ESSAYS ON PSEUDO PANEL DATA AND TREATMENT EFFECTS
By
Fei Jia

A DISSERTATION
Submitted
to Michigan State University
in partial fulfillment of the requirements
for the degree of
Economics – Doctor of Philosophy
2017

ABSTRACT
ESSAYS ON PSEUDO PANEL DATA AND TREATMENT EFFECTS
By
Fei Jia
This dissertation is composed of three chapters that study two suitable estimation methods
for identifying causal relationships in the presence of (pseudo) panel data. The first and
the second chapters are devoted to minimum distance estimation for pseudo panel models,
whereas the third chapter is concerned with the estimation of controlled direct effects in
causal mediation analyses using panel data.
The first chapter focuses on finite sample properties of minimum distance estimators
in pseudo panel models. Previous research shows theoretically that the minimum distance
asymptotic theory is a natural fit for pseudo panel models when cohort sizes are large.
However, little is known about how minimum distance estimation performs with a realistic
sample size. In a carefully designed simulation study that mimics the sampling scheme
of repeated cross sections, we compare the optimal minimum distance estimator to the
fixed effects estimator which is identical to the minimum distance estimators using identity
weighting matrix. The results show that both estimators perform well in realistic finite
sample setups. The results also confirm that the optimal minimum distance estimator is
generally more efficient than the fixed effect estimator. In particular, we find that cohortwise heteroskedasticity and varying cohort size are the two typical scenarios that call for the
use of optimal weighting. For the fixed effects estimator, we find that the minimum distance
inference is more suitable than the naive inference which incorrectly ignores the estimation
errors in the pseudo panel of variable cohort means.
The second chapter extends the basic pseudo panel models in the first chapter by adding
extra instrumental variables. The additional instruments, if non-redundant, can improve
estimation efficiency. To have the efficiency gain result in a general form, we derive it in

a non-separable minimum distance framework developed in this chapter. Along with the
efficiency gain result, consistency, asymptotic normality, and optimal weighting theorems
are also established. This efficiency gain result echoes the property of generalized methods
of moments that more moment conditions do not hurt. After developing the results in the
non-separable minimum distance framework, we apply them to the extended pseudo panel
models. we show that the minimum distance estimators in the extended pseudo panels are
generalized least squares estimators, and the optimal weighting matrix is block diagonal.
Because of the last fact, the use of optimal weighting becomes more important than in basic
pseudo panels. Simulation evidence confirms the theoretical findings in realistic finite sample
setups. For an empirical illustration, we apply the method to estimate returns to education
using data from the Current Population Survey in the US.
The third chapter, coauthored with Zhehui Luo and Alla Sikorskii, proposes a flexible
plug-in estimator for controlled direct effects in mediation analyses using the potential outcome framework. A controlled direct effect is the direct treatment effect on an outcome when
the indirect treatment effect through a mediator is shut off by holding the mediator fixed.
The flexible plug-in estimator for controlled direct effects is a parametric g-formula with
an additional partially linear assumption on the outcome equation. Compared to simulation based method in the literature, this estimator avoids estimation of conditional densities
and numerical evaluation of expectations. We compare the flexible plug-in estimator to the
sequential g-formula estimator, and prove theoretically and via simulation that they are numerically equivalent under certain settings. We also discuss a sensitivity analysis to check
the robustness of the flexible plug-in estimator to a particular violation of the sequential ignorability assumption. We illustrate the use of the flexible plug-in estimator in a secondary
analysis of a random sample of low birthweight and normal birthweight infants to estimate
the controlled direct effect of low birth weight on reading scores at age 17 when a behavior
problem index is used as the mediator.

Copyright by
FEI JIA
2017

ACKNOWLEDGMENTS

The Ph.D. journey at Michigan State University has been a life-changing experience for me.
Along this journey, I have been fortunate to meet so many amazing people who help me to
gain little advantages every day. I thank Jeffrey Wooldridge who served as the chair of my
dissertation committee. His advising throughout my studies at Michigan State University
shaped two thirds of this dissertation and my current research. I thank Zhehui Luo who
supervised my research assistantship in the Department of Epidemiology and Biostatistics
at Michigan State University. The interdisciplinary collaboration with her and Alla Sikorskii
makes the the third chapter of this dissertation possible. I also thank my other committee
members, Peter Schmidt and Timothy Vogelsang for their help at various stages of my
research. Their feedback had an important positive impact on the quality of my work.
Finally, I thank Lori Jean Nichols and Margaret Lynch who work in the administrative staff
of the department of economics. Their continuous support for five years helped a lot in
completing this degree.
The work in the third chapter was supported by National Institute of Mental Health
grant RC4MH092737 (Luo) and the data came from the grant R01MH44586 (Breslau).

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xii

CHAPTER 1 FINITE SAMPLE PROPERTIES OF THE MINIMUM DISTANCE
ESTIMATOR FOR PSEUDO PANEL DATA . . . . . . . . . . . .
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 The population models . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Discussion on exogeneity . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Minimum distance estimation . . . . . . . . . . . . . . . . . . . . .
1.2.3.1 Limiting distribution of cohort-time cell means . . . . . .
1.2.3.2 Minimum distance approach for pseudo panels . . . . . . .
1.2.3.3 Closed-form MD estimators for pseudo panels . . . . . . .
1.2.3.4 Discussion on the difference between MD FE and naive FE
inference . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Simulation and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.2 Deterministic aggregate time effects . . . . . . . . . . . . . . . . . .
1.3.3 Covariate distributions . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.4 Cohort-wise heteroskedasticity in the idiosyncratic error . . . . . .
1.3.5 Cohort-time cell size . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4 Conclude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CHAPTER 2 EXPLORING ADDITIONAL MOMENT CONDITIONS IN NONSEPARABLE MINIMUM DISTANCE ESTIMATION WITH AN
APPLICATION TO PSEUDO PANELS . . . . . . . . . . . . . . .
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 The NMD framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Asymptotic normality . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.3 Optimal weighting matrix . . . . . . . . . . . . . . . . . . . . . . .
2.2.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 More conditions do not hurt . . . . . . . . . . . . . . . . . . . . . .
2.3 Pseudo panels with additional IVs . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Population model and structural equations . . . . . . . . . . . . . .
2.3.2 Useful notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

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46

The partial derivatives L and B and the inverse optimal weighting
matrix M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ˆ . . . . . . . . . . . . . . . . . . . . . . .
2.3.4.1 Asymptotics of π
2.3.4.2 Estimation of L . . . . . . . . . . . . . . . . . . . . . . . .
ˆ and the FE estimator θ
ˇ . . . . .
2.3.4.3 The general estimator θ
opt
ˆ
2.3.4.4 Estimation of B, M and θ
. . . . . . . . . . . . . . . .
ˆ θ
ˇ and θ
ˆ opt .
2.3.4.5 Estimation of the asymptotic variances of θ,
2.3.5 The GLS perspective . . . . . . . . . . . . . . . . . . . . . . . . . .
ˇ . . . . . . . . . . . . . . . . . . . .
2.3.6 Naive variance estimators for θ
2.4 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Simulation design . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Simulation results for the small pseudo panel . . . . . . . . . . . . .
2.4.3 Simulation results for the middle sized pseudo panel . . . . . . . . .
2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX A
PROOFS AND ALGEBRA . . . . . . . . . . . . . . . . . . . .
APPENDIX B
ADDITIONAL TABLES . . . . . . . . . . . . . . . . . . . . . .
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3

. 47
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. 49
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. 53
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. 55
. 59
. 60
. 64
. 67
. 69
. 72
. 73
. 83
. 104

CHAPTER 3 A FLEXIBLE PLUG-IN G-FORMULA FOR CONTROLLED DIRECT EFFECTS IN MEDIATION ANALYSIS . . . . . . . . . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Existing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 The g-Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.2 The Sequential g-formula estimator . . . . . . . . . . . . . . . . . . .
3.4 The Flexible Plug-in g-formula estimator . . . . . . . . . . . . . . . . . . . .
3.4.1 The Partial Linearity Assumption and the Plug-in g-formula estimator
3.4.2 Estimation Procedure for the Flexible Plug-in g-formula estimator
of CDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3 plim of Parametric g-Formula is the Flexible Plug-in g-formula estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.4 Flexible Plug-in g-formula estimator Is Numerically Equivalent to
Sequential g-formula estimator . . . . . . . . . . . . . . . . . . . . . .
3.4.5 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.6 Comparison of Flexible plug-in g-formula estimator with Sequential
g-formula estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
APPENDIX A
PROOFS AND ALGEBRA ON G-FORMULA . . . . . . . . . .

vii

107
107
109
111
111
114
117
117
118
119
119
120
123
125
127
128
131
132

APPENDIX B
APPENDIX C
APPENDIX D
BIBLIOGRAPHY

PROOFS AND ALGEBRA ON SEQUENTIAL
NUMERICAL EQUIVALENCE . . . . . . . . .
SENSITIVITY ANALYSIS . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .

viii

G-ESTIMATOR
. . . . . . . . . .
. . . . . . . . . .
. . . . . . . . . .

135
147
152
153

LIST OF TABLES

Results for benchmark. G = 6, T = 4, ngt ≈ 40, sampling rate = .2%;
R denotes number of replications; Monte Carlo averages on top, Monte
Carlo standard deviations in parentheses. . . . . . . . . . . . . . . . . . .

19

Results for benchmark. G = 6, T = 4, ngt ≈ 200, sampling rate = .2%;
R denotes number of replications; Monte Carlo averages on top, Monte
Carlo standard deviations in parentheses. . . . . . . . . . . . . . . . . . .

20

Results for benchmark. G = 6, T = 4, ngt ≈ 1000, sampling rate = .2%;
R denotes number of replications; Monte Carlo averages on top, Monte
Carlo standard deviations in parentheses. . . . . . . . . . . . . . . . . . .

21

Results for different aggregate time effect processes. G = 6, T = 4, ngt ≈
200, sampling rate = .2%; 10,000 replications; Monte Carlo averages on
top, Monte Carlo standard deviations in parentheses. . . . . . . . . . . . .

24

Results for distribution of x2 . G = 6, T = 4, ngt ≈ 200, sampling rate
= .2%; 10,000 replications; Monte Carlo averages on top, Monte Carlo
standard deviations in parentheses. . . . . . . . . . . . . . . . . . . . . . .

26

Results for cohort-wise heteroskedasticity in error term. G = 6, T =
4, ngt ≈ 200, sampling rate = .2%; 10,000 replications; Monte Carlo
averages on top, Monte Carlo standard deviations in parentheses. . . . . .

28

Table 1.7

Cohort-time cell sizes for the two sampling schemes . . . . . . . . . . . .

29

Table 1.8

Results for varying cohort size. G = 6, T = 4; ngt follows the three
specifications given in section 1.3.5 and is generated by varying the sampling rate; 10,000 replications; Monte Carlo averages on top, Monte Carlo
standard deviations in parentheses. . . . . . . . . . . . . . . . . . . . . . .

30

Table 1.1

Table 1.2

Table 1.3

Table 1.4

Table 1.5

Table 1.6

Table 2.1

x

x

x

Variance-covariance and correlation matrix of (µgt2 , µgt3 , µgt4 ); correlation
x

x

coefficients in parentheses. µgt3 = sin(gt), µgt4 = (1 + exp[1.5 ∗ sin(gt/2)])−1 . 62
Table 2.2

Table 2.3

Table 2.4

Finite sample properties of various estimators of β2 and its standard
error, G = 6, T = 4. Case 3. x2it ∼ N (gt/6, 1) + zi . . . . . . . . . . . . .

65

Finite sample properties of various estimators of β2 and its standard
error, G = 6, T = 4. Case 4. x2it ∼ N (gt/6, 1) + zi + fi . . . . . . . . . .

66

Finite sample properties of various estimators of β2 and its standard
error, G = 30, T = 20. Case 3. x2it ∼ N (gt/150, 1) + zi . . . . . . . . . .

67

ix

Finite sample properties of various estimators of β2 and its standard
error, G = 30, T = 20. Case 4. x2it ∼ N (gt/150, 1) + zi + fi . . . . . . . .

68

Table B.1 Small panel with G = 6, T = 4. Case 1.a: x2it ∼ N (gt/6, 1), ngt = 200,
sampling rate= 1%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively. . . . . . . . . . . . . . . . . . .

84

Table B.2 Small panel with G = 6, T = 4. Case 1.b: x2it ∼ N (gt/6, 1), ngt = 1000,
sampling rate= 1%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively. . . . . . . . . . . . . . . . . . .

85

Table B.3 Small panel with G = 6, T = 4. Case 1.1: x2it ∼ N (gt/6, 1), ngt = 200,
sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively. . . . . . . . . . . . . . . . . . .

86

Table B.4 Small panel with G = 6, T = 4. Case 1.1: x2it ∼ N (gt/6, 1), ngt = 200,
sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively. . . . . . . . . . . . . . . . . . .

87

Table B.5 Small panel with G = 6, T = 4. Case 2.a: x2it ∼ N (gt/6, 1) + fi ,
ngt = 200, sampling rate= 1%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

88

Table B.6 Small panel with G = 6, T = 4. Case 2.b: x2it ∼ N (gt/6, 1) + fi ,
ngt = 1000, sampling rate= 1%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

89

Table B.7 Small panel with G = 6, T = 4. Case 2.1: x2it ∼ N (gt/6, 1) + fi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

90

Table B.8 Small panel with G = 6, T = 4. Case 2.1: x2it ∼ N (gt/6, 1) + fi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

91

Table B.9 Small panel with G = 6, T = 4. Case 3.a: x2it ∼ N (gt/6, 1) + zi ,
ngt = 200, sampling rate= 1%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

92

Table B.10 Small panel with G = 6, T = 4. Case 3.b: x2it ∼ N (gt/6, 1) + zi ,
ngt = 1000, sampling rate= 1%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

93

Table B.11 Small panel with G = 6, T = 4. Case 3.1: x2it ∼ N (gt/6, 1) + zi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

94

Table 2.5

x

Table B.12 Small panel with G = 6, T = 4. Case 3.1: x2it ∼ N (gt/6, 1) + zi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

95

Table B.13 Small panel with G = 6, T = 4. Case 4.a: x2it ∼ N (gt/6, 1) + zi + fi ,
ngt = 200, sampling rate= 1%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

96

Table B.14 Small panel with G = 6, T = 4. Case 4.b: x2it ∼ N (gt/6, 1) + zi + fi ,
ngt = 1000, sampling rate= 1%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

97

Table B.15 Small panel with G = 6, T = 4. Case 4.1: x2it ∼ N (gt/6, 1) + zi + fi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

98

Table B.16 Small panel with G = 6, T = 4. Case 4.1: x2it ∼ N (gt/6, 1) + zi + fi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . .

99

Table B.17 Small panel with G = 6, T = 4. Case 5.a: x2it ∼ N (gt/2, 1) + zi + fi ,
ngt = 200, sampling rate= 1%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . . 100
Table B.18 Small panel with G = 6, T = 4. Case 5.b: x2it ∼ N (gt/2, 1) + zi + fi ,
ngt = 1000, sampling rate= 1%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . . 101
Table B.19 Small panel with G = 6, T = 4. Case 5.1: x2it ∼ N (gt/2, 1) + zi + fi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . . 102
Table B.20 Small panel with G = 6, T = 4. Case 5.2: x2it ∼ N (gt/2, 1) + zi + fi ,
ngt = 1000, sampling rate= 0.2%. sen , ser and sec are the non-roust,
robust and cluster-robust standard errors, receptively. . . . . . . . . . . . 103
Table 3.1

Compare the plug-in estimator with the sequential g-estimator under
different specifications for the outcome conditional mean and different
structural nested mean models. . . . . . . . . . . . . . . . . . . . . . . . . 116

Table 3.2

Simulation results: flexible plug-in g-formula v.s. sequential g-estimator . 121

xi

LIST OF FIGURES

Figure 3.1

A directed acyclic graph for a longitudinal study with three time points.
(A, M ) are the intervention nodes, (L0 , L1 , Y ) the non-intervention nodes,
and U0 the unobservables. . . . . . . . . . . . . . . . . . . . . . . . . . . 110

Figure 3.2

Subgraphs for G where an upper bar means arrows pointing to a node
are removed and an under bar means arrows emitting from a node are
removed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Figure 3.3

Controlled direct effect of LBW on reading with bad behavior as mediator, Model 1 to 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xii

CHAPTER 1
FINITE SAMPLE PROPERTIES OF THE MINIMUM DISTANCE
ESTIMATOR FOR PSEUDO PANEL DATA

1.1

Introduction

Repeated cross-sectional data is available when a series of different random samples can
be obtained from the population over time. The Current Population Survey in the U.S.A,
conducted monthly, is an example of such type of data sets. By combining cross sections at
consecutive points in time, repeated cross-sectional data gains the replicability over time in
absence of genuine panel data. Although we still cannot track each individual over time, we
are able to estimate certain panel data models, especially those with fixed individual-specific
effects and those with individual dynamics, under appropriate conditions.
The literature that makes possible the estimation of these panel data models with only
repeated cross sections dates back to the seminal work by Deaton (1985). Deaton’s idea is
to divide individuals into cohorts according to certain predetermined characteristics, such as
year of birth, and then use the cohort means of all relevant variables to construct a panel at
the cohort level. Since the variable cohort means are estimated rather than directly observed,
such a constructed panel is often called a pseudo panel. Common panel data approaches
such as first difference and fixed effects (FE) estimation are readily applicable because of
this panel structure. In this chapter, our focus is on the pseudo panel FE estimator.
Despite the fact that the cohort means are error-ridden estimates, the pseudo panel FE
coefficient estimator is generally consistent. The corresponding standard error estimators
(the naive standard errors hereafter), however, are potentially problematic for ignoring the
estimation errors in the cohort means, whether they are made robust to heteroskedasticity
and/or serial correlation. To make the standard errors right, Imbens and Wooldridge (2007)
propose a minimum distance (MD) approach for pseudo panel models. With asymptotics

1

relying on large cohort sizes, this approach is a natural fit for many microeconomic analyses,
since for microeconomic data the cohort-wise number of observations is often large, and
the number of cohorts and the number of time periods are often small. The MD approach
effectively takes account of the estimated cohort means. More importantly, it provides
an asymptotically efficient way to utilize all the moment conditions through its weighting
procedure. In fact, Imbens and Wooldridge (2007) show that the pseudo panel fixed effect
estimator is exactly the MD estimator that puts equal weights on the moment conditions
via an identity weighting matrix.
The superiority of the MD approach for pseudo panels relies on large sample theory, but
its finite sample properties have not been fully studied. It is possible that the naive FE
standard errors, especially those made robust to heteroskedasticity and/or serial correlation,
can still achieve acceptable accuracy under certain circumstances. Moreover, although the
result on optimal weighting in Imbens and Wooldridge (2007) implies that departures from
identity weighting call for optimal weighting, it is unclear what are the typical causes of
those departures.
In this chapter, we investigate the finite sample properties of the MD approach for pseudo
panels through a carefully designed simulation study. In particular, the attention is paid to
the comparison of the optimal MD estimator and the MD estimator with identity weighting
matrix. We identify two stylized causes, namely varying cohort sizes and cohort-wise error
heteroskedasticity, of departures of the optimal weighting matrix from identity. In presence
of these two features, optimal weighting evidently outperforms identity weighting. As for the
naive FE inference, we find that it is always inferior to the MD FE inference. Therefore, we
should never throw away individual-level data in empirical studies, for they contain useful
information that the sample cohort means do not have.
The MD approach is certainly not the only approach to pseudo panels. Deaton (1985), for
example, treats the estimated cohort means as a measurement error problem, and proposes a
measurement-error corrected ordinary least squares (OLS) estimator. Collado (1997) extends

2

the analysis to dynamic models, and develops a measurement-error corrected GMM estimator
based on the instrument variables (IV) method in Arellano and Bond (1991). Another strand
of researches go beyond pseudo panels and dive into individual level. Moffitt (1993) considers
both dynamic and binary choice models, and proposes an IV estimator that constructs IV
from functions of cohort and/or time. In particular, Moffit points out that aggregating to
the cohort level is equivalent to using a full set of cohort, time, and cohort-time dummies
as IV. Girma (2000) quasi-differences pairs of individuals in the same cohort to circumvent
the problem of missing individual trajectories, and proposes a particular GMM IV method
that uses past and present values of the dependent and explanatory variables within the
same group. Verbeek and Vella (2005) propose an alternative computationally attractive IV
estimator. A more thorough review that also covers important empirical applications can be
found in Verbeek (2008).
The rest of the chapter is organized as follows. In section 2 we set up the notations
and framework. In section 3 we reports and discusses the results from the simulation study.
Section 4 concludes.

1.2

Framework

Deaton (1985) shows the importance of distinguishing between the population model and
the sampling scheme. This distinction, as pointed out by Imbens and Wooldridge (2007),
“is critical for understanding the nature of the identification problem, and in deciding the
appropriate asymptotic analysis”. Therefore we follow this convention in this paper . The
exposition in this section borrows heavily from Imbens and Wooldridge (2007).

1.2.1

The population models

Consider the population model
yit = xit β + ηt + fi + uit , t = 1, . . . , T.
3

(1.1)

in which yit is the dependent random variable, xit is a 1 × K vector of random covariates
with the first entry a constant term, fi is the unobserved time invariant effect , and uit is
the unobserved idiosyncratic error. β is the parameter of practical interest. ηt ’s are the time
varying intercepts and are also treated as parameters to estimate since we are considering
applications with small T . An alternative representation is to include time dummies in xt
and then the ηt ’s are obsorbed in β. The index i refers to the same individual over time
in the population model. Writing the subscript i explicitly helps to indicate whether the
quantities are changing only across t, changing only across i, or changing across both, which
will become useful later. The model (1.1) imposes the same data generating structure for all
T time periods, which assumes a stationary population over time. Later we will see that, by
stationary population, we essentially means that the population cohort means of fi do not
change over time.
Following Deaton (1985), we assume the population can be divided into G predetermined
group. The group designation must be determined before the samples are drawn, and must
be independent of time. Birth year, for example, is one of the most commonly used characteristic to define the group designation. Let gi be the random variable indicating the group
membership of a random draw i. gi takes values in {1, 2, . . . G}. Take expectation of (1.1)
conditional on group membership, we have
E(yit |gi = g) = E(xit |gi = g)β+ηt +E(fi |gi = g)+E(uit |gi = g), t = 1, . . . , T, g = 1, . . . , G.
(1.2)
Define the population cohort means as
y

µgt = E(yit |gi = g)
µx
gt = E(xit |gi = g)

(1.3)

αg = E(fi |gi = g)
δgt = E(uit |gi = g)
for g = 1, . . . , G and t = 1, . . . , T . Note that all the four quantities above are deterministic

4

population cohort means. Then we can rewrite (1.2) as
y

µgt = µx
gt β + ηt + αg + δgt , g = 1, . . . , G, t = 1, . . . , T.

(1.4)

(1.2) and (1.4) are different notations for the population model at the cohort level. The
parameter δgt can be considered as the effect of the cohort-time cell (g, t) net of the cohort
effect αg and the time effect ηt .
y

Even if µgt and µx
gt are known, the system of linear equations in (1.4) is not identified
if we leave δgt vary freely. Therefore, we need certain restrictions on δgt . In a standard
panel data model, a weak exogeneity assumption we usually make is the contemporaneous
exogeneity of xit given fi :
E(uit |xit , fi ) = 0, t = 1, . . . , T.

(1.5)

This condition is, however, is not required here. A weaker condition that is relevant in the
context of (1.1) is
E(uit |fi ) = 0, t = 1, . . . , T.

(1.6)

Note that by iterated expectation, (1.5) implies (1.6). This gives certain flexibility to pseudo
panels on the exogeneity of xit , which will be discussed in more details later.
Because fi summarized all time-invariant unobservables, Imbens and Wooldridge (2007)
argue that (1.6) should be true for not only the lump sum fi but also any time-invariant
factors including gi . In other words, fi should represent any random variable that does not
depend on time. While this thought experiment makes sense, rigorously speaking, it does
impose stronger conditions than (1.6).1 Nevertheless, we keep this treatment in this chapter.
In particular, replacing fi with the group indicator gi , we obtain
E(uit |gi ) = 0, t = 1, . . . , T.
1 The

(1.7)

sigma algebra generated by fi is not necessarily a subset of the sigma algebra
generated by gi

5

Note that E(uit |gi ) is still a random variable. Since gi takes only finitely many values, an
alternative way to write (1.7) is
δgt = E(uit |gi = g) = 0, g = 1, . . . , G, t = 1, . . . , T.

(1.8)

Substitute (1.8) in (1.4), we get
y

µgt = µx
gt β + ηt + αg , g = 1, . . . , G, t = 1, . . . , T.

(1.9)

Let θ = (β , η , α ) be the (K + T + G) × 1 column vector of parameters with η =
(η1 , . . . , ηT ) and α = (α1 , . . . , αG ) . There are, however, only T + G − 2 parameters to
estimate. Since xit includes a constant term, only (G − 1) parameters in αg and (T − 1) in ηt
are separately identifiable. We impose the normalization α1 = 0 and η1 = 0 which is slightly
different from the normalization

G
g=1 αg

= 0 and η1 = 0 in Imbens and Wooldridge (2007).

With this treatment, αg , g = 2, · · · , G and ηt , t = 2, · · · , T represent the net effects relative
y

to the first cohort at the first time period. If µgt and µx
gt are known, GT ≥ K + T + G − 2,
and the equations in (1.9) are linearly independent, then (1.9) contains enough (maybe
over-identified) restrictions to solve for θ.
As pointed out in Imbens and Wooldridge (2007), what (1.7) really imposes is that the
cohort-level equations contain only the set of cohort and time effects but not the cohorttime interaction effects. If for any cohort-time cell (g, t) δgt is nonzero, then there is a
misspecification in the population model (1.1). In the extreme case where the true model
contains a full set of cohort-time net effects, nothing is identified since the identification of
any parameter comes from the variation of its associated variable over cohort and/or time.
Perhaps another representation helps understanding this better. Write the population
model with a full set of cohort-time effects as
yit = xit β + ηt + fi + δgi ,t + uit , t = 1, . . . , T,
where δgi ,t = E(uit |gi ), the cohort-time effect of cell (gi , t), is properly treated as a random
6

variable. Then (1.7) is exactly.
δgi ,t = 0,
i.e. the population model does not contain a full set of cohort-time effects.
Details about some common estimation strategies given (1.9), such as OLS, FE and FD
can be found in Imbens and Wooldridge (2007). They are straightforward after treating the
cohort means as known.

1.2.2

Discussion on exogeneity

We argue that a subtle flexibility is gained thanks to the fact that (1.6) is weaker than (1.5).
Specifically, the weaker condition (1.6) allows the deviation of xit from its cohort mean to be
non-exogenous with respect to the deviation of uit from its cohort mean. Put it differently,
within a given cohort-time cell, xit and uit are allowed to be correlated. But at the cohort
level, the cohort mean of xit must be exogenous with respect to the cohort mean of uit , if we
treat the variation in their cohort means over cohort and time as the source of randomness.
In sum, endogeneity at the individual level is allowed, but exogeneity at the cohort level is
still required.
The first implication of this is that the allowed dependence between xit and uit is not
arbitrary. xit can still contain lagged dependent variables, most commonly yi,t−1 , or explanatory variables that are contemporaneously endogenous, but the dependence cannot be
fundamental, meaning that it exists at the cohort level. In our setup, this is guaranteed by
two restrictions: (i) the specification in the individual level population model (1.1) is correct,
and (ii) the zero cohort mean of uit condition in (1.7) holds. They together translate to the
exclusion of a full set of cohort-time effects.
Another implication is that, if G is large enough so that we can rely on large G asymptotics,2 we do not need the zero cohort mean of uit condition imposed in (1.7) for consistent
we can assume the conditional distribution of δgt given µx
gt is normal and
use maximum likelihood estimation.
2 Alternatively,

7

estimation of β. The condition can be relaxed to some form of exogeneity at the cohort level.
Let g = gi to simplify notation, and denote the cohort-level random explanatory variable
and error by µx
gt and δgt , which treats the cohort dimension as random but still leaves the
time dimension fixed. Then one form of such exogeneity assumption can be expressed as
E(δgt |µx
gt ) = 0, t = 1, . . . , T.

(1.10)

Apparently, the condition (1.7) implies (1.10). The analysis in Deaton (1985) goes a bit
further to treat the time dimension as random as well, and thus relies on large GT asymptotics, but the idea is essentially the same. Nevertheless, this treatment “seems unnatural for
the way pseudo panels are constructed, and the thought experiment about how one might
sample more and more groups is convoluted”, as pointed out by Imbens and Wooldridge
(2007). Therefore, if we do not have large G but only large cohort-time cell size, Ngt , MD
estimation is the way to go, and we need to impose the stronger zero cohort mean of uit
condition (1.7).
The treatment regarding µx
gt and δgt above also breaks the barrier between the view
of constructing cohort-level equations from the individual level, as represented by Deaton
(1985) and Imbens and Wooldridge (2007), and the view of starting the analysis right from
the cohort level. Both views make sense and are unified under this treatment. But when
starting the analysis from the cohort level, we need to make sure that the assumptions are
consistent with the process of construction from the individual level. In particular, attention
should be paid to proper asymptotics.

1.2.3

Minimum distance estimation

Given a repeated cross-sectional data set with large cohort sizes, small number of cohorts
and small number of time periods, the MD estimator is a natural fit. Because of the large
y

cohort sizes, the cohort means µgt and µx
gt in (1.9) can be estimated fairly precisely by their
sample analogs in each cohort-time cell . The system of equations (1.9) is the link between
8

y

the reduced-form parameter {(µgt , µx
gt ), g = 1, . . . , G, t = 1, . . . , T } and the structural
parameter θ. The MD approach is essentially a delta method, recovering structural estimates
from reduced-form estimates.
In the next several subsections, we derive the limiting distribution of the sample cohort
means, present the minimization problem of MD estimators, and give a closed-form expression of the general MD estimator for pseudo panels. In particular, the optimal MD estimator
and the FE estimator as the MD estimator with identity weighting are discussed in detail.

1.2.3.1

Limiting distribution of cohort-time cell means

Specifically, assume we have a random sample on (xit , yit ) of size nt for each t, and we denote
them collectively by{(xit , yit ), i = 1, . . . , nt }. i may refer to different individuals in different
time periods. This notation works fine as long as we keep in mind the in each time period
we have a new random sample.
For each random draw i, let ri = (rit,1 , rit,2 , . . . , rit,G ) be a vector of group indicators
such that rit,g = 1{g =g} , where 1A is the indicator function that takes values in {0, 1}
i
and equals 1 only if A is true. In this way we properly treat the group membership of the
random draw i as a random vector ri . With ri , the sample average of the response variable
in cohort-time cell (g, t) can be written as
y
µ
ˆgt

=

n−1
gt

nt

rit,g yit =

(ngt /nt )−1 n−1
t

i=1

where ngt =

nt
i=1 rit,g

nt

rit,g yit

(1.11)

i=1

is properly treated as a random variable.

y

y

µ
ˆgt is generally consistent for µgt . Specifically, let ρg = P (rit,g = 1), the fraction of
the population in cohort g. We have treated ρg as time invariant because we assume the
population is stationary. Then
p

ρˆgt = (ngt /nt ) −→ ρg ,

9

(1.12)

and thus we have
y
µ
ˆgt

=

−1
ρˆ−1
gt nt

nt

p

y

rit,g yit −→ ρ−1
g E(rit,g yit ) = µgt .

i=1
y

The last equality holds because E(rit,g yit ) = P (rit,g = 1)E(yit |rit,g = 1) = ρg µgt . The same
argument also holds for the other cohort means.
y
ˆ sgt = (ˆ
ˆx
Let sit = (yit , xit ), and define µ
µgt , µ
gt ) as in (1.11). Then the asymptotic

ˆ sgt is
distribution of µ
√

s
ˆ sgt − µsgt ) −→ N ormal(0, ρ−1
nt (µ
g Ωgt )

where
Ωsgt = V ar(sit |g)
is the (K + 1) × (K + 1) variance-covariance matrix for the cohort-time cell (g, t). When
later we stack the means across groups and time periods, it is useful to have the result
√
ˆ sgt − µsgt ) −→ N ormal(0, (ρg κt )−1 Ωsgt )
n(µ
where n =

T
t=1 nt

(1.13)

and κt = limn→∞ (nt /n) is essentially the fraction of all observations

accounted for by cross section t. ρg κt is consistently estimated by ngt /n. A consistent
estimator for Ωsgt is
ˆs
Ω
gt

=

n−1
gt

nt

ˆ sgt )(sit − µ
ˆ sgt ) .
rit,g (sit − µ

(1.14)

i=1

which is the sample variance-covariance matrix of s within the cell (g, t).
Let π = (µs11 , µs12 , . . . , µs1T , µs21 . . . , µsGT ) , the column vector of all cell means. π is a
ˆ by replacing µsgt with µ
ˆ sgt . Now, µ
ˆ sgt
GT (K + 1) vector since each µsgt is K + 1. Define π
are independent across g because we have random sampling for each t. When xit does not
ˆ sgt are independent across t, too. Then, by stacking (1.13) for all
contain lags or leads, µ
(g, t), we have
√

ˆ − π) −→ N ormal(0, Ω),
n(π

10

(1.15)

where Ω is the GT (K + 1) × GT (K + 1) block diagonal matrix with the gt-th block
(ρg κt )−1 Ωsgt . Note that Ω incorporates both different cell variance-covariance matrices as
well as the different frequencies of observations. As we will see in the simulation study, this
is exactly the reason why the optimal MD estimator outperforms other MD estimators when
there are cohort-wise heteroskedasticity and varying cohort sizes.

1.2.3.2

Minimum distance approach for pseudo panels

Classical MD estimation is useful for obtaining structural estimates from reduced form estimates when a known relationship exists between the structural and reduced form parameters
(see, e.g., Wooldridge (2010)). In the pseudo panel setup, the group means in π are the reduced form parameters, θ contains the structural parameters, and the cohort-level equations
embody the known relationship between π and θ.
To facilitate the discussion, we rearrange terms in (1.9) by putting everything on the left
hand side of the equality sign. Write the resulting expression as
h(π, θ) = 0

(1.16)

where h(·, ·) is a GT × 1 vector valued function (recall θ = (β , η , α ) ). The gt-th row of
y

h(π, θ) is −µgt + µx
gt β + ηt + αg , or equivalently,
π g (−1, β ) + ηt + αg

(1.17)

where π g is the g-th T × 1 block of π. The parameters π and θ do not appear in a separable
way directly in h(π, θ), but it can be shown that this is a separable case.
The classical MD estimator is a solution to the minimization problem
ˆ θ) W h(π,
ˆ θ).
min h(π,

θ∈Θ

(1.18)

where Θ is the space of θ and W is a GT × GT weighting matrix. W is needed when the
restrictions in (1.16) over-identifies θ (GT > K + G + T − 2). We focus on the over-identified
11

case because it is usually the case in practice. Chamberlain (Harvard lecture notes) shows
that the optimal weighting matrix is the inverse of
M = ∇π h(π, θ)Ω∇π h(π, θ) ,

(1.19)

where ∇π h(π, θ) is the GT × GT (K + 1) Jacobian of h(π, θ) with respect to π. Use
Kronecker product (notation ⊗) and (1.17), we have
∇π h(π, θ) = IGT ⊗ (−1, β )
where IGT is the GT × GT identity matrix. This last result is exciting because, with Ω
block diagonal, it implies that (1.19) is a GT × GT diagonal matrix with the gt-th diagonal
entry
(ρg κt )−1 (−1, β )Ωsgt (−1, β ) .

(1.20)

But recall that Ωsgt = V ar(sit |g), we have
2 ≡ (−1, β )Ωs (−1, β ) = V ar(y − x β|g),
τgt
it
it
gt
2 is
and therefore, a consistent estimator of τgt
2
τˇgt

=

n−1
gt

Nt

ˇ − ηˇt − α
ˇ g )2
rit,g (yit − xit β
i=1

ˇ is the initial estimator of
which is the sample residual variance within cell (g, t). Here θ
ˇ = θ(
ˇ π).
ˇ is exactly the least squares
ˆ
θ obtained by putting W = IGT . Note that θ
θ
dummy variables (LSDV) estimator of θ on the pseudo panel. Since (ρg κt )−1 is consistently
estimated by (ngt /n)−1 , (1.20) can be consistently estimated by
2.
(ngt /n)−1 τˇgt

(1.21)

.
ˇ −1 the estimated optimal weighting matrix, where the gt-th diagonal entry
Denote by M
2 . Note that M
ˇ = C(θ(
ˇ π))
ˇ −1 is (ngt /n)/ˇ
ˇ = M(θ)
ˆ is a function of the reduced-form
of M
τgt

ˆ Then the minimization problem of the optimal MD estimator is
estimate π.
ˇ −1 h(π,
ˆ θ) M
ˆ θ).
min h(π,

θ∈Θ

12

(1.22)

1.2.3.3

Closed-form MD estimators for pseudo panels

In the pseudo panel setup, h(π, θ) is linear in each argument, and the MD estimator of θ is
in closed form. We derive this expression in this section.
x

Let µgt = (µx
gt , dt , cg ) be the 1 × (K + G + T − 1) row vector of regressors, where dt is
a 1 × (T − 1) vector of time dummies and cg is a 1 × G vector of group dummies. Let


x
 µg1 
 x 
 µ 


x
µg =  .g2  , T × (K + G + T − 1),
 .. 




x
µgT


x
µ1
x
µ2











µx =  .  ,
 .. 




x
µG

GT × (K + G + T − 1).

Then ∇θ h(π, θ) = µx , and the FOC for (1.22) is
ˆ−µ
ˇ −1 (µ
ˆx M
ˆx θ
µ
ˆy ) = 0,
ˆ x = (µ
ˆx
where µ
gt , dt , cg ). Therefore, the optimal MD estimator is
ˆ = (µ
ˇ −1 µ
ˇ −1 µ
ˆ x )−1 µ
ˆx M
ˆx M
θ
ˆy .

(1.23)

which looks like a weighted least squares estimator. Following Chamberlain, the estimated
ˆ is simply
asymptotic variance of θ
ˆ = (µ
ˇ −1 µ
ˆx M
ˆ x )−1 /n.
Avar(θ)

(1.24)

2 , it is easy to weight each cell
ˇ −1 is the diagonal matrix with entries (ngt /n)/ˇ
Because M
τgt

(g, t) by

ˆ and its asymptotic standard errors via a
ngt /n/ˆ
τgt and then compute both θ

2.
weighted regression. In STATA, this can be done by specifying aweight (ngt /n)/ˇ
τgt

13

The FE estimator applied to the pseudo panel of cohort means turns out to be the MD
ˇ −1 with the
estimator with identity weighting matrix. To see that, we simply replace M
identity matrix IGT in (1.23),
ˇ = (µ
ˆ x )−1 µ
ˆx µ
ˆx µ
ˆy
θ

(1.25)

Strictly speaking, (1.25) is the LSDV estimator on the pseudo panel. But since it gives
the same estimates for β, we also call it the FE estimator. The MD asymptotic variance
ˇ is
estimator for θ
ˇ = (µ
ˇ −1 µ
ˆ x )(µ
ˆ x )−1 (µ
ˆ x )−1 /n.
ˆx µ
ˆx M
ˆx µ
Avar(θ)

(1.26)

Apparently, this formula is different from the naive FE asymptotic variance estimators to
be discussed in the next section, whether they are made robust to heteroskedasticity and/or
serial correlation.

1.2.3.4

Discussion on the difference between MD FE and naive FE inference

ˇ can not be
Unlike the optimal MD asymptotic variance, the MD asymptotic variance for θ
estimated directly from a weighted regression. In fact, since the corresponding weighting
matrix for FE is the identity matrix, the correct weight for each cell is simply no weight
(equal weight). Without any weighting, a linear regression gives us the naive asymptotic
variance estimator3
ˇ = (µ
ˆ x )−1 σ
ˆx µ
ˇ2
Avarn (θ)

(1.27)

where
y
ˇ
ˆx
µ
ˆgt − µ
ˇg
gt β − ηˇt − α

σ
ˇ 2 = (GT − 1)−1

2

.

g,t

ˇ −1 equals σ
Clearly, (1.26) and (1.27) coincide if nM
ˇ 2 IGT , which is generally not the case.
Making it robust to heteroskedasticity (White (1980)), we get the naive heteroskedasticityrobust asymptotic variance estimator
3 Proper

adjustment of degrees of freedom can also be proposed, which we do not discuss
for simplicity. It also applies to the two robust naive variance estimators.
14

ˇ = µ
ˆx
ˆx µ
Avarr (θ)

−1

ˆx
µ

ˆ uˇ )
diag(µ

2

ˆx
µ

ˆx
ˆx µ
µ

−1

ˆ uˇ ) is the diagonal matrix created by putting the vector µ
ˆ uˇ on the principal
where diag(µ
ˆ uˇ is the column vector that stacks all cohort-level residuals over g and t, and its
diagonal. µ
[(g − 1)T + t]-th entry is
ˇ
µ
ˆugt

=

n−1
gt

nt

rit,g uˇit .
i=1

ˇ is the sample cohort mean of the individual-level residuals within cell (g, t). The
That is, µ
ˆugt

individual level residual, uˇit , is defined as
ˇ − (ˇ
ηt + α
ˇ g ).
uˇit = yit − xit β
ˇ )2 is different from τˇ2 . The former is the square of the residual cohort mean
Note that (ˆ
µugt
gt

for cell (g, t), which only contains cohort-level information, where as the latter is the sample
variance of the residuals within cell (g, t), which contains individual-level information.
Further making it robust to heteroskedasticity and serial correlation (see, e.g. Wooldridge
(2010)), we get the naive cluster-robust asymptotic variance estimator
−1
ˇ = µ
ˆx µ
ˆ x diagG (µ
ˆx
ˆ uˇ )diagG (µ
ˆ uˇ ) µ
ˆx
Avarc (θ)
µ

ˆx µ
ˆx
µ

−1

ˆ uˇ ) is the block diagonal matrix with the g-th diagonal block µ
ˆ ugˇ for g =
where diagG (µ
1, · · · , G. The subscript G in the notation diagG indicates that the block diagonal matrix
ˇ . Alternatively,
ˆ ugˇ is a T × 1 vector with the t-th entry µ
has G blocks on the diagonal. µ
ˆugt

the middle term can be written as
ˇµ
ˇ )
ˆ uˇ )diagG (µ
ˆ uˇ ) = diag(µ
ˆ u1ˇ µ
ˆ u1ˇ , µ
ˆ u2ˇ µ
ˆ u2ˇ , · · · , µ
ˆ uG
ˆ uG
diagG (µ

ˇ −1 , this is a block diagonal matrix with µ
ˆ zg uˇ on the
ˆ zg uˇ µ
Unlike the diagonal matrix nM
g-th diagonal block.
To summarize, the three naive FE asymptotic variance estimators can be obtained
2
ˇ0 = M
ˇ −1 /n in (1.26) with V
ˇn = σ
ˇ r = diag(µ
ˇc =
ˆ uˇ ) and V
by replacing V
ˇ 2 IGT , V

15

ˆ uˇ )diagG (µ
ˆ uˇ ) , respectively. But the naive FE inference is fundamentally different
diagG (µ
from the MD inference. The former only relies on cohort-level information, where as the latter
ˇ and Avarc (θ)
ˇ
abstracts information from the individual level. The robustness of Avarr (θ)
is also with respect to cohort-level heteroskedasticity and/or serial correlation only, i.e. heteroskedasticity and/or serial correlation in µugt , which requires at least large G asymptotics.
As illustrated by the simulation study in the next section, the naive FE inference is far less
efficient since it discards all individual-level information.

1.3

Simulation and results

We now present the Monte Carlo simulation study that investigates the finite sample properties of the MD approach for pseudo panels. The simulation study focuses on two questions.
First, what are the typical scenarios in which the optimal MD estimator outperforms the
FE estimator? From (1.21), we know that if there is cohort-wise heteroskedasticity and/or
varying the cell sizes, the optimal MD estimator is expected to outperform the FE estimator.
In general, if there is any pattern in the population model that makes the optimal weighting
matrix evidently different from the identity matrix, the optimal MD estimator is supposed
to perform better. We check if it is the case in the simulation study.
Secondly, can the naive FE inference still provide satisfactory accuracy and if it could,
what are these typical scenarios? As discussed in the last section, the naive FE asymptotic
variances and the MD FE asymptotic variance are alike in their formulae. On the other
hand, the naive FE inference is fundamentally different form the MD FE inference in that
the former discards all information at the individual level. The simulation study helps to
understand these two seemingly conflicting facts.
As Imbens and Wooldridge (2007) point out, the simulation design should be careful in
at least two places.
First, data for each cross section should be drawn from the population independently

16

across time, and the group identifier should also be randomly drawn. This is accomplished
by a two step procedure. In the first step, we draw the population using (1.1). The population
cohort sizes are fixed and depending on the design may or may not depend on cohort and/or
time.4 In the second step, we mimic the sampling scheme of repeated cross sections by
drawing independent random samples over time. In each period, we draw a tiny portion of
the population as the cross-sectional sample for that period.
Second, the underlying model should have full time effects to be realistic. If, as in Verbeek
and Vella (2005), we omit the aggregate time effects while let explanatory variables to have
means differ by cohort-time cell, the variation in µx
gt will be relatively rich and thus we may
set up too optimistic a situation for the estimators.
We consider five scenarios. The first is a benchmark scenario in which all things are balanced across cohort-time cells. In the remaining four scenarios, we manipulate four different
features of the population model, namely the time effects, the covariate distribution, the
cohort-wise heteroskedasticity and the varying cohort sizes, one at a time. In this way, it is
easy to isolate the cause. We begin with the benchmark scenario.

1.3.1

Benchmark

In the benchmark scenario, we generate the outcome yit as a linear function of the covariates
(x1it = 1, x2it , x3it , x4it ), the time effect ηt , the individual effect fi and the idiosyncratic
error uit as in
yit = β1 + β2 x2it + β3 x3it + β4 x4it + ηt + fi + uit , i = 1, · · · , Nt , t = 1, · · · , T.

(1.28)

The parameter values used are β = (β1 , β2 , β3 , β4 ) = (1, 1, 1, 1). The time effects are generated by ηt = t − 1, and the cohort effects are generated by αg = g − 1. Individual
4 Ideally,

we would like a population with infinity many observations so that it is infinitely
close to the population distribution defined by (1.1). In reality this is impossible, so we draw
a large number of individuals to approximate the population distribution.

17

fixed effects are generated by adding a random normal disturbance to the cohort effects, i.e.
fi ∼ N (αg , 1). The distribution of idiosyncratic error is given by uit ∼ N (0, 10).
To fix ideas, it might be helpful to think of x2it , x3it and x4it as education, experience
and marital status, respectively. The outcome yit is the log hourly wage, and there is an
individual effect fi representing some unobserved ability. The three explanatory variables
x2it , x3it and x4it are generated as follows
x2it ∼ N (gt/6, 1),
x3it ∼ N (sin(gt), 1),
x4it ∼ Bernouli

1
1 + exp[1.5 ∗ sin(gt/2)]

.

That is, x2it is a continuous variable with population cohort mean gt/6 and with-cell variance
1. x3it is a continuous variable with the population cohort mean sin(gt) and within-cell
1
. The
variance 1, and x4it is a binary variable equal to 1 with probability 1+exp(1.5∗sin(gt/2))

key is to let the three variable cohort means have distinct variation over g and t.
We apply the optimal MD estimator and the MD estimator with identity weighting. The
latter is numerically equivalent to the FE estimator on the pseudo panel of cohort means.
For each estimator, we compute the MD coefficient and standard error estimates. For the
MD estimator with identity weighting, we also compute the three naive FE standard errors
discussed in the last section.
We consider a small panel with G = 6 cohorts and T = 4 time periods. The population
cell sizes Ngt are 2×104 , 105 and 5×105 respectively in the three cases considered. After the
population panel is generated, we fix it over simulation replications. To mimic the sampling
scheme of repeated cross-sectional surveys, we draw .2% of the population in each period.
The resulting sample cell sizes ngt are approximately 40, 200 and 1000, respectively. For
each case, we consider three different numbers of replications. The results are reported in
Table 1.1, Table 1.2 and Table 1.3.
There are several observations worth discussing. First of all, the optimal MD estimator
18

Table 1.1 Results for benchmark. G = 6, T = 4, ngt ≈ 40, sampling rate = .2%; R denotes
number of replications; Monte Carlo averages on top, Monte Carlo standard deviations in
parentheses.
MD Identity
βˇ
R = 1000

x2
x3
x4
c2
d2
cons

R = 5000

x2
x3
x4
c2
d2
cons

R = 10000

x2
x3
x4
c2
d2
cons

ˇ
se(β)

MD Optimal
ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.977
0.353
(0.345) (0.041)
0.999
0.199
(0.191) (0.017)
1.001
0.634
(0.631) (0.063)
0.980
0.433
(0.420) (0.032)
1.025
0.412
(0.417) (0.036)
0.973
0.380
(0.364) (0.034)

0.345
(0.080)
0.194
(0.042)
0.619
(0.139)
0.424
(0.090)
0.404
(0.088)
0.373
(0.079)

0.313
(0.097)
0.185
(0.047)
0.584
(0.163)
0.400
(0.122)
0.387
(0.108)
0.347
(0.116)

0.322
(0.142)
0.201
(0.080)
0.657
(0.249)
0.198
(0.092)
0.469
(0.174)
0.290
(0.108)

0.979
(0.343)
1.001
(0.195)
0.998
(0.633)
0.977
(0.423)
1.023
(0.420)
0.978
(0.365)

0.347
(0.041)
0.196
(0.016)
0.622
(0.061)
0.425
(0.031)
0.405
(0.035)
0.373
(0.032)

0.981
0.351
(0.345) (0.041)
1.003
0.198
(0.193) (0.017)
1.009
0.632
(0.640) (0.062)
0.984
0.432
(0.420) (0.032)
1.020
0.410
(0.414) (0.036)
0.975
0.379
(0.372) (0.033)

0.339
(0.079)
0.192
(0.042)
0.611
(0.136)
0.419
(0.090)
0.398
(0.087)
0.368
(0.079)

0.308
(0.095)
0.182
(0.046)
0.574
(0.159)
0.393
(0.118)
0.380
(0.105)
0.340
(0.113)

0.320
(0.143)
0.194
(0.075)
0.645
(0.243)
0.194
(0.090)
0.459
(0.169)
0.286
(0.109)

0.982
(0.347)
1.003
(0.195)
1.006
(0.644)
0.985
(0.424)
1.021
(0.416)
0.975
(0.374)

0.345
(0.040)
0.195
(0.017)
0.621
(0.060)
0.425
(0.031)
0.403
(0.035)
0.372
(0.032)

0.984
0.350
(0.344) (0.041)
1.003
0.198
(0.194) (0.017)
1.007
0.632
(0.634) (0.061)
0.984
0.432
(0.422) (0.031)
1.017
0.410
(0.411) (0.035)
0.979
0.378
(0.373) (0.032)

0.338
(0.079)
0.191
(0.042)
0.610
(0.137)
0.417
(0.090)
0.397
(0.087)
0.367
(0.079)

0.306
(0.095)
0.182
(0.047)
0.574
(0.159)
0.392
(0.119)
0.380
(0.105)
0.339
(0.113)

0.320
(0.141)
0.193
(0.075)
0.646
(0.246)
0.193
(0.089)
0.458
(0.169)
0.286
(0.108)

0.984
(0.346)
1.004
(0.195)
1.005
(0.637)
0.985
(0.426)
1.018
(0.414)
0.978
(0.375)

0.344
(0.040)
0.195
(0.017)
0.620
(0.059)
0.424
(0.030)
0.402
(0.034)
0.372
(0.031)

19

Table 1.2 Results for benchmark. G = 6, T = 4, ngt ≈ 200, sampling rate = .2%; R denotes
number of replications; Monte Carlo averages on top, Monte Carlo standard deviations in
parentheses.
MD Identity
βˇ
R = 1000

x2
x3
x4
c2
d2
cons

R = 5000

x2
x3
x4
c2
d2
cons

R = 10000

x2
x3
x4
c2
d2
cons

ˇ
se(β)

MD Optimal
ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.990
0.161
(0.165) (0.009)
1.008
0.089
(0.088) (0.003)
1.010
0.286
(0.289) (0.012)
0.996
0.191
(0.194) (0.006)
1.000
0.184
(0.181) (0.007)
0.982
0.168
(0.165) (0.007)

0.157
(0.034)
0.087
(0.018)
0.279
(0.059)
0.186
(0.039)
0.180
(0.038)
0.164
(0.035)

0.140
(0.042)
0.082
(0.021)
0.262
(0.072)
0.175
(0.052)
0.171
(0.047)
0.151
(0.049)

0.143
(0.062)
0.087
(0.033)
0.299
(0.110)
0.088
(0.036)
0.206
(0.075)
0.128
(0.047)

0.990
(0.165)
1.008
(0.087)
1.011
(0.288)
0.995
(0.194)
1.000
(0.182)
0.982
(0.165)

0.160
(0.009)
0.089
(0.003)
0.285
(0.012)
0.190
(0.006)
0.183
(0.007)
0.167
(0.006)

0.991
0.160
(0.161) (0.009)
1.005
0.089
(0.089) (0.003)
1.010
0.286
(0.287) (0.012)
0.991
0.191
(0.192) (0.006)
1.006
0.184
(0.181) (0.007)
0.983
0.168
(0.168) (0.006)

0.157
(0.033)
0.087
(0.018)
0.280
(0.059)
0.187
(0.039)
0.180
(0.038)
0.164
(0.034)

0.140
(0.042)
0.082
(0.020)
0.263
(0.071)
0.174
(0.051)
0.171
(0.046)
0.150
(0.049)

0.142
(0.063)
0.087
(0.032)
0.298
(0.109)
0.088
(0.036)
0.207
(0.074)
0.127
(0.046)

0.992
(0.161)
1.005
(0.089)
1.011
(0.287)
0.991
(0.192)
1.006
(0.181)
0.983
(0.169)

0.160
(0.009)
0.089
(0.003)
0.285
(0.012)
0.190
(0.006)
0.184
(0.007)
0.167
(0.006)

0.995
0.161
(0.161) (0.009)
1.004
0.089
(0.089) (0.003)
1.012
0.286
(0.286) (0.012)
0.990
0.191
(0.191) (0.006)
1.002
0.184
(0.183) (0.007)
0.985
0.168
(0.168) (0.007)

0.157
(0.034)
0.087
(0.018)
0.279
(0.059)
0.186
(0.039)
0.180
(0.038)
0.164
(0.035)

0.140
(0.042)
0.082
(0.020)
0.263
(0.071)
0.174
(0.052)
0.171
(0.046)
0.150
(0.050)

0.143
(0.063)
0.088
(0.032)
0.298
(0.110)
0.088
(0.036)
0.206
(0.074)
0.127
(0.047)

0.995
(0.161)
1.004
(0.089)
1.013
(0.286)
0.990
(0.191)
1.002
(0.183)
0.985
(0.168)

0.160
(0.009)
0.089
(0.003)
0.285
(0.012)
0.190
(0.006)
0.183
(0.007)
0.167
(0.007)

20

Table 1.3 Results for benchmark. G = 6, T = 4, ngt ≈ 1000, sampling rate = .2%; R
denotes number of replications; Monte Carlo averages on top, Monte Carlo standard
deviations in parentheses.
MD Identity

R = 1000

x2
x3
x4
c2
d2
cons

R = 5000

x2
x3
x4
c2
d2
cons

R = 10000

x2
x3
x4
c2
d2
cons

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.003
(0.072)
0.997
(0.040)
1.001
(0.130)
1.004
(0.085)
1.004
(0.083)
1.003
(0.079)

0.072
(0.002)
0.040
(0.001)
0.128
(0.002)
0.085
(0.001)
0.082
(0.001)
0.075
(0.001)

0.070
(0.014)
0.038
(0.008)
0.124
(0.026)
0.082
(0.017)
0.079
(0.016)
0.072
(0.015)

0.062
(0.018)
0.037
(0.009)
0.117
(0.031)
0.076
(0.022)
0.075
(0.019)
0.065
(0.021)

0.062
(0.028)
0.039
(0.014)
0.132
(0.049)
0.039
(0.015)
0.090
(0.031)
0.056
(0.020)

1.003
(0.072)
0.997
(0.040)
1.001
(0.130)
1.004
(0.085)
1.004
(0.083)
1.003
(0.079)

0.072
(0.002)
0.040
(0.001)
0.128
(0.002)
0.085
(0.001)
0.082
(0.001)
0.075
(0.001)

1.003
(0.072)
0.998
(0.040)
1.009
(0.127)
0.998
(0.084)
1.002
(0.081)
1.003
(0.076)

0.072
(0.002)
0.040
(0.001)
0.128
(0.002)
0.085
(0.001)
0.082
(0.001)
0.075
(0.001)

0.071
(0.015)
0.039
(0.008)
0.126
(0.026)
0.084
(0.017)
0.081
(0.017)
0.073
(0.015)

0.063
(0.019)
0.037
(0.009)
0.119
(0.032)
0.077
(0.023)
0.077
(0.021)
0.067
(0.022)

0.064
(0.029)
0.039
(0.015)
0.135
(0.050)
0.040
(0.016)
0.093
(0.033)
0.057
(0.021)

1.003
(0.072)
0.998
(0.040)
1.009
(0.127)
0.999
(0.085)
1.002
(0.081)
1.002
(0.076)

0.072
(0.002)
0.040
(0.001)
0.128
(0.002)
0.085
(0.001)
0.082
(0.001)
0.075
(0.001)

0.999
(0.073)
0.997
(0.040)
1.009
(0.127)
1.002
(0.084)
1.001
(0.082)
1.001
(0.075)

0.072
(0.002)
0.040
(0.001)
0.128
(0.002)
0.085
(0.001)
0.082
(0.001)
0.075
(0.001)

0.071
(0.015)
0.039
(0.008)
0.125
(0.026)
0.083
(0.017)
0.080
(0.017)
0.073
(0.015)

0.063
(0.019)
0.037
(0.009)
0.118
(0.031)
0.077
(0.023)
0.076
(0.020)
0.067
(0.022)

0.064
(0.028)
0.039
(0.015)
0.134
(0.049)
0.039
(0.015)
0.092
(0.033)
0.057
(0.020)

0.999
(0.073)
0.997
(0.040)
1.009
(0.127)
1.002
(0.084)
1.001
(0.082)
1.001
(0.075)

0.072
(0.002)
0.040
(0.001)
0.128
(0.002)
0.085
(0.001)
0.082
(0.001)
0.075
(0.001)

21

has no advantage over the MD estimator with identity weighting in benchmark. This is
under the current specification the optimal weighting matrix is an identity matrix, so the
MD estimator with identity weighting is the optimal MD estimator.
Second, even in the case where the sample cohort size is about 40, the two MD estimators
perform well. The Monte Carlo averages of the coefficient estimates are fairly close to
the true parameter values. For each covariate, the Monte Carlo averages of the standard
error estimates are also fairly close to the Monte Carlo standard deviations the coefficient
estimates. Since the results are fairly stable across the three different numbers of replications,
we will report the results for 10, 000 only in later discussions.
Third, the three naive FE standard errors are much more volatile than the MD FE
standard error. This observation is consistent with the fact that the naive FE inference
relies on cohort-level information and discards all individual-level information. Moreover,
there seems to be downward small-sample bias in the naive FE standard errors than in the
MD FE standard errors. This is mainly due to the small G setting. It is well known that
α
ˇ g ’s are inconsistent under fixed G, which contaminate the residual estimates and in turn
the naive FE standard errors. Another reason is that the degree-of-freedom adjustment used
does not take into account the fact that the cohort means are estimated. Although the FE
MD standard errors seem also biased downwards, the size of the biases is always smaller
across the three tables. Clearly, in the benchmark scenarios the MD FE inference is superior
to the naive FE inference.
Fourth, the cluster-robust naive FE standard errors are severely biased downwards for
the cohort effect α2 . This observation remains valid for all the scenarios considered in this
chapter. The explanation is again the fact that the estimates for the fixed effects obtained via
LSDV are essentially based on only T observations, so the cluster-robust naive FE standard
errors are inconsistent for fixed T .
Lastly, the performance of all estimators improve universally as the the cohort size increases. For the naive FE standard errors, the reason is that the sample cohort means of the

22

residuals approaches zero as cohort size increases. To keep the discussion concise, we will
report the results for ngt ≈ 200 in later discussions.

1.3.2

Deterministic aggregate time effects

The aggregate time effects, i.e. the time intercepts ηt ’s, are treated as parameters in the
population. Therefore, to generate the aggregate time effects properly, only deterministic
functions of time need to be considered. If randomness is otherwise imposed on ηt , the
random disturbance would become part of the idiosyncratic error, which is a separate scenario
considered in section 1.3.4.
In the benchmark scenario, the aggregate time effects are ηt = t − 1 which is linear in t.
In this section, we consider two additional deterministic functions of time: quadratic, and
natural log

ηt = (t − 1)2 ,
ηt = ln(t).
The variation in the quadratic function is greater than that in the natural log function.
The results are reported in Table 1.4. The patterns of the results are similar to those
in the last section. In fact, the two panels in Table 1.4 are exactly the same as the third
panel in Table 1.2 except for the coefficient estimates on the time dummy d2 in the lower
panel where ηt = ln(t). Note that the true coefficient on d2 is ln(2) ≈ .693 in that case.
These results suggest that the aggregate time effect process has little effect on effect on the
performance of the estimators. It only changes the true parameter values of the time effects.
Of course, the fact the models are correct specified also plays a role. Correct specification
implies that both estimators are consistent. As a result, changes in the deterministic process
of the aggregate time effect have little effect on the estimated residuals and thus do not
matter for inference or the estimation of other coefficients.

23

Table 1.4 Results for different aggregate time effect processes. G = 6, T = 4, ngt ≈ 200,
sampling rate = .2%; 10,000 replications; Monte Carlo averages on top, Monte Carlo
standard deviations in parentheses.
MD Identity
βˇ
ηt = (t − 1)2

x2
x3
x4
c2
d2
cons

ηt = ln(t)

x2
x3
x4
c2
d2
cons

ˇ
se(β)

MD Optimal
ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.995
(0.161)
1.004
(0.089)
1.012
(0.286)
0.990
(0.191)
1.002
(0.183)
0.985
(0.168)

0.161
0.157
0.140
0.143
0.995
0.160
(0.009) (0.034) (0.042) (0.063) (0.161) (0.009)
0.089
0.087
0.082
0.088
1.004
0.089
(0.003) (0.018) (0.020) (0.032) (0.089) (0.003)
0.286
0.279
0.263
0.298
1.013
0.285
(0.012) (0.059) (0.071) (0.110) (0.286) (0.012)
0.191
0.186
0.174
0.088
0.990
0.190
(0.006) (0.039) (0.052) (0.036) (0.191) (0.006)
0.184
0.180
0.171
0.206
1.002
0.183
(0.007) (0.038) (0.046) (0.074) (0.183) (0.007)
0.168
0.164
0.150
0.127
0.985
0.167
(0.007) (0.035) (0.050) (0.047) (0.168) (0.007)

0.995
(0.161)
1.004
(0.089)
1.012
(0.286)
0.990
(0.191)
0.695
(0.183)
0.985
(0.168)

0.161
0.157
0.140
0.143
0.995
0.160
(0.009) (0.034) (0.042) (0.063) (0.161) (0.009)
0.089
0.087
0.082
0.088
1.004
0.089
(0.003) (0.018) (0.020) (0.032) (0.089) (0.003)
0.286
0.279
0.263
0.298
1.013
0.285
(0.012) (0.059) (0.071) (0.110) (0.286) (0.012)
0.191
0.186
0.174
0.088
0.990
0.190
(0.006) (0.039) (0.052) (0.036) (0.191) (0.006)
0.184
0.180
0.171
0.206
0.695
0.183
(0.007) (0.038) (0.046) (0.074) (0.183) (0.007)
0.168
0.164
0.150
0.127
0.985
0.167
(0.007) (0.035) (0.050) (0.047) (0.168) (0.007)

24

1.3.3

Covariate distributions

To understand how the distributions of the covariates affect estimation, we manipulate the
distribution of the covariates in this section. In particular, attention is paid to the covariate
x2 .
In addition to the distribution x2it ∼ N (gt/6, 1) considered in the benchmark, we look
at the following two distributions
x2it ∼ N ((gt)2 /6, 1),
x2it ∼ N (ln(gt)/6, 1).
The quadratic product of g and t embodies a greater variation than the product only, and
the product in turn embodies a greater variation than its natural log transformation. All
the other variables are generated as in the benchmark.
The results are summarized in Table 1.5. The pattern is similar to the last section.
The only difference is that the greater variation in the cohort mean of x2it in the first panel
makes the estimation of its coefficient easier, whereas in the second panel the weaker variation
renders estimation harder. Changes in the distribution of x2it have little effect on the two
MD estimators. The explanation is the same as that for the aggregate time effects. Since
both estimators are consistent, the variation in the distribution of x2it does not enter the
residuals. The optimal weighting matrix is still an identity matrix, so the performance of
the two MD estimators are similar.

1.3.4

Cohort-wise heteroskedasticity in the idiosyncratic error

In the benchmark, the idiosyncratic error uit is homoskedastic. In this section, we investigate
how cohort-wise heteroskedasticity in uit would affect estimation. We present the results for
two case in which the variance of uit depends on (g, t). Specifically, we consider

25

Table 1.5 Results for distribution of x2 . G = 6, T = 4, ngt ≈ 200, sampling rate = .2%;
10,000 replications; Monte Carlo averages on top, Monte Carlo standard deviations in
parentheses.
MD Identity
βˇ
x2 ∼

x2

N ((gt)2 /6, 1)

x3
x4
c2
d2
cons

x2 ∼

x2

N (ln(gt)/6, 1)

x3
x4
c2
d2
cons

ˇ
se(β)

MD Optimal
ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.000
0.004
0.004
0.004
0.004
1.000
0.004
(0.004) (0.000) (0.001) (0.001) (0.002) (0.004) (0.000)
1.004
0.086
0.084
0.080
0.083
1.004
0.086
(0.087) (0.003) (0.018) (0.020) (0.032) (0.087) (0.003)
1.011
0.286
0.279
0.263
0.299
1.012
0.285
(0.286) (0.012) (0.059) (0.071) (0.111) (0.286) (0.012)
0.989
0.176
0.172
0.167
0.058
0.988
0.175
(0.176) (0.004) (0.036) (0.050) (0.023) (0.176) (0.004)
1.000
0.157
0.153
0.152
0.189
1.000
0.157
(0.155) (0.004) (0.032) (0.039) (0.065) (0.156) (0.004)
0.986
0.163
0.159
0.153
0.125
0.986
0.162
(0.163) (0.004) (0.033) (0.048) (0.044) (0.163) (0.004)
1.003
1.023
0.954
0.884
0.979
1.003
1.020
(1.002) (0.252) (0.286) (0.303) (0.433) (1.002) (0.251)
1.003
0.090
0.084
0.080
0.086
1.003
0.090
(0.086) (0.009) (0.018) (0.021) (0.033) (0.086) (0.009)
1.014
0.306
0.287
0.268
0.301
1.015
0.305
(0.293) (0.031) (0.063) (0.072) (0.112) (0.294) (0.031)
0.987
0.224
0.210
0.202
0.130
0.987
0.223
(0.216) (0.040) (0.055) (0.065) (0.073) (0.216) (0.040)
0.998
0.201
0.188
0.182
0.213
0.998
0.201
(0.195) (0.038) (0.051) (0.056) (0.085) (0.195) (0.038)
0.987
0.160
0.150
0.146
0.111
0.987
0.159
(0.153) (0.015) (0.033) (0.045) (0.042) (0.153) (0.015)

26

uit ∼ N (0, 10 + (gt)2 ),
uit ∼ N (0, 10 + gt).
The degree of heteroskedasticity is greater in the first case. All the other variables are
generated in the same way as in the benchmark.
f

We note here that introducing variation in the distribution of fi = αg + εi is at most
another way to introduce heteroskedasticity. First of all, it is not interesting to vary the
deterministic process of the cohort effects αg ’s, because they are parameters to estimate.
f

Secondly, it is not interesting to vary the mean of the distribution of εi , because that would
f

only affect the process of αg ’s. Lastly, letting the variance of εi depend on g is the same as
introducing cohort-wise heteroskedasticity in uit . It does not make sense to let the variance
f

of εi depend on t because fi is time invariant.
The results in Table 1.6 show that cohort-wise heteroskedasticity has two major effects.
First, the optimal MD estimator outperforms the MD estimator with identity weighting,
especially in the top panel where uit ∼ N (0, 10 + (gt)2 ). This is because cohort-wise heteroskedasticity makes the optimal weighting matrix non-identity. Secondly, the strict increase in the variance of uit in either case raises the standard errors of both MD estimators
compared to the benchmark. This rise is universal.

1.3.5

Cohort-time cell size

In this section, we let the cohort-time cell size vary by cohort and time. Specifically, we manipulate the sampling rate so that the sample size for cohort g at time t follows approximately
the following two processes
1. ngt ≈ (200 + 180 × 1.5) − 180|g − 3.5| = 470 − 180|g − 3.5|,

g = 1, . . . , G

2. ngt ≈ (200 + 50 × 1.5) − 50|g − (3.5 − (t − 3))| = 275 − 50|g + t − 6.5|,

27

g = 1, . . . , G

Table 1.6 Results for cohort-wise heteroskedasticity in error term. G = 6, T = 4, ngt ≈ 200,
sampling rate = .2%; 10,000 replications; Monte Carlo averages on top, Monte Carlo
standard deviations in parentheses.
MD Identity
βˇ
uit ∼

x2

N (0, 10 + (gt)2 )

x3
x4
c2
d2
cons

uit ∼

x2

N (0, 10 + gt)

x3
x4
c2
d2
cons

ˇ
se(β)

MD Optimal
ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.989
0.651
0.510
0.505
0.509
0.991
0.448
(0.653) (0.051) (0.141) (0.202) (0.256) (0.448) (0.031)
1.012
0.323
0.283
0.280
0.291
1.012
0.214
(0.326) (0.019) (0.077) (0.090) (0.115) (0.214) (0.013)
1.040
1.101
0.910
0.943
1.098
1.002
0.762
(1.100) (0.062) (0.248) (0.320) (0.472) (0.758) (0.048)
0.969
0.488
0.608
0.495
0.343
0.988
0.321
(0.489) (0.040) (0.164) (0.159) (0.163) (0.316) (0.023)
1.000
0.589
0.585
0.541
0.645
1.007
0.326
(0.589) (0.046) (0.159) (0.180) (0.271) (0.324) (0.026)
0.976
0.348
0.534
0.416
0.356
0.979
0.265
(0.349) (0.030) (0.144) (0.149) (0.138) (0.263) (0.016)
0.994
0.221
0.209
0.190
0.193
0.993
0.213
(0.222) (0.013) (0.046) (0.059) (0.087) (0.215) (0.012)
1.006
0.122
0.116
0.111
0.117
1.006
0.119
(0.123) (0.005) (0.025) (0.028) (0.044) (0.120) (0.005)
1.017
0.403
0.372
0.361
0.411
1.012
0.390
(0.403) (0.018) (0.081) (0.102) (0.158) (0.391) (0.018)
0.985
0.233
0.249
0.221
0.122
0.988
0.227
(0.233) (0.009) (0.054) (0.064) (0.051) (0.227) (0.009)
1.001
0.236
0.239
0.223
0.268
1.003
0.226
(0.235) (0.011) (0.052) (0.060) (0.096) (0.226) (0.011)
0.983
0.194
0.218
0.188
0.161
0.983
0.191
(0.195) (0.008) (0.047) (0.060) (0.058) (0.192) (0.008)

28

Table 1.7 Cohort-time cell sizes for the two sampling schemes
ngt : 1

ngt : 2
t

g
1
2
3
4
5
6
col. sum
total

1
20
200
380
380
200
20
1200

2
3
20
20
200 200
380 380
380 380
200 200
20
20
1200 1200
4800

t
g
1
2
3
4
5
6
col. sum
total

4
20
200
380
380
200
20
1200

1
2
3
285 217 149
285 285 217
217 285 285
149 217 285
81
149 217
13
81
149
1030 1234 1302
4800

4
81
149
217
285
285
217
1234

In the first case, the cohort size starts from 20 at cohort 1, increases linearly with step 180
up to 380 at cohorts 3 and 4, and then decreases with the same step down to 20 at cohort
6. The idea is to let the cohorts in the middle have more observations. The overall sample
size is about 4800. The second case has approximately the same overall sample size, but the
middle peak cohorts shifts over time. The highest sample cell size is 285, and the step is 68.
The two schemes are shown in Table 1.7. Note that the changes in the two schemes are both
quite radical.
The results are summarized in Table 1.8. The impact of varying cell size is similar to that
of cohort-wise heteroskedasticity. The optimal MD estimator significantly outperforms the
MD estimator with identity weighting in both cases. This is again due to the non-identity
weighting matrix caused by the varying cell size.
The naive FE inference cannot provide satisfactory standard error estimates. Depend on
the covariate and the robust type, it can either overestimate or underestimate, and the bias
is overall large.

1.4

Conclude

Build upon the theoretical analysis in Imbens and Wooldridge (2007), we study the finite
sample properties of the MD estimator for pseudo panels in this chapter. In particular, we

29

Table 1.8 Results for varying cohort size. G = 6, T = 4; ngt follows the three specifications
given in section 1.3.5 and is generated by varying the sampling rate; 10,000 replications;
Monte Carlo averages on top, Monte Carlo standard deviations in parentheses.
MD Identity
βˇ
ngt : 1

x2
x3
x4
c2
d2
cons

ngt : 2

x2
x3
x4
c2
d2
cons

ˇ
se(β)

MD Optimal
ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.993
(0.443)
1.004
(0.155)
1.005
(0.482)
0.993
(0.468)
1.005
(0.453)
0.983
(0.457)

0.444
0.298
0.328
0.285
1.008
0.282
(0.067) (0.097) (0.127) (0.157) (0.289) (0.027)
0.155
0.166
0.148
0.162
1.002
0.095
(0.017) (0.049) (0.045) (0.068) (0.095) (0.006)
0.480
0.529
0.457
0.538
1.020
0.308
(0.062) (0.156) (0.149) (0.226) (0.311) (0.020)
0.472
0.362
0.383
0.177
0.984
0.412
(0.063) (0.107) (0.146) (0.100) (0.436) (0.039)
0.455
0.347
0.370
0.424
0.987
0.263
(0.063) (0.105) (0.140) (0.207) (0.266) (0.021)
0.461
0.319
0.353
0.232
0.992
0.398
(0.070) (0.094) (0.154) (0.109) (0.422) (0.039)

0.993
(0.306)
1.005
(0.100)
1.008
(0.438)
0.994
(0.444)
1.003
(0.421)
0.983
(0.473)

0.303
0.196
0.205
0.220
0.998
0.220
(0.065) (0.062) (0.090) (0.121) (0.226) (0.018)
0.099
0.108
0.099
0.108
1.004
0.088
(0.008) (0.032) (0.029) (0.045) (0.088) (0.004)
0.431
0.350
0.342
0.415
1.015
0.308
(0.086) (0.106) (0.126) (0.186) (0.312) (0.017)
0.437
0.236
0.286
0.145
0.983
0.271
(0.099) (0.072) (0.140) (0.084) (0.279) (0.016)
0.418
0.228
0.272
0.354
0.996
0.265
(0.095) (0.071) (0.131) (0.194) (0.271) (0.019)
0.466
0.209
0.276
0.205
0.992
0.289
(0.118) (0.065) (0.158) (0.109) (0.298) (0.021)

30

focus on the comparison of the optimal MD estimator and the MD estimator with identity
weighting matrix. The latter is of interest because it coincides with the FE estimator applied to the pseudo panel of cohort means. We find that in cases where there is significant
heteroskedasticity by cohort-time cells, or in cases where the cohort-time cell size varies, the
optimal MD estimator significantly outperform the MD estimator with identity weighting
in that the former’s standard errors are smaller. This finding is consistent with the large
cohort size asymptotics under the MD estimation framework, as the optimal MD estimator
achieves the smallest asymptotic variance.
We also compare the MD FE inference to the naive FE inference. We find that in cases
where the optimal weighting matrix is close to an identity matrix, the naive FE standard
errors are barely satisfactory. But when the optimal weighting matrix is far from identity,
the naive FE standard errors are not acceptable without doubt. In any case, the MD FE
inference is always more efficient than the naive FE inference. This finding is consistent
with the fact that the FE inference relies on large number of cohorts and it discards all
individual-level information. In a setup with small number of cohorts and time periods, the
naive FE inference cannot work well.
The simulation setup in this analysis considers sample cohort sizes in hundreds, and
the results are already promising provided that the variation in covariate cohort means are
rich enough. In practice, sample cohort sizes of repeated cross sections can easily exceed
thousands. Therefore, the results in this chapter should bring confidence to the application
of the MD approach to pseudo panels.
In future studies, we could extend the analysis to dynamic models where we have lagged
dependent or explanatory variables. For robustness check, we should allow correlation covariates and individual fixed effects. Moreover, given the weak exogeneity condition (1.7),
we could also allow covariates that are endogenous at the individual level but not at the
cohort level. Results from these extensions can provide more practical implications.

31

BIBLIOGRAPHY

32

BIBLIOGRAPHY

Arellano, Manuel, and Stephen Bond. 1991. “Some tests of specification for panel
data: Monte Carlo evidence and an application to employment equations.” The review of
economic studies, 58(2): 277–297.
Collado, M Dolores. 1997. “Estimating dynamic models from time series of independent
cross-sections.” Journal of Econometrics, 82(1): 37–62.
Deaton, Angus. 1985. “Panel data from time series of cross-sections.” Journal of econometrics, 30(1): 109–126.
Girma, Sourafel. 2000. “A quasi-differencing approach to dynamic modelling from a time
series of independent cross-sections.” Journal of Econometrics, 98(2): 365–383.
Imbens, Guido, and Jeffrey M Wooldridge. 2007. What’s new in econometrics? NBER.
Moffitt, Robert. 1993. “Identification and estimation of dynamic models with a time series
of repeated cross-sections.” Journal of Econometrics, 59(1): 99–123.
Verbeek, Marno. 2008. “Pseudo-panels and repeated cross-sections.” In The Econometrics
of Panel Data. 369–383. Springer.
Verbeek, Marno, and Francis Vella. 2005. “Estimating dynamic models from repeated
cross-sections.” Journal of econometrics, 127(1): 83–102.
White, Halbert. 1980. “A heteroskedasticity-consistent covariance matrix estimator and
a direct test for heteroskedasticity.” Econometrica: Journal of the Econometric Society,
817–838.
Wooldridge, Jeffrey M. 2010. Econometric Analysis of Cross Section and Panel Data. .
2nd ed., Boston MA:MIT Press.

33

CHAPTER 2
EXPLORING ADDITIONAL MOMENT CONDITIONS IN
NON-SEPARABLE MINIMUM DISTANCE ESTIMATION WITH AN
APPLICATION TO PSEUDO PANELS

2.1

Introduction

Minimum distance (MD) estimation is a useful approach to recover structural estimates from
reduced form estimates when there exists a known relationship between the structural and
reduced form parameters. The known relationship is often in the form of structural equations,
moment conditions,1 or restrictions, which are terminology used interchangeably hereafter.
When applying MD, researchers may encounter situations in which they need to introduce
additional moment conditions into estimation. This could happen, for example, when some
new instrument variables (IVs) become available as the research proceeds. An important
question to ask in such a situation is whether we can always improve asymptotic efficiency
by using all the moment conditions than using just part of them. In this chapter, we provide
an affirmative answer to this question. We show that in MD estimation it never hurts to
have more moment conditions. In particular, when the additional moment conditions are
non-redundant, adding them to estimation strictly improves efficiency. This efficiency gain
result echoes the similar property for generalized method of moments (GMM) in Breusch
et al. (1999).
The motivation for deriving this efficiency gain result comes from the need in pseudo
panel models to incorporate external IVs. A pseudo panel model can estimate an underlying
unobserved effect panel data model with only repeated cross sections. The idea, which
dates back to Deaton (1985), is to divide the population into a number of groups by certain
predetermined group membership such as age cohorts. Then the group averages of the
1 More

precisely, the moment conditions in MD are conditional moment conditions.

34

variables can be used to construct a panel at the group level. Since the group averages
are error ridden estimates, Deaton suggests to treat the estimation as a measurement error
problem. In this chapter, we adopt the MD perspective proposed by Imbens and Wooldridge
(2007). Within the MD framework, the group averages of the variables are the reduced
form estimates, and the group averages of the panel data model are the structural equations
linking the reduced form to the structural parameters. When new IVs become available,
the additional set of structural equations induced by the IVs can be easily added to the
estimation. Clearly, the aim of having more structural equations is to improve estimation
efficiency. However, there is no such theory in MD estimation telling us whether efficiency
gain can be achieved. Therefore, we attempt to derive such a result in this chapter to fill
this gap.
We derive the result within a so called non-separable minimum distance (NMD) framework developed in this chapter. The framework is a special case of the “high level” MD
framework in Newey and McFadden (1994).2 The key difference between NMD and the
high level MD framework is that NMD models the reduced form parameters explicitly. This
feature makes the NMD framework convenient to use when our exact purpose is to recover
structural estimates from reduced form estimates. The qualifier “non-separable” highlights
NMD’s capability to deal with structural equations that are non-separable in the structural and reduced form parameters. Note, however, that the separable framework, i.e. the
Classical Minimum Distance (CMD) framework, is still covered as a special case.
We establish consistency and asymptotic normality within the NMD framework. We also
derive the optimal weighting matrix for the over-identified case in which the the number of
structural equations is greater than that of the structural parameters. The optimal weighting
matrix turns out to be the asymptotic variance of the rescaled structural equations, which
gives an intuitive explanation of the weighting procedure. That is, the optimal weighting
2 In

effect, the MD framework in Newey and McFadden (1994) is so general that both
generalized method of moments (GMM) and classical minimum distance (CMD) are its
special cases.
35

matrix readjusts the relative importance of the conditions according to their own volatility
as well as their correlation with each other. Building on these basic results, we then give the
main efficiency result discussed at the beginning of the chapter.
After the general results are established in the NMD framework, we apply them back to
the case of pseudo panels with external IVs. We show that a pseudo panel NMD estimator with an arbitrary weighting matrix is a generalized least squares (GLS) estimator. The
inverse of the optimal weighting matrix corresponds to the usual unconditional variancecovariance matrix in GLS estimation. As a result of the added structural equations, the
optimal weighting matrix becomes block diagonal. This result generalizes the finding in Imbens and Wooldridge (2007) that the optimal weighting matrix is diagonal in the case without
external IVs. The inclusion of extra IVs in pseudo panel models also highlights a typical case
where the optimal weighting matrix should be used over the naive identity matrix. In the
first chapter, we have shown that varying cohort sizes and cohort-wise heteroskedasticity in
idiosyncratic errors are two typical causes of a non-identity yet diagonal optimal weighting
matrix. When IVs are added, the optimal weighting matrix is usually block diagonal since
within-cohort dependence between structural equations generally exists. As a result, it is
more likely to achieve efficiency gain by using the optimal weighing matrix.
A related question is whether we can estimate pseudo panel models naively by applying fixed effect on the sample cohort means and then making the inference robust to heteroskedasticity and/or serial correlation. In this chapter, we show that the naive fixed effect
coefficient estimator is still valid because it coincides with the NMD estimator using the
identity weighting matrix. But the naive inference, whether made robust or not, is invalid
because it is different from the correct NMD inference. The fundamental reason of the difference is that the naive inference only uses the cohort averages and ignores any individual
level information. In terms of asymptotic theory, the naive inference requires the number
of cohorts tend to infinity and the number of time periods remain fixed (see, e.g., Arellano
(1987); Wooldridge (2010); Hansen (2007a)), or both tend to infinity (Kezdi (2003); Hansen

36

(2007b)). In pseudo panel models, however, we often have large cohort sizes but fixed numbers of cohorts and time periods. This fact makes the MD framework a natural fit to the
pseudo panel models.
As mentioned in Verbeek (2008), repeated cross sections have several advantages over
panel data sets. Because it is usually easier and less costly to collect random samples than
panel data, the sample sizes of repeated cross sections are often much larger than common
panel data sets. Moreover, repeated cross sections are naturally immune to attrition which
is a common issue for panel data. Therefore, the availability of the NMD approach to
pseudo panels potentially opens many new research opportunities in cases where unobserved
individual fixed effects are a concern.
The rest of the chapter is organized as follows. In section 2, we lay out the NMD framework. The consistency and asymptotic normality the NMD estimator, the optimal weighting
matrix, and the property that more moment conditions do not hurt are discussed. In section
3, we apply the NMD framework to pseudo panel models with additional instruments. There
are also two special subsections in which we discuss the GLS perspective and the naive variance estimators. Section 4 contains a simulation study of the pseudo panel NMD estimators.
The last section concludes.

2.2

The NMD framework

Minimum distance is essentially a delta method - it recovers structural estimates from reduced form estimates when there exists a known set of structural equations that links the
structural and reduced form parameters. Formally, let Π × Θ be an subset of RP × RK ,
which is the product space for the reduced form parameter π and the structural parameter
θ. Let h : Π × Θ → RJ be an vector-valued function satisfying
h(π 0 , θ 0 ) = 0

37

(2.1)

for some true parameter value (π 0 , θ 0 ) ∈ Π × Θ. Hereafter, h is referred to as the structural
function, and eq. (2.1) is the set of structural equations. Suppose there is an estimator
p
ˆ of θ 0 is defined as
ˆ → π 0 . Then an NMD estimator θ
π

ˆ = argmin h(π,
ˆ h(π,
ˆ θ).
ˆ θ) W
θ

(2.2)

θ∈Θ
p
ˆ is a J-dimensional positive semi-definite matrix and W
ˆ →
where W
W.

2.2.1

Consistency

A consistency result for NMD is summarized in the following theorem (similar to Theorem
2.6 in Newey and McFadden (1994)):
p
p
ˆ →
ˆ → π0, W
Theorem 1. Suppose that π
W, and (i) (Identification) W is positive semi-

definite and Wh(π 0 , θ) = 0 only if θ = θ 0 ; (ii) (Boundedness) θ 0 ∈ Θ, which is compact;
(iii) (Continuity) hj (π, θ) is continuous on Π and on Θ, for j = 1, · · · , J; (iv) (Uniform
p
p
ˆ→
ˆ θ) − hj (π 0 , θ)| → 0, for j = 1, · · · , J. Then θ
convergence) supθ∈Θ |hj (π,
θ0 .

Proof. See Appendix.

2.2.2

Asymptotic normality

Theorem 3.2 in Newey and McFadden (1994) requires

√

d

ˆ θ 0 ) → N (0, Ω), which denh(π,

mands effort to verify when h(π, θ) takes on some general functional form. If in addition
continuous differentiability of h(π, θ) with respect to π is assumed, a Taylor expansion of
√
d
ˆ θ 0 ) → N (0, Ω) holds. The verification
ˆ θ 0 ) around π 0 can be used to verify that nh(π,
h(π,
step however could be saved with the establishment of the following theorem.
p
p
ˆ satisfies (2.2), θ
ˆ→
ˆ →
Theorem 2. Suppose that θ
θ0 , W
W where W is positive semi-

definite, and
(i) π 0 ∈ interior(Π) and θ 0 ∈ interior(Θ);

38

(ii) h(π, θ) is continuously differentiable with respect to θ in a neighborhood N (θ 0 ) of
θ 0 , and h(π, θ) is continuously differentiable with respect to π in a neighborhood N (π 0 ) of
π0;
(iii)

√

d

ˆ − π 0 ) → N (0, Ω);
n (π
p

ˆ θ) − L(π 0 , θ) → 0, and for B(π, θ) :=
(iv) For L(π, θ) ≡ ∇θ h(π, θ), supθ∈N (θ ) L(π,
0
p

ˆ θ) − B(π 0 , θ) → 0;
∇π h(π, θ), supθ∈N (θ ) B(π,
0
(v) L WL is nonsingular, where L ≡ L(π 0 , θ 0 ).
Let B ≡ B(π 0 , θ 0 ). Then
√
d
−1
−1
ˆ − θ0 →
).
L WBΩB WL L WL
n θ
N (0, L WL

(2.3)

Proof. See Appendix.
With the presence of the added smoothness assumption with respect to π, the theorem
above provides a more constructive and straightforward version of the “high level” theorem
in Newey and McFadden (1994). To obtain the asymptotic variance of the MD estimator,
all we need is to find the two partial derivatives of h and plugging them in eq. (2.3).

2.2.3

Optimal weighting matrix

The asymptotic variance in (2.3) depends on the probability limit W of the weighting matrix
ˆ When W = M−1 where
W.
M = BΩB

(2.4)

the asymptotic variance simplifies to
Avar

√
ˆ − θ0
n θ

= (L (BΩB )−1 L)−1 .

(2.5)

As shown in the following theorem, the inverse of BΩB is the optimal weighting matrix
since (2.5) is the “smallest” asymptotic variance that can be obtained by optimizing over all
possible nonsingular weighting matrices.

39

p
ˆ →
Theorem 3. Suppose M = BΩB is nonsingular. Then an NMD estimator with W

W = M−1 is asymptotically efficient in the class of NMD estimators based on the same set
of structural eqations.
Proof. See Appendix.
The intuition for using an optimal weighting matrix is straightforward. Asymptotically, it
√
is not about over-identification. Rather, it is because the the conditions in nh(π 0 , θ 0 ) = 0
are asymptotically random. More accurate conditions exhibit less volatility, and the conditions are potentially correlated. To use all the conditions optimally, more weights should
be given to less volatile conditions, and the correlation between conditions should also be
accounted for.
The best characterization of the relative volatility of all conditions is the the asymptotic
variance-covariance matrix of the rescaled conditions. It turns out that M is exactly that
variance-covariance matrix . Specifically, the first part of condition (ii) in Theorem 2 and a
Taylor expansion imply that
√
√
ˆ θ 0 ) = B · n (π
ˆ − π 0 ) + op (1),
nh(π,
d

→ N (0, BΩB ).

(2.6)

The optimal weighting operation is essentially a standardization that assigns more loadings to less volatile conditions and untangles the correlation between conditions. It standardizes the asymptotic variance-covariance matrix to an identity matrix. Admittedly, that
the inverse of the optimal weighting matrix is the asymptotic variance is a known result
which can be found in, e.g., Newey and McFadden (1994), and the idea of standardization
by volatility can also be found in the generalized method of moments (GMM) and generalized least squares (GLS) literature. However, this intuitive explanation is often overlooked
when it comes to MD estimation.

40

In the application to pseudo panel, the intuition is even clearer, for M is exactly the
variance-covariance matrix of individual level residuals.

2.2.4

Estimation

ˆ for π 0 , the NMD estimator using the identity weighting
Given a consistent estimator π
matrix, i.e.
ˇ = argmin h(π,
ˆ θ) h(π,
ˆ θ),
θ
θ∈Θ

ˇ follows from Theorem 1. By
can be used as an initial estimator for θ 0 . Consistency of θ
p
ˇ →
ˇ = ∇π h(π,
ˆ θ)
B. Then,
continuity of the partial derivatives, the plug-in estimator B

ˆ for Ω, M
ˆ ≡B
ˇΩ
ˆB
ˇ is a consistent estimator for M, and an
given a consistent estimator Ω
asymptotically efficient for θ 0 can be obtained by
ˆ opt = argmin h(π,
ˆ −1 h(π,
ˆ θ) M
ˆ θ).
θ
θ∈Θ

The corresponding consistent estimator for the asymptotic variance-covariance matrix is
given by
ˆ
Avar(θ

opt

ˆ −1 /n
ˆ (B
ˆΩ
ˆB
ˆ )−1 L)
) = (L

ˇ and L
ˇ
ˆ ≡ ∇π h(π,
ˆ ≡ ∇π h(π,
ˆ θ)
ˆ θ).
where B
The estimator defined above iterates only once. Multiple iterations are also allowed.
They are, however, asymptotically equivalent.

2.2.5

More conditions do not hurt

Partition the restrictions in (2.1) into two parts:


h1 (π 0 , θ 0 )
h(π 0 , θ 0 ) = 
 = 0,
h2 (π 0 , θ 0 )
where h1 is J1 × 1, h2 is J2 × 1, and J1 + J2 = J. Let
˜ = argmin h1 (π,
ˆ θ) M−1
ˆ θ)
θ
1,1 h1 (π,
θ∈Θ

41

(2.7)

with M1,1 = B1 ΩB1 for L1 ≡ L1 (π 0 , θ 0 ) ≡ ∇θ h1 (π 0 , θ 0 ), B1 ≡ B1 (π 0 , θ 0 ) ≡ ∇π h1 (π 0 , θ 0 ).
Then
√

d

˜ − θ 0 ) → N (0, [L (B1 ΩB )−1 L1 ]−1 ).
n(θ
1
1

On the other hand, if all restrictions are used, we have
√
d
ˆ opt − θ 0 →
n θ
N (0, [L (BΩB )−1 L]−1 ).
ˆ is at least as efficient as θ.
˜ The theorem
The following theorem shows that asymptotically θ
as well as the proof is similar to its GMM counterpart in Breusch et al. (1999).
Theorem 4. Let Li ≡ Li (π 0 , θ 0 ) ≡ ∇θ hi (π 0 , θ 0 ) and Bi ≡ Bi (π 0 , θ 0 ) ≡ ∇π hi (π 0 , θ 0 )
for i = 1, 2. Let Mi,j = Bi ΩBj for i = 1, 2 and j = 1, 2. Assume BΩB and B1 ΩB1 are
both nonsingular. Let F = M2,2 − M2,1 M−1
1,1 M1,2 . Then
L (BΩB )−1 L − L1 (B1 ΩB1 )−1 L1
= M2,1 M−1
1,1 L1 − L2

F−1 M2,1 M−1
1,1 L1 − L2

and thus is positive semi-definite.
Proof. See Appendix.
The condition h2 (π 0 , θ 0 ) = 0 is redundant if L2 = M2,1 M−1
1,1 L1 , i.e.
L2 = B2 ΩB1 (B1 ΩB1 )−1 L1 .

(2.8)

We can think of Φ = (B1 ΩB1 )−1 B1 ΩB2 as the coefficient matrix from the GLS of B2 on
B1 with weight Ω. Then h2 (π 0 , θ 0 ) = 0 is redundant if L2 is a linear transformation of L1
with the transformation matrix Φ .
Eq. (2.8) is similar to condition (C) of Theorem 1 in Breusch et al. (1999). We can
also derive a condition similar to condition (B) in that theorem to have a more intuitive
explanation of the redundancy condition. Specifically, define
√

ˆ θ0 ) ≡
nr2 (π,

√

√
ˆ θ 0 ) − M2,1 M−1
ˆ θ 0 ).
nh2 (π,
1,1 nh1 (π,
42

(2.9)

√
ˆ θ 0 ), and M2,1 is the asymptotic
nh1 (π,
√
ˆ θ 0 ) and h1 (π,
ˆ θ 0 ). Therefore, asymptotically, M2,1 M−1
ˆ θ 0 ) is
covariance of h2 (π,
1,1 nh1 (π,
√
√
√
ˆ θ 0 ) on nh1 (π,
ˆ θ 0 ), and nr2 (π,
ˆ θ 0 ) is the residual in
the linear projection of nh2 (π,
By eq. (2.6), M1,1 is the asymptotic variance of

this linear projection. It follows that a redundancy condition that is equivalent to but more
intuitive than eq. (2.8) is
∇θ

√

ˆ θ 0 ) ≡ ∇θ
nr2 (π,

√
√
ˆ θ 0 ) − M2,1 M−1
ˆ θ 0 ) = 0.
nh2 (π,
1,1 nh1 (π,

ˆ θ 0 ) = 0 to be redundant is that r2 (π,
ˆ θ 0 ) is marginally
That is, the condition for h2 (π,
uninformative for θ 0 .

2.3

Pseudo panels with additional IVs

In the case of pseudo panels with additional IVs, the restrictions defined by (2.1) are not
additively separable in π and θ as in the CMD case. Therefore it serves as a good example
to illustrate the NMD framework. Moreover, the first-order condition takes the form of the
normal equation of a GLS estimation. Therefore, the optimal NMD estimator in this case
turns out to be a GLS estimator using the optimal weighting matrix as the unconditional
variance-covariance matrix.
The adoption of the MD perspective in pseudo panel models provides a new way to
deal with errors in variables. In the seminal work of Deaton (1985), this issue is treated
as a measurement error problem. By specifying the measurement error structure, Deaton
proposes a measurement-error corrected estimator. Collado (1997) follows the measurement
error thinking and extends Deaton’s method to a more general measurement-error corrected
GMM estimator. In the MD framework, group averages are treated as estimates for the
reduced form parameters. Since the group sizes are usually large for repeated cross sections,
the MD framework is a natural fit for pseudo panel models.
In the following subsections we go through the derivation of the particular contents of
h, π, θ , L, B and M in the pseudo panel case, discuss estimation, and summarize the
43

asymptotics.

2.3.1

Population model and structural equations

Formally, for outcome yit , covariate xit (1 × K), coefficient β (K × 1), time effect ηt , fixed
effect fi , and idiosyncratic error uit , consider the following model for a generic individual in
the population
yit = xit β + ηt + fi + uit , t = 1, . . . , T.

(2.10)

fi and uit are unobserved. The first entry of xit is unity for notation convenience. Essentially,
we are thinking of the population as a genuine panel data set from which different samples
are drawn each period. The same treatment is also adopted in Verbeek and Vella (2005).
Let zit = (1, z2it , · · · , zP it ) be a 1 × P row vector of instrumental variables satisfying
E(zit uit |fi ) = 0,
Cov(zpit , fi |gi ) = 0,

p = 1, 2, · · · , P,

(2.11)
(2.12)

where, for convenience, z1it is also set to 1.
In a standard panel, the conditional exogeneity of xit given fi is usually assumed:
E(uit |xit , fi ) = 0, t = 1, . . . , T.

(2.13)

This condition is not required here. A weaker condition that suffices is
E(uit |fi ) = 0, t = 1, . . . , T.

(2.14)

Note that by iterated expectation, (2.13) implies (2.14). Because fi aggregates all timeconstant unobservables, we should think of (2.13) and (2.14) as being true for not only the
lump sum fi but also any time constant factors. In particular, replacing fi with the group
indicator gi (i.e. applying iterated expectation) leads to
E(uit |gi ) = 0, t = 1, . . . , T.
44

(2.15)

Let E(·|g) be the shorthanded notation for E(·|gi = g), and let αg = E(fi |g) be the group
fixed effect for group g. By (2.12) and the fact that
E(zpit · fi |g) = Cov(zpit , fi |g) + E(zpit |g) · E(fi |g), p = 1, · · · , P,
the structural model follows as
E(zit yit |g) =E(zit xit |g)β + E(zit |g)ηt + E(zit |g)αg ,

(2.16)

f or t = 1, . . . , T ; g = 1, · · · , G.
Thanks to z1it = 1, the first row in eq. (2.16) represents the cohort level equations without
instruments, which is the basic case studied in Imbens and Wooldridge (2007).
The exogeneity condition (2.15) might appear non-substantial at the first glance, because
it seems we can always make E(uit |gi ) = 0 holds by subtracting E(uit |gi ) from uit and
redefine the deviation as uit . But this subtraction operation is equivalent to the inclusion
of a full set of cohort-time effects in the structural model (2.16). Perhaps the following
equivalent representation of (2.15) makes the explanation clearer
δgt = E(uit |gi = g) = 0, g = 1, . . . , G, t = 1, . . . , T.
If eq.

(2.17)

(2.17) (or equivalently (2.15)) is not imposed, the GT parameters δgt for g =

1, . . . , G, t = 1, . . . , T will enter the structural model (2.16) as the full set of cohort-time
effects. Including the full set of cohort-time effects is equivalent to not imposing (2.17) (or
(2.15)). Therefore, the key assumption disguised by eq. (2.15) together with the specification
in (2.10) is that the structural model (2.16) requires only the set of group and time effects
(ηt and αg ) but not the full set of cohort-time effects (δgt ). If any such cohort-time effect
is required, then, as pointed out in Imbens and Wooldridge (2007), one way to think about
the misspecification is that some δgt = E(uit |gi = g) is not zero.
Note that the structural model with the full set of cohort-time effects is always correctly
specified, but it is not interesting because the variation in the covariate cohort means is
45

absorbed by δgt . As a result, such a the model is only identified up to the GT cohort-time
effects.
A technical point here is that, due to the setup x1it = z1it = 1, there are only (G − 1)
parameters in αg and (T − 1) in ηt to estimate. Imbens and Wooldridge (2007) make the
normalization

G
g=1 αg

= 0 and η1 = 0. This chapter however proceeds with α1 = 0 and

η1 = 0. The purpose of this slightly different normalization is to cope with the estimation
convention that the dummies for the first cohort and the first time period are always dropped.
As a result of the dropout, the sum (β1 + α1 + η1 ) is identified, but β1 , α1 and η1 are not
separately identifiable. The remaining estimated group and time effects are the relative
effects (αg − α1 ) for g = 2, · · · , G and (ηt − η1 ) for t = 2, · · · , T . Setting α1 = η1 = 0 then
conveniently simplifies (β1 + α1 + η1 ), (αg − α1 ) and (ηt − η1 ) to β1 , αg and ηt .

2.3.2

Useful notations

Some notations are useful later. Let µxgt = E(xit |g) denote the population mean of a generic
variable xit conditional on gi = g. For a vector (e.g. xit ) or a matrix (e.g. zit xit ) variable,
zx
bold symbols like µx
gt or µgt will be used. In this notation, eq. (2.16) can be written as
zy

0 = −µgt + µzgtx β + µzgt (ηt + αg ),

(2.18)

f or t = 1, . . . , T ; g = 1, · · · , G.
Also, for a generic variable xit and j = (g − 1)T + t, let µx denote the column “vector” with
µxgt the jth row block. Depending on the dimension of xit , µx can be either a column vector
or a matrix.
Let vit = (yit , xit ) and sit = zit ⊗ vit with ⊗ denotes Kronecker product. sit is a long
row vector. Assume the variance-covariance matrix of sit exists and is denoted by
Ωsgt = V ar(sit |g).
An explicit formula for Ωsgt is given in Appendix.
46

(2.19)

Then the reduced form parameter π in this pseudo panel case can be expressed as
π = µs = (µs11 , µs12 , · · · , µsGT ) .
The structural parameter is θ = (β , η , α ) with α = (α1 , . . . , αG ) and η = (η1 , . . . , ηT ) .
Moreover, the right hand side of eq. (2.18) says the j-th row block of the h function is
zy

hj (π, θ) = −µgt + µzgtx β + µzgt (ηt + αg )

(2.20)

with j = (g − 1)T + t. Note that each hj (π, θ) is P × 1. Let xit = (xit , d, c) with d the
vector of time dummies and c the vector of group dummies. Then a second useful expression
for hj is
zy

zx

hj (π, θ) = −µgt + µgt θ.

(2.21)

Later we will see that the two expressions (2.20) and (2.21) are convenient for calculating
partial derivatives of h.

2.3.3

The partial derivatives L and B and the inverse optimal weighting matrix
M

By eq. (2.21), it is trivial that
L = ∇θ h(π, θ) = µz x

(2.22)
zx

where, as defined in the last section, µz x is the matrix with µgt the j-th row block for
j = (g − 1)T + t.
z

z

On the other hand, recall eq. (2.20). For z1it = 1, µgt1 = 1, define ∇ z1 [µgt1 (ηt + αg )] =
µ
gt

(ηt + αg ). Define β gt by replacing the first entry of β, i.e. β1 , with (β1 + ηt + αg ), and
let bgt (θ) = IP ⊗ (−1, β gt ) with IP the P -dimensional identity matrix. Some algebra (see
Appendix ) then shows that



bgt (θ), if g˜ = g and t˜ = t,

∇π ˜hj (π, θ) =
g˜t


0,

otherwise.
47

Define another block diagonal matrix b(θ) by putting bgt (θ) on its gt-th diagonal block.
Then
B = ∇π h(π, θ) = b(θ).

(2.23)

With the general formula in eq. (2.4) and the particular contents in eq. (2.19) and (2.23),
the inverse of the optimal weighting matrix, M, is given by
M = b(θ)Ωs b(θ) .

(2.24)

In the Appendix, we show that an expansion of the right hand side of eq. (2.24) leads to
M = diagGT [(ρ1 κ1 )−1 b11 (θ)Ωs11 b11 (θ) , (ρ1 κ2 )−1 b12 (θ)Ωs12 b12 (θ) , · · ·
· · · , (ρG κT )−1 bGT (θ)ΩsGT bGT (θ) ].

(2.25)

That is, M is a block diagonal matrix with (ρg κt )−1 bgt (θ)Ωsgt bgt (θ) on the gt-th diagonal
block.
We also show in the Appendix that bgt (θ)Ωsgt bgt (θ) is actually the variance-covariance
matrix of the composite errors within cell (g, t)
bgt (θ)Ωsgt bgt (θ) = Ξgt ≡ V ar[zit yit − zit xit β − zit (ηt + αg )|g].

(2.26)

Therefore another useful expression for M is
M = diagGT (ρ1 κ1 )−1 Ξ11 , (ρ1 κ2 )−1 Ξ12 , · · · , (ρG κT )−1 ΞGT .

2.3.4

(2.27)

Estimation

Assume we have T repeated cross-sectional random samples denoted by
{(yit , xit , zit , git ), i = 1, · · · , nt ; t = 1, · · · T }
where nt is the number of observations for cross section t. Note that in each time period we
have a new random sample, so in general the same index i refers to different individuals in
different time periods, and thus git sees a subscript t added.
48

2.3.4.1

ˆ
Asymptotics of π

Let 1A be the indicator function equal to 1 if A is true and equal to 0 otherwise. Let
rit = (rit,1 , rit,2 , . . . , rit,G ) be a vector of group indicators with rit,g = 1{git = g} where
1{·} is the indicator function equal to one if the event in {·} is ture. In this way the group
membership of the random draw i at time t is properly treated as a random variable. It
follows that the number of observations in cell (g, t) is also a random variable given by
ngt =

nt
i=1 rit,g .

Let µ
ˆxgt denote the sample average within cell (g, t) for a generic variable xit . Let ρg =
p

P (rit,g = 1) be the fraction of the population in cohort g and assume ρˆgt = ngt /nt → ρg .
Let κt = limn→∞ nt /n be the fraction of all observations accounted for by cross section t.
By (essentially) the central limit theorem, for g = 1, · · · , G and t = 1, · · · , T ,
√

d

ˆ sgt − µsgt ) → N ormal(0, (ρg κt )−1 Ωsgt ).
nt (µ

ˆ = (µ
ˆ s11 , µ
ˆ s12 , · · · , µ
ˆ sGT ) and π 0 = (µsgt , µsgt , · · · , µsgt ) . Then the results
Furthermore, let π
above can be stacked in
√
d
ˆ − π 0 ) → N (0, Ωs )
n (π
where Ωs is a block diagonal matrix with (ρg κt )−1 Ωsgt on the gt-th diagonal block.

2.3.4.2

Estimation of L

Eq. (2.22) suggests that a straightforward estimator for L is
ˆ =µ
ˆ z x.
L

(2.28)

ˆ z x is the sample analog of µz x . Recall that xit = (xit , d, c) and that xit contains a
µ
ˆ z x is the matrix of the sample cohort means of the explanatory
constant term. Then µ
variables, the instruments, and their interactions. Its dimension is GP T × (K + G + T − 2).

49

2.3.4.3

ˆ and the FE estimator θ
ˇ
The general estimator θ

There exists an analytical solution to eq. (2.2) in the current setup, which turns out to be
a GLS estimator.
Specifically, given eq.(2.21) and (2.28), the first-order condition to eq. (2.2) in the current
setup can be written as3
ˆ µ
ˆ z xθ − µ
ˆ z y = 0.
ˆ z x) W
(µ
ˆµ
ˆ z x) W
ˆ z x is nonsingular, then the general pseudo panel NMD estimator with
Assume (µ
ˆ is given by
a weighting matrix W
ˆ = (µ
ˆµ
ˆ z x) W
ˆz x
θ

−1

ˆµ
ˆ z x) W
ˆz y
(µ

(2.29)

ˆ serves as the inverse of the
Clearly, (2.29) is of the form of a GLS estimator where W
ˆz x
“unconditional variance-covariance matrix of the error term”, and the cohort means µ
ˆ z y are the matrix of right-hand-side variables and left-hand-side variable, receptively.
and µ
ˆ with the identity matrix gives the fixed effect estimator
In particular, replacing W
ˇ = (µ
ˆz x
ˆ z x) µ
θ

−1

ˆ z x) µ
ˆ z y.
(µ

(2.30)

The standard case without instrument in Imbens and Wooldridge (2007) corresponds to the
case P = 1, i.e. deleting the letter z in eq. (2.30).

2.3.4.4

ˆ opt
Estimation of B, M and θ

ˇ as an initial estimator, an estimator for B follows from eq. (2.23) by substituting θ
With θ
ˇ which leads to
with θ
ˇ
ˆ = b(θ).
B
An obvious estimator for the variance-covariance matrix of s defined in (2.19) is
ˆ s = n−1
Ω
gt
gt

nt

ˆ sgt ) (sit − µ
ˆ sgt ).
rit,g (sit − µ
i=1

3 Note

ˆ z x ) is the transpose of µ
ˆ z x and is not the same as µ
ˆ x z.
that (µ
50

ˆ s , can be defined as the block diagonal matrix with the gt-th
Then an estimator for Ωs , Ω
s

ˆ , i.e.,
diagonal block (ngt /n)−1 Ω
gt
ˆ s ).
ˆ s , · · · , (nGT /n)−1 Ω
ˆ s , (n12 /n)−1 Ω
ˆ s = diagGT ((n11 /n)−1 Ω
Ω
GT
12
11
ˆ and Ω
ˆ s , the following estimator for the inverse of the optimal weighting matrix
Given B
follows from eq. (2.24)
ˇ Ω
ˇ .
ˆ = b(θ)
ˆ s b(θ)
M

(2.31)

Eq. (2.31), however, may involve big matrices in calculation when the number of covariates and/or instruments is large (the dimension of s increase quickly with multiple instruments). Fortunately, eq. (2.27) provides an alternative but numerically equivalent way to
estimate M - all we need is an estimator for Ξgt . By eq. (2.26), Ξgt can be conveniently
ˆ gt , the sample variance-covariance matrix of the residuals in cell (g, t) to be
estimated by Ξ
defined as follows.
ˇ to obtain the individual residual
First, using the fixed effect estimator θ
ˇ − (ˇ
uˇit = yit − xit β
ηt + α
ˇ g ).

(2.32)

The cohort residual is then defined as
zp u
ˇ
µ
ˆgt

=

n−1
gt

nt

rit,g zpit uˇit .

(2.33)

i=1

For p, q = 1, · · · , P , let τˆpq (drop subscript g, t from τ for simplicity) denote the entry on
ˆ gt . Then τˆpq is given by
row p, column q of Ξ
τˆpq =

n−1
gt

nt

zp u
ˇ

rit,g zpit uˇit − µ
ˆgt
i=1

ˆ gt is defined as the matrix with the pq-th entry τˆpq
Finally, Ξ
ˆ gt = (ˆ
Ξ
τpq ).

51

zq u
ˇ

zqit uˇit − µ
ˆgt

.

(2.34)

ˆ gt , the second method to estimate M is via
Given Ξ
ˆ GT ]
ˆ 12 , · · · , (nGT /n)−1 Ξ
ˆ 11 , (n12 /n)−1 Ξ
ˆ = diagGT [(n11 /n)−1 Ξ
M

(2.35)

ˆ gt .
which is the block diagonal matrix with the gt-th diagonal block (ngt /n)−1 Ξ
The numerical equivalence of the two estimators for M is summarized in the following
theorem.
ˆ defined in eq. (2.31) and (2.35) are numerically
Theorem 5. The two ways of computing M
equivalent.
Proof. See Appendix.
When p = q = 1, τˆ11 = n−1
gt

nt
i=1 rit,g

uˇit − µ
ˇugt

2

which is of the same form as the τˆ2

ˆ becomes a diagonal matrix that coincides
defined in Imbens and Wooldridge (2007),4 and M
ˆ in Imbens and Wooldridge (2007).
with the matrix C
ˆ in hand, by replacing W
ˆ with M
ˆ −1 in eq. (2.29), the optimal pseudo panel
With M
NMD estimator is obtained as
ˆ opt = (µ
ˆ −1 µ
ˆz x
ˆ z x) M
θ

−1

ˆ −1 µ
ˆ z x) M
ˆ z y.
(µ

(2.36)

The above formula in its appearance is similar to a GLS estimator on the cohort level data.
ˆ is computed
But the weighting matrix is not the usual one used by a feasible GLS because M
ˆ opt to
from individual level data. For more detail about the connection and difference of θ
GLS, see the next section.
4 The

formula in Imbens and Wooldridge (2007) needs the correction of demeaning. Because in STATA, the command for calculating the sample variance automatically demeans
the residuals.

52

2.3.4.5

ˆ θ
ˇ and θ
ˆ opt
Estimation of the asymptotic variances of θ,

ˆ
With all the pieces worked out, and by Theorem 2, the asymptotic variance estimator for θ
is
ˆ = (µ
ˆµ
ˆ z x) W
ˆz x
Avar(θ)

−1

ˆM
ˆW
ˆµ
ˆ z x) W
ˆz x
(µ

ˆµ
ˆ z x) W
ˆz x
(µ

−1

/n.

ˇ it is
For θ,
ˇ = (µ
ˆ z x) µ
ˆz x
Avar(θ)

−1

ˆµ
ˆ z x) M
ˆz x
(µ

ˆ z x) µ
ˆz x
(µ

−1

/n.

(2.37)

ˆ opt , it is
Finally, for θ
ˆ
Avar(θ

opt

ˆ −1 µ
ˆ z x) M
ˆz x
(µ

) =

−1

/n.

With the presence of additional IVs, dependence between restrictions are introduced since
each cohort repeats itself several times in the restrictions. The optimal weighing matrix is
ˆ gt ). In fact, some
more likely to be non-diagonal (it is block diagonal with block (ngt /n)−1 Ξ
algebra (see Appendix) shows that another expression for Ξgt is
f

Ξgt = E (εi + uit )2 zit zit |g .

(2.38)

where
f

εi ≡ fi − αg

(2.39)

is the deviation of individual effect from its cohort mean. Without further assumptions
f

regarding the correlation between the quadratic terms (εi + uit )2 and zit zit , and the correlation among the IVs in zit , Ξgt is generally non-diagonal. As a result, the use of optimal
weighting matrix becomes more important with the presence of additional IVs.

2.3.5

The GLS perspective

ˆ opt and its relation to GLS, define the
To better understand the relationship between θ
individual composite error as
f

eit ≡ yit − xit β − (ηt + αg ) = εi + uit .
53

The residual uˇit given in (2.32) is obviously an consistent estimator for eit . With eit , the
vector of individual composite errors in cohort g is zit eit , and an alternative expression for
Ξgt in (2.26) is
Ξgt = V ar[zit eit |g].

(2.40)

For a given g, define the cohort composite error as
zp e
µ
ˆgt

=

n−1
gt

nt

rit,g zpit eit .

(2.41)

i=1

ˆ zgte is similarly defined and represents the vector of cohort composite errors in cell (g, t).
µ
ˆ zgte conditional on g is given by
The variance-covariance matrix of µ
ˆ zgte |g] = n−1
V ar[µ
gt Ξgt .

(2.42)

From the MD perspective, ngt is large, and n−1
gt Ξgt → 0 as ngt → ∞. It thus does not
make sense to model and estimate the “cohort composite errors” because they degenerate to
0 asymptotically.
The usual feasible GLS on the pseudo panel of cohort means ignores individual level data
and relies on much stringent assumptions on the unconditional variance-covariance structure
of the cohort composite error. In particular, the underlying asymptotics rely on large G.
The GLS estimator in eq. (2.36) is apparently not the usual feasible GLS. Rather, it is an
GLS imposing the following block diagonal variance-covariance structure of all the cohort
composite errors
−1
−1
diagGT [n−1
11 Ξ11 , n12 Ξ12 , · · · , nGT ΞGT ]

(2.43)

Eq. (2.43) contains GT P (P + 1)/2 parameters, and thus is never feasible if only the GT P
cohort means are observed. But if the individual level data are available, eq. (2.43) can be
well estimated by
−1 ˆ
−1 ˆ
−1 ˆ
ˆ
diagGT [n−1
11 Ξ11 , n12 Ξ12 , nGT ΞGT ] = n M.

54

(2.44)

ˆ in a GLS formula leads to eq. (2.36); the n−1 cancels off. From the GLS
Using n−1 M
ˆ −1 standardizes the sample cohort composite errors so
perspective, the weighting by n−1 M
that they become close to uncorrelated and homoskedastic.
It is worth noting that eq. (2.43) is not the unconditional variance-covariance matrix of
ˆ z e . On each diagonal block is eq. (2.42), the conditional variance-covariance matrix of
µ
ˆ zgte given g.
µ
From the MD perspective, there is no asymptotic variance-covariance matrix for the
ˆ −1 → 0
ˆ z e ] → 0 as ngt → ∞; so is n−1 M
sample cohort composite errors because V ar[µ
as ngt → ∞. Rather, what matters is the following set of relocated and rescaled estimated
structural equations,
0=−
=−

√
√

zy

zy

ˆ gt − µgt ) +
n(µ

√

ˆ zgtx − µzgtx )β +
n(µ

√
ˆ zgt − µzgt )(ηt + αg ),
n(µ

ˆ zgte ,
nµ

(2.45)
(2.46)

f or t = 1, . . . , T ; g = 1, · · · , G.
The above equation is obtained by manipulating eq. (2.18). Asymptotically, eq. 2.46
√ ze
ˆ gt is exactly M.
converges to GT P random restrictions. The asymptotic variance of nµ
Therefore, to use all the random conditions efficiently, the random restrictions need to be
ˆ −1 , but their
weighted by the square root of M−1 . In estimation, M−1 is replaced by M
function is equivalent asymptotically. Given fixed G, T and P , and ngt → ∞, the use of
ˆ −1 is totally legit since M
ˆ −1 does not converge to 0. The weight M
ˆ −1 adjusts the relative
M
importance of each sample restriction according to its level of accuracy. The level of accuracy
ˆ gt .
for the gt-th sample restriction is measured by (ngt /n)−1 Ξ

2.3.6

ˇ
Naive variance estimators for θ

ˇ is the fixed effect estimator on the pseudo panel of the sample cohort means,
Because θ
it is also convenient to compute the usual asymptotic variance estimators for a fixed effect

55

estimator. These naive estimators, however, are generally incorrect because they only make
use of the sample cohort means.
Before listing the formulae for the naive variance estimators, we summarize several possible reasons in repeated random cross sections that may ruin their validity. We cite the
reasons that apply to the breakdown of each estimator in later discussion.
zp u
ˇ

zq u
ˇ

1. µ
ˆgt and µ
ˆgt are generally correlated (dependence over p for fixed g and t)
zp u
ˇ

zp u
ˇ

zq u
ˇ

2. the variance of µ
ˆgt , as well as the covariance of µ
ˆgt and µ
ˆgt , depends on zit (heteroskedasticity)
zp u
ˇ

3. µ
ˆgt depends on g because of either zit or even uit itself depends on g (non-identical
distribution over g)
Among the three items, the last one is the most crucial because all the naive variance
estimators discussed below rely on large G.
We consider three naive asymptotic variance estimators. Their formulae in a standard
model can be found in Wooldridge (2010) as well as other textbooks. The first is the nonrobust variance estimator for which the consistency relies on a scalar (proportional to an
identity matrix) variance-covariance structure of the cohort composite errors. To obtain this
zp u
ˇ

formula, recall the definition of µ
ˆgt

in eq. (2.33). Define the mean squared error for the

pseudo panel as
zp u
ˇ

σ
ˇ 2 = (GT P − K − G − T + 2)−1

(ˆ
µgt )2 .
g,t,p

Then the naive non-robust variance estimator can be written as
−1

x z zp x
ˆ gt 
ˆ gt p µ
µ

ˇ =σ
Avarn (θ)
ˇ2 

.

g,t,p

The subscript n in typewriter font stands for “non-robust”. Its validity hings on i.i.d. samzp u
ˇ

pling over (g, t, p) and homoskedasticity of µ
ˆgt , neither of which holds in a pseudo panel of
sample cohort means due to all three reason listed.
56

The second is the naive heteroskedasticity-robust variance estimator whose formula is
given by
−1 



x z zp x
ˆ gt 
ˆ gt p µ
µ

ˇ =
Avarr (θ)

zp u
ˇ
x z zp x
ˆ gt p µ
ˆ gt  
(ˆ
µgt )2 µ



g,t,p

g,t,p

−1



x z zp x
ˆ gt 
ˆ gt p µ
µ

.

g,t,p

The subscript r stands for “robust”. The estimator is robust to heteroskedasticity in the
zp e

cohort composite error µ
ˆgt . But its validity still relies on i.i.d. sampling over (g, t, p) which
does not hold due to reasons 1 and 3 mentioned above.
The third is the naive cluster-robust variance estimator and its formula is
−1 


x z zp x
ˆ gt 
ˆ gt p µ
µ

ˇ =
Avarc (θ)

zp u
ˇ u
ˇzq x zp zq x
ˆ gt µ
ˆ gr  
µ
ˆgt µ
ˆgr µ


g,t,r,p,q

g,t,p

−1

x z zp x
ˆ gt p µ
ˆ gt 
µ

.

g,t,p

The middle term can also be written as
zp u
ˇ u
ˇzq x zp zq x
ˆ gr =
ˆ gt µ
µ
ˆgt µ
ˆgr µ

xz

z u
ˇ

g,t,p

g,t,r,p,q

ˇzq x zp zq x
zp u
ˇ u
ˆ gt µ
ˆ gr
ˆgr µ
µ
ˆgt µ

z x

p
ˆ gt p µ
ˆ gtp +
(ˆ
µgt )2 µ
g,t=r,p=q

where the first sum is exactly the middle term in the naive heteroskedasticity-robust variance
estimator. The naive cluster-robust variance estimator is robust to arbitrary heteroskedasˆ zg e . But its validity relies on
ticity and serial correlation in the cohort composite errors µ
i.i.d. sampling over g which may not hold due to reason 3 listed above.
Some other equivalent representations of the three naive variance estimators are inforˇ Write the three naive estimators as
mative of their link to Avar(θ).
ˇ = (µ
ˆ z x) µ
ˆz x
Avarn (θ)
ˇ = (µ
ˆ z x) µ
ˆz x
Avarr (θ)
ˇ = (µ
ˆ z x) µ
ˆz x
Avarc (θ)

−1

−1

−1

ˆ z x ) (ˇ
ˆz x
(µ
σ 2 I)µ
ˆ z uˇ )
diag(µ

ˆ z x)
(µ

2

ˆ z x) µ
ˆz x
(µ
ˆz x
µ

−1

,

ˆ z x) µ
ˆz x
(µ

ˆ z x ) diagG (µ
ˆ z uˇ )diagG (µ
ˆ z uˇ ) µ
ˆz x
(µ

−1

,

ˆ z x) µ
ˆz x
(µ

−1

,

ˆ z uˇ ) is the square, diagonal matrix created by putting the vector µ
ˆ z uˇ on the
where diag(µ
ˆ z uˇ ) is the block diagonal matrix with the gth diagonal block
principal diagonal, and diagG (µ
ˆ zg uˇ for g = 1, · · · , G. Then clearly, the three naive variance estimators can be obtained
µ
57

ˆ
by replacing M/n
in eq. (2.37) with (ˇ
σ 2 I),

2

ˆ z uˇ )
diag(µ

ˆ z uˇ )diagG (µ
ˆ z uˇ ) ,
or diagG (µ

respectively.
Yet another set of equivalent representations provide some insights on the large G perspective of the naive estimators. Specifically, write
−1

ˆ gz x )
(µ

ˇ =
Avarn (θ)

−1

ˆ zg x )
(µ

ˆ zg x
µ
g

g

ˇ =
Avarr (θ)

ˆ gz x )
(µ

ˆ zg x
µ

2

ˆ zg uˇ )
diag(µ

ˆ zg x )
(µ

ˆ zg x
µ

ˇ =
Avarc (θ)

−1

ˆ zg x
µ

ˆ gz x )
(µ

ˆ gz x
µ

ˆ zg uˇ (µ
ˆ zg uˇ )
µ

ˆ zg x )
(µ

g

g

,

g

−1

ˆ zg x )
(µ

,
−1

ˆ gz x
µ

g

g

ˆ zg x
µ

g

−1

ˆ gz x )
(µ

ˆ gz x )
(µ

ˆ zg x
(ˇ
σ 2 IT P )µ

ˆ gz x
µ

.

g

(2.47)
ˆ zg uˇ ) is the square, diagonal matrix created by putting the vector µ
ˆ zg uˇ on the
where diag(µ
principal diagonal. In essence, the three naive variance estimators differ in estimating (treat
g as random)
ˆ gz x ) µ
ˆ zg e (µ
ˆ zg e ) µ
ˆ gz x ,
E (µ
i.e. the middle term of the sandwich-form. But they all need i.i.d. sampling over g, which
is not satisfied in the MD framework due to reason 3 listed above. The estimation errors in
the cohort means are also ignored.
ˇ and Avar(θ).
ˇ
The last point we want to make is about the relationship between Avarc (θ)
First, rewrite
−1

ˆ zg x )
(µ

ˇ =
Avar(θ)

−1

ˆ zg x
µ

ˆ gz x )
(µ

ˆ
(n−1 M

ˆ gz x
g )µ

g

g

ˆ zg x )
(µ

ˆ gz x
µ

.

g

(2.48)
ˇ and
ˆ g are the the only difference between Avarc (θ)
ˆ zg uˇ (µ
ˆ zg uˇ ) and n−1 M
It is then clear that µ
ˇ Notice that µ
ˆg=
ˆ zg uˇ (µ
ˆ zg uˇ ) ≈ diag(µ
ˆ zg1uˇ (µ
ˆ zg1uˇ ) , · · · , µ
ˆ zgTuˇ (µ
ˆ zgTuˇ ) ) and that n−1 M
Avar(θ).
zp u
ˇ zq u
ˇ
ˆ g1 , · · · , n−1 Ξ
ˆ gT ). Moreover, notice that µ
ˆ z uˇ (µ
ˆ z uˇ ) = (ˆ
diag(n−1 Ξ
µ µ
ˆ )p,q and that
g1

gt

gT

58

gt

gt

gt

−1 τ ) . Therefore, the comparison boils down to the difference between
ˆ
n−1
pq p,q
gt Ξgt = ngt (ˆ
zp u
ˇ zq u
ˇ
µ
ˆgt µ
ˆgt

=

n−1
gt

nt

n−1
gt

rit,g zpit uˇit
i=1

5 and

n−1
gt τˆpq

=

n−2
gt

nt

rit,g zqit uˇit
i=1

nt

zp u
ˇ

rit,g zpit uˇit − µ
ˆgt

zq u
ˇ

zqit uˇit − µ
ˆgt

.

i=1
zp u
ˇ zq u
ˇ
ˇ uses µ
That is, Avarc (θ)
ˆgt µ
ˆgt to approximate the covariance between the cohort composite
zp e

zq e

errors µ
ˆgt and µ
ˆgt , which uses only cohort-level information, and is not an estimator for
zp e

zq e

zp u
ˇ zq u
ˇ

Cov(ˆ
µgt , µ
ˆgt |g) because µ
ˆgt µ
ˆgt is observed only once for given g, t, p. On the other hand,
ˇ uses τˆpq to estimate the covariance between the individual composite errors zp e and
Avar(θ)
zq e, which uses individual-level information, and is indeed an estimator for Cov(zp e, zq e|g)
because τˆpq averages over ngt observations. The additional n−1
gt then transform it to a
zp e

zq e

legitimate estimator for Cov(ˆ
µgt , µ
ˆgt |g). Apparently, n−1
gt τˆpq is a better estimator for
zp e

zq e

ˇ
zp u
ˇ zq u

ˆgt .
ˆgt µ
ˆgt |g) than µ
Cov(ˆ
µgt , µ
ˇ can only make
What conclusion do we get from this comparison? First of all, Avarc (θ)
sense if we have random sample over g, because eq. (2.47) averages over g. Second, a
relatively large number of groups is also needed for the large G asymptotics to work. In the
ˇ because the residuals are all 0. Third, Avarc (θ)
ˇ
just-identified case, there is no Avarc (θ)
ˆ zgte degenerates to 0. This is
also needs fixed ngt , otherwise the cohort composite error µ
however not too much a problem because in a sample ngt is always finite.

2.4

Simulation

This section contains a simulation study for the optimal NMD estimator and the NMD
estimator with identity matrix (i.e. the FE estimator) in the pseudo panel case with instruˇ
zp u
ˇ zq u

5 Note

zp u
ˇ zq u
ˇ

that we do not need the formula above to calculate µ
ˆgt µ
ˆgt ; µ
ˆgt µ
ˆgt can be
obtained from calculating the cohort-level residuals. The formula is to provide an insight of
its relationship to n−1
gt τˆpq .
59

ments. The major purposes of this simulation study are (i) to illustrate that the formulae
derived in the last section work when the model are correctly specified, and (ii) to show that
valid instruments improve estimation efficiency. We also look at naive ways of computing the
standard errors that only make use of the cohort level data. Their performance is compared
to the NMD standard errors, and explanations for the difference are provided.

2.4.1

Simulation design

Throughout the simulation study, the outcome yit is generated as a linear function of the
covariates (x1it = 1, x2it , x3it , x4it ), the time effect ηt , the individual effect fi , and the
idiosyncratic error uit :
yit = β1 + β2 x2it + β3 x3it + β4 x4it + ηt + fi + uit , i = 1, · · · , Nt , t = 1, · · · , T.

(2.49)

The parameter values used are β = (β1 , β2 , β3 , β4 ) = (1, 1, 1, 1). The time effects are generated by ηt = t − 1, and the cohort effects are generated by αg = g − 1. Individual
fixed effects are generated by adding a random normal disturbance to the cohort effects, i.e.
fi ∼ N (αg , 1). To fix ideas, it might be helpful to think of x2it , x3it and x4it as education,
experience and marital status, respectively. The outcome yit is the log hourly wage, and
there is an individual effect fi representing some unobserved ability.
We focus on estimating the coefficient of x2it for which the distribution is given later.
The distributions of the two auxiliary variables x3it and x4it are given by
x3it ∼ N (sin(gt), 1),
x4it ∼ Bernouli

1
1 + exp[1.5 sin(gt/2)]

.

That is, x3it is a continuous variable with population cohort mean sin(gt) and within cell
1
variance 1; x4it is a binary variable equal to 1 with probability 1+exp(1.5∗sin(gt/2))
. Since the

individual-level disturbance to x3it and x4it are independently generated, they are always
valid IVs. A time-invariant external instrument is generated as zi ∼ N (0, 1).
60

We investigate a small pseudo panel (G = 6, T = 4) and a middle sized one (G = 30,
T = 20). In the small pseudo panel, the idiosyncratic error uit follows N (0, 10), and the
following 5 cases for x2it are considered
1. x2it ∼ N (gt/6, 1),
2. x2it ∼ N (gt/6, 1) + fi ,
3. x2it ∼ N (gt/6, 1) + zi ,
4. x2it ∼ N (gt/6, 1) + zi + fi ,
5. x2it ∼ N (gt/2, 1) + zi + fi .
x

The standard deviation for µgt2 over (g, t) is about 1. Note that x2it is a valid IV in cases 1
through 3, but not valid in cases 4 and 5.
In the middle sized pseudo panel, uit follows N (0, 100) which has a bigger variance than
in the small pseudo panel. The five cases considered for x2it are
1. x2it ∼ N (gt/150, 1),
2. x2it ∼ N (gt/150, 1) + fi ,
3. x2it ∼ N (gt/150, 1) + zi ,
4. x2it ∼ N (gt/150, 1) + zi + fi ,
5. x2it ∼ N (gt/50, 1) + zi + fi ,
x

The standard deviation for µgt2 over (g, t) is about 23. The variance-covariance as well as
x

x

x

correlation matrix of (µgt2 , µgt3 , µgt4 ) are given in Table 2.1.
Case 4 in each setup is the case of major interest. Cases 1 through 3 are used to isolate
the effect of adding fi or zi as part of x2it . Case 5 checks the effect of a larger variation in
the cohort mean of x2it .

61

x

x

x

Table 2.1 Variance-covariance and correlation matrix of (µgt2 , µgt3 , µgt4 ); correlation
x
x
coefficients in parentheses. µgt3 = sin(gt), µgt4 = (1 + exp[1.5 ∗ sin(gt/2)])−1 .
G = 6, T = 4;
x
µgt2 = gt/6
x

x

µgt2
x

µgt2
x

µgt3
x

µgt4

µgt3

G = 30, T = 20;
x
µgt2 = gt/150
x

x

µgt4
x

µgt2

1.078
(1)
-0.142
(-0.195)

0.488
(1)

0.107
(0.444)

0.018
(0.109)

x

µgt3
x

µgt4

0.054
(1)

x

x

µgt2
x

µgt3
x

µgt4

x

µgt3

x

-0.013
(-0.020)

0.507
(1)

0.009
(0.040)

-0.001
(-0.006)

x

µgt4

µgt2
x

µgt2
0.488
(1)

0.322
(0.444)

0.018
(0.109)

x

µgt4

0.056
(1)

G = 30, T = 20;
x
µgt2 = gt/50

9.701
(1)
-0.425
(-0.195)

µgt3

0.834
(1)

G = 6, T = 4;
x
µgt2 = gt/2
µgt2

x

µgt2

x

µgt3
x

µgt4

0.054
(1)

x

µgt3

x

µgt4

7.508
(1)
-0.039
(-0.020)

0.507
(1)

0.026
(0.040)

-0.001
(-0.006)

0.056
(1)

In the ideal situation, we would like to draw a population of size infinity so that the
cohort level population equations hold exactly. Take case 4 as an example, that would mean
the following set of equations holds exactly
E(yit |g) = β1 + β2

gt
1
+ β3 sin(gt) + β4
+ (t − 1) + (g − 1), (2.50)
150
1 + exp[1.5 sin(gt/2)]
g = 1, · · · , G; t = 1, · · · , T.

But drawing an infinite number of observations is obviously infeasible. Therefore, we choose
a relatively large number as the population size. The true distribution of the resulting
population is of course its empirical distribution, but we could think of it as an approximation
62

of the population defined by eq. (2.49). Eq. (2.50) also holds only approximately, but the
difference should be negligible for the purpose of this simulation study.
In the current setup, the population cohort sizes are set equally to Ngt = 105 for all g
and t. That means a population panel of N = 2.4 × 106 individual-time points for G = 6
and T = 4, and N = 6 × 107 for G = 30 and T = 20.
After the population is generated, we fix it over simulations. In each replication, we
draw repeated random cross sections from this fixed population. To have an idea on how
the sample size affects the estimates, we consider two different sampling rates, 0.2% and 1%,
which result in the sample cohort sizes ngt = 200 and 1000, respectively.6
The simulation design above is careful in the two places emphasized by Imbens and
Wooldridge (2007). First, data for each section is drawn from the population independently
across time, and because of the random sampling in each period, the group identifier is also
randomly drawn. Second, eq. (2.49) has full time effects which is more realistic than Verbeek
and Vella (2005) that omits the aggregate time effects, for the variation in µx
gt here is net of
the time effects.
For θ = (β, η, α) in eq. (2.49), we consider the NMD estimator with identity matrix
ˇ and its standard error (s.e.), and the optimal NMD estimator (θ)
ˆ and its s.e.. Because
(θ)
ˇ is the fixed effect estimator on the pseudo panel of sample cohort means, three naive s.e.
θ
estimators, namely the non-robust s.e., the heteroskedasticity-robust s.e. and the clusterrobust s.e., are also computed. They are the usual s.e. estimators routinely computed for
the fixed effect estimator in a true panel, but are naive in a pseudo panel because they
treat the sample cohort means as observations carrying no errors and completely ignore the
individual-level data.
Besides the basic NMD uses no IV, each of z, x2 , x3 and x4 is used one at a time as the
6A

relatively higher sampling rate might introduce too much overlap among the repeated
cross-sectional samples. Therefore, we also consider the setup Nt = 1.5 ∗ 107 with sampling
rate 0.2%. The result shows that there is no essential difference from the setup Nt = 3 ∗ 103
with sampling rate 1%.

63

additional IV. The NMD using all 4 variables as the additional IVs is also estimated.

2.4.2

Simulation results for the small pseudo panel

At the center of the simulation study is Case 4, as cases 1 through 3 are its simplified cases to
pin down the effect of the correlation of x2it with zi and fi , and Case 5 is a variation of Case
4 that increases the variation in the cohort mean of x2 . Therefore we focus on analyzing Case
4 in this section. The Monte Carlo simulation results for case 4 from 1000 replications for
the coefficient and s.e. estimators of x2 are presented in Table 2.3. Two sample cohort sizes,
ngt = 200 and 1000, are considered. For each considered quantity, the Monte Carlo average
and standard deviation over the 1000 replications are reported, with the standard deviation
in parentheses. The estimators with no IV, z as IV, and x2 as IV are picked because they
provide most of the insights. The results on the same quantities in Case 3 are reported in
Table 2.2. Detailed results are in Tables B.1 through B.20 in Appendix B.
Several observations stand out from Table 2.3. First of all, the NMD coefficient estimators
work well in all cases except when the invalid IV x2 is used. Both βˇ2 and βˆ2 are close to
the true value in columns 1, 2, 4 and 5. As the sample cohort size ngt gets bigger, the
slight biases in βˇ2 and βˆ2 get even smaller, and their Monte Carlo standard deviations also
shrink. Second, the NMD s.e. estimators also work well, even when x2 is used as the IV. The
Monte Carlo averages of se(βˇ2 ) and se(βˆ2 ) are close to the standard deviations of βˇ2 and
βˆ2 throughout all columns, and having a bigger cohort size, as expected, reduces se(βˇ2 ) and
se(βˆ2 ) universally. Third, using a valid and relevant IV improves efficiency, but the validity
of IV is crucial. Compared to using no IV (column 1 and 4), using z as IV (columns 2 and 5)
leads to reduced Monte Carlo averages of se(βˇ2 ) and se(βˆ2 ) and smaller finite sample bias in
βˇ2 and βˆ2 . The usage of the invalid IV x2 (columns 3 and 6), however, introduces persistent
biases in βˇ2 and βˆ2 that do not vanish as cohort size gets larger. Note that x2 is not a valid
IV because it is correlated with fi , which violates the condition in (2.12).
A comparison with the results in Table 2.2 confirms that the correlation between x2it and
64

Table 2.2 Finite sample properties of various estimators of β2 and its standard error,
G = 6, T = 4. Case 3. x2it ∼ N (gt/6, 1) + zi
ngt = 200

ngt = 1000

none

z

x2

none

z

x2

βˇ2

.9909
(.1623)

.9980
(.0430)

.9965
(.0421)

.9937
(.0714)

.9982
(.0191)

.9964
(.0191)

se(βˇ2 )

.1590
(.0117)

.0463
(.0012)

.0436
(.0020)

.0720
(.0023)

.0206
(.0002)

.0192
(.0004)

sen (βˇ2 )

.1552
(.0349)

.0457
(.0054)

.0488
(.0092)

.0698
(.0144)

.0205
(.0025)

.0218
(.0039)

ser (βˇ2 )

.1393
(.0425)

.0508
(.0073)

.0490
(.0094)

.0627
(.0181)

.0229
(.0034)

.0217
(.0040)

sec (βˇ2 )

.1423
(.0603)

.0496
(.0159)

.0431
(.0154)

.0639
(.0283)

.0223
(.0073)

.0189
(.0069)

βˆ2

.9911
(.1623)

.9979
(.0437)

.9981
(.0324)

.9936
(.0713)

.9982
(.0191)

.9987
(.0143)

se(βˆ2 )

.1585
(.0117)

.0452
(.0012)

.0327
(.0007)

.0720
(.0023)

.0205
(.0002)

.0148
(.0001)

fi invalidates x2it as IV. In absence of the correlation between x2it and fi , x2it is exogenous
and becomes a valid IV for itself. As a result, no obvious bias is observed in βˇ2 and βˆ2 when
x2 is used as IV in Table 2.2. In effect, x2 is a better IV than z, since Table 2.2 shows that
se(βˇ2 ) and se(βˆ2 ) become smaller on average when the IV is changed from z to x2 . This
makes sense because no IV is more relevant to a variable than the variable itself.
When the IV is changed from z to x2 in Table 2.2, a larger reduction is observed in se(βˆ2 )
than in se(βˇ2 ). This observation highlights a typical situation to use the optimal weighting
matrix - when the IV brings in within-cell heteroskedasticity and correlation. Specifically,
when z is the IV, we show in the Appendix that




1 0 
Ξgt = σe2 
,
0 1
f

(2.51)

where σe2 = E (εi + uit )2 |g . This implies that the optimal weighting matrix is proportional
65

Table 2.3 Finite sample properties of various estimators of β2 and its standard error,
G = 6, T = 4. Case 4. x2it ∼ N (gt/6, 1) + zi + fi
ngt = 200

ngt = 1000

none

z

x2

none

z

x2

βˇ2

1.0153
(.1599)

1.0048
(.0431)

1.2218
(.0947)

.9989
(.0716)

.9996
(.0191)

1.2166
(.0423)

se(βˇ2 )

.1575
(.0143)

.0462
(.0014)

.0951
(.0071)

.0719
(.0029)

.0206
(.0003)

.0421
(.0014)

sen (βˇ2 )

.1537
(.0356)

.0455
(.0054)

.0728
(.0188)

.0697
(.0145)

.0205
(.0025)

.0405
(.0082)

ser (βˇ2 )

.1388
(.0427)

.0506
(.0074)

.0752
(.0191)

.0627
(.0181)

.0229
(.0034)

.0458
(.0106)

sec (βˇ2 )

.1440
(.0631)

.0494
(.0159)

.0947
(.0420)

.0642
(.0284)

.0223
(.0073)

.0711
(.0254)

βˆ2

1.0155
(.1598)

1.0048
(.0437)

1.3194
(.0266)

.9989
(.0715)

.9996
(.0191)

1.3220
(.0120)

se(βˆ2 )

.1569
(.0142)

.0451
(.0014)

.0266
(.0006)

.0719
(.0029)

.0205
(.0003)

.0120
(.0001)

to an identity matrix, which explains why the averages of se(βˆ2 ) and se(βˇ2 ) are close to each
other and to the standard deviations of βˇ2 and βˆ2 in Table 2.2 when z is the IV. On the
other hand, when x2 is used as the IV, we show that


gt
 1

150
Ξgt = σe2 
,
gt
gt 2
150 2 + 150

(2.52)

which has within-cell heteroskedasticity and correlation. The resulting optimal weighting
matrix is distinct from an identity matrix.
The results on the naive s.e. estimators, sen (βˇ2 ), ser (βˇ2 ) and sec (βˇ2 ) are also consistent
with the theory. We leave the discussion to the next subsection because the pattern is more
obvious when G is greater.

66

Table 2.4 Finite sample properties of various estimators of β2 and its standard error,
G = 30, T = 20. Case 3. x2it ∼ N (gt/150, 1) + zi
ngt = 200

ngt = 1000

none

z

x2

none

z

x2

βˇ2

1.0134
(.0839)

1.0022
(.0277)

1.0012
(.0248)

.9980
(.0399)

.9996
(.0121)

.9996
(.0106)

se(βˇ2 )

.0842
(.0011)

.0277
(.0001)

.0239
(.0002)

.0387
(.0002)

.0123
(.0000)

.0105
(.0000)

sen (βˇ2 )

.0842
(.0028)

.0274
(.0006)

.0286
(.0009)

.0388
(.0012)

.0123
(.0003)

.0129
(.0004)

ser (βˇ2 )

.0838
(.0045)

.0281
(.0008)

.0269
(.0009)

.0387
(.0021)

.0125
(.0003)

.0119
(.0004)

sec (βˇ2 )

.0841
(.0143)

.0282
(.0037)

.0241
(.0035)

.0385
(.0067)

.0124
(.0016)

.0105
(.0016)

βˆ2

1.0134
(.0843)

1.0020
(.0279)

1.0016
(.0209)

.9979
(.0400)

.9995
(.0121)

.9993
(.0087)

se(βˆ2 )

.0837
(.0011)

.0269
(.0001)

.0196
(.0001)

.0387
(.0002)

.0123
(.0000)

.0089
(.0000)

2.4.3

Simulation results for the middle sized pseudo panel

The results in Table 2.5 and 2.4 for the middle sized pseudo panel basically tell the same
story as the small pseudo panel. We focus on two points that stand out. These two points
are less clear, although also exist, in the small pseudo panel.
First, the results on the naive s.e. estimators, sen (βˇ2 ), ser (βˇ2 ) and sec (βˇ2 ) are consistent
with the theory. This is best seen from the last column in Table 2.4. Moving down the list
sen (βˇ2 ), ser (βˇ2 ) and sec (βˇ2 ), the bias in the Monte Carlo averages gradually declines. The
Monte Carlo average of sec (βˇ2 ) rounded four decimal places is even identical to that of
se(βˇ2 ). The reason is that, when x2 is used as IV,
−1
ˆ zg e |g] = diag(n−1
V ar[µµ
g1 Ξg1 , · · · , ngT ΞgT )

is indeed block diagonal by eq. (2.52). Among the three naive variance estimators, only the

67

Table 2.5 Finite sample properties of various estimators of β2 and its standard error,
G = 30, T = 20. Case 4. x2it ∼ N (gt/150, 1) + zi + fi
ngt = 200

ngt = 1000

none

z

x2

none

z

x2

βˇ2

1.0496
(.0830)

1.0106
(.0276)

1.1137
(.1376)

1.0061
(.0395)

1.0013
(.0120)

1.0285
(.0642)

se(βˇ2 )

.0826
(.0012)

.0276
(.0002)

.1339
(.0026)

.0386
(.0003)

.0123
(.0000)

.0633
(.0006)

sen (βˇ2 )

.0826
(.0028)

.0273
(.0006)

.0783
(.0037)

.0387
(.0012)

.0123
(.0003)

.0382
(.0016)

ser (βˇ2 )

.0822
(.0045)

.0280
(.0008)

.1249
(.0111)

.0385
(.0020)

.0125
(.0003)

.0594
(.0051)

sec (βˇ2 )

.0824
(.0139)

.0281
(.0037)

.1222
(.0302)

.0384
(.0066)

.0124
(.0016)

.0585
(.0151)

βˆ2

1.0493
(.0835)

1.0104
(.0279)

1.3186
(.0175)

1.0060
(.0395)

1.0013
(.0120)

1.3199
(.0071)

se(βˆ2 )

.0821
(.0012)

.0268
(.0002)

.0162
(.0001)

.0385
(.0003)

.0122
(.0000)

.0073
(.0000)

cluster-robust version correctly accounts for the variance-covariance structure of the cohort
composite error. The heteroskedasticity-robust version only captures the heteroskedasticity
but not the within-cluster correlation. The non-robust version accounts for neither.
As a comparison, in the 4th and 5th columns of Table 2.4, the Monte Carlo averages of
ˆ zg e |g] is
sen (βˇ2 ), ser (βˇ2 ) and sec (βˇ2 ) are all close to that of se(βˇ2 ). This is because V ar[µ
proportional to an identity matrix when none or z is used as IV. As a result, all three versions
ˆ zg e |g]. Moreover,
of the naive variance estimators are correct in their modeling of V ar[µ
from sen (βˇ2 ) through ser (βˇ2 ) to sec (βˇ2 ), the s.e. estimators become less and less efficient,
indicated by greater and greater Monte Carlo standard deviations. This is also consistent
with their well-known relative efficiency property. Of course, se(βˇ2 ) is much more efficient
than any of the naive estimators, for se(βˇ2 ) makes use of the extra information from the
individual-level data.

68

Secondly, the first column in Table 2.5 shows noticeable bias in βˇ2 and βˆ2 . It is finite
sample bias because as ngt gets larger, the bias shrinks quickly. A comparison with the first
column in Table 2.4 confirms that the correlation between x2it and fi contributes to a large
part of the bias.

2.5

Concluding remarks

This chapter develops a general NMD framework that imposes (partial) differentiability on
the structural equations. The differentiability conditions are stronger than the MD framework in Newey and McFadden (1994), but the resulting framework is more convenient to
work with in application. Consistency and asymptotic normality are established, as well as
the optimal weighting matrix expressed as functions of the partial derivatives of the structural equations. A theorem that echoes the GMM property that more moment conditions
do not hurt is given. The general framework is then applied to the special case of pseudo
panel. Simulation results are consistent with the theory.
The property that having more moment conditions could improve efficiency is first noticed
in the exercise of adding external instruments in pseudo panel MD estimation, which is
an extension to the work on pseudo panel by Imbens and Wooldridge (2007). We would
like to establish this property in a more general setup, hence the NMD framework at the
beginning of this chapter is motivated. Having both the general framework and the case
of pseudo panel as an example in the same chapter helps the understanding of the general
concepts. In particular, we find that the inverse optimal weighting matrix is exactly the
variance-covariance of the relocated and rescaled structural equations in the pseudo panel
application, which provides straightforward intuition for why the optimal weighting matrix
works. Essentially, the optimal weighting matrix down-weights the structural equations that
are volatile and give more weights to those that are less volatile, and correlation between
structural equations are also accounted for.

69

The NMD estimation in pseudo panel correctly relies on large ngt but fixed G, T asymptotics. Naive methods like FE on the cohort means are found to be the NMD estimators
using the identity weighting matrix. But the naive s.e. estimators, including the usual s.e.,
the s.e. robust to heteroskedasticity and the cluster-robust s.e., rely on at least large G
asymptotics. Even when G is moderate or large, depending on how complicated the IVs are,
the usual s.e. and the one only robust to heteroskedasticity may not capture the correct
variance-covariance structure. The cluster-robust s.e. though has the potential to work for
large G because it is fully robust. But since it ignores the individual level data completely,
it is always less efficient than the NMD s.e. estimator using identity weighting matrix. The
optimal NMD is always the most efficient among these candidates. we conclude that when
there are extra IVs to explore, it is important to use optimal weighting.
The comments in Imbens and Wooldridge (2007) regarding flexible specifications provide
several ideas we would like to investigate in future research. First, we intend to extend
the application to dynamic models, i.e., to add lagged dependent variables in the list of
explanatory variables. This is an issue that has been studied by Moffitt (1993); Collado
(1997); Girma (2000); Verbeek and Vella (2005); McKenzie (2004) among others. In dynamic
models, the advantage of having the general NMD framework stands out. There is no
need to tailor the framework in any way, since we can still define the vector of reduced
form parameters as before. Because cohort means of the dependent variable do not appear
redundantly in the reduce-form parameters, their asymptotics are well defined. The cohortlevel equations are also of the same form as before; the only difference is that the equations
for the first several periods need to be dropped because of the lags. Second, we intend to
add unit-specific trend in the unobserved heterogeneity as in the random growth model of
Heckman and Hotz (1989). Third, an even more flexible extension is to let the factor loads
on the unobserved heterogeneity be time-varying. These extensions should be easily handled
by the NMD framework. Lastly, we are also interested in an empirical application of the
method. Currently, we am working on applying the pseudo panel method to estimate returns

70

to education using data from the U.S. Current Population Survey.

71

APPENDICES

72

APPENDIX A

PROOFS AND ALGEBRA

A.1

Proof of consistency

Proof. Prove by verifying verifying (i)-(iv) of Theorem 2.1 in Newey and McFadden (1994)

A.2

Proof of asymptotic normality

A sketch of the idea first. By the first part of (ii), a mean value expansion of each comˆ around θ 0 leads to h(π,
ˆ = h(π,
¯ θ
ˆ − θ 0 ) where θ
¯ is a
ˆ θ)
ˆ θ)
ˆ θ 0 ) + L(π,
ˆ θ)(
ponent of h(π,
ˆ θ 0 ) = B(π,
¯ θ 0 )(π
ˆ − π 0 ), follows by the
vector of mean values. A similar expansion, h(π,
second part of (ii) and h(π 0 , θ 0 ) = 0. Substituting the two expansions in the first-order
√ ˆ
ˆ WL(
¯ −1 · L(π,
ˆ WB(
ˆ π,
ˆ
ˆ θ)
ˆ θ)]
ˆ θ)
¯ θ0 ) ·
− θ 0 ) = −[L(π,
π,
condition and solving gives n(θ
√
p
ˆ→
ˆ − π 0 ). By the first part of (iv), θ
n(π
θ 0 and continuity of L(π, θ) on N (θ 0 ) in (ii),
ˆ − L(π 0 , θ)
ˆ +
ˆ − L ≤ L(π,
ˆ θ)
ˆ θ)
we have that, with probability approaching one, L(π,
p
ˆ −L →
ˆ − L ≤ sup
ˆ θ) − L(π 0 , θ) + L(π 0 , θ)
0. Similar, the
L(π 0 , θ)
θ∈N (θ 0 ) L(π,
p
p
p
p
¯ −L →
¯ →
ˆ θ)
¯ θ 0 ) − B → 0 by π
¯ → π0
convergence L(π,
0 follows by θ
θ 0 , and B(π,

and continuity of B(π, θ) on N (θ 0 ). Condition (iv) guarantees that L WL

−1

exists,

ˆ WL(
¯ −1 . The
ˆ π,
ˆ θ)
ˆ θ)]
and thus, with probability approaching one, the existence of [L(π,
conclusion then follows by (iii), Slutsky’s theorem and the asymptotic equivalence theorem.
A full proof with technical details is given below.
Proof. By (i), without of generality N (θ 0 ) (N (π 0 )) can be assumed to be a convex, open
set contained in Θ (Π). Then N (θ 0 ) (N (π 0 )) is also connected since Θ ∈ RP (Π ∈ RK ).

73

ˆ ∈ N (θ 0 )} and A2 =
Let 1A denote the indicator function for an event A. Let A1 = {θ
p
p
p
p
ˆ→
ˆ ∈ N (π 0 )}, and . Note that θ
ˆ → π 0 ) implies 1A1 → 1 (1A2 → 1).
{π
θ 0 (π

(1) By the first part of condition (ii) and the first order condition for a minimum, 1A1 ·
ˆ Wh(
ˆ = 0. The multiplication by 1A is needed because by (ii) L(π, θ) only
ˆ π,
ˆ θ)
ˆ θ)
L(π,
1
ˆ is pointwise defined by θ
ˆ = argmin h(π,
ˆ h(π,
ˆ θ) W
ˆ θ), not by the first
exists on N (θ 0 ). θ
θ∈Θ

ˆ Wh(
ˆ = 0. For some realization of π,
ˆ may
ˆ π,
ˆ θ)
ˆ θ)
ˆ the corresponding θ
order condition L(π,
not lie in N (θ 0 ).
(2) Since N (θ 0 ) is connected, by condition (ii) and mean value expansion theorem,
ˆ = 1A · hj (π,
¯ j )(θ
ˆ − θ 0 ) for j = 1, · · · , J, where θ
¯ j is
ˆ θ)
ˆ θ 0 ) + 1A1 · Lj (π,
ˆ θ
1A1 · hj (π,
1
a random variable equal to the mean value if 1A1 = 1 and equal to θ 0 otherwise. Again,
p
ˆ is not necessarily in N (θ 0 ). Clearly, θ
¯j →
this complication is needed because θ
θ 0 as
p
ˆ→
¯ and let L(π,
¯ be the matrix with
ˆ θ)
θ
θ 0 . Collect all the J mean values in the matrix θ,

¯ j ). Then those expansions can be written collectively as 1A · h(π,
ˆ =
ˆ θ
ˆ θ)
j-th row Lj (π,
1
¯ θ
ˆ − θ 0 ). Substituting in 0 = 1A · L(π,
ˆ Wh(
ˆ leads to
ˆ π,
ˆ θ 0 ) + 1A · L(π,
ˆ θ)(
ˆ θ)
ˆ θ)
1A · h(π,
1

1

1

ˆ Wh(
ˆ WL(
¯ θ
ˆ − θ 0 ).
ˆ π,
ˆ π,
ˆ θ)
ˆ θ 0 ) + 1A1 · L(π,
ˆ θ)
ˆ θ)(
0 = 1A1 · L(π,
ˆ θ 0 ) = 1A2 ·
(3) By a similar reasoning and the fact h(π 0 , θ 0 ) = 0, write 1A2 · h(π,
¯ θ 0 )(π
ˆ − π 0 ), where π
¯ j , the j-th column of the matrix π,
¯ equals to a mean value the if
B(π,
¯ θ 0 ) is the matrix with the j-th row Bj (π
¯ j , θ 0 ).
1A2 = 1 and equal to π 0 otherwise, and B(π,
p

p

p

p

¯ j → π 0 as π
¯ j → π 0 . Also, 1A2 → 1, and B(π,
¯ θ 0 ) → B by the second part of condition
π
ˆ Wh(
ˆ π,
ˆ θ)
ˆ θ 0 )B(π,
¯ θ 0 )(π
ˆ − π 0 ) + 1A ∩A ·
(iv). Substituting again gives 0 = 1A ∩A · L(π,
1

2

1

2

ˆ WL(
¯ θ
ˆ − θ 0 ).
ˆ π,
ˆ θ)
ˆ θ)(
L(π,
ˆ WL(
¯ is nonsingular}. Let V
ˆ π,
¯ be a random variable equal
ˆ θ)
ˆ θ)
(4) Let A3 = {L(π,
ˆ WL(
¯ if 1A = 1 and equal to the K-dimensional identity matrix otherwise.
ˆ π,
ˆ θ)
ˆ θ)
to L(π,
3
p
p
ˆ →
¯ →
ˆ θ)
ˆ θ)
By the first part of condition (iv), L(π,
L and L(π,
L. Then by condition
p
p
p
¯ →
ˆ →
(v) and W
W, 1A3 → 1 and V
L WL. Substituting for another time gives 0 =
ˆ − θ 0 ).
ˆ Wh(
ˆ π,
¯ θ
ˆ θ)
ˆ θ 0 )B(π,
¯ θ 0 )(π
ˆ − π 0 ) + 1A ∩A ∩A · V(
1A ∩A ∩A · L(π,
1

2

3

1

2

3
p

Now, let A0 = A1 ∩A2 ∩A3 . Note that 1A0 = 1A1 ·1A2 ·1A3 → 1. Multiplying by
74

√
n and

√ ˆ
√
ˆ Wh(
¯ −1 L(π,
ˆ π,
ˆ − π 0 ) − (1A0 −
ˆ θ)
ˆ θ 0 )B(π,
¯ θ 0 ) · n(π
solving gives n(θ
− θ 0 ) = −1A0 · V
√ ˆ
− θ 0 ). The conclusion follows by Slutsky’s theorem and the asymptotic equivalence
1) n(θ
theorem.

A.3

Proof of optimal weighting matrix

Proof. For an arbitrary W,
L WL

−1

L WBΩB WL L WL

−1

− (L (BΩB )−1 L)−1
= D (BΩB )−1 D

is positive semi-definite, where
D = L WL

A.4

−1

L W − [L L WL

−1

L]−1 L (BΩB )−1 .

Proof that extra conditions do not hurt

Proof. Notice that

  
∇π h1 (π 0 , θ 0 ) B1 
B = ∇π h(π 0 , θ 0 ) = 
 =  .
∇π h2 (π 0 , θ 0 )
B2
Then
   
B1  B1 
M = BΩB =   Ω  
B2
B2

 

 B1 ΩB1 B1 ΩB2   M1,1 M1,2 
=
=
.
B2 ΩB1 B2 ΩB2
M2,1 M2,2
Also notice that


  
∇θ h1 (π 0 , θ 0 ) L1 
L = ∇θ h(π 0 , θ 0 ) = 
 =  .
∇θ h2 (π 0 , θ 0 )
L2
75

Now, define F = M2,2 − M2,1 M−1
1,1 M1,2 . F is the Schur complement of M1,1 in M. Then
L M−1 L
  
−1  
L1   M1,1 M1,2  L1 
=  
  
L2
M2,1 M2,2
L2
  
 
−1
−1
−1
−1
−1
−1
L1   M + M1,1 M1,2 F M2,1 M1,1 −M1,1 M1,2 F
 L1 
=    1,1
 
−1
L2
−F−1 M2,1 M−1
F
L2
1,1
−1
−1
−1
−1
−1
= L1 M−1
1,1 + M1,1 M1,2 F M2,1 M1,1 − L2 F M2,1 M1,1 , ...
 
L
−1 + L F−1  1 
... −L1 M−1
 
2
1,1 M1,2 F
L2
−1
−1
−1
−1
−1
=L1 M−1
1,1 + M1,1 M1,2 F M2,1 M1,1 L1 − L2 F M2,1 M1,1 L1
−1
−1
− L1 M−1
1,1 M1,2 F L2 + L2 F L2 .

Therefore,
L M−1 L − L1 M−1
1,1 L1
−1
−1
−1
−1
−1
=L1 M−1
1,1 + M1,1 M1,2 F M2,1 M1,1 L1 − L2 F M2,1 M1,1 L1
−1
−1
−1
− L1 M−1
1,1 M1,2 F L2 + L2 F L2 − L1 M1,1 L1 .
−1
−1
−1
−1
=L1 M−1
1,1 M1,2 F M2,1 M1,1 L1 − L2 F M2,1 M1,1 L1
−1
−1
− L1 M−1
1,1 M1,2 F L2 + L2 F L2
−1
−1
= L1 M−1
1,1 M1,2 − L2 F M2,1 M1,1 L1
−1
− L1 M−1
1,1 M1,2 − L2 F L2
−1 M M−1 L − L
= L1 M−1
2,1 1,1 1
2
1,1 M1,2 − L2 F

= M2,1 M−1
1,1 L1 − L2

F−1 M2,1 M−1
1,1 L1 − L2

where the fact M1,2 = M2,1 is used in the last equality. Clearly, the last expression
is in quadratic form and thus is positive semi-definite. The condition for redundancy of
76

h2 (π 0 , θ 0 ) = 0 is
M2,1 M−1
1,1 L1 − L2 = 0,
or
B2 ΩB1 (B1 ΩB1 )−1 L1 − L2 = 0.

Useful expressions for sit and Ωsgt

A.5

For vit = (yit , xit ) and sit = zit ⊗ vit , an explicit expression for sit is
sit = (vit , z2it vit , · · · , zP it vit ).
Hence,



Cov(v
,
z
v
|g)
it P it it




 Cov(z v , v |g)

V
ar(z
v
|g)
·
·
·
Cov(z
v
,
z
v
|g)
2it
it
it
2it
it
2it
it
it
P
it


s
Ωgt = 
.
.
.
.
..


.
.
.
.
.
.
.




Cov(zP it vit , vit |g) Cov(z2it vit , z2it vit |g) · · ·
V ar(zP it vit |g)
z v

V ar(vit |g)

ˆ gtp = n−1
For µ
gt

Cov(vit , z2it vit |g)

nt
i=1 rit,g zpit vit ,

ˆ pq,gt =
Γ

n−1
gt

nt

···

an estimator for Cov(zpit vit , zqit vit |g) is
z v

z v

ˆ gtp ) (zqit vit − µ
ˆ gtq ).
rit,g (zpit vit − µ

i=1

It is also informative to write sit as
sit = (yit , xit , z2it yit , z2it xit , · · · , zP it yit , zP it xit ).
Because xit includes unity, sit contains all yit , xit , zit and the interactions of zit with (yit , xit ).

77

A.6

Derivation for M

M
=b(θ)Ωs b(θ)

b11 (θ)


b12 (θ)

=
...




b11 (θ)


b12 (θ)


...



b11 (θ)Ωs11 b11 (θ)
ρ1 κ1


 s
Ω11
  ρ1 κ 1







bGT (θ)



Ωs12
ρ1 κ2

...
ΩsGT
ρG κT





·












bGT (θ)





=





b12 (θ)Ωs12 b12 (θ)
ρ1 κ2

..

.
bGT (θ)ΩsGT bGT (θ)
ρG κT

78










A.7

ˆ
Equivalence of the two ways of computing M

ˇ Ω
ˇ shows that
ˆ = b(θ)
ˆ s b(θ)
Proof. Expanding M
ˇ Ω
ˇ
ˆ s b(θ)
b(θ)

ˇ
b11 (θ)


ˇ
b12 (θ)

=
..

.



ˇ
b11 (θ)


ˇ
b12 (θ)


..

.


ˇ Ω
ˇ
ˆ s b (θ)
b11 (θ)
11 11

n11 /n

 ˆs
Ω11

n
  11 /n







ˇ
bGT (θ)


ˆs
Ω
12
n12 /n

...
ˆs
Ω
GT
nGT /n










ˇ
bGT (θ)





=









·






ˇ Ω
ˇ
ˆ s b (θ)
b12 (θ)
12 12
n12 /n







..

.

ˇ Ω
ˇ 
ˆs b
bGT (θ)
(
θ)
GT GT
nGT /n

79

For each (g, t),
ˇ Ω
ˇ
ˆ s bgt (θ)
bgt (θ)
gt
ˇ ) [Γ
ˇ )
ˆ pq,gt ]P IP ⊗ (−1, β
= IP ⊗ (−1, β
gt
gt

 
ˇ
ˆ
(−1, β gt )
  Γ11,gt

 

 Γ
ˇ )
ˆ
(−1, β
gt

  21,gt
=
  .
..

  ..
.

 

 
ˇ )
ˆ
(−1, β
gt P ΓP 1,gt


ˇ
(−1, β gt )





ˇ )
(−1,
β
gt




...






ˇ
(−1, β gt )
P
 
 
   
−1 
−1   −1 
  −1  ˆ
ˆ 12,gt 
   Γ11,gt 
Γ






 ˇ
ˇ
ˇ
ˇ
 β gt
β gt
β gt
β gt
 
 
   


−1 
−1   −1 
  −1 
ˆ 22,gt 
ˆ 21,gt 
  Γ
 
    Γ

ˇ
ˇ
ˇ
ˇ
= β
β
β
β
gt
gt
gt
gt


..
..

.
.

 
 
   

 −1
  ˆ
 −1 
 −1   −1  ˆ
  ΓP 1,gt     Γ

P 2,gt 
ˇ
ˇ
ˇ
ˇ
β
β
β
β
gt

gt

gt

gt


ˆ 12,gt · · · Γ
ˆ 1P,gt
Γ


ˆ 22,gt · · · Γ
ˆ P P,gt 
Γ

·
..
.
..
.. 
.
.


ˆ P 2,gt · · · Γ
ˆ P P,gt
Γ







−1  


ˇ
β gt 
 


 −1 
ˆ
ΓP P,gt  

ˇ
β
gt 


..
..

.
.

 
 


 −1 
 −1  ˆ
· · ·   ΓP P,gt  
ˇ
ˇ
β
β
gt
gt
 −1 
···  
ˇ gt
β
 
 −1 
···  
ˇ
β
gt

ˆ 1P,gt 
Γ


For each (p, q; g, t), recall vit = (yit , xit ) and notice that
 
 −1 
ˇ gt )
vit   = −(yit − xit β
ˇ
β gt
ˇ − (ˇ
= −(yit − xit β
ηt + α
ˇ g )) = −ˇ
uit
and that





z v  −1 
ˆ gtp  
µ
ˇ
β
gt

=

n−1
gt



nt



nt

 −1 
rit,g zpit vit   = n−1
rit,g zpit uˇit
gt
ˇ
β gt
i=1
i=1
80



we have








 −1  ˆ
 −1 
  Γpp,gt  
ˇ
ˇ
β
β
gt
gt
 
 
nt
z v
z v  −1 
 −1 
ˆ gtp ) (zqit vit − µ
ˆ gtq )  
=   n−1
rit,g (zpit vit − µ
gt
ˇ
ˇ
β
β
i=1
gt
gt
 
 
 
  

nt
z v  −1 
z v  −1  
 −1 
 −1 

ˆ gtq  
ˆ gtp   zqit vit   − µ
=n−1
rit,g zpit vit   − µ
gt
ˇ gt
ˇ gt
ˇ gt
ˇ gt
β
β
β
β
i=1
=n−1
gt
=n−1
gt

nt

rit,g zpit uˇit − n−1
gt

i=1
nt

nt

rit,g zpit uˇit

zqit uˇit − n−1
gt

i=1
zp u

ˇgt
rit,g zpit uˇit − µ

nt

rit,g zpit uˇit
i=1

zp u

zqit uˇit − µ
ˇgt

= τˆpq .

i=1

Hence the two ways are numerically equivalent.

A.8

Further algebra on Ξgt

In general,

Ξgt = V ar[zit yit − zit xit β − zit (ηt + αg )|g]
f

= V ar[zit (εi + uit )|g]
= E

f

f

zit (εi + uit ) − E[zit (εi + uit )|g]
f

f

f

zit (εi + uit ) − E[zit (εi + uit )|g]

f

f

= E (εi + uit )2 zit zit |g − E zit (εi + uit )|g E (εi + uit )zit |g
f

= E (εi + uit )2 zit zit |g
f

= E (εi + uit )2 |g E zit zit |g if independent
= σ 2f (g) · Ωz if independent of gi & mean zero
ε +u

81

For zit = (1, x2it ) in Case 3,
f

Ξgt = E (εi + uit )2 zit zit |g
f

= E (εi + uit )2 |g E zit zit |g if independent
= σ 2f E (1, x2it ) (1, x2it )|g
ε +u

 
 1 x2it  
= σ 2f E 
 |g 
ε +u
2
x2it x2it


gt/150 
 1
= σ 2f 

ε +u
gt/150 2 + (gt/150)2

82

APPENDIX B

ADDITIONAL TABLES

83

Table B.1 Small panel with G = 6, T = 4. Case 1.a: x2it ∼ N (gt/6, 1), ngt = 200, sampling
rate= 1%. sen , ser and sec are the non-roust, robust and cluster-robust standard errors,
receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0021
(0.1534)
0.9973
(0.0901)
1.0127
(0.2818)

0.1590
(0.0122)
0.0887
(0.0040)
0.2848
(0.0137)

0.1555
(0.0326)
0.0866
(0.0176)
0.2780
(0.0570)

0.1397
(0.0424)
0.0826
(0.0195)
0.2624
(0.0659)

0.1413
(0.0628)
0.0878
(0.0315)
0.2943
(0.1070)

1.0020
(0.1531)
0.9976
(0.0904)
1.0129
(0.2834)

0.1585
(0.0121)
0.0884
(0.0040)
0.2838
(0.0137)

1.0530
(0.1447)
0.9880
(0.0884)
1.0230
(0.2764)

0.1504
(0.0104)
0.0870
(0.0037)
0.2788
(0.0129)

0.1486
(0.0206)
0.0858
(0.0105)
0.2749
(0.0349)

0.1225
(0.0304)
0.0690
(0.0148)
0.2187
(0.0496)

0.1240
(0.0514)
0.0736
(0.0263)
0.2455
(0.0870)

1.0532
(0.1452)
0.9875
(0.0892)
1.0221
(0.2774)

0.1486
(0.0103)
0.0860
(0.0036)
0.2755
(0.0127)

1.2350
(0.1365)
1.0028
(0.1091)
0.9988
(0.3368)

0.1360
(0.0085)
0.1100
(0.0048)
0.3415
(0.0184)

0.0931
(0.0216)
0.0528
(0.0118)
0.1670
(0.0384)

0.1025
(0.0257)
0.0709
(0.0235)
0.2146
(0.0699)

0.1344
(0.0535)
0.0708
(0.0304)
0.2608
(0.1138)

1.4768
(0.0329)
0.9063
(0.0833)
1.1332
(0.2681)

0.0326
(0.0009)
0.0802
(0.0027)
0.2694
(0.0116)

1.0329
(0.1561)
0.9879
(0.0458)
1.0083
(0.2885)

0.1634
(0.0117)
0.0463
(0.0016)
0.2932
(0.0134)

0.1218
(0.0190)
0.0353
(0.0049)
0.2332
(0.0333)

0.1115
(0.0258)
0.0367
(0.0064)
0.2191
(0.0421)

0.1253
(0.0532)
0.0406
(0.0138)
0.2567
(0.0918)

1.0504
(0.1404)
0.9885
(0.0419)
1.0254
(0.2607)

0.1424
(0.0098)
0.0415
(0.0012)
0.2617
(0.0112)

1.0061
(0.1546)
0.9975
(0.0862)
1.0097
(0.1550)

0.1588
(0.0119)
0.0863
(0.0036)
0.1585
(0.0043)

0.1007
(0.0175)
0.0542
(0.0088)
0.1271
(0.0206)

0.1011
(0.0269)
0.0564
(0.0120)
0.1303
(0.0268)

0.1173
(0.0509)
0.0693
(0.0252)
0.1408
(0.0491)

1.0505
(0.1463)
0.9892
(0.0811)
1.0143
(0.1008)

0.1479
(0.0105)
0.0812
(0.0032)
0.1011
(0.0023)

1.2449
(0.1268)
0.9943
(0.0871)
0.9839
(0.3119)

0.1267
(0.0079)
0.0881
(0.0034)
0.3163
(0.0155)

0.0574
(0.0111)
0.0301
(0.0055)
0.1043
(0.0198)

0.0867
(0.0218)
0.0514
(0.0157)
0.1798
(0.0578)

0.1108
(0.0446)
0.0582
(0.0226)
0.2246
(0.0964)

1.4718
(0.0335)
0.9687
(0.0406)
1.0243
(0.0983)

0.0319
(0.0008)
0.0392
(0.0009)
0.0972
(0.0016)

84

Table B.2 Small panel with G = 6, T = 4. Case 1.b: x2it ∼ N (gt/6, 1), ngt = 1000,
sampling rate= 1%. sen , ser and sec are the non-roust, robust and cluster-robust standard
errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9940
(0.0715)
1.0037
(0.0391)
1.0067
(0.1274)

0.0722
(0.0018)
0.0398
(0.0007)
0.1282
(0.0024)

0.0700
(0.0144)
0.0385
(0.0079)
0.1244
(0.0256)

0.0628
(0.0180)
0.0367
(0.0085)
0.1174
(0.0308)

0.0639
(0.0286)
0.0390
(0.0146)
0.1333
(0.0486)

0.9939
(0.0714)
1.0037
(0.0391)
1.0069
(0.1276)

0.0721
(0.0018)
0.0397
(0.0007)
0.1281
(0.0024)

0.9940
(0.0709)
1.0038
(0.0389)
1.0064
(0.1274)

0.0718
(0.0018)
0.0397
(0.0007)
0.1278
(0.0024)

0.0714
(0.0088)
0.0395
(0.0049)
0.1272
(0.0156)

0.0517
(0.0142)
0.0301
(0.0069)
0.0963
(0.0247)

0.0531
(0.0233)
0.0322
(0.0120)
0.1100
(0.0399)

0.9940
(0.0710)
1.0037
(0.0390)
1.0067
(0.1278)

0.0716
(0.0018)
0.0396
(0.0007)
0.1275
(0.0024)

0.9979
(0.0372)
1.0036
(0.0459)
1.0219
(0.1538)

0.0360
(0.0011)
0.0464
(0.0009)
0.1536
(0.0033)

0.0336
(0.0064)
0.0270
(0.0052)
0.0751
(0.0145)

0.0312
(0.0057)
0.0286
(0.0071)
0.0839
(0.0273)

0.0306
(0.0120)
0.0373
(0.0140)
0.1172
(0.0518)

0.9987
(0.0208)
1.0026
(0.0357)
1.0086
(0.1241)

0.0205
(0.0002)
0.0371
(0.0005)
0.1259
(0.0023)

0.9998
(0.0778)
1.0025
(0.0206)
1.0097
(0.1327)

0.0768
(0.0020)
0.0207
(0.0003)
0.1337
(0.0025)

0.0565
(0.0078)
0.0158
(0.0022)
0.1059
(0.0147)

0.0512
(0.0110)
0.0165
(0.0028)
0.0992
(0.0195)

0.0573
(0.0248)
0.0181
(0.0065)
0.1159
(0.0408)

0.9946
(0.0658)
1.0031
(0.0186)
1.0070
(0.1218)

0.0680
(0.0016)
0.0188
(0.0002)
0.1202
(0.0020)

0.9959
(0.0711)
1.0040
(0.0381)
1.0028
(0.0686)

0.0728
(0.0018)
0.0387
(0.0006)
0.0705
(0.0007)

0.0457
(0.0075)
0.0242
(0.0040)
0.0567
(0.0093)

0.0458
(0.0114)
0.0251
(0.0053)
0.0580
(0.0121)

0.0534
(0.0238)
0.0313
(0.0117)
0.0635
(0.0218)

0.9936
(0.0695)
1.0045
(0.0370)
1.0011
(0.0442)

0.0708
(0.0017)
0.0370
(0.0006)
0.0454
(0.0003)

1.0002
(0.0390)
1.0028
(0.0248)
1.0154
(0.1157)

0.0377
(0.0010)
0.0250
(0.0003)
0.1166
(0.0018)

0.0256
(0.0030)
0.0132
(0.0015)
0.0515
(0.0060)

0.0271
(0.0045)
0.0150
(0.0024)
0.0599
(0.0167)

0.0300
(0.0112)
0.0201
(0.0073)
0.0872
(0.0357)

0.9986
(0.0208)
1.0031
(0.0179)
1.0014
(0.0440)

0.0203
(0.0002)
0.0182
(0.0002)
0.0448
(0.0003)

85

Table B.3 Small panel with G = 6, T = 4. Case 1.1: x2it ∼ N (gt/6, 1), ngt = 200, sampling
rate= 0.2%. sen , ser and sec are the non-roust, robust and cluster-robust standard errors,
receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9898
(0.1652)
1.0083
(0.0875)
1.0103
(0.2886)

0.1606
(0.0089)
0.0889
(0.0034)
0.2863
(0.0124)

0.1568
(0.0342)
0.0868
(0.0185)
0.2794
(0.0593)

0.1402
(0.0423)
0.0823
(0.0208)
0.2619
(0.0716)

0.1430
(0.0615)
0.0871
(0.0326)
0.2986
(0.1097)

0.9900
(0.1652)
1.0081
(0.0874)
1.0113
(0.2882)

0.1601
(0.0089)
0.0886
(0.0034)
0.2853
(0.0123)

0.9896
(0.1599)
1.0091
(0.0864)
1.0095
(0.2832)

0.1565
(0.0084)
0.0879
(0.0033)
0.2817
(0.0119)

0.1546
(0.0198)
0.0868
(0.0105)
0.2783
(0.0341)

0.1186
(0.0308)
0.0682
(0.0160)
0.2179
(0.0541)

0.1203
(0.0477)
0.0716
(0.0268)
0.2447
(0.0908)

0.9905
(0.1599)
1.0084
(0.0863)
1.0111
(0.2818)

0.1547
(0.0083)
0.0869
(0.0033)
0.2785
(0.0117)

0.9966
(0.0829)
1.0086
(0.1007)
1.0183
(0.3441)

0.0809
(0.0052)
0.1035
(0.0043)
0.3417
(0.0167)

0.0752
(0.0153)
0.0606
(0.0122)
0.1684
(0.0341)

0.0701
(0.0131)
0.0642
(0.0165)
0.1871
(0.0628)

0.0687
(0.0255)
0.0817
(0.0322)
0.2611
(0.1184)

0.9978
(0.0476)
1.0065
(0.0808)
1.0130
(0.2811)

0.0453
(0.0010)
0.0819
(0.0027)
0.2750
(0.0113)

1.0019
(0.1685)
1.0038
(0.0468)
1.0200
(0.2924)

0.1681
(0.0089)
0.0464
(0.0013)
0.2956
(0.0123)

0.1241
(0.0180)
0.0353
(0.0048)
0.2340
(0.0327)

0.1127
(0.0253)
0.0367
(0.0060)
0.2198
(0.0441)

0.1275
(0.0542)
0.0399
(0.0140)
0.2580
(0.0882)

0.9931
(0.1513)
1.0035
(0.0433)
1.0173
(0.2629)

0.1473
(0.0073)
0.0416
(0.0010)
0.2633
(0.0101)

0.9909
(0.1641)
1.0105
(0.0840)
1.0028
(0.1582)

0.1610
(0.0087)
0.0865
(0.0030)
0.1582
(0.0036)

0.1020
(0.0177)
0.0544
(0.0091)
0.1276
(0.0210)

0.1020
(0.0263)
0.0563
(0.0126)
0.1301
(0.0278)

0.1189
(0.0507)
0.0693
(0.0266)
0.1415
(0.0498)

0.9899
(0.1565)
1.0093
(0.0800)
1.0005
(0.0996)

0.1529
(0.0079)
0.0814
(0.0027)
0.1008
(0.0016)

0.9993
(0.0849)
1.0059
(0.0547)
1.0151
(0.2575)

0.0839
(0.0045)
0.0559
(0.0017)
0.2579
(0.0091)

0.0566
(0.0070)
0.0296
(0.0035)
0.1144
(0.0140)

0.0602
(0.0104)
0.0335
(0.0056)
0.1336
(0.0385)

0.0668
(0.0250)
0.0443
(0.0167)
0.1932
(0.0819)

0.9979
(0.0479)
1.0036
(0.0421)
1.0013
(0.1007)

0.0443
(0.0010)
0.0398
(0.0008)
0.0984
(0.0015)

86

Table B.4 Small panel with G = 6, T = 4. Case 1.1: x2it ∼ N (gt/6, 1), ngt = 200, sampling
rate= 0.2%. sen , ser and sec are the non-roust, robust and cluster-robust standard errors,
receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0027
(0.0722)
0.9971
(0.0401)
1.0010
(0.1304)

0.0722
(0.0017)
0.0398
(0.0007)
0.1279
(0.0024)

0.0697
(0.0145)
0.0384
(0.0079)
0.1235
(0.0256)

0.0618
(0.0181)
0.0366
(0.0086)
0.1167
(0.0306)

0.0622
(0.0281)
0.0390
(0.0138)
0.1325
(0.0490)

1.0027
(0.0722)
0.9972
(0.0401)
1.0012
(0.1304)

0.0721
(0.0017)
0.0398
(0.0007)
0.1278
(0.0024)

1.0028
(0.0715)
0.9971
(0.0400)
1.0009
(0.1304)

0.0718
(0.0017)
0.0397
(0.0007)
0.1275
(0.0024)

0.0709
(0.0087)
0.0392
(0.0048)
0.1259
(0.0152)

0.0510
(0.0143)
0.0300
(0.0069)
0.0957
(0.0245)

0.0516
(0.0229)
0.0323
(0.0114)
0.1092
(0.0403)

1.0029
(0.0715)
0.9973
(0.0401)
1.0009
(0.1302)

0.0716
(0.0017)
0.0396
(0.0007)
0.1272
(0.0024)

1.0030
(0.0353)
0.9962
(0.0461)
1.0016
(0.1567)

0.0360
(0.0011)
0.0465
(0.0009)
0.1535
(0.0034)

0.0332
(0.0066)
0.0267
(0.0053)
0.0742
(0.0148)

0.0308
(0.0058)
0.0284
(0.0072)
0.0832
(0.0290)

0.0301
(0.0115)
0.0359
(0.0137)
0.1166
(0.0554)

0.9992
(0.0204)
0.9980
(0.0374)
0.9999
(0.1278)

0.0205
(0.0002)
0.0372
(0.0005)
0.1256
(0.0023)

1.0005
(0.0754)
0.9989
(0.0211)
1.0006
(0.1349)

0.0769
(0.0018)
0.0207
(0.0003)
0.1335
(0.0025)

0.0565
(0.0075)
0.0158
(0.0021)
0.1057
(0.0139)

0.0511
(0.0112)
0.0165
(0.0026)
0.0993
(0.0182)

0.0574
(0.0255)
0.0180
(0.0060)
0.1159
(0.0404)

1.0006
(0.0674)
0.9998
(0.0190)
0.9981
(0.1211)

0.0680
(0.0015)
0.0188
(0.0002)
0.1198
(0.0021)

1.0030
(0.0731)
0.9965
(0.0391)
1.0027
(0.0713)

0.0729
(0.0017)
0.0388
(0.0006)
0.0705
(0.0007)

0.0456
(0.0074)
0.0242
(0.0039)
0.0564
(0.0091)

0.0452
(0.0114)
0.0251
(0.0053)
0.0579
(0.0120)

0.0518
(0.0234)
0.0310
(0.0117)
0.0628
(0.0222)

1.0031
(0.0707)
0.9970
(0.0369)
1.0032
(0.0461)

0.0708
(0.0017)
0.0370
(0.0006)
0.0454
(0.0003)

1.0022
(0.0369)
0.9980
(0.0252)
1.0025
(0.1185)

0.0378
(0.0010)
0.0250
(0.0004)
0.1164
(0.0019)

0.0255
(0.0029)
0.0132
(0.0015)
0.0512
(0.0059)

0.0268
(0.0046)
0.0149
(0.0024)
0.0595
(0.0175)

0.0299
(0.0113)
0.0197
(0.0069)
0.0863
(0.0377)

0.9991
(0.0203)
0.9998
(0.0183)
1.0026
(0.0460)

0.0203
(0.0002)
0.0182
(0.0002)
0.0449
(0.0003)

87

Table B.5 Small panel with G = 6, T = 4. Case 2.a: x2it ∼ N (gt/6, 1) + fi , ngt = 200,
sampling rate= 1%. sen , ser and sec are the non-roust, robust and cluster-robust standard
errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0021
(0.1534)
0.9973
(0.0901)
1.0127
(0.2818)

0.1590
(0.0122)
0.0887
(0.0040)
0.2848
(0.0137)

0.1555
(0.0326)
0.0866
(0.0176)
0.2780
(0.0570)

0.1397
(0.0424)
0.0826
(0.0195)
0.2624
(0.0659)

0.1413
(0.0628)
0.0878
(0.0315)
0.2943
(0.1070)

1.0020
(0.1531)
0.9976
(0.0904)
1.0129
(0.2834)

0.1585
(0.0121)
0.0884
(0.0040)
0.2838
(0.0137)

1.0530
(0.1447)
0.9880
(0.0884)
1.0230
(0.2764)

0.1504
(0.0104)
0.0870
(0.0037)
0.2788
(0.0129)

0.1486
(0.0206)
0.0858
(0.0105)
0.2749
(0.0349)

0.1225
(0.0304)
0.0690
(0.0148)
0.2187
(0.0496)

0.1240
(0.0514)
0.0736
(0.0263)
0.2455
(0.0870)

1.0532
(0.1452)
0.9875
(0.0892)
1.0221
(0.2774)

0.1486
(0.0103)
0.0860
(0.0036)
0.2755
(0.0127)

1.2350
(0.1365)
1.0028
(0.1091)
0.9988
(0.3368)

0.1360
(0.0085)
0.1100
(0.0048)
0.3415
(0.0184)

0.0931
(0.0216)
0.0528
(0.0118)
0.1670
(0.0384)

0.1025
(0.0257)
0.0709
(0.0235)
0.2146
(0.0699)

0.1344
(0.0535)
0.0708
(0.0304)
0.2608
(0.1138)

1.4768
(0.0329)
0.9063
(0.0833)
1.1332
(0.2681)

0.0326
(0.0009)
0.0802
(0.0027)
0.2694
(0.0116)

1.0329
(0.1561)
0.9879
(0.0458)
1.0083
(0.2885)

0.1634
(0.0117)
0.0463
(0.0016)
0.2932
(0.0134)

0.1218
(0.0190)
0.0353
(0.0049)
0.2332
(0.0333)

0.1115
(0.0258)
0.0367
(0.0064)
0.2191
(0.0421)

0.1253
(0.0532)
0.0406
(0.0138)
0.2567
(0.0918)

1.0504
(0.1404)
0.9885
(0.0419)
1.0254
(0.2607)

0.1424
(0.0098)
0.0415
(0.0012)
0.2617
(0.0112)

1.0061
(0.1546)
0.9975
(0.0862)
1.0097
(0.1550)

0.1588
(0.0119)
0.0863
(0.0036)
0.1585
(0.0043)

0.1007
(0.0175)
0.0542
(0.0088)
0.1271
(0.0206)

0.1011
(0.0269)
0.0564
(0.0120)
0.1303
(0.0268)

0.1173
(0.0509)
0.0693
(0.0252)
0.1408
(0.0491)

1.0505
(0.1463)
0.9892
(0.0811)
1.0143
(0.1008)

0.1479
(0.0105)
0.0812
(0.0032)
0.1011
(0.0023)

1.2449
(0.1268)
0.9943
(0.0871)
0.9839
(0.3119)

0.1267
(0.0079)
0.0881
(0.0034)
0.3163
(0.0155)

0.0574
(0.0111)
0.0301
(0.0055)
0.1043
(0.0198)

0.0867
(0.0218)
0.0514
(0.0157)
0.1798
(0.0578)

0.1108
(0.0446)
0.0582
(0.0226)
0.2246
(0.0964)

1.4718
(0.0335)
0.9687
(0.0406)
1.0243
(0.0983)

0.0319
(0.0008)
0.0392
(0.0009)
0.0972
(0.0016)

88

Table B.6 Small panel with G = 6, T = 4. Case 2.b: x2it ∼ N (gt/6, 1) + fi , ngt = 1000,
sampling rate= 1%. sen , ser and sec are the non-roust, robust and cluster-robust standard
errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9993
(0.0717)
1.0027
(0.0391)
1.0082
(0.1274)

0.0721
(0.0025)
0.0398
(0.0008)
0.1283
(0.0026)

0.0698
(0.0144)
0.0385
(0.0079)
0.1244
(0.0256)

0.0628
(0.0181)
0.0367
(0.0085)
0.1174
(0.0309)

0.0642
(0.0287)
0.0390
(0.0146)
0.1333
(0.0487)

0.9992
(0.0716)
1.0027
(0.0392)
1.0084
(0.1277)

0.0720
(0.0025)
0.0397
(0.0008)
0.1282
(0.0026)

1.0107
(0.0706)
1.0004
(0.0389)
1.0113
(0.1272)

0.0712
(0.0025)
0.0396
(0.0008)
0.1277
(0.0026)

0.0709
(0.0088)
0.0394
(0.0048)
0.1271
(0.0156)

0.0526
(0.0140)
0.0301
(0.0068)
0.0966
(0.0247)

0.0545
(0.0235)
0.0323
(0.0120)
0.1102
(0.0401)

1.0107
(0.0706)
1.0004
(0.0390)
1.0115
(0.1276)

0.0711
(0.0024)
0.0395
(0.0008)
0.1274
(0.0025)

1.2316
(0.0608)
1.0107
(0.0480)
1.0146
(0.1537)

0.0609
(0.0017)
0.0491
(0.0009)
0.1532
(0.0034)

0.0555
(0.0095)
0.0312
(0.0052)
0.0992
(0.0166)

0.0664
(0.0135)
0.0354
(0.0098)
0.1064
(0.0313)

0.1070
(0.0301)
0.0398
(0.0134)
0.1261
(0.0536)

1.4756
(0.0147)
0.9080
(0.0352)
1.1474
(0.1225)

0.0147
(0.0002)
0.0365
(0.0005)
0.1238
(0.0023)

1.0101
(0.0776)
1.0016
(0.0206)
1.0115
(0.1325)

0.0763
(0.0026)
0.0207
(0.0003)
0.1337
(0.0026)

0.0562
(0.0078)
0.0158
(0.0022)
0.1058
(0.0147)

0.0512
(0.0110)
0.0165
(0.0028)
0.0991
(0.0195)

0.0571
(0.0247)
0.0181
(0.0065)
0.1160
(0.0408)

1.0094
(0.0658)
1.0023
(0.0185)
1.0087
(0.1217)

0.0675
(0.0022)
0.0188
(0.0002)
0.1201
(0.0022)

1.0027
(0.0712)
1.0030
(0.0382)
1.0039
(0.0687)

0.0726
(0.0025)
0.0387
(0.0007)
0.0705
(0.0008)

0.0456
(0.0075)
0.0242
(0.0040)
0.0567
(0.0092)

0.0458
(0.0114)
0.0251
(0.0053)
0.0580
(0.0121)

0.0537
(0.0238)
0.0313
(0.0117)
0.0635
(0.0218)

1.0094
(0.0690)
1.0018
(0.0370)
1.0017
(0.0442)

0.0703
(0.0024)
0.0369
(0.0007)
0.0454
(0.0004)

1.2392
(0.0575)
1.0034
(0.0387)
0.9986
(0.1424)

0.0574
(0.0016)
0.0394
(0.0006)
0.1423
(0.0029)

0.0337
(0.0050)
0.0175
(0.0025)
0.0609
(0.0088)

0.0573
(0.0117)
0.0263
(0.0065)
0.0899
(0.0260)

0.0867
(0.0257)
0.0337
(0.0100)
0.1093
(0.0448)

1.4726
(0.0146)
0.9803
(0.0176)
1.0102
(0.0429)

0.0146
(0.0002)
0.0179
(0.0002)
0.0442
(0.0003)

89

Table B.7 Small panel with G = 6, T = 4. Case 2.1: x2it ∼ N (gt/6, 1) + fi , ngt = 200,
sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and cluster-robust
standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0147
(0.1624)
1.0034
(0.0871)
1.0175
(0.2885)

0.1591
(0.0126)
0.0889
(0.0041)
0.2864
(0.0133)

0.1553
(0.0351)
0.0868
(0.0187)
0.2795
(0.0597)

0.1399
(0.0428)
0.0824
(0.0209)
0.2623
(0.0720)

0.1450
(0.0643)
0.0875
(0.0327)
0.2987
(0.1101)

1.0148
(0.1624)
1.0033
(0.0871)
1.0184
(0.2881)

0.1586
(0.0125)
0.0886
(0.0041)
0.2854
(0.0132)

1.0622
(0.1535)
0.9950
(0.0855)
1.0297
(0.2812)

0.1505
(0.0110)
0.0872
(0.0037)
0.2804
(0.0124)

0.1490
(0.0202)
0.0863
(0.0106)
0.2775
(0.0341)

0.1216
(0.0303)
0.0687
(0.0159)
0.2190
(0.0543)

0.1253
(0.0516)
0.0725
(0.0269)
0.2455
(0.0905)

1.0638
(0.1538)
0.9942
(0.0855)
1.0314
(0.2796)

0.1487
(0.0109)
0.0862
(0.0037)
0.2772
(0.0122)

1.2409
(0.1366)
1.0163
(0.1082)
1.0187
(0.3458)

0.1363
(0.0084)
0.1100
(0.0046)
0.3425
(0.0175)

0.0920
(0.0235)
0.0525
(0.0126)
0.1662
(0.0403)

0.1010
(0.0265)
0.0702
(0.0238)
0.2122
(0.0745)

0.1331
(0.0548)
0.0707
(0.0311)
0.2587
(0.1200)

1.4705
(0.0328)
0.9150
(0.0793)
1.1427
(0.2719)

0.0324
(0.0009)
0.0804
(0.0026)
0.2708
(0.0113)

1.0486
(0.1633)
0.9996
(0.0465)
1.0287
(0.2912)

0.1630
(0.0119)
0.0463
(0.0016)
0.2948
(0.0129)

0.1211
(0.0189)
0.0352
(0.0049)
0.2336
(0.0328)

0.1114
(0.0259)
0.0366
(0.0061)
0.2195
(0.0442)

0.1277
(0.0546)
0.0399
(0.0140)
0.2580
(0.0887)

1.0592
(0.1457)
1.0002
(0.0430)
1.0250
(0.2612)

0.1422
(0.0099)
0.0415
(0.0012)
0.2626
(0.0107)

1.0223
(0.1602)
1.0058
(0.0836)
1.0077
(0.1577)

0.1584
(0.0123)
0.0864
(0.0036)
0.1581
(0.0043)

0.1008
(0.0184)
0.0545
(0.0092)
0.1276
(0.0212)

0.1012
(0.0268)
0.0564
(0.0127)
0.1302
(0.0279)

0.1200
(0.0525)
0.0696
(0.0268)
0.1417
(0.0498)

1.0598
(0.1503)
0.9974
(0.0793)
1.0033
(0.0991)

0.1478
(0.0108)
0.0812
(0.0032)
0.1008
(0.0022)

1.2502
(0.1276)
1.0080
(0.0864)
1.0039
(0.3193)

0.1271
(0.0078)
0.0881
(0.0032)
0.3170
(0.0145)

0.0568
(0.0120)
0.0299
(0.0059)
0.1038
(0.0206)

0.0854
(0.0223)
0.0510
(0.0159)
0.1782
(0.0619)

0.1108
(0.0458)
0.0579
(0.0234)
0.2232
(0.1021)

1.4657
(0.0335)
0.9815
(0.0415)
1.0100
(0.0996)

0.0318
(0.0008)
0.0392
(0.0009)
0.0970
(0.0016)

90

Table B.8 Small panel with G = 6, T = 4. Case 2.1: x2it ∼ N (gt/6, 1) + fi , ngt = 200,
sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and cluster-robust
standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0080
(0.0721)
0.9961
(0.0401)
1.0026
(0.1304)

0.0719
(0.0024)
0.0398
(0.0008)
0.1279
(0.0027)

0.0694
(0.0145)
0.0384
(0.0079)
0.1235
(0.0255)

0.0615
(0.0179)
0.0366
(0.0086)
0.1166
(0.0305)

0.0619
(0.0280)
0.0390
(0.0138)
0.1322
(0.0489)

1.0080
(0.0721)
0.9961
(0.0401)
1.0028
(0.1303)

0.0719
(0.0024)
0.0398
(0.0008)
0.1278
(0.0027)

1.0191
(0.0714)
0.9939
(0.0400)
1.0057
(0.1302)

0.0711
(0.0024)
0.0396
(0.0008)
0.1273
(0.0026)

0.0702
(0.0086)
0.0391
(0.0047)
0.1258
(0.0151)

0.0514
(0.0138)
0.0300
(0.0068)
0.0957
(0.0244)

0.0520
(0.0227)
0.0323
(0.0115)
0.1090
(0.0403)

1.0193
(0.0715)
0.9940
(0.0400)
1.0057
(0.1300)

0.0709
(0.0023)
0.0395
(0.0008)
0.1270
(0.0026)

1.2376
(0.0602)
1.0016
(0.0485)
0.9972
(0.1544)

0.0610
(0.0018)
0.0492
(0.0010)
0.1535
(0.0035)

0.0545
(0.0092)
0.0308
(0.0051)
0.0979
(0.0163)

0.0645
(0.0132)
0.0350
(0.0098)
0.1051
(0.0310)

0.1023
(0.0293)
0.0389
(0.0128)
0.1255
(0.0525)

1.4775
(0.0147)
0.9031
(0.0364)
1.1404
(0.1251)

0.0146
(0.0002)
0.0365
(0.0006)
0.1235
(0.0023)

1.0108
(0.0749)
0.9980
(0.0211)
1.0025
(0.1349)

0.0763
(0.0025)
0.0207
(0.0003)
0.1334
(0.0027)

0.0562
(0.0076)
0.0158
(0.0021)
0.1057
(0.0139)

0.0509
(0.0111)
0.0165
(0.0026)
0.0993
(0.0182)

0.0571
(0.0253)
0.0180
(0.0060)
0.1158
(0.0403)

1.0152
(0.0672)
0.9991
(0.0189)
0.9998
(0.1210)

0.0674
(0.0021)
0.0188
(0.0003)
0.1198
(0.0022)

1.0096
(0.0730)
0.9955
(0.0391)
1.0038
(0.0713)

0.0725
(0.0024)
0.0388
(0.0008)
0.0704
(0.0009)

0.0454
(0.0074)
0.0241
(0.0039)
0.0564
(0.0091)

0.0450
(0.0114)
0.0251
(0.0053)
0.0578
(0.0120)

0.0515
(0.0233)
0.0310
(0.0116)
0.0627
(0.0221)

1.0191
(0.0703)
0.9943
(0.0368)
1.0038
(0.0460)

0.0701
(0.0023)
0.0370
(0.0007)
0.0454
(0.0005)

1.2440
(0.0568)
0.9957
(0.0391)
0.9832
(0.1436)

0.0575
(0.0017)
0.0394
(0.0007)
0.1426
(0.0030)

0.0332
(0.0048)
0.0172
(0.0025)
0.0601
(0.0086)

0.0555
(0.0114)
0.0260
(0.0065)
0.0886
(0.0260)

0.0824
(0.0251)
0.0329
(0.0093)
0.1086
(0.0443)

1.4744
(0.0147)
0.9777
(0.0180)
1.0106
(0.0448)

0.0145
(0.0002)
0.0179
(0.0002)
0.0442
(0.0003)

91

Table B.9 Small panel with G = 6, T = 4. Case 3.a: x2it ∼ N (gt/6, 1) + zi , ngt = 200,
sampling rate= 1%. sen , ser and sec are the non-roust, robust and cluster-robust standard
errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9764
(0.1533)
1.0024
(0.0902)
1.0051
(0.2823)

0.1586
(0.0116)
0.0887
(0.0038)
0.2850
(0.0131)

0.1548
(0.0325)
0.0865
(0.0175)
0.2779
(0.0569)

0.1389
(0.0425)
0.0825
(0.0195)
0.2626
(0.0663)

0.1403
(0.0621)
0.0874
(0.0315)
0.2944
(0.1072)

0.9765
(0.1528)
1.0027
(0.0903)
1.0055
(0.2839)

0.1581
(0.0116)
0.0884
(0.0038)
0.2840
(0.0131)

1.0018
(0.0460)
0.9980
(0.0841)
1.0087
(0.2738)

0.0463
(0.0012)
0.0826
(0.0027)
0.2766
(0.0118)

0.0456
(0.0055)
0.0813
(0.0098)
0.2723
(0.0341)

0.0508
(0.0077)
0.0667
(0.0150)
0.2202
(0.0495)

0.0499
(0.0154)
0.0713
(0.0257)
0.2462
(0.0869)

1.0020
(0.0462)
0.9976
(0.0850)
1.0080
(0.2751)

0.0452
(0.0011)
0.0816
(0.0026)
0.2734
(0.0116)

1.0042
(0.0431)
0.9967
(0.1076)
0.9932
(0.3591)

0.0440
(0.0020)
0.1092
(0.0046)
0.3584
(0.0192)

0.0487
(0.0088)
0.0679
(0.0126)
0.1885
(0.0365)

0.0489
(0.0092)
0.0707
(0.0175)
0.2042
(0.0643)

0.0422
(0.0154)
0.0881
(0.0335)
0.2764
(0.1203)

1.0041
(0.0331)
0.9978
(0.0846)
1.0090
(0.2779)

0.0329
(0.0007)
0.0815
(0.0026)
0.2734
(0.0117)

0.9840
(0.1582)
0.9923
(0.0460)
0.9989
(0.2890)

0.1633
(0.0111)
0.0464
(0.0014)
0.2941
(0.0129)

0.1215
(0.0189)
0.0353
(0.0049)
0.2333
(0.0332)

0.1104
(0.0257)
0.0367
(0.0064)
0.2192
(0.0421)

0.1236
(0.0529)
0.0403
(0.0137)
0.2552
(0.0917)

0.9850
(0.1420)
0.9919
(0.0421)
1.0189
(0.2617)

0.1423
(0.0091)
0.0416
(0.0011)
0.2625
(0.0107)

0.9740
(0.1543)
1.0025
(0.0862)
1.0045
(0.1553)

0.1582
(0.0113)
0.0863
(0.0034)
0.1586
(0.0036)

0.1002
(0.0173)
0.0541
(0.0087)
0.1270
(0.0205)

0.1004
(0.0270)
0.0563
(0.0119)
0.1303
(0.0269)

0.1162
(0.0502)
0.0689
(0.0252)
0.1406
(0.0490)

0.9828
(0.1441)
1.0010
(0.0810)
1.0111
(0.1014)

0.1474
(0.0096)
0.0812
(0.0030)
0.1011
(0.0017)

1.0052
(0.0399)
0.9925
(0.0559)
0.9959
(0.2635)

0.0404
(0.0013)
0.0565
(0.0017)
0.2653
(0.0101)

0.0296
(0.0037)
0.0312
(0.0039)
0.1211
(0.0157)

0.0335
(0.0051)
0.0348
(0.0058)
0.1423
(0.0383)

0.0379
(0.0126)
0.0458
(0.0160)
0.2010
(0.0807)

1.0043
(0.0333)
0.9914
(0.0414)
1.0133
(0.1012)

0.0322
(0.0007)
0.0397
(0.0008)
0.0985
(0.0015)

92

Table B.10 Small panel with G = 6, T = 4. Case 3.b: x2it ∼ N (gt/6, 1) + zi , ngt = 1000,
sampling rate= 1%. sen , ser and sec are the non-roust, robust and cluster-robust standard
errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9937
(0.0714)
1.0038
(0.0391)
1.0066
(0.1273)

0.0720
(0.0023)
0.0398
(0.0007)
0.1283
(0.0025)

0.0698
(0.0144)
0.0385
(0.0079)
0.1244
(0.0256)

0.0627
(0.0181)
0.0366
(0.0085)
0.1174
(0.0308)

0.0639
(0.0283)
0.0391
(0.0146)
0.1332
(0.0487)

0.9936
(0.0713)
1.0038
(0.0391)
1.0068
(0.1276)

0.0720
(0.0023)
0.0397
(0.0007)
0.1282
(0.0025)

0.9982
(0.0191)
1.0029
(0.0356)
1.0077
(0.1244)

0.0206
(0.0002)
0.0372
(0.0005)
0.1261
(0.0023)

0.0205
(0.0025)
0.0371
(0.0045)
0.1258
(0.0154)

0.0229
(0.0034)
0.0295
(0.0067)
0.0982
(0.0243)

0.0223
(0.0073)
0.0317
(0.0121)
0.1118
(0.0403)

0.9982
(0.0191)
1.0029
(0.0357)
1.0080
(0.1249)

0.0205
(0.0002)
0.0371
(0.0005)
0.1259
(0.0023)

0.9964
(0.0191)
1.0049
(0.0489)
1.0179
(0.1651)

0.0192
(0.0004)
0.0489
(0.0009)
0.1624
(0.0036)

0.0218
(0.0039)
0.0307
(0.0055)
0.0859
(0.0154)

0.0217
(0.0040)
0.0320
(0.0077)
0.0921
(0.0290)

0.0189
(0.0069)
0.0406
(0.0153)
0.1246
(0.0553)

0.9987
(0.0143)
1.0027
(0.0355)
1.0084
(0.1239)

0.0148
(0.0001)
0.0370
(0.0005)
0.1258
(0.0023)

0.9995
(0.0773)
1.0026
(0.0206)
1.0097
(0.1327)

0.0763
(0.0025)
0.0207
(0.0003)
0.1338
(0.0025)

0.0562
(0.0078)
0.0158
(0.0022)
0.1059
(0.0147)

0.0510
(0.0109)
0.0165
(0.0028)
0.0992
(0.0195)

0.0570
(0.0245)
0.0180
(0.0065)
0.1159
(0.0408)

0.9945
(0.0654)
1.0031
(0.0186)
1.0070
(0.1218)

0.0675
(0.0021)
0.0188
(0.0002)
0.1202
(0.0020)

0.9956
(0.0710)
1.0041
(0.0382)
1.0028
(0.0686)

0.0726
(0.0023)
0.0387
(0.0007)
0.0705
(0.0007)

0.0456
(0.0075)
0.0242
(0.0040)
0.0567
(0.0093)

0.0457
(0.0114)
0.0251
(0.0054)
0.0580
(0.0121)

0.0533
(0.0236)
0.0314
(0.0117)
0.0634
(0.0219)

0.9933
(0.0692)
1.0046
(0.0371)
1.0011
(0.0442)

0.0703
(0.0022)
0.0369
(0.0006)
0.0454
(0.0003)

0.9979
(0.0171)
1.0037
(0.0252)
1.0120
(0.1207)

0.0178
(0.0003)
0.0252
(0.0004)
0.1203
(0.0019)

0.0133
(0.0016)
0.0141
(0.0017)
0.0553
(0.0067)

0.0150
(0.0022)
0.0158
(0.0026)
0.0646
(0.0176)

0.0168
(0.0058)
0.0208
(0.0076)
0.0910
(0.0377)

0.9986
(0.0144)
1.0031
(0.0179)
1.0014
(0.0440)

0.0147
(0.0001)
0.0182
(0.0002)
0.0448
(0.0003)

93

Table B.11 Small panel with G = 6, T = 4. Case 3.1: x2it ∼ N (gt/6, 1) + zi , ngt = 200,
sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and cluster-robust
standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9909
(0.1623)
1.0079
(0.0870)
1.0113
(0.2880)

0.1590
(0.0117)
0.0890
(0.0037)
0.2868
(0.0127)

0.1552
(0.0349)
0.0868
(0.0185)
0.2798
(0.0594)

0.1393
(0.0425)
0.0825
(0.0210)
0.2625
(0.0714)

0.1423
(0.0603)
0.0873
(0.0329)
0.2991
(0.1095)

0.9911
(0.1623)
1.0077
(0.0870)
1.0123
(0.2877)

0.1585
(0.0117)
0.0887
(0.0037)
0.2858
(0.0127)

0.9980
(0.0430)
1.0070
(0.0804)
1.0131
(0.2794)

0.0463
(0.0012)
0.0827
(0.0027)
0.2780
(0.0115)

0.0457
(0.0054)
0.0820
(0.0097)
0.2757
(0.0336)

0.0508
(0.0073)
0.0670
(0.0159)
0.2210
(0.0535)

0.0496
(0.0159)
0.0704
(0.0269)
0.2490
(0.0888)

0.9979
(0.0437)
1.0065
(0.0803)
1.0138
(0.2780)

0.0452
(0.0012)
0.0818
(0.0027)
0.2749
(0.0113)

0.9965
(0.0421)
1.0109
(0.1051)
1.0141
(0.3642)

0.0436
(0.0020)
0.1090
(0.0048)
0.3598
(0.0184)

0.0488
(0.0092)
0.0688
(0.0129)
0.1914
(0.0358)

0.0490
(0.0094)
0.0712
(0.0177)
0.2046
(0.0669)

0.0431
(0.0154)
0.0886
(0.0343)
0.2767
(0.1251)

0.9981
(0.0324)
1.0064
(0.0801)
1.0140
(0.2797)

0.0327
(0.0007)
0.0816
(0.0026)
0.2749
(0.0113)

1.0028
(0.1629)
1.0036
(0.0467)
1.0203
(0.2919)

0.1633
(0.0111)
0.0465
(0.0014)
0.2961
(0.0125)

0.1213
(0.0187)
0.0353
(0.0049)
0.2343
(0.0328)

0.1105
(0.0252)
0.0367
(0.0061)
0.2202
(0.0442)

0.1241
(0.0517)
0.0400
(0.0140)
0.2588
(0.0881)

0.9952
(0.1465)
1.0034
(0.0433)
1.0180
(0.2629)

0.1426
(0.0092)
0.0417
(0.0010)
0.2638
(0.0103)

0.9917
(0.1605)
1.0102
(0.0836)
1.0034
(0.1576)

0.1583
(0.0112)
0.0865
(0.0033)
0.1584
(0.0037)

0.1006
(0.0182)
0.0545
(0.0091)
0.1277
(0.0210)

0.1008
(0.0264)
0.0564
(0.0128)
0.1303
(0.0278)

0.1172
(0.0498)
0.0695
(0.0268)
0.1417
(0.0498)

0.9920
(0.1498)
1.0088
(0.0795)
1.0010
(0.0992)

0.1477
(0.0098)
0.0813
(0.0029)
0.1010
(0.0017)

0.9980
(0.0376)
1.0071
(0.0550)
1.0122
(0.2666)

0.0402
(0.0013)
0.0564
(0.0018)
0.2656
(0.0100)

0.0296
(0.0037)
0.0314
(0.0039)
0.1223
(0.0153)

0.0335
(0.0050)
0.0350
(0.0059)
0.1432
(0.0408)

0.0379
(0.0125)
0.0458
(0.0171)
0.2011
(0.0858)

0.9981
(0.0330)
1.0036
(0.0421)
1.0013
(0.1007)

0.0321
(0.0007)
0.0397
(0.0008)
0.0984
(0.0015)

94

Table B.12 Small panel with G = 6, T = 4. Case 3.1: x2it ∼ N (gt/6, 1) + zi , ngt = 200,
sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and cluster-robust
standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0027
(0.0721)
0.9971
(0.0401)
1.0010
(0.1305)

0.0720
(0.0023)
0.0398
(0.0008)
0.1280
(0.0025)

0.0695
(0.0146)
0.0384
(0.0080)
0.1235
(0.0256)

0.0618
(0.0182)
0.0366
(0.0086)
0.1166
(0.0305)

0.0623
(0.0281)
0.0390
(0.0138)
0.1324
(0.0489)

1.0027
(0.0722)
0.9972
(0.0401)
1.0012
(0.1304)

0.0720
(0.0023)
0.0398
(0.0008)
0.1279
(0.0025)

1.0012
(0.0196)
0.9975
(0.0372)
1.0004
(0.1286)

0.0206
(0.0002)
0.0373
(0.0005)
0.1258
(0.0023)

0.0203
(0.0024)
0.0369
(0.0045)
0.1245
(0.0150)

0.0227
(0.0034)
0.0295
(0.0069)
0.0977
(0.0240)

0.0222
(0.0073)
0.0320
(0.0115)
0.1110
(0.0398)

1.0012
(0.0197)
0.9976
(0.0372)
1.0005
(0.1283)

0.0205
(0.0002)
0.0372
(0.0005)
0.1256
(0.0023)

1.0005
(0.0186)
0.9963
(0.0476)
1.0004
(0.1665)

0.0192
(0.0004)
0.0490
(0.0010)
0.1621
(0.0038)

0.0216
(0.0041)
0.0305
(0.0057)
0.0849
(0.0161)

0.0216
(0.0040)
0.0317
(0.0081)
0.0912
(0.0305)

0.0189
(0.0069)
0.0389
(0.0149)
0.1240
(0.0578)

1.0000
(0.0145)
0.9978
(0.0372)
1.0000
(0.1274)

0.0148
(0.0001)
0.0371
(0.0005)
0.1255
(0.0023)

1.0004
(0.0751)
0.9989
(0.0211)
1.0005
(0.1350)

0.0765
(0.0024)
0.0207
(0.0003)
0.1335
(0.0025)

0.0563
(0.0076)
0.0158
(0.0021)
0.1057
(0.0139)

0.0511
(0.0113)
0.0165
(0.0026)
0.0993
(0.0182)

0.0573
(0.0254)
0.0180
(0.0060)
0.1158
(0.0403)

1.0003
(0.0673)
0.9998
(0.0190)
0.9980
(0.1212)

0.0675
(0.0019)
0.0188
(0.0002)
0.1199
(0.0021)

1.0028
(0.0730)
0.9965
(0.0391)
1.0027
(0.0713)

0.0727
(0.0023)
0.0388
(0.0007)
0.0705
(0.0007)

0.0455
(0.0075)
0.0242
(0.0039)
0.0564
(0.0091)

0.0451
(0.0115)
0.0251
(0.0053)
0.0579
(0.0120)

0.0518
(0.0233)
0.0310
(0.0116)
0.0628
(0.0221)

1.0026
(0.0699)
0.9971
(0.0369)
1.0032
(0.0461)

0.0703
(0.0022)
0.0370
(0.0006)
0.0454
(0.0003)

1.0004
(0.0172)
0.9983
(0.0251)
1.0008
(0.1228)

0.0178
(0.0003)
0.0252
(0.0004)
0.1200
(0.0020)

0.0132
(0.0016)
0.0140
(0.0017)
0.0549
(0.0066)

0.0149
(0.0022)
0.0156
(0.0025)
0.0640
(0.0181)

0.0170
(0.0057)
0.0203
(0.0072)
0.0902
(0.0386)

1.0000
(0.0145)
0.9998
(0.0183)
1.0026
(0.0460)

0.0147
(0.0001)
0.0182
(0.0002)
0.0449
(0.0003)

95

Table B.13 Small panel with G = 6, T = 4. Case 4.a: x2it ∼ N (gt/6, 1) + zi + fi ,
ngt = 200, sampling rate= 1%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0020
(0.1521)
0.9972
(0.0901)
1.0131
(0.2825)

0.1570
(0.0139)
0.0888
(0.0043)
0.2854
(0.0140)

0.1533
(0.0329)
0.0866
(0.0176)
0.2783
(0.0571)

0.1380
(0.0425)
0.0825
(0.0196)
0.2628
(0.0662)

0.1399
(0.0620)
0.0875
(0.0314)
0.2947
(0.1072)

1.0020
(0.1516)
0.9975
(0.0903)
1.0133
(0.2839)

0.1565
(0.0138)
0.0884
(0.0043)
0.2844
(0.0139)

1.0084
(0.0456)
0.9967
(0.0841)
1.0104
(0.2735)

0.0460
(0.0014)
0.0825
(0.0027)
0.2764
(0.0119)

0.0454
(0.0055)
0.0813
(0.0098)
0.2722
(0.0342)

0.0505
(0.0078)
0.0667
(0.0150)
0.2201
(0.0495)

0.0495
(0.0154)
0.0712
(0.0257)
0.2462
(0.0869)

1.0086
(0.0459)
0.9963
(0.0850)
1.0097
(0.2747)

0.0450
(0.0013)
0.0816
(0.0027)
0.2732
(0.0117)

1.2218
(0.0963)
1.0029
(0.1114)
0.9978
(0.3442)

0.0954
(0.0072)
0.1121
(0.0048)
0.3470
(0.0185)

0.0741
(0.0173)
0.0529
(0.0122)
0.1676
(0.0396)

0.0775
(0.0191)
0.0726
(0.0237)
0.2207
(0.0705)

0.0979
(0.0426)
0.0728
(0.0311)
0.2681
(0.1154)

1.3252
(0.0276)
0.9357
(0.0841)
1.0932
(0.2715)

0.0267
(0.0006)
0.0803
(0.0026)
0.2699
(0.0115)

1.0304
(0.1521)
0.9880
(0.0459)
1.0083
(0.2888)

0.1585
(0.0128)
0.0463
(0.0016)
0.2938
(0.0135)

0.1187
(0.0190)
0.0352
(0.0049)
0.2335
(0.0333)

0.1090
(0.0251)
0.0367
(0.0064)
0.2194
(0.0422)

0.1223
(0.0512)
0.0405
(0.0137)
0.2572
(0.0922)

1.0463
(0.1369)
0.9887
(0.0420)
1.0252
(0.2608)

0.1377
(0.0107)
0.0415
(0.0012)
0.2622
(0.0112)

1.0055
(0.1523)
0.9975
(0.0861)
1.0098
(0.1552)

0.1557
(0.0133)
0.0863
(0.0038)
0.1587
(0.0043)

0.0990
(0.0177)
0.0542
(0.0088)
0.1272
(0.0206)

0.0995
(0.0271)
0.0564
(0.0120)
0.1304
(0.0268)

0.1154
(0.0500)
0.0691
(0.0252)
0.1410
(0.0490)

1.0488
(0.1402)
0.9894
(0.0805)
1.0140
(0.1008)

0.1426
(0.0110)
0.0811
(0.0033)
0.1013
(0.0023)

1.1804
(0.0720)
0.9944
(0.0890)
0.9829
(0.3180)

0.0713
(0.0056)
0.0897
(0.0033)
0.3207
(0.0151)

0.0412
(0.0074)
0.0309
(0.0054)
0.1070
(0.0196)

0.0544
(0.0116)
0.0520
(0.0160)
0.1838
(0.0584)

0.0698
(0.0279)
0.0576
(0.0234)
0.2289
(0.0978)

1.3193
(0.0285)
0.9766
(0.0416)
1.0171
(0.0999)

0.0263
(0.0006)
0.0393
(0.0009)
0.0975
(0.0015)

96

Table B.14 Small panel with G = 6, T = 4. Case 4.b: x2it ∼ N (gt/6, 1) + zi + fi ,
ngt = 1000, sampling rate= 1%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9989
(0.0716)
1.0027
(0.0391)
1.0081
(0.1274)

0.0719
(0.0029)
0.0398
(0.0009)
0.1283
(0.0027)

0.0697
(0.0145)
0.0385
(0.0079)
0.1244
(0.0256)

0.0627
(0.0181)
0.0366
(0.0085)
0.1174
(0.0309)

0.0642
(0.0284)
0.0391
(0.0146)
0.1332
(0.0488)

0.9989
(0.0715)
1.0028
(0.0392)
1.0083
(0.1277)

0.0719
(0.0029)
0.0397
(0.0009)
0.1282
(0.0027)

0.9996
(0.0191)
1.0026
(0.0356)
1.0082
(0.1244)

0.0206
(0.0003)
0.0372
(0.0005)
0.1261
(0.0023)

0.0205
(0.0025)
0.0371
(0.0045)
0.1258
(0.0154)

0.0229
(0.0034)
0.0295
(0.0067)
0.0982
(0.0243)

0.0223
(0.0073)
0.0317
(0.0121)
0.1118
(0.0403)

0.9996
(0.0191)
1.0026
(0.0357)
1.0085
(0.1249)

0.0205
(0.0003)
0.0371
(0.0005)
0.1259
(0.0023)

1.2166
(0.0423)
1.0113
(0.0491)
1.0149
(0.1568)

0.0421
(0.0014)
0.0500
(0.0009)
0.1559
(0.0035)

0.0405
(0.0082)
0.0286
(0.0057)
0.0910
(0.0182)

0.0458
(0.0106)
0.0357
(0.0104)
0.1077
(0.0325)

0.0711
(0.0254)
0.0402
(0.0145)
0.1274
(0.0546)

1.3220
(0.0120)
0.9385
(0.0353)
1.1027
(0.1224)

0.0120
(0.0001)
0.0365
(0.0005)
0.1241
(0.0023)

1.0097
(0.0771)
1.0017
(0.0206)
1.0115
(0.1325)

0.0758
(0.0030)
0.0207
(0.0003)
0.1337
(0.0026)

0.0559
(0.0079)
0.0158
(0.0022)
0.1058
(0.0147)

0.0509
(0.0110)
0.0165
(0.0028)
0.0991
(0.0195)

0.0569
(0.0243)
0.0181
(0.0065)
0.1159
(0.0408)

1.0091
(0.0653)
1.0024
(0.0186)
1.0087
(0.1216)

0.0671
(0.0025)
0.0188
(0.0003)
0.1201
(0.0022)

1.0024
(0.0710)
1.0031
(0.0382)
1.0038
(0.0687)

0.0724
(0.0029)
0.0387
(0.0008)
0.0705
(0.0008)

0.0455
(0.0076)
0.0242
(0.0040)
0.0567
(0.0093)

0.0457
(0.0114)
0.0251
(0.0053)
0.0580
(0.0121)

0.0536
(0.0236)
0.0314
(0.0117)
0.0635
(0.0219)

1.0090
(0.0689)
1.0019
(0.0371)
1.0017
(0.0442)

0.0698
(0.0027)
0.0369
(0.0007)
0.0454
(0.0004)

1.1734
(0.0317)
1.0038
(0.0395)
0.9986
(0.1451)

0.0314
(0.0011)
0.0401
(0.0006)
0.1446
(0.0028)

0.0235
(0.0031)
0.0174
(0.0023)
0.0608
(0.0081)

0.0345
(0.0060)
0.0252
(0.0070)
0.0877
(0.0268)

0.0533
(0.0158)
0.0304
(0.0111)
0.1063
(0.0455)

1.3182
(0.0121)
0.9883
(0.0182)
1.0073
(0.0444)

0.0120
(0.0001)
0.0180
(0.0002)
0.0444
(0.0003)

97

Table B.15 Small panel with G = 6, T = 4. Case 4.1: x2it ∼ N (gt/6, 1) + zi + fi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0153
(0.1599)
1.0031
(0.0867)
1.0184
(0.2879)

0.1575
(0.0143)
0.0890
(0.0043)
0.2869
(0.0135)

0.1537
(0.0356)
0.0868
(0.0187)
0.2799
(0.0597)

0.1388
(0.0427)
0.0825
(0.0211)
0.2628
(0.0718)

0.1440
(0.0631)
0.0876
(0.0331)
0.2990
(0.1099)

1.0155
(0.1598)
1.0029
(0.0866)
1.0194
(0.2877)

0.1569
(0.0142)
0.0887
(0.0043)
0.2859
(0.0135)

1.0048
(0.0431)
1.0057
(0.0804)
1.0149
(0.2792)

0.0462
(0.0014)
0.0827
(0.0027)
0.2779
(0.0115)

0.0455
(0.0054)
0.0820
(0.0097)
0.2756
(0.0336)

0.0506
(0.0074)
0.0670
(0.0159)
0.2210
(0.0535)

0.0494
(0.0159)
0.0704
(0.0270)
0.2488
(0.0889)

1.0048
(0.0437)
1.0052
(0.0803)
1.0157
(0.2778)

0.0451
(0.0014)
0.0818
(0.0027)
0.2748
(0.0113)

1.2218
(0.0947)
1.0175
(0.1094)
1.0190
(0.3531)

0.0951
(0.0071)
0.1119
(0.0048)
0.3481
(0.0175)

0.0728
(0.0188)
0.0522
(0.0129)
0.1655
(0.0410)

0.0752
(0.0191)
0.0715
(0.0243)
0.2170
(0.0755)

0.0947
(0.0420)
0.0722
(0.0320)
0.2634
(0.1217)

1.3194
(0.0266)
0.9439
(0.0790)
1.1004
(0.2737)

0.0266
(0.0006)
0.0806
(0.0026)
0.2714
(0.0112)

1.0478
(0.1584)
0.9997
(0.0465)
1.0287
(0.2908)

0.1585
(0.0127)
0.0463
(0.0016)
0.2953
(0.0130)

0.1185
(0.0191)
0.0352
(0.0049)
0.2339
(0.0328)

0.1092
(0.0256)
0.0366
(0.0061)
0.2199
(0.0441)

0.1239
(0.0524)
0.0400
(0.0139)
0.2586
(0.0884)

1.0572
(0.1415)
1.0003
(0.0431)
1.0252
(0.2615)

0.1379
(0.0106)
0.0415
(0.0012)
0.2631
(0.0108)

1.0224
(0.1568)
1.0056
(0.0832)
1.0083
(0.1572)

0.1557
(0.0136)
0.0864
(0.0038)
0.1583
(0.0044)

0.0994
(0.0187)
0.0545
(0.0092)
0.1278
(0.0212)

0.1000
(0.0266)
0.0565
(0.0128)
0.1303
(0.0279)

0.1179
(0.0514)
0.0698
(0.0270)
0.1418
(0.0499)

1.0573
(0.1437)
0.9976
(0.0789)
1.0035
(0.0988)

0.1430
(0.0116)
0.0811
(0.0033)
0.1010
(0.0022)

1.1794
(0.0707)
1.0090
(0.0875)
1.0036
(0.3253)

0.0710
(0.0057)
0.0896
(0.0032)
0.3215
(0.0143)

0.0407
(0.0078)
0.0306
(0.0057)
0.1062
(0.0198)

0.0531
(0.0114)
0.0514
(0.0163)
0.1814
(0.0624)

0.0677
(0.0273)
0.0573
(0.0243)
0.2260
(0.1035)

1.3142
(0.0275)
0.9894
(0.0424)
1.0073
(0.1018)

0.0262
(0.0006)
0.0393
(0.0009)
0.0974
(0.0015)

98

Table B.16 Small panel with G = 6, T = 4. Case 4.1: x2it ∼ N (gt/6, 1) + zi + fi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0079
(0.0721)
0.9961
(0.0401)
1.0026
(0.1305)

0.0718
(0.0028)
0.0398
(0.0009)
0.1279
(0.0027)

0.0693
(0.0146)
0.0384
(0.0079)
0.1235
(0.0255)

0.0615
(0.0181)
0.0365
(0.0086)
0.1165
(0.0305)

0.0620
(0.0280)
0.0390
(0.0138)
0.1321
(0.0488)

1.0079
(0.0721)
0.9961
(0.0400)
1.0027
(0.1304)

0.0717
(0.0028)
0.0398
(0.0009)
0.1279
(0.0027)

1.0025
(0.0196)
0.9972
(0.0372)
1.0008
(0.1285)

0.0205
(0.0003)
0.0373
(0.0006)
0.1258
(0.0023)

0.0203
(0.0024)
0.0369
(0.0045)
0.1244
(0.0150)

0.0226
(0.0034)
0.0295
(0.0069)
0.0977
(0.0240)

0.0221
(0.0073)
0.0320
(0.0115)
0.1109
(0.0398)

1.0026
(0.0196)
0.9973
(0.0372)
1.0009
(0.1283)

0.0204
(0.0003)
0.0372
(0.0006)
0.1255
(0.0023)

1.2219
(0.0420)
1.0015
(0.0490)
0.9986
(0.1587)

0.0422
(0.0015)
0.0501
(0.0010)
0.1561
(0.0036)

0.0397
(0.0079)
0.0281
(0.0056)
0.0895
(0.0179)

0.0445
(0.0105)
0.0352
(0.0104)
0.1057
(0.0324)

0.0677
(0.0250)
0.0389
(0.0138)
0.1275
(0.0540)

1.3238
(0.0121)
0.9336
(0.0367)
1.0948
(0.1257)

0.0120
(0.0001)
0.0366
(0.0005)
0.1238
(0.0023)

1.0106
(0.0747)
0.9980
(0.0211)
1.0024
(0.1349)

0.0759
(0.0029)
0.0207
(0.0003)
0.1335
(0.0027)

0.0560
(0.0076)
0.0158
(0.0021)
0.1057
(0.0139)

0.0509
(0.0111)
0.0165
(0.0026)
0.0992
(0.0181)

0.0570
(0.0251)
0.0180
(0.0060)
0.1158
(0.0402)

1.0148
(0.0670)
0.9991
(0.0189)
0.9997
(0.1211)

0.0669
(0.0024)
0.0188
(0.0003)
0.1198
(0.0023)

1.0094
(0.0729)
0.9955
(0.0391)
1.0038
(0.0713)

0.0723
(0.0028)
0.0388
(0.0008)
0.0705
(0.0009)

0.0453
(0.0075)
0.0241
(0.0039)
0.0564
(0.0091)

0.0449
(0.0115)
0.0251
(0.0053)
0.0578
(0.0119)

0.0515
(0.0233)
0.0310
(0.0116)
0.0627
(0.0220)

1.0185
(0.0696)
0.9944
(0.0368)
1.0038
(0.0461)

0.0697
(0.0026)
0.0370
(0.0007)
0.0454
(0.0005)

1.1771
(0.0312)
0.9954
(0.0395)
0.9840
(0.1473)

0.0316
(0.0012)
0.0402
(0.0007)
0.1447
(0.0029)

0.0232
(0.0030)
0.0172
(0.0022)
0.0601
(0.0078)

0.0336
(0.0058)
0.0248
(0.0071)
0.0861
(0.0270)

0.0513
(0.0157)
0.0294
(0.0103)
0.1059
(0.0456)

1.3198
(0.0123)
0.9849
(0.0185)
1.0069
(0.0458)

0.0120
(0.0001)
0.0180
(0.0002)
0.0444
(0.0003)

99

Table B.17 Small panel with G = 6, T = 4. Case 5.a: x2it ∼ N (gt/2, 1) + zi + fi ,
ngt = 200, sampling rate= 1%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9951
(0.0524)
1.0007
(0.0906)
1.0074
(0.2817)

0.0539
(0.0021)
0.0887
(0.0033)
0.2843
(0.0129)

0.0527
(0.0106)
0.0866
(0.0175)
0.2777
(0.0569)

0.0470
(0.0142)
0.0825
(0.0194)
0.2625
(0.0661)

0.0472
(0.0211)
0.0875
(0.0316)
0.2945
(0.1070)

0.9951
(0.0522)
1.0010
(0.0908)
1.0078
(0.2834)

0.0537
(0.0021)
0.0884
(0.0032)
0.2834
(0.0128)

1.0029
(0.0360)
0.9966
(0.0863)
1.0107
(0.2754)

0.0359
(0.0010)
0.0847
(0.0028)
0.2777
(0.0120)

0.0354
(0.0043)
0.0833
(0.0101)
0.2733
(0.0345)

0.0358
(0.0053)
0.0687
(0.0143)
0.2200
(0.0494)

0.0352
(0.0117)
0.0727
(0.0256)
0.2460
(0.0865)

1.0030
(0.0361)
0.9962
(0.0871)
1.0099
(0.2765)

0.0352
(0.0010)
0.0836
(0.0028)
0.2744
(0.0118)

1.0240
(0.0782)
0.9991
(0.1104)
0.9839
(0.3425)

0.0787
(0.0044)
0.1118
(0.0048)
0.3461
(0.0188)

0.0380
(0.0098)
0.0487
(0.0126)
0.1403
(0.0369)

0.0456
(0.0120)
0.0692
(0.0237)
0.2007
(0.0720)

0.0469
(0.0193)
0.0668
(0.0319)
0.2496
(0.1127)

1.2598
(0.0252)
0.8485
(0.0867)
1.2129
(0.2806)

0.0247
(0.0007)
0.0824
(0.0028)
0.2737
(0.0119)

1.0004
(0.0552)
0.9908
(0.0461)
1.0021
(0.2886)

0.0572
(0.0022)
0.0463
(0.0013)
0.2932
(0.0129)

0.0422
(0.0059)
0.0353
(0.0049)
0.2331
(0.0332)

0.0381
(0.0087)
0.0368
(0.0063)
0.2190
(0.0420)

0.0426
(0.0187)
0.0405
(0.0136)
0.2554
(0.0911)

1.0026
(0.0496)
0.9907
(0.0422)
1.0212
(0.2612)

0.0502
(0.0018)
0.0415
(0.0010)
0.2617
(0.0107)

0.9950
(0.0534)
1.0009
(0.0867)
1.0061
(0.1549)

0.0544
(0.0022)
0.0863
(0.0029)
0.1583
(0.0036)

0.0343
(0.0056)
0.0541
(0.0087)
0.1269
(0.0205)

0.0343
(0.0091)
0.0563
(0.0119)
0.1303
(0.0269)

0.0395
(0.0173)
0.0689
(0.0253)
0.1407
(0.0491)

1.0018
(0.0515)
0.9971
(0.0818)
1.0121
(0.1011)

0.0522
(0.0020)
0.0815
(0.0027)
0.1008
(0.0017)

1.0226
(0.0740)
0.9952
(0.1000)
0.9774
(0.3333)

0.0745
(0.0040)
0.1014
(0.0041)
0.3369
(0.0176)

0.0227
(0.0052)
0.0284
(0.0065)
0.0857
(0.0199)

0.0388
(0.0101)
0.0565
(0.0189)
0.1775
(0.0634)

0.0409
(0.0166)
0.0575
(0.0265)
0.2239
(0.1009)

1.2477
(0.0255)
0.9575
(0.0421)
1.0247
(0.1011)

0.0240
(0.0006)
0.0399
(0.0009)
0.0986
(0.0016)

100

Table B.18 Small panel with G = 6, T = 4. Case 5.b: x2it ∼ N (gt/2, 1) + zi + fi ,
ngt = 1000, sampling rate= 1%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9985
(0.0239)
1.0034
(0.0391)
1.0072
(0.1273)

0.0241
(0.0004)
0.0397
(0.0006)
0.1282
(0.0024)

0.0233
(0.0048)
0.0385
(0.0079)
0.1244
(0.0256)

0.0209
(0.0060)
0.0366
(0.0085)
0.1174
(0.0308)

0.0213
(0.0095)
0.0390
(0.0146)
0.1333
(0.0486)

0.9985
(0.0239)
1.0034
(0.0391)
1.0074
(0.1276)

0.0241
(0.0004)
0.0397
(0.0006)
0.1281
(0.0024)

0.9991
(0.0150)
1.0031
(0.0368)
1.0075
(0.1251)

0.0160
(0.0002)
0.0382
(0.0006)
0.1268
(0.0023)

0.0160
(0.0019)
0.0380
(0.0047)
0.1264
(0.0156)

0.0161
(0.0022)
0.0304
(0.0065)
0.0981
(0.0243)

0.0157
(0.0052)
0.0324
(0.0120)
0.1116
(0.0404)

0.9992
(0.0151)
1.0030
(0.0369)
1.0078
(0.1256)

0.0160
(0.0002)
0.0381
(0.0006)
0.1265
(0.0023)

1.0251
(0.0355)
1.0089
(0.0489)
1.0059
(0.1551)

0.0351
(0.0008)
0.0498
(0.0009)
0.1552
(0.0035)

0.0193
(0.0041)
0.0247
(0.0052)
0.0715
(0.0150)

0.0217
(0.0052)
0.0312
(0.0104)
0.0902
(0.0330)

0.0229
(0.0093)
0.0307
(0.0149)
0.1118
(0.0517)

1.2599
(0.0109)
0.8485
(0.0368)
1.2361
(0.1265)

0.0111
(0.0001)
0.0374
(0.0006)
0.1259
(0.0023)

1.0010
(0.0260)
1.0023
(0.0206)
1.0104
(0.1325)

0.0257
(0.0005)
0.0207
(0.0002)
0.1337
(0.0025)

0.0189
(0.0026)
0.0158
(0.0022)
0.1059
(0.0147)

0.0171
(0.0037)
0.0165
(0.0028)
0.0991
(0.0195)

0.0191
(0.0083)
0.0180
(0.0065)
0.1159
(0.0408)

0.9998
(0.0221)
1.0028
(0.0186)
1.0076
(0.1217)

0.0227
(0.0004)
0.0188
(0.0002)
0.1201
(0.0020)

0.9993
(0.0238)
1.0037
(0.0381)
1.0032
(0.0686)

0.0243
(0.0004)
0.0387
(0.0006)
0.0705
(0.0007)

0.0153
(0.0025)
0.0242
(0.0040)
0.0567
(0.0093)

0.0153
(0.0038)
0.0251
(0.0053)
0.0579
(0.0121)

0.0178
(0.0079)
0.0313
(0.0117)
0.0635
(0.0218)

0.9995
(0.0233)
1.0036
(0.0370)
1.0013
(0.0442)

0.0237
(0.0004)
0.0369
(0.0005)
0.0454
(0.0003)

1.0231
(0.0337)
1.0052
(0.0444)
0.9991
(0.1511)

0.0333
(0.0008)
0.0452
(0.0008)
0.1512
(0.0033)

0.0117
(0.0021)
0.0147
(0.0026)
0.0445
(0.0080)

0.0188
(0.0043)
0.0257
(0.0082)
0.0801
(0.0290)

0.0197
(0.0081)
0.0267
(0.0123)
0.1005
(0.0463)

1.2498
(0.0109)
0.9681
(0.0183)
1.0142
(0.0447)

0.0109
(0.0001)
0.0182
(0.0002)
0.0448
(0.0003)

101

Table B.19 Small panel with G = 6, T = 4. Case 5.1: x2it ∼ N (gt/2, 1) + zi + fi ,
ngt = 200, sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

0.9993
(0.0550)
1.0066
(0.0874)
1.0131
(0.2886)

0.0539
(0.0021)
0.0888
(0.0032)
0.2859
(0.0124)

0.0527
(0.0113)
0.0868
(0.0185)
0.2794
(0.0595)

0.0471
(0.0141)
0.0823
(0.0208)
0.2623
(0.0718)

0.0479
(0.0206)
0.0871
(0.0326)
0.2988
(0.1102)

0.9994
(0.0550)
1.0064
(0.0874)
1.0141
(0.2882)

0.0537
(0.0021)
0.0885
(0.0032)
0.2849
(0.0124)

1.0017
(0.0342)
1.0058
(0.0826)
1.0151
(0.2798)

0.0360
(0.0010)
0.0848
(0.0028)
0.2793
(0.0117)

0.0355
(0.0042)
0.0841
(0.0100)
0.2767
(0.0339)

0.0359
(0.0051)
0.0688
(0.0154)
0.2209
(0.0534)

0.0353
(0.0115)
0.0715
(0.0270)
0.2483
(0.0897)

1.0017
(0.0347)
1.0052
(0.0828)
1.0159
(0.2784)

0.0353
(0.0010)
0.0838
(0.0028)
0.2761
(0.0115)

1.0280
(0.0797)
1.0159
(0.1086)
1.0101
(0.3527)

0.0785
(0.0042)
0.1117
(0.0048)
0.3470
(0.0178)

0.0376
(0.0107)
0.0483
(0.0134)
0.1393
(0.0385)

0.0449
(0.0125)
0.0684
(0.0242)
0.1977
(0.0766)

0.0466
(0.0206)
0.0662
(0.0324)
0.2463
(0.1195)

1.2559
(0.0244)
0.8574
(0.0821)
1.2250
(0.2813)

0.0245
(0.0006)
0.0825
(0.0027)
0.2751
(0.0116)

1.0062
(0.0569)
1.0022
(0.0468)
1.0234
(0.2921)

0.0571
(0.0022)
0.0464
(0.0013)
0.2950
(0.0123)

0.0420
(0.0059)
0.0353
(0.0049)
0.2338
(0.0328)

0.0381
(0.0086)
0.0367
(0.0061)
0.2197
(0.0443)

0.0432
(0.0184)
0.0399
(0.0139)
0.2577
(0.0886)

1.0058
(0.0514)
1.0023
(0.0432)
1.0203
(0.2627)

0.0502
(0.0018)
0.0416
(0.0010)
0.2628
(0.0102)

1.0004
(0.0550)
1.0088
(0.0839)
1.0047
(0.1579)

0.0543
(0.0021)
0.0864
(0.0028)
0.1580
(0.0037)

0.0344
(0.0058)
0.0544
(0.0091)
0.1276
(0.0211)

0.0344
(0.0087)
0.0563
(0.0127)
0.1303
(0.0279)

0.0400
(0.0171)
0.0694
(0.0267)
0.1418
(0.0502)

1.0051
(0.0536)
1.0049
(0.0800)
1.0017
(0.0993)

0.0522
(0.0020)
0.0815
(0.0026)
0.1007
(0.0017)

1.0262
(0.0755)
1.0115
(0.0984)
1.0035
(0.3431)

0.0743
(0.0038)
0.1012
(0.0040)
0.3378
(0.0165)

0.0225
(0.0056)
0.0283
(0.0069)
0.0852
(0.0207)

0.0382
(0.0106)
0.0559
(0.0194)
0.1750
(0.0674)

0.0409
(0.0176)
0.0572
(0.0271)
0.2212
(0.1072)

1.2446
(0.0250)
0.9700
(0.0428)
1.0149
(0.1026)

0.0239
(0.0006)
0.0399
(0.0009)
0.0985
(0.0017)

102

Table B.20 Small panel with G = 6, T = 4. Case 5.2: x2it ∼ N (gt/2, 1) + zi + fi ,
ngt = 1000, sampling rate= 0.2%. sen , ser and sec are the non-roust, robust and
cluster-robust standard errors, receptively.
MD Identity

IV: none

x2
x3
x4

IV: z

x2
x3
x4

IV: x2

x2
x3
x4

IV: x3

x2
x3
x4

IV: x4

x2
x3
x4

IV: all

x2
x3
x4

MD Optimal

βˇ

ˇ
se(β)

ˇ
sen (β)

ˇ
ser (β)

ˇ
sec (β)

βˆ

ˆ
se(β)

1.0015
(0.0241)
0.9967
(0.0401)
1.0016
(0.1304)

0.0241
(0.0004)
0.0398
(0.0006)
0.1279
(0.0024)

0.0232
(0.0048)
0.0384
(0.0079)
0.1235
(0.0255)

0.0206
(0.0060)
0.0366
(0.0086)
0.1166
(0.0305)

0.0207
(0.0094)
0.0390
(0.0138)
0.1324
(0.0489)

1.0015
(0.0241)
0.9968
(0.0401)
1.0018
(0.1304)

0.0241
(0.0004)
0.0398
(0.0006)
0.1278
(0.0024)

1.0018
(0.0151)
0.9966
(0.0383)
1.0017
(0.1293)

0.0160
(0.0002)
0.0383
(0.0006)
0.1265
(0.0023)

0.0158
(0.0019)
0.0378
(0.0046)
0.1251
(0.0151)

0.0160
(0.0022)
0.0304
(0.0066)
0.0975
(0.0239)

0.0157
(0.0054)
0.0327
(0.0114)
0.1107
(0.0399)

1.0019
(0.0151)
0.9967
(0.0384)
1.0018
(0.1290)

0.0159
(0.0002)
0.0382
(0.0006)
0.1262
(0.0023)

1.0291
(0.0343)
0.9982
(0.0489)
0.9876
(0.1574)

0.0351
(0.0008)
0.0500
(0.0010)
0.1554
(0.0036)

0.0190
(0.0039)
0.0245
(0.0050)
0.0708
(0.0146)

0.0214
(0.0050)
0.0309
(0.0106)
0.0891
(0.0336)

0.0219
(0.0086)
0.0296
(0.0143)
0.1111
(0.0533)

1.2617
(0.0109)
0.8421
(0.0381)
1.2310
(0.1299)

0.0111
(0.0001)
0.0375
(0.0006)
0.1255
(0.0023)

1.0014
(0.0253)
0.9985
(0.0212)
1.0012
(0.1350)

0.0257
(0.0004)
0.0207
(0.0003)
0.1334
(0.0025)

0.0189
(0.0025)
0.0158
(0.0021)
0.1057
(0.0139)

0.0171
(0.0037)
0.0165
(0.0026)
0.0993
(0.0181)

0.0192
(0.0085)
0.0180
(0.0060)
0.1158
(0.0403)

1.0019
(0.0226)
0.9995
(0.0190)
0.9987
(0.1211)

0.0227
(0.0003)
0.0188
(0.0002)
0.1198
(0.0021)

1.0017
(0.0244)
0.9962
(0.0392)
1.0031
(0.0713)

0.0243
(0.0004)
0.0388
(0.0006)
0.0705
(0.0007)

0.0152
(0.0025)
0.0241
(0.0039)
0.0564
(0.0091)

0.0151
(0.0038)
0.0251
(0.0053)
0.0579
(0.0120)

0.0173
(0.0078)
0.0309
(0.0117)
0.0628
(0.0221)

1.0028
(0.0237)
0.9961
(0.0369)
1.0034
(0.0460)

0.0237
(0.0004)
0.0370
(0.0005)
0.0454
(0.0004)

1.0270
(0.0325)
0.9952
(0.0445)
0.9816
(0.1535)

0.0333
(0.0008)
0.0453
(0.0008)
0.1514
(0.0033)

0.0116
(0.0020)
0.0146
(0.0026)
0.0440
(0.0078)

0.0185
(0.0042)
0.0254
(0.0084)
0.0790
(0.0297)

0.0188
(0.0074)
0.0258
(0.0118)
0.0997
(0.0478)

1.2514
(0.0110)
0.9645
(0.0187)
1.0142
(0.0461)

0.0109
(0.0001)
0.0182
(0.0002)
0.0448
(0.0003)

103

BIBLIOGRAPHY

104

BIBLIOGRAPHY

Arellano, Manuel. 1987. “PRACTITIONERS’ CORNER: Computing Robust Standard
Errors for Within-groups Estimators.” Oxford bulletin of Economics and Statistics,
49(4): 431–434.
Breusch, Trevor, Hailong Qian, Peter Schmidt, and Donald Wyhowski. 1999.
“Redundancy of moment conditions.” Journal of econometrics, 91(1): 89–111.
Collado, M Dolores. 1997. “Estimating dynamic models from time series of independent
cross-sections.” Journal of Econometrics, 82(1): 37–62.
Deaton, Angus. 1985. “Panel data from time series of cross-sections.” Journal of econometrics, 30(1): 109–126.
Girma, Sourafel. 2000. “A quasi-differencing approach to dynamic modelling from a time
series of independent cross-sections.” Journal of Econometrics, 98(2): 365–383.
Hansen, Christian B. 2007a. “Asymptotic properties of a robust variance matrix estimator
for panel data when T is large.” Journal of Econometrics, 141(2): 597–620.
Hansen, Christian B. 2007b. “Generalized least squares inference in panel and multilevel
models with serial correlation and fixed effects.” Journal of Econometrics, 140(2): 670–694.
Heckman, James J, and V Joseph Hotz. 1989. “Choosing among alternative nonexperimental methods for estimating the impact of social programs: The case of manpower
training.” Journal of the American statistical Association, 84(408): 862–874.
Imbens, Guido, and Jeffrey M Wooldridge. 2007. What’s new in econometrics? NBER.
Kezdi, Gabor. 2003. “Robust standard error estimation in fixed-effects panel models.”
Available at SSRN 596988.
McKenzie, David J. 2004. “Asymptotic theory for heterogeneous dynamic pseudo-panels.”
Journal of Econometrics, 120(2): 235–262.
Moffitt, Robert. 1993. “Identification and estimation of dynamic models with a time series
of repeated cross-sections.” Journal of Econometrics, 59(1): 99–123.
Newey, Whitney K, and Daniel McFadden. 1994. “Large sample estimation and hypothesis testing.” Handbook of econometrics, 4: 2111–2245.
Verbeek, Marno. 2008. “Pseudo-panels and repeated cross-sections.” In The Econometrics
of Panel Data. 369–383. Springer.
Verbeek, Marno, and Francis Vella. 2005. “Estimating dynamic models from repeated
cross-sections.” Journal of econometrics, 127(1): 83–102.
105

Wooldridge, Jeffrey M. 2010. Econometric Analysis of Cross Section and Panel Data. .
2nd ed., Boston MA:MIT Press.

106

CHAPTER 3
A FLEXIBLE PLUG-IN G-FORMULA FOR CONTROLLED DIRECT
EFFECTS IN MEDIATION ANALYSIS

3.1

Introduction

In the literature of epidemiology and biostatistics, the term g-methods (e.g., Westreich et al.,
2012) are often used to collectively refer to g-formula (Robins, 1986), g-estimation of structural nested models (Robins, 1998), and inverse probability weighting of marginal structural
models (Horvitz and Thompson, 1952; Robins, 1989; Hernan and Robins, 2015), all of which
are useful approaches in estimating the effects of time-varying treatments in the presence of
time-varying confounders. The g-formula in its original form is non-parametric and is the
foundation for the other two. While non-parametric g-formula is flexible in its model specification, it is also quite demanding on data. Therefore, we often introduce semi-parametric or
parametric modeling. Despite the fact that parametric models are almost always misspecified, the parametric g-formula often yields satisfactory estimates as long as the specified
models are reasonably flexible. However, most applications of the parametric g-formula (e.g.,
Westreich et al., 2012; Taubman et al., 2009; Young et al., 2011; Danaei et al., 2013; Lajous
et al., 2013; Garcia-Aymerich et al., 2014) still use Monte Carlo integration to calculate because closed-form expressions of the treatment effects of interest are either non-existent or
tedious to derive.
The application of these g-method to mediation analysis is straightforward as mediation
analysis is conceptually equivalent to a sequential treatment of two periods. Since Robins
and Greenland (1992) conceptualize the natural and controlled effects in mediation analysis
using the potential outcome (counterfactual) framework, several mediation analysis methods are developed from g-methods. These methods include, among others, the parametric
g-formula in Daniel, De Stavola and Cousens (2011) and Valeri and Vanderweele (2013),

107

and the parametric version of the sequential g-estimation of structural nested mean models
(SNMMs) studied by Vansteelandt (2009). As in the case of g-methods, these methods also
rely on Monte Carlo integration to calculate the treatment effects of interest, which can be
computationally demanding. Meanwhile, since there is no closed-form expressions, we almost
always need to bootstrap the standard errors, which raises the computation intensity rapidly.
When the estimation itself is time-consuming, the problem gets amplified even further. This
includes, but not limited to, maximum likelihood estimation when it converges slowly and
most semi-parametric or non-parametric techniques that require cross validation for tuning
parameter selection.
In view of this limitation, in this chapter we propose a so called flexible plug-in g-formula
for controlled direct effects (CDE) in mediation analysis. The key assumption needed is
that the conditional expectation of the outcome is linear in time-varying confounders. This
partial linearity allows us to replace the confounders with their fitted values, which results
in a plug-in estimator for CDE. At the same time, it also relaxes the fully linear assumptions
that are commonly used in empirical studies, which gives us more flexibility in choosing the
functional form of the outcome conditional mean. As a result, we have a better chance to
be closer to the true underlying model.
Besides the partial linearity assumption, another necessary condition for the consistency
of the flexible plug-in g-formula is the sequential ignorability assumption (Robins, 1986). To
check the robustness of the estimator to a particular violation of the sequential ignorability
assumption, we present a sensitivity analysis that is similar in spirit to that proposed by
Imai, Keele and Tingley (2010). The proposed estimator is evaluated in a small simulation
and its use is illustrated in a longitudinal cohort study.
The rest of this chapter is organized as follows. We first set up the counterfactual
mediation analysis framework in the second section. In the third section, we review the
general g-formula as well as the sequential g-estimation. In the fourth section, we present
in detail the flexible plug-in g-formula. In particular, it is compared to the sequential g-

108

estimation in some commonly used linear specifications. The next two sections outline the
sensitivity analysis and an empirical application, respectively. The last section concludes.

3.2

Framework

A causal mediation analysis is typically guided by a directed acyclic graphs (DAG) (Pearl,
2009). We use the DAG G in Figure 1 for illustration, but our method can be applied in
similar models that satisfy the assumptions below. Throughout this paper, a DAG is viewed
as a graphical representation of an underlying non-parametric structural equation model with
independent errors (NPSEM-IE) (Pearl, 2009).1 A DAG and the associated NPSEM-IE are
related as usual: “there is an equation for each variable in the model, specifying that variable
as a function of its parents in the graph” (Richardson and Robins, 2013). The counterfactual
outcomes, defined by intervening on certain variables in the NPSEM-IE model, are used in
constructing the CDEs of interest.
Specifically, assume we have a longitudinal study in which each respondent was interviewed 3 times at k = 0, 1, 2, with a one-time treatment A at k = 0. Each interview
generates data Lk . The purpose is to learn to what extent the treatment effect of A on Y
is mediated by a mediator M . (A, M ) are the intervention nodes, and {Lk : k = 0, 1, 2} are
observed non-intervention nodes or confounders. L0 contains all baseline information, and
L1 contains all post-treatment non-intervention nodes that occur before M and confounds
the mediator-outcome relationship. As noted in Pearl (2014), we cannot identify natural
mediation effects non-parametrically because of the existence of L1 . Hence, we focus on the
CDEs in this paper. The model allows for a type of harmless unobserved variables, collec1 Although the NPSEM-IE model makes many more counterfactual independence assump-

tions than, and is a strict submodel of, the finest fully randomized causally interpretable
structured tree graph (FFRCISTG) model of (Robins, 1986; Robins and Richardson, 2010),
the generality of the latter does not play an essential role in our paper. The adoption of
NPSEM-IE, however, makes the description of the model straightforward and easy to follow,
since it is built on the traditional structural equation models and imposes simple distribution
assumptions on the errors (exogenous variables).
109

Figure 3.1 A directed acyclic graph for a longitudinal study with three time points. (A, M )
are the intervention nodes, (L0 , L1 , Y ) the non-intervention nodes, and U0 the
unobservables.

G
tively denoted by U0 that are parents of the observed non-intervention nodes. Based on the
back-door criterion the observed non-intervention nodes block the confounding effects of U0
Pearl (2009).
Each node (including the unobservable U0 ) has an exogenous error (not shown in the
graph) attached solely to itself. For example, εY is the exogenous error for Y , as εU0 is for
U0 . All errors are unobserved and are assumed to be jointly independent in NPSEM-IE. The
independence assumption will be relaxed in our sensitivity analysis in which a more general
NPSEM allowing correlations between ε’s will be used.
Let Y am be the counterfactual outcome where A and M are intervened to be fixed at
a and m. In the context of NPSEM-IEs, such an intervention would correspond to the
operation of deleting the equations for A and M from the system and substituting A = a
and M = m in the system. This operation is called the do operation in Pearl (2009). In
this paper, we consider a binary A so that a ∈ {0, 1}. An extension to multi-valued A
is trivial. For a fixed m, we are interested in the CDE(m) defined as E(Y 1m − Y 0m ), or
E(Y (1, m) − Y (0, m)).

110

3.3

Existing Methods

In the context of DAG G, we discuss two existing methods for estimating the CDE. The first
method is the g-formula for the marginal distribution of Y am introduced in Robins (1986) and
revisited in Richardson and Robins (2013), which is the basis for the parametric g-formula
and in particular for the flexible plug-in estimator. The second method is the sequential
g-estimation Vansteelandt (2009), which is, as we will show, numerically equivalent to the
flexible plug-in estimator in certain linear cases.

3.3.1

The g-Formula

In Richardson and Robins (2013), the term g-formula refers to the unextended g-formula
of Robins (1986), the extended g-formula of Robins, Hernán and SiEBERT (2004), or the
g-formula for a sequence of treatments and a single response. Among the three, the last one
is of interest in the context of DAG G and the estimation of CDEs.
Let P (Y am = y) be the distribution of the potential response Y under the sequence
of interventions (A = a, M = m), from which we can construct the CDE. Assume the
consistency rule holds Robins (1994), that is, a potential response under a hypothetical
condition that happened to take place is precisely the observed response. In addition, the
following form of sequential ignorability condition from Robins (2000) is imposed:
Y am ⊥
⊥ A | L0 ,

f or all a, m

(3.1)

Y am ⊥
⊥ M | L0 , A, L1 ,
where ⊥
⊥ represents distributional independence. Note that condition (3.1) summarizes the
set of independence conditions for all possible values of a and m, which are needed to identify
CDEs for all possible values of M .
A logically equivalent algorithm of finding the right conditioning set of variables as the
sequential ignorability conditions in (3.1) is the sequential back-door criterion developed in

111

Figure 3.2 Subgraphs for G where an upper bar means arrows pointing to a node are
removed and an under bar means arrows emitting from a node are removed.

GAM

G(M )

112

Pearl and Robins (1995), which states
(Y ⊥
⊥ A | L0 )G

AM

,
(3.2)

(Y ⊥
⊥ M | L0 , A, L1 )G ,
M
where G(M ) denotes the subgraph obtained by removing from G all arrows emerging from M
(Figure 3.2, B), and GAM denotes the removal of both incoming arrows to M and outgoing
arrows from A (Figure 3.2, A).
Under either (3.1) or (3.2), the g-formula for the expected counterfactual outcome is
ˆ ˆ ˆ
E(Y am )

=
ˆ ˆ
=

yfY |L ,A,L ,M (y|l0 , a, l1 , m)fL |L (l1 |l0 , a)fL0 (l0 )dydl1 dl0
0
1
1 0
E(y|l0 , a, l1 , m)fL |L (l1 |l0 , a)fL0 (l0 )dl1 dl0
1 0

(3.3)
(3.4)

where, e.g, P (y|l0 , a, l1 , m) is shorthand for P (Y = y|L0 = l0 , A = a, L1 = l1 , M = m). See
Appendix A for detail. Then the CDE is
E(Y 1m − Y 0m )
ˆ ˆ
=
E(y|l0 , 1, l1 , m)fL |L (l1 |l0 , 1)fL0 (l0 )dl1 dl0
1 0
ˆ ˆ
−
E(y|l0 , 0, l1 , m)fL |L (l1 |l0 , )fL0 (l0 )dl1 dl0 .
1

0

(3.5)

Equation (3.3) is non-parametric in the sense that no parametric assumptions are made
yet for fY |L ,A,L ,M (y|l0 , a, l1 , m), fL |L (l1 |l0 , a) and fL0 (l0 ). Equation (3.4) adds an
0
1
1 0
additional assumption that the conditional mean E(y|l0 , a, l1 , m) exits and is finite.
Two straightforward estimation strategies to estimate E(Y am ) follow from equations
(3.3) and (3.4). The first strategy exploits equation (3.3), with either non-parametric or
parametric specification of the distributions. This strategy generally involves Monte Carlo
simulation and numerical integration for calculating CDEs (Daniel, De Stavola and Cousens,
2011; Imai, Keele and Tingley, 2010; Hicks and Tingley, 2011).
The second strategy exploits equation (3.4), which avoids the estimation of the density function fY |L ,A,L ,M (y|l0 , a, l1 , m). Instead, we only estimates the conditional mean
0
1
113

E (Y |l0 , a, l1 , m). We can keep the non-parametric feature or choose suitable parametric
models to reduce computation burdens. Many applications have shown that a properly chosen parametric method can provide good approximations (Westreich et al., 2012; Taubman
et al., 2009; Young et al., 2011; Danaei et al., 2013; Lajous et al., 2013; Garcia-Aymerich
et al., 2014). The estimator proposed in this chapter falls in the second estimation strategy
and imposes a particular parametric assumption on E (Y |l0 , a, l1 , m) that simplifies equation
(3.4) even further (details below).

3.3.2

The Sequential g-formula estimator

The sequential g-estimation for CDEs is a two-step estimator based on an SNMM (Vansteelandt, 2009). The idea is to first partial out the effect of the mediator on the outcome and
then regress the adjusted outcome on the treatment, the confounders, and possibly their
interactions to identify the direct effect. The sequential g-formula estimator assumes an
additive separable functional form in the conditional mean equation for Y
E(Y |l0 , a, l1 , m) = qA (l0 , a, l1 ; γ) + qM (l0 , a, l1 , m; γ)

(3.6)

where qA (·) and qM (·) are arbitrary known functions with finite dimensional parameter γ,
satisfying qM (l0 , a, l1 , m = 0; γ) = 0. For example, we can assume qA = γ0 + γA a + γL0 l0 +
γL1 l1 and qM = γM m. In addition, assume an SNMM for E(Y am − Y 0m |l0 ) = ϕA a where
ϕA is the CDE. Then the sequential g-estimation procedure is:
1. regress Y on (1, L0 , A, L1 , M ) and obtain the ordinary least square (OLS) estimator
γˆM for γM and generate Yˆ−M ≡ Y − γˆM M , and
2. regress Yˆ−M on (1, L0 , A). Denote by ϕˆA the OLS estimator for the coefficient of A.
It can be shown ϕˆA is a consistent estimator for ϕA .2
2 Under

the sequential ignorability conditions and additive separability (3.6), Vansteelandt (2009) gave the key identification result E [Y − qM (L0 , A, L1 , M ; γ)|L0 = l0 , A = a] =

114

See Appendices C and D for the validity of the estimation procedure.
Table 3.1 lists five typical examples of SNMMs compatible with DAG G that can be
estimated using the sequential g-formula estimator. The simple example above is Model 1 in
Table 3.1. The first three models were discussed in Vansteelandt (2009), and we added the
latter two models with more flexible specifications. One difficulty in applying the sequential
g-formula estimator is to find the proper qA (·) and qM (·) functions for a given SNMM, as
can been seen in Table 3.1. The derivation of standard errors of the estimator is also no
mean feat.
The sequential g-estimation is not always as simple as it looks in Model 1. For example,
in a model with up to two-way interactions (Model 5 in Table 3.1), we need to estimate
ϕAM , ϕAL0 and ϕAM L0 in E(Y 1m − Y 0m ) = ϕA + ϕAM m + ϕAL0 E(L0 ) + ϕAM L0 mE(L0 ).
It turns out that in this case only E(L1 |L0 = l0 , A = 0) = π0 + πL0 l0 is not enough to
identify ϕAM or ϕAM L0 . What is needed is the stronger assumption
E(L1 |L0 = l0 , A = a) = π0 + πL0 l0 + πA a + πAL0 a × l0
which implies (by setting γAM L0 = γAM L1 = 0 in Appendix B)
ϕAM = γAM + γM L1 πA ,
ϕAM L0 = γM L1 πAL0 ,
Clearly, the above equations show that ϕAM and ϕAM L0 cannot be obtained directly from
the first-step regression, since now an additional regression for E(L1 |L0 = l0 , A = a) is
needed to estimate π.

E Y a0 |L0 = l0 . We show that, in addition to the assumptions in Vansteelandt (2009), a
necessary and sufficient condition for the validity of the second-step regression is f (l0 , a) ≡
E(L1 |L0 = l0 , A = 0) = π0 + πL0 l0 . See Appendix B for detail.
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Table 3.1 Compare the plug-in estimator with the sequential g-estimator under different specifications for the outcome
conditional mean and different structural nested mean models.
Structural Nested Mean
Model (SNMM)
E(Y am − Y 0m |l0 )
1

ϕA a

2

ϕA a + ϕAM a · m

3

4

5

ϕA a + ϕAL0 a · l0

ϕA a + ϕAM a · m
+ϕAL0 a · l0

ϕA a + ϕAM a · m+
+ϕAL0 a · l0 + ϕAL0 a · l0
+ϕAM L0 a · m · l0

Sequential g-formula estimator
E(Y |l0 , a, l1 , m) =
qA (l0 , a, l1 ; γ) + qM (l0 , a, l1 , m; γ)
qA = γ0 + γA a + γL0 l0 + γL1 l1
qM = γM m
f (l0 , 0) = π0 + πL0 l0
qA = γ0 + γA a + γL0 l0 + γL1 l1
qM = γM m + γAM a · m
f (l0 , 0) = π0 + πL0 l0

Flexible Plug-in g-formula estimator
E(Y |l0 , a, l1 , m) =
h0 (a, m)l0 + h1 (a, m)l1 + h(a, m)
h0 = γL0 , h1 = γL1
h = γ0 + γA a + γM m
f (l0 , a) = π0 + πL0 l0 + πA a
h0 = γL0 , h1 = γL1
h = γ0 + γA a + γM m + γAM a · m
f (l0 , a) = π0 + πL0 l0 + πA a

qA = γ0 + γA a + γL0 l0 + γL1 l1
+γAL0 a · l0
qM = γM m
f (l0 , 0) = π0 + πL0 l0
qA = γ0 + γA a + γL0 l0 + γL1 l1
+γAL0 a · l0
qM = γM m + γAM a · m
f (l0 , 0) = π0 + πL0 l0

h0 = γL0 + γAL0 a, h1 = γL1
h = γ0 + γA a + γM m
f (l0 , a) = π0 + πL0 l0 + πA a
+πAL0 a · l0
h0 = γL0 + γAL0 a, h1 = γL1
h = γ0 + γA a + γM m + γAM a · m
f (l0 , a) = π0 + πL0 l0 + πA a
+πAL0 a · l0

qA = γ0 + γA a + γL0 l0 + γL1 l1
+γAL0 a · l0 + γAL1 a · l1
qM = γM m + γAM a · m + γM L0 m · l0
+γM L1 m · l1
f (l0 , a) = π0 + πL0 l0 + πA a
+πAL0 a · l0

h0 = γL0 + γAL0 a + γM L0 m
h1 = γL1 + γAL1 a + γM L1 m
h = γ0 + γA a + γM m + γAM a · m
f (l0 , a) = π0 + πL0 l0 + πA a
+πAL0 a · l0

Notation: qA ≡ qA (l0 , a, l1 ; γ), qM ≡ qM (l0 , a, l1 , m; γ), f (l0 , a) ≡ E(L1 |L0 = l0 , A = a).

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3.4
3.4.1

The Flexible Plug-in g-formula estimator
The Partial Linearity Assumption and the Plug-in g-formula estimator

Although the idea of using linear outcome conditional mean is not particularly new (Robins,
2000; Van der Wal et al., 2009), to the best of our knowledge, the flexible plug-in g-formula
estimator proposed here is the first to make full use of this idea. This parametric g-formula
has a closed-form expression for CDE and thus does not require numerical integration.
Specifically, let the conditional expectation of Y given (L0 , A, L1 , M ) be linear in L0 and
L1 , namely
E(Y |l0 , a, l1 , m) = h0 (a, m; γ)l0 + h1 (a, m; γ)l1 + h(a, m; γ)

(3.7)

where the hk ’s are arbitrary known functions of (a, m) known up to certain parameters γ,
for k = {0, 1, ∅}. This is of course a strong parametric assumption since it ignores any
interaction among confounders. We should think of Equation (3.7) as the first order Taylor
approximation to any function of (L0 , L1 ). The plug-in g-formula estimator can be extended
to include higher order terms of confounders.
Equations (3.4) and (3.7) lead to the proposed flexible plug-in g-formula estimator for
E(Y am ):
E(Y am ) = h0 (a, m; γ)E(L0 ) + h1 (a, m; γ)E [E(L1 |L0 , A = a)] + h(a, m; γ).

(3.8)

See Appendix A for proof.
The last column of Table 3.1 shows that, by varying the specifications of hk ’s, equation
(3.7) can provide the same specification on E(Y |l0 , a, l1 , m) as equation (3.6) in all the five
models there. The separability in qA and qM for the sequential g-formula estimator and the
estimating equation that is linear in confounders for the plug-in g-formula estimator do not
nest within each other. Neither estimator is strictly more flexible than the other. Note that
the unknown parameters γ in (3.7) are not the structural parameters that would appear
in the structural equation Y = fY (L0 , A, L1 , M, Y ) in the NPSEM-IE. Equation (3.7) is
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an estimating equation only. The parameters in this equation are not of direct interest in
general, but they eventually identify E(Y am ).
Note also that as long as equation (3.7) holds, the presence of U0 does not affect the
identification of CDE. However, we do need a model for E(L1 |L0 , A). This model needs not
be linear, and can potentially be semi- or non-parametric if the dimension of the conditioning
set is low. For example, if L1 is binary, a logistic model can be used. But in the application
below, we use the following linear model for simplicity:
E(L1 |L0 , A) = π0 + πA A + πL0 L0 + πAL0 A × L0 ,

(3.9)

E [E(L1 |L0 , A = a)] = π0 + πA a + πL0 E(L0 ) + πAL0 a × E(L0 ).

(3.10)

and thus

3.4.2

Estimation Procedure for the Flexible Plug-in g-formula estimator of
CDE

Given the discussion above, the CDE can be estimated as follows:
1. estimate γ in equation (3.7) using a proper method, e.g., a quasi-maximum likelihood
estimator, which is consistent given correctly specified conditional mean (Wooldridge,
2010); and
2. estimate E(L0 ) using a proper method, e.g., the sample mean, and get E(L0 ).
3. Estimate E(L1 |L0 , A) using a proper method and get E [E(L1 | L0 , A = a)]. For example, regress L1 on (1, A, L0 , AL0 ).
4. Plug E(L0 ) and E [E(L1 | L0 , A = a)] into (3.8) to obtain E(Y am ). Then CDE(m) =
E(Y am ) − E(Y 0m ).
CDE(m) is the flexible plug-in estimator for the CDE evaluated at m. Bootstrap can be
used to obtain the standard errors for the estimated CDEs.

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3.4.3

plim of Parametric g-Formula is the Flexible Plug-in g-formula estimator

By the law of large numbers, the flexible plug-in g-formula estimator of CDE is the plim of
the corresponding parametric g-formula in section 3.1. This is simply because the former is
an analytical solution for the integral which the latter is trying to evaluate via Monte Carlo
simulation. We verify this claim using Model 5 in Table 3.1 through a simulation study with
the aid of the Stata command gformula developed in Daniel, De Stavola and Cousens (2011).
In the simulation study, we let the number of Monte Carlo simulations used by the gformula
command increase towards infinity. The results show that the estimates obtained using the
gformula command indeed come closer and closer to those obtained using the flexible plug-in
g-formula as the simulations increase.
One reason for using Model 5 is that the specification is complex enough to make noticeable difference in computation time between the two methods. Obviously, if we are only
interested in a point estimate of the controlled direct effect, the flexible plug-in g-formula
does not gain us much. However, since we also need to obtain the standard errors for the
estimators, and bootstrap is often inevitable in g-methods, the flexible plug-in g-formula can
save considerable amount of computation time.

3.4.4

Flexible Plug-in g-formula estimator Is Numerically Equivalent to Sequential g-formula estimator

We show that, in each of the five models in Table 3.1, the flexible plug-in g-formula estimator
and the sequential g-formula estimator are numerically identical. See Appendix C for proof.
It is worth emphasizing the following two conditions that are met by each of the five models
in Table 3.1.
First, the SNMM used by the sequential g-formula estimator must be compatible with the
specification on E(Y |l0 , a, l1 , m). Note that a given SNMM and a compatible specification
on E(Y |l0 , a, l1 , m) implies a specification on E(L1 |L0 , A = 0) (or on E(L1 |L0 , A) in Model
5) as shown in Table 3.1. Second, the specification on E(L1 |A, L0 ) used by the flexible
119

plug-in estimator must be properly chosen. By “properly chosen”, we mean the specification
on E(L1 |A, L0 ) must be of the same level of flexibility as the second-step regression of the
sequential g-formula estimator. See the remark in Appendix C for more detail.
Later when we discuss the issue of “one single parameter”, the two estimators are not
identical except for the no interaction case. The sequential g-formula estimator forces
E(Y am − Y 0m ) to be ϕa when it is actually not, a typical case in which the incompatibility issue arises.

3.4.5

Simulation

We use a simple simulation study to evaluate the equivalency. Assume the data generating
process (DGP) is as follows:
U0 = εU0
L0 = U0 + εL0
A = 1[expit(L0 ) ≥ εA ]
L1 = U0 + L0 + A + εL1
M = 50 × expit(L0 + A + L1 + εM )
Y

= L0 × log(1 + A + M 2 ) + L1 × (A + M ) + U0 + εY

where all ε’s except εA are standard normal, εA is uniform on (0, 1), and expit(x) =
exp(x)/(exp(x) + 1) is the inverse of the logit transformation. All ε’s are independent
of each other. For simplicity, all coefficients are set to unity. The resulting true CDE is
CDE(m) = m + 1 for any fixed value m. Under this DGP, the following estimating equation
that is linear in L0 and L1 holds:
1
1
E(Y |L0 , A, L1 , M ) = log(1 + A + M 2 ) L0 + ( + A + M )L1 + − A .
3
3
h0 (a,m)

h1 (a,m)

120

h(a,m)

(3.11)

Table 3.2 Simulation results: flexible plug-in g-formula v.s. sequential g-estimator

m
1

ture CDE
2

25

26

50

51

m
1

ture CDE
2

25

26

50

51

FPG
57.0846
(8.5109)
40.1442
(5.1294)
22.4979
(7.1983)

A: Simulation Results for Model 4
E(L1 |A, L0 ) =
E(L1 |A, L0 ) =
π0 + πA A + πL0 L0
π0 + πA A + πL0 L0 + πAL0 A × L0
SG
Difference
FPG
SG
Difference
−3
57.0781
57.0781
0 (0)
57.0781
−6.481e
(.1090)
(8.5106)
(8.5106)
(8.5106)
40.1377
40.1377
0 (0)
40.1377
−6.481e−3
(.1090)
(5.1285)
(5.1285)
(5.1285)
−3
22.4914
22.4914
0 (0)
22.4914
−6.481e
(.1090)
(7.1973)
(7.1973)
(7.1973)

B: Simulation Results for Model 5
E(L1 |A, L0 ) =
E(L1 |A, L0 ) =
π0 + πA A + πL0 L0
π0 + πA A + πL0 L0 + πAL0 A × L0
FPG
SG
Difference
FPG
SG
Difference
−4
.0426 (.4303)
.04219 (.4303)
−4.951e
.0419 (.4303)
.0419 (.4303)
0 (0)
(.0236)
24.9052
24.9047
−4.951e−4
24.8994
24.8994
0 (0)
(2.3424)
(2.3425)
(.0236)
(2.3411)
(2.3411)
−4
50.8037
50.8032
−4.951e
50.7926
50.7926
0 (0)
(4.6871)
(4.6871)
(.0236)
(4.6847)
(4.6847)

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For illustration purposes, we only show the estimation results for the two estimators in
Models 4 and Model 5 in Table 3.1. The following two specifications for E(L1 |A, L0 ) which
differ in their flexibility are considered for both Model 4 and Model 5:
E(L1 |A, L0 ) = π0 + πA A + πL0 L0 ,

(3.12)

E(L1 |L0 , A) = π0 + πA A + πL0 L0 + πAL0 A × L0 .

(3.13)

The specifications for E(L1 |A, L0 ) affect the estimates of the flexible plug-in estimator in
both Model 4 and Model 5. As for the sequential g-formula estimator, the specifications for
E(L1 |A, L0 ) have no effect in Model 4 since no estimates involve the estimation of π, which
is in turn a result of the specification that L1 does not interact with M ; but they do have
an effect in Model 5 since both ϕAM and ϕAM L0 need the estimation of π. When the same
specification on E(Y |L0 , A, L1 , M ) and the same proper specification on E(L1 |L0 , A) are
used, the flexible plug-in g-formula estimator and the sequential g-formula estimator must
give exactly the same estimates on each occasion.
The simulation consists of 1000 runs with 500 observations in each sample. To save space,
we report in Table 3.2 the results for CDE(1), CDE(25) and CDE(50) only, m ranges from
1 to 50. For each estimator, the average and standard deviation (in parenthesis) over the
1000 simulations are reported. We also calculate the difference of the two estimates in each
simulation run and report the average and standard deviation of the difference over the 1000
simulations.
There are two important observations from the simulation results in Table 3.2. First, the
flexibility of the specification on E(Y |L0 , A, L1 , M ) is important. Although Models 4 and
5 both misspecify the true estimating equation, however, compared to Model 5, Model 4 is
more restrictive, leading to larger biases in both estimators. Note that in terms of qA (·) and
qM (·), Model 4 is typical in causal mediation analysis, partly because researchers usually
think the two-way interactions A × L1 and M × L1 are unnecessary. On the other hand, if
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the flexible plug-in g-formula estimator is used, the complete set of two-way interactions in
Model 5 becomes typical.
Second, when E(L1 |A, L0 ) = π0 + πA A + πL0 L0 , the two estimates are close but not the
same. Because the CDE is linear in m for both estimators, the difference does not depend on
m. When the proper specification E(L1 |A, L0 ) = π0 + πA A + πL0 L0 + πAL0 A × L0 is used,
the two estimates become identical. This supports the numerical equivalence claim made in
the last section.

3.4.6

Comparison of Flexible plug-in g-formula estimator with Sequential gformula estimator

In cases where the two estimators are identical, the flexible plug-in g-formula estimator
inherits everything the sequential g-formula estimator has. However, the difference in the
estimation procedures grants the former several advantage over the latter in applications.
We discuss several points of importance in this respect.
First, when applying the sequential g-formula estimator, one needs to make sure the
specifications for qA (·) and qM (·) are compatible with the chosen SNMM. For example,
E(Y am − Y 0m ) = ϕa is not compatible with the qM (·) in Model 2. As qA (·) and qM (·) become more complex, it becomes more difficult to find the corresponding SNMM. The flexible
plug-in g-formula estimator avoids this issue, because it starts from assuming the specifications on E(Y |l0 , a, l1 , m) and E(L1 |a, l0 ), and the model for CDEs follows naturally. The
resulting SNMM can even be nonlinear depending on the specifications on E(Y |l0 , a, l1 , m)
and E(L1 |a, l0 ).
Second, unless the SNMM is forced to be E(Y am − Y 0m ) = ϕa, the sequential g-formula
estimator does not always depend on “one single parameter” for CDE Vansteelandt (2009).
For example, if there is a strong belief that there is treatment-mediator interaction in qM (·)
as in Model 2, then E(Y am − Y 0m ) = ϕA a + ϕAM a × m is a function of two parameters.
The aforementioned compatibility issue will arise if, in order to force CDE to depend on “one

123

single parameter”, the SNMM is assumed to be E(Y am − Y 0m ) = ϕa. In an incompatible
case it is difficult to interpret what effect the single parameter ϕ captures. Without further
investigation, all we can say is that it is some average of the CDE evaluated at different
values of m, and it is unknown whether it is practically relevant. As a result, the test of
existence of CDE based on this average becomes less useful than a test evaluated at different
values of m.
Third, there are interesting specifications on E(Y |l0 , a, l1 , m) that the sequential gestimation does not allow. The feasibility of the sequential g-formula estimator hinges on
the additive separability between qA (·) and qM (·), and clearly, not all specifications for
E(Y |l0 , a, l1 , m) satisfy this restriction. Practically interesting examples include cases where
h = log(γ0 + γA a + γM m2 ) or h = exp(γ0 + γA a + γM m2 ), i.e. we use the link function idea
of generalized linear models to enrich specifications on the hk functions. To be fair, however,
we also note that there are specifications that the flexible plug-in g-formula estimator cannot
handle. For example, if the conditional expectation of Y is nonlinear in L0 and L1 , equation
(3.7) will not hold, but equation (3.6) may still be satisfied provided that the nonlinearity
does not interfere with the additive separability requirement. Extensions of equation (3.7)
to be nonlinear in L0 and L1 are possible but complicated, in which case the original parametric g-formula might be a better choice. In sum, the sequential g-formula estimator has
the potential of allowing nonlinearity in confounders but generally not in the treatment and
mediator, and for the flexible plug-in g-formula estimator the converse is true. In this sense
these two estimators complement each other.
Finally, the sequential g-estimation procedure changes in a nontrivial way as the specification for qA (·) and qM (·) changes, unless one does not care about compatibility and always
uses E(Y am − Y 0m ) = ϕa. The two steps of the procedure must be derived and tailored
individually, and become more complex as we move from Model 1 to Model 5 (see Appendix
B). In particular, in Model 5 (or whenever there are M L1 and/or AM L1 interactions),
ϕAM in SNMM can not be estimated simply by the first-step regression anymore, and the
124

derivation of the formula is not trivial. Moreover, there is an additional parameter ϕAM L0
to be estimated. These scenarios make the sequential g-formula estimator inconvenient to
use when one would like to build more flexibility into the model. On the other hand, the
estimation procedure for the plug-in g-formula works uniformly across different settings, and
there is no derivation by hand because the work is done by the computer.

3.5

Sensitivity Analysis

The untestable sequential ignorability conditions in (3.1) is crucial for any g-formula driven
estimators. The second part of the assumption is particularly vulnerable in mediation analyses since sequential randomization is not always the case in practice. In this section, we
provide a sensitivity analysis for one type of violation of the sequential ignorability conditions. For illustration purposes, this section only shows the sensitivity analysis for Model 1.
Similar procedures can be derived for other models in Table 3.1.
Recall the exogenous parents ε’s omitted from DAG G are assumed to be jointly independent in NPSEM-IE. Suppose now εM and εY are correlated, then the original g-formula
and consequently the flexible plug-in g-formula estimator do not work any more. To identify
the CDE in the analysis, we use the following conditions to perform a sensitivity analysis.
1. The unobservable U0 does not enter the structural equation of Y , i.e. the arrow from
U0 to Y in Figure 1 is deleted.
2. The structural equation for Y is linear in its coefficients. If we use fY (L0 , A, L1 , M ) to
denote the structural equation for Y , then in Model 1, the linearity assumption means
fY (·) = γ0 + γL0 L0 + γA A + γL1 L1 + γM M + εY , which is stronger than a linear
estimation equation.
3. The structural equation for M is additive separable in εM , i.e. M = fM (L0 , A, L1 ) +
εM for some function fM .

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Given these assumptions, the counterfactual Y am is
Y am = γ0 + γL0 L0 + γA a + γL1 L1 + γM m + εY ,
and the CDE is
E Y 1m − Y 0m = γA .
To consistently estimate γA , let Z = (1, L0 , A, L1 , M ) and γ Z = (γ0 , γL0 , γA , γL1 , γM ).
Then we can write
Y = Zγ Z + εY .
Define γ OLS
= E(Z Z)
Z

−1

E(Z Y ). Because εM and εY are correlated, in general γ OLS
is
Z

not equal to γ Z . But under conditions (1-3), γ Z can be derived through a bias correction
term:



γ Z = γ OLS
− E(Z Z) −1 
Z


0
σ εM εY




where σεM εY is the covariance between εM and εY , and 0 is a 4 × 1 vector. (See Appendix
D for derivation.)
To see how sensitive the flexible plug-in g-formula estimator is to this particular violation
of sequential ignorability conditions, we let σεM εY vary within some range and estimate the
CDE accordingly. As a rule of thumb, the covariance between M and Y could provide a
reference on the choice of the range for σεM εY , since εM (εY ) only represents part of the
variation in M (Y ) if one believes that the chosen model is a sound one.
Compared to the sensitivity analysis in Imai, Keele and Tingley (2010), the sensitivity
analysis in this section relaxes the structural assumption on the mediator. One major reason
that this relaxation can be made is that CDE is the parameter of interest in this paper
instead of the natural direct effect, which is not nonparametrically identified in a model with
post-treatment confounders.

126

3.6

An Application

In a longitudinal study by Breslau, Johnson and Lucia (2001) and Breslau, Paneth and Lucia
(2004), a random sample of low birthweight (LBW, < 2500 grams) and normal birthweight
(NBW, ≥ 2500 grams) infants is selected from two socioeconomically disparate populations
in southeast Michigan and followed over 17 years. The goal is to study the long term impact
of LBW on academic achievements. The first assessment occurs when the children are 6
years old, the second assessment occurs when the children are 11, and the last assessment
when the children are 17. A test from the Woodcock-Johnson Psychoeducational BatteryRevised (WJ-R) by Woodcock, Johnson and Mather (1990) is used to measure their academic
achievement in reading at ages 11 and 17. The WJ-R tests are age standardized with a mean
of 100 and a standard deviation of 15. An earlier paper by Breslau, Johnson and Lucia (2001)
found that the reading score for LBW children at age 11 is 3.6 points lower than those of NBW
children. However, the difference became trivial and insignificant after adjusting for their IQ,
visual-motor-integration (VMI) function from Beery (1989) and phonologic awareness (PA)
from Rosner and Simon (1971) at age 6. Their conclusion is thus that the deficit in reading
score in LBW children at age 11 relative to NBW children is accounted for (mediated) mostly
by the deficit in their cognitive skills at age 6. In the follow-up study Breslau, Paneth and
Lucia (2004), a similar conclusion is obtained for reading score at age 17.
In this application, we estimate the CDE of LBW on reading scores at age 17 when a
behavior problem index is used as the mediator. The behavior problem index is constructed
by summing up 8 binary indicators for different behavior problems at age 17, including ever
smoked a cigarette, ever smoked cigarettes daily, ever used alcohol, ever used marijuana,
ever used cocaine, ever used crack, ever used any hallucinogen, and ever used inhalants. The
index ranges from 0 to 8. Based on Breslau, Johnson and Lucia (2001); Breslau, Paneth
and Lucia (2004); Luo et al. (2014), we put the subject’s gender and residence at birth and
mother’s IQ, education and marital status as the baseline confounders in L0 . For post-

127

treatment confounders in L1 , the subject’s IQ, VMI and PA at age 6 were used. After
deleting observations that contain missing values, we obtain a sample of 704 complete cases
out of the total 713 who were assessed at age 17. The five models in Table 3.1 were applied,
and the results are presented in Figure 3.3.
The results for Model 1 show a negative constant CDE estimate, and the effect is not
statistically significant according to the normal-based bootstrap confidence intervals. When
the interaction A × M is added as in Model 2, the CDE estimates show a downward trend
as the behavior problem became more severe. Specifically, the CDE estimate decreases
from −.75 to −7.62 as the mediating behavior problem changes from 0 to 8. The effect
is significant when the number of behavior problems is greater than 1. Model 3 includes
the M × L0 interaction, but the result is almost identical to Model 1 because the resulting
SNMM is not a function of m. Model 4 has both A × M and M × L0 interactions, and its
result is similar to Model 2 but with slightly wider confidence intervals. Finally, Model 5
included M × L1 interactions on top of Model 4 but led to similar results.
The downward trends in Model 2, 4 and 5 mainly comes from the negative effect of the
A × M interaction, although this interaction is not statistically significant. Ignoring other
channels, this negative interaction effect essentially indicated that even if the immediate
effect of the behavior problem on reading is shut down by controlling the behavior problem,
a more severe behavior problem would still exacerbate the negative effect of LBW on reading
by altering the mechanism through which LBW exerts its effect.

3.7

Conclusion

In this chapter, wee formalize the idea of using partially linear conditional mean models of
the outcome and propose a flexible plug-in g-formula estimator in for controlled direct effects
causal mediation analysis. Partial linearity of outcome conditional expectation is of interest
because under this linear assumption, we can replace the confounders in the conditional

128

Figure 3.3 Controlled direct effect of LBW on reading with bad behavior as mediator,
Model 1 to 5

1

2

0

2

4
Behavior Problem as Mediator

CDE
CDE if Cov(eM,eY)=-1000

6

A=1 if LBW. Sample Size 704.

Controlled Direct Effect of LBW on Reading
10
-5
0
5
-20 -15 -10

Controlled Direct Effect of LBW on Reading
0
-20 -15 -10
10
-5
5

A=1 if LBW. Sample Size 704.

8

0

95% normal-based CI
CDE if Cov(eM,eY)=1000

2

CDE
CDE if Cov(eM,eY)=-1000

3

2

4
Behavior Problem as Mediator

6

8

0

95% normal-based CI
CDE if Cov(eM,eY)=1000

Controlled Direct Effect of LBW on Reading
-5
5
-20 -15 -10
10
0

A=1 if LBW. Sample Size 704.

2

4
Behavior Problem as Mediator

CDE
CDE if Cov(eM,eY)=-1000

6

2

95% normal-based CI
CDE if Cov(eM,eY)=1000

4
Behavior Problem as Mediator

CDE
CDE if Cov(eM,eY)=-1000

5

0

8

A=1 if LBW. Sample Size 704.

Controlled Direct Effect of LBW on Reading
-20 -15 -10
-5
0
5
10

Controlled Direct Effect of LBW on Reading
-5
5
-20 -15 -10
10
0

CDE
CDE if Cov(eM,eY)=-1000

6

4

A=1 if LBW. Sample Size 704.

0

4
Behavior Problem as Mediator

8

95% normal-based CI
CDE if Cov(eM,eY)=1000

129

6

95% normal-based CI
CDE if Cov(eM,eY)=1000

8

mean model for the outcome by properly fitted values of the confounders, which results in
a plug-in estimator for the controlled direct effects. The flexible plug-in g-formula is closedform and thus can save some computation time by avoiding Monte Carlo integration that
the traditional parametric g-formula usually relies on to evaluate integrals.
We also show that under certain conditions the flexible plug-in g-formula estimator is
numerically equivalent to the sequential g-formula estimator in the literature. Although the
sequential g-formula estimator is supposed to be a parametric version of the g-estimation of
structural nested mean models, this equivalence result indicates that it can also be viewed as
a particular parametric g-formula. Indeed, since the g-estimation of structural nested models
is a semiparametric version of the original g-formula, when stronger parametric assumptions
are imposed, we should expect it to come close to the parametric g-formula. Therefore the
equivalence result provides a new insight of the connections between parametric g-formula
and g-estimation of structural nested mean models.
The interest in the flexible plug-in g-formula estimator is manifold. First, in the linear
case, the flexible plug-in g-formula estimator provides an closed-form expression without
introducing additional assumptions than those commonly made in empirical studies. In view
of the fact that linear regression is often the first choice for modeling continuous outcomes in
parametric g-formula, the flexible plug-in g-formula actually imposes no stronger parametric
assumption than some of the parametric g-formulae that already exist in the literature.
Second, the flexible plug-in g-formula estimator connects to the sequential g-estimation
but may be more straightforward to use. The two estimators also complement each other
in giving practitioners choices of reasonable specifications in their context. For example, if
there is reason to believe the functional form for Y should be nonlinear in A and M , we
can use the flexible plug-in estimator; and on the other hand, if the nonlinearity lies in the
confounding factors, we can use the sequential g-formula estimator. In any case, if the results
from both methods are similar, the estimates are more likely to be robust.

130

APPENDICES

131

APPENDIX A

PROOFS AND ALGEBRA ON G-FORMULA

A.1

A direct proof of the g-formula for fY am (y)

The proof for the g-formula has been given in a series of paper by Robins and his colleagues.
Here we repeat it for the continuous outcome case for easy reference. We adopt the convention
to use upper case letters for random variables and lower case letters for realizations.
Under sequential ignorability and consistency,
fY |L ,A,L ,M (y|l0 , a, l1 , m)fL |L ,A (l1 |l0 , a)fL0 (l0 )
0
1
1 0

fY am (y) =
l0 ,l1

where, e.g., fY |L ,A,L ,M (y|l0 , a, l1 , m) is the shorthanded notation for the conditional den0
1
sity function of Y given (L0 , A, L1 , M ).
Proof. It can be shown that
ˆ
fY am (y) =

ˆ

fY am |L (y|l0 )fL0 (l0 )dl0
0

=

fY am |L ,A (y|l0 , a)fL0 (l0 )dl0
0
ˆ ˆ
=
fY am |L ,A,L (y|l0 , a, l1 )fL |L (l1 |l0 , a)fL0 (l0 )dl1 dl0
1 0
0
1
ˆ ˆ
=
fY am |L ,A,L ,M (y|l0 , a, l1 , m)fL |L (l1 |l0 , a)fL0 (l0 )dl1 dl0
0
1
1 0
ˆ ˆ
=
fY |L ,A,L ,M (y|l0 , a, l1 , m)fL |L (l1 |l0 , a)fL0 (l0 )dl1 dl0
0

1

1

0

where the first and the third equality uses law of total probability, the second uses Y am ⊥
⊥
A|L0 , the fourth uses (Y am ⊥
⊥ M |L0 , A = a, L1 ), and the last uses the consistency axiom.

132

Remark 6. Based on the g-formula for fY am (y), it follows that the g-formula for E(Y am ) is
ˆ
ˆ ˆ
am
E(Y ) =
y
fY |L ,A,L ,M (y|l0 , a, l1 , m)fL |L (l1 |l0 , a)fL0 (l0 )dl1 dl0 dy
0
1
1 0
ˆ ˆ ˆ
=
yfY |L ,A,L ,M (y|l0 , a, l1 , m)fL |L (l1 |l0 , a)fL0 (l0 )dydl1 dl0
0
1
1 0
ˆ ˆ ˆ
=
yfY |L ,A,L ,M (y|l0 , a, l1 , m)dy fL |L (l1 |l0 , a)fL0 (l0 )dl1 dl0
0
1
1 0
ˆ ˆ
=
E(y|l0 , a, l1 , m)fL |L (l1 |l0 , a)fL0 (l0 )dl1 dl0 .
1

A.2

0

The flexible plug-in g-formula for E(Y am )

If sequential ignorability and consistency hold, and the outcome conditional mean is given
by
E(Y |l0 , a, l1 , m) = h0 (a, m)l0 + h1 (a, m)l1 + h(a, m),
then
E(Y am ) = h0 (a, m)E(L0 ) + h1 (a, m)E [E(L1 |L0 , A = a)] + h(a, m).
Proof. Under the assumptions, we get
˚
am
E(Y ) =
yf (y|l0 , a, l1 , m)f (l1 |l0 , a)f (l0 )dydl1 dl0
ˆ ˆ ˆ
=
yf (y|l0 , a, l1 , m)dy f (l1 |l0 , a)f (l0 )dl1 dl0
ˆ ˆ
=
E(Y |l0 , a, l1 , m)f (l1 |l0 , a)f (l0 )dl1 dl0
ˆ ˆ
=
E(Y |l0 , a, l1 , m)f (l1 |l0 , a)dl1 f (l0 )dl0
ˆ ˆ
=
[h0 (a, m)l0 + h1 (a, m)l1 + h(a, m)] f (l1 |l0 , a)dl1 f (l0 )dl0
ˆ
=
[h0 (a, m)l0 + h1 (a, m)E(L1 |l0 , a) + h(a, m)] f (l0 )dl0
ˆ
= h0 (a, m)E(L0 ) + h1 (a, m) E(L1 |l0 , a)f (l0 )dl0 + h(a, m)
= h0 (a, m)E(L0 ) + h1 (a, m)E [E(L1 |L0 , A = a)] + h(a, m)

133

Remark 7. Note that E [E(L1 |L0 , A = a)] = E(L1 |A = a). For example, L1 = exp(L0 ) + A.
Then E(L1 |L0 , A = a) = exp(L0 ) + a, and the LHS is E [exp(L0 ) + a] = E [exp(L0 )] + a.
But the RHS is E [exp(L0 ) + A|A = a] = E [exp(L0 )|A = a] + a. If A ⊥
⊥ L0 , the equality
holds.
In general, let the structural model for L1 be L1 = f (L0 , A, εL1 ). Then the LHS is
E f (L0 , a, εL1 )|L0 . The RHS is E f (L0 , a, εL1 )|A = a . If we assume εL1 is independent
of (L0 , A), the LHS becomes E f (L0 , a, εL1 ) . If in addition A ⊥
⊥ L0 , the RHS becomes
E f (L0 , a, εL1 ) and the equality holds.

134

APPENDIX B

PROOFS AND ALGEBRA ON SEQUENTIAL G-ESTIMATOR

B.1

Validity of the second step of sequential g-estimator

The proof of the validity of the sequential g-estimator in the Appendix of Vansteelandt
(2009) needs to be extended because L0 is included in our analysis. Specifically, we want
to show that, in presence of L0 , and under additive separability, qM (l0 , a, l1 , 0; γ) = 0, and
sequential ignorability,
E [Y − qM (L0 , A, L1 , M ; γ)|L0 , A] = E Y A0 |L0 ,
i.e. for any (l0 , a), show
E [Y − qM (L0 , A, L1 , M ; γ)|L0 = l0 , A = a] = E Y a0 |L0 = l0 .

(B.1)

Proof. Under the assumptions, we have
E [Y − qM (l0 , a, l1 , m; γ)|L0 = l0 , A = a, L1 = l1 , M = m] = qA (l0 , a, l1 ; γ).
The last equality holds for any m. Therefore
E [Y − qM (l0 , a, l1 , M ; γ)|L0 = l0 , A = a, L1 = l1 , M ] = qA (l0 , a, l1 ; γ).
Take expectation of both sides conditional on L0 = l0 , A = a, L1 = l1 and use iterated
expectation, we get
E [Y − qM (l0 , a, l1 , M ; γ)|L0 = l0 , A = a, L1 = l1 ] = qA (l0 , a, l1 ; γ).

135

Now, sequential ignorability implies
E Y a0 |L0 = l0 , A = a, L1 = l1
=E Y a0 |L0 = l0 , A = a, L1 = l1 , M = 0
=E [Y |L0 = l0 , A = a, L1 = l1 , M = 0]
=E [qA (L0 , A, L1 ; γ) + qM (L0 , A, L1 , M ; γ)|L0 = l0 , A = a, L1 = l1 , M = 0]
=E [qA (l0 , a, l1 ; γ) + qM (l0 , a, l1 , 0; γ)|L0 = l0 , A = a, L1 = l1 , M = 0]
=qA (l0 , a, l1 ; γ)
The equality holds for any l1 . Therefore
E [Y − qM (l0 , a, L1 , M ; γ)|L0 = l0 , A = a, L1 ] = E Y a0 |L0 = l0 , A = a, L1 .
Take expectation of both sides conditional on (L0 = l0 , A = a), we get
E [Y − qM (l0 , a, L1 , M ; γ)|L0 = l0 , A = a] = E Y a0 |L0 = l0 , A = a .
Lastly, notice that Y a0 ⊥
⊥ A|L0 , we have
E [Y − qM (l0 , a, L1 , M ; γ)|L0 = l0 , A = a] = E Y a0 |L0 = l0 .

We provide an alternative proof below.
Proof. (alternative proof) First, we show that
E [Y − qM (L0 , A, L1 , M ; γ)|L0 , A, L1 , M ] = E Y A0 |L0 , A, L1 .

136

Specifically,
E [Y − qM (L0 , A, L1 , M ; γ)|L0 , A, L1 , M ]
=qA (L0 , A, L1 ; γ)
=E [qA (L0 , A, L1 ; γ)|L0 , A, L1 , M = 0]
=E [qA (L0 , A, L1 ; γ) + qM (L0 , A, L1 , 0; γ)|L0 , A, L1 , M = 0]
=E [qA (L0 , A, L1 ; γ) + qM (L0 , A, L1 , M ; γ)|L0 , A, L1 , M = 0]
=E [Y |L0 , A, L1 , M = 0]
=E Y A0 |L0 , A, L1 , M = 0
=E Y A0 |L0 , A, L1
where the first equality holds by the definition of qA (L0 , A, L1 ; γ), the second equality holds
since qA is a function of L0 , A, L1 , the third equality holds because qM (L0 , A, L1 , 0; γ) = 0,
the fourth and fifth are simply rewriting, the sixth by consistency, and the last by Y am ⊥
⊥
M |L0 , A, L1 .
Then, take expectation of both sides conditional on (L0 , A), we get
E [Y − qM (L0 , A, L1 , M ; γ)|L0 , A] = E Y A0 |L0 , A .
Lastly, by Y am ⊥
⊥ A|L0 , we have
E [Y − qM (L0 , A, L1 , M ; γ)|L0 , A] = E Y A0 |L0 .

B.2

Estimation procedures and interpretations for the sequential
g-estimator in Model 1 and in a general setup with up to threeway interactions in E(Y |L0 , A, L1 , M )

We discuss Model 1 and Model 5. Model 1 is the simplest specification, so it is used as an
example to illustrate the idea. Model 5 is the most general specification, so the discussion
137

on Model 5 shows that the conclusion applies to all 5 models.

B.2.1

Model 1 (No interaction)

Model assumptions
1. The NPSEM-IE associated with DAG G. (Thus consistency and sequential ignorability
hold.)
2. Structural Nested Mean Model:
E(Y am − Y 0m |l0 ) = ϕA a.

(B.2)

3. Conditional mean of the outcome:
E(Y |L0 , A, L1 , M ) = γ0 + γL0 L0 + γA A + γL1 L1 + γM M,

(B.3)

qA (L0 , A, L1 ; γ) = γ0 + γL0 L0 + γA A + γL1 L1 ,

(B.4)

so that

qM (L0 , A, L1 , M ; γ) = γM M.
4. Conditional mean of the post-treatment confounder at A = 0:
f (l0 , 0) ≡ E(L1 |L0 = l0 , A = 0) = π0 + πL0 l0 .

(B.5)

Estimation procedure
1. Regress Y on (1, L0 , A, L1 , M ) and obtain the OLS estimator γˆM for γM . Generate
Yˆ−M ≡ Y − γˆM M .
2. Regress Yˆ−M on (1, L0 , A). Denote by ϕˆA the OLS estimator for the coefficient of A.
ˆ
Then ϕˆA is a consistent estimator for ϕA . Hence, CDE(m)
= ϕˆA .

138

Validity of the second-step regression (i.e. the consistency of ϕˆA for ϕA )
Proof. (i) First of all, we show
E(Y 00 |L0 = l0 ) = γ0 + γL0 l0 + γL1 f (l0 , 0).

(B.6)

Under sequential ignorability, consistency, and the specification in (B.3), we have
E(Y a0 |L0 = l0 , A = a, L1 = l1 )
=E(Y a0 |L0 = l0 , A = a, L1 = l1 , M = 0)
=E(Y |L0 = l0 , A = a, L1 = l1 , M = 0)
=γ0 + γA a + γL0 l0 + γL1 l1 .
The equality holds for any (l0 , a, l1 ), and therefore we can write
E(Y A0 |L0 , A, L1 ) = γ0 + γA A + γL0 L0 + γL1 L1 + γAL1 AL1 .
Take expectation of both sides conditional on (L0 , A), we get
E(Y A0 |L0 , A)
=γ0 + γA A + γL0 L0 + γAL0 AL0 + γL1 E(L1 |L0 , A) + γAL1 AE(L1 |L0 , A)
=γ0 + γA A + γL0 L0 + γAL0 AL0 + γL1 f (L0 , A) + γAL1 Af (L0 , A),
i.e. for any (l0 , a),
E(Y a0 |L0 = l0 , A = a)
=γ0 + γA a + γL0 l0 + γAL0 al0 + γL1 f (l0 , a) + γAL1 af (l0 , a).
Then the first part of the sequential ignorability implies
E(Y a0 |L0 = l0 )
=γ0 + γA a + γL0 l0 + γAL0 al0 + γL1 f (l0 , a) + γAL1 af (l0 , a).

139

(B.7)

Set a = 0, we get
E(Y 00 |L0 = l0 ) = γ0 + γL0 l0 + γL1 f (l0 , 0).
(ii) Secondly, we show
E(Y − qm (L0 , A, L1 , M ; γ)|L0 = l0 , A = a)
=ϕA a + γ0 + γL0 l0 + γL1 f (l0 , 0).

(B.8)

By setting m = 0 in (B.2), we have
E(Y a0 − Y 00 |l0 ) = ϕA a.
Then, (B.1) and (B.6) imply
E(Y − qm (L0 , A, L1 , M ; γ)|L0 = l0 , A = a)
=E(Y a0 |L0 = l0 )
=E(Y a0 − Y 00 |L0 = l0 ) + E(Y 00 |L0 = l0 )
=ϕA a + γ0 + γL0 l0 + γL1 f (l0 , 0).
(iii) Lastly, given (B.5), we have
E(Y − qm (L0 , A, L1 , M ; γ)|L0 = l0 , A = a)
=ϕA a + γ˜0 + γ˜L0 l0 ,

(B.9)

where γ˜0 = γ0 + γL1 π0 and γ˜L0 = γL0 + γL1 πL0 . Therefore, the OLS estimator ϕˆA in the
second-step regression is consistent for ϕA .

Necessity and sufficiency of (B.5) given all the other model assumptions and that
L0 is not binary
Proof. The sufficiency of (B.5) given all the other model assumptions have been shown by
(B.9).

140

To show its necessity, first we rewrite (B.8) as
Y−M = ϕA A + γ¨0 + γL0 L0 + v

(B.10)

where
Y−M ≡ Y − qm (L0 , A, L1 , M ; γ) = Y − γM M,
γ¨0 ≡ γ0 + γL1 E [f (L0 , 0) − L0 ] ,
f¯(L0 , 0) ≡ [f (L0 , 0) − L0 ] − E [f (L0 , 0) − L0 ] ,
ξ ≡ Y−M − E [Y−M |A, L0 ] ,
v ≡ γL1 f¯(L0 , 0) + ξ.
Since L0 is not binary, equation (B.10) indicates that f (l0 , 0) must be linear for ϕˆA to be
consistent for ϕA . We prove this by contradiction. If f (l0 , 0) is nonlinear in l0 , in general
we have
Cov(L0 , v) = E(L0 v) = γL1 E L0 f¯(L0 , 0) = 0
−1
unless, e.g, f¯(L0 , 0) = L−1
0 and E(L0 ) = 0, or some other particular conditions hold by

fluke. But then, unless a regression of Y−M on (1, A, L0 ) would yield inconsistent estimators
for all coefficients.

B.2.2

General model with a three-way interaction in E(Y |L0 , A, L1 , M )

Model assumptions
1. The NPSEM-IE associated with DAG G. (Thus consistency and sequential ignorability
hold.)
2. Structural Nested Mean Model:
E(Y am − Y 0m |l0 ) = ϕA a + ϕAM am + ϕAL0 al0 + ϕAM L0 aml0
.
141

(B.11)

3. Conditional mean of the outcome:
E(Y |L0 , A, L1 , M )
=γ0 + γA A + γM M + γAM AM
+ γL0 L0 + γAL0 AL0 + γM L0 M L0 + γAM L0 AM L0
+ γL1 L1 + γAL1 AL1 + γM L1 M L1 + γAM L1 AM L1 ,

(B.12)

so that
qA (L0 , A, L1 ; γ)
=γ0 + γA A + γL0 L0 + γAL0 AL0 + +γL1 L1 + γAL1 AL1

qM (L0 , A, L1 , M ; γ)
=γM M + γAM AM + γM L0 M L0 + γAM L0 AM L0
+ γM L1 M L1 + γAM L1 AM L1
4. Conditional mean of the post-treatment confounder, which is given by the following
equation
f (l0 , a) ≡ E(L1 |l0 , a) = π0 + πL0 l0 + πA a + πAL0 al0 .

(B.13)

Estimation procedure
1. Regress Y on (1, A, M, AM, L0 , AL0 , M L0 , AM L0 , L1 , AL1 , M L1 , AM L1 ) and obtain
the OLS estimator γˆ for γ. Generate
Yˆ−M ≡ Y − γˆM M − γˆAM AM − γˆM L0 M L0 − γˆAM L0 AM L0
−ˆ
γM L1 M L1 − γˆAM L1 AM L1 .
2. Regress Yˆ−M on (1, L0 , A, AL0 ) and obtain the OLS estimators, ϕˆA and ϕˆAL0 , for the
coefficients of A and AL0 , respectively.
142

3. Regress L1 on (1, L0 , A, AL0 ) and obtain the OLS estimator π
ˆ for π. Then the sequential g-formula estimator for the controlled direct effect at M = m is
¯ 0 + ϕˆAM L mL
¯ 0,
E(Y 1m − Y 0m ) = ϕˆA + ϕˆAM m + ϕˆAL0 L
0
where
ϕˆAM = γˆAM + γˆM L1 π
ˆA + γˆAM L1 (ˆ
π0 + π
ˆA ) ,
ˆL0 + π
ˆAL0 .
ϕˆAM L0 = γˆAM L0 + γˆM L1 π
ˆAL0 + γˆAM L1 π

Validity of the second step (i.e. the consistency of ϕˆA and ϕˆAL0 for ϕA and ϕAL0 )
Proof. (i) First of all, we show
E(Y 00 |L0 = l0 ) = γ0 + γL0 l0 + γL1 f (l0 , 0).

(B.14)

which is exactly the same as (B.6).
Under sequential ignorability, consistency, and the specification in (B.12), and using a
similar argument to that for (B.7), we have
E(Y a0 |L0 = l0 , A = a, L1 = l1 )
=E(Y a0 |L0 = l0 , A = a, L1 = l1 , M = 0)
=E(Y |L0 = l0 , A = a, L1 = l1 , M = 0)
=γ0 + γA a + γL0 l0 + γAL0 al0 + γL1 l1 + γAL1 al1 .
The equality holds for any (l0 , a, l1 ), and therefore we can write
E(Y A0 |L0 , A, L1 )
=γ0 + γA A + γL0 L0 + γAL0 AL0 + γL1 L1 + γAL1 AL1

143

(B.15)

Take expectation of both sides conditional on (L0 , A), we get
E(Y A0 |L0 , A)
=γ0 + γA A + γL0 L0 + γAL0 AL0 + γL1 E(L1 |L0 , A) + γAL1 AE(L1 |L0 , A)
=γ0 + γA A + γL0 L0 + γAL0 AL0 + γL1 f (L0 , A) + γAL1 Af (L0 , A),
i.e. for any (l0 , a),
E(Y a0 |L0 = l0 , A = a)
=γ0 + γA a + γL0 l0 + γAL0 al0 + γL1 f (l0 , a) + γAL1 af (l0 , a).
Then the first part of the sequential ignorability implies
E(Y a0 |L0 = l0 )
=γ0 + γA a + γL0 l0 + γAL0 al0 + γL1 f (l0 , a) + γAL1 af (l0 , a).
Set a = 0, we get
E(Y 00 |L0 = l0 ) = γ0 + γL0 l0 + γL1 f (l0 , 0).
(ii) Secondly, we show
E(Y − qm (L0 , A, L1 , M ; γ)|L0 = l0 , A = a)
=ϕA a + γ0 + γL0 l0 + γL1 f (l0 , 0).
By setting m = 0 in (B.11), we have
E(Y a0 − Y 00 |l0 ) = ϕA a + ϕAL0 al0 .
Then, (B.1) and (B.14) imply
E(Y − qm (L0 , A, L1 , M ; γ)|L0 = l0 , A = a)
=E(Y a0 |L0 = l0 )
=E(Y a0 − Y 00 |L0 = l0 ) + E(Y 00 |L0 = l0 )
=ϕA a + ϕAL0 al0 + γ0 + γL0 l0 + γL1 f (l0 , 0).
144

(B.16)

(iii) Lastly, given (B.5), we have
E(Y − qm (L0 , A, L1 , M ; γ)|L0 = l0 , A = a)
=ϕA a + ϕAL0 al0 + γ˜0 + γ˜L0 l0 ,

(B.17)

where γ˜0 = γ0 + γL1 π0 and γ˜L0 = γL0 + γL1 πL0 . Therefore, in the second-step regression
the OLS estimators ϕˆA and ϕˆAL0 are consistent for ϕA and ϕAL0 , respectively.

Necessity and sufficiency of (B.5) for the validity of the second step, given all
the other model assumptions and that L0 is not binary
The same argument as that in D.1 can be used.

Validity of the third step (i.e. the consistency of ϕˆAM and ϕˆAM L0 for ϕAM and
ϕAM L0 )
Proof. Given equation (B.12) and Y am ⊥
⊥ M |L0 , A, L1 , we have
E(Y am |L0 , a, L1 )
=γ0 + γA a + γM m + γAM am
+ γL0 L0 + γAL0 aL0 + γM L0 mL0 + γAM L0 amL0
+ γL1 L1 + γAL1 aL1 + γM L1 mL1 + γAM L1 amL1 .
Take expectation conditional on (L0 , a), and notice Y am ⊥
⊥ A|L0 , we have
E(Y am |L0 )
=γ0 + γA a + γM m + γAM am
+ γL0 L0 + γAL0 aL0 + γM L0 mL0 + γAM L0 amL0
+ γL1 E(L1 |L0 , a) + γAL1 aE(L1 |L0 , a)
+ γM L1 mE(L1 |L0 , a) + γAM L1 amE(L1 |L0 , a).

145

Hence,
E(Y am − Y 0m |L0 )
=γA a + γAM am + γAL0 aL0 + γAM L0 amL0
+ γL1 [E(L1 |L0 , a) − E(L1 |L0 , a = 0)] + γAL1 aE(L1 |L0 , a)
+ γM L1 m [E(L1 |L0 , a) − E(L1 |L0 , a = 0)] + γAM L1 amE(L1 |L0 , a).

(B.18)

Assume E(L1 |L0 , A) = π0 + πL0 L0 + πA A + πAL0 AL0 , we have
E(Y am − Y 0m |L0 )
=γA a + γAM am + γAL0 aL0 + γAM L0 amL0
+ γL1 πA a + πAL0 aL0 + γAL1 a π0 + πL0 L0 + πA a + πAL0 aL0
+ γM L1 m πA a + πAL0 aL0 + γAM L1 am π0 + πL0 L0 + πA a + πAL0 aL0
= γA + γL1 πA + γAL1 (π0 + πA ) a
+ γAM + γM L1 πA + γAM L1 (π0 + πA ) am
+ γAL0 + γL1 πAL0 + γAL1 πL0 + πAL0

aL0

+ γAM L0 + γM L1 πAL0 + γAM L1 πL0 + πAL0

amL0 .

Compare the last equation with B.11, we see that
ϕA = γA + γL1 πA + γAL1 (π0 + πA )

(B.19)

ϕAM = γAM + γM L1 πA + γAM L1 (π0 + πA )

(B.20)

ϕAL0 = γAL0 + γL1 πAL0 + γAL1 πL0 + πAL0

(B.21)

ϕAM L0 = γAM L0 + γM L1 πAL0 + γAM L1 πL0 + πAL0

(B.22)

Therefore, we can estimate ϕAM and ϕAM L0 consistently by
ϕˆAM = γˆAM + γˆM L1 π
ˆA + γˆAM L1 (ˆ
π0 + π
ˆA ) ,
ϕˆAM L0 = γˆAM L0 + γˆM L1 π
ˆAL0 + γˆAM L1 π
ˆL0 + π
ˆAL0 ,
where γˆ is from the first step regression, and π
ˆ is from the third step regression.

146

APPENDIX C

NUMERICAL EQUIVALENCE

We prove the numerical equivalence for Model 5 which is the most flexible model in all the
five models considered. The proof for the other four models can be obtained in a similar
fashion.
To present the results rigorously, we first need to express the two estimators using the
same set of notation. Let X1 = (1, A, L0 , AL0 ), X2 = (L1 , AL1 ), W = (M, AM, M L0 , M L1 ),
Z = (X, W ) and e = Y − E(Y |Z). Let
γ = (γ0 , γA , γL0 , γAL0 , γL1 , γAL1 , γM , γAM , γM L0 , γM L1 ) .
Assume the sample size is n. Stack all observations in the matrix denoted by the corresponding bold letters. Under these notations, Model 5 says that
Y = Zγ + e.
ˆ . Then
Denote the OLS estimator for γ by γ
ˆ, where Z e
ˆ = 0.
Y = Zˆ
γ+e
Both the flexible plug-in estimator and the SG estimator use the above linear regression in
the first step. In addition, the sequential g-formula estimator generates the fitted outcome
Yˆ−M = Y − γˆM M − γˆAM AM − γˆM L0 M L0 − γˆM L1 M L1
= Y − X3 γˆW
which is free of the effect of M .
Let β 1 = (β0 , βA , βL0 , βAL0 ) and β 2 = (βL1 , βAL1 ) . Combine them in β = (β 1 , β 2 ) .
Let π = (π0 , πA , πL , πAL ) . Define ε = Yˆ−M − E(Yˆ−M |X), u = Yˆ−M − E(Yˆ−M |X1 ), and
0

0

147

v = L1 − E(L1 |X1 ). Then
ˆ −M = Xβ + ε,
Y
ˆ −M = X1 β 1 + u,
Y
L1 = X1 π + v.
ˆ β
˜ 1 and π,
ˆ
Denote the linear projection coefficients in the above three equations by β,
respectively. Then
ˆ +ε
ˆ −M = Xβ
ˆ,
Y

ˆ=0
Xε

˜1 + u
ˆ −M = X1 β
˜,
Y
ˆ +v
ˆ.
L 1 = X1 π

˜ =0
X1 u
ˆ=0
X1 v

(C.1)
(C.2)
(C.3)

Note that (C.2) represents the second-step regression of the sequential g-estimation.
ˆ denote the sequential g-formula estiFinally, let ϕ = (ϕA , ϕAL0 , ϕAM , ϕAM L0 ) . Let ϕ
ˇ the flexible plug-in estimator. The two estimators are the same in estimating
mator, and ϕ
ϕAM and ϕAM L0 :
ϕˆAM = ϕˇAM = γˆAM + γˆM L1 π
ˆA ,

(C.4)

ˆL0 + π
ˆAL0 ,
ϕˆAM L0 = ϕˇAM L0 = γˆAM L0 + γˆM L1 π
ˆAL0 + γˆAM L1 π

(C.5)

but they differ in estimating ϕA and ϕAL0 :
ϕˆA = β˜A ,

(C.6)

ϕˆAL0 = β˜AL0 .

(C.7)

ϕˇA = γˆA + γˆL1 π
ˆA + γˆAL1 (ˆ
π0 + π
ˆA ) ,

(C.8)

ϕˇAL0 = γˆAL0 + γˆL1 π
ˆAL0 + γˆAL1 π
ˆL0 .

(C.9)

Theorem 8. (Numerical Equivalence in Model 5) The flexible plug-in g-formula and the
sequential g-estimation are numerically equivalent in Model 5. That is
ϕˆA = ϕˇA ,
ϕˆAL0 = ϕˇAL0 .
148

Proof. Regarding those linear projection coefficients, the first fact is that
ˆ =γ
ˆX
β

(C.10)

ˆ X is the sub-vector of γ
ˆ associated with X. To see why, let γ
ˆ W be the sub-vector
where γ
ˆ associated with W . Then the regression of Yˆ−M on X can equivalently be cast as the
of γ
ˆ W , and the later
restricted regression of Y on Z = (X, W ) with the constraint γ W = γ
ˆX .
restricted regression is known to yield γ
The second fact is that the orthogonality condition following each of the three equations
(C.1), (C.2) and (C.3) is definitional for the corresponding linear projection coefficient vector.
Note that the rank condition always hold in practice unless by fluke. Therefore, given a data
ˆ β˜1 and π
set, β,
ˆ are uniquely defined by their respective orthogonality conditions.
Now we are ready to prove the equivalence. Plug equation (C.3) into (C.1), we have
ˆ −M = Xβˆ + ε
ˆ
Y
ˆ
= βˆ0 + βˆA A + βˆL0 L0 + βˆAL0 AL0 + βˆL1 L1 + βˆAL1 AL1 + ε
= βˆ0 + βˆA A + βˆL0 L0 + βˆAL0 AL0 + βˆL1 (ˆ
π0 + π
ˆA A + π
ˆL0 L0 +
ˆ ) + βˆAL1 A(ˆ
ˆ) + ε
ˆ
π
ˆAL0 AL0 + v
π0 + π
ˆA A + π
ˆL0 L0 + π
ˆAL0 AL0 + v
= (βˆ0 + βˆL1 π
ˆ0 ) + βˆA + βˆL1 π
ˆA + βˆAL1 (ˆ
π0 + π
ˆA ) A + (βˆL0 + βˆL1 π
ˆL0 )L0
ˆ
+ βˆAL0 + βˆL1 π
ˆAL0 + βˆAL1 (ˆ
πL0 + π
ˆAL0 ) AL0 + (βˆL1 + βˆAL1 )ˆ
v+ε
∗ A + βˆ∗ L + βˆ∗ AL + ε
ˆ∗
= βˆ0∗ + βˆA
0
L0 0
AL0

ˆ∗ + ε
ˆ∗
= X1 β
1
where we define
βˆ0∗ = βˆ0 + βˆL1 π
ˆ0 ,
∗ = βˆ + βˆ π
ˆ
βˆA
π0 + π
ˆA ),
A
L1 ˆA + βAL1 (ˆ

(C.11)

∗ = βˆ + βˆ π
βˆL
L0
L1 ˆL0
0
∗
βˆAL
= βˆAL0 + βˆL1 π
ˆAL0 + βˆAL1 (ˆ
πL0 + π
ˆAL0 ),
0

149

(C.12)

ˆ ∗ = (β ∗ , β ∗ , βˆ∗ , βˆ∗ ) ,
β
1
0 A L0 AL0
ˆ∗ = (βˆL1 + βˆAL1 )ˆ
ˆ.
ε
v+ε
But (C.3) and (C.2) imply that
ˆ∗ = X1 (βˆL1 + βˆAL1 )ˆ
ˆ
X1 ε
v+ε
ˆ + X1 ε
ˆ
= (βˆL1 + βˆAL1 )X1 v
= 0.

(C.13)

˜1 = β
ˆ ∗ . In particular, we have
Hence, by definition and uniqueness, β
1
∗,
β˜A = βˆA
∗ .
β˜AL0 = βˆAL
0

It follows immediately from (C.6), (C.7), (C.10), (C.11) and (C.12) that
ϕˆA = ϕˇA ,
ϕˆAL0 = ϕˇAL0 .

ˆ∗ = 0, it shows why we need the regressors be the same in the
Remark 1: In proving X1 ε
model of L1 as in the model of the second-step regression.
Remark 2: If the statistical model for E(L1 |A, L0 ) is of the same level of flexibility as
the second step regression of the SG estimator, we say the model is properly chosen. For
example, in Model 1, the second step of the SG estimator is to regress Yˆ−M,i on (1, Ai , L0i ),
and the proper model for E(L1 |A, L0 ) is E(L1 |A, L0 ) = πC + πA A + πL0 L0 . In Model 5,
the second step of the SG estimator is to regress Yˆ−M,i on (1, Ai , L0i , AL0i ), and the proper
model for E(L1 |A, L0 ) should be E(L1 |A, L0 ) = πC + πA A + πL0 L0 + πAL0 AL0 .
If the second-step regression of the SG estimation contains AL0 , but the linear model
for E(L1 |A, L0 ) excludes AL0 , then the numerical equivalence does not hold any more. The
150

intuition is that when πAL0 is forced to be zero, it will alter the the estimates for π0 , πA and
πL0 , which can be viewed as some restricted estimates. The math is explained in (C.13):
ˆ is not zero anymore.
X1 v

151

APPENDIX D

SENSITIVITY ANALYSIS

In this section we provide the derivation details for the sensitivity analysis in section 5 of
main text. Specifically, we want to show that


γ Z = γ OLS
− E(Z Z) −1 
Z


0
σ εM εY


.

Proof. Recall that
Y = Zγ Z + εY ,

(D.1)

Z = (1, L0 , A, L1 , M ),
σεM εY = Cov(εM , εY ) = 0.
Premultiply both sides of equation (D.1) by X , take expectation„ we have
E(Z Y ) = E(Z Z)γ Z + E(Z εY ).
Assume E(Z Y ) has full rank, then
γ Z = E(Z Z) −1 E(Z Y ) − E(Z Z) −1 E(Z εY )
Now notice that the three additional assumptions in Section 5 imply
E[(1, L0 , A, L1 ) εY ] = 0.

(D.2)

where 0 is a 4×1 zero vector. Meanwhile recall that γ OLS
= E(Z Z) −1 E(Z Y ). It follows
Z
immediately that


γ Z = γ OLS
− E(Z Z) −1 
Z

152


0
σ εM εY


.

BIBLIOGRAPHY

153

BIBLIOGRAPHY

Beery, Keith E. 1989. Developmental test of visual-motor integration: Administration and
scoring manual. Cleveland, OH:Modern Curriculum Pr.
Breslau, Naomi, Eric O Johnson, and Victoria C Lucia. 2001. “Academic achievement
of low birthweight children at age 11: the role of cognitive abilities at school entry.” Journal
of Abnormal Child Psychology, 29(4): 273–279.
Breslau, Naomi, Nigel S Paneth, and Victoria C Lucia. 2004. “The lingering academic
deficits of low birth weight children.” Pediatrics, 114(4): 1035–1040.
Danaei, Goodarz, An Pan, Frank B Hu, and Miguel a Hernán. 2013. “Hypothetical
midlife interventions in women and risk of type 2 diabetes.” Epidemiology, 24(1): 122–8.
Daniel, Rhian M, Bianca L De Stavola, and Simon N Cousens. 2011. “gformula:
Estimating causal effects in the presence of time-varying confounding or mediation using
the g-computation formula.” Stata Journal, 11(4): 479.
Garcia-Aymerich, Judith, Raphaëlle Varraso, Goodarz Danaei, Carlos A. Camargo, and Miguel A. Hernán. 2014. “Incidence of adult-onset asthma after hypothetical interventions on body mass index and physical activity: An application of the
parametric G-Formula.” American Journal of Epidemiology, 179(1): 20–26.
Hernan, Miguel A, and ames M Robins. 2015. Causal Inference. Boca Raton:Chapman
& Hall/CRC.
Hicks, Raymond, and Dustin Tingley. 2011. “Causal mediation analysis.” Stata Journal,
11(4): 605.
Horvitz, Daniel G, and Donovan J Thompson. 1952. “A generalization of sampling
without replacement from a finite universe.” Journal of the American Statistical Association, 47(260): 663–685.
Imai, Kosuke, Luke Keele, and Dustin Tingley. 2010. “A general approach to causal
mediation analysis.” Psychological Methods, 15(4): 309.
Lajous, Martin, Walter C. Willett, James Robins, Jessica G. Young, Eric Rimm,
Dariush Mozaffarian, and Miguel A. Hernán. 2013. “Changes in fish consumption
in midlife and the risk of coronary heart disease in men and women.” American Journal
of Epidemiology, 178(3): 382–391.
Luo, Zhehui, Joshua Breslau, Joseph C Gardiner, Qiaoling Chen, and Naomi
Breslau. 2014. “Assessing interchangeability at cluster levels with multiple-informant
data.” Statistics in medicine, 33(3): 361–375.

154

Pearl, Judea. 2009. Causality: Models, Reasoning and Inference. Cambridge University
Press.
Pearl, Judea. 2014. “Interpretation and identification of causal mediation.” Psychological
methods, 19(4): 459.
Pearl, Judea, and James Robins. 1995. “Probabilistic evaluation of sequential plans
from causal models with hidden variables.” 444–453, Morgan Kaufmann Publishers Inc.
Richardson, Thomas S, and James M Robins. 2013. “Single World Intervention Graphs
(SWIGs): A unification of the counterfactual and graphical approaches to causality.”
Center for the Statistics and the Social Sciences, University of Washington Series. Working
Paper, 128.
Robins, James M. 1986. “A new approach to causal inference in mortality studies with a
sustained exposure period - application to control of the healthy worker survivor effect.”
Mathematical Modelling, 7(9-12): 1393–1512.
Robins, James M. 1989. “The control of confounding by intermediate variables.” Statistics
in Medicine, 8(6): 679–701.
Robins, James M. 1994. “Correcting for non-compliance in randomized trials using
structural nested mean models.” Communications in Statistics-Theory and methods,
23(8): 2379–2412.
Robins, James M. 1998. “Structural nested failure time models.” Encyclopedia of Biostatistics.
Robins, James M. 2000. “Marginal structural models versus structural nested models as
tools for causal inference.” In Statistical models in epidemiology, the environment, and
clinical trials. 95–133. Springer.
Robins, James M, and Sander Greenland. 1992. “Identifiability and exchangeability
for direct and indirect effects.” Epidemiology, 143–155.
Robins, James M, and Thomas Richardson. 2010. “Alternative graphical causal models and the identification of direct effects.” Causality and psychopathology: Finding the
determinants of disorders and their cures, 103–158.
Robins, J M, M A Hernán, and U W E SiEBERT. 2004. “Effects of multiple interventions.” Comparative quantification of health risks: global and regional burden of disease
attributable to selected major risk factors, 1: 2191–2230.
Rosner, Jerome, and Dorothea P Simon. 1971. “The Auditory Analysis Test An Initial
Report.” Journal of Learning Disabilities, 4(7): 384–392.
Taubman, Sarah L, James M Robins, Murray A Mittleman, and Miguel A
Hernán. 2009. “Intervening on risk factors for coronary heart disease: an application
of the parametric g-formula.” International Journal of Epidemiology, 38(6): 1599–1611.
155

Valeri, Linda, and Tyler J Vanderweele. 2013. “Mediation analysis allowing for
exposure-mediator interactions and causal interpretation: theoretical assumptions and
implementation with SAS and SPSS macros.” Psychological Methods, 18(2): 137–50.
Van der Wal, W M, M Prins, B Lumbreras, and R B Geskus. 2009. “A simple
G-computation algorithm to quantify the causal effect of a secondary illness on the progression of a chronic disease.” Statistics in Medicine, 28(18): 2325–2337.
Vansteelandt, Stijn. 2009. “Estimating direct effects in cohort and case-control studies.”
Epidemiology, 20(6): 851–860.
Westreich, Daniel, Stephen R Cole, Jessica G Young, Frank Palella, Phyllis C
Tien, Lawrence Kingsley, Stephen J Gange, and Miguel A Hernán. 2012. “The
parametric g-formula to estimate the effect of highly active antiretroviral therapy on incident AIDS or death.” Statistics in Medicine, 31(18): 2000–2009.
Woodcock, Richard W, M Bonner Johnson, and Nancy Mather. 1990. WoodcockJohnson psycho-educational battery–Revised. DLM Teaching Resources.
Wooldridge, Jeffrey M. 2010. Econometric Analysis of Cross Section and Panel Data. .
2nd ed., Boston MA:MIT Press.
Young, Jessica G, Lauren E Cain, James M Robins, Eilis J O’Reilly, and
Miguel A Hernán. 2011. “Comparative effectiveness of dynamic treatment regimes: an
application of the parametric g-formula.” Statistics in biosciences, 3(1): 119–143.

156