ESSAYS ON HETEROGENEITY IN ECONOMETRIC MODELS
By
Shengwu Shang

A DISSERTATION
Submitted
to Michigan State University
in partial fulfillment of the requirements
for the degree of
Economics–Doctor of Philosophy
2013

ABSTRACT
ESSAYS ON HETEROGENEITY IN ECONOMETRIC MODELS
By
Shengwu Shang
The dissertation consists of three parts and the theme is to deal with heterogeneity in
econometrics models for positive response variables. The first part studies the models with
multiplicative heterogeneity for cross sectional data; the multiplicative heterogeneity can
be transformed from the log linear model with additive heterogeneity. We introduce the
notion of Average Partial Effect (APE) and Conditional APE (CAPE); the estimators and
their asymptotic distribution are proposed. In order to catch the positivity of the unknown
conditional expectation function of the unobserved heterogeneity, we borrow the idea of
power series approximation of unknown function in Newey (1993, 1994) and develop an
“exponential sieves” estimator for CAPE suggested in Wooldridge (1992a).
The second part of the dissertation pertains to extending results for CAPE in chapter 1
for panel data sets. First, Using the models in Wooldridge (1999), We compare three main
estimation methods for positive response variable– FE method for log linear model (LFE),
Poisson Quasi-Maximum Likelihood (PQML) and Generalized Method of Moment (GMM)
– by Monte Carlo Simulation and real life data set. It is not surprising that LFE estimator
is not consistent when PQML is; however, we do find circumstance where both LFE and
PQML estimators are consistent plus LFE is more efficient. With this regard, we introduce
GMM to improve the efficiency of PQML estimator as well as keeping the consistency; this
way also finds a solution to the problem raised in Wooldridge (1999). From the simulation
results, we find that GMM can reduce the standard error of PQML estimator by almost
a half. Second, an “exponential sieves” estimator for CAPE is proposed under panel data
setting; the result automatically extends the results in Ai and Norton (2008) from cross
sectional setting to panel data models. Third, We also apply the GMM to a US domestic

airlines data set and the result shows that GMM improves the efficiency by 10% compared
with PQML.
The third part investigates the effect of spatial correlation for fractional response variable.
By a MEAP data of Michigan in 2009/2010 school year, we investigate again the effect of
school financing reform on school performance which is studied by Papke (2005, 2008), Pake
and Wooldridge (2008); we use both level math test pass rate (linear case)and its log odds
ratio (nonlinear) as dependent variable to run OLS and GLS regression; Conley (1999)’s
spatial dependence corrected standard errors are calculated and find that the statistical
significance for some regressors hinges on the choice of cut off points ; however there do exist
other factors whose statistical significance is robust to the choice. This way we shed some
light on how to pick the right window size. Moreover, by transforming LOR back to level
rate, we find the spending effect estimated from linear model is about 4 ∼ 6% higher than
from nonlinear one.

This thesis is dedicated to my mom, Jinxiang Mao, and my dad, Chuchuan Shang.

iv

ACKNOWLEDGEMENTS

Understanding econometrics, in my opinion, is like peeling an onion. Each time you peel
away one layer you discover that another awaits. Each time you think you reach some
understanding of intuition, estimation and application, you later discover that much more
needs to be understood. I fear that I will never reach the core of the onion. But this is what
makes the subject of econometrics so exciting.

I would never been able to finish my dissertation without guidance, suggestions and
support from several people. First, my deepest appreciation goes to my advisor, Professor
Jeffrey M. Wooldridge. Albeit lots of challenges and struggles on my studies at Michigan
State University, I have never regretted working with him. He accepted me as one of his
students when I felt lost and hopeless four years ago and helped me get through the most
difficult time in my life. He provided me his perspective and helpful comments all the time
and his contributions to this dissertation were invaluable. I could never have reached the
depths of this dissertation without his mentorship and insightful advice. In my mind, he
is a genuine scientist; He sets an example of a world-class researcher and teacher for me to
model myself after in my career life.
I also want to express my deep appreciation to Professor Peter Schmidt for his guidance
in my research. I did not take any courses from him, which is really regretful; but his office
door was always open for me whenever I had questions. His comments such as “econometric
research is to tell a story” really were eye opening for me. He amazed me with his thorough
understanding of theories and ability to expressing them in simple words. In addition, I
cannot say enough thanks to Professor Steve Woodbury, who was always sharp on comments
and had an open mind, providing formidable advice that greatly improved this dissertation.
My special gratitude goes to Professor Lijian Yang, who has been a big brother for me and
v

ready for help throughout my entire graduate career. Beside them, I would like to sincerely
thank the following faculty and staff members at MSU: Leslie E. Papke, Jack Meyer, Byron
Brown, Timothy Vogelsang, Richard Baillie, Todd Elder, Tony Doblas Madrid, Kun Ho
Kim, Dorothy Pathak, Pramod Pathak, Ed Mahoney, Ashton Shortridge, Andrew O. Finley,
Michael Frazier, R. V. Ramamoorthi and Margaret Lynch. I have been very fortunate to
meet them at MSU and they all helped me at various stages of my doctoral study.
While I cannot mention everyone, I would like to thank my fellow graduate students,
particularly Jeff Brown, Quentin Brummet, Paul Burkander, Dooyeon Cho, Sanders Chang,
Simon Chang, Hon Foong Cheah, Yu-Wei Chu, Myoung-Jin Keay, Do Won Kwak, Jinyoung Lee, Cuicui Lu, Tan Lu, Kritkorn Nawakitphaitoon, Ilya Rahkovsky, Seunghwa Rho,
Monthien Satimanon, Valentin Verdier, Wei-Siang Wang, Yali Wang. We had numerous
discussions and I appreciate their thoughtful comments and enjoyable conversation.
I thank my parents Jinxiang Mao and Chuchuan Shang for their unwavering love and
support throughout this endeavor; without their encouragement, I could have never accomplished this. I would like to express my gratitude to my elder brother Shengwen Shang ,
elder sister Shenghua Shang and younger brother Shenggang Shang who took my part of the
responsibility for my parents, which made it possible for me to stay single-minded on my
studies for so long. Last but not least, I would like to thank my wife who was always there
cheering me up and stood by me through the good and bad times.

vi

TABLE OF CONTENTS

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

ix

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

xi

CHAPTER 1
1.1
1.2
1.3
1.4
1.5
1.6

ON ESTIMATING PARTIAL EFFECTS AFTER RETRANSFORMATION . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Model and Partial Effects . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimating the APEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimating Conditional APEs . . . . . . . . . . . . . . . . . . . . . . . . . .
Application to Treatment Effects . . . . . . . . . . . . . . . . . . . . . . . .
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
3
7
10
16
19

CHAPTER 2 ON THE USE OF EXPONENTIAL VERSUS LOG-LINEAR
MODELS FOR PANEL DATA . . . . . . . . . . . . . . . . . . 21
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Model 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 PQML VS. LFE: a Simulation Approach . . . . . . . . . . . . . . . . . . . . 24
2.3.1 Simulation 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.2 Simulation 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.3 Simulation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Estimating APE: Exponential Sieve Estimator . . . . . . . . . . . . . . . . . 28
2.4.1 Estimating APE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4.2 Estimating CAPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5 More Efficient Estimator: GMM . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.1 Model 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.5.2 GMM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5.3 Simulation 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5.4 Optimal IV Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.6 Empirical Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
CHAPTER 3 A SPATIAL ANALYSIS OF SPENDING EFFECT ON
MEAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 Data Characteristics and Sources . . . . . . . . . . . . . . . . . . . . 52
3.3.2 Spatial Dependence Measurement . . . . . . . . . . . . . . . . . . . . 54
3.4 Linear Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.4.1 Ordinary Least Square (OLS) . . . . . . . . . . . . . . . . . . . . . . 56
vii

3.5

3.6

3.4.2 Generalized Least Square (GLS) .
Nonlinear Model . . . . . . . . . . . . .
3.5.1 Regression of Log Odds Ratio . .
3.5.2 Estimate the APE for Level Rates
Conclusions . . . . . . . . . . . . . . . .

APPENDICES
Appendix A .
Appendix B .
Appendix C .

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. . . 100

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

viii

LIST OF TABLES

Table B.1 Estimation results: xtreg . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

Table B.2 Simulation results where Vit has Gamma distribution

79

. . . . . . . . . . .

Table B.3 Simulation results where Vit has Gamma distribution(Continued)

. . . .

79

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80

Table B.5 Simulation results where Vit has log-normal distribution . . . . . . . . . .

80

Table B.6 Vit = exp(a ∗ x2it + b ∗ xit ∗ zit ) with N = 500 . . . . . . . . . . . . . . . .

81

Table B.7 Vit = exp(a ∗ x2it + b ∗ xit ∗ zit ) with N = 1000 . . . . . . . . . . . . . . .

81

Table B.4 Special case

2

Table B.8 Simulation results with Vit = exp(−.125Xi + .5Xi ∗ zit ) . . . . . . . . . .

82

Table B.9 Simulation results for four estimators . . . . . . . . . . . . . . . . . . . .

82

Table B.10 Summary statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83

Table B.11 Dependent variable, passen

83

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

Table C.1 Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Table C.2 OLS Regression, dependent variable=math4 . . . . . . . . . . . . . . . . . 102
Table C.3 OLS Regression with Conley S.E., dependent variable=math4 . . . . . . . 103
Table C.4 OLS Regression with Conley S.E., dependent variable=math4 . . . . . . . 103
Table C.5 OLS Regression with Conley S.E., dependent variable=math4 . . . . . . . 104
Table C.6 OLS Regression with Conley S.E., dependent variable=math4 . . . . . . . 104
math4 ) . . . . . . . . . . . . . 105
Table C.7 OLS Regression, dependent variable=log( 1−math4
math4 ) . . . 105
Table C.8 OLS Regression with Conley S.E., dependent variable=log( 1−math4
math4 ) . . . 106
Table C.9 OLS Regression with Conley S.E., dependent variable=log( 1−math4
math4 ) . . . 106
Table C.10 OLS Regression with Conley S.E., dependent variable=log( 1−math4

Table C.11 OLS Regression with Conley S.E. in Nonlinear Model . . . . . . . . . . . 107
ix

Table C.12 APEs with Bootstrap S.E. in Nonlinear Model . . . . . . . . . . . . . . . 108
Table C.13 QGLS with Conley S.E., dependent variable=math4 . . . . . . . . . . . . 121
Table C.14 QGLS with Conley S.E., dependent variable=math4 in Year 2010 . . . . . 122
Table C.15 SAR GLS, dependent variable=math4 . . . . . . . . . . . . . . . . . . . . 122
Table C.16 SAR GLS, dependent variable=math4 in Year 2010 . . . . . . . . . . . . 123
Table C.17 SAR GLS, dependent variable=math4, contiguity . . . . . . . . . . . . . . 123
Table C.18 SAR GLS, dependent variable=math4, inverse distance . . . . . . . . . . 124
Table C.19 Summary of Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Table C.20 Summary of Correlation(div) . . . . . . . . . . . . . . . . . . . . . . . . . 126

x

LIST OF FIGURES

Figure 1.1

Picture of Sieve Estimator. . . . . . . . . . . . . . . . . . . . . . . . . . .

11

Figure B.1 Bias of LFE and PQML with change of ρ . . . . . . . . . . . . . . . . . .

84

Figure B.2 Std. error of LFE and PQML with change of ρ . . . . . . . . . . . . . .

85

Figure B.3 Bias of LFE and PQML with change of ρ, N=1000 . . . . . . . . . . . .

86

Figure B.4 Std. error of LFE and PQML with change of ρ, N=1000 . . . . . . . . .

87

Figure B.5 Histogram of Passengers . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Figure C.1 All school districts of Michigan in 2010. . . . . . . . . . . . . . . . . . . . 110
Figure C.2 All Colleges of Michigan in 2010. . . . . . . . . . . . . . . . . . . . . . . 111
Figure C.3 MEAP math pass rate for 4th graders of Michigan in 2010. . . . . . . . . 112
Figure C.4 Colleges and math pass rate for 4th graders of Michigan in 2010. . . . . . 113
Figure C.5 Selection of 96 School Districts. . . . . . . . . . . . . . . . . . . . . . . . 114
Figure C.6 Selection of 96 School Districts with centroids. . . . . . . . . . . . . . . . 115
Figure C.7 Selection of 96 School Districts with centroids in grids. . . . . . . . . . . 116
Figure C.8 Conley Coordinates of 96 School Districts. . . . . . . . . . . . . . . . . . 117
Figure C.9 Histogram for all covariates . . . . . . . . . . . . . . . . . . . . . . . . . 118
Figure C.10 APE w.r.t average expenditure . . . . . . . . . . . . . . . . . . . . . . . 119
Figure C.11 APE w.r.t enroll . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Figure C.12 APE w.r.t lunch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xi

Chapter 1
ON ESTIMATING PARTIAL EFFECTS AFTER RETRANSFORMATION

1.1

Introduction

Strictly positive response variables are very common in economics and other social sciences.
Just a few examples include prices, populations, and firm sales. If Y > 0 is the response
that we would like to explain, the most common approach for modeling Y is to use a linear
model for its natural log, log(Y ), and then estimate the linear model using an appropriate
technique – usually ordinary least squares (OLS) or instrumental variables (IV).
There are at least two reasons modeling log(Y ) may not be sufficient. First, one might
wish to predict Y , not log(Y ). When the prediction of Y is based on its expected value
conditional on a vector of covariates – say, X – in general there is no way to recover E(Y |X)
from E[log(Y )|X]. The difficulty in predicting Y given a model for log(Y ) has long been
recognized, and solutions are available under varying levels of assumptions. Duan (1983)
covers the case where an additive error in the model for log(Y ) is assumed independent of
X, and this method is covered even in some introductory econometrics texts [for example,
Wooldridge (2009, Chapter 6)]. Under distributional assumptions, such as assuming Y given
X follows a lognormal distribution, parametric heteroskedasticity in Var[log(Y )|X] is easily
allowed.
More recently, Ai and Norton (2008) provide a semiparametric approach that produces
consistent predictions under weak assumptions, although the approach they use allows the
possibility that predictions of Y can be negative. Wooldridge (1992) proposes direct estimation of E(Y |X) via quasi-likelihood methods using flexible functional forms that ensure
nonnegative predictions. Of course, nonparametric methods, of the type covered in Li and
Racine (2007), can be used, too.

1

Related to the prediction issue is the calculation of partial effects. Wooldridge (1992)
makes a case for basing partial effects on E(Y |X) (in cases where the elements of X are
appropriately “exogenous”). Such an approach underlies the work by Ai and Norton (2008),
who begin with a linear model for log(Y ) but then employ nonparametric methods to recover an estimate E(Y |X) – without placing further restrictions on D(Y |X), the conditional
distribution of Y given X. But basing partial effects on E(Y |X) (or even some other feature
of the conditional distribution, such as the median) is not the only possibility. Lately, the
notion of an average partial effect (APE) has become important in applied econometrics;
see, for example, Wooldridge (2005, 2010). The APE is closely tied to Blundell and Powell’s
(2003) average structural function (ASF), which is defined by averaging out unobservables
from a “structural” model. One potentially important implication of the ASF approach has
largely gone unnoticed: the ASF approach can provide very different partial effects than
that based on E(Y |X). A heteroskedastic probit example is provided by Wooldridge (2005):
the partial effects based on E(Y |X) = P (Y = 1|X) and those based on the ASF need not
even have the same sign, let alone similar magnitudes.
One reason the APE/ASF concept is appealing is because it can be easily applied to
cases where explanatory variables should be treated as endogenous. Consequently, the focus
on the ASF has led to a widely applicable class of control function estimators in nonlinear
models – see Blundell and Powell (2003, 2004). Further, in a broad class of models, the sign
of an APE and an underlying parameter on the covariate of interest are the same. In this
paper we highlight another useful feature of the APE approach: it provides justification for
Duan-type retransformation estimators even when Duan’s key assumption – independence
between the underlying error U and the covariates X – is violated.
We also consider the notion of a “conditional” average partial effect (CAPE) (for example, Wooldridge, 2004, 2005), which is considered more generally as the “local average
response” in Altonji and Matzkin (2005). Interestingly, general consideration of CAPEs
leads to essentially the same estimation problem described in Ai and Norton (2008), except

2

that our approach here allows for endogenous explanatory variables. Plus, when on restricts
the nature of the condititioning set, then simple strategies are available that do not require
complicated nonparametric estimation.
After presenting the model and definitions of partial effects in Section 2, Section 3 discusses estimation of APEs – which, in the current seeting, turns out to be nothing more
than extending Duan’s (1983) “smearing” estimate to more general settings. We consider
the estimation of CAPEs in Section 4, and Section 5 contains a brief conclusion. Technical
derivations are contained in an appendix.

1.2

The Model and Partial Effects

The setup we consider is a standard linear model with log(Y ) as the dependent variable.
Initially assume that, in the population,

log(Y ) = Xβ + U
E(U |X) = 0,

(1.2.1)
(1.2.2)

where X is a 1 × K vector of covariates with first element X unity. We could consider
nonlinear regression functions in place of Xβ, but retransformation methods are almost
always applied when the transformed variable follows a linear-in-parameters model. As in
Duan (1983), we could also consider other strictly monotonic transformations of Y but the
natural logarithm is by far the most popular.
Given equation (1.2.1), we can write

Y = exp(Xβ + U ) = exp(Xβ) exp(U ).

(1.2.3)

Following the discussion in the introduction, we base the partial effects of interest on this
equation because we are interested in how the Xj affect Y , not log(Y ).

3

Ai and Norton (2008) focus on the conditional mean E(Y |X), which can be written
generally as

E(Y |X) = exp(Xβ)E[exp(U )|X] ≡ exp(Xβ)r(X)

(1.2.4)

where r(X) ≡ E[exp(U )|X]. If Xj is a continuous variable then the partial effect on µ(x) ≡
E(Y |X = x) is
∂E(Y |X = x)
∂r(x)
∂µ(x)
=
= β j exp(xβ)r(x) + exp(xβ)
.
∂xj
∂xj
∂xj

(1.2.5)

To estimate this partial effect we need to estimate r(·) in addition to β – the problem
considered by Ai and Norton (2008). We reconsider this problem from a different perspective
in Section 4.
Equation (1.2.5) clearly shows that the partial effect of xj on E(Y |X = x) need not even
have the same sign as β j , and the magnitude of the partial effect can depend on r(·) in a rather
complicated way. In the special case E[exp(U )|X] = exp(Xδ), E(Y |X) = exp[X(β + δ)],
and so the partial effects of xj on E(Y |X = x) is the same sign as β j + δ j .
There is another definition of partial effects that is easier to summarize and, as it turns
out, also easier to estimate. Following Blundell and Powell (2003), the average structural
function (ASF) is defined as

ASF (x) = E[exp(xβ) exp(U )] = exp(xβ)E[exp(U )] ≡ η exp(xβ),

(1.2.6)

where

η ≡ E[exp(U )].
The definition of the ASF is related to the notion of a “Marshallian structural function”
defined in Heckman (2001). In defining the ASF it is important to see that the covariates
are held fixed at specified values, with U averaged out. Once the ASF is obtained, we can see

4

how this function changes as the xj change. As discussed in Wooldridge (2005), the ASF is
closely tied to the notion of an average partial effect (APE). From (1.2.3), the partial effect
of Xj on Y is
∂Y
= β j exp(Xβ) exp(U ).
∂Xj
To get the APE we average U out of the partial effect for given covariate values x. In other
words, the APE at x is

AP Ej (x) = E β j exp(xβ) exp(U ) = η[β j exp(xβ)],
and this is easily seen to be the partial derivative of the ASF. A similar argument works if
we use discrete changes in xj rather than a calculus approximation.
An attractive feature of the ASF is that its definition does not require us to take a stand
on possible dependence between U and X. The definition is unchanged even if U and X are
correlated. When U and X are independent, µ(x) = ASF (x), but generally these quantities
differ when D(U |X) depends on X – even if X is exogenous in the sense of assumption
(1.2.2). The potential difference between average partial effects and partial effects based
on E(Y |X = x) has been pointed out by Wooldridge (2005), who uses a probit model with
heteroskedasticity to illustrate that when U and X are not independent, the APEs and partial
effects based on E(Y |X) will be different – perhaps very different. Unfortunately, it does
not seem possible to resolve the issue of how one should compute partial effects. The choice
between E(Y |X = x) and ASF (x) is essentially one of preference. The main contribution
of the current paper is to show that it is easy to estimate the ASF in the retransformation
context without taking a stand on D(U |X).
One shortcoming with APEs is that the heterogeneity is averaged across the entire population whereas the covariates are set at specific values. Altonji and Matzkin (2005) argue
that finding partial effects conditional on specific outcomes x is generally more useful – what

5

they call a “local average response” (LAR). Wooldridge (2005) also discusses partial effects
average over a subset of the population, based on observable characteristics.
As a general statement, suppose
Y = g(X, U ),
so that, for a continuous covariate, the partial effect at X = x is ∂g(x, U )/∂xj . Now suppose
we wish to compute the expected value of this partial effect not across the entire distribution
of U , but for the subset of the population with X = x. Then we can define a conditional
average partial effect (CAPE) as

CAP Ej (x) = E

∂g(x, U )
X=x
∂xj

In the exponential case, the CAPE can be expressed as

CAP Ej (x) = β j exp(xβ)E[exp(U )|X = x]
≡ β j exp(xβ)r(x)
Note that this is different than basing partial effects on E(Y |X = x), yet we need to estimate
the same function, r(x). From an interpretation standpoint CAPE has the convenient feature
that it is the same sign as β j ; we simply need to compute the scale factor, r(x), that multiplies
β j exp(xβ). The function AP Ej (x) replaces r(x) with the mean value η = E[r(X)] =
E[exp(U )].
As with the APE, the CAPE makes sense even if X includes endogenous elements. In
fact, the CAPE includes as a special case average treatment effects for various subpopulations
when treatment assignment is endogenous. (With binary treatments we would use changes,
not derivatives.)

6

1.3

Estimating the APEs

√
The expression for ASF (x) makes it clear that a N -consistent estimator of ASF (x) is
√
available if N -consistent estimators of β and η are available. By contrast, the dependence
of µ(x) on the nonparametric function r(x) means that partial effects based on E(Y |X = x)
√
are not generally estimable at the N -rate (because nonparametric rates of convergence
√
are slower – often much slower – than N ). Thus, we can estimate the ASF much more
precisely than E(Y |X = x).
Because µ(x) and ASF (x) both depend on β, we first need to consistently estimate β. It
is very common to use OLS as the estimator of β, even though it may not be asymptotically
efficient under E(U |X) = 0. (For example, a weighted least squares estimator that attempts
to exploit nonconstant Var(U |X) could be more efficient.)
As is well known, the assumption E(U |X) = 0 does not substantively restrict r(x) ≡
E[exp(U )|X = x]: it can be virtually any positive function of x. Of course, if we suitably
restrict D(U |X) then we can usually find r(x). A useful assumption is that U and X are
independent, a case considered by Duan (1983); see also Wooldridge (2009, Section 6.4).
Then

r(X) = E[exp(U )|X] = E[exp(U )] = η,

(1.3.1)

E(Y |X = x) = η exp(xβ) = ASF (x).

(1.3.2)

and it follows that

[If we specify the distribution of U , then we can sometimes write η in terms of higher moments
of U . For example, if U ∼ N ormal(0, σ 2 ) then η = exp(σ 2 /2); see Wooldridge (2009, Section
6.4).]
In the case considered by Duan (1983) – where U and X are assumed to be independent
– estimation of η is straightforward. First, by the law of large numbers,

7

N

N −1

p

exp(Ui ) → η.

(1.3.3)

i=1

The average is not an estimator because we do not observe the Ui . Instead, given a random
ˆ from the OLS regression
sample {(Xi , Yi ) : i = 1, ..., N }, obtain β

log(Yi ) on Xi , i = 1, ..., N

(1.3.4)

p
ˆ be the OLS residuals. Because β
ˆ→
and then let Uˆi = log(Yi ) − Xi β
β it is not suprising

that
N

η=

N −1

exp(Uˆi )

(1.3.5)

i=1

is generally consistent for η. Wooldridge (2010, Lemma 12.1) contains a general result that
implies consistency under weak regularity conditions. Because U and X are independent,
we estimate µ(x) and ASF (x) in exactly the same way:

ˆ = ASF (x).
µ
ˆ (x) = ηˆ exp(xβ)
Wooldridge (2010, Problem 12.17) can be used to show

√

(1.3.6)

N (ˆ
η − η) has a limiting normal

distribution and to find its asymptotic variance. Below we consider the problem of estimating
the joint asymptotic variance under very weak assumptions.
For estimating the ASF, there is an important point about Duan’s estimator in (1.3.6):
it is a consistent estimator of η = E[exp(U )] even when U and X are dependent. In fact,
the most we need to assume is

E(X U ) = 0

(1.3.7)

as this is sufficient for OLS to consistently estimate β. Duan (1983) was interested in recovering E(Y |X), which is why he assumed independence between U and X; see also Abrevaya

8

(2002), who obtained the asymptotic variance of the predictions under the assumption of independence. But when we view Duan’s estimator as estimating the scale factor that appears
ˆ is consistent for β. In settings
in the ASF, the estimator is generally consistent provided β
where we intend X to be exogenous in (1.2.1), it suffices to assume (1.3.7) – in which case
the OLS estimator from (1.3.3) is consistent but not necessarily unbiased.
In Section 2 we mentioned how the definition of the ASF is unchanged regardless of
dependence between U and X. As can be seen from equation (1.2.6), we consistently esimate
ASF (x) = η exp(xβ) provided we have consistent estimators of η and β, and (1.3.5) shows
we just need a consistent estimator of β to consisently estimate η. Having some elements of
X endogenous in (1.2.1) in the sense that Cov(X, U ) = 0 causes no problems for estimating
ASF (x) provided we have suitable instrumental variables. In particular, suppose we have a
1 × L vector satisfying
E(Z U ) = 0
rank E(Z X) = K

(1.3.8)

rank E(Z Z) = L
Under these assumptions, the 2SLS estimator (as well as other generalized method of moments estimators) is consistent for β; see, for example, Wooldridge (2010, Chapter 5). Then,
we can let β be the 2SLS estimator from
log(Yi ) = Xi β + Ui

(1.3.9)

using IVs Zi . The Ui are now the 2SLS residuals, and η is still computed from (1.3.5).
Notice that it would be meaningless in this context to base partial effects on E(Y |X) or
E(Y |X, Z), whereas the ASF can have a causal interpretation. We summarize the above as
the following theorem.
Theorem 1.3.1. As assumptions in equation (1.3.8) and Cov(X, U ) = 0, β be the 2SLS

9

estimator from equation (1.3.9) using IVs Zi and η is defined in equation (1.3.5), then


√ β − β  d
N
 → N (0, Ω)
η−η




 Si 
Where Ω ≡ E   ∗ Si Qi
Qi

PROOF: See the appendix.

1.4

Estimating Conditional APEs

Estimation of CAPEs is more difficult due to the need to estimate the function r(x), which is
the same problem faced by Ai and Norton (2008). The motivation for their general approach
is straightforward. If we could observe the Ui then we could use nonparametric regression
of exp(Ui ) on Xi . Because we do not know β we replace Ui with the OLS residuals Ui and
use exp(Ui ) as the dependent variable in a nonparametric regression.
Because of technical complications, Ai and Norton propose linear series estimation, where
exp(Ui ) is regressed on various functions of Xi . As in all nonparametric contexts, the rate
√
of convergence of r(·) to r(·) is slower than N , and much slower when the dimension of X
is large. See Ai and Norton (2008) for details.
From equation (1.2.4), we have:
r(x) ≡ E

Y
X=x
exp(Xβ)

(1.4.1)

From now on, the focus is how to estimate r(x). In the literature, we do have at
least two ways to estimate r(x): one way is the traditional parametric method; we assume r(x) = g(xα), where g(.) is a real function satisfying certain conditions. The remained
work is to estimate the parameter α. If we assume g(.) is a linear function of xα, α can
be estimated by the classic ordinary least square; if g(.) is nonlinear, then nonlinear least
10

r(x)

1

r∗ (x)

3
2
r(x)

Figure 1.1: Picture of Sieve Estimator: For interpretation of the references to color in this
and all other figures, the reader is referred to the electronic version of this dissertation.

squared can be applied. Details can be referred to Wooldridge (2010). The other way is
noparametric method which does not impose any parametric assumptions on r(x). As for
the comparison of those two methods, refer to a recent paper by Ackerberg et al. (2011).
Here the nonparametric assumption is used, and we will propose a sieve estimator for r(x);
the idea of sieves can be vividly explained by the above picture Figure1.1.
In Figure1.1, r(x) is unknown and is our target, we approximate it with r∗ (x); we know
more terms, the number of which is denoted as M , can lead to more close approximations.
However,with more terms to approximate, the estimating bias from r(x) to r(x) is increasing.
The triangle picture shows that we need to choose a balanced M . This can also be seen from
the order of the mean square error in theorem 2. Theoretically, as long as r(x) is smooth
enough, we can always find a reasonable M . Newey (1994) and Ai and Norton (2008) suggest
a sample based method to decide M -cross validation.
Note that r(x) > 0, we cannot guarantee it if we follow the method in Ai and Norton
(2008). At the end of his paper, Wooldridge (1992a) suggests “exponential sieves”to replace
linear ones to span the unknown space composed of positive functions. Similarly, Hirano
et al.(2003) propose a “logit series” estimator for the probability distribution functions.
Inspired by previous work, we develop an “exponential sieve” estimator.

11

Let B = [trace(B B)]1/2 be the Euclidian norm of a matrix B; ζ(M ) = sup GM (x) ;
x∈Ξ

the power series GM (x) we use here are the same as in Hirano et al (2003).First, we list
some assumptions:
Assumption 1: (Y1 , X1 ), . . . , (YN , XN ) are i.i.d and Var(Y |X) is bounded and bounded
Y
− r(X)
away from zero; Var exp(Xβ)

4

is bounded.

Assumption 2:(i) the smallest eigenvalue of E[GM (X) GM (X)] is bounded away from
zero uniformly in M ;(ii) there is a sequence of constants ζ(M ) and M (N ) such that ζ(M )2 M/N −→
0 as N −→ ∞

These assumptions are as usual as in the literature; since we use power series here,ζ 0 (M ) =
CM , later on we use this equivalence lots of times in the derivation.
Assumption 3: If f : RK −→ R is s times continuously differentiable and gM (x) =
[1, x, ..., xn ] , M = (n + 1), Note that gM (x) has powers in x at least up to n; there is a
M-vector γ M such that for GM (x) = AM gM (x), and on the compact set Ξ ⊆ RK ,
sup f (x) − GM (x)γ M < C1 n−s ≤ C2 M −s ,

(1.4.2)

x∈Ξ

This assumption is used as a fact in Hirano et al (2003), while Newey (1997) puts it as
assumption. To ensure that the approximation of r(x) is positive, we first approximate the
log of r(x):
sup log(r(x)) − GM (x)π M < CM −s ,

(1.4.3)

x∈Ξ

So the exponential sieves estimator of r(x) is r(x) = exp(GM (x)π M ), where M is fixed
and

N

Yi

π M = arg min
π

exp(Xi β)

i=1

2

− exp(GM (X

i )π)

(1.4.4)

p

ˆ M − π ∗M −→ 0, where
For N −→ ∞, we have π
π ∗M = arg min E
π

2
Yi
− exp(GM (Xi )π)
exp(Xi β)

12

(1.4.5)

Lemma 1: Suppose that:
1. the support Ξ of x is a compact set of RK
2. r(x) is s times continuously differentiable, that is, r(x) ∈ C s ,with s ≥ 4
3. r(x) is bounded away from zero
4. the density of X is bounded away from zero on Ξ
Then:
r(x) − exp(GM (x)π ∗M ) = O(CM −s ),
PROOF: See appendix.

Lemma 1 is corresponding to the step 1 in the picture; as long as the M is large enough,
the deviation term will vanish. With a higher M , the converge rate is faster.

Lemma 2: Suppose that same four conditions as in Lemma 1 hold. In addition, suppose
that: (5) M (N ) is a sequence of values of M satisfying M (N ) → ∞, and ζ(M (N ))4 /N → 0.
Then

π M (N ) − π ∗M (N ) = Op

M (N )
N

PROOF: See appendix.

Theorem 1.4.1. Given all the assumptions in Lemma 1 and 2, then

[r(x) − r(x)]2 dF (x) = O(M/N + M −2s )

13

PROOF: Note that,
[r(x) − r(x)]2 dF (x)
=

[r(x) − exp(GM (xi )π ∗M ) + exp(GM (xi )π ∗M ) − exp(GM (xi )π M )]2 dF (x)

≤

[r(x) − exp(GM (xi )π ∗M )]2 + [exp(GM (xi )π ∗M ) − exp(GM (xi )π M )]2 dF (x)

= O(M/N + M −2s )

The last equality follows from Lemma 1 and 2. From this theorem, we can see the two
steps in the picture and the tradeoff of increase the M
Let
ΣM ≡ E[GM (Xi ) GM (Xi )

2
Yi
− r(Xi ) exp(2GM (Xi )π ∗M )],
exp(Xi β)

QM ≡ E[GM (Xi ) GM (Xi ) exp(2GM (Xi )π ∗M )],
−1 Σ Q−1 (x)GM (x) exp(2GM (x)π ∗ ).
VM (x) ≡ GM (x)QM
M M
M

Lemma 3: Suppose that the same four conditions as in Lemma 1 hold, then
√

−1/2

N VM
−1/2

Note that VM

d

(x) r(x) − r(x) → N (0, 1)

(1.4.6)

(x) ≤ O(M −1/2 ), so this result gives an upper bound Op ((M/N )1/2 )

converge rate, which is lower than Op ((1/N )1/2 ); Ai and Norton (2008) find the similar result
for linear case. This result also coincides many other results in semi/no parametric literature.

The structure of the APE estimator is interesting: it is a scale factor times the estimator
of the parameter of interest. The ideal is originally from the so-called average structure
function in Blundell and Powell (1994), and Papke and Wooldridge (2008) advance further
to estimate APE of parameter of interest in a panel data probit model with and without

14

endogenous explanatory variables.
From section 2, the CAPE can be expressed as

CAP Ej (x) = β j exp(xβ)E[exp(U )|X = x]
≡ β j exp(xβ)r(x)
So the estimator of CAP E follows as :
CAP Ej (x) = β j exp(xβ)r(x) = β j exp(xβ) exp(GM (x)π M )
Theorem 1.4.2. Given N (1/2) M −(s+1) → 0 as N → ∞ and β j = 0, Assumptions 1-3 and
all the assumptions in Lemma 1 and 2, then
√
d
−1/2
N VM (x) CAP Ej (x) − CAP Ej (x) → N (0, V )
Where V = β 2j exp(2xβ)
PROOF: See appendix.

Note that, theorem 3 requires that at least one element of β is not zero: if β is zero
completely, the model does not make any practical use; but it is possible that some elements
of β are zero. If there is some element in β is zero, without loss of generality, let’s put it as
β j =0, then:
Corollary: Given all the assumptions in theorem 3 except β j =0, then
√

d

N CAP Ej (x) − CAP Ej (x) → N (0, W )

Where CAP Ej (x) ≡ β j exp(xβ)r(x); as for the detailed form of W , refer to proof in appendix.
PROOF: See appendix.

15

This is an interesting result: if some element of β is zero, then the estimator for the
√
corresponding CAP E is N consistent, which is faster than the rates of other CAPEs of
√
−1/2
nozero β. From theorem 3, we can see that under N VM (x), CAP Ej (x) degenerates to
zero when β j = 0; so it is not surprising that we need faster rate to get a non-degenerating
asymptotic distribution. There are several such examples in Ferguson (1996), e.g., refer to
example 3 in Chapter 7; also, theorem 5.4 in Lee (2004).

1.5

Application to Treatment Effects

As an example of where we might want to apply retransformation after IV estimation,
consider the modern treatment effect literature. Let D denote a binary treatment and
denote the strictly positive countfactual outcomes as Y (0) and Y (1). We observe D and

Y = (1 − D)Y (0) + DY (1).
Assume we have covariates W such that

Y (0) = exp(α0 + Wβ 0 + U )
Y (1) = exp(α1 + Wβ 1 + U )
where, for simplicity, we assume only one source of heterogeneity, U , and E(U ) = 0. Also,
assume that U and W are independent. then the average treatment effect, as a function of
w, defined by

τ ate (w) = E[Y (1) − Y (0)|W = w],
is easily seen to be

τ ate (w) = η[exp(α1 + wβ 1 ) − exp(α0 + wβ 0 )]

16

where η = E[exp(U )]. Furthermore, in terms of the observed outcome Y , τ ate (w) can be
written in terms of its ASF, ASF (d, w). To see how, write
Y = Y (0)(1−D) Y (1)D
and so
Y = exp(α0 + γD + Wβ 0 + D · Wδ + U ),
where γ = α1 − α0 and δ = β 1 − β 0 . So the ASF for Y is

ASF (d, w) = exp(α0 + γd + wβ 0 + d · wδ)E[exp(U )]
= η exp(α0 + γd + wβ 0 + d · wδ)
If we evaluate the ASF at d = 1 and d = 0 and difference we get

ASF (1, w) − ASF (0, w) = η[exp(α0 + γ + wβ 0 + wδ) − exp(α0 + wβ 0 )]
= η[exp(α1 + wβ 1 ) − exp(α0 + wβ 0 )]
= τ ate (w)
We can now apply a simple 2SLS strategy if we assume

E(U |Z) = 0
for IVs Z that include W and at least one element not in W that predicts treatment status,
D. Write

log(Yi ) = α0 + γDi + Wi β 0 + Di · Wi δ + Ui
and use instruments, say, (1, Zi , Zi ⊗ Wi ), or one can be selective with the interactions. Or,
as described by Wooldridge (2010, Chapter 21), probit or logit fitted values can be used as
IVs, in which case the list would look like
17

ˆ i , Wi , G
ˆ i · Wi ),
(1, G
ˆ i are the fitted probabilities from a binary response model of Di on Zi .
where the G
Given the 2SLS residuals Uˆi we compute ηˆ exactly as in equation (1.3.5), and then the
estimated ASF is

ˆ 0 + d · wˆ
ASF (d, w) = ηˆ exp(ˆ
α0 + γˆ d + wβ
δ)
For any w we estimate τˆate (w) = ASF (1, w) − ASF (0, w), and the unconditional average
treatment effect, τ ate = E[Y (1) − Y (0)], is consistently estimated as

N

τˆate = N −1

[ASF (1, Wi ) − ASF (0, Wi )]
i=1
N

= N −1

ˆ 0 )]
ˆ 1 ) − exp(ˆ
α 0 + Wi β
ηˆ[exp(ˆ
α 1 + Wi β
i=1

The average treatment effect on the treated is

τ att (w) = E[Y (1) − Y (0)|D = 1, W = w]
= [exp(α1 + wβ 1 ) − exp(α0 + wβ 0 )]E[exp(U )|D = 1, W = w]
≡ [exp(α1 + wβ 1 ) − exp(α0 + wβ 0 )]r(1, w),
where

r(d, w) = E[exp(U )|D = d, W = w].
Now we use nonparametrics of exp(Uˆi ) on Wi for Di = 1.

ˆ 1 ) − exp(ˆ
ˆ 0 )]ˆ
τˆatt (w) = [exp(ˆ
α1 + w β
α0 + w β
r(1, w)
Often reported is
18

τ att = E[Y (1) − Y (0)|D = 1]
and a consistent estimator is
N

N1−1

ˆ 1 ) − exp(ˆ
ˆ 0 )]ˆ
Di [exp(ˆ
α 1 + Wi β
α 0 + Wi β
r(1, Wi ).
i=1

1.6

Concluding Remarks

We have shown that a common retransformation method due to Duan (1983) can be used
generally to estimate the parameters in the average structural function under weak assumptions on the dependence between the error and the covariates. In a standard regression
setting, even a zero correlation assumption suffices. Further, the method easily applies when
instrumental variables are needed to consistently estimate the coefficients in the log-linear
model.
Our derivation of the joint asymptotic distribution of the parameters indexing the ASF
holds under weak assumptions – much weaker than the independence assumption used in
the standard Duan (1983) setting where errors and explanatory variables are independent.
In the case where X does not contain what we traditionally think of as endogenous
variables – so that the dependence of D(U |X) comes through moments other than E(U |X)
– one might question whether partial effects defined through the ASF are more useful than
those obtained from E(Y |X = x). After all, for prediction purposes we prefer E(Y |X = x)
to ASF (x) because the former is the minimum mean square predictor of Y . But if we are
interested in getting the best predictor of Y then we might just model E(Y |X) – such as
a flexible exponential function – and estimate it directly. Of course, we would get partial
effects directly, too.
We also considered estimation of conditional APEs, which generally differ from the APEs
when U and X are dependent (even though they may be mean independent). Here, we are

19

led to an estimation problem essentially the same as Ai and Norton (2008). However, the
calculation of partial effects differs, and the CAPEs have the same signs is the coefficients
on the log-linear model. Some of this extends immediately to other transformations, such
as log[Y /(1 − Y )] when 0 < Y < 1. More work is needed in treatment effect examples with
more than one unobservable. Heckman switching regression. “Simple” solution is to make
distributional assumptions.

20

Chapter 2
ON THE USE OF EXPONENTIAL VERSUS LOG-LINEAR MODELS FOR
PANEL DATA

2.1

Introduction

As we all know, more and more data are collected over time for the same cross section units
with the advance of technology; a big chunk of them are to deal with positive response
variables, just name a few, the observed wage rate for workers over 5 years, number of
patents applied for the firms in the last several years and etc. The immediate extension
of the results in chapter one to the panel data setting would be that model the logarithm
transformation of positive response variable in linear functional form of covariates still and
assume, at a minimum, the error in each time period was assumed to be uncorrelated with
the explanatory variables in the same time period. Basically, we just stack the observations
of all time periods for each cross section unit and repeat the analysis in chapter one again; as
we can see, everything just follows with a bigger sample size in terms of application. However,
we know that assumption is too strong for certain panel data applications.In fact, a primary
motivation for using panel data is to solve the omitted variables problem, and error terms in
each time period are hardly uncorrelated most of the time. So a more interesting extension
would be modeling logarithm transformation of positive response variable in a “modern”
panel data setting, which explicitly contain a time-constant unobserved effect and treat
it as random variable, drawn from the population along with the observed explained and
explanatory variables. We can refer to the “linear panel data model (LPD)” in Wooldridge
(2010) for more details.
While, the problem of modeling the logarithm transformation of positive response variable
in a LPD still exists and it can become even much worse because that unobserved time
invariant heterogeneity can either cause sever correlation among the error terms, or itself

21

is correlated with covariates; even with it removed as in the standard Fixed Effect (FE)
transformation, the problem can hardly be solved. For example, Blackbrun (2007) compares
Poisson Quasi-Maximum Likelihood (PQML) method to wage equation and first difference
to log wage equation; he finds that the latter overestimates the union coverage effect on
wage by almost 14% using panel data of 1989 and 1993 NLSY. We will also see this from
an analysis of a data set about the American domestic airline market from year 1997 to
2000. The results of usual FE method estimation are in Table B.1. Based on the results, we
would say the price elasticity of demand of the market is over one in magnitude. We know
this number is way too big and does not make any sense intuitively. For example, Park et.
al.(2007) finds that most city pairs airline routes for 12 main US carriers are inelastic in short
run. We will say more about it in the application section. This prompts our question of the
estimation method itself. Considering the retransformation problem we have done in chapter
1, it is natural to ask the question: Why should we use the logarithmic transformation of
the response variables instead of modeling them directly?
Several attempts are made in the literature. Most of the work uses nonlinear panel data
models and relies on the method of conditional maximum likelihood (CML) , where a sufficient statistic (the sum of the explained variable across time) is conditioned on to remove
the unobserved effect. Examples include Chamberlain (1980, 1984) for binary responses and
Hausman et al.(1984) for count data. Wooldridge (1999) investigates robustness properties of these CMLEs to misspecification of the initially specified joint distribution and has
shown CMLEs to be consistent when only the conditional mean in the unobserved effects
multiplicative panel data model is correctly specified, which means that the consistency is
robust to arbitrary patterns of serial correlation. Moreover, the results hold not only for
binary or count variables, but also for any nonnegative variables. So it is not surprising
that the estimation method is widely used in empirical applications, especially after Simcoe (2008) writes a STATA code titled “xtpqml ”1 . It sheds new light on how to estimate
1 the

code is updated to xtpoisson in STATA12

22

the conditional mean parameters consistently for positive response variable without taking
logarithmic transformation in panel data models.
So, this chapter starts with Wooldridge (1999). First, we maintain the conditional mean
specification condition (equation (3.1) in Wooldridge (1999)) and show what consequence
of logarithm transformation can cause for the usual FE estimation in LPM by Mont Carlo
simulation; Second, we extend the “exponential sieve” estimator in chapter 1 to the panel
data model- estimate the average partial effect (APE) of the respective interesting explanatory variables; Third, we use “exponential sieve” estimator to construct the optimal IV as
in Newey (1993, 1994) to improve the the efficiency for the GMM estimator suggested in
Wooldridge (1999). Section 2 introduces model; under which the simulations are constructed
to compare PQML and LFE estimators in section 3. A “exponential sieve” estimator of APE
in the panel data model is formulated in section 4 .Section 5 discusses the GMM method
for models with conditional mean and variance functions specified. The GMM can be understood as double PQML: one is for original scale of dependent variable; the other is for
squared scale. We also propose the optimal instrument variables (OIV) estimator used in
Newey (1993, 1994). An empirical application to airfare data is provided in section 6, and
some remarks are contained in section 7.

2.2

Model 1

The model we are considering is the following:
Yit = exp(Xit β)Ci Vit ,

(2.2.1)

where Ci is the unobserved heterogeneity and Vit is the multiplicative idiosyncratic error
term; we assume both Ci and Vit are positive. The immediate question is how to estimate
β. It is tempting to take log transformation for both sides of equation (2.2.1):
log(Yit ) = Xit β + log(Ci ) + log(Vit ),

23

(2.2.2)

This is typical linear panel data model as in Wooldridge (2010) if the relevant assumptions
are made. Justifications for this log transformation include to deal with a dependent variable
badly skewed to the right, and to interpret β as semi elasticity of Y with respect to X, for
example in Manning (1998), Ai and Norton (2000). As suggested in Wooldridge (2010), it
is natural to do the usual FE transformation to remove the heterogeneity:
¨ )=X
¨ ),
¨ it β + log(V
log(Y
it
it
¨ it = Xit − T −1
Where, X

T
t=1 Xit ,

(2.2.3)

¨ ) and log(V
¨ ). So pooled ordinary
similarly for log(Y
it
it

least square can be used to estimate β, which is exactly the FE method in Wooldridge
(2010); we denote this method as LFE. Whether LFE is consistent or not depends on the
¨ ) are uncorrelated. Instead, we assume:
¨ it and log(V
assumptions, one of which is that X
it
E(Vit |Xi , Ci ) = 1,

(2.2.4)

E(Yit |Xi , Ci ) = exp(Xit β + log(Ci )),

(2.2.5)

Note that, we can easily get:

So the correct conditional mean specification condition as in Wooldridge (1999) is satisfied;
We can use Poission Qausi Maximum Likelihood (PQML) to estimate β consistently. It
interesting to compare LFE estimator of β with PQML under equation (2.2.1) and (2.2.4).
It is not difficult to know that LFE estimator of β is not consistent since equation (2.2.4)
¨ ), are uncorrelated to the
can not guarantee that error terms in equation (2.2.3), log(V
it
¨ it . Simulations to compare LFE with PQML are in the following section.
covariates, X

2.3

PQML VS. LFE: a Simulation Approach

In this section, we compare PQML with LFE by 2 Monte Carlo Simulations. The key point
here is how to generate Vit which are positive and satisfy equation (2.2.4). In the first
simulation, we specify that Vit has Gamma distribution and find that PQML estimator is
24

consistent while LFE is not; while in the second simulation, we specify that Vit has log
normal distribution; we find both are consistent and LFE is more efficient than PQML.

2.3.1

Simulation 1

We follow the model in section 2, i. e., equation (2.2.3). The data generating process is
specified as the following:
• i = 1, 2, ..., N ; t = 1, 2, ..., T ,
i.i.d

• Xit ∼ N(0, 1),
• Ci = exp(Xi + N(0, 1)), where Xi = T −1

T
t=1 Xit

• β = .1,
i.i.d

• Vit ∼ Gamma(α, γ), where γ = 1/α = exp(a ∗ X2it + b ∗ Xit ),
• T = 5, N =500 if it is not specified,
• Number of simulations =1000.
We have two reasons to specify the Gamma distribution of Vit that way: 1. it guarantees Vit s are positive; 2. It makes that the condition of equation (2.2.4) is satisfied since
E(Vit |Xi , Ci ) = α ∗ γ = 1, ∀a, b. As for the values of a and b, they will be specified in the
specific setting.
Table B.2 and Table B.3 show the results for PQML and LFE methods; Note the true
value of β is .1. The PQML estimates are very close to it for all the values of a and b.
However, for LFE, the bias is big; such as a = .01, b = .1, a = .05, b = .1, and a = .1, b = .1.
These strengthen the findings in Blackburn (2007); they tell us that we should be very careful
to use LFE to deal with positive response variables. On the other hand, as we can see from
¨ ) biases the LFE estimator; so
¨ and log(V
equation (2.2.3) that the correlation between X
it
whenever the value of b is relatively high compared to a, the correlation between X and
25

log(Vit ), which is denoted as ρx, lv , is high, and forces the LFE away from its target. We
consider an extreme case here: let a = .1, b = 0 and do the simulations for three different
N values: 500, 1000 and 2000; and the results are in table titled “special case”. We can see
that all three LFE estimates beat PQML in terms of bias and efficiency. The bias of PQML
decreases with creasing of sample size, but for small sample, it may not perform as well as
LFE does.

2.3.2

Simulation 2

Compared with simulation 1, simulation 2 is the same as simulation 1 except for the distribution of Vit . Here, Vit has a log-normal distribution. The data generating process is
specified as the following:
• i = 1, 2, ..., N ; t = 1, 2, ..., T ,
i.i.d

• Xit ∼ N(0, 1),
• Ci = exp(Xi + N(0, 1)),
• β = .1,
i.i.d

• Vit = exp(a ∗ X2it + b ∗ Xit ∗ zit ), where zit ∼ N(0, 1) if not specified.
• T = 5, N =500 or 1000,
• Number of simulations =1000.
For the distribution of Vit , the positive issue is obviously satisfied; but the values of a
and b cannot be arbitrary since equation (2.2.4) is required to be satisfied. By algebra, we
need have b2 = −2a.
Table B.4 shows that LFE estimator is as good as PQML in terms of consistency; moreover, PQML estimator is more sensitive to the values of a and b. On the other hand, the

26

standard errors of LFE estimator is only half of PQML. This result is very interesting since
this adds one more advantage to the log transformation-more efficient.

2.3.3

Simulation 3

Compared with simulation 1, simulation 2 is the same as simulation 1 except for the distribution of Vit . Here, Vit has a log-normal distribution. The data generating process is
specified as the following:
• i = 1, 2, ..., N ; t = 1, 2, ..., T ,
i.i.d

• Xit ∼ N(0, 1),
• Ci = exp(Xi + N(0, 1)),
• β = .1,






2
• Vit = exp(a∗Xit +b∗Xit ∗zit ), where Z = 





.

zi1










zi2 





zi3  ∼N(0, Σ), Σ = 





zi4 



zi5

ρ

ρ2 ρ3 ρ4

ρ

1

ρ

ρ2

ρ2

ρ

1

ρ

ρ3 ρ2

ρ

1

ρ4 ρ3 ρ2

ρ

1






2
ρ 



ρ 

1

ρ3

• T = 5, N =500 and ρ = {−.95, −.5, −.1, .1, .5, .95},
• Number of simulations =1000.
The focus of this simulation is setting of Vit : note that equation (2.2.4) only specifies
relationship between Vit , Xi and ci, it does not exclude any correlation among error terms
of all the time periods for a certain cross section unit. The variance covariance matrix of Z,
Σ is the popular exchangeable working matrix in generalized estimating equation literature.
The reason that name is called is every observation in an individual is equally correlated

27



with every other observation in that individual. The degree of correlation is measured by the
intraclass correlation coefficient. Here we can understand that Corr(zit , zit+j ) = ρj , j = 0,
1, 2, · · · , T −1 and |ρ| < 1. All the estimation results are in Table B.6; overall, both methods
perform well in terms of bias and efficiency. However, we do see some fluctuation when ρ
varies in its ranges: from Figure B.3, we can see that the biases of LFE are fluctuating
around zero while all PQML estimators display downward bias; but there is no monotonic
relationship between the bias and the ρ. When it comes to standard errors, the story is
different: Figure B.4 shows that the standard errors of both estimators are increasing with
crease of ρ; and LFE always beats PQML. We repeat the analysis for N = 1000, and similar
patterns apply. So the serial correlation can cause problem to PQML, especially when sample
size is not that big.

2.4

Estimating APE: Exponential Sieve Estimator

After Wooldridge (1999) proves that PQML is appropriate for any non-negative dependent
variable-not just count data that follow a Poisson distribution and the estimator is robust
to arbitrary patterns of serial correlation, the method is widely used in all kinds of empirical
works, especially after Simcoe (2008) writes a STATA command xtpqml. However, since most
of the time, the model is nonlinear in observed explanatory variables with multiplicative
unobserved heterogeneity, it is difficulty to interpret the estimates of parameters, which are
the popular so called semi-elasticities in log linear models; this is one of the key drawbacks
of the method compared with OLS/IV to the log linear models. Inspired by what we have
done in chapter 1, we propose a nonparametric estimation method for APE in model as in
equation (2.2.1). This way, we extend the analysis in chapter 1 from cross section to panel
data settings.

28

2.4.1

Estimating APE

Refer to equation (2.2.1), let Uit =Ci Vit be the whole error term. For the simplicity of
denotation, we drop i for the time being. Similarly to what we have done in Chapter One,
we define APE as follows:
AP Ej (xt ) = E β j exp(xt β)Ut = η[β j exp(xt β)],

(2.4.1)

Here the idea is the same as in Chapter One: the expectation in equation (2.4.1) is taken
only with respect to Ut . In order to estimate AP Ej (xt ), the key is to estimate η; from
equation (2.2.5), we have,
Yit
X
exp(Xit β) i

E

= E(Uit |Xi ),

(2.4.2)

So,
η ≡ E(Uit ) = E

Yit
exp(Xit β)

,

Then the straight forward estimator for η is:
ηˆ =

1
NT

N

T

Yit

ˆ
exp(Xit β)
i=1 t=1

,

(2.4.3)

Where βˆ is the PQML estimator of β in equation (2.2.1). Similarly to Theorem 1 in Chapter
one, we have the following:
Theorem 2.4.1. Let β be the PQML estimator from equation (2.2.1) with condition in
equation (2.2.5)and η is defined in equation (2.4.3), then


√ β − β  d
N
 → N(0, Ω)
η−η




 Si 
Where Ω ≡ E   × Si Qi
Qi

29

PROOF: See appendix.

With delta method, it is natural to have the asymptotic result for the device in equation
(2.4.1):
Corollary 2.4.2.
√

d

N AP E j (xt ) − AP Ej (xt ) → N(0, Θ ΩΘ)

Where
ˆ
AP E j (xt ) = ηˆβ j exp(Xit β)

Θ = ∇θ {η[β j exp(xt β)}, θ = β , η

2.4.2

Estimating CAPE

First, we assume:
D(Ci |Xi ) = D(Ci |Xi ),
where Xi = T −1

T
t=1 Xit ,

(2.4.4)

which is the same as in the simulations in the previous section,

represents the time-averaged Xit over the various panel periods. This assumption can date
back to Mundlak (1978) and Chamberlain (1980, 1982, 1984), and more recently appears in
Papke and Wooldridge (2008). Of course, we don’t need the normal distribution assumption
as they do.
Combine equations (2.2.1), (2.2.4) and (2.4.4), and we get the following by iterated conditional expectation:
E(Yit |Xi ) = E[(Yit |Xi , Ci )|Xi ]
= exp(Xit β)E(Ci |Xi )
= exp(Xit β)E(Ci |Xi )
30

(2.4.5)

So, we define:
E

Yit
X
exp(Xit β) i

= E(Ci |Xi ) ≡ r(Xi ),

(2.4.6)

Here, the r(Xi ) is our target to estimate and assume it has the same properties as in chapter
1. The only difference is that the independent variable is Xi instead of Xi , and variables in
the left hand side of above equation has one more layer of subscript, t, other than i. So the
ˆ M ), where
exponential sieves estimator of r(Xi ) is rˆ(Xi ) = exp(GM (Xi )π
N

T

Yit

ˆ M = arg min
π
π

i=1 t=1

ˆ
exp(Xit β)

2

− exp(GM (X

i )π)

(2.4.7)

p

ˆ M − π ∗M −→ 0, where
For N −→ ∞, we have π
T

π ∗M = arg min E
π

t=1

2
Yit
− exp(GM (Xi )π)
exp(Xit β)

(2.4.8)

Lemma 2.4.3. Suppose that:
1. the support Ξ of X is a compact set of RJ
2. r(X) is s times continuously differentiable, that is, r(X) ∈ C s , with s ≥ 4
3. r(X) is bounded away from zero
4. the density of X is bounded away from zero on Ξ
Then:
r(X) − exp(GM (X)π ∗M ) = Op (CM −s ),

(2.4.9)

PROOF: See appendix.

Lemma 2.4.4. Suppose that same three conditions as in Lemma 1 hold. In addition, suppose
that: (iv) M (N ) is a sequence of values of M satisfying M (N ) → ∞, and ζ(M (N ))4 /N → 0.
Then

31

ˆ M (N ) − π ∗M (N ) = Op
π

M (N )
N

(2.4.10)

PROOF: See appendix.
These two lemmas are almost the same as lemma 1 and 2 in chapter 1 expect here is for Xi .

Theorem 2.4.5. Given all the assumptions in Lemma 2.4.3 and Lemma 2.4.4, then

[r(X) − rˆ(X)]2 dF (X) = O(M/N + M −2s )

(2.4.11)

PROOF: the proof is the same as Theorem 2 in chapter 1.

From this theorem, we can see that the same kind of trade off argument as in chapter 1
happens here: we know more terms, which means higher value of M , can lead to more close
approximations. However, with more terms to approximate, the estimating bias from r(X)
to rˆ(X) is increasing. Refer to chapter 1 for detailed illustration.
The next question is how to estimate the CAPE. First, we need to define the CAPE in
the new device; usually, we need to find the functional form of E(Yit |Xit ). From equation
(2.2.1):
∂Yit
= β j exp(Xit β)Ci Vit ,
∂xij
By iterated expectation and combine equation (2.4.5) and equation (2.4.6)

E

∂Yit
Xi
∂xij

= β j exp(Xit β)r(Xi )

So

CAP Ej (x) ≡ E

∂Yit
Xi = x
∂xij

32

= β j exp(xt β)r(x),

Where, x = (x1 , · · · , xT ), x = T −1

T
t=1 xt .

The natural estimator for CAPE is:

CAP E j (x) = β j exp(xt β)r(x),
Since β can be easily obtained from PQML method, the next will focus on how to estimate
r(x).
Let


T

GM (Xi ) GM (Xi )

ΣM ≡ E 
t=1

Yit
− r(Xi )
exp(Xit β)

2


exp(2GM (Xi )π ∗M ) ,

QM ≡ E[GM (Xi ) GM (Xi )exp(2GM (Xi )π ∗M )],

−1 x)GM (¯
VM (¯
x) ≡ GM (¯
x)Q−1
x) exp(2GM (¯
x)π ∗M ).
M ΣM QM (¯

Lemma 2.4.6. Suppose that the same four conditions as in Lemma 2.4.3 hold, then
√
d
−1/2
N VM (x) (r(x) − r(x)) → N(0, 1)
Theorem 2.4.7. Given N (1/2) M −(s+1) → 0 as N → ∞ and β j = 0 , assumptions 1-3 and
all the assumptions in Lemma 2.4.3 and Lemma 2.4.4, then
√

−1/2

N VM

d

(x) CAP E j (x) − CAP Ej (x) → N(0, V )

Where V = β 2j exp(2xβ)
PROOF: See appendix.

Note that, theorem 2.4.7 requires that at least one element of β is not zero: if β is zero
completely, the model does not make any practical use; but it is possible that some elements
of β are zero. If there is some element in β is zero, without loss of generality, let’s put it as
β j =0, then:
33

Corollary 2.4.8. Given all the assumptions in theorem 2.4.7 except β j =0, then
√

d

N CAP E j (x) − CAP Ej (x) → N (0, W )

Where CAP Ej (x) ≡ β j exp(xβ)r(x) as in equation ; as for the detailed form of W , refer to
proof in appendix.
PROOF: See appendix.

The above results are the analog extensions as in Chapter 1. But we do have a special
result for panel data case. From equation (2.4.5), we know that E(Yit |Xi ) only depends on
Xit and Xi with assumption of equation (2.4.4); so we do the following manipulation:
E(Yit |Xit , Xi ) = E[E(Yit |Xi )|Xit , Xi ]
= E[exp(Xit β)r(Xi )|Xit , Xi ]
= exp(Xit β)r(Xi )

(2.4.12)

Following the same derivation as in Blundell and Powell (2003), we have:
ASF (Xt ) =

E(Yit |Xit = Xt , Xi )dFX

= exp(Xt β)

r(Xi )dFX

i

i

If we let E(Yit |Xit , Xi ) = H(Xit , Xi ), then the RHS of equation (2.4.13) is

(2.4.13)
H(Xit , Xi )dFX ,
i

which is exactly the so called average structural function in Blundell and Powell (2003); as
in chapter 1, we define APE basing on ASF. Denote the average partial effect with respect
to Xt as:
∂ASF (Xt )
∂Xt
∂exp(Xt β)
=
∂Xt

r(Xi )dFX

= β exp(Xt β)

r(Xi )dFX

λ ≡

34

i
i

(2.4.14)

The corresponding estimator is:




N

ˆ pqml exp(Xt β
ˆ pqml ) N −1
λ=β

ˆ M )
exp(GM (Xi )π

(2.4.15)

i=1

The structure of λ is interesting: it only depends on Xt ; the effect of cross sectional heterogeneity is coorperated as a scalar factor. Note that it is between AP E and CAP E(x) :
the former measures the overall effect of covariates and heterogeneity; the latter considers
both cross section and time dimensions. While λ considers the effect of covariates on both
dimensions, but only overall effect of heterogeneity. Papke and Wooldridge (2008) get a similar device in a panel data probit model with and without endogenous explanatory variables.
To obtain asymptotic result for λ, we go further following Papke and Wooldridge (2008):


N



T

ˆ pqml (N T )−1
τ =β

ˆ pqml + GM (Xi )π
ˆ M )
exp(Xit β
i=1 t=1

Note that λ can be considered as a special case of τ :



N



T

ˆ pqml (N T )−1
β

ˆ pqml + GM (Xi )π
ˆ M )
exp(Xit β
i=1 t=1





N

ˆ pqml ) N −1
ˆ pqml exp(Xt β
= β

ˆ M )
exp(GM (Xi )π
i=1

Where Xit = (Xt , 0, · · · , 0). So as long as we obtain asymptotic results for τ , the result
T −1

for λ can be easily achieved:
Theorem 2.4.9. Given N (1/2) M −s → 0 as N → ∞, and all the assumptions in Lemma
2.4.3 and 2.4.4 , then
√

d

N (ˆ
τ − τ ) → N(0, V)

PROOF: See appendix.

35

(2.4.16)

The structure of the variance is a little complicated, the derivations and estimation are
in the appendix.
Note that, it follows from equation (2.4.6):
Yit
X
exp(Xit β) i

E r(Xi ) ≡ E E(Ci |Xi ) = E E

=E

Yit
exp(Xit β)

Then there is an “naive” estimator of APE automatically:
ˆ pqml exp(Xt β
ˆ pqml ) 1
β
NT

N

T

i=1 t=1

Yit
ˆ pqml )
exp(Xit β

With all these solved, we can propose the following estimating procedure:
Step 1: do the PQMLE of Yit on Xit , and denote the estimator of β as β pqml ;
Step 2: use power series to approximate r(Xi ), denote as rˆ(Xi );
Step 3: estimate the APE (or CAPE) of parameter of interest and corresponding standard
error.

2.5

More Efficient Estimator: GMM

As we find that in simulation 2 in section 3, when both PQML and LFE estimators are
consistent, the latter is more efficient than the former. Wooldridge (1999) suggests GMM
method to improve the efficiency of PQML estimator. The key step of GMM is to derive a
moment condition from equation (2.2.3).

2.5.1

Model 2

As we all known, we need a class to compare efficiency; the model we are considering here
as follows:

E(Yit |Xi , φi , ψ i ) = φi µ(Xit , β 0 ),

(2.5.1)

Var(Yit |Xi , φi , ψ i ) = ψ 2i [E(Yit |Xi , φi , ψ i )]2 = ψ 2i [φi µ(Xit , β 0 )]2

(2.5.2)

36

Cov(Yit , Yir |Xi , φi , ψ i ) = 0, t = r

(2.5.3)

This model is initiated in Wooldridge (1999) who put it as a question. As we can see
that the function µ(., .) can be any positive functional form, such as exponential in section
2. Compared with model 1, we have one more equation to specify the relationship between
the conditional variance and the square of conditional mean. Note that ψ i is also a random
variable, so it is the extension of constant coefficient of variation model.

2.5.2

GMM

From the above two equations, we can easily get the following equation:

E(Yit2 |Xi , φi , ψ i ) = (1 + ψ 2i )[E(Yit |Xi , φi )]2 = (1 + ψ 2i )φ2i [µ(Xit , β 0 )]2

(2.5.4)

if we put the Yit2 as the the Yit , (1 + ψ 2i )φ2i as the exp(Ci ) and [µ(Xit , β 0 )]2 as exp(Xit β)
in the equation (2.2.5), we can apply the PQML method to Yit2 . From here, we can derive
the moment conditions. We define:
T

(Yit )j = nji , j = 1, 2
t=1

pjt (Xi , β 0 ) ≡

µj (Xit , β 0 )
T
j
r=1 µ (Xir , β 0 )

, j = 1, 2

uji (β) ≡ (Yi )j − pj (Xi , β)njt , j = 1, 2
where, pj (Xi , β) ≡ [pj1 (Xi , β), ..., pjT (Xi , β)] , Yi = [Yi1 , ..., YiT ]
By iterated expectation, we have:
E(uji (β 0 )|Xi , φi , ψ i ) = E((Yi )j |Xi , φi , ψ i ) − pj (Xi , β)E(nji |Xi , φi , ψ i ) =0, j = 1, 2
(2.5.5)
So,
E(Dj (Xi , β) uji (β 0 )) = 0, j = 1, 2
37

(2.5.6)

Where Dj (Xi , β) is any appropriate function of Xi .
Wooldridge (1999) suggests the following:

Dj (Xi , β) = [Wj (Xi , β)∇β pj (Xi , β)|∇β pj (Xi , β)].

(2.5.7)

where, Wj (Xi , β) ≡ [diag{pj1 (Xi , β), ..., pjT (Xi , β)}]−1
If we define:




0
 D1 (Xi , β)

D(Xi , β) = 

0
D2 (Xi , β)




 u1i (β) 
ui (β) = 

u2i (β)
the two moment conditions can be combined as:
E(D(Xi ) ui (β 0 )) = 0

(2.5.8)

GMM follows easily:


N

ˆ gmm = arg min 
β

 

N

Di ui ui Di 

Di ui (β) 
i=1

−1 

i=1

N


Di ui (β)



(2.5.9)

i=1

ˆ pqml ), ui = ui (β
ˆ pqml ).
Where Di = D(Xi , β
As pointed out by Wooldridge (1999), GMM with only D1 (Xi , β) and u1i (β) is identical
to the PQML.
Theorem 2.5.1. As consistent estimators for β 0 in model 2, GMM is more efficient than
PQML.
PROOF: From Wooldridge (2002, section 8.3.3), we know:

ˆ gmm ) = C −1 C
A Var(β

38

ˆ pqml ) = C Λ−1 C1
A Var(β
1 1
Where,
Λ = E Di ui ui Di
C = E Di ∇β ui (β)
Λ1 and C1 are similar to Λ and C with Di and ui replaced by D1i and u1i respec−1 C is p.s.d.; then the result follows
tively. From White (1984), the matrix C1 Λ−1
1 C1 − C

immediately.

2.5.3

Simulation 3

The data generating process is specified as the following:
i.i.d

• Xit ∼ N(0, 1),
• Ci = Xi + N(0, 1),
• β = .1,
i.i.d

2

• Vit = exp(−.125 ∗ Xi + .5 ∗ Xi ∗ zit ), where zit ∼ N(0, 1).
• Yit = exp(Xit β + Ci )Vit ,
• T=5, Number of simulations =5000 .
Here, Vit has the similar form as in simulation 1 and 2; however, since Vit has to satisfy
equation (2.2.2), we use Xi instead of Xit . We only use a = −.125, b = .5 here since other
values have the similar results.
Table B.8 has the results for all the three methods. All the three estimators are consistent.
But the differences of standard errors are big. Compared with PQML with GMM, standard
error of the latter is about a half of the former; so the extra moment condition does matter
here and GMM is the right direction to improve the efficiency. However, the standard errors

39

of LFE are smaller than GMM. Theoretically, there are cases where LFE is not consistent
while GMM and PQML are; but we fail to find a simulation .

2.5.4

Optimal IV Estimator

Considering Model 2, we can derive the so-called optimal instrumental variables (OIV) estimator in Newey (1990, 1993) to improve efficiency. For the moment condition, u1i (β) ≡
Yi − p1 (Xi , β)n1i , we first need to find its variance structure. By the conditions equations
(2.5.1)-(2.5.3)from model 2, we can get:

Var(u1i |Xi ) = E(u1i ∗ u1i |Xi ) = E[ψ 2i φ2i |Xi ] ∗ Ωi

(2.5.10)

Where,

Ωi (r, s) =




1−













(

2∗µ(Xis ,β)
T µ(X ,β)
it
t=1

T µ(X ,β)2
it
t=1
T µ(X ,β))2
it
t=1

+

−

(

T µ(X ,β)2
it
t=1
T µ(X ,β))2
it
t=1

µ(Xis ,β)+µ(Xir ,β)
T µ(X ,β)
it
t=1

µ(Xis , β)2 ,

if r = s;

µ(Xis , β)µ(Xir , β), if r = s.

In fact, if we use matrix algebra, we can do the following:
 



 pi1 pi1 · · · pi1  


 

 p
 
p
·
·
·
p
i2
i2  

 i2
u1i (β) = IT −  .

..
. 

 ..

. · · · .. 


 


 
piT piT · · · piT



Yi1 

Yi2 

.. 
. 


YiT

so,we can get the following:


 Yi1

 Y
 i2
Var(u1i (β)|Xi , φi , ψ i ) = A Var  .
 ..


YiT

40






Xi , φi , ψ i  A = ψ 2i φ2i ABA




where,





 pi1 pi1



 p

 i2 pi2
A = IT −  .
..

 ..
.




piT piT

2
0
 µ(Xi1 , β)


0
µ(Xi2 , β)2

B=

0
0


0
0



· · · pi1 


· · · pi2 



.

· · · .. 


· · · piT
···

0

···
...

0
0

· · · µ(XiT , β)2











So,
Var(u1i (β)|Xi ) = E(ψ 2i φ2i |Xi )ABA
Note that,
Ωi = ABA
The remained part is to find E(ψ 2i φ2i |Xi ); note that:
Var(u1it |Xi ) =

E[ψ 2i φ2i |Xi ]

1−

2 ∗ µ(Xit , β)
T
t=1 µ(Xit , β)

+

(

T
2
t=1 µ(Xit , β)
T
2
t=1 µ(Xit , β))

µ(Xit , β)2 ,

So,

E[ψ 2i φ2i |Xi ] = E 

1−

u21it
T µ(X ,β)2
2∗µ(Xit ,β)
it
t=1
2
+
T µ(X ,β) ( T µ(X ,β))2 µ(Xit ,β)
it
it
t=1
t=1


Xi  ,

If we assume that:
E[ψ 2i φ2i |Xi ] = E[(1 + ψ 2i )φ2i |Xi ] = h(Xi )

(2.5.11)

Here, function h(.) is a unknown function. As we know that h(.) is the conditional
expectation of positive random variables, so it is positive. Hence the main problem is still
how to catch the positivity. Newey (1994) uses the truncation, which works in the large

41

sample case. Since the truncation threshold is very arbitrary, then it may not work well in
finite sample case. Next, we try to find the D(Xi ) as in Newey (1990, 1993):
D(Xi ) ≡ E(∇β u1i (β)|Xi )
= E(∇β [Yi − p1 (Xi , β)n1i ]|Xi )
= −∇β [p1 (Xi , β)]E(n1i |Xi )
T

= −∇β [p1 (Xi , β)]

µ(Xit , β)E(φi |Xi )
t=1
T

= −∇β [p1 (Xi , β)]

µ(Xit , β)r(Xi )
t=1

Here again we will use “exponential sieve” to estimate h(Xi ) and r(Xi ) as in section 3; note
that the method shows up two times for OIV estimator. We use cross validation as in Newey
(1993) to choose the number of terms M .
We propose the following estimating method:
Step 1: do the PQML of Yit on Xit , and denote the estimator of β as β pqml ; also define,
uˆ1it = Yit − n1i pit (β pqml ) ;
ˆ i ); define B(Xi ) = D(X
ˆ i) ∗
Step 2: use power series to approximate h(Xi ), denote as h(X
ˆ i );
ˆ i )−1 /h(X
(Ω
Step 3: do the following GMM,


 

N

ˆ gmm = arg min 
β

N

Bi u1i (β) 
i=1

−1 
Bi Bi 

i=1

N


Bi u1i (β)



(2.5.12)

i=1

We use a simulation to end this section:
Simulation 4
We follow the model 2 in section 5, i.e, equations (2.5.1)-(2.5.3). The data generating
process is specified as the following:
• i = 1, 2, ..., N ; t = 1, 2, ..., T ,
42

i.i.d

• Xit ∼ N(0, 1),
• Ci = exp(Xi + N(0, 1)), where Xi = T −1

T
t=1 Xit

• β = .1,
i.i.d

2

i.i.d

• Vit ∼ Gamma(α, γ), where γ = 1/α = exp(−.125 ∗ Xi + .5 ∗ Xi ∗ zit ), where zit ∼
N(0, 1),
• T = 5, N =500 if it is not specified,
• Yit = exp(Xit β)Ci Vit ,
• T=5, Number of simulations =5000 .

The estimation results are in Table B.9; for the LFE, PQML and GMM, estimates are
almost the same as Table B.8: all of them are very close to the true value of parameter;
in terms of efficiency, LFE is best and PQML is the worst. Now with the new estimation
method, OIV, we can see that it performs very well considering the standard errors, which
are close to those of LFE. Finally, we do find a method that can compete with LEF.

2.6

Empirical Application

In this section we illustrate GMM estimator as well as PQML by estimating APE of price
on demand for airline market. The market is defined the same as in Park et. al. (2007),
which is a trip between origin and destination cities. For each route, i,e, an pair of cities, the
measurement is taken everyday. In order to wash out the daily fluctuations, like weekend
and holidays, the yearly average is taken from year 1997 to 2000. Of course, for some of the
routes, they do not have records for these consecutive four years and have to be dropped;
but the drop rate is less than 10%-we have over 1000 routes. So the data set is a typical
balanced panel data set. The description of the key variables as follows:

43

For each route,the variable passen measures the average number of daily passengers for
the year; lf are is log transformation of average one-way fare in US dollars; concen is the
market share of the biggest carrier along the route. Refer to Table B.10 for details.
The Table B.11 shows the estimation results for the three methods. The GMM method
reduce the standard errors by about 10% compared with PQML. The estimates of PQML
and GMM are close to each other. As for the practice purpose, from equation (2.4.12) in
section 4, we know the APE for lf are
ˆ pqml ) 1
β j exp(xt β
NT

N

T

i=1 t=1

Yit
≈ 512
ˆ pqml )
exp(Xit β

(2.6.1)

N

βˆ j ∗ N −1

ˆ pqml + GM (Xi )π
ˆ M ) ≈ 507
exp(xt β

(2.6.2)

i=1

ˆ pqml + GM (x)π
ˆ M ) ≈ 421
βˆ j ∗ exp(xt β

(2.6.3)

Where xt is evaluated at the median values of lf are and concen and year dummy is 1998;
x is evaluated at at mean values of lf are and concen. For the interest variable lf are,
a 1% price rise in air tickets will exclude about 500 passengers overall while about 400
at the specific setting. Compare those result with LFE: a 1% price rise in air tickets will
exclude about 6 passengers overall (here we evaluate passen at its mean). By calculating the
corresponding standard errors, all the estimates are statistically significant at 5% level. From
Figure B.5, we can see that passen is skewed to the right and is not symmetric even after
taking logarithmic transformation. In terms of partial effect of airfare on market demand,
the advantage of equation 2.6.3 is that it can give arbitrary partial effect of airfare on demand
for any value of x. For example, we can consider the APE of airfare on market demand for
the route which has the highest demand of 8497: in Year 1998, a 1% price rise in air tickets
will exclude about 545 passengers from the route; this way, the price elasticity of demand is
about 6%.

44

2.7

Conclusions

We summarize the three usual estimation methods for positive response variables. By simulation, we list the disadvantages and advantages for each of them-basically the trade off
between consistency and efficiency. As long as the conditional mean function is correctly
specified,the PQML is robust to any distribution of the error terms. On the other hand, it
suffers for less preciseness for error with certain distribution. If we can assume more on the
distribution, we can use GMM to reduce the standard errors of PQML estimator. By the
application of the airfare data set, we show that it works. As for the estimation of APE,
it keeps the property of distribution free and amplify the positivity of the conditional mean
function of the heterogeneity.
Many issues can be studied in the future research. For one, whether it is possible to
develop a test to distinguish PQML and LFE, like the Hausman test for RE and FE estimators. Another interest problem is what the consequence if the conditional mean function is
misspecified.

45

Chapter 3
A SPATIAL ANALYSIS OF SPENDING EFFECT ON MEAP

3.1

Introduction

As we all know, public schools of the K-12 educational system in the U.S. are financed
mostly by local revenue, primarily by taxes levied on property. One of the disadvantages
of this policy is that this can potentially lead to economic inequality across school districts
within a state since, as is often argued, demand for (and affordability of) a good education
increases with parental income and educational attainment. Take Michigan as an example,
in year 1992, the per pupil expenditure in a rich school district (the name is Bloomfield
Hills School District, DCODE 63080)could reach as more than 9 times as it in a poor school
district (the name is Ionia Township S/D #2, DCODE 34360).1 It is under this kind of
background, in 1994, Michigan initiated a school finance reform which is called Proposal
A, aimed at equalization of school finances among school districts within state boundaries.
A great body of research has been done for its impacts. Papke(2005, 2008)use panel data
sets, either school level or school district level, and find that there is statistically significantly
positive relationship between student performance which is measured in the pass rate of math
test for fourth graders and finance expenditure with linear regression models. Moreover, as
pointed out in Papke (2008), the magnitude of the effect of initially high-performing districts
are lower than the initially low-performing ones 2 , which suggests clearly the nonlinearity
of the relationship. Under this circumstance, Papke and Wooldridge (2008) extend the
analysis further to a nonlinear model with a panel data set of school district but with less
1 We get this from the data set used in Papke and Wooldridge (2008);

In terms of household
income, Chakrabarti and Roy (2012) find that the median income in a rich school district is
more than three times of it in a poor one.
2 and Roy (2011) even claims that the finance reform may have had negative effect on
student’s performance in the highest spending districts.

46

time span compared with Papke (2008). One of the challenges in Papke and Wooldridge
(2008) is to deal with the unobserved heterogeneity in probit setting; with strictly exogenous
explanatory variables, they propose a conditional normal assumption following Mundlak
(1978) and Chamberlain (1980) device. The usual shortcoming for nonlinear models, to
make inference about the average partial effect, is elegantly overcome. More recently with
even boarder data coverage, Chakrabarti and Roy (2012) study the impact of Proposal
A on spatial segregation of housing market and find that there is continued high demand
for residence in highest-spending districts, suggesting the importance of neighborhood peer
effects (“local” social capital). However, none of them accounts for the spatial dependence
in the unobserved cross sectional fixed effect, either building level or school district level.
This is the first reason that the problem is revisited.
On the other hand, as pointed out in Papke (2008), using a data set with time span of 10
years, Papke (2008) finds that although spending inequality was reduced in the years of immediately after Proposal A, equalization has slowed considerably since year 2000. Chakrabarti
and Roy (2012) also notice that there is continued high demand for residence in the highestspending communities, implying that even a comprehensive government aid program can fail
to make a large impact on residential segregation. So considering the data set we use is in
Year 2010, it is interesting to investigate the effect Proposal A after 15 years of implementation, especially with spatial effects controlled.
Spatial econometrics can be understood as a parallel extension of time series with time
index replaced by space. In applied literature, issues relating to geographic proximity, transportation, spillover effects, etc., are important. Indeed, in recent years the spatial analysis
in economics is booming: refer to Case (1991), Anselin and Florax (1995), Kelejian and
Prucha (1999), Anselin (2010) and literatures therein. Modeling spatial interactions that
arises in spatially referenced data is traditionally done by incorporating the spatial dependence into the covariance structure via an autoregressive model. For example, Wall (2004)
analyzed the SAT scores of all 48 contiguous states of America for the year of 1999 by two

47

mostly used models in spatial statistics: conditional autoregressive model (CAR) and simultaneously autoregressive model (SAR). Both of models cooperate spatial dependence in the
covariance structure as a function a neighbor matrix and often a fixed unknown spatial correlation parameter; refer to the paper and Banerjee et.al.(2004) for more details. Obviously,
that approach is not robust to the misspecfication of the covariance functional form; what’s
more, as it states in Conley (1999), whenever there are errors in the measurement of spatial
dependence, which is happening very often in application, we cannot estimate parameters of
interest consistently without assuming distributions of the errors; as we all know, the distribution assumptions are most of the time naive in reality. Conley and Molinari (2007) show
how poorly MLE could perform when the distribution is misspecified; while the method in
Conley (1999) works well. Moreover, he extends the spatial dependence to a broad economics
system, which is general dependence within a cross section and not necessarily related to
geographical features; for example, he creates a notion of “economics distance” among the
some countries in the world in Conley and Ligon (2002); the physical distance definitely will
contribute to it but it also has some type of “border effect” as in Engle and Roger (1996).
On the other hand, he also pioneers a nonparametric approach for covariance structure estimation; the basic idea of that estimator, as it is pointed out in Keller and Shiue (2007), it
is spatial version of Newey-West (1987) type heteroskedasticity and autocorrelation robust
covariance estimator. The method is nicely used in many areas of economics, such as development, international economics and labor, etc; refer to Conley and Ligon (2002), Conley
and Topa (2002), Conley and Dupor (2003).
However, Conley (1999) does not elaborate about how to choose the cutoff points other
than the asymptotic condition of one third order of the sample size(or the width of sample
region). By simulation, Conley and Molinari (2007) argue that the choice is relevant; so
it might be hard for an empirical researcher to apply the method. Hence, the purpose of
this paper is twofold. First, we use the Michigan MEAP data of year 2010 to retrieve the
studies in Papke (2005, 2008) and in a nonlinear regression model; to investigate the effect of

48

spatial dependence specified by Conley (1999), we propose two ways to choose cutoff points
and show how the spatial dependence corrected standard errors are related to the choice.
Second, in order to get the average partial effect (APE) as in Papke and Wooldridge (2008),
an average structure function (ASF) device in Blundell and Powell (2003) is used to get
the estimates of the spending effect as well as other ones. The remainder of this paper is
organized as follows. Section 2 introduces the linear and nonlinear models; data description
and related issues are contained in section 3. Section 4 presents the linear regression with all
kinds of standard error calculations. Section 5 provides the results of nonlinear model and
how the APE is estimated correspondingly. Conclusions and discussions are in Section 6.

3.2

Model

Let (Yi , Xi ) be a sequence of observations for area Ai ; first following Papke (2005, 2008),
the linear model is considered:
Yi = Xi β + Ci , E(Ci |Xi ) = 0

(3.2.1)

where Yi is scalar, Xi is 1 × K vector and β is K × 1 vector of unknown parameters; for
area Ai : Yi and Xi can be understood as Y (Ai ) and X(Ai ) respectively; but for simpler denotation, we just put them as Yi and Xi . Ci or C(Ai ) is the unobserved spatial heterogeneity
for each area Ai . If there are total areas of N , then i = 1, 2, · · · , N .
Under general conditions [e.g., Wooldridge (2010, Chapter 4)], the OLS estimator of β
based on N observations looks like:
ˆ = (X X)−1 X Y,
β

(3.2.2)

Where Y ≡ [Y1 , Y2 , · · · , YN ] is N × 1, X ≡ [X1 , X2 , · · · , XN ] is N × K. With little
algebra [refer to Wooldridge (2010, Chapter 4)], we can get:

−1 

N
N
√
ˆ − β) =  1
N (β
Xi Xi  N −1/2
Xi C i  ,
N
i=1

i=1

49

(3.2.3)

ˆ is weakly consistent for β and asymptotically normally disSo with general conditions, β
tributed; note that the asymptotic distribution really hinges on the second term in equation
(3.2.3).
If we assume the following conditions:
V ar(Ci |Xi ) = σ 2 , i = 1, 2, · · · , N ;

(3.2.4)

{Xi Ci : i = 1, 2, · · · , N } is an uncorrelated sequence.

(3.2.5)

Condition (3.2.4) is the appropriate homoskedasticity assumption and (3.2.5) is no correlation
requirement for the sequence; it can be also extended to be independent. Then the covariance
ˆ is σ 2 (X X)−1 . Let σ
matrix of β
ˆ 2 = SSR/(N − K) be the usual square of the standard
error of the regression; the usual standard error of jth OLS estimator βˆ j is the square root
of the jth diagonal element of σ
ˆ 2 (X X)−1 . That is also the ‘standard error’ printed out by
all regression packages.
Without homoskedasticity [condition (3.2.4)] but with no correlation [condition (3.2.5)],
we can allow arbitrary heteroskedasticity of conditional variance C on X by Huber (1976)
and White (1980) robust standard errors, which are square root of diagonal terms of matrix
(X X)−1 (

N
ˆ2
i=1 Xi Ci Xi )(X

V ar N −1/2

N
i=1 Xi Ci

X)−1 . Note that N1 (

N
ˆ2
i=1 Xi Ci Xi )

is a consistent estimator of

, the second term of equation (3.2.3). Algebra details are referred

to Wooldridge (2010, Chapter 4).
For those two types of standard errors, they allow no correlation among the sequence
{Xi Ci : i = 1, 2, · · · , N }. To relax that assumption, we can try two directions. One is to
ˆ g = (Cˆi , · · · , Cˆj ) ,
divide the whole N observations into G groups: Wg = (Xi , · · · , Xj ) , v
ˆ are square root
where g = 1, · · · , G; then the group corrected standard errors of elements of β
of diagonal terms of matrix

−1
G
g=1 Wg Wg

N
ˆg v
ˆ g Wg
g=1 Wg v

−1
G
,
g=1 Wg Wg

which allow arbitrary correlation within each group while independence is assumed across
groups; that is similar to the idea for cluster corrected robust standard errors in Wooldridge
(2010, Chapter 20); and Wang et. al (2013) also adopt that idea: split the whole sample into

50

many groups, one of which has two observations with arbitrary dependence; and a bivariate
Probit method is proposed. Note that when G = 1, then the cluster corrected standard
errors are H-W standard errors.
Conley (1999) goes that way further: he divides the sequence {Xi Ci : i = 1, 2, · · · , N }
into two groups with one group is lagged the other in two non-opposing directions and allow
arbitrary correlation between the two groups; with different lag, we can get two different
groups and corresponding correlation between them-that is, the sequence is regrouped with
times of the number of lags; then final estimator of V ar

N
i=1 Xi Ci

, which is denoted as

ˆ is the weighted sum of all the covariances. Let Cˆi be the OLS residual in equation (3.2.1)
V,
, that is:
L1 (N ) L2 (N )

ˆ =
V

N

N

(Xs−i Cˆs−i Cˆt−j Xt−j + Xt−j Cˆt−j Cˆs−i Xs−i )

W (i, j)
j=0

i=0
N

s=i+1 t=j+1

N

(Xs Cˆs Cˆt Xt )

−

(3.2.6)

s=1 t=1

Where W (i, j) are the weights for covariance with one direction lagged of i and the other
j; Without lags, that is i = j = 0, then W (i, j) = 1. L1 (N ) and L2 (N ) are the cutoff
points of the two direction and both of them converge to ∞ with order of L1 (N ) = o(N 1/3 )
ˆ can be understood as the Newey and
and L2 (N ) = o(N 1/3 ) as N −→ ∞. Intuitively, V
West (1987) estimator of V ar

N
i=1 Xi Ci

in two non-opposing directions summed up with

proper weights; so it is basically a nonparametric estimator. Hence, Conley’s estimator of
ˆ is (X X)−1 V(X
ˆ
variance-covariance matrix of β
X)−1 .
As for the nonlinear model, we follow Papke and Wooldridge (1996):
Yi = G(Xi β + Ci ),

(3.2.7)

Where G(·) is any continuous function with range of (0, 1) in the real line; for example, we
exp(x)
x
1
2
can have G(x) = 1+exp(x) or G(x) = Φ(x), where Φ(x) = −∞
2π exp(−t /2)dt. So it is

natural to have:
G−1 (Yi ) = Xi β + Ci ,
51

(3.2.8)

The equation (3.2.8) is very appealing since its right hand side has the linear form as
equation (3.2.1) does. For example, if G(·) takes the logistic function form, then equation
(3.2.8) turns into:
log

Yi
1 − Yi

= Xi β + C i ,

(3.2.9)

This is the popular linear model for log-odds ratio. As in Papke and Wooldridge (1996),
Y

i
with usual assumption like, E(Ci |Xi ) = 0, we can get E log 1−Y
i

Xi = Xi β. So we

can model transformed response variable instead of original one. This is the advantage of
equation (3.2.7); but we know the responses should be strictly in the interval of (0, 1) since
0 and 1 will explode the transform. Note that the model in 3.2.9 is different from the model
exp(X β+C )

i
i , in which case the responses can be 0 and 1.
specification E(Yi |Xi , Ci ) = 1+exp(X
i β+Ci )
Another issue is the average partial effect, which is hard to be estimated in most nonlinear

models. But since we have equation (3.2.7), from which we can get:
∂Yi
= g(Xi β + Ci )β j ,
∂xij
where g(x) =

∂G(x)
∂x .

(3.2.10)

The rest of the paper will keep models (3.2.1) and (3.2.8), and

assume E(Ci |Xi ) = 0 and necessary rank condition, then OLS estimators of β in both
models are consistent, we will check standard errors under different conditional covariance
ˆ in (3.2.2) and the corresponding APE estimates.
structures of β

3.3
3.3.1

Data
Data Characteristics and Sources

All the data are from Michigan Department of Education: expenditure data are from Bulletin
1014 ; enrollment and free and reduced price lunch data are from the Center for Educational
Performance and Information (CEPI); the test results data are from Michigan Educational
Assessment Program (MEAP). The data resource is the same as in Papke (2005, 2008)
and Papke and Wooldridge (2008), but there is some change for the name of the program
since we use the data of Year 2010. Papke (2005) uses building level data but we use
52

school district level ones, which are the same as in Papke (2008). Other than the reasons
explained in Papke and Wooldridge (2008), we pick the school districts for the convenience
of spatial analysis. As for the dependent variable, math4, the pass rate of math test for
forth graders in each school district of year 2010, which is defined the same in Papke (2005,
2008). The variable of interest, average expenditure per pupil in each school district, is
collected differently than in Papke (2005, 2008); instead, we follow the definition in Papke
and Wooldridge (2008)-take the average per pupil real expenditure in the last four years:
avgexp = (avgexp+avgexp−1 +avgexp−2 +avgexp−3 )/4. Since both papers find significant
effect of the previous expending on school performance, it is meaningful to collect expenditure
variables this way-the reason we take last 4 years is that we are considering the test pass
rates for forth graders. The real dollars are calculated in the price of year 2010 Midwest
Urban Price index from Bureau of Labor Statistics. For the purpose of comparison, we also
use the average per pupil expenditure of year 2010. Two other explanatory variables are
the same in previous papers except in year 2010: enroll is the student enrollment for each
school district in academic year 2009/10 and lunch is percentage of students who are eligible
for free lunch or reduced price lunch program. scdist is the distance in kilometers for each
school district to its nearest 2-year or 4-year college. Inspired by Kane and Rouse (1995),
we are trying to investigate the effect the higher education has on the K-12 system, which
is not done in previous studies. Intuitively, the schools in a school district which is closer
to a college should have a higher chance to perform well, like it is easier for a forth grader
to find a tutor since the college students are around; or the students can take advantages of
the facilities in the colleges. On the other hand, most locations of colleges are in the areas
with higher social economic status, which has positive externality to public schools. The
summary statistics for these variables are in Table C.1.
There are 551 school districts in shape file which is from Michigan Geographic Data
Library. Combined with MEAP data, there are only 518 school districts because of data
missing. With help of ArcMAP, we plot the the map of the all the school districts in Figure

53

C.1. As for the colleges in Michigan, I download all the 131 Michigan 2-year and 4-year
colleges from National Center for Education Statistics website, and plot into the map(Figure
C.2). If we plot the test pass rates into the map (Figure C.3), we can find some areas are
clustered with high rates while others with low; that is, the there is possibility that high
rates are congested together; so do the lower rates. That is the what the spatial dependence
stems from. When add college data in and get Figure C.4, we do find that some colleges
are in or close to the areas with high math test pass rates; while other colleges are in the
areas with low test rates. How to measure the spatial dependence among those 518 school
districts is the main part of application. Since the shape of each district is not regular, and
the area sizes of them vary from 3.7km2 to 3317.7km2 , we should treat them as area data
instead of point ones. The next subsection will explain how to set up the spatial dependence
among these available 518 school districts.

3.3.2

Spatial Dependence Measurement

Since we treat each school district as an irregular lattice, the physical distance between
the centroids of any two districts cannot be a perfect measure for spatial dependence among
them; for example, for some districts, their centroids are outside of the polygons. So distance
based spatial weighting matrix, which is perfect for points data, is not that appropriate here;
on the other hand, the dependence which is measured only by contiguity for any two areas
ignores the size variations among those school districts. Keller and Shiue (2007) apply
both methods and have excellent descriptions about them. Since Conley (1999) initiates a
measurement for spatial dependence, which is robust to measurement error (Conley, 2007),
we will adopt their method. First, we get the projected latitude and longitude coordinates for
the centroid of each school districts3 ; then we use the smallest distance among all those 518
centroids to construct little squares in the whole state; whenever a centroid is in a square, the
3 the

reason we use projected lattitude/longitude instead of original ones is that we want
to use coordinates in a plane rather a sphere

54

coordinates of the upright corner the lattice are the new coordinates for each school district.
A picture is worth one thousand words: we choose part of the 518 school districts and make
a picture to explain the process: as in figure C.5, we have 96 school districts and we plot
the 96 centroids into the map (Figure C.6); then we divide the whole area into 70 × 70 little
squares in Figure C.7 with unit of about 2.5 kilometers, which is about the smallest distance
among those 96 points. Now, we know the new coordinates for each centroid are decided by
the little square where it lies and those are called Conley coordinates for each school district
in Figure C.8. From there, we can see that one of the advantage of this measurement is that
as long as a centroid is in a square, it will have the same coordinates no matter where it stay
in the square; this way, it can cover the damage of measurement error while the traditional
ways cannot. Thereafter, we define the spatial weighting matrix, W = (wij ), this way:


 (1 − |i| )(1 − |j| ), for |i| < L1 , |j| < L2
L1
L2
wij =

 0,
else
We can see that the weight function is a Barlett window in each direction. The choice of
L1 and L2 is based on theory and practice: In theory, it should not be bigger than 1/3
root of the sample region; among those 518 school districts, the sample region is about 1000
horizontally and vertically; so we can take as about 10, which is about 10 kilometers since
we know each unit of the coordinate is about one kilometer. While from practice, we know
we should have a distance longer than that. We will explain more about that in the analysis.
For the coordinates beyond L1 and L2 , we would not consider the dependence any more. Of
course, we can set L1 and L2 as very large and so that we will include all the sample districts.
the The missing data definitely will not cause big problems for the matrix defined this way,
while the 0 − 1 measure of spatial dependence based on contiguity will: some element would
be one were data for all 551 districts available. This is another advantage compared with
conventional method. With this spatial weighting matrix, we will use the Conley (1999)
estimator described in section 2 to account for dependence among school districts.

55

3.4

Linear Model

As in Papke (2005, 2008), we run the following regression:
math4i = β 0 + β 1 log(avgexp)i + β 2 log(enroll)i + β 3 lunchi + β 4 log(scdist)i + Ci ,
math4i = β 0 + β 1 log(exp10)i + β 2 log(enroll)i + β 3 lunchi + β 4 log(scdist)i + Ci ,
(4.1a, 4.1b)
The only difference of those two models [equations (4.1a, 4.1b)] is the first explanatory
variable: we replace avgexp in the first regression with exp10. From the description, we know
that some differences do exist between them and we wonder how those difference matter for
the test rate. The reason we take logarithms transformation for explanatory variables avgexp
and scdist is that their distributions are skewed: It can be easily seen from the histograms
(Figure C.9); and Manning (1998) claims that logarithms transformation is a good way to
model those kinds of variables. Such analysis also applies to enroll.

3.4.1

Ordinary Least Square (OLS)

Here, we first assume that usual assumptions of linear regression as in Wooldrdige (2010,
Chapter4) hold; the OLS results are in the first two columns of Table C.2. Note that the
usual standard errors are calculated under assumption of homoskedasticity as well as other
assumptions for consistency. As we all know, the magnitudes of standard errors are crucial to
the inference: they will decide whether the estimates of parameters are significant or not at a
given significance level. From Table C.2, we can see that all the explanatory variables except
scdist are significant at 5% level: the effect of student expenditure as well as enrollment on
school performance is positive and lunch program is negative; interestingly, the effect of a
school district’s distance to its nearest college is positive which means the longer the better
and it is against usual intuition. Fortunately, the effect is not significant at 5% level or even
at 10%; and we see the same pattern for the second regression and the size of the effect of the

56

last four years’ expenditure on school performance is slightly bigger than year 2010. However,
if we abandon the homoskedasticity assumption and allow arbitrary functional form of the
conditional variance of the error term Ci on explanatory variables and we will get the the
White standard errors; then the positive effective of student enrollment is not significant at
5% level any more; so does for the second regression. Further, if we calculate the robust
standard errors corrected for the intermediate school districts (ISD), even the positive effect
of year 2010’s expenditure is not significant at 5% level. Considering Michigan geographical
feature, we divide the whole sample into two clusters: the Upper peninsula and lower; if we
correct for the peninsula clusters, neither the last four years’ average expenditure nor year
2010’s expenditure has the significant positive effect on school performance. Now, we know
it is important to have a reasonable method to calculate standard errors.
Now we calculate Conley (1999) standard errors; Based on the description in section 2
and 3, we can get the coordinates for each centroid of 518 school districts. The key step is
how to choose the cutoff points. Since the coordinates range from 0 to 758 horizontally, and
from 0 to 880 vertically, from theory the cutoff points should not bigger than 10; while we
think the number should be larger; we will design two schemes to see how the significance
level changes with respect to its corresponding cutoff points. First, we have 11 pairs of the
cutoff points, which increases by 50 or 100 by each step; and results are in Table C.3 and
C.4. We can see that the overall trend of the standard error magnitudes is decreasing with
increasing of the window size; but we do see some local fluctuations: for example in Table
C.3, for the variable exp10, the Conley standard errors of the estimator of its parameter
are increasing for the first three pairs of cutoff points. At 5% level, the estimators of β 1
and β 3 are all statistically significant under all the cutoff points; but for β 2 , its estimator
is not significant at 5% level if the first three pairs are chosen. In Table C.4, the similar
pattern holds for the regression with exp10 replaced by avgexp, while the corresponding
errors are a little inflated. Secondly, we have 7 pairs of points, which are the the 5%, 15%,
25%, 50%, 75%, 95% and 100% percentiles of the coordinates respectively; and results are

57

in Table C.5 and C.6. Note that, with each of the pair, we can make that many of the 518
sample covered in the weight calculation. Surprisingly, the similar story happens as the first
scheme; and there is no difference of signifiance between only 5% samples are considered for
positive weights and all samples are included for β 1 and β 3 at 5% level; while we do see the
difference for β 2 : more sample considered for positive weights more significant. Considering
each unit of the coordinates is equal to about 2 kilometers, it is not that hard to have a
reasonable choice; there is a catch here: if the decreasing trend of the Conley standard errors
holds, we make them as small as possible: for example, I make a pair of 106 , even the β 4 can
be statistically significant at 5% level; but we know that number does not make any sense
considering none of the coordinates is greater than 900.
3.4.2

Generalized Least Square (GLS)

The matrix form of equation (3.2.1) [or equations (4.1a, 4.1b)]:

Y = Xβ + C

(3.4.1)

Where Y and X are as in equation (3.2.2), C = [C1 , C2 , · · · , CN ] .
If we assume the Spatial Autoregressive (SAR) structure in the error term C:
C = ρWC + ε, ε ∼ N(0, IN σ 2 )

(3.4.2)

Where ρ is the unknown spatial correlation parameter and W is the weighting matrix; that
is a conventional assumption in the spatial econometrics literature. Based on theoretical
results of maximum likelihood estimator (MLE) and QMLE in Lee (2004), STATA has the
build-in function of “spreg ml” to implement MLE or QMLE given a weighting matrix.
The first interesting result is that estimates of covariate parameters β are close while the
estimates of spatial correlation parameter ρ are different for different weighting matrix. From
Table C.17 and Table C.18, we can see that the magnitude of ρ estimates under weighting
matrix of contiguity is only one third of it under weighting matrix of inverse distance; if we
investigate the weighting matrix further, we find that contiguity weighting matrix is much
58

smaller than the inverse distance weighting matrix element by element; the usual summary
statistics of the correlation matrix of error terms are in Table C.19 and Table C.20.
One strong assumption of MLE or QMLE for SAR model is the homoskedasticity; without
it, MLE (or QMLE) are not consistent. That result has been explored thoroughly in the
literature , such as Arraiz at. el. (2010), Kelejian and Prucha (2010), Lin and Lee (2010).
They all come up with some new methods to cover heteroskedasticity. On the other hand,
the SAR structure might be misspecified. Here we combine the idea of SAR structure and
Conley standard errors, and a new form of standard errors is introduced. From equation
(3.4.2):

C = (IN − ρW)−1 ε
Plug into queation (3.4.1) :

Y = Xβ+ (IN − ρW)−1 ε

(3.4.3)

With the homoskedasticity, GLS in euqation (3.4.3) is the same as MLE with ε ∼ N(0,
IN σ 2 ) (or QMLE without normality asumption); that is aslo what “spreg ml” reports in
STATA. Moreover, GLS is also equivalent to the OLS in the following model:

(IN − ρW) Y = (IN − ρW) Xβ+ε

(3.4.4)

Since ρ is unkown, by the idea of quasi-GLS, we can replace ρ with its GLS estimator,
ρ, in equation (3.4.3). Then we apply Conley’s method to the following model:

Y∗ = X∗ β+ε

(3.4.5)

Where Y∗ = (IN − ρW) Y, X∗ = (IN − ρW) X∗ .
Note that OLS in equation (3.4.5) wtih Conley standard errors has at least two advantanges: first, it allows hetroskedastictiy by H-W standard errors and robust to condition in
59

equation (3.2.5); second, it is robust to the misspecification of SAR struction. Note that even
if the SAR structure is wrong, this transformation adds one more possible structure for the
error terms; and we know that Quasi-GLS is still consistent as long as E (ε | X) = 0, which
means strictly exogenous covariates. From the results in Table C.13, the heteroskedasticity
does make a difference for magnitude of standard errors of expenditure and enrollment; the
similar patterns for the Conley standard errors as in OLS case. Since the efficiency gain
from Conley’s method only works for high cutoff points in which case correlation is weak,
we can see that the SAR structure is not a good choice for error term.

3.5

Nonlinear Model

3.5.1

Regression of Log Odds Ratio

We follow the model in equation (3.2.7) with the following form :
Yi =

exp(Xi α + ei )
,
1 + exp(Xi α + ei )

(3.5.1)

One of the advantages of the above equation (3.5.1), compared with equation (3.2.1), is
that logistic function of its right hand side is strictly in the range of (0,1), which will make
prediction in the interval; however, it comes at the expense of the difficulty of estimating
parameters and causal effect. Fortunately, the equation (3.5.1) can be easily transformed as
follows:
log

math4
1 − math4

= α0 + α1 log(avgexp)i + α2 log(enroll)i + α3 lunchi + α4 log(scdist)i + ei ,

log

math4
1 − math4

= α0 + α1 log(exp10)i + α2 log(enroll)i + α3 lunchi + α4 log(scdist)i + ei ,
(4.5.1a, 4.5.1b)

The left hand side of the above two equations (4.5.1a, b) are the logarithm transformation
of relative risk of passing the MEAP math test for each school district; it is popularly called as
“log odds ratio” in literature. The right hand sides are exactly the same as in equations (4.1a,
60

b). Avoiding abusing of notation, we use α for the parameters of interests; more important
is that interpretations of the parameters of interest are different: the β in equations (4.1a, b)
are the average partial effect (APE) of covariates on pass rate, while α’s in equations (5.1a,
b) are APE of covariates on log ratio of each school’s pass rate to its failure rate. Of course,
we can retrieve the first APE and it is in the next subsection.
First, we regress the log odds ratio under four types of assumptions for the conditional
covariance of ei on the regressors and the estimates are in the first column of Table C.7; as
in previous section, we consider two sets of expenditure variable: avgexp and exp10. The
signs of the estimates are as expected: the effect of expenditure and enroll on the odds ratio
is positive; although the lscdist shows some positive effect, it is not statistically significant
at even 10% level, let alone 5%. we remember those results show up in the level pass rate
regression in the previous section. However, the significance picture is different. Under iid
assumption, increasing avgexp by 1% will increase the odds ratio by about 5% and about 4%
for exp10; and the effects are statistically significant at 5% level. If we correct for arbitrary
correlation among ei s or between ei s and explanatory variables, the significance claim does
not hold any more; nor do the corrected clusters of ISD and peninsula cases. That gives
us motivation to investigate further about the assumptions for the conditional covariance
structure. On the contrary, the effect of enrollment on pass rate ratio is robust to the
assumptions: if we increase the enrollment by 5%, then the odds ratio will increase by 9%
with either avgexp or exp10 controlled; and the effect is statistically significant at 5% level.
The same pattern holds for lunch: every 1% point increase in the percentage of students in
free lunch or reduced price lunch for a school district, it will lead to reduce the odds ratio
by almost 3%; and the effect is statistically significant at 5% level.
Then, we consider the Conley (1999) standard errors. As in the level pass rate case, it
is important to find an appropriate window size for the weights; we apply the same scheme.
For the first 11 pairs of cutoff points and corresponding results are in Table C.8 and C.9, the
global trend of the magnitude of standard errors for each variable is decreasing with increase

61

of cutoff points; while for some variable, like log(exp10), there is some local distortion, and
for other variables, like lunch and log(enroll), the decreasing trend is strict. Given at 5%
level, the statistical significance of OLS estimators of lunch and log(enroll) is robust to the
choice of cutoff points: all of them are significant; while for log(scdist), the OLS estimator
of its parameter is never significant at 5% level even at 1000 of the window size. So we are
very confident to say that there positive effect of enroll on the odds ratio of math test pass
rate, while negative for lunch at 5% significant level; and scdist is not significant factor to
explain the ratio. As for log(exp10), its effect only becomes statistically significant at 5%
level when we set the window at size of 1000; the similar story is told when log(exp10) is
replaced by log(avgexp), whose positive effect on the odds ratio is changed to be significant
at the cutoff points of 600 and 700 and over. When the percentiles are specified for the cutoff
points, judging from the results in Table C.10 and C.11 the same thing happens for lunch,
log(enroll) and log(scdist): the statistical significance of their effects on the odds ratio at
5% level holds the same situation as first scheme of cutoff points. For log(exp10), it effect is
not statistically significant at 5% level even though all the points are included in the weights
to correct for spatial dependence; the case of log(avgexp) is slightly different: when 95% or
above of the sample are included for weights, its effect is significant. The same comment
shows up as for the level test pass rate: as long as we keep the cutoff size large enough, all the
variables are statistically significant factors to explain the ratio although the magnitude of
the window does not make any practical sense. In a word, if we try to explain the odds ratio
of the test instead of rates themselves, only two factors are statistically significant at 5%
level: lunch and log(enroll); the former keeps its significance while the latter is changed-it
replace the expenditure variables as in level pass rates case.

3.5.2

Estimate the APE for Level Rates

As we all know, economists are interested in what a model would suggest for a policy as well
as estimating parameters in the model itself. After we go to the estimation of parameters,

62

we notice that one of disadvantages for the nonlinear model as in equation (3.5.1) is that
we cannot estimate APE as smoothly as in linear case, where the estimates of parameters
themselves speak everything; Plus, there is another issue-the difference between APE and
PAE, refer to Wooldridge (2005, 2010) for more detailed discussion about this. In this paper,
we define the APE with respect to xj evaluated at X as follows:
+∞

AP Ej (X) = αj

−∞

exp(Xα + e)
(1 + exp(Xα + e))2

f (e)de,

(3.5.2)

Note that equation (3.5.2) follows the idea of average structure function(ASF) in Blundell
and Powell (2003); one of the advantages of ASF is that it can specify partial effect with
respect to individual change of X, not just average. Refer to equations (5.1a, b), Xα =
α0 + α1 log(avgexp) + α2 log(enroll) + α3 lunch + α4 log(scdist); for example, if we want to
know the expenditure effects on school performance, then APE with respect to log(avgexp) is
+∞
of our interest; by equation (3.5.2), AP E1 (X) = α1 −∞

exp(Xα+e)
f (e)de.
(1+exp(Xα+e))2

The notion

of that is the partial effect defined in equation (3.2.10) with the heterogeneity averaged out
ˆ be parameter estimator and eˆi be the residuals from log
over its whole population. Let α
odds ratio OLS regression, it is straight forward to estimate the APE this way:
1
AP E j (X) = α
ˆj
N

N

ˆ + eˆi )
exp(Xα

ˆ ˆi ))2
i=1 (1 + exp(Xα + e

,

(3.5.3)

The idea behind this method is the method of moment; it can also be understood as
Duan’s(1983) swearing estimator. But we know, we do not need to assume the independence
of Xi and ei , which can be an attractive feature. As for the asymptotic variance of AP E j (X),
it can be obtained by delta method, which is conveniently implemented using the method
of moments approach in Newey and McFadden (1994). Bootstrapping methods can also be
readily applied.
Continuing the avgexp example, we can estimate the APE at the mean value of all X
as .1151 with standard error of .0663. The results for other explanatory variables evaluated
at mean and 25%, 50%, 75%, 95% percentiles are in Table C.12. We can see that the
magnitudes of the APEs do correlate to the values at which the X are evaluated; take the
63

avgexp example, the trend of its APE is decreasing with higher percentile of X and the
difference of APE between 95% and 25% is about 2%: the average expenditure in the lower
end of school districts kicks in at a higher rate than in the higher ones. This coincides the
results in Papke (2008) who divides the sample into two groups by the median of average
expenditure; thanks to the nonlinear model, we do not need to split the data and we can
investigate it in an even finer setting: technically, we can get the APE for any value of X,
which definitely is attractive to the practitioners. We can also find the similar trend for
APEs of lunch program and enrollment albeit the change is smaller. As for inference, the
APEs of enrollment and lunch are statistically significant at 5% level when avgexp is used to
control for expenditure, whose significance level is 10%; when the expenditure of year 2010
is used, the significance story for enrollment and lunch is the same while the expenditure is
not statistically significant at 10% level any longer. This is another difference compared with
line model: if there is a temporally lagged effect for the expenditure, the linear model cannot
catch it while nonlinear does. To dig into the question deeper, we compare the estimates of
linear and nonlinear models and put the results in figure C.10-C.12. First, for the average
expenditure, the linear estimate is about 4 ∼ 6% higher than nonlinear ones, with the gap
wider for the higher spending school districts. But for enrollment, the trend is reversed:
the effect of enrollment estimated in the nonlinear model is higher than in the linear one,
although the difference is about half percentage and the gap shrinks with more students
enrolled. Positive effect of enrollment coincides with the “peer effect” in Lin (2010). And
for free lunch program, its effects on school performance both in linear model and nonlinear
are almost the same.

3.6

Conclusions

We have investigated the effects of Proposal A in Michigan on school performance in a broader
setting compared with the previous literature, such as the nonlinear model, the interaction
between K12 public schools and colleges; To control the spatial dependence among all school
64

districts, we adopt the method in Conley (1999) and explore the effect of different cutoff
points. Considering the statistical significance varies with standard errors, we find that the
OLS estimates in the linear level test rates are statistically significant at 5% level given a
reasonable window size; however, the picture is different for the nonlinear model, which is
the linear in the log odds ratio of the level test rates: even for a very large cutoff points,
the spending effect is not statistically significant at 5% level any more. What’s more, after
transformed back from log odds ratio into level rates, both the magnitudes and statistical
significance of APEs are changed; one of the advantages of the nonlinear model is that it
can catch the effect variation with specific part of population. Also, the difference between
some estimates from the linear and nonlinear models is not negligible, this raises a question
about issues of specifications of regression functional form.
As for the future work, the interaction between public schools and charter schools, as
in Imberman(2011), is an interesting extension. Also, to make use of panel data is also a
promising direction.

65

APPENDICES

66

Appendix A

PROOF OF THEOREM 1: Note that here we prove the 2SLS case; and OLS is special
case. Abrevaya (2002) mentions how the delta method can be used for the Duan estimator,
but considers the case where U , X are independent, which is not necessarily in our case
Let

θ = (β , η) , θ = (β , η) .


√





N −1/2

√

β − β  
N
=
N −1/2
η−η
 
N
 Si 
≡ N −1/2
  + op (1)
i=1 Qi

N (θ − θ) =

N
−1
i=1 A Zi Ui

N
i=1 (PSi

+ exp (Ui ) − η)



 + op (1)

where

A ≡ (C D−1 C)−1 C D−1
C ≡ E(Zi Xi ), D ≡ E(Zi Zi ), P ≡ E(Xi exp(Ui ))
Si ≡ A−1 Zi Ui
Qi ≡ PSi + exp (Ui ) − η
So by central limit theorem, this finishes the proof. The above process is similar to the
solution to question 12.17 in Wooldridge (2010). For details, refer to Wooldridge (2011).

67

PROOF OF LEMMA 1: From (1.4.3), and by monotonicity of logarithmic function,
exp(−CM −s + GM (x)π M ) − exp(GM (x)π M )
< r(x) − exp(GM (x)π M )

(A.1)

< exp(CM −s + GM (x)π M ) − exp(GM (x)π M ),
By the mean value theorem applied to the lower and upper bound:
exp(−CM −s + GM (x)π M ) − exp(GM (x)π M )
= −CM −s exp(ξ 1 ),

ξ 1 ∈ [−CM −s + GM (x)π M , GM (x)π M ]

exp(CM −s + GM (x)π M ) − exp(GM (x)π M )
= CM −s exp(ξ 2 ),

ξ 2 ∈ [GM (x)π M , CM −s + GM (x)π M ]

So, for the π M that satisfies (A.1), we have:
sup r(x) − exp(GM (x)π M ) < CM −s ,

x∈Ξ

So,
E r(Xi ) − exp(GM (Xi )π M )

2

≤ CM −2s

Note that:
E

2
Yi
− exp(GM (Xi )π)
exp(Xi β)

= E Var

Yi
X
exp(Xi β) i

+ E r(Xi ) − exp(GM (Xi )π)

2

Considering equation (1.4.5) and the first term in the above equation is constant w.r.t π, we
get
π ∗M = arg min E r(Xi ) − exp(GM (Xi )π)
π

68

2

So,
E r(Xi ) − exp(GM (Xi )π ∗M )

2

≤ E r(Xi ) − exp(GM (Xi )π)

2

≤ CM −2s

Hence,
r(x) − exp(GM (x)π ∗M ) = O(CM −s )

PROOF OF LEMMA 2:
this proof drives heavily from proof of Lemma 2 in Hirano et.al.(2003) In the sequel we
write M for M (N ).By definition of GM (x),
1
SM =
N

N

GM (Xi )GM (Xi )
i=1

has expecatation equal to IM .By Newey(1997), it satisfies
SM − IM = Op

ζ(M )

M
N

,

which converges to zero in probability by condition (5). Hence the probability that the
smallest eigenvalue of SM is larger than 1/2 goes to one. Let
N

LN (π) = −
i=1

Yi
exp(Xi β)

2

− exp(GM (X

i )π)

Next, we will show that
1 ∂LN ∗
(π M ) = Op
N ∂π

69

M
N

,

(A.2)

Consider
1 ∂LN ∗ 2
(π M )
N ∂π


2
Yi
1
trE 
=
− exp(GM (Xi )π ∗M ) exp(2GM (Xi )π ∗M )GM (Xi ) GM (Xi ))
N
exp(Xi β)
E

=

1
trE
N

Var

2
Yi
Xi + r(Xi ) − exp(GM (Xi )π ∗M ) + op (1)
exp(Xi β)

exp(2GM (Xi )π ∗M )GM (Xi ) GM (Xi )
≤

C
trE GM (Xi ) GM (Xi )
N

≤

CM
N

and Markov inequality implies (A.2). Next, let
η=

inf

x∈Ξ,M

2 exp(GM (x π ∗M ) − r(x)) exp(2GM (x)π ∗M ))

,which by assumptions and Lemma 1 is positive. For any ε > 0, choose C such that for N
large enough
P

1 ∂LN ∗
(π M ) < ηC
N ∂π

M
N

≥1−

ε
2

Note that,
exp(2GM (x)π) − exp(2GM (x)π ∗M )

sup
x∈Ξ,|π−π ∗ |<ηC

≤

M
N

|CGM (x)(π − π ∗ )|

sup
x∈Ξ,|π−π ∗ |<ηC

≤ ζ(M )C

M
N

M
N

which goes to zero, so that for large enough N
70

(A.3)

2 exp(GM (x)π) − r(x)) exp(2GM (x)π) ≥ 4η

inf
x∈Ξ, π−π ∗ <ηC

M
N

Choose N large enough so that this inequality holds. that (A.3) holds with probability
at least 1 − ε/2, and the the probability that the smallest eigenvalue of SM is larger than
1/2 is at least 1 − ε/2. Then the probability that both of these hold is at least 1 − ε, then
for every π with π − π ∗ =

M,
N

a second order expansion gives

∂ 2 LN
1
1
1 ∂LN ∗
1
¯
LN (π) = LN (π ∗M ) +
(π M )(π − π ∗ ) +
(π − π ∗ )
(π)(π
− π ∗ ), (A.4)
N
N
N ∂π
2N
∂π∂π
¯ − π∗ ≤ π − π∗ =
where π

M.
N

We have

1 ∂ 2 LN
¯
(π)
2N ∂π∂π
1
= −
2N
= −

N

¯ −
2 exp(GM (Xi )π)
i=1

1
E
2N

Yi
exp(Xi β)

¯ M (Xi )) GM (Xi ))
exp(GM (Xi )π)G

¯ − r(Xi ) exp(GM (Xi )π)G
¯ M (Xi )) GM (Xi )) + op (1)
2 exp(GM (Xi )π)

≤ −2η SM + op (1)

with its eigenvalues bounded away from zero in absolute value by η. Then, rearranging (A.4)
and using the triangle inequality, with probability greater than 1 − ε, for π − π ∗ =
1
1
1 ∂LN ∗
LN (π) − LN (π ∗M ) ≤
(π M )(π − π ∗ ) − η π − π ∗ 2 + op (1)
N
N
N ∂π
≤

=

1 ∂LN ∗
(π M )
N ∂π

π − π ∗ − η π − π ∗ 2 + op (1)

1 ∂LN ∗
(π M ) − η
N ∂π

< 0
71

M
N

π − π ∗ + op (1)

M,
N

That is, we have with probability greater than 1 − ε, N1 LN (π) < N1 LN (π ∗M ) for all π
with π − π ∗ =
π : π − π∗ ≤

M
N

Since LN (π) is continuous, it has a maximum on the compact set

M . By the last inequality, this maximum must
N
M . Hence the first order conditions are satisfied
N

πM − π∗ <

occur for some π M with
at π M and by concavity

of LN (π), π M maximize LN (π) over all of GM . Because the probability of this is greater
than 1−ε with ε arbitrary, we conclude that π M exists and satisfies the first order conditions
with probability approaching one, and that π M (N ) − π ∗M (N ) = Op

M (N )
N

.

PROOF OF LEMMA 3:

√

−1/2

N VM
√

−1/2

N VM

=

r(x) − r(x)
r(x) − exp(GM (x)π ∗M ) +

√
−1/2
N VM
exp(GM (x)π ∗M ) − r(x)

≡ T 1 + T 2,

(A.5)

By mean value theorem:
−1/2 √

T 1 = VM

−1/2

= VM

N exp(GM (x)π M ) − exp(GM (x)π ∗M )

√
ˇ M ) N π M − π ∗M ,
GM (x) exp(GM (x)π

From equation (1.4.4), we know:
1
N

N
i=1

Yi
exp(Xi β)

− exp(GM (Xi )π M ) GM (Xi ) exp(GM (Xi )π M ) = 0

72

(A.6)

Taylor expansion around π ∗M for the left hand side of the above equation:

N
i=1

1
N

Yi
exp(Xi β)

N
i=1

1
N

Yi
exp(Xi β)

− 2 exp(GM (Xi )π M ) GM (Xi ) GM (Xi ) exp(GM (Xi )π M ) π M − π ∗M
N
i=1

= N1

1
N

Yi
exp(Xi β)

− exp(Xi β)
i

Y

Yi
exp(Xi β)

− r(Xi ) GM (Xi ) exp(GM (Xi )π ∗M ) +

N
i=1

N
i=1

r(Xi ) − exp(GM (Xi )π ∗M ) GM (Xi ) exp(GM (Xi )π ∗M ) +

− exp(GM (Xi )π M ) GM (Xi ) GM (Xi ) exp(GM (Xi )π M ) π M − π ∗M

N
M
M
i=1 exp(G (Xi )π M )G (Xi )

− N1

GM (Xi ) exp(GM (Xi )π ∗M ) +

1
N

Yi
exp(Xi β)

N
i=1

1
N

− exp(GM (Xi )π ∗M ) GM (Xi ) exp(GM (Xi )π ∗M ) +

GM (Xi ) exp(GM (Xi )π M ) π M − π ∗M
=0

So
√

N
M
M
i=1 exp(G (Xi )π M )G (Xi )

1
N
1
N

N
i=1

Yi
exp(Xi β)
√1
N
√1
N

√1
N

N π M − π ∗M =

N
i=1

GM (Xi ) exp(GM (Xi )π M )−

− exp(GM (Xi )π M ) GM (Xi ) GM (Xi ) exp(GM (Xi )π M )}−1

N
i=1
N
i=1

Yi
exp(Xi β)
Yi
exp(Xi β)

Y

− exp(Xi β)
i

GM (Xi ) exp(GM (Xi )π ∗M )+

− r(Xi ) GM (Xi ) exp(GM (Xi )π ∗M )+

r(Xi ) − exp(GM (Xi )π ∗M ) GM (Xi ) exp(GM (Xi )π ∗M )
≡ {E − F}−1 {G + H + J}

73

So from equation (A.6),
−1/2

T 1 = VM

√
¯ M ) N π M − π ∗M
GM (x) exp(GM (x)π

ˇ M ){E − F}−1 {G + H + J},
= GM (x) exp(GM (x)π
Let
ΣM ≡ E[GM (Xi ) GM (Xi )

2
Yi
− r(Xi ) exp(2GM (Xi )π ∗M )],
exp(Xi β)

QM ≡ E[GM (Xi ) GM (Xi ) exp(2GM (Xi )π ∗M )],
−1 Σ Q−1 (x)GM (x) exp(2GM (x)π ∗ ).
VM (x) ≡ GM (x)QM
M M
M

so
−1/2

T 1 = VM

˘ M ){E − F}−1 {G + H + J}
GM (x) exp(GM (x)π

Note that
E

F

=

≤

1
N

1
N

N

GM (Xi ) GM (Xi ) exp(2GM (Xi )π M ) ≤ Op (M 2 )
i=1
N

Yi

i=1

exp(Xi β)

− exp(GM (Xi )π M ) exp(GM (Xi )π M )

N

GM (Xi ) GM (Xi ) ≤ Op (M 2−s )
i=1
N

G

≤

1
√
N i=1

N

Yi

Yi
−
exp(Xi β) exp(Xi β)

exp(GM (X

≤

GM (Xi )

≤ Op (M )

i=1

N

J

∗
i )π M )

1
√
r(Xi ) − exp(GM (Xi )π ∗M ) exp(GM (Xi )π ∗M )
N i=1

≤ Op (N 1/2 M 1−s )

74

N

GM (Xi )
i=1

p

Since s > 2, so F → 0 as M −→ ∞. Hence {E − F}−1 is equivalent to E−1 as M −→ ∞;
while
E−1 G

≤

E−1

G ≤ Op (M −2 )Op (M ) = Op (M −1 )

E−1 J

≤

E−1

J ≤ Op (M −2 )Op (N 1/2 M −s ) = Op (N 1/2 M −(s+2) )

Here, we assume N 1/2 M −(s+1) −→ 0 as N −→ ∞; so
T1
−1/2

¯ M ){E}−1 {H} + op (1)
GM (x) exp(GM (x)π

−1/2

¯M)
GM (x) exp(GM (x)π

= VM

= VM


1 N
N

i=1

−1

M
M
M
M
exp(G (Xi )π M )G (Xi ) G (Xi ) exp(G (Xi )π M )



 1 N
√
 N
i=1
−1/2

= VM




Yi
− r(Xi ) GM (Xi ) exp(GM (Xi )π ∗M ) + op (1)

exp(Xi β)

¯ M )Q−1 ×
GM (x) exp(GM (x)π


 1 N
√
 N
i=1




Yi
− r(Xi ) GM (Xi ) exp(GM (Xi )π ∗M ) + op (1)

exp(Xi β)

Next, Let
= [ 1, · · · , N ] ,
i

=

Yi
− r(Xi ),
exp(Xi β)

1
O(π M ) = √ [exp(GM (X1 )π M )GM (X1 ) , . . . , exp(GM (XN )π M )GM (XN ) ],
N
−1/2

ZiN = VM

√
¯ M ){O(π M )O(π M ) }−1 O(π ∗M )i i / N
GM (x) exp(GM (x)π

75

so that
N

−1/2

ZiN = VM

√
¯ M ){O(π M )O(π M ) }−1 O(π ∗M ) / N
GM (x) exp(GM (x)π

i=1

Note that for each N , ZiN (i = 1, · · · , N ) is i.i.d. Also, E[ZiN ] = 0,

N
2
i=1 E[ZiN ]

= 1; and

∀ε > 0
2 ] = N ε2 E[1(|Z /ε| > 1)(Z /ε)2 ]
N E[1(|ZiN | > ε)ZiN
iN
iN

≤ N ε2 E[(ZiN /ε)4 ]
−2
≤ N ε 2 VM

¯ M ) 2 E[ O(π ∗M ) 2 E[ 4i |xi ]]/(N 2 ε4 )
GM (x) exp(GM (x)π

≤ Cζ 0 (M )2 M/N −→ 0

Then by Lindbergh-Feller central limit theorem,

d
N
i=1 ZiN →

N (0, 1),i.e.

d

T 1 → N (0, 1)
As for the second term, T2 , in equation(A.5)
√

−1/2

N VM

exp(GM (x)π ∗M ) − r(x)

≤

√
−1/2
N VM

exp(GM (x)π ∗M ) − r(x)

√
≤ O( N M −(2+s) ) −→ 0
So,
√

−1/2

N VM

d

r(x) − r(x) → N (0, 1)

QED.

PROOF OF THEOREM 3:

76

Note that
CAP Ej (x) − CAP Ej (x)
= β j exp(xβ) exp(GM (x)π M ) − β j exp(xβ)r(x)
= β j exp(xβ) exp(GM (x)π M ) − β j exp(xβ)r(x)
+β j exp(xβ)r(x) − β j exp(xβ)r(x)
= β j exp(xβ) r(x) − r(x) + r(x) β j exp(xβ) − β j exp(xβ)
From result in theorem 1, we know
√

√

N (β j − β j ) = Op (1); so by delta method

N β j exp(xβ) − β j exp(xβ) = Op (1);

so
√

−1/2

N VM

r(x) β j exp(xβ) − β j exp(xβ)

−1/2

≤ C VM

≤ CM −1/2 −→ 0

While From Lemma 3 and Slutsky theorem
√
d
−1/2
N VM β j exp(xβ) r(x) − r(x) → β j exp(xβ)N (0, 1)
Hence
√

−1/2

N VM

d

CAP Ej (x) − CAP Ej (x) → β j exp(xβ)N (0, 1)

QED.
PROOF OF COROLLARY: Note that,
√

N CAP Ej (x) − CAP Ej (x)
√
=
N β j exp(xβ)r(x)
From theorem 1, we know that

√

N βj =

√
d
N (β j − β j ) → N (0, Ω ), where

(0, · · · , 1, 0, · · · , 0). So:
√

d

N CAP Ej (x) − CAP Ej (x) → N (0, r2 (x) exp(2xβ) Ω )

QED.
77

=

Appendix B

Tables and Figures
Table B.1: Estimation results: xtreg
Variable
lfare
concen
y98
y99
y00
Intercept

Coefficient
-1.163
0.145
0.045
0.104
0.197
11.769

78

(Std. Err.)
(0.023)
(0.040)
(0.006)
(0.006)
(0.006)
(0.116)

Table B.2: Simulation results where Vit has Gamma distribution

b = .01
.0934
(.0289)*

a=0
b = .05
.0675
(.0283)

b = .1
.0350
(.0284)

.1001
(.0415)

.0988
(.0404)

se(βˆ lf e )
se(βˆ pqml )

.0286
.0388

ρX,V ∗ ∗
ρX,lv ∗ ∗
mean(lv)
sd(lv)

βˆ lf e
βˆ pqml

b = .01
.0939
(.0290)

a = .01
b = .05
.0666
(.0230)

b = .1
.0315
(.0294)

.0967
(.0421)

.1011
(.0410)

.1002
(.0411)

.0994
(.0439)

.0287
.0388

.0291
.0397

.0292
.0390

.0294
.0393

.0296
.0405

.0008
-.0057

-.0004
-.0250

-.0003
-.0450

.0000
-.0046

-.0004
-.0263

-.0014
-.0523

-.5781
1.2816

-.5767
1.2839

-.5800
1.2914

-.5836
1.2919

-.5855
1.2945

-.5884
1.3021

* Monte Carlo Standard Deviations in parentheses
** ρX,V = Corr(X, V ), ρX,lv = Corr(X, lv),lv = log(V )

Table B.3: Simulation results where Vit has Gamma distribution(Continued)
a = .05
b = .01 b = .05 b = .1
.0921
.0630
.0222
(.0327) (.0319) (.0318)
.0997
(.0449)

.1023
(.0454)

se(βˆ lf e )
se(βˆ pqml )

.0320
.0414

ρX,V
ρX,lv
mean(lv)
sd(lv)

βˆ lf e
βˆ pqml

b = .01
.0922
(.0365)

a = .1
b = .05
.0495
(.0376)

b = .1
.0032
(.0373)

.0998
(.0454)

.0996
(.0484)

.0972
(.0471)

.1006
(.0516)

.0321
.0421

.0327
.0427

.0374
.0448

.0380
.0453

.0387
.0467

-.0000
-.0057

.0004
-.0277

.0004
-.0575

-.0001
-.0062

-.0011
-.0350

.0002
-.0672

-.6145
1.3358

-.6138
1.3398

.6169
1.3492

-.6572
1.4140

-.6592
1.4197

-.6604
1.4309

79

Table B.4: Special case
a=.1,b=0
500
1000
.0997
.1004
(.0372)
(.0266)

2000
.1006
(.0185)

.1017
(.0461)

.1021
(.0334)

.1011
(.0248)

se(βˆ lf e )
se(βˆ pqml )

.0375
.0445

.0266
.0329

.0189
.0240

ρx,v
ρx,lv

.0001
-.0003

.0005
.0009

.0006
.0004

mean(lv)
sd(lv)

-.6574
1.4120

-.6558
1.4126

-.6558
1.4134

N
βˆ

lf e

βˆ pqml

Table B.5: Simulation results where Vit has log-normal distribution

N
βˆ lf e

500
0.10001
(.01998)
βˆ pqml
0.09837
(.04476)
ˆ
se(β lf e )
0.01969
se(βˆ pqml ) 0.03758
ρX,V
-0.0012
ρX,lv
0.00003
mean(lv) -0.1249
sd(lv)
0.52993

a=-.125, b=.5
1000
0.09990
(.01411)
0.09655
(.03261)
0.01396
0.02778
-0.0010
0.0003
-0.1254
0.5305

2000
0.09991
(.01015)
0.09834
(.02412)
0.00987
0.02084
-0.0005
-0.0003
-0.1251
0.53047

80

500
0.09827
(.04807)
0.07994
(.13131)
0.04838
0.08555
-0.0004
-0.0011
-0.5004
1.2255

a=-.5, b=1
1000
0.10144
(.03315)
0.09138
(.11863)
0.03424
0.07361
0.00025
0.00088
-0.5001
1.2254

2000
0.10127
(.02410)
0.09515
(.09854)
0.02418
0.05789
0.00051
0.00074
-0.4998
1.2243

Table B.6: Vit = exp(a ∗ x2it + b ∗ xit ∗ zit ) with N = 500

ρ∗
βˆ

a=-.5, b=1, N=500
-0.1
0.1

-0.95

-0.5

0.5

0.95

lf e

0.10010
(.04456)

0.09669
(.04563)

0.09978
(.04795)

0.10090
(.04927)

0.09768
(.05510)

0.10261
(.06163)

βˆ pqml

0.07839
(.10740)

0.08340
(.13876)

0.08476
(.12536)

0.08303
(.13958)

0.08036
(.12433)

0.08497
(.13743)

se(βˆ lf e )
se(βˆ pqml )

0.04448
0.06785

0.04558
0.08034

0.04751
0.08202

0.04909
0.08652

0.05334
0.08887

0.06217
0.09637

ρx,v
ρx,lv
mean(lv)
sd(lv)

-0.00018
-0.00005
-0.49941
1.22230

-0.00247
-0.00268
-0.49949
1.22259

0.00054
-0.00007
-0.49859
1.22251

0.00014
0.00087
-0.49969
1.22435

-0.00177
-0.00163
-0.49972
1.22324

0.00190
0.00235
-0.49948
1.22320

*zi ∼ N (I5 , Σ), ρ = Corr(zit , zit+1 )

Table B.7: Vit = exp(a ∗ x2it + b ∗ xit ∗ zit ) with N = 1000

ρ
βˆ lf e

a=-.5, b=1, N=1000
-0.1
0.1

-0.95

-0.5

0.5

0.95

0.09984
0.03032

0.09959
0.03281

0.09845
0.03553

0.10023
0.03549

0.10137
0.03643

0.10145
0.04594

βˆ pqml

0.08465
0.10493

0.08509
0.09423

0.08018
0.10190

0.08802
0.12003

0.08665
0.11121

0.08614
0.11349

se(βˆ lf e )
se(βˆ pqml )

0.03152
0.06084

0.03231
0.06522

0.03370
0.06652

0.03469
0.07327

0.03764
0.07415

0.04395
0.08087

ρx,v
ρx,lv
mean(lv)
sd(lv)

-0.00129
-0.00039
-0.50118
1.22425

0.00053
-0.00024
-0.49880
1.22294

-0.00198
-0.00124
-0.49996
1.22432

-0.00088
0.00013
-0.49942
1.22375

0.00101
0.00090
-0.49900
1.22204

-0.00066
0.00087
-0.50027
1.22413

81

2

Table B.8: Simulation results with Vit = exp(−.125Xi + .5Xi ∗ zit )
N
βˆ pqml

2000
0.0998
( 0.0070)

1000
0.1001
(0.0095)

500
0.1001
(0.0132)

250
0.0998
(0.0183)

100
0.1001
(0.0278)

βˆ lf e

.10004
( 0.0025)

.09997
(0.0035)

.09989
(0.0050)

.09995
(0.0071)

.09991
(0.0112)

βˆ gmm

0.1001
(0.0042)

0.1003
( 0.0058)

0.1001
(0.0079)

0.1002
(0.0107)

0.1003
(0.0168)

se(βˆ pqml )

0.0074

0.0102

0.0142

0.0205

0.0384

se(βˆ lf e )

0.0025

0.0035

0.0050

0.0070

0.0110

se(βˆ gmm )

0.0044

0.0057

0.0073

0.0094

0.0136

Table B.9: Simulation results for four estimators
N
βˆ pqml

2000
0.0999
( 0.0068)

1000
0.1002
(0.0093)

500
0.1003
(0.0146)

250
0.0995
(0.0179)

100
0.999
(0.0265)

βˆ lf e

.10002
( 0.0027)

.09989
(0.0038)

.09998
(0.0052)

.09992
(0.0069)

.09995
(0.0120)

βˆ gmm

0.9999
(0.0044)

0.1002
( 0.0055)

0.9998
(0.0080)

0.1003
(0.0110)

0.1004
(0.0172)

βˆ oiv

0.1001
(0.0026)

0.9999
( 0.0038)

0.1001
(0.0066)

0.9998
(0.0077)

0.1006
(0.0118)

se(βˆ pqml )

0.0077

0.0110

0.0139

0.0207

0.0379

se(βˆ lf e )

0.0028

0.0039

0.0052

0.0073

0.0109

se(βˆ gmm )

0.0045

0.0060

0.0071

0.0090

0.0140

se(βˆ oiv )

0.0024

0.0034

0.0053

0.0069

0.0106

82

Table B.10: Summary statistics

passen
lfare
concen

Obs
4596
4596
4596

Mean
636.8242
5.095601
0.6101149

Std. Dev.
812
0.4363999
0.196435

Min
Max
2
8497
3.610918 6.257668
0.1605
1

Table B.11: Dependent variable, passen
lf are

concen

y88

y89

y00

βˆ pqml

-0.8658

-0.1289

0.0427

0.1093

0.1899

βˆ lf e

-1.1632

0.1455

0.0454

0.1038

0.1970

βˆ gmm

-0.8515

-0.1450

0.0431

0.1081

0.1911

se(βˆ pqml )

0.0366

0.0544

0.0037

0.0054

0.0085

se(βˆ lf e )

0.1101

0.0890

0.0049

0.0063

0.0101

se(βˆ gmm )

0.0336

0.0538

0.0035

0.0049

0.0069

83

.005
0
−.005
−.01
−.015
−.02
−1

−.5

0
rou
bias_fe

.5
bias_pqml

Figure B.1: Bias of LFE and PQML with change of ρ

84

1

.1
.08
.06
.04
−1

−.5

0
rou
se_fe

.5
se_pqml

Figure B.2: Std. error of LFE and PQML with change of ρ

85

1

0
−.005
−.01
−.015
−.02
−1

−.5

0
rou
bias_fe

.5
bias_pqml

Figure B.3: Bias of LFE and PQML with change of ρ, N=1000

86

1

.08
.07
.06
.05
.04
.03
−1

−.5

0
rou
se_fe

.5
se_pqml

Figure B.4: Std. error of LFE and PQML with change of ρ, N=1000

87

1

15
10

30

5

Percent

20
Percent

0

10
0
0

2000
4000
6000
avg. passengers per day

8000

0

2

Figure B.5: Histogram of Passengers

88

4
6
log(passen)

8

10

Proofs
PROOF OF Theorem 2.4.1

√
N

β−β
η−η

=

N
−1
i=1 A Vi

N −1/2

N
i=1 (PSi

N −1/2
N

≡ N −1/2
i=1

+ Ui − η)

+ op (1)

Si
+ op (1)
Qi

Where
Vi = Yi − p(Xi , β)ni
A =E(ni ∇β p(Xi , β) W(Xi , β)∇β p(Xi , β))

p(Xi , β) =

W(Xi , β) = diag

exp(Xi1 β)
,
T
t=1 exp(Xit β)
exp(Xi1 β)
T
t=1 exp(Xit β)

··· ,

, ··· ,

exp(XiT β)
T
t=1 exp(Xit β)
exp(XiT β)

−1

T
t=1 exp(Xit β)

P =E(Xit Ui ), Qi = PSi + Ui − η
PROOF OF LEMMA 2.4.3: From equation (1.4.3), and by monotonicity of logarithmic function,

exp(−CK −s + GM (X)π M ) − exp(GM (X)π M )
< r(X)) − exp(GM (X)π M )
< exp(CK −s + GM (X)π M ) − exp(GM (X)π M ),

89

By the mean value theorem applied to the lower and upper bound:
exp(−CK −s + GM (X)π M ) − exp(GM (X)π M )
= −CK −s exp(ξ 1 ),

ξ 1 ∈ [−CK −s + GM (X), GM (X)π M ]

exp(CK −s + GM (X)π M ) − exp(GM (X)π M )
= CK −s exp(ξ 2 ),

ξ 2 ∈ [GM (X)π M , CK −s + GM (X)]

So, for the π M that satisfies equation (1.4.3), we have:
sup |r(X) − exp(GM (X)π M )| < CK −s ,

X∈Ξ

So,
E r(Xi ) − exp(GM (Xi )π M )

2

≤ CK −2s

Note that:
T

E
t=1
T

=

Var
t=1

2
Yit
− exp(GM (Xi )π)
exp(xit β)

Yit
X
exp(Xit β) i

+ T E r(Xi ) − exp(GM (Xi )π)

2

So,
π M = arg min E r(Xi ) − exp(GM (Xi )π)
π

2

So,
E r(Xi ) − exp(GM (Xi )π ∗M )

2

≤ E r(Xi ) − exp(GM (Xi )π)

So, similarly,
r(Xi ) − exp(GM (Xi )π ∗M = Op (CK −s )

90

2

≤ CK −2s

PROOF OF LEMMA 2.4.4: this proof drives heavily from proof of Lemma 2 in
Hirano et.al.(2003) In the sequel we write M for M (N ). By definition of GM (X),
1
SˆM =
N

N

GM (Xi )GM (Xi )
i=1

has expecatation equal to IM . By Newey (1997), it satisfies
SˆM − IM = Op

ζ(M )

M
N

,

which converges to zero in probability by condition (iv). Hence the probability that the
smallest eigenvalue of SˆM is larger than 1/2 goes to one. Let
N

T

Yit

LN (π) = −
i=1 t=1

ˆ
exp(Xit β)

2

− exp(GM (X

i )π)

Next, we will show that
1 ∂LN ∗
(π M ) = Op
N ∂π

M
N

,

(B.1)

Consider
1 ∂LN ∗ 2
(π M )
N ∂π


2
T
1
Yit
=
− exp(GM (Xi )π ∗M ) exp(2GM (Xi )π ∗M )GM (Xi ) GM (Xi )
tr E 
ˆ
N
t=1 exp(Xit β)
E

=

T
tr E
N

Var

Yit
|X
exp(Xit β) i

+ r(Xi ) − exp(GM (Xi )π ∗M )

exp(2GM (Xi )π ∗M )GM (Xi ) GM (Xi )
≤

C
tr E GM (Xi ) GM (Xi )
N

≤

CK
N

91

2

+ op (1)

and Markov inequality implies (B.1). Next, let
η=

inf

X∈Ξ,M

2 exp(GM (X)π ∗M ) − r∗ (X)) exp(2GM (X)π ∗M )

,which by assumptions and Lemma 1 is positive. For any ε > 0, choose C such that for N
large enough
1 ∂LN ∗
(π M ) < ηC
N ∂π

P

M
N

≥1−

ε
2

(B.2)

Note that,
exp(2GM (X)π) − exp(2GM (X)π ∗M )

sup
X∈Ξ,|π−π ∗ |<ηC

≤

M
N

|CK(X)(π − π ∗ )|

sup
X∈Ξ,|π−π ∗ |<ηC

≤ ζ(M )C

M
N

M
N

which goes to zero,so that for large enough N
2 exp(GM (X)π) − r∗ (X)) exp(2GM (X)π) ≥ 4η

inf
X∈Ξ, π−π ∗ <ηC

M
N

Choose N large enough so that this inequality holds. that (B.2) holds with probability
at least 1 − ε/2, and the the probability that the smallest eigenvalue of SˆM is larger than
1/2 is at least 1 − ε/2. Then the probability that both of these hold is at least 1 − ε, then
for every π with π − π ∗ =

M,
N

a second order expansion gives

1
1
1 ∂LN ∗
1
∂ 2 LN
(π)(π − π ∗ ) (B.3)
LN (π) = LN (π ∗M ) +
(π M )(π − π ∗ ) +
(π − π ∗ )
N
N
N ∂π
2N
∂π∂π

92

where π − π ∗ ≤ π − π ∗ =

M.
N

We have

1 ∂ 2 LN
(π)
2N ∂π∂π
= −

= −

1
2N

N

T

2 exp(GM (Xi )π) −
i=1 t=1

1
E
2N

Yit
ˆ
exp(Xit β)

exp(GM (Xi )π)GM (Xi )) GM (Xi ))

2 exp(GM (Xi )π) − r∗ (Xi ) exp(GM (Xi )π)GM (Xi )) GM (Xi )) + op (1)

≤ −2η SˆM + op (1)

with its eigenvalues bounded away from zero in absolute value by η. Then, rearranging (B.3)
and using the triangle inequality, with probability greater than 1 − ε, for π − π ∗ =

M,
N

1
1 ∂LN ∗
1
LN (π) − LN (π ∗M ) ≤
(π M )(π − π ∗ ) − η π − π ∗ 2 + op (1)
N
N
N ∂π
≤

1 ∂LN ∗
(π M )
N ∂π

=

π − π ∗ − η π − π ∗ 2 + op (1)

1 ∂LN ∗
(π M ) − η
N ∂π

M
N

π − π ∗ + op (1)

< 0
That is, we have with probability greater than 1 − ε, N1 LN (π) < N1 LN (π ∗M ) for all π
with π − π ∗ =
π : π − π∗ ≤
ˆ M − π∗ <
π

M
N

Since LN (π) is continuous, it has a maximum on the compact set

M . By the last inequality, this maximum must
N
M . Hence the first order conditions are satisfied
N

ˆ M with
occur for some π
ˆ M and by concavity
at π

ˆ M maximize LN (π) over all of GM . Because the probability of this is greater
of LN (π), π
ˆ M exists and satisfies the first order conditions
than 1−ε with ε arbitrary, we conclude that π
ˆ M (N ) − π ∗M (N ) = Op
with probability approaching one, and that π
PROOF OF THEOREM 2.4.9:

93

M (N )
N

.

let
ω it = (Xit , GM (Xi )),


 β 
θ = 

π ∗M
T

j(wit , θ) =

exp(Xit β + GM (Xi )π ∗M )β

T −1
t=1

So,
√
√
√
N (ˆ
τ − τ ) = N (ˆ
τ − Ej(wit , θ)) + N (Ej(wit , θ) − τ ) ≡ T 1 + T 2

94

Note that,
N
√
−1
T1 =
N (N T )

T

ˆ M βˆ pqml
exp Xit βˆ pqml + GM (Xi )π

i=1 t=1

− exp Xit β + GM (Xi )π ∗M β
N
√
−1
+
N (N T )

T

exp Xit β + GM (Xi )π ∗M ) β − Ej(wit , θ)

i=1 t=1
N
√
−1
=
N (N T )

T

exp(wit ˆθ)βˆ pqml − exp(wit θ)β

i=1 t=1



N
T
T
√
+
T −1
N (N )−1
exp(wit θ) − ET −1
exp(wit θ) β
t=1

i=1
N

=

N −1

T

T −1
i=1
N

√
exp(wit θ) N (βˆ pqml − β)

t=1





T

∇θ (T )−1

+ N −1

t=1

exp(wit θ) (θ) ˆθ − θ
t=1

i=1



N
√
j(wit , θ) − Ej(wit , θ) + op (1)
+
N N −1
i=1
N

=

N −1

√
j(wit , θ) N (βˆ pqml − β)

i=1
N

+

N −1

∇θ (j(wit , θ)) (θ) ˆθ − θ
i=1



N
√
+
N N −1
j(wit , θ) − Ej(wit , θ) + op (1)
i=1

While,

N −1

N
i=1 ∇θ (j(wit , θ)) (θ)

= Op (ζ(M )), ˆθ − θ
95

= Op

M (N )
N

So the

M (N )
N

second term of the above equation is of order Op ζ(M )

, which vanishes by as-

sumptions in Lemma 2.
On the other hand,


T
√
√
T 2 ≡ N (Ej(wit , θ) − τ ) = N T −1
E exp(Xit β)(exp(GM (Xi )π ∗M ) − r(Xi )) β
t=1

From Lemma 1, we know the term in the parasynthesis is of order Op (CK −s ); as long
as N (1/2) M −s → 0, it can be ignored too.
Hence,
√

N

N (ˆ
τ −τ ) = N −1



N
√
√
j(wit , θ) N (βˆ pqml −β)+ N N −1
j(wit , θ) − Ej(wit , θ)+op (1)

i=1

i=1

From Wooldridge (1999), the first term follows as:
√

N

N (βˆ pqml − β) = N −1/2

∇2β p1 (Xi , β)W1 (Xi , β)(Yi − p1 (Xi , β)n1i )
i=1

And the second term:


N
N
√
N N −1
j(wit , θ) − Ej(wit , θ) = N −1/2
(j(wit , θ) − Ej(wit , θ))
i=1

i=1

Therefore:
√

N

N (ˆ
τ − τ ) = N −1/2

{Ej(wit , θ)∇2β p1 (Xi , β)W1 (Xi , β)(Yi − p1 (Xi , β)n1i )
i=1

+ (j(wit , θ) − Ej(wit , θ))} + op (1)

We follow that:
√

N (ˆ
τ − τ ) ⇒ N(0, V )

Where,

96

V = Var Ej(wit , θ)∇2β p1 (Xi , β)W1 (Xi , β)(Yi − p1 (Xi , β)n1i ) + (j(wit , θ) − Ej(wit , θ))

Q.E.D.
As for the estimation of V is straight forward:
let,




N

Vˆi1 = N −1

j(wit , ˆθ) ∇2β p1 (Xi , βˆ pqml )W1 (Xi , βˆ pqml )(Yi − p1 (Xi , βˆ pqml )n1i )
i=1





N

+ j(wit , ˆθ) − N −1

j(wit , ˆθ)
i=1

then,
N

Vˆ = N −1

Vˆi1 Vˆi1
i=1

Note, for the denotations here, please refer to section 2 and 3.
GMM Simulation Setup:
• we do the following setting up:
T

yit = ni
t=1
T
2 =n
yit
i2
t=1

pt (xi , β) ≡

pt2 (xi , β) ≡

97

exp(βxit )
T
r=1 exp(βxir )

exp(2βxit )
T
r=1 exp(2βxir )

p(xi , β) ≡ [p1 (xi , β), ..., pT (xi , β)] ,

p2 (xi , β) ≡ [p12 (xi , β), ..., pT 2 (xi , β)] ,

u1i (β) ≡ Yi − p(xi , β)ni , where Yi = [Yi1 , · · · , YiT ]

2]
u2i (β) ≡ Yi2 − p2 (xi , β)ni2 , where, Yi2 = [Yi12 , · · · , YiT

D1 (xi , β) = xi1 −

D2 (xi , β) = 2xi1 −

T
r=1 xir exp(βxir ) ,
T
r=1 exp(βxir )

T
r=1 2xir exp(2βxir ) ,
T
r=1 exp(2βxir )

T
r=1 exp(βxir )

,

T
r=1 2xir exp(2βxir )
T
r=1 exp(2βxir )

, ··· ,

T
r=1 xir exp(βxir )
T
r=1 exp(βxir )

pT (xi , β) xiT −

=

..., 2xiT −

T
r=1 xir exp(βxir )
T
r=1 exp(βxir )

D3 (xi , β) = ni p1 (xi , β) xi1 −

ni xi1 exp(βxi1 )

..., xiT −

T
r=1 xir exp(βxir )
T
r=1 exp(βxir )

−

T
r=1 xir exp(βxir ) ,
T
2
r=1 exp(βxir ))

ni exp(βxi1 )
(

ni xiT exp(βxiT ) ni exp(βxiT ) Tr=1 xir exp(βxir )
−
T
exp(βx
)
( Tr=1 exp(βxir ))2
ir
r=1

98

··· ,

,





0
 D1 (xi , β)

Di (xi , β) = 

0
D2 (xi , β)




 u1i (β) 
ui (β) = 

u2i (β)
• Step 1: PQML
N

T

βˆ pqml = arg max

Yit log(pt (xi , β))
i=1 t=1



se(βˆ pqml ) = 

−1 

N

D3 D1 



N

D1 ui1 u1i D1  



i=1

i=1

i=1

N

−1 1/2

D3 D1  

Where,
D1 = D1 (xi , βˆ pqml ), D3 = D3 (xi , βˆ pqml ), u1i = u1i (βˆ pqml )
• Step 2: GMM



N

βˆ gmm = arg min N −1

 
Di ui (β) N −1


se(βˆ gmm ) = 

 

N

N

Di ∇βˆ ui  
i=1

Di ui ui Di 

−1 
Di ui ui Di 

i=1

N


i=1

Where,
Di = Di (xi , βˆ pqml ), ui = ui (βˆ pqml )

99



N

N −1

Di ui (β)
i=1

i=1

i=1



−1 

N

−1/2

Di ∇βˆ ui 

∇βˆ ui

ˆ
 ∇β u1i (β gmm )

= 
∇β u2i (βˆ gmm )






exp(βxi1 ) Tr=1 xir exp(βˆ gmm xir )
xi1 exp(βˆ gmm xi1 )
−
T
ˆ
exp(
β
x
)
( Tr=1 exp(βˆ gmm xir ))2
ir
gmm
r=1

=

··· ,
xiT exp(βˆ gmm xiT )
exp(βxiT ) Tr=1 xir exp(βˆ gmm xir )
−
T
ˆ
exp(
β
x
)
( Tr=1 exp(βˆ gmm xir ))2
ir
gmm
r=1

ni ,

ni

2xi1 exp(2βˆ gmm xi1 )
exp(2βˆ gmm xi1 ) Tr=1 2xir exp(2βˆ gmm xir )
−
T
ˆ
( Tr=1 exp(2βˆ gmm xir ))2
r=1 exp(2β gmm 2xir )
··· ,
ˆ
exp(2βˆ gmm xiT ) Tr=1 2xir exp(2βˆ gmm xir )
2xiT exp(2β gmm xiT )
−
T
ˆ
( Tr=1 exp(2βˆ gmm xir ))2
r=1 exp(2β gmm 2xir )

n2i ,

n2i

• Step 3: OIV


 

N

βˆ oiv = arg min N −1

Bi u1i (β) N −1
i=1



se(βˆ gmm ) = 

−1 

N

N

Bi Bi 
i=1

 
Bi ∇βˆ u1i  

i=1

Bi Bi 
i=1

N −1

Bi u1i (β)
i=1

−1 

N

N


i=1

−1/2

Bi ∇βˆ u1i 

Where,
ui = u1i (βˆ pqml )
ˆ i ) ∗ (Ω
ˆ i )−1 /ˆ
B(Xi ) = D(X
g (Xi )
T

ˆ i = −∇β [p1 (Xi , βˆ pqml )]
D

exp(βˆ pqml Xit )ˆ
r(Xi )
t=1

100



N

∇βˆ u1i
≡ ∇β u1i (βˆ oiv )
=

exp(βˆ oiv xi1 ) Tr=1 xir exp(βˆ oiv xir )
xi1 exp(βˆ oiv xi1 )
−
T
ˆ
( Tr=1 exp(βˆ oiv xir ))2
r=1 exp(β oiv xir )

ˆ M ),
exp(GM (Xi )π

··· ,
exp(βˆ oiv xiT ) Tr=1 xir exp(βˆ oiv xir )
xiT exp(βˆ oiv xiT )
−
T
ˆ
( Tr=1 exp(βˆ oiv xir ))2
r=1 exp(β oiv xir )

101

ˆM)
exp(GM (Xi )π

Appendix C

Tables
Table C.1: Summary Statistics
Mean

Median

Standard
deviation

Minimum

Maximum

Sample
size

math4
37.73
avgexp
9037
exp10
9251
enroll
2769.4
lunch
39.087
scdist
20.303

38.05
8646
8800
1596.5
37.792
16.935

15.107
1644.950
1862.571
4473.652
16.049
16.250

3.1
7258
6890
64
5.993
.134

81.3
28611
30379
75263
87.815
80.952

518
518
518
518
518
518

Table C.2: OLS Regression, dependent variable=math4
Coef.
Estimate

Usual
Std. Err.

H-W
Std. Err.

ISD cluster
Std. Err.

log(avgexp)
lunch
log(enroll)
log(scdist)
constant

0.1595
-0.0058
0.0147
0.0030
-0.9639

0.0405*
0.0004*
0.0068*
0.0068
0.3744*

0.0558*
0.0004*
0.0076
0.0068
0.4973

0.0671*
0.0005*
0.0086
0.0080
0.6036

log(exp10)
lunch
log(enroll)
log(scdist)
constant

0.1289
-0.0058
0.0145
0.0017
-0.6848

0.0373*
0.0004*
0.0068*
0.0068
0.3451*

0.0552*
0.0004*
0.0077
0.0068
0.4895

0.0699
0.0005*
0.0087
0.0080
0.6259

* Significant at 5% level

102

Table C.3: OLS Regression with Conley S.E., dependent variable=math4

Coef.
H-W
Conley
cut1
50
100
150
200
350
400
500
600
700
800
1000

Estimate
Std. Err.
Std. Err.
cut2
100
150
200
250
400
500
600
700
800
900
1000

log(exp10)

log(enroll)

lunch

log(scdist)

constant

0.1289
0.0552*

0.0145
0.0077

-0.00576
0.00038

0.0017
0.0068

-0.6848
0.4895

0.0606*
0.0636*
0.0640*
0.0633*
0.0586*
0.0552*
0.0517*
0.0482*
0.0455*
0.0430*
0.0404*

0.0086
0.0085
0.0077
0.0072*
0.0062*
0.0063*
0.0060*
0.0054*
0.0050*
0.0046*
0.0042*

0.00044*
0.00044*
0.00042*
0.00037*
0.00032*
0.00030*
0.00027*
0.00025*
0.00024*
0.00022*
0.00021*

0.0080
0.0079
0.0075
0.0075
0.0069
0.0062
0.0057
0.0054
0.0051
0.0049
0.0048

0.5507
0.5743
0.5737
0.5696
0.5322
0.4981
0.4682
0.4366
0.4134
0.3926
0.3709

Table C.4: OLS Regression with Conley S.E., dependent variable=math4

Coef.
White
Conley
cut1
50
100
150
200
350
400
500
600
700
800
1000

Estimate
Std. Err.
Std. Err.
cut2
100
150
200
250
400
500
600
700
800
900
1000

log(avgexp)

log(enroll)

lunch

log(scdist)

constant

0.1595
0.0558

0.0147
0.0076

-0.00579
0.00036

0.0030
0.0068

-0.9639
0.4973

0.0616*
0.0638*
0.0634*
0.0625*
0.0591*
0.0559*
0.0529*
0.0497*
0.0474*
0.0452*
0.0430*

0.0086
0.0084
0.0076
0.0071*
0.0061*
0.0062*
0.0059*
0.0053*
0.0049*
0.0046*
0.0041*

0.00042*
0.00042*
0.00040*
0.00035*
0.00029*
0.00028*
0.00025*
0.00023*
0.00022*
0.00021*
0.00019*

0.0080
0.0080
0.0075
0.0075
0.0070
0.0063
0.0058
0.0055
0.0052
0.0050
0.0048

0.5640
0.5802
0.5710
0.5655
0.5412
0.5112
0.4856*
0.4569*
0.4371*
0.4178*
0.3991*

103

Table C.5: OLS Regression with Conley S.E., dependent variable=math4

Coef.
H-W
Conley
cut1
242
364
435
546.5
649
719
758

Estimate
Std. Err.
Std. Err.
cut2
21
41
88
167.5
284
637
880

log(exp10)

log(enroll)

lunch

log(scdist)

constant

0.1289
0.0552

0.0145
0.0077

-0.00576
0.00038

0.0017
0.0068

-0.6848
0.4895

0.0553*
0.0570*
0.0620*
0.0602*
0.0587*
0.0491*
0.0437*

.00758
.00760
0.0080
0.0072*
0.0059*
0.0052*
0.0047*

.00044*
.00049*
.00050*
.00045*
.00033*
.00025*
.00023*

0.0073
0.0078
0.0089
0.0093
0.0088
0.0057
0.0050

0.4942
0.5102
0.5602
0.5541
0.5454
0.4506
0.3976

Table C.6: OLS Regression with Conley S.E., dependent variable=math4

Coef.
H-W
Conley
cut1
242
364
435
546.5
649
719
758

Estimate
Std. Err.
Std. Err.
cut2
21
41
88
167.5
284
637
880

log(avgexp)

log(enroll)

lunch

log(scdist)

constant

0.1595
0.0558

0.0147
0.0076

-0.00579
0.00036

0.0030
0.0068

-0.9639
0.4973

0.0561*
0.0578*
0.0629*
0.0617*
0.0609*
0.0515*
0.0457*

.00755
.00760
0.0080
0.0072*
0.0059*
0.0051*
0.0047*

.00042*
.00046*
.00047*
.00042*
.00030*
.00023*
.00021*

0.0073
0.0077
0.0089
0.0093
0.0089
0.0058
0.0051

0.5037
0.5226
0.5740
0.5747
0.5713
0.4783*
0.4218*

104

math4 )
Table C.7: OLS Regression, dependent variable=log( 1−math4

Coef.
Estimate

usual
Std. Err

H-W
Std. Err

ISD cluster
Std. Err.

Peninsula
cluster S.E.

log(avgexp)
log(enroll)
lunch
log(scdist)
constant

0.5277
0.0919
-0.0275
0.0235
-5.0362

0.1976*
0.0332*
0.0018*
0.0332
1.8271*

0.2946
0.0387*
0.0018*
0.0328
2.6355

0.3505
0.0446*
0.0023*
0.0398
3.1478

0.0937
0.0021*
0.0007*
0.0166
0.8732

log(exp10)
log(enroll)
lunch
log(scdist)
constant

0.4152
0.0911
-0.0273
0.0192
-4.0088

0.1815*
0.0333*
0.0018*
0.0331
1.6817*

0.2811
0.0389*
0.0019*
0.0324
2.5017

0.3554
0.0450*
0.0024*
0.0396
3.1720

0.0631
0.0000*
0.0007*
0.0167
0.5806

math4 )
Table C.8: OLS Regression with Conley S.E., dependent variable=log( 1−math4

Coef.
H-W
Conley
cut1
50
100
150
200
350
400
500
600
700
800
1000

Estimate
Std. Err.
Std. Err.
cut2
100
150
200
250
400
500
600
700
800
900
1000

log(exp10)

log(enroll)

lunch

log(scdist)

constant

0.4152
0.2811

0.0911
0.0389

-.0273
.00193

0.0192
0.0324

-4.0088
2.5017

0.3094
0.3235
0.3244
0.3209
0.2907
0.2737
0.2596
0.2443
0.2320
0.2208
0.2079*

0.0432*
0.0430*
0.0397*
0.0376*
0.0331*
0.0335*
0.0319*
0.0287*
0.0263*
0.0245*
0.0222*

.00224*
.00225*
.00216*
.00192*
.00159*
.00152*
.00135*
.00124*
.00116*
.00109*
.00101*

0.0396
0.0407
0.0397
0.0400
0.0373
0.0333
0.0309
0.0291
0.0278
0.0267
0.0257

2.8198
2.9325
2.9270
2.9141
2.6737
2.5026
2.3823
2.2454
2.1394
2.0409*
1.9330*

105

math4 )
Table C.9: OLS Regression with Conley S.E., dependent variable=log( 1−math4

Coef.
H-W
Conley
cut1
50
100
150
200
350
400
500
600
700
800
1000

Estimate
Std. Err.
Std. Err.
cut2
100
150
200
250
400
500
600
700
800
900
1000

log(avgexp)

log(enroll)

lunch

log(scdist)

constant

0.5277
0.2946

0.0919
0.0387

-.0275
.00187

0.0235
0.0328

-5.0362
2.6355

0.3248
0.3345
0.3323
0.3292
0.3037
0.2856
0.2720
0.2576*
0.2468*
0.2361*
0.2250*

0.0431*
0.0428*
0.0394*
0.0373*
0.0330*
0.0334*
0.0319*
0.0287*
0.0263*
0.0245*
0.0223*

.00218*
.00217*
.00208*
.00184*
.00147*
.00141*
.00125*
.00116*
.00109*
.00102*
.00095*

0.0401
0.0411
0.0400
0.0405
0.0381
0.0342
0.0317
0.0298
0.0285
0.0273
0.0263

2.9776
3.0533
3.0183
3.0113
2.8254
2.6533
2.5345*
2.4021*
2.3058*
2.2103*
2.1153*

math4 )
Table C.10: OLS Regression with Conley S.E., dependent variable=log( 1−math4

Coef.
H-W
Conley
cut1
242
364
435
546.5
649
719
758

Estimate
Std. Err.
Std. Err.
cut2
21
41
88
167.5
284
637
880

log(exp10)

log(enroll)

lunch

log(scdist)

constant

0.4152
0.2811

0.0911
0.0389

-0.0273
0.0019

0.0192
0.0324

-4.0088
2.5017

0.2787
0.2855
0.3076
0.3007
0.3026
0.2478
0.2238

0.0389*
0.0387*
0.0412*
0.0374*
0.0314*
0.0278*
0.0250*

0.0022*
.00236*
.00237*
0.0021*
0.0016*
0.0012*
0.0011*

0.0368
0.0400
0.0473
0.0506
0.0479
0.0308
0.0268

2.5073
2.5739
2.8016
2.8079
2.8503
2.3079
2.0654

106

Table C.11: OLS Regression with Conley S.E. in Nonlinear Model

Coef.
H-W
Conley
cut1
242
364
435
546.5
649
719
758

Estimate
Std. Err.
Std. Err.
cut2
21
41
88
167.5
284
637
880

log(avgexp)

log(enroll)

lunch

log(scdist)

constant

0.5277
0.2946

0.0919
0.0387

-0.0275
0.0019

0.0235
0.0328

-5.0362
2.6355

0.2952
0.3024
0.3230
0.3192
0.3274
0.2665*
0.2386*

0.0389*
0.0387*
0.0413*
0.0376*
0.0317*
0.0279*
0.0251*

0.0021*
0.0023*
0.0022*
0.0020*
0.0015*
0.0011*
0.0010*

0.0369
0.0399
0.0472
0.0511
0.0487
0.0315
0.0275

2.6643
2.7459
2.9710
3.0140
3.1091
2.5120*
2.2302*

107

Table C.12: APEs with Bootstrap S.E. in Nonlinear Model

log(avgexp)
log(enroll)
lunch
log(scdist)

log(exp10)
log(enroll)
lunch
log(scdist)

APE Evaluated at
50%
75%

Mean

25%

0.1151*
(.0663)
0.0200**
(.0083)
-0.0060**
(.0004)
0.0051
(.0071)

0.1193*
(.0683)
0.0208**
(.0085)
-0.0062**
(.0004)
0.0053
(.0073)

0.1156*
(.0663)
0.0201**
(.0084)
-0.0060**
(.0004)
0.0052
(.0072)

0.1100*
(.0636)
0.0191**
(.0082)
-0.0057**
(.0004)
0.0049
(.0068)

0.0982*
(.0590)
0.0171**
(.0078)
-0.0051**
(.0004)
0.0044
(.0063)

0.0905
(.0631)
0.0199**
(.0084)
-0.0060**
(.0004)
0.0042
(.0070)

0.0940
(.0652)
0.0206**
(.0086)
-0.0062**
(.0004)
0.0043
(.0072)

0.0910
(.0632)
0.0200**
(.0084)
-0.0060**
(.0004)
0.0042
(.0070)

0.0863
(.0603)
0.0189**
(.0082)
-0.0057**
(.0004)
0.0040
(.0067)

0.0767
(.0553)
0.0168**
(.0077)
-0.0051**
(.0004)
0.0035
(.0061)

* Significant at 10% level
** Significant at 5% level
Bootstraps standard errors are in parenthesis

108

95%

Figures

109

Ü

Legend
MI SD
N/A

Figure C.1: All school districts of Michigan in 2010: For interpretation of the references
to color in this and all other figures, the reader is referred to the electronic version of this
dissertation.
110

Ü
!!
!

!

!

!

!

!

!
!

!!
!
!
!

!

!

!!

!!

!
!

!

!
!
!

!

Legend
!

!
!!

MI College
MI SD
N/A

!!
!!!
!!
!
! !

!!!
!
!!!

!

!!

! !!
!
!
!

!!
!!

!

!!
!
!!

!
! !

!
!!
!
!
!
!!
!

!
!

!
!

!!

Figure C.2: All Colleges of Michigan in 2010.

111

!

!!!
!
!!
!!
!
! !
!! ! !
!
!!
!
!!
! !!
!
!!
!

Ü

math4

3.1-21.3

21.4 - 33.3
33.9- 44.4

44.6 - 57.6
58.0 - 81.3
N/A

Figure C.3: MEAP math pass rate for 4th graders of Michigan in 2010.

112

Ü
!!
!

!

!

!

!

!

!
!

!!
!

math4

!
!

!

3.1-21.3

!

!
!

21.4 - 33.3

!

!

!

!
!!

44.6 - 57.6
N/A

!

!
!

33.9- 44.4

58.0 - 81.3

!!

!!

!
!!
!!
!!
!
! !

!!!
!
!!!

!
! !!

!
!
!

!!
!!

!

!!
!
!!

!
! !

!!

!!!
!
!!
!!
!
! !
!
! ! !! ! !
!
! !!
!
!
!
!
! !
!
! !!
! !!

!
!

!
!

!!

!

MI College

Figure C.4: Colleges and math pass rate for 4th graders of Michigan in 2010.
113

Ü

Legend
selection
MI SD
N/A

Figure C.5: Selection of 96 School Districts.
114

Ü

Figure C.6: Selection of 96 School Districts with centroids.
115

Ü

Figure C.7: Selection of 96 School Districts with centroids in grids.

116

Ü

Figure C.8: Conley Coordinates of 96 School Districts.

117

Percent
0 10 20 30 40 50

10 20 30 40
0

Percent

5K

10K

15K

20K

25K

30K

5K

15
0

0

Percent
5
10

Percent
20 40 60

80

exp

10K 15K 20K 25K 30K
exp10

0

20K

40K
enroll

60K

80K

0

20

Figure C.9: Histogram for all covariates

118

40
scdist

60

80

APE for Average Expenditure
25%
20%
APE

15%
10%
5%
0
Mean

25%

50%
75%
APE Evaluated at

95%

APE Nonlinear
OLS Linear
90% confidence interval − APE
Figure C.10: APE w.r.t average expenditure

119

APE for Enrollment
4%

APE

3%
2%
1%
0
Mean

25%

50%
75%
APE Evaluated at

APE Nonlinear
95% confidence interval − APE

95%

OLS Linear

Figure C.11: APE w.r.t enroll

APE for Free Lunch
−.45%

APE

−.50%
−.55%
−.60%
−.65%
−.70%
Mean

25%

50%
75%
APE Evaluated at

95%

APE Nonlinear
OLS Linear
95% confidence interval − APE
Figure C.12: APE w.r.t lunch

120

Table C.13: QGLS with Conley S.E., dependent variable=math4

Coef.
Usual
H-W
Conley
cut1
50
100
150
200
350
400
500
600
700
800
1000

Estimate
Std. Err.
Std. Err.
Std. Err.
cut2
100
150
200
250
400
500
600
700
800
900
1000

log(avgexp)

log(enroll)

lunch

0.1690
0.0416*
0.0577*

0.0171
0.00705*
0.00778*

-0.00589 0.00255
0.00038* 0.00707
0.00038* 0.00712

-1.0588
0.3862*
0.5137*

0.0621*
0.0634*
0.0626*
0.0615*
0.0596*
0.0560*
0.0534*
0.0502*
0.0479*
0.0457*
0.0433*

0.0083
0.0079
0.0070
0.0065
0.0058*
0.0056*
0.0053*
0.0048*
0.0044*
0.0041*
0.0037*

0.00041*
0.00040*
0.00038*
0.00033*
0.00027*
0.00026*
0.00024*
0.00022*
0.00021*
0.0002*
0.00018*

0.5669
0.5739
0.5615
0.5550
0.5469
0.5131*
0.4907*
0.4614*
0.4405*
0.4205*
0.4007*

121

log(scdist)

0.0083
0.0083
0.0079
0.0079
0.0077
0.0067
0.0063
0.0059
0.0057
0.0055
0.0054

constant

Table C.14: QGLS with Conley S.E., dependent variable=math4 in Year 2010

Coef.
Usual
H-W
Conley
cut1
50
100
150
200
350
400
500
600
700
800
1000

Estimate
Std. Err.
Std. Err.
Std. Err.
cut2
100
150
200
250
400
500
600
700
800
900
1000

log(exp10)

log(enroll)

lunch

log(scdist)

constant

0.13496
0.03828
0.05759

0.0169
0.00708
0.00785

-0.00585
0.00039
0.00039

0.00142
0.00708
0.00709

-0.74907
0.35662
0.51117

0.0616*
0.0638*
0.0639*
0.0631*
0.0584*
0.0556*
0.0522*
0.0486*
0.0458*
0.0433*
0.0406*

0.0083*
0.0080*
0.0071*
0.0066*
0.0056*
0.0057*
0.0054*
0.0048*
0.0044*
0.0041*
0.0037*

0.00043*
0.00043*
0.0004*
0.00035*
0.00031*
0.00029*
0.00026*
0.00024*
0.00023*
0.00021*
0.0002*

0.0083
0.0084
0.0080
0.0080
0.0074
0.0066
0.0062
0.0059
0.0057
0.0055
0.0053

0.5580
0.5729
0.5696
0.5648
0.5295
0.5010
0.4724
0.4395
0.4149
0.3929
0.3694*

Table C.15: SAR GLS, dependent variable=math4
Weight Matrix

Contiguity Weight

GLS Estimates

Coef.

log(avgexp)
log(enroll)
lunch
log(scdist)
constant

Std. Err.

Inverse Dist Weight
Coef.

Std. Err.

0.1602579 0.0420131*
0.015429
0.0069654*
-0.005761 0.0004002*
0.0034104 0.0072557
-0.9785768 0.3874754

0.1690079
0.0170857
-0.0058864
0.00255
-1.058793

0.0426937*
0.0070499*
0.0003816*
0.0070406
0.3950128

ρ

0.2677935

0.0590989*

0.8456347

0.1394274*

σ2

0.0133044

0.0008314*

0.013609

0.0008473*

122

Table C.16: SAR GLS, dependent variable=math4 in Year 2010
Weight Matrix

Contiguity Weight

GLS Estimates

Coef.

log(exp10)
log(enroll)
lunch
log(scdist)
constant

Inverse Dist Weight

Std. Err.

Coef.

Std. Err.

0.1245523 0.0387077*
0.0153031 0.0069908*
-0.0057036 0.0004028*
0.0023392 0.0072584
-0.6548259 0.3581062

0.1349613
0.0168974
-0.005847
0.0014194
-0.7490704

0.0399863*
0.0070563*
0.0003847*
0.0070466
0.3715918

ρ

0.2638944

0.0592924*

0.8401031

0.1468289*

σ2

0.0134163

0.0008383*

0.0137166

0.000854*

Table C.17: SAR GLS, dependent variable=math4, contiguity
math4

Coef.

lexp
lenroll
lunch
lscdist
cons

Std. Err.

z

[95% Conf.

Interval ]

0.1602579 0.0420131 3.81 0
0.015429
0.0069654 2.22 0.027
-0.005761 0.0004002 -14.4 0
0.0034104 0.0072557 0.47 0.638
-0.9785768 0.3874754 -2.53 0.012

0.0779136
0.001777
-0.0065453
-0.0108106
-1.738015

0.2426021
0.029081
-0.0049766
0.0176313
-0.219139

rho

0.2677935

0.0590989

4.53

0

0.1519618

0.3836253

sigma2

0.0133044

0.0008314

16

0

0.0116749

0.014934

123

P>z

Table C.18: SAR GLS, dependent variable=math4, inverse distance
math4

Coef.

Std. Err.

z

P>z

[95% Conf.

Interval]

lexp
lenroll
lunch
lscdist
cons

0.1690079
0.0170857
-0.0058864
0.00255
-1.058793

0.0426937
0.0070499
0.0003816
0.0070406
0.3950125

3.96
2.42
-15.43
0.36
-2.68

0
0.015
0
0.717
0.007

0.0853298
0.0032681
-0.0066343
-0.0112493
-1.833003

0.252686
0.0309033
-0.0051385
0.0163492
-0.2845826

rho

0.8456345

0.1394278

6.07

0

0.572361

1.118908

sigma2

0.013609

0.0008473

16.06

0

0.0119483

0.0152696

124

Table C.19: Summary of Correlation
Overall
Overall
Overall
Overall

Max
Min
mean
variance

1
3.33E-38
0.00346437
0.002071544

Row mean
1%
5%
10%
25%
50%

Percentiles
0.0030197
0.0031132
0.0031967
0.0033227
0.0034535

75%
90%
95%
99%

0.0035881
0.0037387
0.0038485
0.004069

Smallest
2.85E-03
0.0028733
0.0029348
0.0029758
Largest
0.004259
0.0042819
0.0043556
0.0057957

Obs
Sum of Wgt.
Mean
Std. Dev.

518
518
0.0034644
0.0002449

Variance
Skewness
Kurtosis

6.00E-08
1.967204
18.49351

Obs
Sum of Wgt.
Mean
Std. Dev.

518
518
1.35E-21
1.11E-20

Variance
Skewness
Kurtosis

1.24E-40
13.54964
209.841

Obs
Sum of Wgt.
Mean
Std. Dev.

518
518
0.0020755
0.0000389

Variance
Skewness
Kurtosis

1.51E-09
2.174566
10.42997

Row min
1%
5%
10%
25%
50%

Percentiles
5.22E-38
1.34E-36
9.47E-36
9.24E-34
1.87E-29

75%
90%
95%
99%

1.49E-25
8.06E-23
1.47E-21
3.52E-20

Smallest
3.33E-38
3.33E-38
4.15E-38
5.04E-38
Largest
4.02E-20
4.70E-20
1.34E-19
1.92E-19

Row variance
1%
5%
10%
25%
50%
75%
90%
95%
99%

Percentiles
0.0020325
0.0020376
0.0020417
0.002049
0.002062
0.0020927
0.0021256
0.0021453
0.0022243

Smallest
2.03E-03
0.0020309
0.0020312
0.0020322
Largest
0.0022539
0.002289
0.0022999
0.0023159

125

Table C.20: Summary of Correlation(div)
Overall
Overall
Overall
Overall

Max(2nd largest)
Min
mean
variance

1(.266828657)
0.062464033
0.079143268
0.001712584

Row mean
1%
5%
10%
25%
50%
75%
90%
95%
99%

Percentiles
0.0694387
0.0710834
0.0731629
0.0765987
0.0794727
0.0817451
0.0848623
0.0865209
0.087731

Smallest
0.0692969
0.0693381
0.0693386
0.0693432
Largest
0.0879126
0.0879339
0.0881153
0.0881707

Obs
Sum of Wgt.
Mean
Std. Dev.

518
518
0.0791433
0.0042475

Variance
Skewness
Kurtosis

0.000018
-0.2346403
2.719421

Obs
Sum of Wgt.
Mean
Std. Dev.

518
518
0.0658829
0.0016574

Variance
Skewness
Kurtosis

2.75E-06
0.5907457
2.952739

Obs
Sum of Wgt.
Mean
Std. Dev.

518
518
0.0016978
0.0000361

Variance
Skewness
Kurtosis

1.30E-09
1.809396
5.520294

Row min
1%
5%
10%
25%
50%

Percentiles
0.0625941
0.06363
0.063966
0.0647366
0.0656149

75%
90%
95%
99%

0.0668986
0.0682221
0.0692942
0.0701836

Smallest
0.062464
0.062464
0.0624661
0.0625628
Largest
0.070315
0.0703266
0.070412
0.0704638

Row variance
1%
5%
10%
25%
50%

Percentiles
0.0016646
0.0016661
0.0016686
0.0016759
0.0016843

75%
90%
95%
99%

0.0017045
0.0017555
0.0017896
0.0018151

Smallest
0.0016636
0.0016644
0.0016644
0.0016645
Largest
0.0018191
0.0018207
0.0018216
0.0018218

126

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