variables for non-monotone-I unit using the imputation methods in 2.4.3.
Third, by treating imputed non-monotone-I units as complete case, we combine complete case
and imputed non-monotone-I data and apply IPW. We obtain MTE αj and its covariance
matrix Vj . Finally, we iterate the second and third procedures multiple(M ) times. By
extending Rubin’s formula(Rubin [1987]) for MI method to MI-IPW method, we obtain
MI-IPW estimator for coeﬃcient and variance matrix as in (2.38).

1
α=
M
where B = M1
−1

M
j=1 (αj

M
j=1

1
αj , V =
M

− α)(αj − α)
83

M

Vj +
j=1

M +1
B
M

(2.38)

As probability weight in IPW method eliminates bias due to diﬀerential missing while MI
for missing variables enhances the eﬃciency. Both IPW and MI method use MAR assumption
in the speciﬁcation of probability of selection (missing) and missing variables.

2.5

Monte Carlo experiment

We adopt data generating processes (DGPs) with monotone missing data(i.e. missing by
attrition) so that data missing occurs simultaneously for outcome and all covariates for a
cross-section unit. The covariates in outcome model include both discrete and continuous
variables. We are primarily interested in obtaining consistent MTE using IPW method.

2.5.1

DGP I: Correct speciﬁcation of both outcome and probability of selection models

We start with a DGP I which satisﬁes the correct speciﬁcations for both outcome and
selection models. Conditional on auxiliary variables c2it , selection is not correlated with
unobserved errors in outcome equation so that MAR is satisﬁed for DGP I.

Model for data generating process

We use a linear model with single treatment and

single covariate for DGP. We generate data with n from 100 to 5,000 and t is 4 to mimic the
actual size of Project STAR data.
yit = dit · α + xit · β + c1i + uit
where i = 1, 2, ..., n, dit is a binary variable and xit is continuous variable.
si1 = 1; sit = 1 if fi + c2it + it > a and sit−1 = 1, for i = 1, 2, .., n and t = 2, 3, 4
84

where a is the cut-oﬀ value that determines the fraction of missing sample to full sample
and a=0, -1, -2 are chosen and corresponding fraction of missing data to full data is 75, 50,
and 25 percent, respectively.

Variables generation for DGP I
1. dit = 1 if fi + qit > 0 and dit = 0 otherwise, where fi ∼ iidU (0, 1) , qit ∼ iidN (0, 1)
and α =1
2. xit ∼ iidU (0, 1), β = −0.2
3. c1i = 0
4. uit = rit + vit where rit ∼ iidU (0, 1) and
  





0  
 
 
0  
 
 , 
 
0  
 
 
 
0

1 ρ ρ ρ 


ρ 1 ρ ρ 



ρ ρ 1 ρ 



ρ ρ ρ 1













vi1 






vi2 


 ∼ M V N 



vi3 






vi4

where M V N is multivariate normal and ρ=0.7 in the simulation.
5. si1 = 1 and sit = 1 if sit−1 = 1 and fi + c2it + it > a where it ∼ iidN (0, 1), and
c2it = rit + qit .

Selection process and the potential source of bias The correlation among selection,
treatment and unobserved errors are induced to be strictly positive in DGP. First, for a
balanced data, dit is not correlated with unobserved errors so that balanced panel unweighted
estimators are consistent. The correlation between selection and treatment is induced by
85

fi and qit while the correlation between selection and unobserved errors are induced by
rit . Therefore, there is strictly positive diﬀerential correlation between selection and errors
across dit . Therefore, all pooled LS, FE and FD estimators should show ﬁnite sample bias.
However, under MAR(i.e. conditional on c2it = rit +qit and fi ), the selection is not correlated
with unobserved errors so both pooled LS and FD estimators with correct speciﬁcation of
probability of selection should not show bias more than sampling error.

Estimation of probability weight for IPW method

Estimation of probability weight

is based on (2.39) and (2.40). In Stata, we estimate pit for t = 2, 3, 4 using a command (probit
sit fi c2it if sit−1 =1 or logit sit fi c2it if sit−1 =1)

πi1 = 1; pit (sit = 1|zit , sit−1 = 1) = πit = Φ(zit · γ + eit > 0)

(2.39)

where eit is standard normal for i = 1, 2, .., n and t = 2, 3, 4.

pit = πit · πit−1 . . . πi2 , ∀i and for t = 2, 3, 4

(2.40)

pit for t = 2, 3, 4 using a command in Stata as follows:
probit s f c2 if s_t-1=1
In this DGP, probit model with predictors, fi and c2it , provides the correct speciﬁcation
for probability of selection.

Imputation of missing variables using MICE
for y, d, x using following speciﬁcations.

86

In Stata, we impute missing variables

Table 2.7. Speciﬁcation of models for missing variables
variable
yM it
dM it
xM it

distributional assumption
Normal
Logistic
Normal

command
regress
logit
regress

prediction equation
dOit xOit cO2it fOi
yOit xOit cO2it fOi
yOit dOit cO2it fOi

M represents missing variable and O represent observed variable. In implementation, we
type in a Stata command as follows (Royston [2005]): ice y d x c2 f using temp.dta, m(10)
cmd(d: logit).

Simulation result: Correctly speciﬁed outcome and selection models
Table 2.8 reports unweighted estimates for pooled LS, FE and FD with DGP I. It shows
that unweighted estimates are all biased since MCAR is violated. The biases of unweighted
estimators do not depend on n but the biases for unweighted estimators decrease as a (missing
fraction from full data) falls. Interestingly, all unweighted estimators underestimate the
marginal treatment eﬀect (MTE). The magnitude of biases among unweighted pooled LS ,
FE and FD estimators are not much diﬀerent from each other.
Table 2.9 shows that IPW estimates for pooled LS and FD obtained using data from DGP
I. As theory predicts, both IPW-pooled LS and IPW-FD estimators show no signiﬁcant ﬁnite
sample bias as the mean of MC point estimates converges to true value 1 as n increases for
all a. For the inference with IPW LS estimator, the rejection frequency converges to true
nominal level 0.05 as n increases and the coverage rate of rejection frequency contains 0.05
for all n and a ≤0.5. On the other hand, IPW-FD estimates are getting close to true MTE
as n increases while the rejection frequency does not converges to 0.05 and the coverage rate
of rejection frequency does not contain 0.05 for all n and a. There is over-rejection in the
inference of IPW-FD estimator even for n=5,000 and all a. For example, the mean of MC

87

rejection frequency is .09 at the nominal level of .05.
Table 2.10 reports MI estimates for pooled LS, FE and FD with DGP I. The mean of
MC point estimate for MTE of pooled LS is quite close to true value 1 while rejection rate
coverage fails to contain 0.05 for all a and n. The means of MC point estimates for MTE
and associated rejection frequency of FD are all biased. MI estimates for LS, FE and FD
provide incorrect standard errors in ﬁnite sample.

88

Table 2.8. DGP I: MTE estimates for unweighted estimators
Unweighted
number of n missing fraction
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25
5,000
0.75
5,000
0.5
5,000
0.25

LS

Mean(α)
0.968
0.974
0.987
0.963
0.974
0.985
0.963
0.974
0.988
0.961
0.974
0.987

FE

SD(α)
.152
.111
.090
.099
.071
.058
.071
.048
.041
.031
.022
.018

Mean(α)
.951
.968
.986
.951
.969
.984
.951
.968
.985
.951
.969
.985

FD

SD(α)
.142
.082
.065
.086
.053
.040
.062
.037
.028
.026
.016
.013

Mean(α)
.952
.967
.987
.951
.970
.984
.951
.967
.985
.951
.969
.985

*Note: Missing fraction is the fraction of missing sample to full sample.

89

SD(α)
.155
.093
.073
.095
.059
.046
.068
.042
.033
.029
.018
.015

Table 2.9. DGP I: MTE estimates for IPW estimators
IPW
n
% missing
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25
5,000
0.75
5,000
0.5
5,000
0.25

LS

Mean(α)
1.001
.997
1.001
.996
1.001
.998
.997
1.001
1.000
.994
1.001
1.001

SD(α)
.246
.142
.096
.179
.091
.061
.137
.065
.043
.070
.030
.019

FD

r
coverage of r
.086
(.074,.099)
.060
(.050,.070)
.050
(.040,.060)
.080
(.067,.091)
.047
(.037,.056)
.049
(.040,.058)
.078
(.066,.089)
.047
(.037,.056)
.049
(.039,.058)
.055
(.045,.064)
.048
(.038,.057)
.046
(.036,.055)

Mean(α)
.986
.993
1.000
.985
.998
.997
.986
.996
.998
.992
.997
.999

SD(α)
.238
.125
.081
.169
.079
.050
.127
.057
.036
.065
.026
.016

r
coverage of r
.143
(.128,.159)
.090
(.077,.103)
.090
(.077,.103)
.122
(.107,.136)
.087
(.075,.099)
.086
(.074,.098)
.106
(.092,.119)
.096
(.083,.109)
.086
(.074,.098)
.087
(.074,.099)
.082
(.070,.094)
.092
(.079,.105)

*Note: r ≡ 1(p < .05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; Cluster robust standard error has
been used in all estimations.

90

Table 2.10. DGP I: MTE for estimators with multiple imputations
MI
n
% missing
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.006
1.007
1.002
1.013
1.003
1.004
1.011
1.006
1.000

FE

SD(α)
.168
.112
.094
.106
.071
.059
.074
.051
.040

Mean(α)
.980
.994
.996
.988
.991
.999
.987
.994
.998

SD(α)
.161
.093
.068
.101
.059
.042
.071
.042
.029

FD

r
coverage of r
.551
(.529,.573)
.261
(.242,.280)
.128
(.113,.142)
.563
(.541,.585)
.277
(.257,.297)
.108
(.095,.122)
.559
(.537,.581)
.274
(.255,.294)
.096
(.083,.108)

Mean(α)
.947
.973
.984
.951
.968
.986
.952
.970
.985

SD(α)
.157
.091
.074
.097
.058
.046
.066
.041
.033

r
coverage of r
.089
(.077,.102)
.065
(.054,.076)
.064
(.053,.074)
.093
(.080,.106)
.093
(.080,.106)
.065
(.054,.076)
.110
(.096,.124)
.107
(.094,.121)
.082
(.070,.094)

*Note: r ≡ 1(p < .05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; Cluster robust standard error has
been used in all estimations.

91

2.5.2

DGP II: The correct probability of selection model with
endogenous treatment eﬀect

Consider the same data generating process as DGP I except ci1 = fi + ai so that, even
for a balanced panel data, the treatment, dit , is correlated with unobserved heterogeneity.
However, FE or FD transformation eliminates the source of bias in balanced panel data. In
unbalanced panel data, unweighted pooled LS, FE and FD estimators are biased due to the
correlations between selection and the treatment and between selection and time-varying
errors. However, under MAR for probability of selection, IPW-FD estimator can eliminate
the source of diﬀerential correlation between selection and unobserved heterogeneity across
dit . Once MAR assumption holds and the probability of selection is correctly speciﬁed,
IPW-FD estimator should not show any bias more than ﬁnite sample error in simulation.

Simulation results
In table 2.11, unweighted pooled LS, FE and FD estimators show ﬁnite sample bias for the
data with all n and a > 0. Unweighted Pooled LS estimator overestimates MTE while FE
and FD estimators underestimate MTE for all n and a. The biases do not depend on n for
all unweighted pooled LS, FE and FD estimators while the biases decrease as a (the fraction
of missing sample to full sample) decreases.
Table 2.12 reports that IPW LS is biased and overestimates MTE. Compared to unweighted LS, IPW LS estimates has much larger variance but somewhat smaller bias. The
bias for IPW LS estimator does not depend on n and a and the size distortion is prohibitively
large for all n and a. On the other hand, IPW FD estimates are very close to true marginal

92

eﬀect 1 while rejection frequency shows some evidence of over-rejection. Although the size
distortion decreases as n increases, over-rejection problem remain even for n=5,000 and all
a. The size distortion for IPW FD estimator does not disappear even for n=20,000.
Table 2.13 shows pooled LS FE and FD with MI . MC estimates show ﬁnite sample bias
for all n and a. Pooled LS estimator with MI overestimates the marginal treatment eﬀect
while MI FD estimators underestimate the marginal treatment eﬀect for all n and a. The
biases do not depend on n for both MI LS and MI FD estimators while the biases decrease
as a (fraction of missing sample to full sample) decreases. The mean of MC point estimates
for MI FE estimator shows that the bias is quite small for all n and a ≤0.5 while the size
distortion is large and is negatively correlated to a.

93

Table 2.11. DGP II: MTE for unweighted estimators
Unweighted
number of n missing fraction
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25
5,000
0.75
5,000
0.5
5,000
0.25

LS

Mean(α)
1.144
1.142
1.136
1.150
1.145
1.138
1.148
1.143
1.137
1.148
1.144
1.138

FE

SD(α)
.171
.122
.098
.110
.077
.060
.075
.054
.043
.034
.024
.020

Mean(α)
.949
.968
.985
.952
.969
.985
.951
.970
.985
.952
.969
.985

FD

SD(α)
.140
.083
.064
.088
.051
.040
.059
.037
.028
.027
.016
.013

Mean(α)
.949
.968
.985
.952
.969
.984
.952
.970
.985
.952
.969
.985

SD(α)
.154
.094
.064
.098
.058
.046
.065
.041
.031
.030
.018
.014

*Note: Missing fraction is the fraction of missing sample to full sample. Sensitivity analysis is conducted for unweighted LS,
FE and FD estimators.

94

Table 2.12. DGP II: MTE estimates for IPW estimators
IPW
n
% missing
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25
5,000
0.75
5,000
0.5
5,000
0.25

LS

Mean(α)
1.129
1.131
1.132
1.131
1.134
1.133
1.136
1.132
1.133
1.136
1.133
1.134

SD(α)
.272
.150
.104
.196
.077
.063
.147
.072
.046
.074
.033
.021

FD

r
coverage of r
.151
(.136,.167)
.181
(.164,.198)
.261
(.242,.280)
.175
(.158,.191)
.344
(.323,.365)
.542
(.520,.564)
.253
(.233,.272)
.538
(.516,.560)
.831
(.815,.847)
.620
(.598,.641)
.972
(.965,.979)
.999
(.997,1)

Mean(α)
.983
.993
.997
.987
.995
.997
.992
.998
.998
.991
.997
.999

SD(α)
.235
.118
.080
.170
.079
.050
.127
.058
.035
.064
.026
.015

r
coverage of r
.154
(.139,.171)
.098
(.084,.110)
.094
(.081,.106)
.129
(.114,.143)
.090
(.077,.103)
.091
(.078,.104)
.085
(.072,.097)
.084
(.071,.096)
.083
(.070,.095)
.085
(.073,.097)
.073
(.062,.084)
.076
(.064,.088)

*Note: r ≡ 1(p < .05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; Cluster robust standard error has
been used in all estimations.

95

Table 2.13. DGP II: MTE estimates for estimators with multiple imputations
MI
n
% missing
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.141
1.142
1.139
1.145
1.141
1.136
1.143
1.137
1.136

FE

SD(α)
.176
.121
.100
.111
.076
.060
.078
.053
.045

Mean(α)
.982
.993
.998
.989
.994
.997
.986
.991
.997

SD(α)
.163
.096
.067
.107
.060
.041
.073
.043
.030

FD

r
coverage of r
.547
(.525,.569)
.270
(.251,.290)
.107
(.094,.121)
.578
(.557,.600)
.259
(.239,.278)
.100
(.087,.113)
.572
(.550,.594)
.289
(.269,.309)
.099
(.087,.113)

Mean(α)
.950
.968
.985
.953
.970
.985
.949
.968
.984

SD(α)
.154
.094
.072
.097
.059
.046
.066
.040
.032

r
coverage of r
.073
(.062,.085)
.075
(.064,.087)
.056
(.046,.066)
.087
(.075,.100)
.084
(.072,.096)
.057
(.047,.067)
.117
(.102,.131)
.114
(.100,.128)
.080
(.068,.092)

*Note: r ≡ 1(p < .05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; Cluster robust standard error has
been used in all estimations.

96

2.5.3

The robustness of IPW estimator to misspeciﬁcation of the
probability of selection

This section provides the simulation results for the sensitivity of probability of selection
model misspeciﬁcation. One of main criticism for IPW method is its instability of estimates
to the misspeciﬁcation of the probability of selection. We test the sensitivity of IPW estimator to the misspeciﬁcation of probability of selection model using a DGP in which an
omitted variable induces misspeciﬁcation of probability of selection. For instance, in STAR
empirical analysis, it is possible that we possibly omit some important family characteristics
in the selection model.

DGP with selection model misspeciﬁcation We use a linear model with a treatment
and a covariate. We generate data with n from 100 to 5,000 and t=4 to mimic the actual
size of Project STAR data.
yit = dit · α + xit · β + c1i + uit , for i = 1, 2, .., n and t = 1, 2, 3, 4
where dit a binary variable and xit is continuous variable.
Variable generation conducted as follows.
1. dit = 1 if fi + qit > 0 and dit = 0 otherwise, where fi ∼ iidU (0, 1) , qit ∼ iidN (0, 1)
and α =1
2. xit ∼ iidU (0, 1), β = −0.2
3. c1i ∼ iidU (0, 1)

97

4. uit = rit + vit where rit = fi + iidU (0, 1) and

  













vi1 







vi2 

 ∼ M V N 



vi3 






vi4

0  
 
 
0  
 
 , 
 
0  
 
 
 
0


1 ρ ρ ρ 


ρ 1 ρ ρ 



ρ ρ 1 ρ 



ρ ρ ρ 1

where M V N is multivariate normal and ρ=0.7 in the simulation.
The selection is complicated by including an interaction term which is omitted in the logit
model estimation.
si1 = 1 and sit = 1 if sit−1 = 1 and fi + δyit−1 ∗ c2it + c2it + it > a
where it ∼ iidN (0, 1) and c2it = rit + qit and where a is the cut-oﬀ value that determines
the fraction of missing sample to full sample and a=0, -1, -2 are chosen and corresponding
fraction of missing data are about 75, 50, and 25 percents, respectively. For correctly speciﬁed selection model, IPW FD estimator is consistent since the unique source of diﬀerential
correlation, fi , can be eliminated by FD transformation.
The probability weights for IPW are obtained using following model speciﬁcation for selection process, πit , which is given by:
πit =

exp(zit γ)
, t = 2, 3, 4 for sit−1 = 1
1 + exp(zit γ)

where zit =(fi , c2it ) and the omission of interaction term and logit estimation instead of
probit lead to misspeciﬁcation of selection model.

Simulation Result: Sensitivity of IPW estimators to probability of selection
model misspeciﬁcation We simulate with four values for the parameter of misspeciﬁ98

cation, δ=0, 0.2, 0.5, and 1 where δ=0 implies correct speciﬁcation. As the values of δ
increases, the degree of misspeciﬁcation increases. For instance, in DGP with δ=0.2, the
variation by the omitted variable (yit−1 ∗ c2it ) represents about 3 percent of total variation.
Moreover, as δ increases to .5 and 1, the fraction of variation due to the variation of the
omitted variable represents 10 and 25 percent of total variation respectively. This simulation
intends to clarify the impact of misspeciﬁcation of selection since we possibly do not observe
some important variables in the selection model. Tables from 2.14 to 2.21 show simulation
results. Even with the misspeciﬁcation for the probability of selection model, the bias of
IPW estimators is no greater than the bias of unweighted estimators with δ ≤0.5. Figure
2.1 compares biases between unweighted and IPW methods for LS and FD estimators. Especially if missing fraction is less than or equal to 0.5, the biases for IPW estimators are less
than those of unweighted estimators in the presence of misspeciﬁcation of selection model
for δ ≤0.5. In sum, simulation results show that the estimates from IPW FD estimator does
not exacerbate the bias even if selection model is misspeciﬁed when it is compared to the
estimates from unweighted FD estimator with δ ≤0.5.

99

Table 2.14. Selection model misspeciﬁcation: MTE for unweighted estimators, δ=0
Unweighted
number of n missing fraction
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.265
1.267
1.269
1.257
1.258
1.261
1.251
1.253
1.257
1.259
1.259
1.260

FE

SD(α)
.246
.169
.154
.179
.115
.106
.111
.072
.066
.081
.049
.045

Mean(α)
.955
.973
.981
.951
.976
.983
.949
.973
.982
.951
.975
.982

FD

SD(α)
.197
.102
.092
.136
.073
.064
.086
.044
.040
.063
.032
.028

Mean(α)
.953
.975
.981
.951
.977
.984
.948
.974
.982
.951
.974
.982

SD(α)
.219
.118
.104
.147
.082
.073
.096
.052
.046
.070
.036
.032

*Note: Missing fraction is the fraction of missing sample to full sample. δ represents the degree of misspeciﬁcation of the
selection model.

100

Table 2.15. Selection model misspeciﬁcation: MTE for IPW estimators, δ=0
IPW
n
% missing
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.271
1.271
1.275
1.250
1.261
1.265
1.243
1.263
1.263
1.255
1.266
1.266

SD(α)
.354
.193
.165
.275
.134
.115
.199
.086
.073
.153
.058
.049

FD

r
coverage of r
.209
(.181,.236)
.342
(.313,.371)
.413
(.383,.444)
.233
(.207,.260)
.552
(.521,.583)
.663
(.634,.692)
.384
(.354,.414)
.876
(.856,.896)
.955
(.942,.968)
.553
(.522,.584)
.990
(.984,.996)
.998
(.995,1)

Mean(α)
.989
.994
.994
.982
.998
1.000
.984
.998
.998
.990
.998
.998

SD(α)
.299
.139
.113
.216
.097
.083
.160
.064
.051
.116
.043
.035

r
coverage of r
.171
(.146,.196)
.090
(.072,.108)
.081
(.064,.098)
.120
(.100,.140)
.087
(.070,.104)
.079
(.062,.096)
.116
(.096,.136)
.088
(.070,.106)
.077
(.060,.094)
.097
(.079,.115)
.084
(.067,.101)
.074
(.058,.090)

*Note: r ≡ 1(p < .05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; Cluster robust standard error has
been used in all estimations.

101

Table 2.16. Selection model misspeciﬁcation: MTE for unweighted estimators, δ=.2
Unweighted
number of n missing fraction
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.401
1.303
1.288
1.417
1.317
1.295
1.418
1.316
1.293
1.415
1.314
1.292

FE

SD(α)
.240
.157
.146
.169
.112
.104
.107
.072
.067
.075
.050
.047

Mean(α)
.938
.973
.984
.941
.973
.982
.941
.974
.982
.939
.975
.983

FD

SD(α)
.165
.095
.089
.115
.069
.064
.075
.043
.040
.050
.030
.028

Mean(α)
.926
.966
.978
.929
.966
.977
.932
.968
.977
.929
.969
.979

*Note: Missing fraction is the fraction of missing sample to full sample.

102

SD(α)
.183
.110
.102
.126
.078
.073
.084
.049
.045
.057
.035
.032

Table 2.17. Selection model misspeciﬁcation: MTE for IPW estimators, δ=.2
IPW
n
% missing
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.381
1.288
1.281
1.405
1.305
1.289
1.401
1.304
1.286
1.395
1.302
1.286

SD(α)
.300
.158
.144
.216
.116
.105
.140
.074
.068
.095
.050
.047

r ≡ 1(p < .05)
.318
.458
.488
.533
.780
.809
.855
.992
.989
.980
1
1

FD

coverage of r
(.289,.347)
(.426,.489)
(.455,.521)
(.502,.564)
(.754,.806)
(.784,.883)
(.833,.877)
(.986,.998)
(.983,.995)
(.971,.989)
(1,1)
(1,1)

Mean(α)
.930
.970
.981
.937
.970
.981
.942
.973
.981
.940
.974
.982

SD(α)
.231
.115
.103
.163
.081
.074
.109
.051
.046
.073
.036
.033

r ≡ 1(p < .05)
.141
.114
.106
.129
.116
.107
.141
.137
.129
.185
.181
.134

coverage of r
(.119,.163)
(.094,.134)
(.086,.126)
(.108,.150)
(.096,.136)
(.088,.126)
(.119,.163)
(.116,.158)
(.108,.150)
(.161,.209)
(.157,.205)
(.113,.155)

*Note: 1(P<.05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; Cluster robust standard error has been
used in all estimations.

103

Table 2.18. Selection model misspeciﬁcation: MTE for unweighted estimators, δ=.5
Unweighted
number of n missing fraction
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.416
1.303
1.292
1.424
1.307
1.288
1.424
1.306
1.289
1.423
1.305
1.287

FE

SD(α)
.215
.147
.143
.146
.101
.097
.098
.071
.068
.067
.048
.046

Mean(α)
.936
.971
.980
.936
.974
.982
.939
.974
.982
.941
.976
.983

FD

SD(α)
.142
.094
.089
.097
.064
.060
.062
.042
.039
.042
.028
.027

Mean(α)
.916
.962
.976
.918
.967
.977
.922
.965
.976
.924
.968
.978

*Note: Missing fraction is the fraction of missing sample to full sample.

104

SD(α)
.157
.110
.105
.110
.074
.070
.070
.047
.045
.048
.033
.031

Table 2.19. Selection model misspeciﬁcation: MTE for IPW estimators, δ=.5
IPW
n
% missing
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.379
1.297
1.290
1.389
1.300
1.285
1.385
1.299
1.286
1.384
1.297
1.284

SD(α)
.259
.154
.145
.177
.105
.099
.113
.073
.069
.079
.049
.046

r ≡ 1(p < .05)
.400
.505
.495
.656
.798
.799
.942
.991
.992
.995
1
1

FD

coverage of r
(.370,.431)
(.472,.539)
(.454,.535)
(.627,.685)
(.773,.823)
(.773,.825)
(.927,.957)
(.985,.997)
(.986,.998)
(.991,.999)
(1,1)
(1,1)

Mean(α)
.914
.962
.977
.910
.968
.978
.918
.967
.977
.920
.969
.979

SD(α)
.178
.112
.104
.124
.075
.071
.077
.048
.045
.053
.033
.031

r ≡ 1(p < .05)
.142
.120
.114
.174
.109
.099
.257
.169
.138
.405
.215
.161

coverage of r
(.120,.164)
(.098,.141)
(.088,.140)
(.150,.198)
(.090,.129)
(.080,.119)
(.230,.284)
(.146,.192)
(.117,.160)
(.375,.435)
(.189,.241)
(.138,.184)

*Note: 1(P<.05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; Cluster robust standard error has been
used in all estimations.

105

Table 2.20. Selection model misspeciﬁcation: MTE for unweighted estimators, δ=1.0
Unweighted
number of n missing fraction
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.378
1.314
1.298
1.382
1.313
1.297
1.374
1.313
1.299
1.376
1.311
1.298

FE

SD(α)
.207
.156
.151
.150
.114
.110
.090
.067
.065
.064
.049
.047

Mean(α)
.929
.961
.966
.936
.966
.973
.931
.964
.972
.933
.964
.972

FD

SD(α)
.129
.093
.091
.093
.067
.064
.057
.042
.040
.038
.029
.027

Mean(α)
.905
.944
.955
.910
.952
.963
.905
.948
.960
.907
.949
.962

*Note: Missing fraction is the fraction of missing sample to full sample.

106

SD(α)
.150
.107
.106
.107
.077
.072
.064
.047
.045
.045
.034
.032

Table 2.21. Selection model misspeciﬁcation: MTE for IPW estimators, δ=1.0
IPW
n
% missing
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

LS

Mean(α)
1.402
1.323
1.305
1.412
1.323
1.305
1.402
1.322
1.307
1.406
1.321
1.305

SD(α)
.248
.160
.154
.176
.118
.112
.105
.069
.066
.074
.050
.047

r ≡ 1(p < .05)
.442
.529
.527
.730
.807
.817
.981
.997
.996
1
1
1

FD

coverage of r
(.410,.473)
(.496,.563)
(.488,.565)
(.702,.758)
(.783,.832)
(.792,.841)
(.973,.989)
(.994,1)
(.992,1)
(1,1)
(1,1)
(1,1)

Mean(α)
.902
.943
.955
.903
.952
.964
.896
.947
.960
.897
.948
.962

SD(α)
.167
.108
.107
.115
.077
.072
.069
.048
.045
.049
.034
.032

r ≡ 1(p < .05)
.170
.133
.127
.208
.149
.129
.416
.257
.204
.614
.423
.297

coverage of r
(.146,.193)
(.110,.156)
(.102,.153)
(.183,.233)
(.127,.171)
(.108,.150)
(.385,.447)
(.230,.284)
(.179,.229)
(.584,.644)
(.392,.454)
(.269,.325)

*Note: 1(P<.05) is the rejection rate for the null of H0 :α=1 with nominal value 0.05.; Cluster robust standard error has been
used in all estimations.

107

2.5.4

The robustness of ﬁrst-diﬀerencing estimator with small
within-variation

In most experiments, program participants ideally maintain their treatment or control groups
status from the start to the end of a program. This inevitably induces very small withinvariation for treatment variable and this is one of main criticisms for ﬁxed eﬀects and ﬁrst
diﬀerencing estimators in the program evaluation studies. Thus, we examine the sensitivity
of IPW ﬁrst-diﬀerencing estimator to small within-variation of treatment variable. The
magnitude of within-variation is induced randomly and we assume correct speciﬁcation for
the probability of selection.

Motivation of sensitivity test for IPW FD estimator to small within variation
Fixed eﬀects and ﬁrst diﬀerencing estimators are very eﬀective methods to eliminate the
source of bias when across treatment and control groups, response variables are systematically diﬀerent for missing data individual units and its diﬀerence is due to unobserved
heterogeneity. However, identiﬁcation of ﬁrst diﬀerencing estimator is based only on within
variation.

DGP for dit with small within variation
Three diﬀerent magnitudes of within-variation for treatment status dit are induced randomly
in the simulation.

No restriction on treatment status L = 0

First type use the generation of treatment

variable without any restriction for within-variation imposed on for treatment variable. No

108

Figure 2.1. Bias of unweighted and IPW LS estimator

(Note) For ﬁgures in ﬁrst row, six sets of bar represent LS (a=.25), IPW-LS (a=.25), LS
(a=.5), IPW-LS (a=.5), LS(a=.75), and IPW-LS (a=.75) estimates from the left to the
right where true value is 0. The colors of bar represent degree of mis-speciﬁcation - blue
(correctly speciﬁed), Red (δ=.2), green (δ=.5), and Purple (δ=1.0)- where δ is the
magnitude of misspeciﬁcation coeﬃcient in the selection model and a is fraction of missing
data. For ﬁgures in second row, six sets of bar represent FD (a=.25), IPW-FD (a=.25), FD
(a=.5), IPW-FD (a=.5), FD(a=.75), and IPW-FD (a=.75) estimates from the left to the
right. In both columns, the left panel reports simulation results for n=100 and right panel
for n=1000.

109

restriction induces dit for which about 68 percent of sample change status during four time
periods.
dit = 1 if fi + qit > 0 and dit = 0 otherwise, where fi ∼ iidU (0, 1) , qit ∼ iidN (0, 1)

Small within variation treatment variable

Smaller within-variations for the treat-

ment status were induced with the following DGP. di1 = 1 if fi + qi1 > 0 and di1 = 0
otherwise. dit = dit−1 if −L < fi + qit < L and dit = 0 if fi + qit < −L and dit = 1 if
fi + qit > L. Therefore, the value L determines the magnitude of fraction of within-variation
for a treatment variable.

si1 = 1 and sit = 1 if sit−1 = 1 and fi +c2it + it > a where it ∼ iidN (0, 1), and c2it = rit +qit .
where a is cutoﬀ value which is adjusted from DGP I.
The model for selection process is assumed to be speciﬁed correctly so that πit is obtained
from the following:
πit =

exp(zit γ)
, t = 2, 3, 4 for sit−1 = 1
1 + exp(zit γ)

where zit =(fi , c2it ).

Sensitivity of IPW FD estimator to small within variation for a treatment variable
We test the sensitivity of IPW FD estimator to small within variation of a treatment variable.
DGPs are a treatment variable with full variation, with 50 % variation of units and with 15
% variation of units. The simulation intends to clarify the impact of small within variation
on the consistency of IPW FD estimator.
110

Simulation results Tables from 2.22 to 2.24 show that the bias of FD and IPW-FD
estimators to small within variation treatment. The key is that for missing fraction less than
50 % and within variation more than 50 %, IPW-FD estimate quite consistently estimate
MTE. Figure 2.2 compares the biases between FD and IPW-FD methods. Especially if
missing fraction is less than 0.5, IPW-FD method works quite well. Unless variation is less
than 25 percent, the bias by random small within variation is not substantial.

111

Table 2.22. Sensitivity analysis to small within-variation: MTE with full within-variation
FD

number of n missing fraction
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

Mean(α)
.946
.967
.984
.949
.970
.987
.948
.967
.987
.947
.968
.986

IPW-FD

SD(α)
.219
.130
.101
.150
.090
.072
.096
.057
.046
.068
.040
.032

Mean(α)
.974
.992
.994
.974
.998
1.000
.979
.993
.998
.985
.994
.999

SD(α)
.290
.155
.107
.224
.111
.078
.169
.079
.049
.124
.055
.036

r ≡ 1(p < .05)
.147
.104
.083
.126
.076
.083
.122
.089
.095
.093
.084
.100

*Note: Missing fraction is the fraction of missing sample to full sample.

112

coverage of r
(.123,.171)
(.085,.122)
(.066,.101)
(.105,.146)
(.060,.092)
(.066,.100)
(.101,.142)
(.071,.107)
(.077,.113)
(.080,.106)
(.067,.101)
(.081,.119)

Table 2.23. Sensitivity analysis: MTE with within-variation of 50% sample only
FD

number of n missing fraction
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

Mean(α)
.941
.962
.987
.952
.964
.983
.941
.961
.984
.933
.965
.983

IPW-FD

SD(α)
.302
.177
.134
.210
.120
.095
.137
.077
.058
.091
.053
.041

Mean(α)
.964
.993
1.001
.987
.995
.998
.979
.998
.999
.983
1.001
1.000

SD(α)
.392
.218
.145
.294
.156
.106
.223
.100
.067
.164
.072
.045

r ≡ 1(p < .05)
.190
.109
.087
.145
.100
.081
.142
.071
.063
.115
.063
.071

*Note: Missing fraction is the fraction of missing sample to full sample.

113

coverage of r
(.163,.216)
(.090,.129)
(.069,.105)
(.123,.167)
(.081,.119)
(.064,.098)
(.120,.164)
(.055,.087)
(.048,.078)
(.095,.135)
(.048,.078)
(.055,.087)

Table 2.24. Sensitivity analysis: MTE with within-variation of 15% sample only
FD

number of n missing fraction
100
0.75
100
0.5
100
0.25
200
0.75
200
0.5
200
0.25
500
0.75
500
0.5
500
0.25
1,000
0.75
1,000
0.5
1,000
0.25

Mean(α)
.733
.935
.959
.904
.940
.973
.927
.942
.976
.902
.946
.971

IPW-FD

SD(α)
.735
.418
.280
.571
.273
.186
.314
.164
.116
.208
.115
.082

Mean(α)
.731
.968
.981
.922
.960
.994
.973
.974
1.003
.919
.986
1.004

SD(α)
.793
.479
.299
.669
.343
.212
.485
.233
.135
.385
.170
.096

r ≡ 1(p < .05)
.566
.221
.117
.412
.155
.090
.251
.116
.060
.191
.097
.055

*Note: Missing fraction is the fraction of missing sample to full sample.

114

coverage of r
(.532,.599)
(.195,.247)
(.097,.137)
(.381,.442)
(.132,.177)
(.077,.102)
(.224,.278)
(.096,.136)
(.045,.075)
(.167,.215)
(.079,.115)
(.041,.070)

Figure 2.2. Bias of unweighted and IPW FD estimator to small within variation

(Note) Six sets of bar represent FD and IPW-FD estimator according to covariate
variation. From left to right, sets of bar represent FD (v=1), IPW-FD (v=1), FD (v=.5),
IPW-FD (v=.5), FD (v=.15), and IPW-FD (v=.15) where v represents fraction of
covariates which vary over time. Missing fraction is represent by the color of bar - Blue:25
% , Red: 50 %, and Green: 75 %. Left panel reports the estimates for n=200 and right
panel reports the estimates for n=1,000.

2.6

Empirical application: The return to class size
reduction (CSR) on SAT score

In this section, we estimate the eﬀect of CSR on SAT score for grades in K-3 using Tennessee’s STAR experiment data which is accumulated for a single cohort of students from
1985 to 1989. Data for class size, student characteristics, teacher characteristics, classroom
peer characteristics, and student achievement scores are collected. However, substantial proportion of data is missing so that only about 50 percent of sample is complete-case. Out
of 11,601 students, for each grade from K to 3, about 6,800 students are in the program in
each grade.
We start by testing MCAR using Hausman-type test and variable addition test to deter-

115

mine whether we can use unweighted estimator with complete-case. As we reject the null
of MCAR, we apply IPW and MI-IPW methods using Becker [1993]’s parent school choice
model for the speciﬁcation of the probability of selection.
This section proceeds as follows. We ﬁrst brieﬂy describe Tennessee’s STAR experiment
and provide detailed information on noncompliance and missing patterns of data.19Second,
we provide Hausman-type test and variable addition test of MCAR. Third, we report IPW
estimates and compare this to unweighted estimates.

2.6.1

Background information on Project STAR experiment of
CSR

Origination of experiment Observational data has not been able to ﬁnd consistent class
size reduction eﬀect on student performance (Hanushek [1986] and Hanushek [1997]) while
experimental studies of class size reduction show strong and positive eﬀect.20 Therefore,
the Tennessee legislature funded a randomized experiment, Tennessee’s Project STAR (Student/Teacher Achievement Ratio), to provide the deﬁnitive answer to the eﬀectiveness of
CSR on student achievement in 1985.21

Experiment design It was a randomized experiment that assigned students and teachers
to three diﬀerent class types, a small-size class (target of 13-17 students), a regular-size
19 More detailed description of STAR experiment is provided in Word et al. [1990].
20 For instance, experimental studies such as Tennessee’s Project STAR (Student/Teacher

Achievement Ratio), Wisconsin’s Student Achievement Guarantee in Education (SAGE)
Program and Israel’s Maimonides’ Rule reported a strong link between class size reduction
and improvement in student achievement.
21 The experiment funded by the Tennessee State Legislature under Lama Alexander at
a total cost of $12 million with $3 million for the ﬁrst year of program.

116

class (target of 22-25 students), or a regular-size class with a full-time teacher’s aide. The
randomization was done within school for both students and teachers.22 Each year about
6,800 students, 330 classes, and 79 schools in 46 districts participated in the program from
1985 to 1989. A single cohort of students who entered a participating school in the 19851986 school year participated in the experiment for grades in K-3. Thus, Project STAR
experiment possess essential features of a controlled experiment designed to produce reliable
evidence about the eﬀects of reducing class size.23

Major ﬁndings in previous studies The evidence showed that the students in the
smaller classes outperformed the students in the regular classes, whether or not the regular
class teachers had a full-time aide helping them. In particular, the positive eﬀect of smaller
classes are greater for black and inner-city poor students. Moreover, students who had been
in small class performed better even after treatment is over (Krueger and Whitmore [2001]).

2.6.2

Evidence of experimental violation

The validity of an experimental analysis depends on initial randomization and no diﬀerential
attrition and refreshment across treatment and control groups. It is quite well supported by
some evidences that initial class types were randomly assigned among students and teachers
within school, but the Project STAR experiment suﬀers from implementation problems of
22 All

participating schools implemented at least one of each of the three types of classes
in order to cancel out the possible inﬂuences coming from variations in the quality of the
participating schools that might aﬀect the quality of the classroom activity.
23 See previous studies such as Word et al. [1990] and Krueger [1999] for more details of
description on experiment design and implementation.

117

Table 2.25. Students in each class types for grades in K-3
small

regular/aide

Total

1900
1900

4425
4425

6325
6325

1293
384
248
248
1925

2867
1929
108
108
4904

4160
2313
356
356
6829

1273
366
377
192
2016

3402
1313
109
47
4824

4675
1679
486
239
6840

1276
373
525
207
2174

3567
908
153
72
4628

4843
1281
678
279
6802

Kindergarten
Randomly assigned
First grade
Previously randomly assigned
New entrant
Switchers
Switchers from previous years
Total
Second grade
Previously randomly assigned
New entrant
Switchers
Switchers from previous years
Total
Third grade
Previously randomly assigned
New entrant
Switchers
Switchers from previous years
Total

source: author’s own calculation from the sample of Project STAR data.

118

noncompliance and diﬀerential missing across treatment and control groups.24

Noncompliance
Size of noncompliance sample There were sizable non-compliance of initial class type
assignment of treatments and control. Among the complete samples of 24,275, about 2,200
samples were reassigned to diﬀerent class types from initial assignment. For instance, table
2.25 shows the number of reassignments from small to other types of class or vice versa. For
instance, ﬁrst grade, out of 4,515 remaining students, 356 (7.9%) students switched class
types from small to aide/regular or vice versa. For second and third grades, out of 5,049
and 5,413, remaining students 239 (4.9%) and 279 (5.2%) students switched class types,
respectively. In sum, about 10 % of students were moved from one class type to another in
a non-random manner. 25

The solution to noncompliance If initial assignments of treatments and control were
completely random, noncompliance problem can be overcome quite easily in linear panel
data model by using Instrumental Variable(IV) method in which one uses initial treatments
as IV for actual treatments.

Missing Data
Diﬀerential missing Ignoring missing causes problem in the estimation of MTE if missing
is systematically diﬀerent across treatment and control groups. Typically, attrition can
24 [Krueger,

1999, p.502] for details of initial random assignment of class types for kindergartners.
25 Krueger [1999] reported that most of these moves were due to student misbehavior,
and were not typically the result of parental request for small class reassignment.

119

depend on some observed and unobserved characteristics and diﬀerent characteristics across
treatment and control groups can lead to diﬀerential attrition. Table 2.26 and 2.27 provide
an indirect evidence for possible diﬀerential attrition across class types by comparing simple
mean of SAT score across groups. Table 2.26 summarizes the frequency and fraction of
missing for each class types and table 2.27 compares average SAT score for attritors and
stayers of the program across class types. The diﬀerence of score between attritors and
remainders is greater for students in regular classes and that in small classes.

Size of missing data There was sizable missing from the prior year’s treatment and
control groups. Table 2.27 shows that about 6,000 to 7,000 students for each grade, 1,809
students left the program in ﬁrst grade, 1,608 students left the program in second grade, and
1,319 students left the program in third grade. For instance, among those students in the
program at the beginning, about 45 percent of students left Project STAR program before
ﬁnishing third grade. Overall, out of 11,601 students, about 48 % of all participant students
either left the program before it ends or enter the program after kinder grade.

120

Table 2.26. Attrition and reassignments of class types (noncompliance)
A. Kindergarten to ﬁrst grade
Kindergarten First grade
small regular regular with aide
leaving sample (All-K)
fraction of attrition
small
1293
60
48
380 (1,900)
0.20
regular
126
737
663
668 (2,194)
0.31
aide
122
761
706
642 (2,231)
0.29
new sample
384
1,026
903
2,313(new)/1,809(leaving)
B. First grade to second grade
First grade
Second grade
small regular regular with aide leaving sample (All-G1) fraction of attrition
small
1,435
23
24
443 (1,925)
0.23
regular
152
1,498
202
732 (2,584)
0.28
aide
40
115
1,560
650 (2,320)
0.28
new sample
389
693
709
1,791(new)/1,668(leaving)
C. Second grade to third grade
Second grade Third grade
small regular regular with aide leaving sample (All-G2) fraction of attrition
small
1,564
37
35
380 (2,016)
0.188
regular
167
1,485
152
525 (2,329)
0.225
aide
40
76
1,857
522 (2,495)
0.209
new sample
403
487
481
1,371(new)/1,319(leaving)
source: Project STAR and beyond: Database User’s Guide(2007)

121

Table 2.27. SAT percentile scores at t − 1 for Leavers and stayers
1st grader
Frequency
Stayer of small class at ﬁrst grade
1,319
Leave small sample at ﬁrst grade
453(25.6%)
Stayer of regular or regular with aide class at ﬁrst grade
2949
Leave regular or regular with aide class at ﬁrst grade
1182(28.6%)
Stayer of all class types at ﬁrst grade
4267
Leaver of all class types at ﬁrst grade
1635
2nd grader
Frequency
Stayer of small class at second grade
1173
Leave small sample at second grade
276(19.0%)
Stayer of regular or regular with aide class at second grade
2187
Leave regular or regular with aide class at second grade
631(22.4%)
Stayer of all class types at second grade
3360
Leaver of all class types at second grade
907

Mean
59.19
45.55
54.56
42.26
55.99
43.18
Mean
61.35
46.94
57.63
44.28
58.93
45.09

SE
Diﬀ(small-reg/aide)
25.40
13.64
28.22
25.26
12.30
26.54
25.40
12.81
27.05
SE
24.69
14.41
27.13
24.26
13.35
25.17
24.27
13.84
25.80

Note: Average score for the group are reported in mean column. Diﬀ(small-reg/aide) column report the average score
diﬀerence between stayers and leavers for each class type.

122

Basic statistics of missing data
Table 2.28 shows data used in the regression analysis for grades in K-3. Overall, 48 percent
of data is missing and there are 16 diﬀerent types of missing patterns appear in the data as
in table 2.29. Among those students in the STAR program, about 5 participating students
have missing variables as shown in table 2.28, but only about 52 percent of students have
complete-case data. Therefore, there is severe attrition problem. Missing occurs in Project
STAR due to students’ attrition and missing variables of participating students.

123

Table 2.28. Data availability for covariates

Small class
Aide class
Regular class
White/Asian
Female
Free Lunch
Teacher white
Teacher experience
Teacher master degree and above
Fraction of free lunch in class
Fraction of kinder attendee in class
outcome (yit ) available
all covariates (xit ) available
Both yit and all covariates (xit ) available (≡ A)
Number of students in the program (≡ B)
A
B

124

K
6325
6325
6325
11467
11581
6300
6286
6304
6304
6325
6325
5907
6256
5840
6325
0.923

G1
6829
6829
6829
11467
11581
6650
6810
6810
6788
6781
6829
6684
6584
6430
6829
0.942

G2
6840
6840
6840
11467
11581
6496
6780
6739
6744
6645
6840
6559
6288
5919
6840
0.865

G3
6802
6802
6802
11467
11581
6520
6737
6751
6736
6652
6801
6464
6428
6086
6802
0.895

Table 2.29. Data availability by selection indicator
K
case1
case2
case3
case4
case5
case6
case7
case8
case9
case10
case11
case12
case13
case14
case15
case16
Total
(Note)

G1

G2

G3
×

×
×
×
×
×
×
×

×
×

×
×
×
×
×
×
×

×

×
×

×
×

×
×
×

×
×
×
×
×
×

×

# of sample
2418
513
1079
898
328
935
1582
847
492
1154
101
53
52
223
124
802
11601

implies sit = 1 and × implies sit = 0.

125

Table 2.30. Basic statistics: The mean and standard deviation
K
52.44
(26.49)
0.30
(0.46)
0.35
(0.48)
0.35
(0.48)
0.67
(0.47)
0.49
(0.50)
0.48
(0.50)
0.84
(0.37)
9.26
(5.81)
0.34
(0.47)
0.48
(0.29)

G1
G2
SAT score
52.82
52.40
(27.46) (26.99)
Small class
0.28
0.29
(0.45) (0.46)
Aide class
0.34
0.37
(0.47) (0.48)
Regular class
0.38
0.34
(0.49) (0.47)
White/Asian
0.67
0.65
(0.47) (0.48)
Female
0.48
0.48
(0.50) (0.50)
Free Lunch
0.52
0.51
(0.50) (0.50)
Teacher white
0.83
0.80
(0.38) (0.40)
Teacher experience
11.63
13.14
(8.94) (8.65)
Teacher master degree and above
0.34
0.36
(0.47) (0.48)
Fraction of free lunch in class
0.52
0.51
(0.28) (0.29)
Fraction of kinder attendee in class
0.66
0.54
(0.18) (0.20)
(Note) Standard deviation is in parenthesis.

126

G3
51.85
(27.17)
0.32
(0.47)
0.37
(0.48)
0.31
(0.46)
0.67
(0.47)
0.48
(0.50)
0.51
(0.50)
0.79
(0.41)
13.93
(8.61)
0.43
(0.49)
0.51
(0.28)
0.47)
(0.19)

Selection indicator, sit : Selection indicator sit has value 1 only if both response variables and all covariates [small class (dummy), class with aide (dummy), student (race, gender, free lunch), teacher (race, experience, degree), class (fraction of free lunch, fraction of
kindergarten attendees)] are observed, otherwise we assign sit to be 0. And we also denote
3

Ti =

sit . Table 2.29 shows the number of students for each type of data based on using

t=K

selection indicator sit . Cases of data from 11 to 15 in table 2.29 are non-monotone II type
missing so we assume MCAR of these missing patterns in the application of MI-IPW method.

2.6.3

The impact of CSR on SAT score for grades in K-3

We start by replicating complete-case estimations in previous studies under MCAR.

Model for the eﬀect of CSR on SAT score
Using the model speciﬁcation of Krueger [1999], we adopt linear panel data model as in
(2.41).

Model Operationally ignoring missing in estimation, we can apply pooled LS or pooled
reduced-form (IV) estimators. 26

yit = Dit · β + x1it δ + x2i γ + ft + ci + uit

(2.41)

i = student, t = K, G1, G2, G3, where Dit : two treatments
yit = xit θ + ci + uit , where xit = (Dit x1it x2i ft ), θ = (β δ γ , 1)
26 A

reduced-form estimation uses initial treatments and control as explanatory variables
in model speciﬁcation.

127

where yit is student academic achievement ,
Dit is vector of class type assignments- small and regular with aide,
Observed covariates include student, teacher, peer characteristics
x1it time varying observable covariates including school dummy,
x2i time-ﬁxed observable covariates,
ci unobserved heterogeneity, ft time eﬀect, and uit unobserved errors

y (Achievement Score) Achievement score we use is Standard Achievement Test (SAT).
It measures student achievement in reading, math, and word skills in grade K-3. Following Krueger [1999], we scaled the test score into percentile ranks. Percentile ranking was
calculated using the scores for those students who are assigned to regular and regular with
full-time aide class types only. And three separate percentile rankings for each subject were
generated. Response variable y is obtained as average of these three percentile rankings.
For those students who are not available all three percentile rankings, we use the average
percentile ranking of available percentile rankings. If one of three subject scores is missing,
we use only two subject test scores to obtain y and if two subject scores are missing, then
we use only available one subject’s test score for y. 27

Covariates x include: (i) student characteristics: race, gender, free lunch status (ii)
teacher characteristics: years of experience, highest degree dummy, race (iii) peer characteristics: fraction of classmates with free lunch, fraction of female classmates, fraction of
27 Project

STAR Public Access Data is obtained from www.heros-inc.org/data.html. All
the following descriptions of variables used in this study are based on Project STAR and
Beyond: Database User’s Guide (2007).

128

classmates of kindergarten attendees, and fraction of minority classmates, and (iv) school
variables: school indicator, school location indicator.

Unweighted estimations Results
Cross-section estimation Table 2.31 reports unweighted cross-section estimates for both
LS and IV. The estimated cross-section return of small class is about 5 to 7 percentile points
for grades in K-3. It is about 1 of standard deviation of SAT score. The estimated cross5
section return of the class with full-time aide is not statistically diﬀerent from 0 for grades
in K-3.

129

Table 2.31. Cross-section unweighted LS and reduced-form estimates
OLS
K
G1
G2
G3
Small
5.205 7.683 5.821 4.964
(.749) (.740) (.776) (0.822)
Aide
.259
1.866 1.642 -.492
(.698) (.710) (.733) (.770)
R-squared
.309
.299
.278
.221
# of sample
5840
6430
5919
6086
Other covariates School dummy, student and
teacher characteristics
OLS
K
G1
G2
G3
Small
5.224 7.599 5.440 4.644
(.749) (.747) (.782) (0.836)
Aide
.434
2.012 1.445 -.791
(.698) (.716) (.738) (.777)
R-squared
.311
.300
.280
.222
# of sample
5840
6430
5919
6086
Other covariates School dummy, student, teacher
peer(classmates) characteristics

reduced-form
K
G1
G2
G3
5.205 6.621 5.279 5.247
(.749) (.758) (.786) (.814)
.259
1.509 1.175 -.072
(.698) (.679) (.700) (.726)
.309
.295
.276
.220
5840
6430
5919
6086
School dummy, student and
teacher characteristics
reduced-form
K
G1
G2
G3
5.224 6.490 5.064 4.979
(.749) (.767) (.786) (.817)
.434
1.560 1.156 -.078
(.698) (.680) (.699) (.728)
.311
.296
.279
.221
5840
6430
5919
6086
School dummy, student, teacher
peer(classmates) characteristics

(Note) Students cluster-robust standard errors are provided in parentheses. Student characteristics include race, gender, and
free lunch status and teacher characteristics include race, years of experience and highest degree dummy. Peer characteristics
include the fraction of classmates with free lunch, fraction of minority classmates and fraction of female classmates.
Reduced-form estimation use initial assignments of treatments as independent variables.

130

Pooled estimation of LS and FD estimation

Table 2.33 reports unweighted FD

estimates for the return of CSR on SAT score. First, the FD estimates for the return of
CSR are about 6.4 percent (1.6 · 4 = 6.4) while the return of CSR for unweighted pooled
LS estimate are about 7.5 percent in table 2.32. Second, unweighted FD estimates imply
that the class with full-time aide has statistically signiﬁcant impact on the improvement of
SAT score when compared to regular class without full-time aide while pooled estimates for
the return of class with full-time aide are not statistically diﬀerent from zero. Overall, with
unweighted methods, the eﬀect of CSR on SAT score is between 5(pooled IV) and 7.5(pooled
LS) percents.

131

Table 2.32. Unweighted pooled LS and reduced-form
LS: full
Initial small
3.243
(.806)
Initial aide
.495
(.671)
Cumulative years in small 1.003
(.388)
Cumulative years in aide .262
(.376)
R-squared
.228
# of sample
24,275
Student characteristics
Yes
teacher characteristics
Yes
peer characteristics
No
School, grade,PP-year
Yes

sample IV: full
3.201 4.991
(.806) (.507)
.518
.782
(.670) (.507)
.978
(.388)
.254
(.377)
.229
.226
24,275 24,275
Yes
Yes
Yes
Yes
Yes
No
Yes
Yes

sample
4.903
(.507)
.796
(.507)

.227
24275
Yes
Yes
Yes
Yes

LS: attrition sample IV: attrition sample
3.474 3.441
4.856 4.806
(.980) (.980)
(.698) (.699)
-.022
-.004
.205
.244
(.797) (.797)
(.663) (.663)
.746
.743
(.436) (.436)
.221
.241
(.426) (.427)
.227
.229
.226
.227
16,222 16,222
16,222 16,222
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
No
Yes
No
Yes
Yes
Yes
Yes
Yes

(Note) Students cluster-robust standard errors are provided in parentheses. PP-year implies program participation year
dummy. Grade implies dummies for grades in K-3. IV represents the estimation method for which initial assignments of class
type were used as treatment variables.

132

Table 2.33. Unweighted FD estimates
FD with full sample
Cumulative years in small 1.738 1.680 1.651
(.362) (.364) (.364)
Cumulative years in aide 1.492 1.370 1.359
(.358) (.363) (.364)
R-squared
.016
.020
.020
# of sample
13214 12924 12924
Student characteristics
Yes
Yes
Yes
teacher characteristics
No
Yes
Yes
peer characteristics
No
No
Yes
School, grade,PP-year
Yes
Yes
Yes

FD with attrition sample
1.827 1.641 1.613
(.416) (.413) (.414)
1.738 1.692 1.689
(.426) (.427) (.428)
.015
.019
.020
9883
9793
9793
Yes
Yes
Yes
No
Yes
Yes
No
No
Yes
Yes
Yes
Yes

(Note) Students cluster-robust standard errors are provided in parentheses. Grade implies dummies for grades in K-3.

133

2.6.4

Test of strict exogeneity assumption

We formally test strict exogeneity using Hausman-type test which is based on unweighted
FD and FE estimators. We perform Hausman-type II test and regression based variable
addition test of Wooldridge [2009] for unbalanced panel data.

Test Results
Table 2.34 shows the rejection of the null hypothesis for strict exogeneity at nominal level
of 0.05. This implies that the violation of strict exogeneity and unweighted FE and FD estimators with complete-case would produce bias and incorrect standard error. Furthermore,
table 2.35 shows the results for a regression based variable addition tests and H0 : λ = 0 in
(2.19)-(2.21) are rejected for FD and FE estimation models. Tests imply MCAR is violated
and, therefore, both unweighted FD and FE estimators are not valid.

Indirect evidence of weak exogeneity from LS and reduced-form(IV)

Unfortu-

nately, we cannot directly test conditional contemporaneous exogeneity(i.e. weak exogeneity
conditional on selection E(sit · dit · errorit ) =0) for pooled LS and reduced-form(IV) estimators. However, if we assume that E(sit |·) and E(sit+1 |·) are serially correlated, we can
conjecture the sign of E(sit · dit · errorit ) from the sign of E(sit+1 · dit · errorit ). LS and IV
columns in table 2.35 report that E(sit+1 · dit · errorit ) >0 and, thus for instance, if both
E(sit · dit · uit ) and E(sit+1 · dit · errorit ) have the same sign, both LS and IV estimators are
biased upward.

134

Table 2.34. Hausman-type test II: strict exogeneity

Cumulative year in
small class
Cumulative year in
aide class
# of sample
H0 : θpols = θF E , k = 2
P-values
Student characteristics
teacher characteristics
peer characteristics
School, grade,PP-year

FD
FE
1.569 .544
(.391) (.321)
1.514 .809
(.386) (.319)
31228
W2 = 4.82
{.089}
No
No
No
Yes

FD
FE
1.741 .477
(.403) (.324)
1.501 .774
(.400) (.322)
30309
W2 = 6.70
{.035}
Yes
No
No
Yes

135

FD
FE
1.682 .458
(.363) (.326)
1.377 .705
(.361) (.323)
29960
W2 = 6.37
{.041}
Yes
Yes
No
Yes

FD
FE
1.648 .442
(.363) (.327)
1.360 .692
(.362) (.324)
29960
W2 = 6.16
{.045}
Yes
Yes
Yes
Yes

Table 2.35. Variable addition test: strict exogeneity
PLS
PIV
Selection (S it+1 )
10.439 10.474
(.420) (.420)
Initial small
2.702 5.047
(.919) (.575)
Initial aide
.076
.922
(.749) (.513)
Cumulative year in
1.602
small class
(.522)
Cumulative year in
.802
aide class
(.510)
R-squared
.269
.269
# of sample
18386 18386
Student characteristics Yes
Yes
teacher characteristics No
No
peer characteristics
No
No
School, grade,PP-year Yes
Yes

FE
FD
PLS
4.571 3.889 10.975
(.514) (.544) (.418)
2.848
(.923)
.185
(.753)
1.276 2.054 .951
(.474) (.491) (.545)
1.866 2.085 .668
(.468) (.492) (.516)
.037
.272
18386 8538
18189
Yes
Yes
Yes
No
No
Yes
No
No
Yes
Yes
Yes
Yes

136

PIV
FE
10.287 4.185
(.419) (.510)
4.113
(.598)
.886
(.516)
1.249
(.478)
1.756
(.471)
.272
18189 18189
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes

FD
3.604
(.540)

2.084
(0.494)
2.110
(.498)
.037
8390
Yes
Yes
Yes
Yes

2.6.5

Estimation with inverse probability weighted(IPW) and
multiple imputation(MI)

The two formal tests in previous section show the evidence of diﬀerential correlation among
unobserved heterogeneity, ci , and idiosyncratic error, ui = (ui1 , ui2 , ...., uiT ) and selection
indicator, si across dit and this implies that MCAR is violated. Therefore, we apply IPW
and MI-IPW methods using the probability of selection which satisﬁes MAR assumption as
weights.

IPW estimation
In the estimation of the probability of selection (i.e. E[sit |Wit−1 ]), we adopt the method in
Robins et al. [1994] and Wooldridge [2010] for which the probability of selection is estimated
in sequential manner for unbalanced panel data. We base our choice of the predictors for
the probability of selection on parents’s school choice between private and public. At the
beginning of period, all students start in public school.

Parents’s school choice and attrition
We use parents’s school choice model in Becker [1993] to describe students attrition in STAR
Program. Using this model, we specify the model for the probability of selection which
satisﬁes MAR assumption. Estimated probability of selection is used as inverse probability
weights in IPW estimation.

Assumptions
1. Parents maximize utility directly for two periods and indirectly inﬁnite periods
137

2. heterogeneity takes two forms which are parent resource (h1i ) and student individual
ability (h2i )
3. Each period choice is made for the resource allocation among (i) consumption (ii)
spending on human capital of children (Et )
4. Initially, as all students start in public school, we assume that the marginal cost of
moving to private school is greater than the marginal beneﬁt.

Model: Parent’s optimization problem

Following Becker [1993], at time t, parent’s

utility depends both on the utility of themselves and their children as in (2.42).
Ut = u(ct ) + ρ · u(ct+1 )

(2.42)

where ct is the consumption of parent, ρ is a constant which measures the relative weight
for child’s utility, ct+1 is children’s consumption when they become adult.
If the preference (2.42) is the same for all generations and consumption during childhood
can be ignored, then the utility of parent can be written as in (2.43).
∞

ρi · u(ct+i ).

Ut =

(2.43)

i=1

The utility of parent depends directly only on the utility of their own children and indirectly
on all descendants.
Resource constraint for parent and children at period t is given by
yt = ct + Et ; yt = h1i ; yt+1 = f (h2i , Et ); with linearity yt+1 = a(h2i ) · Et

(2.44)

where h1i is parent’s income, Et parents’ human capital investment for their children at t
and a >0 is the marginal return for human capital investment which varies across students as
138

their ability varies. For the sake of simplicity of illustration, we assume a linear production
function for children in (2.44). 28
At the optimum, with linearity assumption on children’s production function, we have the
following optimal condition.
u (h1i − Et ) = ρ · Et (a(h2i ) · u (ct+1 )|It )

(2.45)

where Et (·|It ) is conditional expectation at period t using all available information, It , at
the beginning of period t. LHS of (2.45) is marginal cost of moving to private school and
RHS of (2.45) is expected marginal beneﬁt of moving to private school. Parent does not
have complete information on their children’s ability so they update their children’s ability
as they obtain new information. At the beginning of program, for all participating students,
LHS is greater than RHS in (2.45). At the end of each grade, parents update their expected
marginal beneﬁt of moving to private school as new information on student’s score, class,
teacher, and school quality become available. Therefore, attrition by school choice occurs as
RHS becomes larger than LHS in (2.45) after new information arrives at the end of period.
There are other reasons for students move and leave STAR program such as parent job
reallocation, the move due to sibling concerns, and others. We assume that the attrition of
students by these reasons should not be causing diﬀerential attrition across class types since
it is very hard to think that these reasons to move only occur to one type of class. Therefore,
we assume that diﬀerential attrition across class types only occurs through parent’s decision
of the moving.
28 We

assume no transfer from parents to children other than human capital investment
and bequest transfer. Bequest transfer is determined at the level which expected marginal
returns from bequest transfer is equal to marginal return from human capital investment.

139

The probability of selection model

The attrition of student is determined by updated

information Et (·|It ) between at the end of t − 1 and at the beginning of t. It includes
student’s score (yit−1 ) and updated class, teacher, and school characteristics(z1it−1 ) where
z1it−1 include class average on SAT score, teacher’s experience, teacher’s highest degree,
the fraction of free lunch students and so on. In the application, we use yit−1 and (z1it−1 )
as predictors for the probability of selection model. In this setting, MAR is satisﬁed as in
(2.46) and MAR implies that, once we know parent information It , we can ﬁgure out parent’s
school choice and attrition status. We use logit model in the application.

P (sit = 1|Wit ) = P (sit = 1|zit )P (sit = 1|Wit , sit−1 = 1) = P (sit = 1|zit , sit−1 ) = πit
(2.46)
where zit = (z1it−1 , yit−1 ).
πit = G(zit ; γ)Λ(zit · γ)

(2.47)

exp(·)

where wit =(xit−1 , yit−1 ) and Λ(·) = 1+exp(·) .

2.6.6

IPW estimation

We illustrate IPW estimation with complete-case. MI-IPW estimation is conducted with
pseudo-complete case data.29
We construct pit in sequential manner using conditional expectation.
pit = p(sit = 1|zit , sit−1 = 1)·p(sit−1 = 1|zit−1 , sit−2 = 1)····p(si2 = 1|zi2 , si1 = 1), t = 1, 2, 3
(2.48)
29

Pseudo-complete case is the case that combines complete case and non-monotone II
which become complete case after imputations.
140

pit = πit · πit−1 · · · πi1
Using the sequential approach as in (2.48), and assuming the sequence of binary response
models is correctly speciﬁed as in (B.1), we can consistently γ. Once we obtain γ from MLE,
we use G(zit , γ) to construct pit for all i and t with sit = 1.

IPW estimates: The return of CSR on SAT score
For cross-section estimation, we estimate pit = πit = p(sit = 1|zit , sit−1 = 1) using logit
model. zit−1 include class types, student characteristics (gender, race, free lunch), teacher
characteristics (race, experiences, highest degree), school ﬁxed eﬀects, and SAT percentile
score of period t − 1.

Cross-section estimation of probability weight Figure 2.3 show estimated probability weight for cross-section estimation in grades K-3. One of practical criticism for IPW
estimation is that estimates can be very unstable if certain subsets of population have very
low non-attrition probabilities.(Little and Rubin [2002]) However, as wee see in ﬁgure 2.3, all
of estimated pit is strictly greater than 0.4 so that our assumption of pit > 0 is well satisﬁed.
The overlap assumption of pit for treatment and control groups is also well satisﬁed as we
see in ﬁgure 2.4.

IPW estimates: cross-section Table 2.36 reports cross-section IPW estimates of both
LS and reduced-form for grades in 1-3. The estimated cross-section return of small class
is about 5 to 7 percentile points. Students in small classes obtain about 5 to 7 percentile

141

1
point higher test scores than students in regular class for grades in K-G3. It is about 5 of

standard deviation of SAT score. The estimated cross-section return of full-time aide class
is not statistically diﬀerent from 0. The implication of IPW estimates is no diﬀerent from
the one of unweighted estimates. In all grades, the diﬀerence between unweighted and IPW
estimates is within 1 percent percentile.

142

Figure 2.3. Estimated πit for t = 1, 2, 3

(Note) For all ﬁgures, x-axis represents πit and y-axis represents fraction. For the ﬁgure in
ﬁrst row and ﬁrst column, πi1 in x-axis is ranged from 0.4, to 0.9 and its unit interval is .1.
y-axis is ranged from 0 to 0.05 and its unit interval is 0.01. For the ﬁgure in ﬁrst row and
second column, πi2 in x-axis is ranged from 0.4, to 1 and its unit interval is 0.2. y-axis is
ranged from 0 to 0.06 and its unit interval is 0.02. For the ﬁgure in second row and ﬁrst
column, πi3 in x-axis is ranged from 0.6, to 1 and its unit interval is 0.1. y-axis is ranged
from 0 to 0.06 and its unit interval is 0.02.

143

Figure 2.4. Estimated πit conditional on sit = 0 or sit = 1 for t = 1, 2, 3

(Note) x-axis represents πit and y-axis represents fraction. We denote ﬁgure in ith row and
jth column as F(i,j) . F(i,1) shows πit conditional on sit = 0 and F(i,2) shows πit conditional
on sit = 1. F(1,1) shows πi1 conditional on si1 = 0, x-asix is ranged from 0.4 to 0.9 and
y-asix is ranged from 0 to 0.06 with interval of 0.02. F(2,1) shows πi2 conditional on
si2 = 0, x-asix is ranged from 0.4 to 0.9 and y-asix is ranged from 0 to 0.05 with interval of
0.01. F(3,1) shows πi3 conditional on si3 = 0, x-asix is ranged from 0.6 to 1 and y-asix is
ranged from 0 to 0.06 with interval of 0.02. F(1,2) shows πi1 conditional on si1 = 1, x-asix
is ranged from 0.4 to 0.9 and y-asix is ranged from 0 to 0.06 with interval of 0.02. F(2,2)
shows πi2 conditional on si2 = 1, x-asix is ranged from 0.4 to 1 and y-asix is ranged from 0
to 0.08 with interval of 0.02. F(3,1) shows πi3 conditional on si3 = 1, x-asix is ranged from
0.6 to 1 and y-asix is ranged from 0 to 0.08 with interval of 0.02.
144

Table 2.36. Cross-section IPW LS and IPW reduced-form estimation
LS
K
G1
G2
G3
Small
5.205 7.309 6.025 5.491
(.749) (.926) (.928) (0.953)
Aide
.259
1.090 2.375 .289
(.698) (.950) (.913) (.904)
R-squared
.309
.318
.294
.234
# of sample
5840
4052
4338
4533
Other covariates School dummy, student and
teacher characteristics
OLS
K
G1
G2
G3
Small
5.224 7.282 5.595 5.355
(.749) (.935) (.940) (0.969)
Aide
.434
1.442 2.235 .092
(.698) (.955) (.921) (.910)
R-squared
.311
.320
.296
.235
# of sample
5840
4052
4338
4533
Other covariates School dummy, student, teacher
peer(classmates) characteristics

reduced-form
K
G1
G2
G3
5.205 5.613 5.536 6.224
(.749) (.960) (.953) (.945)
.259
.013
1.480 .820
(.698) (.898) (.848) (.833)
.309
.313
.293
.235
5840
4052
4338
4533
School dummy, student and
teacher characteristics
reduced-form
K
G1
G2
G3
5.224 5.508 5.312 6.072
(.749) (.973) (.955) (.949)
.434
.076
1.544 .728
(.698) (.897) (.847) (.834)
.311
.316
.293
.235
5840
4052
4338
4533
School dummy, student, teacher
peer(classmates) characteristics

(Note) Students cluster-robust standard errors are reported in parentheses. Student characteristics include race, gender, and
free lunch status and teacher characteristics include race, years of experience and highest degree dummy. Peer characteristics
include the fraction of classmates with free lunch, fraction of minority classmates and fraction of female classmates. Reduced
form estimation use initial assignments of treatments as independent variables.

145

Panel probability weights estimation

πit is estimated using equation (2.49) by con-

ditional MLE.

πit = Λ(zit · γ) for sit−1 = 1.

(2.49)

where j = 1, 2, 3.
The probability weight estimate, pit is sequentially constructed as in (2.50).
pit = πit · πit−1 · · · ·πi1 .

(2.50)

Figure 2.5 shows the histogram of estimated probability weight, πit and pit πit and pit are
greater than .15 and .35 for all i and t respectively.

146

Figure 2.5. Estimated πit and pit for t = 1, 2, 3 with attrition sample

(Note) F(i,j) identiﬁes the location of ﬁgure. i identiﬁes the row and j identiﬁes column.
Figures in left panel of rows represent πit . First, second, and third row show ﬁgures for πi1 ,
πi2 , and πi3 respectively. Figures in right panel of rows represent πit . First, second, and
third row show ﬁgures for pi1 , pi2 , and pi3 respectively. For πi1 in F(1,1) , x-axis is ranged
from 0.4 to 0.9 and y-axis is ranged from 0 to 0.05 with interval of 0.01. For πi2 in F(2,1) ,
x-axis is ranged from 0.2 to 1 and y-axis is ranged from 0 to 0.06 with interval of 0.02. For
πi3 in F(3,1) , x-axis is ranged from 0.5 to 1 and y-axis is ranged from 0 to 0.08 with interval
of 0.02. For pi1 in F(1,2) , x-axis is ranged from 0.4 to 0.9 and y-axis is ranged from 0 to
0.05 with interval of 0.01. For pi2 in F(2,2) , x-axis is ranged from 0.2 to .8 and y-axis is
ranged from 0 to 0.05 with interval of 0.01. For πi3 in F(3,2) , x-axis is ranged from 0 to .8
and y-axis is ranged from 0 to 0.05 with interval of 0.01.

147

Panel IPW estimation results Table 2.37 provides IPW pooled estimates. The estimated eﬀects of IPW LS for the eﬀect of CSR on SAT score is about 6.4 % while the
estimated eﬀect of unweighted LS is 7.3 % in table 2.32. Moreover, the estimated eﬀects
of IPW IV for the eﬀect of CSR on SAT score is about 4.5 % while the estimated eﬀect of
unweighted IV is 5.5 %. Table 2.38 reports IPW FD estimates. The estimated eﬀects of IPW
FD estimator is 5.5 % while the eﬀect for unweighted FD estimates is about 7 %. Again,
the eﬀect is large for the unweighted estimator. Thus, for all three estimates, unweighted
estimates for the return of CSR on score are greater than IPW estimates.

Summary of estimation results: Unweighted and IPW estimates in unbalanced
panel data
Unweighted estimates are greater than IPW estimates for pooled LS, pooled reducedform(IV), FD estimators. However, the diﬀerence of estimates is very small for pooled LS,
pooled reduced-form, and FD estimator and is within 2%. If IPW estimates are consistent,
unweighted LS, reduced-form, and FD estimators overestimate the return of CSR about 1
to 2 percent. In sum, we conclude that the return of CSR on SAT score for grades in K-3 is
within the range from 4 to 6.5 percent.

148

Table 2.37. IPW pooled LS, pooled IV
Pooled LS
Initial small
2.152
2.374
(1.163) (1.163)
Initial aide
-.601
-.414
(.897) (.905)
Cumulative years in small 1.154
1.033
(.506) (.508)
Cumulative years in aide .360
.275
(.471) (.476)
R-squared
.228
.229
# of sample
15395
15241
Student characteristics
Yes
Yes
teacher characteristics
No
Yes
peer characteristics
No
No
School, grade,PP-year
Yes
Yes

149

Pooled
2.328
4.500
(1.163) (.813)
-.390
-.221
(.903) (.762)
1.038
(.508)
.297
(.476)
.231
.228
15241
15395
Yes
Yes
Yes
No
Yes
No
Yes
Yes

IV
4.497 4.448
(.815) (.815)
-.123 -.074
(.768) (.768)

.229
15241
Yes
Yes
No
Yes

.230
15241
Yes
Yes
Yes
Yes

Table 2.38. Weighted FD estimates
IPW FD
Cumulative years in small 1.317 1.380
(.415) (.416)
Cumulative years in aide 1.697 1.572
(.433) (.435)
R-squared
.014
.018
# of sample
9512
9401
Student characteristics
Yes
Yes
teacher characteristics
No
Yes
peer characteristics
No
No
School, grade,PP-year
Yes
Yes

150

unweighted FD
1.361 1.827 1.641
(.416) (.416) (.413)
1.572 1.738 1.692
(.435) (.426) (.427)
.019
.015
.019
9401
9883
9793
Yes
Yes
Yes
Yes
No
Yes
Yes
No
No
Yes
Yes
Yes

1.613
(.414)
1.689
(.428)
.020
9793
Yes
Yes
Yes
Yes

2.6.7

MI-IPW method

The major disadvantage of imputation method in our application is the necessity of modeling
the joint distribution of missing variables which include both categorical and continuous
variables. For instance, our application of project STAR data includes score (continuous),
free lunch (binary), teacher experience (continuous), teacher race (binary), many other class
peer (fractional variables with having values between zero and one). However, we mitigate
this problem of high dimensional covariates by focusing on non-monotone II type of missing
in imputation. It is easy to implement MI method since built-in command for MI method
exists with Stata, SAS and R.30
.

Models of missing variables with multiple imputation
Participating students with missing observations are about 5 percent at each grade and
missing observations mostly occur to SAT score. Therefore, most of missing data comes
from attrition of students for Project STAR data.

Imputation model for missing variables using MICE In Stata, we imputed missing
variables which include outcome, treatments and covariates using following predictors and
distributional assumption in table 2.39.

30 We

implement multiple imputation for the estimation of the return of CSR on score
using ”ice” Stata command with the simplest assumption on the distribution of missing
variables. Further details on imputation methods under MAR have been studied extensively
in theory and practice(Rubin [1987] and Little and Rubin [2002])
151

Table 2.39. Covariates and distributional assumption used for MICE
variables
SAT score
small class
aide class
free lunch
teacher’s race
teacher’s highest degree
teacher’s years of experience
fraction of students with freelunch
fraction of female students
fraction of black students

distribution
Normal
Logistic
Logistic
Logistic
Logistic
Logistic
Logistic
Normal
Normal
Normal

command
regress
logit
logit
logit
logit
logit
logit
regress
regress
regress

SAT
SAT
SAT
SAT
SAT
SAT
SAT

prediction equation
treatments, student, teacher, peer characteristics
SAT score, student, teacher, peer characteristics
SAT score, student, teacher, peer characteristics
score, treatments, student, teacher, peer characteristics
score, treatments, student, teacher, peer characteristics
score, treatments, student, teacher, peer characteristics
score, treatments, student, teacher, peer characteristics
score, treatments, student, teacher, peer characteristics
score, treatments, student, teacher, peer characteristics
score, treatments, student, teacher, peer characteristics

(Note) (i) Treatments are class-type indicators. (ii) Students characteristics include race, sex and free lunch status. (iii)
Teacher characteristics include race, degree, and experience. (iv) Peer characteristics include the fraction of student with free
lunch, fraction of female and fraction of black classmates. For panel data MI estimation, grade, program participation grade
and school dummies are additionally included in predictions of all missing variables.

152

Estimates with MI-IPW
Using the set of imputed variables for STAR participating students as complete case, we
apply IPW estimation.
Tables 2.40 and 2.41 reports pooled LS, pooled IV and pooled FD estimates for the return
of CSR on score using MI-IPW method. There are about 4.5 percent more sample observations are used in MI-IPW method than IPW method. For instance, in the case of pooled LS
estimator, MI-IPW method uses 17,468 observations while IPW method 15,395 observations
in the estimation. The rest of missing data which is more than 40 percent of full-sample is
due to attrition. The diﬀerence of estimates between IPW and MI-IPW is very small while
standard errors are small for MI-IPW method so that MI-IPW estimates provide the same
implications as those of IPW estimates. In sum, unweighted estimator overestimates CSR
on SAT score about 1 to 2 percentage points.

Summary of results
The return of CSR on SAT score is about 4 to 6.5 percent for MI-IPW estimator while it
is about 5 to 7.5 percent for unweighted estimator. Unweighted estimator overestimates the
eﬀect of CSR by about 1 to 2 percentage point. The return of aid class on SAT score is not
statistically signiﬁcant for pooled LS and IV estimates while it is statistically signiﬁcant for
FD estimator. The statistical signiﬁcance of the return for both small class and aide class
does not change whether we use either unweighted or IPW (MI-IPW) methods.

153

Table 2.40. MI-IPW pooled LS, pooled IV
Pooled LS
Initial small
2.584
2.202
(1.084) (1.087)
Initial aide
.170
-.127
(.840) (.839)
Cumulative years in small .997
1.065
(.453) (.455)
Cumulative years in aide .090
.172
(.421) (.423)
R-squared
.201
.200
# of sample
17468
17468
Student characteristics
Yes
Yes
teacher characteristics
No
Yes
peer characteristics
No
No
School, grade,PP-year
Yes
Yes

Pooled
2.267
4.788
(1.080) (.747)
-.094
.275
(.843) (.716)
.905
(.488)
.164
(.435)
.206
.201
17468
17468
Yes
Yes
Yes
No
Yes
No
Yes
Yes

IV
4.511 4.416
(.756) (.760)
.066
.080
(.716) (.716)

.200
17468
Yes
Yes
No
Yes

.206
17468
Yes
Yes
Yes
Yes

Table 2.41. MI-IPW FD estimates
MI-IPW FD
Cumulative years in small .982
.894
.901
(.411) (.413) (.413)
Cumulative years in aide 1.241 1.358 1.460
(.425) (.430) (.430)
R-squared
.016
.020
.020
# of sample
11,146 10,761 10,761
Student characteristics
Yes
Yes
Yes
teacher characteristics
No
Yes
Yes
peer characteristics
No
No
Yes
School, grade,PP-year
Yes
Yes
Yes

154

unweighted FD
1.827 1.641
(.416) (.413)
1.738 1.692
(.426) (.427)
.015
.019
9883
9793
Yes
Yes
No
Yes
No
No
Yes
Yes

1.613
(.414)
1.689
(.428)
.020
9793
Yes
Yes
Yes
Yes

2.7

Direction (sign) of the bias for unweighted estimators

In this section, we try to obtain information about the direction of the bias for unweighted
estimators by making assumption that E(vit sit |zit ) and E(vit sit+j |zit ) has the same sign.
This is because the source of bias for unweighted estimators is E(vit sit |zit ) and we cannot
directly estimate it. Although we can not obtain an estimate for equation (2.51), we can
obtain an estimate for (2.52) and we infer E(vit sit |zit ) from E(vit sit+j |zit ).

E(vit sit |xit , zit ) = E(vit sit |zit ) where xit ⊆ zit and zit is instrument

(2.51)

where vit = ci + uit
E(vit sit+j |zit ) where |j| ≥ 1

(2.52)

Under the assumption that E(vit sit |zit ) and E(vit sit+j |zit ) has the same sign, we can
calculate the direction of the bias of unweighted IV estimates and thereby we can obtain the
bound for true return of class size reduction.

2.7.1

Bound for the eﬀects of small class

First, we consider an estimator for E(vit sit+j |zit ).

Cross-section IV regression
E(uit sit+j |zit ) =

155

1
n

n

vit sit+j
i=1

(2.53)

where vit be residual from unweighted IV estimator 31 can rewrite IV estimator as follows.
We
θIV

sit zit zit

= (

)−1 (

sit zit zit

= θ+(

zit sit vit )
i=1

n

sit zit zit )−1 (
i=1

sit zit (zit θ + vit ))
i=1

i=1

)−1 (

i=1
n

)−1 (

n

n

(

sit zit zit

sit zit yit ) = (
i=1

i=1

n

n

n

n

zit sit vit ) = (
i=1

1
n

n

sit zit zit )−1 (
i=1

p

→ E(sit zit zit

1
n

)−1 E(z

n

zit sit vit )
i=1
it sit vit )

E(sit zit zit )−1 E(zit )E(sit vit )

=

Since Z exogenous

Thus, ﬁnally, we obtain cross-section IV estimator as follows.

θIV

= θ + E(sit zit zit )−1 E(zit )E(sit vit )

θIV

> θ if sign(E(sit vit )) is positive
< θ if sign(E(sit vit )) is positive

Therefore, θIV becomes the upper bound for θtrue if sign(E(sit vit )) is positive and lower
bound if sign(E(sit vit )) is negative.

pooled IV regression
G3

uit sit+j |zit ) =

E(
t=K

1
nT

n

G3

vit sit+j
i=1 t=K
n

G3

vit = yit − zit θpooledIV , where θpooledIV = (

n

sit zit zit
i=1 t=K

31 This

(2.54)
G3

)−1 (

sit zit yit )
i=1 t=K

is not conventional IV estimator. We deﬁne IV estimator here to be equivalent
estimator as in Krueger [1999] where he called this estimator as reduced form estimator.

156

Similarly, as in cross-section IV case, we obtain an pooled IV estimator as
G3

G3

θpooledIV

sit zit zit

= θ + E(
t=K

θIV

)−1 E(

zit sit vit )
t=K

> θ if sign(E(sit vit )) is positive
< θ if sign(E(sit vit )) is positive

This is because E(zit ) > 0 and zit is exogenous in our data.

Estimation results
Pairwise correlation coeﬃcient estimates for (2.53) is reported in table 2.42. The estimates
1
for E(vit sit+j |zit ) = n

n
i=1 vit sit+j

is positive and statistically signiﬁcantly diﬀerent from

zero ∀ j ≥ 1 in all grades. (i.e. residuals from cross-section IV estimators and lead of selection is positively correlated in statistically signiﬁcant way.) However, pairwise correlation
coeﬃcients for residuals and lag of selection are not statistically diﬀerent from zero. In other
word, we simply cannot reject the null hypothesis of E(vit sit−j |zit ) = 0 for j ≥ 1.
1
Thus, given that sample estimates for E(vit sit+j |zit ) = n

n
i=1 vit sit+j (≥

0) is non-

negative ∀ j and j ==0. It is positive for leads of selection and zero for lags of selection.
Therefore, if we assume the of sign(E(vit sit |zit ))=sign(E(vit sit+j |zit )) ∀ j≥ 1, we can infer
that θIV is biased upward. Thus, under this assumption, θIV overestimate true eﬀects of
small class. Table 2.43 shows the upper bound cross-section IV estimates.

157

Table 2.42. corr(vit , sit+j ) ∀ j≥ 0

siK
siG1
siG2
siG3

siK
siG1
siG2
siG3

covariates(1)
uiK
uiG1
.0011
.1954*
.2539* .2583*
.2708* .2810*
covariates(3)
uiK
uiG1
.0049
.1692*
.2156* .2235*
.2215* .2481*

uiG2
-.0019
.0115

uiG3
-.0020
.0075
.0074

.1409*
uiG2
.0014
.0220

uiG3
.0022
.0099
.0038

.1073*

covariates(2)
uiK
uiG1
.0018
.1733*
.2196* .2261*
.2282* .2561*
covariates(4)
uiK
uiG1
.0048
.1696*
.2156* .2241*
.2217* .2485*

uiG2
.0003
.0205

uiG3
-.0007
.0097
-.0007

.1157*
uiG2
.0012
.0224

uiG3
.0022
.0099
.0041

.1066*

(Note) *: statistically signiﬁcant at the level .05. Covariates (1) include treatments and
school dummies, covariates (2) include covariates (1) and student characteristics.
Covariates (3) include covariates(2) and teacher characteristics and, ﬁnally, covariates (4)
include covariates (3) and peer characteristics.

Table 2.43. Upper bound estimates for cross-section IV estimator

Initial small
Initial aide
R-squared
# of sample
Other covariates

reduced-from
K
G1
G2
5.205 6.621 5.279
(.749) (.758) (.786)
.259
1.509 1.175
(.698) (.679) (.700)
.309
.295
.276
5840
6430
5919
School dummy, student
teacher characteristics

G3
5.247
(.814)
-.072
(.726)
.220
6086
and

K
G1
G2
G3
5.224 6.490 5.064 4.979
(.749) (.767) (.786) (.817)
.434
1.560 1.156 -.078
(.698) (.680) (.699) (.728)
.311
.296
.279
.221
5840
6430
5919
6086
School dummy, student, teacher
peer(classmates) characteristics

(Note) Students cluster-robust standard errors are provided in parentheses. Student
characteristics include (i) race (ii) gender (iii) free lunch status and teacher characteristics
include (i) race (ii) years of experience (iii) highest degree. Peer characteristics include (i)
fraction of classmates with free lunch (ii) fraction of minority classmates (iii) fraction of
female classmates.

158

2.8

conclusion

Two formal tests of the missing completely at random (MCAR) assumption are introduced in
unbalanced panel data. The powers of tests for strict exogeneity increase with cross-section
size n and missing proportion of data. For instance, the power of Hausman-type tests which
is based on the diﬀerence of pooled LS(or RE) and FE estimators with a data set of n ≥5,000
and T = 4 is 1. The power of the variable addition test is also 1 with a data set of n ≥5,000
and T = 4.
We compare the estimates of unweighted, IPW and MI-IPW methods in unbalanced panel
data using a Monte Carlo simulation. The results show that the IPW method under MAR
works well if the probability of selection is correctly speciﬁed. We also study the robustness
of the IPW estimator with the probability of selection model misspeciﬁcation. Simulation
shows that ﬁnite sample bias of the IPW estimator is smaller than that of an unweighted
estimator unless misspeciﬁcation is severe.
We provide the MI-IPW method under MAR to improve eﬃciency by combining the MI
and IPW methods together in an unbalanced panel data model. The estimated eﬀect of
class size reduction (CSR) on student academic achievement for students in grades K-3 is
about 4 to 6 percent with both IPW LS and FD and MI-IPW LS and FD estimators while
the estimated eﬀect for unweighted LS and FD estimators with complete-case is about 6 to
7 percent. Estimators ignoring missing data overestimate the eﬀect of CSR on student score
by about 1 to 2 percentage points.

159

Chapter 3
Cluster robust inference in
unbalanced panel data when T is large

3.1

Introduction

A robust inference method in the linear panel data model with a long T has become more
relevant recently as more panel data are available with long time dimensions. An inference
method which ignores heterogeneity and dependence can severely distort the inference about
parameters of interest. For instance, if errors are correlated across observations either in the
time or cross-section dimension, a covariance matrix using OLS standard error is biased.
Recently, Wooldridge [2003], Hansen [2007], Bester et al. [2009], Vogelsang [2008] and Ibragimov and Muller [2009] provide robust inference methods with dependent data in a balanced
panel data model.1 However, attrition in a panel data and unbalancedness in a two-level data
1 Arellano

[1987] provides asymptotic properties of a covariance matrix in the linear panel
models with random sampling (over cross-section unit) but it is limited only to the case
when T ﬁxed and n large. Moreover, cluster covariance estimators are used with data that

160

occur frequently in a large T data or a two-level data. For instance, unbalancedness of data
occurs when each school has diﬀerent numbers of classes or each region has diﬀerent numbers of states or starting periods of available data that can diﬀer across cross-section units.
Therefore, this paper focuses on robust inferences in unbalanced panel data and the eﬀect
of an unbalancedness on the validity for the inference methods in balanced panel data. In
particular, unbalancedness causes heterogeneity even if attrition or unbalancedness occurs
completely at random and heterogeneity can cause distortion in inference. For example,
the methods in Hansen [2007] and Vogelsang [2008] require homogeneity in cross-section
units to use t-distribution and F -distribution as reference distributions for hypothesis tests.
Therefore, unfortunately, robust inference methods in balanced panel data cannot be directly
extended to unbalanced panel data.
Among those robust inference methods in balanced panel data, the Ibragimov and Muller
[2009] approach provides a t-statistic based inference method. However, this t-statistic based
inference method uses only between variations so there is room for improving eﬃciency.
Bester et al. [2009] shows simulation results that the t-statistic based method of Ibragimov
and Muller [2009] performs poorly if the estimator presents substantial ﬁnite sample bias
using the example of 2SLS simulation with weak IV.
On the other hand, Hansen [2007] provides methods based on constructing t and Wald
statistics using a cluster covariance matrix estimator (CCE). Under homogeneity in both
covariates and errors, t and Wald statistics based on CCE follow standard t and F distrihas a group structure with independence across groups. In our panel data setting, crosssection units represent groups and time series represents group members and we assume
independence between groups. See Wooldridge [2003], Liang and Zeger [1986] and Bertrand
et al. [2004] for more details.

161

bution with scale factors and provide valid inference. Under homogenous covariates and
heterogenous variance, t statistic using CCE with scaling factors converges t-distribution so
the t-test provides conservative inference in Bakirov and Szekely [2006]. When heterogeneity exists for both covariates and errors, the t statistic using CCE converges to a limiting
distribution which is neither standard nor pivotal in both Hansen [2007] and Ibragimov and
Muller [2009]. Hansen [2007]’s method without homogeneity can not perform valid robust inference using standard reference distributions of t and F while Ibragimov and Muller [2009]’s
approach provides conservative inference of Bakirov and Szekely [2006] in the presence of
heterogenous variance.
Vogelsang [2008] provides a robust inference method based on constructing t and Wald
statistics with HACs of a covariance matrix where the HAC smoothing parameter grows
proportionally to the sample size (this is called ﬁxed-b inference). Vogelsang [2008] also
shows convergence of t and Wald statistics with HACs covariance matrix based on ﬁxed-b
smoothing parameter to t and F distribution with scaling factors under homogeneity. Both
in Hansen [2007] and Vogelsang [2008], we need to calculate limiting distributions for t and
Wald statistics treating the covariance matrix as random variables in the limit. This causes
heterogeneous variance for the data with diﬀerent time periods of attrition across diﬀerent
cross-section units.
Our objective, in this paper, is to propose a method which provides a robust inference
with heterogenous variance and autocorrelation in an unbalanced linear panel data model.
We compare our method to those in Hansen [2007] and Ibragimov and Muller [2009] which
are developed for balanced panel data. Our proposed inference method is an extension of the
OLS method in Hansen [2007] to weighted least squares (WLS) in constructing t and Wald
162

statistics using CCE where attrition periods are used as weights. We show that, using the
result of Bakirov and Szekely [2006], t-test with WLS provide conservative inference result.
To our knowledge, there has been no study that focuses on the asymptotic properties of a
variance matrix in unbalanced panel data in the presence of heterogeneity and autocorrelation. For the sake of simplicity, when doing the simulation we focus on random attrition for
panel data and random unbalancedness for two-level data.
In the MC experiment, heterogeneity in variance and covariates are induced through random attrition in unbalanced panel data. Our proposed inference method based on WLS
which extends Hansen [2007] is compared to the t-statistic based method of Ibragimov and
Muller [2009] and the method of Hansen [2007] with OLS. The WLS method which uses the
attrition periods information as weights, in comparison to the OLS method in Hansen [2007],
shows signiﬁcantly smaller size distortion. Simulation results also reveal that our method
shows power improvement over the t-statistic based method in Ibragimov and Muller [2009].
We also extend our proposed method with WLS to the case where the heterogeneity by both
attrition and heteroskedastic variance is present. We compare the power of our method to
that of Ibragimov and Muller [2009] and provide evidence for improvement of power performance through a MC simulation.
Unbalancedness which is caused by attrition introduces heterogeneity and this forces conservative t-tests for the method in Ibragimov and Muller [2009]. Thus, we examine the
tradeoﬀ for the t-tests based on the method in Ibragimov and Muller [2009] as we drop data
on purpose to make the data balanced. Dropping available data inevitably introduces the
loss of power while the balancedness of data eliminates heterogeneity so it leads to no size
distortion in inference. Simulation results show that the cost of losing power dominates the
163

beneﬁt of eliminating size distortion as the loss of power is substantial while size distortion
is not signiﬁcant.
The rest of the paper is organized as follows. In section 2, we introduce models with
individual ﬁxed eﬀects in unbalanced panel data and propose a robust inference method
which extends the method of Hansen [2007] with WLS in unbalanced panel data. Section
3 contains MC experiment results and, in section 4, we provide a brief summary of results
and the conclusion.

3.2

Model

This section presents a method for conducting robust inference with attrition in linear unbalanced panel data model.

yit = zit β0 + ci + ft + uit

(3.1)

with i = 1, 2, ..., n and t = 1, 2, ..., T .
yit is continuous and zit is a vector of covariates, ci is individual ﬁxed eﬀects, ft is time
ﬁxed eﬀects and uit is weakly dependent error. We assume random sampling for cross-section
units but allow weak dependence in time dimension.2 For the sake of simplicity of notation,
2 The

application of balanced panel data model of (3.1) with large T has been appeared
in empirical ﬁnance literature, in equity returns and earnings surprises, and oftentimes in
these models standard errors were estimated with the adjustment to reﬂect correlation across
observations. Typical standard error adjustments was based on Fama and MacBeth [1974]
procedure or the method of clustering across time is most common. Recently, the use of
panel data and linear model with outcome of serial dependence becomes quite common in
ﬁnance and macroeconomics applications but Petersen [2009] surveys that the methods each
researcher uses to adjust the bias of standard errors are quite various.

164

we rewrite (3.1) as in (3.2).
yit = xit β + ci + uit

(3.2)

where xit = (zit , ft ) and β = (β0 , 1) .
Now we deﬁne selection indicator (sit ) to express the model with unbalanced panel data
by random attrition. sit = 1 if both dependent variable (yit ) and all of covariates (xit ) are
available and sit = 0 otherwise.
Consider the following ﬁxed-eﬀects (FE) transformation of the original model of (3.1) to
(3.3) with balanced panel data.

˜
yit = xit β + uit
˜
˜

(3.3)

where
1
yit = yit −
˜
Ti

T

T

sit · yit , Ti =
t=1

t=1

1
˜
sit , xit = xit −
Ti

T
t=1

1
sit · xit , uit = uit −
˜
Ti

T

sit uit
t=1

(3.4)
Correspondingly, ﬁxed-eﬀects (FE) transformed model for unbalanced panel data can be
expressed with selection indicator as follows.

˜
sit yit = sit xit β + sit uit
˜
˜

(3.5)

The focus is on inference about β. OLS estimator for β in (3.5) is given by (3.6).(We
denote this FE estimator as OLS hereafter.)

n

T

ˆ
βols = (

n

˜ ˜
sit xit xit
i=1 t=1

T

)−1

˜ ˜
sit xit yit
i=1 t=1

165

(3.6)

where we use s2 = sit .
it
n

T

ˆ
βols − β = (

n

˜ ˜
sit xit xit

T

)−1

i=1 t=1

n

T

˜ ˜
sit xit uit = (
i=1 t=1

n

˜ ˜
sit xit xit
i=1 t=1

T

)−1

˜
sit xit uit
i=1 t=1

(3.7)

3.2.1

Robust inference with t-test in unbalanced panel data

In this section, we present ﬁxed-n, large-T asymptotic properties of the CCE for unbalanced
panel data. Under ﬁxed-n and large T , the CCE is not consistent but converges in distribution to a limiting random variable. In the CCE with attrition, Ti plays the role of the
HAC smoothing parameter as Ti determines the bandwidth for each cross-section unit i. In
T

particular, smoothing parameter, λi = Ti which changes for each cross-section unit is equivalent to b in HAR covariance estimator in ﬁxed-b asymptotic theory (Kiefer and Vogelsang
[2002] and Kiefer and Vogelsang [2005]).
For balanced panel data, Hansen [2007] shows that we can perform t and F tests with scaled
t and Wald statistics under the assumption that covariates and error terms are independent
of cross-section unit (hereafter we call this assumption as cross-section homogeneity).

Theorem 4 (Hansen 2007) Under cross-section homogeneity and balanced panel, standard t-statistic for βols with cluster robust standard error converges to a t-distribution with
n − 1 degree of freedom, scaled by

n
n−1 ,

under the null hypothesis.

However, the methods which ignore attrition cannot be directly applied with unbalanced
panel data since attrition introduces heterogeneity in both covariates and variance so this
leads to distortion in inference for unbalanced panel data.

166

We extend the result of theorem 4 to unbalanced panel data using weights and the result
of Bakirov and Szekely [2006]. We use weight for each cross-section unit to recover original
homogeneity in covariates of balanced panel data. The idea of recovering homogeneity is
that cross-section units with fewer time-series get higher weights for each observation to
contribute evenly across cross-section.
With the following assumptions, under cross-section homogeneity of original balanced
panel data, t-test with weighted least squares (WLS) of (3.8) and CCE of Vw−clus in (3.9)
provides a conservative inference result of Bakirov and Szekely [2006] for unbalanced panel
data.
n

T

βwls = (
i=1 t=1
n

T

Vw−clus = (
i=1 t=1

x x
sit √it √it )−1
λi λi

n

T

i=1 t=1

1
s x x )−1 Ωwls (
λi it it it

n

y
x
sit √it √it
λi λi
T

i=1 t=1

1
s x x )−1 ,
λi it it it

(3.8)

(3.9)

where
T

n

[(

Ωwls =

i=1 t=1

1
s x u )(
λi it it it

T
t=1

1
s x u )]
λi it it it

and uit = yit − xit βW LS .
Assumption 5

1. Independent sampling across cross-section, n is ﬁxed and Ti goes in-

ﬁnity for each i
2. Weak dependence for xit , uit
1
3. (WLLN) T
1
4. (FCLT) √

T
t=1

p

T

sit xit xit → λi Qi and Q−1 exists, where Ti = λi λi ∈ (0, 1]
i

T

s x u
T t=1 it it it

⇒ Λi W (λi ), where W (λi ) is brownian motion with mean zero

and variance λi

167

5. (Homogeneity across cross-section) Qi = Q and Λi = Λ
6. (Missing Completely At Random) E(uit |xit , si ) = 0 where si = (si1 , si2 , ...., siT )
Consider testing linear hypothesis about β of the form
H0 : Rβ = r, H1 : Rβ = r

(3.10)

where R is a q × k matrix of known constants with full rank and r is a q × 1 vector of
known constants.
In the case q = 1 we can deﬁne the t-statistics
√
t=

T (Rβols − r)

, t∗ =
wls

R · T · Vclus R

√
T (Rβwls − r)

(3.11)

R · T · Vw−clus R

Using following two theorems 6 and 7, we can obtain corollary 8 which allows conservative
inference for t-test with weight.
Theorem 6 (Bakirov and Szekely [2006]) Let X1 , X2 , ....., Xn be an i.i.d. sample from
normal scale mixture, and let y1 , y2 , ....., yn be independent normal (0, σi ) random variables,
1
y=n

n
i=1

1
2
yi , and Sy = n−1

n

(yi − y)2 . Then

i=1

√ X −µ
P{ n
> x} =
sup
SX
σ1 ,σ2 ,...σn

Rn

n
√ y
P{ n
> x}
dF (σi )
Sy
i=1

Proof. See Bakirov and Szekely [2006].
Theorem 7 (Bakirov and Szekely [2006]) Let Cn−1 (α) be the critical value of the usual
two-sided t-test based on (3) of level α (i.e. P (|Tn−1 | > Cn−1 (α)) = α)
√
nX
Tn−1 =
where X1 , X2 , ....., Xn be an i.i.d.sample from normal
SX
168

(3.12)

Then, using theorem 6,
√
(i) If α ≤ 2 · (1 − Φ( 3)) = 0.0832...., then ∀n ≥ 2,
sup
σ1 ,σ2 ,...σn

P (|Tn−1 | > Cn−1 (α)) ≤ P (|Tn−1 | > Cn−1 (α)) = α

(3.13)

(i.e. α ≤ 0.0832 and ∀n ≥ 2, Tn−1 ⇒ Tn−1 where Tn−1 is t-distribution with n − 1 degree
of freedom)
(ii) Usual one-sided t-test of nominal level 0.05 or lower remains valid as long as n ≤14.
For n >15, usual two-sided tests of nominal level 0.1(or one-sided tests of nominal level
0.05) are not automatically conservative for large n.
(iii) For α ≤ 0.0832 and ∀n ≥ 2, as n → ∞ ,

sup
σ1 ,σ2 ,...σn

P (|Tn−1 | > Cn−1 (α)) ≤ P (Z > Cn−1 (α)) = α

where Z ∼ N (0, I)
Proof. See Bakirov and Szekely [2006].
Corollary 8 Under assumptions 5 and attrition, the inference based on t∗ in (3.11) scaled
wls
by

n
n−1

converges to Tn−1 -distribution in theorem 7, under the null hypothesis.

Proof. Here we provide the sketch of the proof. See appendix for details. First, consider the
numerator of t∗ . βwls = (
wls

n

T

xit x
sit √ √it )−1
λi

i=1 t=1

λi

n

T

xit y
sit √ √it . Under assumptions 5

i=1 t=1

λi

λi

and null hypothesis, we can rewrite numerator as
√

√
T (Rβwls − Rβ) = R T (βwls − β)

n
√
√
R T (βwls − β) = R T (

T

i=1 t=1

x x
sit √it √it )−1
λi λi

169

n

T

i=1 t=1

x u
sit √it √it
λi λi

n
√
√
R T (βwls − β) = R T (

T

i=1 t=1
n

=> R(
i=1

1
λ Q )−1
λi i i

n

x x
sit √it √it )−1
Ti Ti

1
Λ W (λi ) =
λi i

i=1

n

T

i=1 t=1
n
R(
Qi )−1
i=1
i=1

x u
sit √it √it
Ti Ti

n

1
Λ W (λi )
λi i

where W (λi ) is wiener process with mean zero and variance λi .
Now consider denominator T · Vw−clus
n

T · Vw−clus = (
i=1

1
Ti

T
t=1

n

[(

Ωwls =
i=1

1 1
√
λi T

T
t=1

i=1

1 1
√
λi T

T

1 1
sit xit uit = √
λi T

n

sit xit xit )−1 Ωwls (

sit xit uit −
t=1

1 1
λi T

T

sit xit uit )(
t=1
T

1
Ti

T

sit xit xit )−1
t=1
T

1 1
√
λi T

sit xit uit ) ]
t=1

√
sit xit xit T (βW LS − β) + op (1)

t=1

1
=⇒ Λi W (λi ) − Qi (
λi

n

n

Qj

)−1 (

j=1

j=1

1
Λ W (λj ))
λj j

Let’s denote W (λj ) as Wj for simplicity of notation.
n

Under cross-section homogeneity,
(nQ)−1 Ωwls (nQ)−1 and Ωwls ⇒ Λ[

T · Vw−clus

n W W
i i
λi λi
i=1

−

⇒

(

Qi )−1 Ωwls (

i=1
n W
n W
j
j
1
n(
λj )(
λj )]Λ
j=1
j=1

n

Qi )−1

=

i=1

.

W (λ )

Let’s deﬁne yi as yi ≡ RQ−1 Λ λ i . Since Wi ≡ W (λi ) is brownian motion with mean
i
0 and the variance λi , yi is mean zero and variance of

RQ−1 Λvar(W (λi ))Λ Q−1 R
λ2
i

. As the

variance of W (λi ) is λi ∗ Iq , for the case of q = 1, y1 , y2 , y3 , ....., yn is independent and each
−1 ΛΛ Q−1 R
λi

yi is normal (0, RQ

). Then, in the case of q = 1, we have

n

yi
t∗
wls

⇒

i=1
n
i=1

=

1
2
yi − n

n
·
n−1

n

n

yi
i=1

1
n−1

ny

=

n

yj
j=1

√
n

i=1

ny

≡

(yi − y)2

i=1

170

2
yi − ny 2

=

ny
n

(yi − y)2

i=1

√ y
n
Tn−1 where Tn−1 ≡ n
n−1
Sy

Thus, the proper weight leaves λi as the only source of heterogeneity so that we can apply
Bakirov and Szekely [2006] result and obtain conservative inference.

Remark 9 Heterogeniety is induced by attrition. Attrition leads to heterogeneity in sit xit xit
and sit xit uit for each i. However, using the ratio of the length of time-series to full length
T

of time series as weight, λi = Ti , we apply WLS and the weight eliminates heterogeneity in
covariates.

Under cross-section homogeneity assumption, as attrition is the unique source of heterogeneity, the use of weights recovers the homogeneity in covariates. Moreover, since empirically most relevant case is two-sided tests with α ≤ 0.05, we can conduct inference with t∗wls statistics by multiplying

n−1 ,
n

n
which is equivalent to replace Vw−clus with n−1 · Vw−clus .

For a signiﬁcant level .05 or lower, the t-test with t∗wls -statistic scaled by

n−1
n

remains

conservative.

3.2.2

WLS with heterogenous covariance matrix in unbalanced
panel data

In addition to heterogeneity by attrition, we introduce heteroskedastic variance for each
cross-section unit in unbalanced panel data. For balanced panel data, t-statistic based test
in Bester et al. [2009] and Ibragimov and Muller [2009] provide conservative inference results
of Bakirov and Szekely [2006] in the presence of heteroskedastic variance. For unbalanced
panel data, if homogeneity only holds for covariates but not for variance so that Qi = Q
and Λi = Λ hold because of heteroskedastic variance, we can still obtain the conservative

171

inference result in Bakirov and Szekely [2006] with t-test as we use t∗wls -statistic with scale
factor.
Under heteroskedastic variance and homogenous covariates for unbalanced panel data, we
have two sources of heterogeneity but we can still apply corollary 8 with βW LS and Vw −clus
and perform t-test. Only change from homoskedastic variance is that we deﬁne yi in the
W (λ )

proof of corollary 8 as RQ−1 Λi λ i so that its variance is
i

RQ−1 Λi Λi Q−1 R
λi

. The key for

the determination of weights is making each cross-section unit to contribute evenly as in
the case of no heteroskedastic variance. Units with fewer time series observation and with
smaller variance receive more weights than the units with more time series or large variance.
This weighting eliminates heterogeniety in covariates but two sources of heteroskedasticity
in error still remains after weighting.
In sum, we show that robust inference method in balanced panel data can be extended to
unbalanced panel data with proper choice of weights for cross-section units. Corollary 8 shows
that t-test with WLS in unbalanced panel data provides the bound result of conservative
inference under heterogenous variance and attrition.

3.3

Monte Carlo experiment

In this section, we examine ﬁnite sample properties of proposed WLS method in unbalanced
linear panel data model with dependent data. We provide two sets of simulation. The only
source of heterogeneity is attrition in the ﬁrst simulation while there are two sources of
heterogeneity, attrition and heteroskedastic variance in the second simulation. Simulation
results for the size distortion and the power of test are provided. In each set of simulation,

172

we compare the size distortion and the power of our method to those of Hansen [2007]
and Ibragimov and Muller [2009] methods which were developed for balanced panel data.
Attrition time periods are generated in simulation not to be correlated with either outcome
or covariates. Thus, each cross-section unit has diﬀerent attrition time period and this leads
to diﬀerent time-series length which introduces heterogenous variance for each cross-section
unit.

3.3.1

The t-test using t∗ -statistic with scaled factor under homewls
geniety and attrition

DGP I: Attrition is the only source of heterogeneity
We consider following data generating process for balanced panel data.
Yit = xit · β + ci + ft + uit , for i = 1, 2, .., n and t = 1, 2, ..., T
where xit is continuous variable.

Variables generation
1. xit ∼ iidN (0, 1), β = 1
2. Homogeneous error, ui , with t=T



 ui1 







 u 

 i2 


 ∼ M V N 
 . 

 . 

 . 







uiT

 



0   1
ρ . . . ρT −1 
 

 

  ρ
0  
1
ρT −2 

 , 

.   .
. 
.   .
. 
.   .
1
. 
 

 

0
ρT −1
...
1

where M V N is multivariate normal and ρ=0.75 in the simulation.
173

3. ci = ai + xi , ai ∼ iidN (0, 1) and xi =
¯
¯

T
t=1 sit

· xit

4. ft = 0
The determination of attrition for unbalanced panel data is presented in table 3.1. Two
key features in the determination of attrition are: (i) the proportion of available data is
about 60% and (ii) the length of available time-series for each cross-section units vary so
the variation of time length for each cross-section unit induces heterogeneity. We consider
various length of time-series, T =50, 100, 300, and 1,000 for balanced panel data. T =50
has relevance for yearly data of 40 years such as data from 1970 to 2009, while T =100 has
relevance for quarterly data of 25 years such as from 1985 to 2009.

174

Table 3.1. selection process
T=50
n=4
n=5
n=6
n=7
n=8
n=9
n=10
n=11
n=16
T=100
n=4
n=5
n=6
n=7
n=8
n=16
T=300
n=4
n=5
n=6
n=7
n=8
n=16

missing fraction
50(2), 10(2)
0.4
50(2), 30(1), 10(2)
0.4
50(2), 40(1), 20(1), 10(2)
0.4
50(2), 40(1), 30(1), 20(1), 10(2)
0.4
50(2), 40(2), 20(2), 10(2)
0.4
50(2), 40(1), 35(1), 30(1), 25(1), 20(1), 10(2)
0.4
50(2), 40(1), 35(1), 30(2), 25(1), 20(1), 10(2)
0.4
50(2), 40(1), 35(1), 30(3), 25(1), 20(1), 10(2)
0.4
50(2), 40(2), 35(2), 30(4), 25(2), 20(2), 10(2)
0.4
missing fraction
100(2), 20(2)
0.4
100(2), 60(1), 20(2)
0.4
100(2), 70(1), 50(1), 20(2)
0.4
100(2), 70(1), 60(1), 50(1), 20(2)
0.4
100(2), 70(1), 60(2), 50(1), 20(2)
0.4
100(2), 80(2), 70(2), 60(4), 50(2), 40(2), 20(2)
0.4
missing fraction
300(2), 100(2)
0.33
300(2), 100(3)
0.4
300(2), 140(2), 100(2)
0.4
300(2), 180(1), 140(2), 100(2)
0.4
300(2), 200(2), 160(1), 140(1), 100(2)
0.4
300(4), 240(1), 210(2), 150(2), 125(2), 90(4)
0.4

(Note) The length of time is reported and number of cross-section is reported in parentheses.

175

Simulation result: Homogeneity and attrition
In all simulation, we consider inference about parameter β in panel data model of (3.3) using
OLS in (3.6) and WLS in (3.8). Table 3.2 shows the size of t-test with nominal level 0.05.
The ﬁrst column shows the dimension of panel data. The second column reports the rejection
probability of t-test for balanced panel data using t-statistic in Hansen [2007]. As the
homegeneity assumption is satisﬁed, t-test for balanced panel data shows no size distortion.
Coverage rate for rejection probability contain nominal level 0.05 for all n. Third column
reports the rejection probability for t-test of complete case from unbalanced panel data.
For all simulated cross-section dimension n ≤8, rejection probability shows substantial size
distortion by over-rejection and the size distortion is greater for unbalanced panel data with
smaller n. Therefore, simulation results verify that t-test using t-statistic in Hansen [2007]
show size distortion with unbalanced panel data. The fourth column shows the rejection
probability from t-test of Ibragimov and Muller [2009] method. t-statistics in Ibragimov and
Muller [2009] is obtained from (3.14).

¯
ˆ
√ β
tβ = n
sβ
ˆ
¯ 1
ˆ
where β= n

n
ˆ
j=1 βj

1
and s2 = n−1
ˆ
β

n
j=1 (βj

(3.14)

¯
ˆ
− β)2

The rejection probability from t-test of Ibragimov and Muller [2009] method provides the
evidence of conservative inference when there is heterogeneity. It also shows that the size
distortion tapers oﬀ as n increases. Eventually, Ibragimov and Muller [2009] method with
heterogeneity and n=16 shows no size distortion at all.
The ﬁfth column in table 3.2 provides the rejection probability of t-test with t∗ -statistics
wls

176

where the weights for each cross-section is calculated by

T
Ti .

As WLS put more weights

to cross-section unit with shorter length so that eliminate heterogeneity across cross-section.
Simulation results for t-test with t∗ -statistics in table 3.2 verify the conservative inference
wls
of corollary 8 and the size distortion with WLS is substantially smaller than that of OLS
especially for small n ≤8. In sum, WLS method reduces type I error. The bound from WLS
is very close to .05 and much closer to .05 than the bound from Ibragimov and Muller [2009]
method.
Table 3.3 reports the size adjusted power of t-test for T =50. Power is greater for t-test
of Hansen [2007] with WLS than for that of Ibragimov and Muller [2009]. An inference of
Hansen [2007] with WLS dominates that of Ibragimov and Muller [2009] in all n and the
power advantage of Hansen [2007] with WLS comes from using both within and between
variations while Ibragimov and Muller [2009] only use between variation. All these imply
that a robust inference of Hansen [2007] with WLS is preferred method especially with small
n ≤8 when attrition is the sole source of heterogeneity. We obtain the same qualitative
implication in table 3.4 with T =100. As n and T increases, the power of tests also increases
for all four methods. The order of higher power is balanced data, complete case with OLS,
complete case with WLS and complete case with Ibragimov and Muller [2009] method.

177

Table 3.2. The size of test, 1(p < 0.05)
T=50

balanced
complete-case
complete-case
OLS
OLS
t-stat inference of IM
n=4
.051(.047,.055) .090(.084,.096)
.032(.028,.035)
n=5
.053(.048,.057) .074(.069,.079)
.034(.030,.037)
n=6
.053(.048,.057) .068(.063,.073)
.039(.035,.043)
n=7
.053(.048,.057) .065(.060,.070)
.039(.035,.043)
n=8
.049(.045,.053) .058(.053,.062)
.045(.041,.049)
n=16
.050(.046,.054) .049(.045,.054)
.050(.044,.054)
T=100
balanced
complete-case
complete-case
OLS
OLS
t-stat inference of IM
n=4
.050(.046,.054) .083(.078,.089)
.035(.031,.038)
n=5
.052(.047,.056) .090(.085,.096)
.024(.021,.027)
n=6
.048(.044,.053) .079(.073,.084)
.029(.026,.032)
n=7
.049(.045,.054) .068(.063,.073)
.028(.025,.031)
n=8
.051(.046,.056) .061(.056,.066)
.042(.038,.045)
n=16
.051(.045,.057) .050(.044,.056)
.051(.045,.058)
T=300
balanced
complete-case
complete-case
OLS
OLS
t-stat inference of IM
n=4
.051(.046,.055) .075(.069,.080)
.034(.030,.037)
n=5
.049(.044,.053) .067(.062,.071)
.040(.036,.044)
n=6
.052(.047,.056) .067(.062,.072)
.042(.038,.046)
n=7
.047(.043,.051) .062(.058,.067)
.041(.037,.045)
n=8
.049(.044,.053) .061(.056,.066)
.042(.038,.045)
n=16
.048(.044,.052) .055(.050,.059)
.047(.043,.051)

complete-case
WLS
.051(.046,.055)
.051(.046,.055)
.050(.044,.054)
.049(.045,.054)
.049(.044,.053)
.046(.041,.051)
complete-case
WLS
.044(.040,.048)
.045(.041,.049)
.045(.041,.049)
.043(.040,.047)
.045(.041,.049)
.050(.044,.056)
complete-case
WLS
.042(.038,.046)
.045(.041,.049)
.047(.043,.051)
.045(.041,.049)
.045(.041,.049)
.049(.045,.054)

(Note) The rejection probability with nominal level .05 is reported. 95 % coverage rate for
rejection probability is reported in parenthesis. The number of replications in simulation is
10,000. Balanced OLS and complete case OLS is based on the methods in Hansen [2007].

178

Table 3.3. The size adjusted power of test
T=50
n=4
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=50
n=5
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=50
n=6
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=50
n=7
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=50
n=8
bias=0
bias=2%
bias=5%
bias=10%
bias=20%

balanced
OLS
.051(.047,.055)
.056(.051,.060)
.077(.071,.082)
.147(.140,.153)
.395(.385,.404)
balanced
OLS
.053(.048,.057)
.057(.052,.061)
.084(.078,.089)
.188(.181,.196)
.551(.541,.561)
balanced
OLS
.053(.048,.057)
.055(.050,.060)
.091(.086,.097)
.227(.219,.235)
.668(.658,.677)
balanced
OLS
.053(.048,.057)
.058(.053,.062)
.107(.101,.113)
.278(.269,.287)
.763(.754,.771)
balanced
OLS
.049(.045,.053)
.059(.054,.063)
.118(.111,.124)
.320(.311,.330)
.835(.827,.842)

complete-case
OLS
.090(.084,.096)
.089(.083,.094)
.103(.097,.109)
.152(.145,.159)
.347(.337,.356)
complete-case
OLS
.074(.069,.079)
.082(.077,.087)
.098(.092,.104)
.171(.164,.178)
.429(.420,.439)
complete-case
OLS
.068(.063,.073)
.074(.069,.079)
.096(.090,.101)
.188(.180,.195)
.498(.488,.507)
complete-case
OLS
.065(.060,.070)
.072(.067,.077)
.100(.094,.106)
.206(.198,.214)
.576(.567,.586)
complete-case
OLS
.058(.053,.062)
.066(.061,.071)
.105(.099,.111)
.230(.221,.238)
.641(.631,.650)

complete-case
t-stat inference of IM
.032(.028,.035)
.033(.029,.037)
.039(.035,.042)
.061(.057,.066)
.157(.149,.164)
complete-case
t-stat inference of IM
.034(.030,.037)
.037(.033,.041)
.048(.044,.053)
.087(.082,.093)
.238(.229,.246)
complete-case
t-stat inference of IM
.039(.035,.042)
.045(.041,.049)
.059(.054,.063)
.107(.100,.113)
.307(.298,.316)
complete-case
t-stat inference of IM
.039(.035,.042)
.045(.040,.049)
.064(.059,.069)
.131(.125,.138)
.372(.363,.382)
complete-case
t-stat inference of IM
.045(.041,.049)
.046(.042,.050)
.075(.069,.080)
.158(.151,.165)
.465(.455,.474)

complete-case
WLS
.051(.046,.055)
.050(.045,.054)
.060(.056,.065)
.096(.091,.102)
.244(.236,.253)
complete-case
WLS
.051(.046,.055)
.054(.050,.059)
.067(.062,.072)
.119(.113,.126)
.345(.226,.354)
complete-case
WLS
.050(.045,.054)
.056(.051,.060)
.072(.067,.077)
.149(.142,.156)
.430(.420,.440)
complete-case
WLS
.049(.045,.054)
.056(.052,.061)
.080(.075,.086)
.172(.165,.179)
.513(.503,.523)
complete-case
WLS
.049(.044,.053)
.056(.052,.061)
.089(.084,.095)
.201(.193,.209)
.601(.591,.610)

(Note) The rejection probability is reported. 95 % coverage rate for rejection probability is
reported in parenthesis. Generated errors are from AR(1) process with correlation
coeﬃcient 0.75. The number of replications is 10,000. Balanced OLS and complete case
OLS is based on the methods in Hansen [2007].
179

Table 3.4. The size adjusted power of test, 1(p < 0.05), T = 100
T=100
n=4
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100
n=5
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100
n=6
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100
n=7
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100
n=8
bias=0
bias=2%
bias=5%
bias=10%
bias=20%

balanced
OLS
.050(.046,.054)
.056(.051,.060)
.097(.091,.104)
.228(.220,.237)
.646(.635,.657)
balanced
OLS
.052(.047,.056)
.063(.056,.069)
.117(.109,.125)
.320(.309,.330)
.821(.813,.828)
balanced
OLS
.048(.044,.053)
.058(.054,.063)
.140(.131,.148)
.406(.396,.416)
.916(.909,.923)
balanced
OLS
.049(.045,.054)
.073(.067,.080)
.172(.161,.182)
.487(.474,.500)
.964(.960,.968)
balanced
OLS
.051(.046,.056)
.070(.064,.075)
.188(.178,.198)
.554(.544,.564)
.986(.982,.989)

complete-case
OLS
.083(.078,.089)
.083(.078,.089)
.117(.109,.124)
.223(.215,.231)
.549(.537,.560)
complete-case
OLS
.090(.085,.096)
.097(.089,.104)
.141(.132,.149)
.281(.271,.290)
.678(.669,.687)
complete-case
OLS
.079(.073,.084)
.089(.083,.094)
.142(.133,.150)
.315(.306,.324)
.766(.755,.777)
complete-case
OLS
.068(.063,.073)
.086(.078,.093)
.142(.132,.151)
.366(.354,.378)
.850(.842,.858)
complete-case
OLS
.061(.056,.066)
.078(.072,.083)
.163(.153,.173)
.408(.398,.418)
.893(.883,.902)

complete-case
t-stat inference of IM
.035(.031,.038)
.036(.032,.039)
.044(.039,.049)
.097(.091,.103)
.289(.278,.299)
complete-case
t-stat inference of IM
.024(.021,.027)
.027(.023,.031)
.040(.035,.045)
.096(.090,.103)
.285(.276,.293)
complete-case
t-stat inference of IM
.029(.026,.032)
.030(.027,.033)
.056(.050,.061)
.131(.124,.138)
.399(.387,.411)
complete-case
t-stat inference of IM
.028(.025,.031)
.038(.033,.043)
.067(.059,.069)
.171(.161,.180)
.495(.484,.505)
complete-case
t-stat inference of IM
.042(.038,.045)
.049(.044,.053)
.095(.086,.104)
.248(.238,.258)
.670(.659,.680)

complete-case
WLS
.044(.040,.048)
.049(.044,.053)
.067(.061,.073)
.140(.133,.146)
.407(.396,.418)
complete-case
WLS
.045(.041,.049)
.046(.040,.051)
.080(.073,.087)
.176(.168,.185)
.543(.533,.553)
complete-case
WLS
.045(.041,.049)
.056(.052,.060)
.094(.087,.102)
.233(.225,.241)
.679(.667,.691)
complete-case
WLS
.043(.039,.047)
.060(.052,.061)
.102(.093,.111)
.293(.282,.304)
.789(.780,.798)
complete-case
WLS
.045(.041,.049)
.060(.052,.071)
.128(.118,.139)
.348(.338,.358)
.855(.844,.865)

(Note) The rejection probability is reported. 95 % coverage rate for rejection probability is
reported in parenthesis. Generated errors are from AR(1) process with correlation
coeﬃcient 0.75. The number of replications is 10,000. Balanced OLS and complete case
OLS is based on the methods in Hansen [2007].
180

Table 3.5. Conﬁguration of heteroskedastic variance

n=4
n=5
n=6
n=7
n=8

3.3.2

a1 a2
1.5 .5
1.5 .5
1.5 .5
1.5 .5
1.5 .5

a3
2
2
2
2
2

a4
.8
.8
.8
.8
.8

a5

a6

a7

a8

.9
.9
.9
.9

1.1
1.1
1.1

1.2
1.2

.7

The t-test using t∗ -statistic with scaled factor under hetwls
erogenous variance and attrition

DGP II: Attrition and heterokedastic variance
We use the same data generating process as in DGP I where attrition is the only source of
heterogeneity and attrition conﬁgurations as in table 3.1. In DGP II, the only diﬀerence
in DGP is coming from error process ui . As in DGP I, each uit is initially drawn from
standard normal with AR(1) process of serial correlation coeﬃcient, ρ=0.75. Then, we
introduce heteroskedastic variance by replace uit with ai ∗ uit where, for the case of n=4,
we assign a1 =1.5, a2 =.5, a3 =2 and a4 =.8. Heteroskedastic variance conﬁgurations in DGPs
are provided in table 3.5.

Simulation result: Heteroskedastic variance and attrition
Table 3.6 shows the size of t-test at the nominal level .05 for data with n=4,5,6,7,8,16 and
T =100. The t-test inference of Hansen [2007] results in balanced panel data are reported
in the second column and it shows conservative inference for all n. As n increases, the
rejection frequency approaches nominal level .05 so that size distortion disappears. Third
column shows the results of t-test with complete case in unbalanced panel data. There are
181

two sources of heterogeneity for the t-test with complete case: attrition and heteroskedastic
variance. Both sources of heterogeneity are not controlled for t-test with complete case.
Results in third column shows that two sources of heterogeneity may work against each
other as attrition mitigates standard error while heteroskedastic variance increases standard
error used in t-statistic. Thus, there are no consistent pattern of over-rejection or underrejection as we change n. The fourth column reports the results of t-tests using the method
in Ibragimov and Muller [2009] with complete case in unbalanced panel data. The t-tests
provide conservative inference for all n while rejection frequency approaches to true nominal
level .05 as n increases up to n=16. The mean value of MC rejection frequency for Ibragimov
and Muller [2009] is lower than that for Hansen [2007] in balanced panel data. Fifth column
reports the results of t-test with complete case in unbalanced panel using WLS which is
proposed in this paper. For this method, weights used in WLS eliminate heterogeneity
in covariates due to attrition while heterogeneity in errors by two sources still remains.
The results in ﬁfth column shows conservative inference for all n. The rejection frequency
approaches to true nominal level .05 as n increases up to n=16. The mean value of MC
rejection frequency is lower than that in balanced panel data for all n.
Table 3.7 shows the power of t-tests for the method in Hansen [2007] with balanced panel,
the method in Hansen [2007] with unbalanced panel, the method in Ibragimov and Muller
[2009] with unbalanced panel, and our suggested WLS method with unbalanced panel. The
power increases with n for all methods. Our WLS method has higher power than the method
in Ibragimov and Muller [2009] for all n and all magnitude of bias. The third column shows
the power for the methods in Hansen [2007] while the ﬁfth column reports the power for our
WLS method in unbalanced panel. The WLS method has lower power than the method in
182

Table 3.6. The size of test, 1(p < 0.05)
T=100
n=4
n=5
n=6
n=7
n=8
n=16

balanced
complete-case
complete-case
OLS
OLS
t-stat inference of IM
.033(.030,.037) .031(.028,.034)
.017(.014,.020)
.036(.031,.040) .049(.044,.054)
.027(.023,.031)
.038(.033,.042) .051(.045,.056)
.027(.023,.031)
.042(.038,.046) .051(.047,.056)
.029(.025,.032)
.042(.038,.046) .055(.050,.060)
.036(.032,.040)
.047(.041,.053) .058(.052,.064)
.042(.036,.048)

complete-case
WLS
.018(.015,.020)
.028(.024,.031)
.033(.028,.037)
.037(.033,.040)
.042(.037,.046)
.046(.040,.052)

(Note) The rejection probability with nominal level .05 is reported. 95 % coverage rate for
rejection probability is reported in parenthesis. The number of replications in simulation is
2,000. Balanced OLS and complete case OLS is based on the methods in Hansen [2007].

Hansen [2007] but the diﬀerence is quite small overall. However, compared to homogeneous
case, the introduction of heteroskedastic variance leads to loss of power for all methods we
considered in simulation.

3.3.3

Trade-oﬀ calculation: Dropping observation to make balanced panel data

As unbalancedness of panel data causes heterogeneity and it forces conservative t-tests for
the methods in Ibragimov and Muller [2009] and Bester et al. [2009] and leads to t-tests
invalid when these methods use t-distribution as reference distribution. Thus, we examine
the tradeoﬀ of dropping some observations on purpose to make data balanced.3 Dropping
observations inevitably introduces the loss of power while balanced panel data eliminates
3

We observe this practice of balancing data occasionally especially in international
macroeconomic data. Many data are available from 1970 and the number of countries analyzed sometimes less than 50. It is quite common that researchers have shorter time-series
for developing countries and they use only balanced data from 1980 or later period.

183

Table 3.7. The size adjusted power of test, 1(p < 0.05), T = 100
T=100
n=4
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100
n=5
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100
n=6
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100
n=7
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100
n=8
bias=0
bias=2%
bias=5%
bias=10%
bias=20%

balanced
OLS
.033(.030,.037)
.037(.028,.046)
.069(.061,.077)
.190(.182,.198)
.497(.489,.505)
balanced
OLS
.036(.031,.044)
.045(.037,.053)
.081(.073,.089)
.260(.252,.268)
.681(.673,.689)
balanced
OLS
.038(.033,.042)
.051(.043,.059)
.114(.106,.122)
.316(.308,.324)
.786(.778,.794)
balanced
OLS
.042(.038,.046)
.059(.051,.067)
.137(.129,.144)
.391(.383,.399)
.870(.863,.878)
balanced
OLS
.042(.038,.046)
.070(.064,.075)
.188(.178,.198)
.554(.544,.564)
.986(.982,.989)

complete-case
OLS
.031(.028,.034)
.033(.030,.037)
.059(.051,.064)
.153(.145,.161)
.422(.414,.430)
complete-case
OLS
.049(.044,.054)
.097(.089,.104)
.141(.133,.149)
.281(.273,.289)
.678(.670,.686)
complete-case
OLS
.051(.045,.056)
.058(.053,.063)
.107(.100,.114)
.254(.246,.262)
.622(.614,.630)
complete-case
OLS
.051(.047,.056)
.063(.056,.071)
.117(.110,.125)
.311(.303,.319)
.717(.709,.725)
complete-case
OLS
.055(.050,.060)
.078(.072,.083)
.163(.153,.173)
.408(.398,.418)
.893(.883,.902)

complete-case
t-stat inference of IM
.017(.014,.020)
.017(.014,.020)
.028(.024,.033)
.064(.059,.069)
.208(.203,.213)
complete-case
t-stat inference of IM
.027(.023,.031)
.027(.023,.031)
.040(.035,.045)
.096(.091,.101)
.285(.276,.290)
complete-case
t-stat inference of IM
.027(.023,.031)
.035(.030,.040)
.051(.047,.056)
.111(.106,.116)
.349(.344,.354)
complete-case
t-stat inference of IM
.029(.025,.032)
.036(.031,.040)
.060(.055,.065)
.134(.129,.140)
.409(.404,.415)
complete-case
t-stat inference of IM
.036(.032,.040)
.049(.044,.053)
.095(.086,.104)
.248(.238,.258)
.670(.659,.680)

complete-case
WLS
.018(.015,.020)
.019(.016,.021)
.034(.027,.041)
.093(.087,.100)
.300(.293,.307)
complete-case
WLS
.028(.024,.031)
.046(.040,.053)
.080(.073,.087)
.176(.169,.184)
.543(.535,.550)
complete-case
WLS
.033(.028,.037)
.043(.037,.050)
.077(.070,.084)
.181(.174,.188)
.565(.558,.572)
complete-case
WLS
.037(.033,.041)
.045(.038,.052)
.085(.078,.092)
.241(.234,.248)
.655(.648,.662)
complete-case
WLS
.042(.037,.046)
.060(.052,.071)
.128(.118,.139)
.348(.338,.358)
.855(.844,.865)

(Note) The rejection probability is reported. 95 % coverage rate for rejection probability is
reported in parenthesis. The number of replications is 2,000. Balanced OLS and complete
case OLS is based on the methods in Hansen [2007].

184

heterogeneity and the size distortion in inference. Thus, we can quantify the tradeoﬀ between
the beneﬁt of eliminating the rejection probability bias and the cost of losing power using
DGP I.
Table 3.8 reports the size of t-tests for a method in Ibragimov and Muller [2009]. The
ﬁrst column shows cross-section dimension of data. The second column reports the rejection
probability of t-test for balanced panel data without attrition. The fourth and ﬁfth columns
report the rejection probability of t-test for balanced panels by dropping observations (reduced balanced panel) from complete case for time dimension and cross-section dimension
respectively. These three columns show that there is no size distortion of inferences for
balanced panels in simulation. The third column reports the rejection probability of t-test
for complete case of unbalanced panel data. Diﬀerent length of data due to missing process
conﬁguration of table 3.1 induces heterogeneity and heteroskedastic variance leads to conservative inference of t-test as theory prediction of Ibragimov and Muller [2009] approach
indicates. As n increases the degree of conservativeness of inference is mitigated and, for
n=16, inference exhibits no size distortion regardless of size of time dimensions.
Table 3.9 and appendix I report the size adjusted power of t-tests in Ibragimov and Muller
[2009] for T =50, 100, 300 and 1,000 respectively. Regardless of T , the loss of power is quite
signiﬁcant for reduced balanced panel. Especially for large n ≥8 and bias=20%, the power
of t-test for reduced balanced panel is less than 50% of the power of unbalanced panel of
complete case. This is noticeable decline of power by dropping observations. As reported in
tables 3.3 and 3.4, the diﬀerence of the power of t-tests of between complete case with OLS
and complete case with WLS is negligibly small. The power analysis for reduced balanced
panel data shows that the cost of losing power overwhelms the beneﬁt of no size distortion.
185

Table 3.8. The size of test, 1(p < 0.05) where AR(1) error with correlation coeﬃcient 0.75
T=50
n=4
n=5
n=6
n=7
n=8
n=9
n=10
n=11
n=16
T=100
n=4
n=5
n=6
n=7
n=8
n=16
T=300
n=4
n=5
n=6
n=7
n=8
n=16
T=1,000
n=4
n=5
n=6
n=7
n=8
n=9
n=16

balanced

complete-case

.053(.045,.059)
.054(.046,.060)
.047(.040,.054)
.046(.039,.053)
.051(.044,.058)
.052(.045,.059)
.047(.040,.054)
.049(.042,.056)
.050(.043,.057)
balanced
.048(.041,.054)
.050(.043,.056)
.046(.040,.053)
.054(.047,.061)
.048(.042,.055)
.048(.041,.055)
balanced
.046(.039,.053)
.051(.044,.058)
.053(.046,.060)
.045(.038,.052)
.053(.042,.055)
.046(.041,.055)
balanced
.056(.049,.063)
.051(.044,.058)
.050(.043,.057)
.047(.040,.053)
.046(.038,.055)
.051(.041,.058)
.050(.043,.057)

.031(.025,.036)
.034(.029,.040)
.037(.031,.043)
.042(.036,.048)
.039(.033,.045)
.040(.034,.046)
.042(.036,.048)
.048(.041,.055)
.048(.041,.055)
complete-case
.028(.023,.033)
.023(.019,.028)
.027(.022,.032)
.030(.025,.036)
.038(.032,.044)
.049(.040,.058)
complete-case
.029(.022,.035)
.032(.026,.038)
.039(.032,.045)
.038(.032,.044)
.042(.035,.049)
.050(.043,.057)
complete-case
.033(.026,.040)
.039(.033,.046)
.040(.034,.047)
.038(.032,.044)
.037(.031,.044)
.043(.036,.051)
.046(.039,.053)

balanced
keep ∀i drop t
.050(.042,.056)
.045(.039,.051)
.049(.042,.056)
.048(.041,.054)
.050(.043,.056)
.047(.040,.053)
.049(.042,.056)
.051(.044,.058)
.054(.047,.062)
balanced-drop t
.046(.040,.053)
.051(.044,.058)
.046(.039,.053)
.047(.040,.054)
.050(.043,.057)
.050(.042,.057)
balanced-drop t
.053(.042,.063)
.051(.045,.058)
.050(.043,.057)
.051(.044,.059)
.049(.040,.057)
.050(.043,.057)
balanced-drop t
.042(.034,.049)
.045(.038,.052)
.050(.043,.057)
.047(.040,.053)
.052(.045,.058)
.049(.042,.056)
.050(.043,.057)

balanced
keep ∀t drop i
.050(.043,.056)
.055(.048,.063)
.050(.043,.056)
.049(.042,.055)
.052(.045,.059)
.052(.045,.059)
.053(.046,.060)
.052(.045,.059)
.048(.041,.054)
balanced -drop i
.053(.046,.060)
.047(.040,.053)
.048(.041,.054)
.048(.041,.054)
.053(.046,.060)
.050(.043,.057)
balanced -drop i
.048(.039,.057)
.046(.040,.053)
.052(.046,.059)
.049(.041,.056)
.050(.042,.058)
.048(.041,.055)
balanced -drop i
.057(.050,.064)
.049(.042,.055)
.050(.043,.057)
.053(.046,.060)
.055(.048,.063)
.045(.038,.052)
.048(.041,.055)

(Note) The rejection probability is reported. 95 % coverage rate for rejection probability is
reported in parenthesis. The number of replications is 4,000.

186

Table 3.9. The size adjusted power of test, 1(p < 0.05), T = 50
T=50, n=4

balanced

complete-case

bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=50, n=8
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=50, n=16
bias=0
bias=2%
bias=5%
bias=10%
bias=20%

.053(.045,.059)
.058(.047,.068)
.074(.063,.085)
.137(.121,.152)
.391(.369,.412)
balanced
.051(.044,.058)
.058(.048,.068)
.114(.100,.128)
.307(.287,.327)
.812(.794,.829)
balanced
.050(.043,.057)
.074(.063,.085)
.191(.173,.208)
.613(.591,.634)
.993(.989,.997)

.031(.025,.036)
.043(.034,.051)
.049(.040,.058)
.072(.060,.083)
.174(.157,.190)
complete-case
.039(.033,.045)
.052(.042,.062)
.071(.059,.082)
.152(.136,.167)
.467(.445,.488)
complete-case
.048(.041,.055)
.053(.043,.062)
.102(.089,.115)
.306(.286,.326)
.810(.792,.827)

balanced
keep ∀i drop t
.050(.042,.056)
.047(.037,.057)
.049(.040,.058)
.058(.047,.068)
.101(.088,.114)
balanced-drop t
.050(.043,.056)
.054(.044,.064)
.060(.049,.070)
.081(.069,.092)
.204(.186,.222)
balanced-drop t
.054(.047,.062)
.053(.043,.063)
.083(.071,.095)
.176(.159,.193)
.503(.481,.525)

balanced
keep ∀t drop i
.050(.043,.056)
.057(.046,.067)
.066(.056,.076)
.082(.070,.094)
.121(.107,.135)
balanced -drop i
.052(.045,.059)
.059(.049,.069)
.064(.053,.074)
.083(.071,.095)
.132(.117,.146)
balanced -drop i
.048(.041,.054)
.054(.049,.064)
.073(.062,.084)
.132(.117,.147)
.383(.361,.404)

(Note) The rejection probability is reported. 95 % coverage rate for rejection probability is
reported in parenthesis. Generated errors are from AR(1) process with correlation
coeﬃcient 0.75. The number of replications is 2,000.

187

3.4

Conclusion

In this paper, we extend the robust inference method with heterogenous variance and autocorrelation in Hansen [2007] to unbalanced panel data. In unbalanced panel data, heterogeneity is induced by attrition and heterogeneity in both covariate and variance makes the
t-test using OLS with scaled t-statistic in Hansen [2007] invalid. We derive robust inference
result using weighted least squares (WLS) for unbalanced panel data where t∗ -statistic with
wls
scaling factor provides conservative inference results with standard t-test in the presence of
both attrition and heteroskedastic variance.
We examine our proposed WLS inference method and compare to the methods in Hansen
[2007] and Ibragimov and Muller [2009] via MC simulation. Simulation results verify the
bound result of t-test for the method with WLS when attrition is the only source of heterogeneity and compared to the method in Hansen [2007], the size distortion is signiﬁcantly
reduced with WLS especially for small n. Moreover, compared to t-statistic based method
of Ibragimov and Muller [2009], WLS method performs better in power and the bound of
type I error is closer to .05. Simulation also shows that the proposed WLS method in an
unbalanced panel provides conservative inference but shows greater power than the method
in Ibragimov and Muller [2009] in the presence of both heterogenous variance and attrition.
Finally, we examine the tradeoﬀ of dropping some observations on purpose to make the data
balanced. Balanced panel data eliminate heterogeneity and the size distortion in inference
but the loss of power due to dropping observations is quite substantial. Overall, the cost of
losing power dominates the beneﬁt of eliminating the rejection probability bias.

188

APPENDICES

Appendix A
Gaussian copula density
Consider following joint logistic distribution for (u1 , u2 , ....., uT ).

F (u1 , u2 , ....., uT ) = C(F1 (u1 ), ..., Ft (ut ), ..., FT (uT ); ρ)
where F (·) is joint logistic distribution and Ft (·) is univariate logistic distribution.
Then, we can obtain joint logistic density as following.

f (u1 , u2 , ....., uT ) =

∂ε
∂ T C(ε1 , ε2 , ...., εT ; ρ) ∂ε1 ∂ε2
··· T
∂ε1 ∂ε2 · · · ∂εT
∂u1 ∂u2
∂uT

∂ε

where Ft (ut ) = εt , ∂ut = ft (ut ) for all t and let
t

∂ T C(ε1 ,ε2 ,....,εT );ρ)
∂ε1 ∂ε2 ···∂εT

(A.1)

= c(ε1 , ε2 , ...., εT ; ρ).

Since C(·) is gaussian copula, we can rewrite c(·) as follow using the probability integral
transformation(Hoel et al. [1972]) which transforms normal CDF to logistic CDF.

c(ε1 , ε2 , ...., εT ; ρ) =

φ(Φ−1 (ε1 ), Φ−1 (ε2 ), ..., Φ−1 (εT ); ρ)
1
2
T
φ1 (Φ−1 (ε1 )) · φ2 (Φ−1 (ε2 )) · · · φT (Φ−1 (εT ))
1
2
T

where φ(·) is joint normal density and φt (·) is a univariate normal density.

189

(A.2)

Using the equations (A.1) and (A.2), we obtain following logistic density. In applications,
we can use equation (A.3) to estimate model parameters by maximum likelihood estimation.

T

f (u1 , u2 , ....., uT ) =

φ(Φ−1 (F1 (u1 )), Φ−1 (F2 (u2 )), ..., Φ−1 (FT (uT )); ρ)
1
2
T

ft (ut )
−1 (F (u )))
t t
t=1 φt (Φt
(A.3)

where joint normal density is given by the following.

φ(g1 , g2 , ..., gT ) = (

1
1
1 T
) 2 |Σ|− 2 exp(− g (|Σ|−1 − IT )g)
2π
2

where g = (g1 , g2 , ..., gT ) and Σ is covariance matrix of g, cov(g).

190

Appendix B
Proof for the consistency of
conditional logit estimator in chapter
1
Likelihood of joint T events yi1 , yi2 ...., yiT can be obtained from the following.
P (yi1 = y1 , ...., yiT = yT |xi , ci , ni = n) =
=
CI−A4

=
SE−A2

P (yi1 = y1 , ...., yiT = yT |xi , ci )
P (ni = n|xi , ci )

P (yi1 = y1 |xi , ci )P (yi2 = y2 |xi , ci ) · · · · · P (yiT = yT |xi , ci )
P (ni = n|xi , ci )

P (yi1 = y1 |xi1 , ci )P (yi2 = y2 |xi2 , ci ) · · · · · P (yiT = yT |xi3 , ci )
P (ni = n|xi , ci )
T

exp(

yit xit β)

t=1

=

(B.1)

T

A3

[exp(
a∈Ri

t=1

191

at xit β)]

T

where a = (a1 , a2 , ...., aT ), at ∈ {0, 1} at = 1 if yit = 1,
RT deﬁned as {a ∈ RT : at ∈ {0, 1} and

at = ni and Ri is the subset of
t=1

T

at = ni }.
t=1
n

βc−log it = arg max
b

li (b)

(B.2)

i=1
T

exp(

yit xit b)

t=1

li (b) = log{

}

T

[exp(
a∈Ri

192

t=1

at xit b)]

Appendix C
Monte Carlo simulation results:
Coeﬃcient β for continuous covariate

193

Table C.1. Unconditional logit estimates for β with no serial correlation
ρ=0, n=100 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.46 1.20 0.86
0.30
(0.28,0.32)
t=3
1.75 0.46 0.44
0.34
(0.32,0.36)
t=4
1.48 0.31 0.29
0.33
(0.31,0.35)
t=5
1.36 0.23 0.22
0.32
(0.30,0.34)
t=6
1.28 0.18 0.18
0.30
(0.28,0.32)
t=7
1.23 0.16 0.16
0.29
(0.27,0.31)
t=8
1.20 0.14 0.14
0.27
(0.25,0.29)
ρ=0, n=200 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.19 0.59 0.54
0.60
(0.58,0.62)
t=3
1.67 0.30 0.29
0.63
(0.61,0.65)
t=4
1.45 0.20 0.20
0.61
(0.59,0.63)
t=5
1.33 0.16 0.15
0.57
(0.54,0.59)
t=6
1.26 0.13 0.13
0.52
(0.50,0.55)
t=7
1.22 0.11 0.11
0.50
(0.47,0.52)
t=8
1.18 0.10 0.10
0.47
(0.45,0.49)
ρ=0, n=400 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.07 0.37 0.36
0.89
(0.88,0.90)
t=3
1.63 0.21 0.20
0.92
(0.90,0.93)
t=4
1.42 0.14 0.14
0.90
(0.89,0.91)
t=5
1.32 0.11 0.11
0.87
(0.85,0.88)
t=6
1.25 0.09 0.09
0.82
(0.81,0.83)
t=7
1.20 0.08 0.08
0.77
(0.75,0.79)
t=8
1.17 0.07 0.07
0.74
(0.71,0.77)
ρ=0, n=600 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.03 0.30 0.29
0.99
(0.98,0.99)
t=3
1.62 0.17 0.16
0.98
(0.98,0.99)
t=4
1.42 0.11 0.11
0.98
(0.98,0.99)
t=5
1.31 0.09 0.09
0.96
(0.95,0.98)
t=6
1.24 0.07 0.07
0.95
(0.94,0.96)
t=7
1.20 0.06 0.06
0.92
(0.91,0.94)
t=8
1.17 0.05 0.05
0.90
(0.88,0.92)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

194

Table C.2. Conditional logit estimates for β with no serial correlation
ρ=0, n=100 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.23 0.60 0.45
0.03
(0.03,0.04)
t=3
1.08 0.26 0.25
0.04
(0.03,0.05)
t=4
1.04 0.20 0.19
0.06
(0.05,0.07)
t=5
1.04 0.16 0.16
0.05
(0.04,0.06)
t=6
1.03 0.14 0.14
0.04
(0.03,0.05)
t=7
1.03 0.13 0.12
0.05
(0.04,0.06)
t=8
1.02 0.11 0.11
0.05
(0.04,0.06)
ρ=0, n=200 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.09 0.29 0.27
0.04
(0.03,0.05)
t=3
1.03 0.17 0.17
0.05
(0.04,0.06)
t=4
1.02 0.13 0.13
0.05
(0.04,0.06)
t=5
1.02 0.11 0.11
0.05
(0.04,0.06)
t=6
1.01 0.10 0.10
0.05
(0.04,0.06)
t=7
1.01 0.09 0.09
0.05
(0.04,0.06)
t=8
1.01 0.08 0.08
0.05
(0.04,0.06)
ρ=0, n=400 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.03 0.19 0.18
0.04
(0.03,0.05)
t=3
1.01 0.12 0.12
0.05
(0.04,0.06)
t=4
1.01 0.09 0.09
0.05
(0.04,0.06)
t=5
1.00 0.08 0.08
0.05
(0.04,0.06)
t=6
1.00 0.07 0.07
0.05
(0.04,0.06)
t=7
1.00 0.06 0.06
0.05
(0.04,0.06)
t=8
1.00 0.05 0.05
0.05
(0.04,0.06)
ρ=0, n=600 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.01 0.15 0.15
0.05
(0.04,0.06)
t=3
1.01 0.10 0.09
0.05
(0.04,0.06)
t=4
1.00 0.07 0.07
0.05
(0.04,0.06)
t=5
1.00 0.06 0.06
0.06
(0.05,0.07)
t=6
1.00 0.05 0.05
0.05
(0.04,0.06)
t=7
1.00 0.05 0.05
0.05
(0.04,0.06)
t=8
1.00 0.05 0.05
0.05
(0.04,0.06)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

195

Table C.3. Unconditional logit estimates for β with serial correlation
ρ=0.2, n=100 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.88 1.89 1.04
0.36
(0.34,0.38)
t=3
1.94 0.53 0.48
0.45
(0.43,0.48)
t=4
1.60 0.34 0.31
0.46
(0.44,0.48)
t=5
1.45 0.24 0.24
0.46
(0.44,0.48)
t=6
1.35 0.20 0.19
0.41
(0.39,0.43)
t=7
1.28 0.17 0.16
0.39
(0.37,0.41)
t=8
1.25 0.15 0.14
0.38
(0.36,0.40)
ρ=0.2, n=200 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.50 0.72 0.62
0.72
(0.70,0.74)
t=3
1.84 0.33 0.32
0.80
(0.78,0.82)
t=4
1.56 0.21 0.21
0.78
(0.76,0.80)
t=5
1.41 0.16 0.16
0.75
(0.73,0.76)
t=6
1.32 0.13 0.13
0.70
(0.68,0.72)
t=7
1.27 0.12 0.11
0.67
(0.65,0.69)
t=8
1.22 0.10 0.10
0.62
(0.60,0.64)
ρ=0.2, n=400 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.37 0.42 0.42
0.97
(0.96,0.98)
t=3
1.80 0.23 0.22
0.98
(0.97,0.99)
t=4
1.54 0.15 0.15
0.98
(0.97,0.99)
t=5
1.40 0.12 0.11
0.97
(0.96,0.98)
t=6
1.32 0.09 0.09
0.95
(0.93,0.96)
t=7
1.26 0.08 0.08
0.92
(0.91,0.93)
t=8
1.22 0.07 0.07
0.90
(0.88,0.92)
ρ=0.2, n=600 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.31 0.35 0.33
0.99
(0.99,1)
t=3
1.79 0.19 0.18
1
(1,1)
t=4
1.53 0.12 0.12
1
(1,1)
t=5
1.40 0.09 0.09
1
(1,1)
t=6
1.31 0.07 0.08
1
(1,1)
t=7
1.25 0.07 0.07
0.99
(0.98,0.99)
t=8
1.22 0.06 0.06
0.98
(0.97,0.99)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

196

Table C.4. Conditional logit estimates for β with serial correlation
ρ=0.2, n=100 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.44 0.94 0.56
0.02
(0.01,0.02)
t=3
1.18 0.30 0.27
0.05
(0.04,0.06)
t=4
1.12 0.22 0.20
0.07
(0.06,0.08)
t=5
1.10 0.17 0.17
0.07
(0.06,0.08)
t=6
1.08 0.15 0.14
0.07
(0.06,0.08)
t=7
1.07 0.13 0.13
0.07
(0.06,0.08)
t=8
1.06 0.12 0.12
0.07
(0.06,0.08)
ρ=0.2, n=200 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.25 0.36 0.32
0.05
(0.04,0.06)
t=3
1.14 0.19 0.18
0.07
(0.06,0.08)
t=4
1.09 0.14 0.14
0.08
(0.07,0.09)
t=5
1.08 0.11 0.12
0.08
(0.07,0.09)
t=6
1.06 0.10 0.10
0.08
(0.07,0.09)
t=7
1.06 0.09 0.09
0.09
(0.08,0.10)
t=8
1.05 0.09 0.08
0.09
(0.08,0.10)
ρ=0.2, n=400 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.18 0.21 0.21
0.08
(0.06,0.09)
t=3
1.11 0.13 0.13
0.11
(0.10,0.13)
t=4
1.08 0.10 0.10
0.12
(0.10,0.13)
t=5
1.07 0.08 0.08
0.13
(0.12,0.15)
t=6
1.06 0.07 0.07
0.12
(0.10,0.14)
t=7
1.05 0.06 0.06
0.10
(0.08,0.11)
t=8
1.04 0.06 0.06
0.10
(0.08,0.12)
ρ=0.2, n=600 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.16 0.17 0.17
0.10
(0.08,0.12)
t=3
1.11 0.11 0.10
0.15
(0.13,0.17)
t=4
1.08 0.08 0.08
0.16
(0.13,0.18)
t=5
1.06 0.06 0.07
0.14
(0.12,0.17)
t=6
1.05 0.06 0.06
0.14
(0.12,0.16)
t=7
1.04 0.05 0.05
0.13
(0.11,0.15)
t=8
1.03 0.05 0.05
0.13
(0.11,0.15)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

197

Table C.5. Unconditional logit estimates for β with serial correlation
ρ=0.4, n=100 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
3.59 2.65 1.32
0.44
(0.42,0.46)
t=3
2.26 0.65 0.57
0.60
(0.58,0.63)
t=4
1.81 0.40 0.36
0.63
(0.61,0.65)
t=5
1.60 0.27 0.26
0.65
(0.63,0.67)
t=6
1.47 0.21 0.21
0.62
(0.60,0.65)
t=7
1.38 0.18 0.18
0.57
(0.54,0.59)
t=8
1.33 0.16 0.15
0.57
(0.55,0.59)
ρ=0.4, n=200 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.99 0.92 0.77
0.84
(0.83,0.86)
t=3
2.12 0.39 0.38
0.93
(0.91,0.94)
t=4
1.75 0.25 0.24
0.93
(0.91,0.94)
t=5
1.56 0.18 0.18
0.92
(0.91,0.94)
t=6
1.44 0.15 0.14
0.89
(0.88,0.91)
t=7
1.36 0.12 0.12
0.87
(0.86,0.89)
t=8
1.31 0.11 0.11
0.83
(0.81,0.84)
ρ=0.4, n=400 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.79 0.54 0.50
0.99
(0.99,1)
t=3
2.07 0.27 0.26
1
(1,1)
t=4
1.73 0.17 0.17
1
(1,1)
t=5
1.54 0.13 0.13
1
(1,1)
t=6
1.42 0.10 0.10
1
(1,1)
t=7
1.35 0.09 0.09
0.99
(0.99,1)
t=8
1.30 0.08 0.08
0.98
(0.98,0.99)
ρ=0.4, n=600 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
2.72 0.43 0.40
1
(1,1)
t=3
2.06 0.21 0.21
1
(1,1)
t=4
1.72 0.14 0.14
1
(1,1)
t=5
1.53 0.10 0.10
1
(1,1)
t=6
1.42 0.08 0.08
1
(1,1)
t=7
1.35 0.07 0.07
1
(1,1)
t=8
1.30 0.06 0.06
1
(1,1)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

198

Table C.6. Conditional logit estimates for β with serial correlation
ρ=0.4, n=100 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.80 1.32 0.77
0.004
(0.01,0.02)
t=3
1.35 0.35 0.32
0.11
(0.04,0.06)
t=4
1.25 0.25 0.22
0.14
(0.06,0.08)
t=5
1.20 0.19 0.18
0.15
(0.06,0.08)
t=6
1.17 0.16 0.15
0.16
(0.14,0.18)
t=7
1.13 0.14 0.14
0.15
(0.13,0.16)
t=8
1.13 0.13 0.13
0.15
(0.13,0.16)
ρ=0.4, n=200 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.50 0.46 0.39
0.12
(0.10,0.13)
t=3
1.28 0.21 0.21
0.21
(0.19,0.23)
t=4
1.21 0.16 0.15
0.25
(0.23,0.27)
t=5
1.18 0.13 0.12
0.26
(0.23,0.27)
t=6
1.15 0.11 0.11
0.25
(0.22,0.26)
t=7
1.13 0.09 0.10
0.24
(0.21,0.26)
t=8
1.11 0.09 0.09
0.23
(0.21,0.25)
ρ=0.4, n=400 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.39 0.27 0.25
0.27
(0.25,0.29)
t=3
1.26 0.15 0.14
0.40
(0.38,0.42)
t=4
1.20 0.11 0.11
0.45
(0.42,0.47)
t=5
1.16 0.09 0.09
0.46
(0.43,0.48)
t=6
1.14 0.08 0.08
0.41
(0.38,0.44)
t=7
1.12 0.07 0.07
0.42
(0.39,0.45)
t=8
1.10 0.06 0.06
0.40
(0.37,0.43)
ρ=0.4, n=600 Mean SD SE r ≡ 1(P<0.05)
95-CI
t=2
1.36 0.21 0.20
0.40
(0.37,0.43)
t=3
1.26 0.12 0.12
0.59
(0.56,0.62)
t=4
1.19 0.09 0.09
0.64
(0.61,0.67)
t=5
1.16 0.07 0.07
0.61
(0.58,0.64)
t=6
1.13 0.06 0.06
0.59
(0.56,0.62)
t=7
1.11 0.06 0.05
0.56
(0.52,0.59)
t=8
1.10 0.05 0.05
0.53
(0.50,0.56)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

199

Table C.7. Unconditional logit estimates for β with serial correlation
ρ=0.6, n=100 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
5.32 10.91 1.89
0.57
(0.55,0.59)
t=3
2.87
0.94 0.76
0.79
(0.77,0.81)
t=4
2.21
0.49 0.45
0.86
(0.84,0.87)
t=5
1.90
0.33 0.32
0.88
(0.87,0.90)
t=6
1.71
0.25 0.25
0.87
(0.85,0.88)
t=7
1.58
0.21 0.20
0.85
(0.84,0.87)
t=8
1.49
0.18 0.17
0.83
(0.82,0.85)
ρ=0.6, n=200 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
3.94
1.66 1.09
0.91
(0.90,0.93)
t=3
2.66
0.52 0.49
0.99
(0.98,0.99)
t=4
2.13
0.32 0.30
0.99
(0.99,1)
t=5
1.84
0.22 0.22
0.99
(0.99,1)
t=6
1.67
0.17 0.17
0.99
(0.99,1)
t=7
1.55
0.15 0.14
0.99
(0.99,1)
t=8
1.47
0.13 0.12
0.98
(0.98,0.99)
ρ=0.6, n=400 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
3.55
0.74 0.67
1
(1,1)
t=3
2.56
0.35 0.33
1
(1,1)
t=4
2.09
0.21 0.21
1
(1,1)
t=5
1.82
0.15 0.15
1
(1,1)
t=6
1.64
0.12 0.12
1
(1,1)
t=7
1.53
0.10 0.10
1
(1,1)
t=8
1.46
0.09 0.09
1
(1,1)
ρ=0.6, n=600 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
3.43
0.58 0.52
0.99
(0.99,1)
t=3
2.55
0.27 0.27
1
(1,1)
t=4
2.08
0.16 0.17
1
(1,1)
t=5
1.80
0.12 0.12
1
(1,1)
t=6
1.64
0.10 0.10
1
(1,1)
t=7
1.53
0.08 0.08
1
(1,1)
t=8
1.45
0.07 0.07
1
(1,1)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

200

Table C.8. Conditional logit estimates for β with serial correlation
ρ=0.6, n=100 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
2.86 11.06 3048013
0.001
(0,0.03)
t=3
1.68
0.50
0.41
0.27
(0.25,0.29)
t=4
1.49
0.29
0.27
0.39
(0.37,0.41)
t=5
1.40
0.22
0.21
0.45
(0.42,0.47)
t=6
1.34
0.18
0.18
0.47
(0.44,0.49)
t=7
1.29
0.16
0.15
0.45
(0.43,0.47)
t=8
1.25
0.14
0.14
0.44
(0.42,0.46)
ρ=0.6, n=200 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
1.97
0.83
0.58
0.23
(0.21,0.25)
t=3
1.57
0.28
0.26
0.59
(0.57,0.61)
t=4
1.44
0.19
0.18
0.72
(0.70,0.74)
t=5
1.36
0.15
0.14
0.73
(0.71,0.75)
t=6
1.31
0.13
0.12
0.75
(0.73,0.77)
t=7
1.27
0.11
0.11
0.71
(0.69,0.73)
t=8
1.24
0.10
0.09
0.71
(0.69,0.73)
ρ=0.6, n=400 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
1.77
0.37
0.34
0.68
(0.67,0.70)
t=3
1.52
0.19
0.18
0.89
(0.88,0.90)
t=4
1.42
0.13
0.13
0.95
(0.94,0.96)
t=5
1.35
0.10
0.10
0.96
(0.95,0.97)
t=6
1.30
0.09
0.08
0.95
(0.94,0.96)
t=7
1.26
0.07
0.07
0.95
(0.94,0.96)
t=8
1.23
0.07
0.07
0.96
(0.94,0.97)
ρ=0.6, n=600 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
1.71
0.29
0.27
0.85
(0.83,0.87)
t=3
1.51
0.15
0.14
0.98
(0.98,0.99)
t=4
1.41
0.10
0.10
1
(0.99,1)
t=5
1.34
0.08
0.08
1
(0.99,1)
t=6
1.29
0.07
0.07
0.99
(0.99,1)
t=7
1.25
0.06
0.06
0.99
(0.99,1)
t=8
1.22
0.06
0.05
0.99
(0.98,0.99)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

201

Table C.9. Unconditional logit estimates for β with serial correlation
ρ=0.8, n=100 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
8.70 17.11 3.05
0.67
(0.65,0.68)
t=3
6.53 25.88 1.67
0.91
(0.89,0.92)
t=4
3.34
0.90 0.75
0.98
(0.98,0.99)
t=5
2.72
0.52 0.50
1
(0.99,1)
t=6
2.36
0.39 0.36
1
(0.99,1)
t=7
2.13
0.31 0.29
1
(0.99,1)
t=8
1.96
0.25 0.24
1
(0.99,1)
ρ=0.8, n=200 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
7.12 12.38 2.33
0.93
(0.92,0.94)
t=3
4.05
0.92 0.83
1
(1,1)
t=4
3.13
0.53 0.48
1
(1,1)
t=5
2.61
0.34 0.33
1
(1,1)
t=6
2.26
0.25 0.24
1
(1,1)
t=7
2.06
0.20 0.20
1
(1,1)
t=8
1.91
0.17 0.17
1
(1,1)
ρ=0.8, n=400 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
5.50
2.28 1.25
1
(0.99,1)
t=3
3.81
0.58 0.54
1
(1,1)
t=4
3.03
0.34 0.33
1
(1,1)
t=5
2.55
0.23 0.23
1
(1,1)
t=6
2.24
0.18 0.17
1
(1,1)
t=7
2.04
0.14 0.14
1
(1,1)
t=8
1.89
0.12 0.12
1
(1,1)
ρ=0.8, n=600 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
5.09
1.07 0.92
1
(1,1)
t=3
3.74
0.46 0.43
1
(1,1)
t=4
3.00
0.27 0.26
1
(1,1)
t=5
2.52
0.18 0.18
1
(1,1)
t=6
2.23
0.14 0.14
1
(1,1)
t=7
2.03
0.11 0.11
1
(1,1)
t=8
1.89
0.09 0.09
1
(1,1)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

202

Table C.10. Conditional logit estimates for β with serial correlation
ρ=0.8, n=100 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
5.52 29.96 6296147
0
(0,0)
t=3
4.03 19.14 380015
0.61
(0.59,0.63)
t=4
2.12
0.50
0.42
0.89
(0.87,0.90)
t=5
1.91
0.32
0.30
0.95
(0.94,0.97)
t=6
1.78
0.26
0.24
0.96
(0.95,0.97)
t=7
1.69
0.22
0.20
0.96
(0.95,0.96)
t=8
1.61
0.19
0.18
0.97
(0.96,0.97)
ρ=0.8, n=200 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
3.69 11.54 1053342
0.27
(0.25,0.29)
t=3
2.29
0.48
0.44
0.98
(0.98,0.99)
t=4
2.01
0.29
0.27
1
(1,1)
t=5
1.84
0.21
0.20
1
(1,1)
t=6
1.72
0.17
0.16
1
(1,1)
t=7
1.64
0.14
0.14
1
(1,1)
t=8
1.57
0.13
0.12
1
(1,1)
ρ=0.8, n=400 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
2.75
1.14
0.68
0.98
(0.98,0.99)
t=3
2.17
0.30
0.28
1
(1,1)
t=4
1.95
0.19
0.18
1
(1,1)
t=5
1.81
0.14
0.14
1
(1,1)
t=6
1.70
0.12
0.11
1
(1,1)
t=7
1.63
0.10
0.10
1
(1,1)
t=8
1.56
0.09
0.08
1
(1,1)
ρ=0.8, n=600 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
2.55
0.54
0.47
1
(1,1)
t=3
2.13
0.24
0.22
1
(1,1)
t=4
1.94
0.15
0.15
1
(1,1)
t=5
1.79
0.11
0.11
1
(1,1)
t=6
1.69
0.09
0.09
1
(1,1)
t=7
1.62
0.08
0.08
1
(1,1)
t=8
1.56
0.07
0.07
1
(1,1)
*Note: r is the rejection rate for the null of H0 :β=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

203

Appendix D
Monte Carlo simulation results: APEs
for continuous covariate

204

Table D.1. LPM, unconditional logit, CRE logit APE estimates for β
n=100
t=2
t=2
t=2
t=3
t=2
t=3
t=4
t=4
t=4
t=5
t=5
t=5
t=6
t=6
t=6
t=7
t=7
t=7
t=8
t=8
t=8

ρ=0
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit

Mean
0.157
0.348
0.158
0.161
0.276
0.162
0.163
0.235
0.163
0.164
0.215
0.164
0.165
0.202
0.165
0.166
0.194
0.166
0.166
0.194
0.166

SD
0.040
0.081
0.040
0.028
0.044
0.028
0.023
0.032
0.023
0.020
0.025
0.020
0.018
0.021
0.018
0.017
0.019
0.017
0.016
0.017
0.015

SE
r ≡ 1(P<0.05)
0.040
0.058
0.048
0.869
0.040
0.059
0.028
0.058
0.034
0.833
0.028
0.057
0.023
0.049
0.027
0.692
0.023
0.053
0.020
0.056
0.022
0.586
0.019
0.060
0.018
0.050
0.019
0.453
0.018
0.069
0.016
0.053
0.017
0.392
0.016
0.059
0.015
0.051
0.015
0.332
0.015
0.056

95-CI
(0.047,0.068)
(0.854,0.884)
(0.048,0.068)
(0.048,0.068)
(0.816,0.849)
(0.056,0.068)
(0.040,0.058)
(0.672,0.712)
(0.043,0.063)
(0.046,0.066)
(0.048,0.068)
(0.050,0.071)
(0.040,0.059)
(0.432,0.475)
(0.040,0.058)
(0.041,0.060)
(0.370,0.413)
(0.049,0.069)
(0.041,0.060)
(0.311,0.352)
(0.046,0.066)

*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.162(t = 2), 0.162(t = 3), 0.165(t = 4), 0.165(t = 5), 0.167(t = 6), 0.166(t = 7), and 0.166(t = 8).

205

Table D.2. LPM, unconditional logit, CRE logit APE estimates for β
n=200
ρ=0
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
LPM
0.157 0.028 0.028
0.059
(0.049,0.069)
t=2
uncon logit 0.349 0.050 0.032
0.979
(0.973,0.985)
t=2
CRE logit 0.157 0.028 0.028
0.060
(0.049,0.070)
t=3
LPM
0.161 0.020 0.020
0.053
(0.043,0.063)
t=3
uncon logit 0.273 0.031 0.024
0.973
(0.966,0.980)
t=3
CRE logit 0.161 0.020 0.020
0.055
(0.045,0.065)
t=4
LPM
0.163 0.016 0.016
0.053
(0.043,0.062)
t=4
uncon logit 0.236 0.022 0.019
0.927
(0.915,0.938)
t=4
CRE logit 0.163 0.016 0.016
0.058
(0.048,0.068)
t=5
LPM
0.164 0.014 0.014
0.057
(0.047,0.067)
t=5
uncon logit 0.214 0.028 0.015
0.851
(0.834,0.866)
t=5
CRE logit 0.164 0.014 0.014
0.056
(0.045,0.066)
t=6
LPM
0.165 0.013 0.013
0.054
(0.044,0.063)
t=6
uncon logit 0.201 0.015 0.013
0.753
(0.734,0.772)
t=6
CRE logit 0.165 0.013 0.013
0.055
(0.045,0.065)
t=7
LPM
0.166 0.011 0.011
0.047
(0.037,0.056)
t=7
uncon logit 0.194 0.013 0.012
0.658
(0.637,0.678)
t=7
CRE logit 0.166 0.011 0.011
0.049
(0.039,0.058)
t=8
LPM
0.166 0.011 0.011
0.060
(0.049,0.070)
t=8
uncon logit 0.188 0.011 0.011
0.622
(0.601,0.643)
t=8
CRE logit 0.166 0.011 0.011
0.059
(0.049,0.069)
*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.163(t = 2), 0.163(t = 3), 0.164(t = 4), 0.165(t = 5), 0.165(t = 6), 0.166(t = 7), and 0.164(t = 8).

206

Table D.3. LPM, unconditional logit, CRE logit APE estimates for β
n=400
ρ=0
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
LPM
0.156 0.020 0.020
0.063
(0.051,0.073)
t=2
uncon logit 0.347 0.033 0.023
1
(1,1)
t=2
CRE logit 0.156 0.020 0.020
0.063
(0.052,0.073)
t=3
LPM
0.160 0.014 0.013
0.060
(0.049,0.070)
t=3
uncon logit 0.273 0.022 0.017
1
(1,1)
t=3
CRE logit 0.160 0.014 0.014
0.055
(0.045,0.065)
t=4
LPM
0.163 0.011 0.011
0.056
(0.046,0.066)
t=4
uncon logit 0.235 0.015 0.013
0.997
(0.994,0.999)
t=4
CRE logit 0.163 0.011 0.011
0.056
(0.046,0.066)
t=5
LPM
0.164 0.010 0.010
0.056
(0.046,0.067)
t=5
uncon logit 0.214 0.012 0.011
0.986
(0.980,0.991)
t=5
CRE logit 0.164 0.010 0.010
0.056
(0.046,0.067)
t=6
LPM
0.165 0.009 0.009
0.044
(0.035,0.060)
t=6
uncon logit 0.202 0.010 0.009
0.962
(0.953,0.970)
t=6
CRE logit 0.165 0.009 0.009
0.048
(0.039,0.062)
t=7
LPM
0.165 0.008 0.008
0.054
(0.044,0.064)
t=7
uncon logit 0.193 0.009 0.008
0.847
(0.831,0.865)
t=7
CRE logit 0.165 0.008 0.008
0.056
(0.046,0.066)
t=8
LPM
0.166 0.007 0.007
0.050
(0.040,0.060)
t=8
uncon logit 0.188 0.008 0.008
0.800
(0.780,0.820)
t=8
CRE logit 0.166 0.007 0.007
0.053
(0.043,0.063)
*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.161(t = 2), 0.162(t = 3), 0.166(t = 4), 0.166(t = 5), 0.165(t = 6), 0.167(t = 7), and 0.167(t = 8).

207

Table D.4. LPM, unconditional logit, CRE logit APE estimates for β
n=600
ρ=0
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
LPM
0.156 0.016 0.016
0.101
(0.082,0.119)
t=2
uncon logit 0.346 0.028 0.018
1
(1,1)
t=2
CRE logit 0.156 0.016 0.016
0.097
(0.078,0.115)
t=3
LPM
0.160 0.012 0.011
0.062
(0.047,0.076)
t=3
uncon logit 0.273 0.018 0.014
1
(1,1)
t=3
CRE logit 0.160 0.012 0.011
0.062
(0.047,0.076)
t=4
LPM
0.163 0.009 0.009
0.056
(0.041,0.072)
t=4
uncon logit 0.235 0.013 0.011
1
(1,1)
t=4
CRE logit 0.163 0.009 0.009
0.060
(0.045,0.076)
t=5
LPM
0.164 0.008 0.008
0.059
(0.044,0.073)
t=5
uncon logit 0.214 0.010 0.009
1
(1,1)
t=5
CRE logit 0.164 0.008 0.008
0.059
(0.045,0.073)
t=6
LPM
0.165 0.007 0.007
0.047
(0.034,0.060)
t=6
uncon logit 0.202 0.008 0.007
0.998
(0.995,1)
t=6
CRE logit 0.165 0.007 0.007
0.047
(0.034,0.060)
t=7
LPM
0.165 0.007 0.007
0.055
(0.041,0.069)
t=7
uncon logit 0.194 0.007 0.007
0.997
(0.967,0.986)
t=7
CRE logit 0.165 0.007 0.007
0.058
(0.044,0.073)
t=8
LPM
0.166 0.006 0.006
0.049
(0.036,0.062)
t=8
uncon logit 0.188 0.006 0.006
0.931
(0.915,0.947)
t=8
CRE logit 0.166 0.006 0.006
0.055
(0.040,0.069)
*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.165(t = 2), 0.163(t = 3), 0.167(t = 4), 0.165(t = 5), 0.164(t = 6), 0.167(t = 7), and 0.167(t = 8).

208

Table D.5. LPM, unconditional logit, CRE logit APE estimates for β
n=100
t=2
t=2
t=2
t=3
t=2
t=3
t=4
t=4
t=4
t=5
t=5
t=5
t=6
t=6
t=6
t=7
t=7
t=7
t=8
t=8
t=8

ρ=0.2
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit

Mean
0.156
0.362
0.157
0.160
0.289
0.161
0.162
0.246
0.163
0.164
0.223
0.164
0.165
0.208
0.165
0.165
0.198
0.165
0.166
0.193
0.166

SD
0.039
0.083
0.039
0.027
0.047
0.027
0.023
0.033
0.023
0.020
0.025
0.020
0.018
0.021
0.018
0.016
0.018
0.016
0.015
0.016
0.015

SE
r ≡ 1(P<0.05)
0.038
0.068
0.047
0.906
0.038
0.070
0.027
0.056
0.034
0.889
0.027
0.057
0.023
0.057
0.027
0.802
0.023
0.057
0.020
0.053
0.022
0.713
0.019
0.053
0.018
0.055
0.019
0.614
0.018
0.063
0.016
0.050
0.017
0.499
0.016
0.056
0.015
0.046
0.015
0.426
0.015
0.050

95-CI
(0.057,0.078)
(0.893,0.918)
(0.060,0.081)
(0.045,0.066)
(0.875,0.903)
(0.046,0.067)
(0.046,0.067)
(0.785,0.820)
(0.047,0.067)
(0.043,0.064)
(0.693,0.733)
(0.043,0.063)
(0.045,0.065)
(0.477,0.521)
(0.052,0.074)
(0.040,0.060)
(0.477,0.521)
(0.046,0.067)
(0.036,0.055)
(0.404,0.448)
(0.040,0.059)

*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.163(t = 2), 0.162(t = 3), 0.164(t = 4), 0.165(t = 5), 0.165(t = 6), 0.166(t = 7), and 0.166(t = 8).

209

Table D.6. LPM, unconditional logit, CRE logit APE estimates for β
n=200
ρ=0.2
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
LPM
0.157 0.027 0.028
0.059
(0.049,0.069)
t=2
uncon logit 0.363 0.050 0.032
0.979
(0.973,0.985)
t=2
CRE logit 0.157 0.028 0.028
0.060
(0.049,0.070)
t=3
LPM
0.161 0.019 0.019
0.049
(0.039,0.058)
t=3
uncon logit 0.289 0.031 0.024
0.995
(0.992,0.998)
t=3
CRE logit 0.161 0.019 0.019
0.054
(0.044,0.064)
t=4
LPM
0.163 0.016 0.016
0.051
(0.041,0.061)
t=4
uncon logit 0.246 0.022 0.019
0.978
(0.972,0.984)
t=4
CRE logit 0.163 0.016 0.016
0.049
(0.039,0.058)
t=5
LPM
0.164 0.014 0.014
0.049
(0.040,0.058)
t=5
uncon logit 0.222 0.017 0.015
0.946
(0.936,0.955)
t=5
CRE logit 0.164 0.014 0.014
0.050
(0.040,0.059)
t=6
LPM
0.165 0.013 0.013
0.050
(0.040,0.059)
t=6
uncon logit 0.208 0.014 0.013
0.858
(0.842,0.872)
t=6
CRE logit 0.165 0.013 0.012
0.055
(0.045,0.065)
t=7
LPM
0.166 0.011 0.011
0.051
(0.041,0.060)
t=7
uncon logit 0.199 0.013 0.012
0.776
(0.758,0.794)
t=7
CRE logit 0.166 0.011 0.011
0.056
(0.046,0.066)
t=8
LPM
0.166 0.011 0.011
0.054
(0.044,0.064)
t=8
uncon logit 0.192 0.012 0.011
0.664
(0.643,0.685)
t=8
CRE logit 0.166 0.011 0.011
0.056
(0.046,0.066)
*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.163(t = 2), 0.162(t = 3), 0.164(t = 4), 0.165(t = 5), 0.166(t = 6), 0.166(t = 7), and 0.164(t = 8).

210

Table D.7. LPM, unconditional logit, CRE logit APE estimates for β
n=400
ρ=0.2
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
LPM
0.156 0.019 0.019
0.056
(0.046,0.066)
t=2
uncon logit 0.364 0.032 0.020
1
(1,1)
t=2
CRE logit 0.156 0.019 0.019
0.056
(0.046,0.066)
t=3
LPM
0.160 0.014 0.014
0.053
(0.043,0.063)
t=3
uncon logit 0.288 0.022 0.017
1
(1,1)
t=3
CRE logit 0.160 0.014 0.013
0.052
(0.042,0.062)
t=4
LPM
0.163 0.011 0.011
0.055
(0.045,0.065)
t=4
uncon logit 0.246 0.015 0.013
1
(1,1)
t=4
CRE logit 0.163 0.011 0.011
0.058
(0.048,0.068)
t=5
LPM
0.164 0.010 0.010
0.052
(0.041,0.063)
t=5
uncon logit 0.222 0.012 0.011
0.998
(0.996,1)
t=5
CRE logit 0.164 0.010 0.010
0.056
(0.046,0.067)
t=6
LPM
0.165 0.010 0.009
0.052
(0.038,0.066)
t=6
uncon logit 0.208 0.010 0.009
0.990
(0.984,0.996)
t=6
CRE logit 0.165 0.009 0.009
0.054
(0.039,0.068)
t=7
LPM
0.165 0.008 0.008
0.050
(0.040,0.060)
t=7
uncon logit 0.198 0.009 0.008
0.972
(0.964,0.979)
t=7
CRE logit 0.165 0.008 0.008
0.052
(0.042,0.062)
t=8
LPM
0.166 0.008 0.008
0.058
(0.044,0.072)
t=8
uncon logit 0.192 0.008 0.008
0.923
(0.907,0.939)
t=8
CRE logit 0.166 0.008 0.007
0.056
(0.042,0.070)
*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.162(t = 2), 0.162(t = 3), 0.166(t = 4), 0.165(t = 5), 0.165(t = 6), 0.165(t = 7), and 0.166(t = 8).

211

Table D.8. LPM, unconditional logit, CRE logit APE estimates for β
n=600
ρ=0.2
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
LPM
0.155 0.015 0.016
0.062
(0.047,0.077)
t=2
uncon logit 0.363 0.028 0.017
1
(1,1)
t=2
CRE logit 0.155 0.015 0.015
0.060
(0.045,0.075)
t=3
LPM
0.160 0.011 0.011
0.052
(0.039,0.066)
t=3
uncon logit 0.288 0.018 0.014
1
(1,1)
t=3
CRE logit 0.160 0.011 0.011
0.052
(0.039,0.066)
t=4
LPM
0.163 0.009 0.009
0.060
(0.045,0.075)
t=4
uncon logit 0.246 0.013 0.011
1
(1,1)
t=4
CRE logit 0.163 0.009 0.009
0.060
(0.045,0.075)
t=5
LPM
0.164 0.008 0.008
0.046
(0.033,0.059)
t=5
uncon logit 0.222 0.010 0.009
1
(1,1)
t=5
CRE logit 0.164 0.008 0.008
0.050
(0.036,0.064)
t=6
LPM
0.165 0.007 0.007
0.046
(0.033,0.059)
t=6
uncon logit 0.208 0.008 0.008
1
(1,1)
t=6
CRE logit 0.165 0.007 0.007
0.049
(0.036,0.062)
t=7
LPM
0.165 0.007 0.007
0.057
(0.042,0.071)
t=7
uncon logit 0.198 0.008 0.007
0.990
(0.984,1)
t=7
CRE logit 0.165 0.007 0.007
0.057
(0.042,0.071)
t=8
LPM
0.166 0.006 0.006
0.045
(0.032,0.058)
t=8
uncon logit 0.192 0.006 0.006
0.975
(0.965,0.985)
t=8
CRE logit 0.166 0.006 0.006
0.045
(0.032,0.058)
*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.165(t = 2), 0.163(t = 3), 0.165(t = 4), 0.164(t = 5), 0.164(t = 6), 0.167(t = 7), and 0.167(t = 8).

212

Table D.9. LPM, unconditional logit, CRE logit APE estimates for β
n=100
t=2
t=2
t=2
t=3
t=2
t=3
t=4
t=4
t=4
t=5
t=5
t=5
t=6
t=6
t=6
t=7
t=7
t=7
t=8
t=8
t=8

ρ=0.4
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit

Mean
0.155
0.379
0.155
0.160
0.310
0.161
0.163
0.262
0.163
0.164
0.235
0.164
0.165
0.218
0.165
0.165
0.206
0.165
0.166
0.198
0.166

SD
0.037
0.094
0.037
0.027
0.047
0.027
0.022
0.034
0.022
0.019
0.025
0.019
0.018
0.021
0.018
0.016
0.018
0.016
0.015
0.017
0.015

SE
r ≡ 1(P<0.05)
0.037
0.074
0.047
0.936
0.036
0.078
0.026
0.057
0.033
0.955
0.026
0.058
0.022
0.058
0.027
0.910
0.022
0.059
0.019
0.055
0.022
0.859
0.019
0.059
0.018
0.053
0.019
0.762
0.018
0.058
0.016
0.045
0.017
0.669
0.016
0.050
0.015
0.055
0.016
0.561
0.015
0.057

95-CI
(0.063,0.086)
(0.925,0.946)
(0.066,0.089)
(0.047,0.068)
(0.946,0.964)
(0.048,0.068)
(0.048,0.068)
(0.897,0.923)
(0.049,0.070)
(0.045,0.065)
(0.844,0.874)
(0.049,0.069)
(0.043,0.063)
(0.743,0.781)
(0.049,0.069)
(0.036,0.054)
(0.648,0.690)
(0.041,0.060)
(0.045,0.065)
(0.539,0.583)
(0.047,0.067)

*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.161(t = 2), 0.162(t = 3), 0.164(t = 4), 0.165(t = 5), 0.166(t = 6), 0.165(t = 7), and 0.167(t = 8).

213

Table D.10. LPM, unconditional logit, CRE logit APE estimates for β
n=200
ρ=0.4
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
LPM
0.157 0.026 0.026
0.057
(0.046,0.067)
t=2
uncon logit 0.378 0.048 0.027
0.998
(0.996,1)
t=2
CRE logit 0.157 0.026 0.025
0.059
(0.048,0.069)
t=3
LPM
0.160 0.018 0.019
0.048
(0.039,0.057)
t=3
uncon logit 0.308 0.031 0.023
0.995
(0.996,1)
t=3
CRE logit 0.160 0.018 0.018
0.053
(0.043,0.063)
t=4
LPM
0.162 0.016 0.016
0.060
(0.049,0.070)
t=4
uncon logit 0.262 0.023 0.019
0.998
(0.982,0.995)
t=4
CRE logit 0.162 0.016 0.016
0.057
(0.047,0.067)
t=5
LPM
0.164 0.014 0.014
0.055
(0.040,0.070)
t=5
uncon logit 0.235 0.018 0.016
0.989
(0.982,0.995)
t=5
CRE logit 0.164 0.014 0.013
0.051
(0.047,0.076)
t=6
LPM
0.165 0.012 0.012
0.058
(0.047,0.068)
t=6
uncon logit 0.218 0.015 0.013
0.953
(0.943,0.962)
t=6
CRE logit 0.165 0.012 0.012
0.056
(0.046,0.076)
t=7
LPM
0.165 0.011 0.011
0.053
(0.043,0.063)
t=7
uncon logit 0.206 0.013 0.012
0.906
(0.893,0.919)
t=7
CRE logit 0.165 0.011 0.011
0.051
(0.041,0.060)
t=8
LPM
0.166 0.011 0.011
0.055
(0.045,0.065)
t=8
uncon logit 0.198 0.012 0.011
0.834
(0.818,0.851)
t=8
CRE logit 0.166 0.011 0.011
0.059
(0.049,0.070)
*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.163(t = 2), 0.161(t = 3), 0.165(t = 4), 0.163(t = 5), 0.166(t = 6), 0.165(t = 7), and 0.164(t = 8).

214

Table D.11. LPM, unconditional logit, CRE logit APE estimates for β
n=400
ρ=0.4
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
LPM
0.156 0.019 0.018
0.068
(0.056,0.079)
t=2
uncon logit 0.379 0.032 0.018
1
(1,1)
t=2
CRE logit 0.156 0.019 0.018
0.066
(0.056,0.076)
t=3
LPM
0.160 0.013 0.013
0.050
(0.040,0.060)
t=3
uncon logit 0.307 0.022 0.016
1
(1,1)
t=3
CRE logit 0.160 0.013 0.013
0.054
(0.044,0.064)
t=4
LPM
0.163 0.011 0.011
0.046
(0.037,0.055)
t=4
uncon logit 0.262 0.016 0.013
1
(1,1)
t=4
CRE logit 0.163 0.011 0.011
0.048
(0.038,0.057)
t=5
LPM
0.164 0.010 0.010
0.054
(0.044,0.064)
t=5
uncon logit 0.234 0.012 0.011
1
(1,1)
t=5
CRE logit 0.164 0.010 0.010
0.056
(0.046,0.067)
t=6
LPM
0.165 0.009 0.009
0.048
(0.035,0.060)
t=6
uncon logit 0.217 0.010 0.010
1
(1,1)
t=6
CRE logit 0.165 0.009 0.009
0.048
(0.035,0.060)
t=7
LPM
0.165 0.008 0.008
0.050
(0.040,0.060)
t=7
uncon logit 0.206 0.009 0.008
1
(1,1)
t=7
CRE logit 0.165 0.008 0.008
0.051
(0.041,0.061)
t=8
LPM
0.166 0.008 0.008
0.056
(0.046,0.076)
t=8
uncon logit 0.199 0.008 0.008
0.982
(0.972,0.991)
t=8
CRE logit 0.166 0.008 0.007
0.055
(0.045,0.075)
*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.162(t = 2), 0.162(t = 3), 0.166(t = 4), 0.165(t = 5), 0.166(t = 6), 0.166(t = 7), and 0.165(t = 8).

215

Table D.12. LPM, unconditional logit, CRE logit APE estimates for β
n=600
ρ=0.4
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
LPM
0.155 0.015 0.015
0.071
(0.055,0.086)
t=2
uncon logit 0.378 0.026 0.014
1
(1,1)
t=2
CRE logit 0.155 0.015 0.015
0.074
(0.057,0.090)
t=3
LPM
0.160 0.011 0.011
0.052
(0.039,0.066)
t=3
uncon logit 0.309 0.017 0.013
1
(1,1)
t=3
CRE logit 0.160 0.011 0.011
0.054
(0.041,0.068)
t=4
LPM
0.163 0.009 0.009
0.045
(0.032,0.057)
t=4
uncon logit 0.262 0.012 0.011
1
(1,1)
t=4
CRE logit 0.163 0.009 0.009
0.048
(0.035,0.061)
t=5
LPM
0.164 0.008 0.008
0.045
(0.032,0.057)
t=5
uncon logit 0.234 0.010 0.009
1
(1,1)
t=5
CRE logit 0.164 0.008 0.008
0.048
(0.035,0.061)
t=6
LPM
0.165 0.007 0.007
0.046
(0.033,0.059)
t=6
uncon logit 0.217 0.008 0.008
1
(1,1)
t=6
CRE logit 0.165 0.007 0.007
0.043
(0.030,0.056)
t=7
LPM
0.165 0.007 0.007
0.054
(0.040,0.068)
t=7
uncon logit 0.206 0.007 0.007
1
(1,1)
t=7
CRE logit 0.165 0.007 0.007
0.057
(0.042,0.071)
t=8
LPM
0.166 0.006 0.006
0.044
(0.031,0.057)
t=8
uncon logit 0.199 0.006 0.006
1
(1,1)
t=8
CRE logit 0.166 0.006 0.006
0.043
(0.031,0.055)
*Note: r is the rejection rate for the null of H0 :β=1 . ρ is AR(1) serial correlation coeﬃcient for the errors. True APE is
0.165(t = 2), 0.162(t = 3), 0.163(t = 4), 0.165(t = 5), 0.164(t = 6), 0.166(t = 7), and 0.166(t = 8).

216

Appendix E
Monte Carlo simulation results:
Coeﬃcient δ for binary covariate

217

Table E.1. Unconditional logit for δ with low serial correlation for n = 100
ρ=0
Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
2.57
3.04 1.22
0.16
(0.13,0.18)
t=3
1.65
0.63 0.62
0.13
(0.11,0.15)
t=4
1.41
0.43 0.42
0.14
(0.12,0.16)
t=5
1.29
0.33 0.34
0.12
(0.10,0.14)
t=6
1.26
0.29 0.29
0.14
(0.12,0.16)
t=7
1.20
0.25 0.25
0.11
(0.09,0.13)
t=8
1.18
0.23 0.23
0.11
(0.00,0.13)
ρ=0.2 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
3.05
3.80 7.93
0.18
(0.16,0.21)
t=3
1.83
0.69 0.66
0.18
(0.16,0.21)
t=4
1.51
0.45 0.45
0.17
(0.15,0.20)
t=5
1.37
0.35 0.35
0.15
(0.13,0.17)
t=6
1.32
0.30 0.30
0.17
(0.15,0.19)
t=7
1.26
0.26 0.36
0.16
(0.13,0.18)
t=8
1.22
0.24 0.24
0.14
(0.12,0.16)
ρ=0.4 Mean SD
SE r ≡ 1(P<0.05)
95-CI
t=2
4.69 30.20 1.73
0.21
(0.18,0.24)
t=3
2.12
0.77 0.75
0.26
(0.24,0.29)
t=4
1.69
0.50 0.48
0.27
(0.24,0.30)
t=5
1.50
0.37 0.37
0.24
(0.22,0.26)
t=6
1.41
0.32 0.32
0.25
(0.23,0.28)
t=7
1.35
0.28 0.28
0.23
(0.20,0.25)
t=8
1.29
0.25 0.25
0.20
(0.18,0.23)
*Note: r is the rejection rate for the null of H0 :δ=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

218

Table E.2. Unconditional logit for δ with high serial correlation for n = 100
ρ=0.6 Mean
SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
7.43 35.86 6.24
0.27
(0.24,0.30)
t=3
2.69
1.52 0.92
0.40
(0.37,0.43)
t=4
2.04
0.57 0.56
0.45
(0.42,0.48)
t=5
1.76
0.43 0.42
0.42
(0.39,0.45)
t=6
1.62
0.35 0.34
0.43
(0.40,0.46)
t=7
1.52
0.30 0.30
0.41
(0.38,0.44)
t=8
1.43
0.26 0.26
0.37
(0.34,0.40)
ρ=0.8 Mean
SD
SE
r ≡ 1(P<0.05)
95-CI
t = 2 146.09 46.19 20.32
0.34
(0.31,0.38)
t=3
6.15 23.83 1.77
0.63
(0.60,0.66)
t=4
3.03
1.37 0.81
0.78
(0.75,0.80)
t=5
2.45
0.58 0.56
0.80
(0.77,0.82)
t=6
2.18
0.47 0.44
0.80
(0.77,0.83)
t=7
2.00
0.38 0.37
0.79
(0.76,0.81)
t=8
1.86
0.33 0.32
0.77
(0.74,0.80)
*Note: r is the rejection rate for the null of H0 :δ=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

219

Table E.3. Conditional logit for δ with low serial correlation for n = 100
ρ=0
Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
1.29
1.55 19.40
0.03
(0.01,0.04)
t=3
1.06
0.39 0.38
0.05
(0.03,0.06)
t=4
1.04
0.31 0.30
0.06
(0.04,0.07)
t=5
1.02
0.26 0.26
0.04
(0.03,0.05)
t=6
1.04
0.23 0.23
0.05
(0.04,0.07)
t=7
1.02
0.21 0.21
0.05
(0.03,0.06)
t=8
1.02
0.19 0.19
0.04
(0.03,0.06)
ρ=0.2 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
1.53
1.94 38.55
0.02
(0.01,0.03)
t=3
1.17
0.43 0.18
0.06
(0.04,0.07)
t=4
1.10
0.32 0.14
0.05
(0.04,0.06)
t=5
1.08
0.27 0.27
0.05
(0.03,0.06)
t=6
1.07
0.24 0.24
0.06
(0.04,0.07)
t=7
1.06
0.22 0.22
0.05
(0.04,0.06)
t=8
1.06
0.21 0.20
0.05
(0.03,0.06)
ρ=0.4 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
4.32 75.84 68.47
0.02
(0.01,0.03)
t=3
1.34
0.46 0.45
0.07
(0.05,0.08)
t=4
1.23
0.35 0.34
0.08
(0.07,0.10)
t=5
1.17
0.28 0.29
0.08
(0.06,0.10)
t=6
1.16
0.25 0.25
0.09
(0.07,0.10)
t=7
1.14
0.23 0.23
0.08
(0.07,0.10)
t=8
1.12
0.21 0.21
0.08
(0.06,0.10)
*Note: r is the rejection rate for the null of H0 :δ=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

220

Table E.4. Conditional logit for δ with high serial correlation for n = 100
ρ=0.6 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t=2
5.23 45.89 326.60
0.02
(0.01,0.02)
t=3
1.67
0.82
3.24
0.17
(0.14,0.19)
t=4
1.47
0.39
0.38
0.19
(0.17,0.22)
t=5
1.37
0.32
0.32
0.19
(0.17,0.22)
t=6
1.33
0.28
0.27
0.20
(0.17,0.23)
t=7
1.28
0.25
0.25
0.19
(0.17,0.21)
t=8
1.24
0.23
0.23
0.17
(0.14,0.19)
ρ=0.8 Mean SD
SE
r ≡ 1(P<0.05)
95-CI
t = 2 253.9 80.97 1255596
0.003
(0.00,0.006)
t=3
3.48 11.39 155.10
0.38
(0.35,0.41)
t=4
2.11
0.78
2.58
0.56
(0.53,0.59)
t=5
1.88
0.41
0.40
0.59
(0.56,0.62)
t=6
1.76
0.36
0.34
0.61
(0.58,0.64)
t=7
1.68
0.31
0.30
0.63
(0.60,0.66)
t=8
1.60
0.28
0.27
0.62
(0.59,0.65)
*Note: r is the rejection rate for the null of H0 :δ=1 with nominal value 0.05.
ρ is AR(1) serial correlation coeﬃcient for the errors.

221

Appendix F
Monte Carlo simulation results: APEs
for binary covariate

222

Table F.1. LPM, unconditional logit, CRE logit APE estimates for δ
n=100
t=2
t=2
t=2
t=3
t=2
t=3
t=4
t=4
t=4
t=5
t=5
t=5
t=6
t=6
t=6
t=7
t=7
t=7
t=8
t=8
t=8

ρ=0
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit

Mean
0.184
0.454
0.174
0.176
0.311
0.168
0.174
0.260
0.168
0.167
0.231
0.162
0.170
0.221
0.166
0.163
0.205
0.160
0.164
0.198
0.160

SD
SE
r ≡ 1(P<0.05)
0.089 0.084
0.063
0.373 1834.10
0.564
0.084 0.036
0.065
0.059 0.059
0.050
0.100 0.078
0.483
0.055 0.055
0.052
0.049 0.049
0.061
0.069 0.059
0.391
0.046 0.046
0.053
0.041 0.041
0.056
0.053 0.048
0.331
0.039 0.039
0.056
0.038 0.037
0.051
0.046 0.041
0.306
0.036 0.035
0.052
0.033 0.034
0.043
0.039 0.037
0.253
0.031 0.031
0.048
0.031 0.031
0.051
0.035 0.033
0.246
0.030 0.030
0.058

95-CI
(0.048,0.086)
(0.534,0.595)
(0.049,0.080)
(0.036,0.063)
(0.453,0.513)
(0.038,0.065)
(0.046,0.076)
(0.361,0.421)
(0.038,0.068)
(0.042,0.070)
(0.302,0.360)
(0.042,0.070)
(0.037,0.065)
(0.277,0.335)
(0.038,0.066)
(0.030,0.056)
(0.226,0.280)
(0.035,0.061)
(0.037,0.065)
(0.219,0.273)
(0.043,0.073)

*Note: r is the rejection rate for the null of H0 :APE for δ=true APE . ρ is AR(1) serial correlation coeﬃcient for the errors.
True APE is 0.172(t = 2), 0.166(t = 3), 0.168(t = 4), 0.164(t = 5), 0.166(t = 6), 0.162(t = 7), and 0.160(t = 8).

223

Table F.2. LPM, unconditional logit, CRE logit APE estimates for δ
n=100
t=2
t=2
t=2
t=3
t=2
t=3
t=4
t=4
t=4
t=5
t=5
t=5
t=6
t=6
t=6
t=7
t=7
t=7
t=8
t=8
t=8

ρ=0.2
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit

Mean
0.184
0.501
0.174
0.176
0.335
0.168
0.172
0.273
0.168
0.167
0.243
0.163
0.170
0.228
0.166
0.164
0.213
0.160
0.164
0.204
0.160

SD
SE
0.081 0.080
0.397 13231
0.077 0.075
0.057 0.057
0.104 0.080
0.054 0.053
0.047 0.047
0.069 0.060
0.044 0.044
0.040 0.041
0.053 0.049
0.038 0.038
0.037 0.037
0.046 0.042
0.035 0.035
0.032 0.033
0.040 0.037
0.031 0.031
0.031 0.031
0.036 0.033
0.029 0.029

r ≡ 1(P<0.05)
0.068
0.616
0.068
0.050
0.547
0.050
0.051
0.439
0.058
0.050
0.280
0.056
0.056
0.360
0.051
0.046
0.298
0.051
0.049
0.294
0.051

95-CI
(0.052,0.084)
(0.586,0.647)
(0.052,0.084)
(0.036,0.064)
(0.516,0.578)
(0.036,0.064)
(0.038,0.065)
(0.408,0.470)
(0.043,0.073)
(0.037,0.064)
(0.350,0.410)
(0.043,0.070)
(0.037,0.070)
(0.330,0.390)
(0.037,0.065)
(0.033,0.059)
(0.270,0.326)
(0.037,0.065)
(0.036,0.062)
(0.266,0.322)
(0.037,0.065)

*Note: r is the rejection rate for the null of H0 :APE for δ=true APE . ρ is AR(1) serial correlation coeﬃcient for the errors.
True APE is 0.172(t = 2), 0.166(t = 3), 0.168(t = 4), 0.164(t = 5), 0.166(t = 6), 0.161(t = 7), and 0.159(t = 8).

224

Table F.3. LPM, unconditional logit, CRE logit APE estimates for δ
n=100
t=2
t=2
t=2
t=3
t=2
t=3
t=4
t=4
t=4
t=5
t=5
t=5
t=6
t=6
t=6
t=7
t=7
t=7
t=8
t=8
t=8

ρ=0.4
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit

Mean
0.182
0.566
0.173
0.176
0.372
0.169
0.173
0.298
0.167
0.167
0.261
0.162
0.169
0.242
0.165
0.164
0.225
0.161
0.163
0.213
0.159

SD
SE
0.076 0.075
0.528 17430
0.072 0.070
0.053 0.054
0.105 0.082
0.050 0.050
0.045 0.045
0.071 0.061
0.042 0.042
0.038 0.039
0.054 0.050
0.036 0.037
0.036 0.036
0.046 0.042
0.034 0.034
0.032 0.032
0.041 0.038
0.031 0.031
0.031 0.030
0.036 0.034
0.029 0.029

r ≡ 1(P<0.05)
0.062
0.677
0.062
0.050
0.652
0.049
0.058
0.555
0.058
0.044
0.477
0.058
0.059
0.436
0.059
0.046
0.417
0.050
0.051
0.379
0.051

95-CI
(0.047,0.077)
(0.647,0.706)
(0.047,0.077)
(0.039,0.061)
(0.627,0.677)
(0.038,0.060)
(0.043,0.073)
(0.524,0.586)
(0.043,0.073)
(0.031,0.057)
(0.446,0.508)
(0.034,0.061)
(0.044,0.074)
(0.405,0.467)
(0.044,0.074)
(0.033,0.059)
(0.386,0.448)
(0.036,0.064)
(0.037,0.065)
(0.349,0.409)
(0.037,0.065)

*Note: r is the rejection rate for the null of H0 :APE for δ=true APE . ρ is AR(1) serial correlation coeﬃcient for the errors.
True APE is 0.172(t = 2), 0.166(t = 3), 0.168(t = 4), 0.164(t = 5), 0.166(t = 6), 0.162(t = 7), and 0.158(t = 8).

225

Table F.4. LPM, unconditional logit, CRE logit APE estimates for δ
n=100
t=2
t=2
t=2
t=3
t=2
t=3
t=4
t=4
t=4
t=5
t=5
t=5
t=6
t=6
t=6
t=7
t=7
t=7
t=8
t=8
t=8

ρ=0.6
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit

Mean
0.182
0.695
0.172
0.176
0.432
0.169
0.173
0.341
0.168
0.167
0.294
0.163
0.169
0.268
0.164
0.164
0.247
0.161
0.162
0.232
0.158

SD
SE
0.069 0.069
0.729 95297
0.065 0.064
0.050 0.050
0.163 0.781
0.047 0.047
0.041 0.042
0.072 0.063
0.022 0.022
0.037 0.037
0.058 0.052
0.035 0.035
0.034 0.034
0.048 0.044
0.032 0.032
0.031 0.031
0.042 0.039
0.029 0.029
0.029 0.029
0.037 0.035
0.028 0.028

r ≡ 1(P<0.05)
0.058
0.737
0.066
0.049
0.785
0.052
0.050
0.737
0.053
0.056
0.673
0.054
0.052
0.633
0.056
0.054
0.591
0.054
0.054
0.568
0.056

95-CI
(0.043,0.073)
(0.708,0.765)
(0.050,0.082)
(0.036,0.062)
(0.759,0.810)
(0.038,0.066)
(0.037,0.064)
(0.710,0.765)
(0.039,0.067)
(0.042,0.070)
(0.644,0.702)
(0.040,0.068)
(0.038,0.065)
(0.603,0.663)
(0.042,0.070)
(0.040,0.068)
(0.560,0.621)
(0.040,0.068)
(0.040,0.068)
(0.536,0.599)
(0.041,0.070)

*Note: r is the rejection rate for the null of H0 :APE for δ=true APE . ρ is AR(1) serial correlation coeﬃcient for the errors.
True APE is 0.172(t = 2), 0.166(t = 3), 0.168(t = 4), 0.164(t = 5), 0.166(t = 6), 0.162(t = 7), and 0.158(t = 8).

226

Table F.5. LPM, unconditional logit, CRE logit APE estimates for δ
n=100
t=2
t=2
t=2
t=3
t=2
t=3
t=4
t=4
t=4
t=5
t=5
t=5
t=6
t=6
t=6
t=7
t=7
t=7
t=8
t=8
t=8

ρ=0.8
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit
LPM
uncon logit
CRE logit

Mean
0.174
0.892
0.165
0.176
0.638
0.168
0.175
0.434
0.168
0.168
0.366
0.163
0.169
0.329
0.165
0.164
0.301
0.161
0.162
0.279
0.159

SD
SE
r ≡ 1(P<0.05)
0.061 0.061
0.060
1.112 173194
0.739
0.058 0.058
0.055
0.045 0.045
0.045
0.600 40290
0.902
0.042 0.042
0.050
0.037 0.039
0.052
0.127 0.329
0.946
0.032 0.032
0.046
0.034 0.034
0.044
0.061 0.054
0.934
0.032 0.032
0.044
0.032 0.032
0.051
0.052 0.046
0.911
0.031 0.030
0.054
0.029 0.029
0.047
0.044 0.041
0.906
0.028 0.028
0.052
0.028 0.028
0.050
0.040 0.037
0.867
0.026 0.026
0.048

95-CI
(0.043,0.077)
(0.708,0.770)
(0.038,0.073)
(0.032,0.058)
(0.883,0.920)
(0.036,0.064)
(0.038,0.066)
(0.933,0.961)
(0.033,0.059)
(0.032,0.057)
(0.919,0.950)
(0.032,0.057)
(0.037,0.067)
(0.893,0.929)
(0.039,0.069)
(0.034,0.060)
(0.888,0.924)
(0.038,0.065)
(0.037,0.064)
(0.845,0.888)
(0.035,0.061)

*Note: r is the rejection rate for the null of H0 :APE for δ=true APE . ρ is AR(1) serial correlation coeﬃcient for the errors.
True APE is 0.171(t = 2), 0.166(t = 3), 0.168(t = 4), 0.164(t = 5), 0.166(t = 6), 0.162(t = 7), and 0.159(t = 8).

227

Appendix G
Proof of Hausman-type test in
chapter 2
We start with rewriting stacked estimator to obtain asymptotic variance.


n

−1

sit xit xit

 i=1 t=1
θa = 

0

0
n

T

.. ..

i=1 t=1


√



n(θa − θa ) = 




√

1
n

N

sit xit xit








T

sit xit yit

 i=1 t=1
 n T

.. ..
sit xit y it

n



T

i=1 t=1

2k×2k

2k×1

−1 

T

i=1 t=1

sit xit xit

0
1
n

0




n

T

i=1 t=1
n

.. ..

sit xit xit
T
















1
( √n
1
( √n

N

i=1 t=1
n T
i=1 t=1
n



T

T

sit xit uit ) 



.. ..
sit xit uit )

1
1
√
sit xit xit )−1 ( √n
sit xit uit )
 n(θpls − θ)   ( n

i=1 t=1
i=1 t=1

=
n(θa − θa ) = 
 
n T
n T
√
.. ..
.. ..
1
1
n(θF E − θ)
sit xit xit )−1 ( √n
sit xit uit )
(n
i=1 t=1

228

i=1 t=1







θa is a consistent estimator for θa by rank conditions and exogeneity conditions for pooled
LS and FE estimators.
1
(p lim
n
1
(p lim
n

n

T

T

sit xit xit

)−1

i=1 t=1
n

T

t=1
.. ..
sit xit xit )−1

T

(p lim

1
n

n

.. ..

sit xit xit )−1 ≡ A−1
22

= E(
t=1

i=1 t=1

1
(p lim
n

sit xit xit )−1 ≡ A−1
11

= E(

T

T

sit xit uit ) = E(
i=1 t=1
n T

sit xit uit ) = 0
t=1
T

.. ..

sit xit uit ) = E(
i=1 t=1

.. ..

sit xit uit ) = 0
t=1

Consider an estimator for the variance of

√

n(θa − θ). we use following notations and

obtain the asymptotic variance estimator as in (G.1).
−1



 A11 0 

G−1 = 
a


0 A22
n

n

D21 =

1
n

T

T

[(

sit xit uit )(

i=1 t=1
n

T

[(
i=1 t=1

t=1
..

T

sit xit xit , A22 =
i=1 t=1

n



 D11 D12 

Da = 


D21 D22

T

A11 =
1
D11 =
n



i=1 t=1

1
sit xit uit ) ], D12 =
n

T

sit xit uit )(

.. ..

sit xit xit

sit xit uit ) ], and D22 =
t=1

n

T

T

[(
i=1 t=1

1
n

..

sit xit uit )((

n

T

[(
i=1 t=1

sit xit uit ) ]
t=1

..

T

sit xit uit )(

..

sit xit uit ) ]
t=1

where uit : residual from FE estimation and uit residual from pooled LS estimation.

−1 

−1
 A11 0 
√

var( n(θa − θa )) = 


0 A22

229

D11 D12   A11 0 






D21 D22
0 A22

(G.1)

D11 =

D12 =

D21 =

D22 =

1
N
1
N
1
N
1
N

n

T

T

t=1
T

i=1 t=1
n
T
i=1 t=1
n
T

[(

..

t=1
T

i=1 t=1
n
T

[(

..

t=1
T

p

t=1
T

t=1
T

p

..

t=1

..

..

sit xit uit ) ] = D21
t=1
T

sit xit uit )(
t=1

..

sit xit uit ) ] = D12

t=1
T

sit xit uit )(

sit xit uit ) ] → E[(

sit xit uit )(

i=1 t=1

sit xit uit )((

sit xit uit ) ] → E[(

sit xit uit )(

sit xit uit ) ] = D11
t=1
T

t=1
T

p

..

sit xit uit )(

sit xit uit ) ] → E[(

sit xit uit )((

[(

T

T

p

sit xit uit ) ] → E[((

sit xit uit )(

[(

..

sit xit uit ) ] = D22
t=1

We obtain (23) from (21) and (22).
p

p

p

A−1 → A−1 , A−1 → A−1 where → denote probability limit
11
11
22
22




D11 D12  p D11 D12 

→





D21 D22
D21 D22
 
−1 

  A11 0   D11 D12   A11 0
√

 

N (θa − θ) ⇒ N 0, 
 

 
0 A22
D21 D22
0 A22

(G.2)

(G.3)
−1 
 
 
 


(G.4)

where ⇒ implies weak convergence (i.e. convergence in distribution)

−1

−1 
 A11 0


0 A22


 D11 D12   A11 0 






D21 D22
0 A22








−1
0   D11 D12
 A11


= 

D21 D22
0
A−1
22


−1
0 
  A11




0
A−1
22


−1
−1
−1
−1
 A11 D11 A11 Aa1 D12 A22 

= 


A−1 D21 A−1 A−1 D22 A−1
22
11
22
22

Thus, we obtain (24) from (23).
 
√
√
 
R N (θa − θ) = R N θa ⇒ N 0, R
 
Rθ=0

A−1 D11 A−1
11
11

A−1 D12 A−1
a1
22

A−1 D21 A−1
22
11

A−1 D22 A−1
22
22

230



R 


(G.5)

Denote following notations ,


−1

 A11 0 

G−1 = 
a


0 A22





 D11 D12 

Da = 


D21 D22

Then, we have (25) from (20) and (23).
√
p
var( N (θa − θ)) = G−1 Da G−1 = G−1 Da G−1 + op (1), (i.e. G−1 Da G−1 → G−1 Da G−1 )
a
a
a
a
a
a
a
a
(G.6)
where


G−1 Da G−1
a
a



−1
0   D11 D12
 A11


= 

0
A−1
D21 D22
22






−1
0 
  A11




0
A−1
22


−1
−1
−1
−1
 A11 D11 A11 Aa1 D12 A22 


= 

A−1 D21 A−1 A−1 D22 A−1
22
11
22
22

√
√
Thus, from (25) we can construct a statistic W1 = [R N θa ] [RG−1 Da G−1 R ]−1 R N θa
a
a
which converges to chi-squared distribution.
√
a
R N θa ∼ N (0, RG−1 Da G−1 R )
a
a
√
√
[R N θa ] [RG−1 Da G−1 R ]−1 R N θa ∼ χ2 .
a
a
k

231

Appendix H
Proof of Corollary 3 in Chapter 3:
Robust inference in unbalanced panel
data
Proof First, we consider WLS estimator.
n

T

βwls = (

1
1
sit √ xit √ xit )−1 (
λ
λ

n

T

1
1
sit √ xit √ yit )
λ
λ

i
i
i
i
i=1 t=1
i=1 t=1
T
n T
1
1
1
1
(
sit √ xit √ xit )−1 (
sit √ xit √ yit ).
Ti
Ti
Ti
Ti
i=1 t=1
i=1 t=1
n

=

Then we have

√

T (βwls −β) =

√

n

T(

T

1
1
sit √ xit √ xit )−1 (
T
T
i

T

1
1
sit √ xit √ uit )
T
T

i
i
i=1 t=1
i=1 t=1
n
T
T
n
1
√1
sit xit xit )−1 (
=
(
s x u )
Ti
Ti λi t=1 it it it
t=1
i=1
i=1
√
n
T
n
T
T 1
1
√ T √
sit xit xit )−1 (
s x u )
(
T T
Ti λi T t=1 it it it
i=1
i=1 i t=1
n
n
n
n
1 Λ W (λ )) = (
1
1 λ Q )−1 (
⇒ (
Qi )−1 (
λi i i
λi i i i
λi Λi W (λi ))
i=1
i=1
i=1
i=1
n
n
1
(
Qi )−1 (
λi Λi Wi )
i=1
i=1

232

i

n

=

=

n
n
√
1
−1 (
T (βwls − β) ⇒ (
Qi )
ΛW)
λi i i
i=1

n

T · var(βwls ) = T · Vw−clus = (
i=1

1
Ti

(H.1)

i=1

T

n

sit xit xit

)−1 Ω

t=1

T

1
√
sit xit xit )−1
wls (
T t=1
i=1

where
1
Qi =
Ti

1 1
√
λi T

T

n

sit xit xit , Ωwls =
t=1

i=1

T

T

t=1

1 1
sit xit uit = √
λi T

1
1
=⇒ Λi Wi − λi Qi (
λi
λi

n

Qj

j=1

n

[(
i=1

1 1
√
λi T

T
t=1

1 1
sit xit uit )( √
λi T

1 1
sit xit uit −
λi T

)−1 (

j=1

Ωwls =

t=1
n

1 1
[( √
λi T

T

T

sit xit uit ) ]
t=1

√
sit xit xit T (βwls − β) + op (1)

t=1

1
1
Λj Wj ) = Λi Wi − Qi (
λj
λi

n

n

Qj

)−1 (

j=1

T

sit xit uit )(
t=1

1 1
√
λi T

j=1

1
Λ W )
λj j j

T

sit xit uit ) ]
t=1

n
n
1 Λ W )]{[ 1 Λ B − Q (
1
Qj )−1 (
j j
i i
i
λj
λi
λj Λj Wj )] }
i
i=1
j=1
j=1
j=1
j=1
n
n
n
1
1
1
1
=
[ λ Λi Wi Wi Λi λ
−
Qi (
Qj )−1 (
−
λj Λj Wj ) λi Wi Λi
i
i=1 i
j=1
j=1
n
n
n
n
n
n
1ΛW(
1 W Λ )(
1 Λ W )(
1 W Λ )(
Qj )−1 Qi +Qi (
Qj )−1 (
Qj )−1 Qi ]
i i
j j
λi
λj j j
λj
λj j j
j=1
j=1
j=1
j=1
j=1
j=1
n

=⇒

1
[ λ Λi Bi − Qi (

n

Qj )−1 (

n

Using homogeneity, Λj = Λ and Qj = Q for all j = 1, 2, .., n
n

=

1
1
1
[ λ ΛWi Wi Λ λ − n (Λ
i
i

i=1

n

1
1
λj Wj ) λi Wi Λ
j=1

n

1
− Λ λ Wi (
i

n

1
1
λj Wj )Λ ) n
j=1

n
1 W )(
1
1
λj j
λj Wj )Λ n ]
j=1
j=1
n W
n W
n W
n W
n W W
n W
n
n W
j
j 1
j
j 1
1(
i i − 1(
i)−(
i )(
= Λ[
)(
)n+
)(
n
n
λi λi
λj
λi
λi
λj
λj
λj ) n ]Λ
j=1
i=1
i=1
j=1
i=1 j=1
j=1
i=1
n W W
n W
n W
j
1
i i
i
= Λ[
λi λi − n (
λj )(
λi )]Λ
i=1
j=1
i=1
1
+ n Λ(

233

n

Ωwls =⇒ Λ[
i=1
n

Wi Wi
1
− (
λi λi
n

i=1
n

= (

Wj n Wj
)(
)]Λ
λj
λj

j=1

j=1

T

T

n

1
Ti

t=1

Qi

T · Vw−clus = (

n

)−1 Ω

sit xit xit )−1 Ωwls (

1
√
sit xit xit )−1
T t=1
i=1

n

Qi )−1

wls (

i=1

i=1
n

1
Wi Wi
− (
λi λi
n

⇒ (nQ)−1 Λ[

n

j=1

i=1

Wj n Wj
)(
)]Λ (nQ)−1
λj
λj

(H.2)

j=1

Now consider t-statistic based on WLS which we use for t-test.

√
t∗ =
wls

T (Rβwls − r)

=

√
R T (βwls − β)

R · T · Vw−clus R

R · T · Vw−clus R

For the numerator, using (H.1), we have
n
n
√
1
Wi
1
−1 Λ(
−1 (
R T (βwls − β) ⇒ R(nQ)
ΛWi ) = RQ
)
λi
n
λi

=

1
(
n

i=1
−1 ΛW
RQ
i

n

λi

i=1

i=1

)≡

1
(
n

n

yi )

(H.3)

i=1

For the denominator, using (H.2), we have
n

R · T · Vw−clus R

⇒ R(nQ)−1 Λ[
i=1

=

=

1
[
n2
1
[
n2

n
i=1
n

n
j=1

Wj n Wj
)(
)]Λ (nQ)−1 R
λj
λj

1
RQ−1 ΛWi Wi Λ Q−1 R
− (
λi
λi
n
yi yi −

i=1

Wi Wi
1
− (
λi λi
n

1
n

n

i=1

RQ−1 ΛWi
)(
λi

n

j=1

Wj Λ Q−1 R
λj

)]

n

yi
i=1

j=1
n

yj ]
j=1

Using (H.3) and (H.4) together, we have

234

(H.4)

1
n(

√

t∗
wls

=

R T (βwls − β)
R · T · Vw−clus R

=

ny
n

(yi − y)2

i=1

=

⇒

n

1 [
yy
n2 i=1 i i

n
·
n−1

(

yi )
i=1
1
−n

√
ny

1
n−1

n

n

n

235

yi
i=1

(yi − y)2

i=1

=

n

n

j=1

≡

yj ]

yi )

i=1
n

[
i=1

n
T
n − 1 n−1

1
yi yi − n

n

n

yi
i=1

j=1

yj ]

Appendix I
Monte Carlo simulation results in
chapter 3: Size-adjusted power for
t-test in Ibragimov and Muller [2009]

236

Table I.1. The size adjusted power of test, 1(p < 0.05), T = 100
T=100, n=4

balanced

complete-case

bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100, n=8
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=100, n=16
bias=0
bias=2%
bias=5%
bias=10%
bias=20%

.048(.041,.054)
.059(.048,.069)
.092(.079,.105)
.219(.200,.237)
.626(.605,.647)
balanced
.048(.042,.055)
.073(.061,.084)
.167(.150,.183)
.517(.495,.539)
.977(.970,.983)
balanced
.048(.041,.055)
.110(.096,.123)
.362(.340,.383)
.887(.873,.901)
1(1,1)

.028(.023,.033)
.038(.029,.046)
.052(.042,.061)
.100(.087,.113)
.302(.282,.322)
complete-case
.038(.032,.044)
.054(.044,.063)
.099(.085,.112)
.243(.224,.262)
.671(.650,.692)
complete-case
.050(.043,.057)
.069(.057,.080)
.191(.174,.208)
.574(.552,.595)
.984(.978,.989)

balanced
balanced
keep ∀i drop t
keep ∀t drop i
.046(.040,.053)
.053(.046,.060)
.048(.039,.057)
.063(.052,.073)
.050(.040,.060)
.070(.059,.081)
.063(.052,.074)
.093(.080,.105)
.136(.120,.151)
.159(.143,.175)
balanced-drop t balanced -drop i
.050(.040,.059)
.053(.046,.060)
.047(.037,.056)
.052(.042,.061)
.062(.051,.072)
.060(.049,.070)
.108(.094,.122)
.089(.077,.101)
.311(.290,.331)
.156(.140,.171)
balanced-drop t balanced -drop i
.050(.043,.057)
.048(.041,.055)
.057(.047,.067)
.059(.048,.069)
.124(.109,.138)
.094(.081,.106)
.330(.309,.351)
.225(.206,.243)
.839(.823,.855)
.628(.607,.649)

(Note) The rejection probability is reported. 95 % coverage rate for rejection probability is
reported in parenthesis. Generated errors are from AR(1) process with correlation
coeﬃcient 0.75. The number of replications is 2,000.

237

Table I.2. The size adjusted power of test, 1(p < 0.05), T = 300
T=300, n=4

balanced

complete-case

bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=300, n=8
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=300, n=16
bias=0
bias=2%
bias=5%
bias=10%
bias=20%

.046(.036,.055)
.067(.056,.078)
.169(.153,.185)
.520(.498,.541)
.968(.960,.976)
balanced
.053(.046,.060)
.122(.107,.136)
.435(.413,.457)
.941(.931,.951)
1(1,1)
balanced
.046(.036,.055)
.187(.170,.204)
.798(.780,.815)
1(1,1)
1(1,1)

.029(.022,.037)
.034(.026,.041)
.088(.075,.100)
.237(.218,.255)
.642(.621,.663)
complete-case
.042(.034,.050)
.089(.076,.101)
.270(.250,.289)
.748(.729,.767)
.999(.998,1)
complete-case
.050(.040,.060)
.135(.120,.150)
.502(.480,.523)
.960(.951,.969)
1(1,1)

balanced
balanced
keep ∀i drop t
keep ∀t drop i
.058(.047,.068)
.048(.039,.077)
.053(.043,.063)
.055(.046,.065)
.072(.061,.083)
.079(.067,.090)
.143(.127,.158)
.145(.130,.160)
.367(.346,.388)
.262(.243,.281)
balanced-drop t balanced -drop i
.049(.040,.059)
.050(.043,.058)
.070(.059,.081)
.053(.043,.063)
.179(.162,.196)
.070(.059,.081)
.548(.526,.569)
.126(.111,.141)
.986(.980,.991)
.249(.230,.268)
balanced-drop t balanced -drop i
.050(.040,.059)
.048(.038,.057)
.084(.072,.096)
.060(.050,.071)
.294(.274,.313)
.180(.163,.197)
.782(.763,.800)
.530(.508,.551)
1(.999,1)
.971(.963,.978)

(Note) The rejection probability is reported. 95 % coverage rate for rejection probability is
reported in parenthesis. Generated errors are from AR(1) process with correlation
coeﬃcient 0.75. The number of replications is 2,000.

238

Table I.3. The size adjusted power of test, 1(p < 0.05) , T = 1, 000
T=1,000, n=4

balanced

complete-case

bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=1,000, n=8
bias=0
bias=2%
bias=5%
bias=10%
bias=20%
T=1,000, n=16
bias=0
bias=2%
bias=5%
bias=10%
bias=20%

.056(.045,.066)
.136(.121,.151)
.484(.462,.506)
.942(.931,.952)
1(1,1)
balanced
.046(.038,.054)
.273(.253,.292)
.906(.893,.918)
1(1,1)
1(1,1)
balanced
.050(.045,.059)
.523(.046,.060)
1(1,1)
1(1,1)
1(1,1)

.033(.025,.041)
.067(.056,.078)
.220(.201,.238)
.603(.581,.624)
.959(.950,.967)
complete-case
.036(.027,.044)
.144(.129,.159)
.577(.555,.598)
.963(.955,.971)
1(1,1)
complete-case
.046(.037,.055)
.285(.265,.305)
.939(.929,.949)
1(1,1)
1(1,1)

balanced
keep ∀i drop t
.042(.033,.050)
.047(.038,.057)
.050(.040,.060)
.072(.061,.083)
.147(.131,.163)
balanced-drop t
.052(.042,.061)
.092(.079,.104)
.281(.261,.300)
.740(.721,.759)
.999(.997,1)
balanced-drop t
.050(.040,.060)
.187(.169,.204)
.739(.721,.758)
.999(.997,1)
1(1,1)

balanced
keep ∀t drop i
.061(.050,.071)
.076(.064,.087)
.130(.115,.144)
.250(.231,.269)
.480(.458,.502)
balanced -drop i
.055(.045,.065)
.066(.055,.077)
.130(.115,.144)
.240(.221,.258)
.442(.420,.464)
balanced -drop i
.050(.040,.060)
.133(.118,.147)
.469(.447,.491)
.942(.932,.952)
1(1,1)

(Note) The rejection probability is reported. 95 % coverage rate for rejection probability is
reported in parenthesis. Generated errors are from AR(1) process with correlation
coeﬃcient 0.75. The number of replications is 2,000.

239

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