ESSAYS ON AVERAGE TREATMENT EFFECTS
By
Myounggin Keay

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Economics
2012

ABSTRACT
ESSAYS ON AVERAGE TREATMENT EFFECTS
By
Myounggin Keay
This dissertation consists of three essays on estimating average treatment effects (ATE)
under counterfactual framework. In Chapter 1, I compare the performances of single-step and
two-step estimators for estimating the ATE in a linear model when treatment assignment
depends on unobservables. Recent advances in computing technology have enabled the
extensive use of single-step estimators, such as Limited Information Maximum Likelihood
(LIML), instead of 2SLS. In this study I make clear that there are two kinds of singlestep estimators for estimating ATE. LIML-type estimator is the one which uses the control
function method, on which the two-step method is also based, whereas FIML-type estimator
directly uses the joint distribution of underlying errors or endogenous variables. I find that
the relative asymptotic efficiency between two-step Heckit and single-step LIML cannot be
determined in general. However, the relative efficiency of single-step LIML with respect to
two-step Heckit is decreasing as the sample size increases, implying that if the asymptotic
variances are same, then single-step LIML is less efficient in finite samples. On the other
hand the FIML estimator tends to have very small finite sample variances, but it is less
robust to misspecification. Newey-type series estimators are also considered for correcting
the misspecification of error distributions, but it turned out that cost is greater than the
benefit. Under weak many instrument cases, the advantage of LIML in terms of median bias
was not as strong as in the linear models.
Chapter 2 explores the ATE estimator proposed by Terza (2009)’s Nonlinear Full En-

dogenous Treatment (NFES) model, where count dependent and binary treatment variables
are present. When the true conditional mean function takes the form of exponential function, the Heckit-type linear method, while it can be a good approximation, is inconsistent
for the true ATE since it is derived under the assumption of linear conditional mean. The
asymptotic distribution of nonlinear estimators have additional terms in asymptotic variance
of which magnitudes depend on population coefficient. Due to their presence, the asymptotic
variances of nonlinear estimators can be either larger or smaller than the linear counterparts
depending on the values of coefficients. It turns out that they tend to have small variances
when the variance of conditional ATE are small. And Monte Carlo experiments show that
they are fairly robust to various distributional misspecifications. In summary, nonlinear ATE
estimators are robust and consistent with small variance when the treatment effects are not
substantially different across individuals. An application to Botswana fertility is given where
the treatment is seven years of education with the dependent variable fertility.
Chapter 3 presents a method for estimating ATE for the case that the dependent variable
is count variable and the coefficients of covariates are random variables which are correlated
with the binary treatment variable. The identifying assumptions are given and the estimating equation based on them is derived. Simulations show that, in large samples, it has
usually smaller biases and larger variances than the linear methods have. An application on
Botswana fertility is given with same variables as in Chapter 2.

Copyright by
MYOUNGGIN KEAY
2012

To Jiyoung and Un-hyeng

v

ACKNOWLEDGMENTS

I would like to express my deepest gratitude to my advisor Jeff Wooldridge. Without his
support and guidance, this dissertation would never have been able to come to existence. I
also extend my appreciation to Todd Elder, Peter Schmidt, Tim Vogelsang and Hira Koul.
Their comments and insights have helped me a great deal whenever I was stuck on obstacles.
I also thank Ot´vio Bartalotti, Guojun Chen, Cheol Keun Cho, Do Won Kwak, Jin Young
a
Lee, Shengwu Shang, Valentin Verdier, Yali Wang and all other my collegues at Michigan
State University.

vi

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapter 1
1.1
1.2
1.3

1.4
1.5
1.6

1.7

Alternative Estimators of Average Treatment Effect under
Misspecification and Weak Instrumental Variables . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Two-step Heckit and Single-Step Quasi-LIML . . . . . . . . . . . . .
1.3.2 Series Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.3 Quasi-FIML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Asymptotic Variances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Comparison between LIV and Heckit1 . . . . . . . . . . . . . . . . .
1.6.2 Comparison between Heckit2 and QLIML . . . . . . . . . . . . . . .
1.6.3 Comparison between Heckit2 and Series estimator. . . . . . . . . . .
1.6.4 QFIML . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.5 Comparison between LIV and Heckit1 under weak IV . . . . . . . . .
1.6.6 Comparison between LIV and L-LIML and between Heckit2 and QLIML
under weak IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30
30

Chapter 2

2.1
2.2

2.3

2.4

Estimating Average Treatment Effect by Nonlinear Endogenous Switching Regression with an Application in Botswana
Fertility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Nonlinear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Full Information Maximum Likelihood . . . . . . . . . . . . . . . . .
2.3.2 Quai-Maximum Likelihood Estimator . . . . . . . . . . . . . . . . . .
2.3.3 Nonlinear Least Squares Estimator . . . . . . . . . . . . . . . . . . .
2.3.4 Weighted Nonlinear Least Squares Estimator . . . . . . . . . . . . . .
Asymptotic Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
4
9
9
13
15
18
20
21
22
25
27
28
29

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32
32
34
35
40
42
42
43
45
46
48

2.5

2.6
2.7

Monte Carlo Simulation . . . . .
2.5.1 Data Generating Processes
2.5.2 Main Results . . . . . . .
2.5.3 Some Other Results . . . .
Empirical Application . . . . . .
Conclusion . . . . . . . . . . . . .

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69

Chapter 3
3.1
3.2
3.3

3.4

3.5

3.6

3.7
3.8

A Two-Regime CRC Model with Nonnegative Dependent Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Previous Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Various Models of CRC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3.1 Continuous Endogenous Variable . . . . . . . . . . . . . . . . . . . .
3.3.2 Binary Endogenous Variable . . . . . . . . . . . . . . . . . . . . . . .
Nonlinear Two-Regime CRC Model . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Identification of ATE . . . . . . . . . . . . . . . . . . . . . . . . . . .
Specification Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.1 Tests for Endogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Model Selection Test . . . . . . . . . . . . . . . . . . . . . . . . . . .
Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6.1 Data Generating Processes . . . . . . . . . . . . . . . . . . . . . . . .
3.6.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .
An application: the effect of elementary school education on fertility in Botswana
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

APPENDICES . . . . . . . . . . . . . . .

70
70
71
75
75
78
80
80
86
88
88
89
90
90
92
94
98

. . . . . . . . . . . . . . . . . . 100

Appendix A Simulation Results for Series Estimator . . . . . . . . . . . . . . 100
Appendix B Simulation Results for All Other Estimators . . . . . . . . . . . 114
Appendix C Other Tables in Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . 124
Appendix D Figures in Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Appendix E Proofs in Chapter 2 . . . . . . . . . . . . . . . .
E.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . .
E.2 Derivation of Estimating Equation for NFES Model . . .
E.3 Derivation of Conditional Variance for WNLS Estimator

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129
129
132
133

Appendix F Tables in Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
Appendix G Figures in Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 158

viii

Appendix H Tables in Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
References . . . . . . . . . . . . . . .

ix

. . . . . . . . . . . . . . . . . . . 170

LIST OF TABLES

Table 1.1

Relationship among Estimators . . . . . . . . . . . . . . . . . . . . .

13

Table 1.2

Distributional assumptions . . . . . . . . . . . . . . . . . . . . . . .

17

Table A.1

Simulation results for series estimators with ρ = 0.0, ξ: normal . . . 102

Table A.2

Results for ρ = 0.0, ξ: t(5) . . . . . . . . . . . . . . . . . . . . . . . 103

Table A.3

Results for ρ = 0.0, ξ: χ2 (5) . . . . . . . . . . . . . . . . . . . . . . 104

Table A.4

Results for ρ = 0.4, ξ: normal . . . . . . . . . . . . . . . . . . . . . 105

Table A.5

Results for ρ = 0.4, ξ: t(5) . . . . . . . . . . . . . . . . . . . . . . . 106

Table A.6

Results for ρ = 0.4, ξ: χ2 (5) . . . . . . . . . . . . . . . . . . . . . . 107

Table A.7

Results for ρ = 0.5, ξ: normal . . . . . . . . . . . . . . . . . . . . . 108

Table A.8

Results for ρ = 0.5, ξ: t(5) . . . . . . . . . . . . . . . . . . . . . . . 109

Table A.9

Results for ρ = 0.5, ξ: χ2 (5) . . . . . . . . . . . . . . . . . . . . . . 110

Table A.10

Results for ρ = 0.6, ξ: normal . . . . . . . . . . . . . . . . . . . . . 111

Table A.11

Results for ρ = 0.6, ξ: t(5) . . . . . . . . . . . . . . . . . . . . . . . 112

Table A.12

Results for ρ = 0.6, ξ: χ2 (5) . . . . . . . . . . . . . . . . . . . . . . 113

Table B.1

Simulation Results: Strong IV and ρ = 0.0 . . . . . . . . . . . . . . 116

Table B.2

Results for Strong IV and ρ = 0.4 . . . . . . . . . . . . . . . . . . . 117

Table B.3

Results for Strong IV and ρ = 0.5 . . . . . . . . . . . . . . . . . . . 118

Table B.4

Results for Strong IV and ρ = 0.6 . . . . . . . . . . . . . . . . . . . 119

x

Table B.5

Results for Weak IV and ρ = 0.0

Table B.6

Results for Weak IV and ρ = 0.4 . . . . . . . . . . . . . . . . . . . . 121

Table B.7

Results for Weak IV and ρ = 0.5 . . . . . . . . . . . . . . . . . . . . 122

Table B.8

Results for Weak IV and ρ = 0.6

Table C.1

Relative Efficiencies (Ratio of MSEs) . . . . . . . . . . . . . . . . . 125

Table C.2

Ratio of estimations with AvarQLIM L < AvarT wo−step . . . . . . . 126

Table F.1

Simulation Results for rho = 0.4 . . . . . . . . . . . . . . . . . . . . 137

Table F.2

Simulation Results for rho = 0.5 . . . . . . . . . . . . . . . . . . . . 139

Table F.3

Simulation Results for rho = 0.6 . . . . . . . . . . . . . . . . . . . . 141

Table F.4

Simulation Results for FIML estimator . . . . . . . . . . . . . . . . 143

Table F.5

Simulation Results for DGP 1 . . . . . . . . . . . . . . . . . . . . . 144

Table F.6

Simulation Results for DGP 2 . . . . . . . . . . . . . . . . . . . . . 146

Table F.7

Simulation Results for DGP 3 . . . . . . . . . . . . . . . . . . . . . 148

Table F.8

Simulation Results for DGP 4 . . . . . . . . . . . . . . . . . . . . . 150

Table F.9

Variables Description . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Table F.10

Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 152

Table F.11

Regression Results: selection equation . . . . . . . . . . . . . . . . . 152

Table F.12

Regression Results: dependent variable children . . . . . . . . . . . 153

Table F.13

Regression Results: dependent variable ced . . . . . . . . . . . . . . 155

Table F.14

Average Treatment Effects . . . . . . . . . . . . . . . . . . . . . . . 157

xi

. . . . . . . . . . . . . . . . . . . 120

. . . . . . . . . . . . . . . . . . . 123

Table H.1

Simulation Results for ρ = 0.3 . . . . . . . . . . . . . . . . . . . . . 163

Table H.2

Simulation Results for ρ = 0.5 . . . . . . . . . . . . . . . . . . . . . 164

Table H.3

Regression Results for Linear and NFES models . . . . . . . . . . . 165

Table H.4

Regression Results for LTCRC and NTCRC models . . . . . . . . . 167

xii

LIST OF FIGURES

Figure 3.1

Analogy Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure D.1

Sample Cumulative Density Functions under Strong IV with ρ = 0.6
and obs=1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

Figure D.2

Sample Cumulative Density Functions under Weak IV with ρ = 0.6
and obs=1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

Figure G.1

Selected Monte Carlo Simulation Results for DGP 0. . . . . . . . . . 158

Figure G.2

Selected Monte Carlo Simulation Results for DGP 1. . . . . . . . . . 159

Figure G.3

Selected Monte Carlo Simulation Results for DGP 2. . . . . . . . . . 160

Figure G.4

Selected Monte Carlo Simulation Results for DGP 3. . . . . . . . . . 161

xiii

75

Chapter 1
Alternative Estimators of Average
Treatment Effect under
Misspecification and Weak
Instrumental Variables

1.1

Introduction

Although single-step estimators such as LIML has long been recognized as a good alternative to the traditional 2SLS (Anderson et al., 1982), the computational burden has prevented
them from being widely used in applied researches. Rather, people have preferred two-step
methods or, for some nonlinear models, have sought to find appropriate two-step estimation procedure in order to circumvent the computational difficulty (Greene, 1998; Maddala,
1986). Nevertheless recent new findings about LIML along with the advances in computing
technology have reinvigorated researchers’ interest on those single-step estimators. Since
the controversial Angrist and Kruger (1991), we are now more familiar to the properties or
advantages of LIML estimators especially when there are many weak instrumental variables
(Bekker, 1994; Staiger-Stock, 1997; Flores-Lagunes, 2007). Extending our attention to the

1

broad class of single-step estimators to which LIML belongs, we see that there are some
more advantages of single-step estimators other than in weak many IV’s: First, it reduces
the sampling error by skipping the first stage. Second, as in bivariate probit models, there
are some cases where there is no proper consistent two-step method (Wooldridge, 2010);
single-step estimator usually exists under some appropriate distributional assumptions.
One of the leading examples of single-step estimator is LIML in linear simultaneous
models. In LIML, after reduced form transformation, both the structural equation and linear
projection of endogenous explanatory variable are jointly estimated by maximum likelihood
method . Although 2SLS and LIML give different estimates, they can be interpreted as
two different opeartional version of same control function (CF) method; LIML is identical
to the single-step CF method, while 2SLS is to the two-step CF in linear models. If CF
method is available for a particular nonlinear model, then each single-step and two-step CF
estimating procedure can be understood as a generalized LIML and 2SLS respectively in
such nonlinear models (Wooldridge, 2007). In line with this idea, LIML can be viewed as
a joint ML estimation method based on the CF approach; such approach provides greater
generality since LIML can then be applied to the nonlinear models as well as the usual linear
simultaneous equations models.
In linear simultaneous models LIML and FIML are mechnically the same; the only difference is that FIML is a term used when the whole equations in the system are under
consideration, whereas LIML is used for a single equation in the system (Hayashi, 2000).
In the context of endogenous switching regression, FIML indicates an estimation method
through direct modeling of endogenous variables without resorting to control functions (Maddala, 1986). If the selection equation is viewed as a structural equation, such method can be
called FIML because the absence of selection variable in the other structural equations makes
2

the reduced form transformation unnecessary. Therefore there are essentially two kinds of
single-step estimators at hand, which are the ones with and without control function in the
estimating equation. In this article, the former will be called LIML and the later FIML.
Even though LIML was fully robust in linear models, one of the shortcomings of the
single-step estimators in nonlinear models is that they usually require strong distributional
assumptions for constructing the likelihood functions. They can be inefficient or even inconsistent unless the distributions are correctly specified. In order to address this problem,
one can also consider some distribution-free methods such as nonparametric or semiparametric approaches. The objective of this article is to propose those various estimators and
then to investigate their performances for the estimation of average treatment effect (ATE)
under counterfactual settings. One of the most common situations requiring two-step estimation might be the case where an explanatory variable is endogenous. In such a situation,
a nonlinearity can easily be brought by assuming a binary endogenous explanatory variable.
Therefore the discussion starts from estimating ATE in endogenous switching regression
models with linear structural equations.1 As it will be clearer later, the partial effects of
binary endogenous variables is a special case of ATE under some parameter restrictions.
Although Angrist (2001) argues that the linear model is sufficiently fine for the partial effect
estimation, this article attempts to justify the use of nonlinear models, which will make the
whole argument more meaningful.
Section 2 describes the model under consideration. Here the relation between the partial
effects of binary endogenous variable and the ATE under counterfactual setting will be
clarified. Section 3 will provide a detailed description of those four estimators i.e. two1 The case of nonlinear structural model is the topic of the second chapter of this disser-

tation.

3

step Heckit, single-step (Q)LIML, single-step (Q)FIML and Series estimators. Section 4
derives the asymptotic distributions of two-step and single-step estimators with the same
CF estimating equation. As it turned out, the asymptotic variance of one estimator is not
always greater or smaller than that of the other. Threfore the asymptotic analysis does not
provide any guideline as to which estimator is more efficient. Section 5 briefly describes the
data generating process used for the simulation and Section 6 discusses the results. The
focus of the analysis will be on the comparison between single-step and two-step estimators
mainly in terms of consistency and efficiency.

1.2

Model

As it was briefly discussed in introduction, the main objective of this paper is to compare the
estimators for the partial effects of binary variables or ATEs depending on the model specifications. Suppose that we are given a model with a binary endogenous variable. Although the
partial effect can easily be estimated given a set of proper instrumental variables, the main
focus here is how to use the fact that the underlying endogenous variable is binary. Although
it is perfectly legitimate to use usual IV procedure without paying too much attention to the
binary nature of the variable, I attempt to make use of such information in order to make
a better estimation. Therefore the discussion starts from the generic problem of estimating
partial effect of an endogenous explanatory variable. As it will be clarified below, this issue
can obtain generality by extending it to the topic of ATE. Below it will be shown that they
use common estimating equation although there are slight differences between the ATE and
the partial effects of binary endogenous explanatory variable. To state more properly it will
be shown that estimating the partial effects of binary variables is a special case of estimating

4

the ATE.
Consider a model as below.

y = µ + τ w + βx + u
(1.1)
w = zδ + ξ,
In the above equations, w is the binary endogenous explanatory variable where τ is the
partial effect, x is exogenous covariates, and z is a set of all exogenous variables containing
x which are uncorrelated with u. In order for z to be proper IV, it must be δ ̸= 0. There
is absolutely no problem in estimating the τ by the usual IV procedure. If the endogeneity
of w is interpreted as an existence of unobserved variables c included in the error, then the
partial effect τ can be written as E(y|x, c, w = 1) − E(y|x, c, w = 0).
Now we want to estimate better by somehow using the additional information, i.e. the
binary nature of w. One natural possibility is to use any binary choice model for the reduced
form equation instead of linear probability model. A clear account of ATE is warranted at
this point before a further discussion. Suppose a counterfactual setting with a binary choice
variable; there exist two regimes for each unit of observation. Let the response variable only
in regime one is observed under w = 1 while that in regime zero is observed under w = 0.
The exogenous variables x are always observed. The value of w is affected by the values of
response variables for each regime, which creates correlation between response variables in
each regime and w. Thus the binary variable w gets an endogeneity, and the model under
such environment is sometimes called endogenous switching regression. A classic example
of such model is found in job training program analysis. In such model the binary variable
w denotes whether an individual takes the program or not, and the response variables y1
and y0 denote the wages with and without participation in the program (for recent survey
5

on program evaluation, see Imbens and Wooldridge (2009)). It is not hard to imagine
those two regimes for each particular individual although those two response variables are
not observed in the real world at the same time; in line with such an idea the model is
called counterfactual (Rubin, 1974). For a particular individual the difference between y1
and y0 is called the treatment effect. The expectation of the treatment effects over the
whole population is called the average treatment effect, i.e. E(y1 − y0 ). Thus it is obvious
that this ATE is not generally same as the partial effect of the binary treatment variable
E(y|x, c, w = 1) − E(y|x, c, w = 0).
An estimable equation for estimating the ATE is derived below. Since there are two
regimes,
y0 = µ0 + xβ0 + u0

(1.2)

y1 = µ1 + xβ1 + u1 .
In the above equations, the relation between x and y are assumed to be linear. For now let
us use another assumption that β0 = β1 and u0 = u1 , which implies that the treatment has
an intercept shifting effect only. In other words, the treatment shifts the response variable
by exactly same magnitude for all the individuals in population. In addition to that let
E(x) = 0. This is without loss of generality because the intercept can be adjusted so that
E(x) = 0 be true. By those manipulations, it is easy to see that the ATE can be expressed
as
E(y1 − y0 ) = µ1 − µ0
In equations (1.2) y1 is denoted as the response variable when w = 1 while y0 is when w = 0.
In fact the observable response variable is y0 when w = 0 and so on. Let us denote the

6

observed response variable as y, then

y = y0 + (y1 − y0 )w

Therefore using the above equation, the equations (1.2) can be incorporated into a single
equation.
y = µ0 + (µ1 − µ0 )w + βx + u

(1.3)

Thus the partial effect of w becomes the ATE under the restriction above. Notice that the
equation (1.3) is same as the equation (1.1). Remembering that the restrictions β0 = β1
and u0 = u1 were put in equation (1.2) to obtain equation (1.3), we can easily see that the
equation (1.1) is just a special case of those models with counterfactual settings. Although
we started from a special case of binary endogenous explanatory variable in the beginning,
it will be discussed under the broad settings of ATEs for the rest of this article.
Returning to the equation (1.2), let us consider how to identify the ATE in a model
without any restrictions by using the fact that the treatment is binary. To do that there has
to be at least more than one IV for w. Rewriting the equation (1.2) with the reduced form
equation,
y0 = µ0 + xβ0 + u0
y1 = µ1 + xβ1 + u1

(1.4)

w = 1[zδ + ξ > 0]
It can be easily seen that once we drop one of the regimes in the above equation, then it
simply becomes the well-known Heckman correction method (Heckman, 1976). Therefore we
can use the Heckman correction model twice for each regime in order to consistently estimate
the coefficients in each regime. Of course all the assumptions for Heckman correction model
7

are also required in this model: z is mean independent of ug where g = 0, 1, E(ug |ξ) is linear
in ξ, and ,although not necessary for the identification itself, w follows probit model so that
the inverse Mill’s ratio can be used. Therefore the estimating equation can be written as

y = µ0 + (µ1 − µ0 )w + xβ0 + wx(β1 − β0 ) + u0 + (u1 − u0 )w
= µ0 + τ w + xβ0 + wx(β1 − β0 )
+ ρ1 wλ(zδ) + ρ0 (1 − w)λ(−zδ) + e0 + (e1 − e0 )w,

(1.5)

where λ(·) is inverse Mill’s ratio, and τ is denoted as the ATE. Estimating procedure is
basically same as Heckman correction model; in the first stage estimate the selection equation
by probit and put the estimated parameters δ for δ and run the second stage regression just
like usual ordinary least squares. If one believes that β0 = β1 and u0 = u1 , then the
estimating equation simply becomes
(
)
y = µ0 + τ w + xβ0 + ρ wλ(zδ) − (1 − w)λ(−zδ) + e0 .

(1.6)

Running two-stage regression of the above equation (1.6) is an alternative estimation method
to the usual IV regression with linear probability model for selection. Nevertheless, following
discussions will be based on the equation (1.5) with greater generality rather than (1.6) .

8

1.3
1.3.1

Estimation
Two-step Heckit and Single-Step Quasi-LIML

So far we have seen the relationship between partial effect of a dummy endogenous variable
and ATE in the previous section. In what follows the equation (1.5) will be used as a basic
estimating equation for most part of the discussion.
In equation (1.5) the errors eg are the difference between the structural errors ug and
the correction terms for each regime. We needed three important assumptions to be able to
write the model as in equation (1.5). The first two are the conditional mean independence
of z and ug and the linear conditional expectation assumption between the errors in each
equation i.e. E(ug |z, ξ) = E(ug |ξ) = ρξ. A sufficient condition for linearity is the trivariate
normal distribution between ug and ξ; under such assumption the coefficient ρg will then
be the covariance between ug and ξ. However, the linear conditional expectation will be
enough for writing the equation (1.5). Here the distribution of ug can be flexible; it doesn’t
have to follow any particular distribution as far as it is continuous. The next assumption
is that ξ follows normal distribution. This is more important than the former ones since it
warrants the use of inverse Mill’s ratio in the structural equation as well as the use of probit
for the first stage estimation. Without it, the above representation is invalid particularly for
the inverse Mill’s ratio, for which case we can possibly consider semiparametric estimation
methods instead of using the inverse Mill’s ratio for correction terms. And again the first
stage estimation can also be run by semiparametric binary choice model, which will be clarified later. The following three propositions summarize the above discussion.

9

Proposition 1.1. Given equation (1.4), it can be shown that

1
E(y1 |z, w = 1) = µ1 + xβ1 +
Pr(w = 1)

∫ ∫ ∞
−zγ

u1 p(u1 , ξ|z)dξdu

Proposition 1.2. Given equation (1.4), if E(u1 |z, ξ) = E(u1 |ξ) = ρξ, then

E(y1 |z, w = 1) = µ1 + xβ1 + ρ

∫ ∞
−zγ

ξp(ξ|ξ > −zγ)dξ

Corollary 1.1. If ξ ∼ N (0, 1) in addition to the assumption in proposition 1.2, then

E(y1 |z, w = 1) = µ1 + xβ1 + ρλ(zγ),

where λ(·) is the inverse Mill’s ratio. Moreover, if u1 and ξ follow bivariate normal, then
ρ = cov(u1 , ξ)/var(ξ).
One can use the two-step Heckit approach discussed in the previous section in order
to estimate ATE using equation (1.5). Easy to implement as it is, it could be inefficient
after accounting for the first stage sampling error. If we are willing to make some further
assumptions, then there actually exists a single-step ML estimation method using the likelihood function for eg , which can be constructed as below. Let us call the composite error as
e = e0 + (e1 − e0 )w and assume that it follows standard normal distribution and that the
selection mechanism is probit. Then a joint density function of y and w can be written as

10

below.

p(y, w|z) = p(y|w, z) · p(w|z)
= √

1

2π · var(e|w, z)
)
(
{y − µ0 − τ · w − xβ0 − wx(β1 − β0 ) − ρ1 wλ(zδ) + ρ2 (1 − w)λ(−zδ)}2
· exp −
2 · var(e|w, z)
·[Φ(zδ)]w · [1 − Φ(zδ)]1−w ,

(1.7)

where inverse Mill’s ratio is compactly denoted as λ(·). The conditional variance of each eg
is given by Johnson and Kotz (1972) as

2
2
var(e0 |w = 0, z) = σ0 − σ0ξ λ(−zδ){−zδ + λ(−zδ)}
2
2
var(e1 |w = 1, z) = σ1 + σ1ξ λ(zδ){−zδ + λ(zδ)},

2
where σg ≡ var(ug ) and σgξ ≡ cov(ug , ξ). All the parameters are estimable by a ML

estimation with equation (1.7). One of the advantages of this approach is that there is no
more sampling error to account for in the first stage estimation, which greatly simplifies
the computation of the standard errors. Let us now consider the normality assumption of
e which is one of two assumptions used in equation (1.7). In fact our knowledge about e is
in fact very limited, although the normality assumption on e was used in order to construct
a likelihood function. The composite error e was obtained by subtracting the Heckman
correction terms from u. Although we can maintain that u follows normal by the usual
central limit theorem argument, it is very hard to maintain that it still follows normal even
after it is purged of the elements causing the endogeneity. Other than that, one can surely
suspect the possibility of heteroscedasticity because the correction terms are the functions
11

of explanatory variables and again e also is.
However, even for the case where the likelihood is not correctly specified, we can expect
to have at least a consistent estimator as long as the conditional expectation of y is correctly
specified (Gourieroux et al., 1984). Without caring about the conditional variances, let us
just use a standard normal distribution for p(y|w, z). Taking natural log, equation (1.7) can
be written as

ℓ(y, w|z)
= −

{y − µ0 − τ · w − xβ0 − wx(β1 − β0 ) − ρ1 wλ(zδ) + ρ2 (1 − w)λ(−zδ)}2
2

+w · ln[Φ(zδ)] + (1 − w) · ln[1 − Φ(zδ)]

(1.8)

Now suppose that the assumptions in Proposition 1.2. and Corollary 1.1. are all satisfied.
Then it can be seen that the true parameter values maximize the expectation of log-likelihood
function in equation (1.8). If we label the true parameter value by “o” subscript, then δo will
surely maximize the expectation of probit log-likelihood function. Also under the assumption
that E(y|w, z) = µ0o −τo ·w−xβ0o −wx(β1o −β0o )−ρ1o wλ(zδo )+ρ2o (1−w)λ(−zδo ), the true
parameters (µ0o , τo , βgo , ρgo , δo ) will maximize the expectation of the first term in (1.8). The
latter statement is true because the normal density belongs to the linear exponential family.
Since (µ0o , τo , βgo , ρgo , δo ) maximize the first term in (1.8) and δo does the same for the
second and third terms, the whole set of parameters maximize Eℓ(y, w|z), which guarantees
the consistency. The above argument makes it clear that consistency depends on the fact
that the selection error is normally distributed, since it enables us to write the correction
terms as the inverse Mill’s ratio, which is a part of conditional expectation function of y. In
other words, although e does not have to follow standard normal distribution, the selection
12

Table 1.1: Relationship among Estimators

single-step CF
two-step CF

linear
LIML
2SLS

nonlinear
(Q)LIML
Heckit

error has to be normally distributed in order to ensure consistency.
A general form of correction terms without the normality but with linearity assumption on
ξ is shown in the proposition 1.2. Also if the conditional expectation E(ug |ξ) is a nonlinear
function of ξ containing terms of higher degrees, then some additional correction terms
might be needed. As it will be discussed in the following section, one can consider using
nonparametric or semiparametric methods in such cases where it is hard to determine the
validity of those assumptions.
The estimation method using the above log-likelihood function (1.8) will be called quasiLIML or QLIML. A qualifier quasi- is used here due to the ignorance of the correct distribution of e. It is LIML since the likelihood function is constructed by the joint density function
of the response and endogenous explanatory variables as in the linear simultaneous equation
case. The only difference between the QLIML used here and the one in linear model is
that the former uses joint density of normal and Bernouilli while the later uses multivariate
normal.

1.3.2

Series Estimator

According to proposition 1.2 and corollary 1.1, the correction term will take the form of
inverse Mill’s ratio when E(ug |ξ) is linear in ξ and ξ is normally distributed. If ξ is not,
the correction term will not take the form of inverse Mill’s ratio any more. If E(ug |ξ) is a

13

nonlinear function of ξ, then the correction term will have some more additional terms as
well as the leading inverse Mill’s ratio. Although it is still possible to derive the appropriate
formulae for correction terms for each distributional assumption, we usually don’t have
any knowledge about the population whenever the failure of the assumptions is suspected;
in such cases one can instead use some distribution-free methods. Rather than the fully
parametric approach, a semiparametric method can also be applied for the correction terms.
Here I will follow the methods proposed by Powell, Newey and Walker (1990), Newey (1994)
and Newey (2009). The discussions in those papers are mainly about the semiparametric
estimation of sample selection models where the β’s in equation (1.2) are identified. They
are semiparametric because the linear index zγ is always used as an argument of control
function.
The estimation will be carried out by two steps. In the first stage, the selection equation
is estimated by the nonparametric binary choice models as in Powell, Stock, and Stoker
(1989), Ichimura (1993) and Klein and Spady (1993). I use Klein-Spady estimator in this
paper since it is the most efficient one among those. Given a first stage estimate, a series
estimator can be constructed in the second stage as below.

y = µ0 +τ ·w+xβ0 +w·x(β1 −β0 )+w

P
(∑

)

ρp ψ(zδ)p +(1−w)

p=1

P
(∑

)
ηp ψ(−zδ)p +ϵ0 +w·(ϵ1 −ϵ0 )

p=1

(1.9)
The errors appeared here are not the same as eg in Heckit or in QLIML. They are essentially
the sums of the series terms from P +1 to infinity. The function ψ is monotone transformation
which makes the estimate less sensitive to outliers (Newey, 1994). For ψ, Newey (2009)
proposes three monotone functions: identity, inverse Mill’s ratio and CDF of standard normal
distribution.
14

In Heckman correction method the correction term in observed subpopulation is the
inverse Mill’s ratio λ(zγ) under the usual assumptions, whereas that in the other unobserved
subpopulation is −λ(−zγ). Then the unconditional expectation of correction term over
the whole population is equal to zero; otherwise the unconditional expectation of error in
structural equation won’t be zero causing the intercept estimator inconsistent. This condition
is not automatically satisfied as in the original Heckman correction model when we apply
series estimator with powers of degree more than two and with a function ψ(·) other than the
inverse Mill’s ratio. Therefore an adjustment that makes the expectation of the correction
terms over the whole population be zero is required for a consistent estimation of the intercept
and ATE; it can be done by subtracting the sample means of the correction terms.

1.3.3

Quasi-FIML

Before considering this estimator, it should be emphasized that this is not in the class of
control function approaches as those previous three estimators. The basic motivation of using control function approach is that we suspect unobserved variables hidden in error which
are correlated with one of the regressors in the structural equation. Since it is unobserved,
by definition, the distribution that it follows is unknown. Indeed we haven’t put any distributional assumptions for ug or eg in the two-step Heckman model. Even though we used
normal assumption for eg in QLIML, we do not claim that it is truly normal by labeling it
as QLIML. Therefore putting distributional assumption directly to the structural error ug
is not in line with our spirit of viewing this problem. However, if the normality assumption
happens to be true, then a FIML estimator that exploits the joint density function becomes
a very attractive alternative estimator with correctly specified likelihood. If it is not correctly specified, then the FIML with wrong distributional assumption on the errors becomes
15

a quasi-FIML or QFIML that can still be used for estimating the ATE. Below is a description
of FIML: suppose the joint distribution of ug and ξ is multivariate normal as below.












 

2
 0   σ0

u0 

  

  
u1  ∼ N  0  , 

  

  
ξ
0


σ10 σ1ξ 


2 σ
σ1
0ξ 


1

The variance of the selection error is set to be one since the coefficients in probit model can
only be estimated up to a scale factor. Under this assumption, we want to derive the joint
distribution of y and w conditional on the exogenous variables z. We can easily transform u
into y since it is linear. The transformation of ξ into w can be done by the equation below.
(∫ ∞
f (u, w|z) =

−zδ

)1−w
)w ( ∫ −zδ
h(u0 , ξ) dξ
g(u1 , ξ) dξ
·
−∞

(1.10)

Those functions g and h are marginal density functions for ug and ξ conditional on z. For
example g can be obtained by integrating out irrelevant u0 from the joint distribution for all
three variables. Here we can see that the trivariate normal assumption is sufficient condition
for our purpose; the above likelihood function shows that the pairwise bivariate distribution
assumption for each ug and ξ is all that is necessary since we are hardly interested in
estimating σ01 . The integrals on the right hand side of the above equation were needed
in order to get the marginal distributions for ug for the event where w = 1 and w = 0
respectively. It can be shown that the above density function can be written as
[ (
f (u, w|z) = Φ

2
zδ + (σ1a /σ1 )u1
√
1 − (σ1a /σ1 )2

)

( ) ] [{
} ( u0 ) ]1−w
u
(
2 )u )
ϕ σ1 w
ϕ σ
zδ + (σ1a /σ1 0
1
0
·
· 1−Φ √
·
2
σ1
σ0
1 − (σ1a /σ1 )
(1.11)

16

Table 1.2: Distributional assumptions

u
ξ
e

QFIML
normal
normal
NA

QLIML
none
normal
none

Heckit
none
normal
none

Series
none
none
NA

where Φ(·) and ϕ(·) denote the CDF and PDF of standard normal distribution respectively.
By using (1.2) the above joint density can easily be transformed to f (y, w|z). The estimator
using the likelihood function given above is called FIML (Maddala, 1986). In linear simultaneous equations models FIML basically uses the joint distribution of all endogenous variables
conditional on exogenous ones; the only difference in this nonlinear setting is that the joint
distribution of endogenous variables are constructed without reduced form transformation.
Since there is no guarantee that the errors follow trivariate normal, it should be properly
called “quasi”-FIML with a qualifier. The ATE is estimated simply as the difference of estimated intercepts in equation (1.2). If the true distribution is trivariate normal, then this
estimator becomes FIML and will be consistent and efficient. However, if that is not the
case, then it is neither consistent nor efficient. Therefore FIML is less robust than Heckit or
QLIML under various violations of distributional assumptions.
In sum there are the four available estimators for ATE using instrumental variables
in counterfactual setting. Those estimators were constructed under different distributional
assumptions which are summarized below. NA indicates the underlying error term is not in
the model. Even though not in the table, it has to be noted that there is linear conditional
expectation condition between u and ξ for the Heckit and QLIML.

17

1.4

Asymptotic Variances

In this section the asymptotic distributions of Heckit and QLIML, which are applications
of two-step and single-step CF method respectively, are given. The asymptotic distribution
of QFIML is straightforward and that of the series estimator in the following discussion
is derived by using essentially same method as in the two-step Heckit. In what follows a
common structural log-likelihood function for both single-step and two-step methods will
be used. The standard normal distribution will be used as the quasi-likelihood of structural
equation for QLIML estimation. Heckit uses OLS at second step; computing the least square
is equivalent to using the standard normal distribution for e. Therefore one can simply use a
common structural log-likelihood function to derive the asymptotic distribution of the twostep Heckit as well as single-step QLIML. To make the discussion as simple as possible, the
parameters θ1 and θ2 are assumed to be scalars.
Let us first consider the asymptotic distribution of QLIML. Let q 1 (y|w, z; θ1 , θ2 ) be
the log-likelihood or objective function to be considered for the structural equation and
q 2 (w|z; θ2 ) for the reduced form equation. In this particular endogenous switching regression
setup, θ1 = (τ, β ′ )′ and θ2 = δ, but again the discussion below assumes that they are scalar
for the sake of simplicity. Since the two sets of parameters are estimated together, the loglikelihood function for QLIML is expressed in additive form as q 1 (y|w, z; θ1 , θ2 )+q 2 (w|z; θ2 ).

18

i
i
Let qj ≡ ∂q i /∂θj and qjk ≡ ∂ 2 q i /∂θj ∂θk . Then





1 2
 E(q1 )

1 1
E(q1 q2 )

B0 = 




1 1
1
2
E(q1 q2 ) E(q2 )2 + E(q2 )2


1 )
1 )
E(q12

 E(q11
A0 = 

1
1
2
E(q21 ) E(q22 ) + E(q22 )

and

√
n(θn − θ0 ) →d (0, A−1 B0 A−1 ). The asymptotic distribution of θ1 is the first diagonal
0
0

element of the sandwich form as below.
√
Avar n(θ1n − θ10 )

(
)
1
1
1 1
1
1
2
E(q1 )2 K 2 − E(q12 ) 2E(q1 q2 )K − E(q12 )[E(q2 )2 + E(q2 )2 ]
,
=
)2
(
1 )K − [E(q 1 )]2
E(q11
12

1
2
where K = E(q22 ) + E(q22 ) and θ1n denotes QLIML estimator of θ1 .

A two-step approach maximizes q 1 (y|w, z; θ1 , θ2 ) given the first stage estimates θ2 . Then it
can be shown by using the result in Wooldridge (2010, Chapter 12) that
√
Avar n(θ1n − θ10 ) =

(
1 2
2 2 2)
1
1 )2 + [E(q12 )] [E(q2 ) ]
E(q1
1
2
[E(q11 )]2
[E(q22 )]2

where θ1n is the second stage estimate of θ10 . The only case where those two asymptotic
1
variances are same is when E(q12 ) = 0, where one can estimate equation by equation with-

out any interaction between θ1 and θ2 possibly through some control functions. Since our
1
model generally contains the interaction terms, the presence of nonzero E(q12 ) should be

understood as a main cause differentiating the asymptotic variances of single-step and two-

19

√
√
step methods. The magnitudes of Avar n(θ1n − θ10 ) and Avar n(θ1n − θ10 ) cannot be
1
determined in general. However, holding other terms fixed, higher value of E(q12 ) makes the

asymptotic variance of single-step estimator decrease since its denominator and numerator
1
are a polynomial of degree four and two respectively. Thus under a higher value of E(q12 ),

the denominator would dominate the numerator yielding smaller asymptotic variance. On
the other hand the asymptotic variance of two-step estimator would increase smoothly for
1
1
higher value of E(q12 ). However, if E(q12 ) is close to zero, then the fraction can possibly
1
be subject to an abrupt change even by a small change in E(q12 ) making comparison more

difficult.

1.5

Simulation Design

Monte Carlo simulations were carried out in order to compare the finite sample performances
of the four estimators under consideration. The data generating process is

x ∼ uniform[−10, 10]
y1 = 3 + 0.1x + u0
y0 = 2 + 0.15x + u1
w = 1[1.4 − 0.05x − 0.3z + ξ ≥ 0]
z ∼ Binomial(1, 1/2),
where ug and ξ can follow normal, centered t(5) and centered χ2 (5) with variances all
normalized as unity.2 Their correlations are set 0.0, 0.4, 0.5 and 0.6. The ATE, the difference
2 First and second moments alone do not uniquely determine the joint distribution of

nonnormal random variables. The stata code for this simulation will be provided upon
request.

20

of the intercepts for each regime, is equal to one. To see the effect of weak instrumental
variable, a set of 10 randomly created binary variables was used. Because the nonlinear
selection function reduces the information in the domain through the indicator function, the
effect of weak IV is not very perceivable unless a large number of weak IVs is put in the
equation. To measure the quality of IVs as a whole, the concentration parameter is used,
i.e. E(δ ′ Q′ Qδ)/L for y ∗ = 1.4 − .05x − .3z + ξ where the i-th row of Q is (1, xi , zi ) and
L is the number of instrumental variables. Then for the observation number n = 100, the
concentration parameter for y ∗ is 10.575 when a relevant IV is used, and 1.0575 when ten
irrelevant IVs were added.
Vella and Verbeek (1999) already compared the performances of IV and CF under various
error distributions, but what they essentially compared was the two-step CF methods with
and without using the binary nature of treatment variable in the present context. And they
just investigated what they called restricted CF which is the single-regime model such as
equation (1.6). The contribution of this paper is that (1) both two-regime and one-regime
models are considered, (2) both single-step and two-step estimation methods are investigated,
and (3) the misspecifications of both selection and structural errors are allowed for.

1.6

Simulation Results

As it was mentioned in the previous sections, we focus on those four estimators, i.e. Heckit,
QLIML, Series and QFIML, the estimable equations of which are equations (1.5), (1.8),
(1.9) ad (1.11). In addition to those two-regime estimations, we also discuss one-regime
procedures i.e. IV estimator and Heckit with single regime, which are (1.1) and (1.6). In
order to distinguish Heckits in those two settings, it will be referred to as Heckit1 and Heckit2

21

for regime one and two respectively. Heckit1 is essentially same as Heckit2 except for the
fact that the restriction that the partial effects of the covariates across two regimes are equal
was used for Heckit1. Also the IV estimator using equation (1.1) will be called linear IV
estimation or LIV lest one get confused with instrumental variable. Thus the difference
between LIV and Heckit1 is that the selection equation for LIV is the linear probability
model while for Heckit1 is probit. Of course we can think of the optimal IV estimation using
probit, but it will not be considered here. LIV is two-step estimation method; when the
same model is estimated by single-step LIML, then it will be written as LLIML in order to
distinguish that from QLIML above. Thus the focus is on the comparison between linear
and nonlinear control function methods i.e. LIV and Heckit1 for one-regime estimation. For
two-regime estimation, the focus is on the comparisons among Heckit1, QLIML, Series, and
QFIML. We discuss strong IV results from section 6.1 through 6.4. Sections 6.5 and 6.6 are
devoted for weak IV.

1.6.1

Comparison between LIV and Heckit1

To begin with, let us investigate the behavior of LIV and Heckit1 with strong IV under various
misspecifications. Although Vella and Verbeek (1999) dealt with this issue, the discussion
here is more extensive. The results are summarized in Table B.1 through Table B.4. For
each simulation session, the seven summary statistics are provided: Monte Carlo mean,
standard deviation, root mean squared error (RMSE), and median. The simulation was run
for different correlation values such as 0.0, 0.4, 0.5 and 0.6, which makes it easy to find out
patterns of behavior of estimators if any. Since there are three possible error specifications
for both structural and selection errors, we have nine combinations to consider. However, it
turned out that the cases of heavier tails generated by t(5) distribution is very similar to the
22

normal cases and their results are omitted in the tables3 . The horizontal rows are for three
possible selection error specifications and the vertical columns are for structural error.
Let us first compare LIV and Heckit1 in terms of bias. One can see that those one-regime
models are not particularly bad despite the true data generating process has two regimes. The
two estimators also converge well to the true parameter value, which is 1, under sufficiently
large samples. Heckit1 is valid under three assumptions which were already made clear
in Proposition 1.2 and Corollary 1.1. Although the exogenous variable z is independent
in the simulation design, the other two assumptions are designed to violated except for in
normal-normal case, where Heckit1 is truly legitimate. The conditional expectation will not
be linear unless both of the errors follow same distribution. Particularly if one of them is
skewed while the other one is symmetric, then the conditional expectation can be nonlinear.
Given the independent z, the other two assumptions fail unless the error combination is
normal-normal. If those assumptions are not satisfied, then the control functions in the form
of inverse Mill’s ratio will be misspecified ultimately causing an inconsistency. The results
in Tables B.1-B.4 clearly show this: regardless of correlation, when the selection errors are
skewed, the finite sample biases of Heckit1 are greater than those of LIV. Such large biases
are found only when the selection errors are skewed. Moreover, they are persistent even
when the sample size is sufficiently large. Also it shows that the results under misspecified
structural error are not as sensitive as those under misspecified selection errors. Since large
finite sample biases are common among the cases with skewed selection errors, it appears
that it is not so much caused by nonlinearity of E(u|ξ) as caused by nonnormality of ξ.
Nevertheless the results are also affected by nonlinearity of E(u|ξ). 4 One can also see the
3 The results for t(5) will be provided upon request.
4 Even though I did not include the results in this article, I also created an error generating

process where each of them is normal but the conditional expectation is not linear in order
23

behaviors of estimates under various correlation values; given symmetric selection errors,
as the correlation becomes larger, the relative magnitudes of Heckit1 biases compared to
those of LIV decrease. However, under skewed selection errors, the Heckit1 biases are larger
than those of LIV regardless of correlations. To summarize, LIV has smaller bias under
skewed selection errors while Heckit1 does under symmetric ones. LIV is robust under
misspecification.
For efficiency, Tables B.1-B.4 show that Heckit1 has smaller Monte Carlo standard deviation than the LIV does, which is true irrespective of the error specifications. Also in terms
of RMSE, except for some cases where the selection errors are skewed, the RMSE of Heckit1
is smaller than that of LIV overall. Therefore we have good reason to use nonlinear models
such as Heckit1 unless a nonsymmetric selection error is suspected.
IV estimators do not necessarily have finite moments, which is reflected by the considerable magnitudes of standard errors under small sample sizes. Therefore, as suggested by
Angrist et al. (1999), it would be very useful to see the median or MAD as well as those
moment estimators such as mean and standard deviation. In terms of mean, Heckit1 has a
smaller bias than LIV except for the cases under skewed selection errors. In terms of median,
on the contrary, the tendency is just the opposite: one can see that in most cases the median
biases of LIV are smaller than those of Heckit1, and, particularly, they are for all the cases
under skewed selection errors. Nevertheless one can see Heckit1 is more efficient than LIV
in terms of MAD, which is in agreement with the results for standard deviation.
to see the pure effect of nonlinearity. The results show that nonlinearity of E(u|ξ) causes
larger finite sample bias for Heckit1.

24

1.6.2

Comparison between Heckit2 and QLIML

Let us now compare the performances of Heckit2 and QLIML. Those two estimators use same
estimating equation; the difference is that Heckit2 is estimated by two-step procedure while
QLIML is by single-step. It is already known that in linear models with just identification
2SLS and LIML give numerically same estimates(Anderson et al., 1982). If IV or 2SLS
is understood as a special case of control function method, then the comparison between
Heckit2 and QLIML is essentially a comparison between 2SLS and LIML with nonlinear
selection function. However, as it was already discussed in previous sections, unlike linear
models, when the selection equation is nonlinear, it is impossible to determine in general
whether the asymptotic variance of one estimator is same or greater than the other. Therefore
it is not very meaningful to discuss the efficiency simply by comparing the Monte Carlo
standard deviations of two estimators under some sample sizes; it would be very likely to
see the similar tendency also in finite samples if the asymptotic variance of one estimator is
substantially larger or smaller than the other. Rather it would be more interesting to find
out how the relative efficiencies, i.e. the ratio of MSE’s, behave under different sample sizes.
For consistency of those two estimators, it is essential that the conditional expectation
of y on x and w be correctly specified(Gourieroux et al., 1984). It is basically whether the
conditional expectation on x, w can be correctly specified by the inverse Mill’s ratio once
it is taken for granted that the conditional expectation on x is linear. It is again based
on the assumption that the selection equation is probit. If true model is not probit, then
both Heckit2 and LIML might be inconsistent. Although it can be well expected that the
two estimators might be inconsistent under skewed selection errors, it turned out that they
converge well to the true parameter value even under such misspecifications. Therefore

25

misspecification of selection error does not seem to be very critical.
The simulation results for Heckit2 and QLIML are also in Tables B.1-B.4 and Figure
D.1 displays the cumulative density functions only for the case of ρ = 0.6 and n = 1000.
According to the results, there are some cases where the Monte Carlo means are not converging to the true value monotonically. It can be explained by two ways: First, theoretically,
in the definitions of any kinds of convergence including the convergence in probability, the
behaviors of early terms of sequence are not very important; also the monotonicity is never
considered in weak law of large numbers or in uniform weak law of large numbers. The
second reason is practically more relevant. Since IV estimators do not necessarily have finite
moments, the IV estimators can produce many estimates of large magnitude on which the
numerical values of mean or standard deviation can be very sensitive. Thus it should not be
a major concern even though the sequence of mean is not monotonically converging to the
true value.
Let us now discuss the efficiencies of those two estimators. It can be seen that the Monte
Carlo standard deviation of QLIML is very large compared to Heckit2, which is more so
under small sample sizes. In addition to that, although QLIML is very inefficient under
small sample sizes, it nearly catches up with Heckit2 very quickly. Table C.1 shows the
relative efficiencies of some selected pair of estimators including QLIML with respect to
Heckit2. The relative efficiencies were computed as the MSE of each relevent estimators
divided by the MSE of Heckit1 or Heckit2. Thus if the value is greater than one, then it
implies that the MSE of Heckit1 or Heckit2 is relatively smaller. It can be seen from Table
C.1 that the relative efficiencies are decreasing as the sample sizes increase. In other words,
the single-step QLIML is less efficient particularly under small sample sizes.
On the contrary, the estimated asymptotic variances for actual estimation give mislead26

ing information: a lot of cases the estimated asymptotic variances of single-step QLIML are
smaller than those of two-step Heckit2. Table C.2 tabulates the ratio of estimations with
higher estimated asymptotic variances for two-step Heckit2 out of total 1000 repetitions.5
According to Table C.2, roughly 50-60% of total estimations produce misleading information about the true sampling distribution; overall the single-step QLIML underestimates its
asymptotic variance
In so far as the median bias of Heckit2 and quasi-LIML, it is hard to find any evidence in
favor of either estimator; the magnitudes of their median biases are more or less the same.
Like the standard deviation, MAD also shows that QLIML is more dispersed than Heckit2.

1.6.3

Comparison between Heckit2 and Series estimator.

A series estimator is expected to solve the following three major problems: allowing for nonlinear E(u|ξ), consistent estimation of index under nonprobit settings, and thus correction
of the misspecified control function.
There are various forms of series estimator, but here the three leading ones suggested
by Newey (2009) were used. According to the results presented in Tables A.1-A.12, one
can see that the ones with inverse Mill’s ratio have the smallest MSE. As for the degree of
power series of correction terms, the estimator that includes only the terms of degree one is
the best in terms of MSE; it is because larger number of control function terms interacted
by w creates severer multicollinearity. Therefore the only difference between best Series
estimator and Heckit2 is that the former uses Klein-Spady semiparametric estimator instead
of probit in the first stage. They are similar in that they use inverse Mill’s ratio of degree
5 This simulation was designed to generate same pseudo-random numbers across different

estimators enabling a direct comparison among them.

27

one. According to Proposition 1.2, under the linearity of E(u|ξ), the control or correction
terms can be expressed as some function of index with single term. Corollary 1.1 implies
that normality of ξ lets us to write the correction terms as inverse Mill’s ratios. Without
the linearity of E(u|ξ), the correction terms should be expressed not as a single term, but
as a power series of some functions of index. However, Tables A show that such methods
create multicollinearity due to the interaction with w, implying that the series estimators
are not good alternatives for the cases where E(u|ξ) is nonlinear. Unfortunately, among the
three motivations of using series estimators, the only one that the Newey type estimators
can address is the second one, that is the consistent estimation of index.
The Series estimator is expected to do better under the skewed selection error, but there
is no particular evidence that the Series estimator gives smaller mean or median biases than
Heckit2 does; in summary, the KS semiparametric estimator combined with inverse Mill’s
ratio does not do better than the usual Heckit2 does in terms of bias.
There is a tendency that the MSE of series estimator is greater than that of Heckit2.
However, under small sample sizes, there are some cases where the MSE of series estimator is
smaller than the Heckit2. Such phenomena are more often found as the correlation becomes
higher.

1.6.4

QFIML

Heckit2, LIML and Series estimators are easy to compare with each other since they all
use or based on same estimating equation. On the contrary QFIML is different in that the
structural and selection errors are directly modeled without resorting to control function
method. QFIML uses trivariate normal assumption; therefore if the specification is true
then its asymptotic variance becomes simpler by information equality and QFIML becomes
28

FIML. On the other hand QFIML fails to be a consistent estimator if the error distribution
is misspecified. Tables B.1-B.4 show this point: when the structural error is nonnormal,
then the sequence of Monte Carlo mean is passing through the true parameter value and
converging to somewhere else. Under normal structural error, QFIML converges to the true
value relatively well. In other words although a misspecified selection error does not have
strong effect, a misspecified structural error does cause inconsistency. One interesting thing
is that the inconsistency is not so serious under zero correlation.
For efficiency, QFIML has the smallest variance among those four estimators. Such fact is
also reflected and even exaggerated in the asymptotic variance approximations; the estimated
asymptotic variances are so small that the 95% coverage rates are abnormally smaller than
other estimators. The coverage rate converges to zero as the sample size grows bigger. This
is also true even when the errors are correctly specified. Therefore QFIML estimator is not
a desirable choice for test unless bootstrap standard error is used. To summarize QFIML
fails to be consistent under misspecified structural errors, and has extremely small coverage
rates.

1.6.5

Comparison between LIV and Heckit1 under weak IV

All the results for weak IV are presented in Tables B.5-B.8. The cumulative distribution
functions for the case of ρ = 0.6 with n = 1000 are shown in Figure D.2. From the table, it
can be seen that the behaviors of LIV and Heckit1 are not very different from those under
strong IV except for in terms of bias. Under both strong and weak IV, if the selection error is
skewed, then the linear model has smaller bias. However, under weak IV, even if the selection
error is symmetric, then the nonlinear model does not have any noticeable advantage over
linear ones as it used to have under strong IV. Thus the linear model is more robust under
29

weak IV case.
Now let us discuss the results on RMSE. First, under symmetric selection error, the
nonlinear model still has advantage over the linear model in terms of RMSE. It is due to
the fact that the nonlinear model is more efficient than the linear one in terms of standard
deviation. Second, on the other hand, under skewed selection error, the efficiency of nonlinear
model is not strong enough to make the nonlinear model more advantageous than the linear
one in terms of RMSE.

1.6.6

Comparison between LIV and L-LIML and between Heckit2
and QLIML under weak IV

As it was already discussed, we have LIV and L-LIML for linear models and as their nonlinear
counterpart Heckit2 and QLIML. Unless the correlation is zero, under all circumstances the
median biases of L-LIML are smaller than those of LIV regardless of sample size and error
specification. On the other hand, the relation of the median biases of Heckit2 and QLIML
are not as simple as in linear models. Particularly, when ρ = 0.4, the median biases of
QLIML are greater than those of Heckit2 under all circumstances and it is still true in many
cases when ρ = 0.5. Nevertheless it shows a tendency of smaller QLIML median bias than
Heckit2 as the correlation becomes larger.

1.7

Conclusion

In finite samples, especially when the sample size is small, QLIML is relatively less efficient
than the two-step Heckit2 suggesting that there is no efficiency gain by running single-step
procedure which mimics two-step. QFIML is more efficient than Heckit2; however, it is not
30

robust to distributional misspecification. In terms of efficiency, Heckit2 is a middle ground
between those two single-step estimators. Nevertheless Heckit2 is more preferable since it is
not only easy to compute but also robust to arbitrary misspecifications. Newey-type Series
estimators were expected to do relatively better under misspecification, but turned out not
to be good alternatives.
Under weak many instruments, QLIML shows better performance in terms of median
bias only when the degree of endogeneity is strong. Under weak endogeneity, there is no
clear evidence of such an advantage.

31

Chapter 2
Estimating Average Treatment Effect
by Nonlinear Endogenous Switching
Regression with an Application in
Botswana Fertility

2.1

Introduction

In order to estimate the treatment effects of binary variable on count dependent variable,
Terza (1998, 2009) proposed nonlinear models that take into account the limited dependent
variables. As alternatives to those fully nonlinear models, a traditional linear regression
model with probit treatment equation can also be used. Although it seems to be more
sensible to apply the nonlinear models given count outcome variable, the previous literature
have not clearly stated the advantages as well as disadvantages of using nonlinear outcome
models rather than simply applying linear methods to estimate the treatment effects. While
the linear models implemented by Heckman’s (1978) method is already well understood,
large part of the statistical properties of Terza’s nonlinear approaches are still unknown.
The goal of this study is to explore the properties of nonlinear approaches to estimating the

32

treatment effects and to give a guidance that might be useful to the empirical analyses.
Terza (1998) considers a model where the binary treatment variable shifts the intercept
inside the exponential conditional mean function and provides estimating equations that
can be implemented by using the observable variables. Also in later works, Terza (2008,
2009) extends the earlier model by incorporating the counterfactual framework where the
treatment status puts the individual in a different regime. Following the terminology used in
Terza (2009), the former model will be called throughout this paper “Nonlinear Endogenous
Treatment Model” (NET), and the latter “Nonlinear Full Endogenous Switching Model”
(NFES). As it will be shown in subsequent sections, NFES model is an extended version
of NET in the sense that an appropriate restriction on coefficients along with a fairly weak
assumption readily makes NFES and NET equivalent. While NFES is relatively new, NET
has acquired wide popularity among empirical economists. For the last decade it has been
applied to see the effect of founder CEO as incumbent on the active acquisition activity
(Fahlenbrach, 2009), the effect of credit constraint on floating net aquaculture adoption
in Indonesia (Miyata and Sawada, 2007), the effect of firm’s voluntary pollution reduction
program on pollution (Innes and Sam, 2008; Sam, 2010), the effect of duplicate coverage on
the demand for health care in Germany (Vargas and Elhewaihi, 2007), the effect of illicit
drug use on emergency room utilization (McGeary and French, 2000), the effect of physician
advice on alcohol consumption (Kenkel and Terza, 2001), the effect of insurance on demand
for health care (Ko¸, 2005), the effect of higher education on smocking (Miranda and Bratti,
c
2006), the effect of socio-economic factors on completed fertility (Miranda, 2003), the effect
of Mexican families’ migration in US on woman’s domestic power (Parrado, Flippen and
McQuiston, 2005; Parrado and Flippen, 2005), the effect of health maintenance organization
plans on the health care expenditure in private sector (Shin and Moon, 2007) and the fertility
33

differences between married and cohabiting couples (Zhang and Song, 2007) to name a few.
Since most studies enumerated above use the NET model to measure the effect of binary
variables, the validity of their conclusions may be put into question unless the single regime
restrictions are correct. One important exception is Ko¸ (2005) where he estimates two
c
different structural equations for each value of treatment variable. However, he mainly
focuses on the equation in each regime and not paying full attention to comparing the values
of dependent variables that might lead to ATE analysis. Although the first papers proposing
the ATE estimator based on the NFES model is Terza (2008, 2009), it only proposes the
possibility of such methodology in unifying framework with other nonlinear models without
fully discussing the properties of ATE estimator compared to traditional approaches. This
study will show that the ATE estimators based on NFES model can have higher efficiency
and smaller finite sample biases only under certain circumstances.
The rest of the paper is organized as follows. Section 2 introduces various switching
regression models such as NFES, NET, LFES and LET and discuss how the ATE can be
identified for each model. Section 3 characterizes the asymptotic biases when the methods
being used does not reflect the true population. Section 4 describes the various estimation
methods for NFES model. Section 5 is devoted to Monte Carlo simulation and discusses the
results and implications. In Section 6, the proposed approach is applied to a real data set
to estimate ATE and Section 7 presents the concluding remarks.

2.2

Model

In what follows the term nonlinear is exclusively reserved to describe the nature of dependent
variable of structural equation. In this count dependent variable setting, nonlinear models

34

will use the linear index transformed by exponential function as their conditional expectation
function. On the other hand the linear models will be constructed as if the dependent variable
were continuous.

2.2.1

Nonlinear Models

The “Nonlinear Endogenous Treatment Model” (NET) first proposed by Terza (1998) is as
follow.

E[y|x, w, ϵ] = exp(α + xβ + γw + ϵ)
w = 1[zδ + v > 0],

where x is 1 × K vector of covariates, w is binary treatment variable and ϵ is unobserved
heterogeneity. The vector of covariates x and the vector of exogenous variables z are all
assumed to be independent with the structural and selection errors. Usually x is the subset
of z. The value of treatment variable, i.e. either one or zero, is determined by a binary
choice model such as probit. The treatment equation tells that the value of w is determined
by the exogenous variables z and the selection error v. When their sum is greater than
zero, w is equal to one, and zero otherwise. If w is determined purely randomly as in
randomized experiment, then it will be independent with the unobserved heterogeneity ϵ and
the regression will become very simple and straightforward. However, when w is correlated
with the unobserved heterogeneity, then a usual estimation that does not control for the
correlated error might suffer from an endogeneity problem for the estimation of γ. For
example, when the number of children a woman has at the time of observation is set as a
dependent variable y, it will be determined by her age and marriage status and so on that
35

constitute the covariates x. The dependent variable will also be affected by the education
status w that is either zero or one depending on whether she has education at all. Since
the education status is determined by an individual’s utility maximization, the factor that
affects w might also affect y creating an endogeneity. Terza (1998) suggests an estimating
equation in the form of conditional mean function with a correction term that is conditioned
only on the observables.
The above model, however, is restrictive in that it supposes a constant semi-elasticity of
dependent variable with respect to the treatment across all the individuals in population.
This is related to the fact that the coefficient on covariates and the unobserved heterogeneity
are invariant under different treatment status. The model that extends the above one is
proposed by Ko¸ (2005) and Zhang and Song (2007) as below.
c

E[yg |x, w, ϵ1 , ϵ0 ] = E[yg |x, ϵg ] = exp(αg + xβg + ϵg ),

g = 0, 1

(2.1)

w = 1[zδ + v > 0],

where different coefficients on covariates and unobserved heterogeneity depending on the
treatment status are allowed for. In other words, the treatment status puts an individual in
a different regime; if w = 1, then she is in regime 1 with the outcome y1 and similarly for
the other regime. Presumably each individual has her y0 and y1 for each treatment status
but one of them is not observed. The way to recover the unobserved counterfactual will
be discussed later on for estimation, but for the time being let us focus on the population
model itself. If those two outcome variables are known, then yi1 − yi0 would be an individual
treatment effect. Since it might be different from person to person, we might want to know
the averaged individual treatment effect E(yi1 − yi0 ) that is the so-called Average Treatment
36

Effect (ATE). Incidentally the individual semi-elasticity can be computed by (yi1 − yi0 )/yi0
that might not be constant across individuals either. This is the extended Terza model that
will be called throughout this paper “Nonlinear Full Endogenous Switching Model” (NFES).
The quantity of interest will then be the ATE that captures the causal effect of treatment.
Returning to (2.1), the first equality in the upper equation tells that the conditional
expectations of dependent variables for each regime depend neither on switching variable w
nor on unobservables for other regime. The exclusion of w is particularly important; once
the covariates and the unobservables ϵg are controlled for, the knowledge about realized
regime does not provide any additional information on the conditional expectation of dependent variables. In other words, the equality assumes the ignorability (Rubin, 1978) or
unconfoundedness (Imbens, 2005) of w conditional on covariates and unobservables.
Although the treatment equation in (2.1) is expressed by a binary choice model, it is
also possible to use the linear probability model that is essentially a linear projection of
w on z. However, in the present model, the fact that the endogenous variable is binary
is not neglected so that an appropriate binary choice model is used. The implication is
that it can be viewed as a structural equation1 . The treatment equation that describes the
regime switching mechanism can be modeled by any binary choice model, but here let us
assume that it is governed by probit model for the sake of simplicity. The robustness of
this assumption will also be discussed later. Now let the errors in outcome and treatment
1 In this respect the one step estimator that simultaneously maximizes the outcome as

well as treatment equation is called Full Information Maximum Likelihood estimator.

37

equation be denoted by ϵ and v and follow trivariate normal distribution as below.






 

2
 0   σ0

 ϵ0 


  


  
 ϵ1  ∼ N  0  , 


  


  
v
0


ρ0 σ0 

2 ρ σ 
σ1 1 1 


1

This assumption becomes sufficient condition for each error to follow normal distribution. If
there is no correlation between ϵ and v, then the regime switching becomes entirely random.
Unless the covariances are equal to zero, the regime choice will be determined by each
individual’s own idiosyncrasies that create correlation between w and ϵ. Heckman correction
can be used to solve this endogeneity problem in linear model where the dependent variable
is continuous; the difference between Heckman corrected linear model and current one is that
the latter allows for noncontinuous outcome distribution with exponential CEF while Heckit
presupposes a continuous structural error of which the conditional expectation is expressed
as a linear function of v. Nevertheless the basic situation is more or less the same.
Under the above assumption the ATE can be identified as below (Terza, 2009).
(
)
AT E = E[y1 − y0 ] = E E[y1 |x] − E[y0 |x]
[
]
2
2
= E exp(α1 + σ1 /2 + xβ1 ) − exp(α0 + σ0 /2 + xβ0 )

(2.2)

Thus an estimate can be computed by using the sample analogue method, i.e.,

AT E = N −1

N
∑[

]

2
2
exp(α1 + σ1 /2 + xβ1 ) − exp(α0 + σ0 /2 + xβ0 ) .

i=1
2
As it will be shown in equation (2.4), the composite intercepts αg + σg /2 are identified

38

2
2
without separately identifying αg and σg . The term αg + σg /2 are the estimates of the

composite intercepts. Incidentally, the Average Treatment Effects on the Treated (See p.
906, Wooldridge, 2010) is computed by

AT T =

( N
∑
i=1

wi

)−1 N
∑

[
]
2
2
wi exp(α1 + σ1 /2 + xβ1 ) − exp(α0 + σ0 /2 + xβ0 ) .

i=1

The NFES model discussed so far nests NET model shown in the very beginning of this
section. By putting restrictions β0 = β1 and ϵ0 = ϵ1 the two outcome equations in NFES
can be combined to be written as
(
)
E[y|x, w, ϵ] = exp α0 + (α1 − α0 )w + xβ + ϵ ,

where y = y0 + w(y1 − y0 ). The NET model, although having been claimed as a switching
regression in Terza (1998), does not clearly incorporate the two distinct regimes; the regime
changes according to the value of the binary variable, but switching is expressed only by
shifting the intercept term inside the exponential function. In linear model, it is similar
to the case where the coefficients of covariates for two regimes are identical except for the
intercept. Thus it is recommended to run the NFES model first; it is preferable unless test
rejects the hypothesis of β1 = β0 . In ET model, the parameter of interest is usually the
coefficient on w, i.e. α1 of which interpretation is the semi-elasticity of y with respect to the
treatment variable. This is distinct from ATE that we are in many cases interested; ATE
must be computed as in equation (2.2).

39

2.2.2

Linear Models

Angrist (2001, 2010) and Angrist and Pischke (2009) have pointed out that in many cases
a linear model may be sufficiently good for estimating the marginal effect of a model with
binary dependent variable. Angrist and Pischke (2009) also maintain the validity of such
approach even for the general limited dependent variable models on the grounds that the
linear coefficient can provide the linear projection coefficients that might be very close to
the actual causal effect. In line with that approach, the above endogenous switching model
can be expressed in linear form as below despite the nonlinear nature of count dependent
variables.

yg = µg + xβg + ug ,

g = 0, 1

(2.3)

w = 1[zδ + v > 0]

Let the explanatory variables be demeaned, then the ATE is E[y1 − y0 ] = µ1 − µ0 . We
call this model “Linear Full Endogenous Switching Model” (LFES) as a linear counterpart
of NFES. As NFES model nests the NET, LFES does it for “Linear Endogenous Treatment
Model” (LET) under the restriction that β1 = β0 and u1 = u0 , whereby the coefficient on w
becomes the ATE that is constant across all individuals. The treatment equation is modeled
as probit as usual.
When the true model is such that the outcome variable is nonnegative, the outcome
equations, i.e. the equations of which dependent variable is yg , in LFES model cannot be
viewed as the error form of conditional expectation. Rather it is the linear projection of y
on covariates and therefore E(yg ) = µg since all the covariates are already demeaned. Then
the ATE is the difference between the two intercepts for each regime. The problem is that
40

there is no known identifying strategy of those intercepts when the true model is exponential.
For example, one can try using the Heckman correction method (Heckman, 1978) for the
outcome equation. However, since the minimum condition is that E[u|x, v] = E[u|v] =
ρv (Olsen, 1980) and the model does not satisfy the first equality under the exponential
conditional mean, the LFES estimator with Heckman correction does not identify the true
ATE. Although u and x are orthogonal by linear projection, they are not mean independent
under the exponential conditional mean assumption.
Another alternative linear approach is the 2SLS that does not explicitly model the two
distinct regimes. As it was already shown by Angrist et al. (1996), the coefficient on w in
the regression without any other covariates will identify the Local Average Treatment Effect
(LATE). With covariates, the coefficient on w identifies the weighted averages of LATE for
each covariate cell (Hirano et al., 2000; Mealli et al., 2004). Although LATE will generally
be different from ATE, Angrist and Pischke (2009) support its usefulness on the grounds
that it does not require any distributional assumptions and its estimates end up being very
similar to ATE estimated by nonlinear modeling. In later sections, we will see the validity
of this claim through simulations.
Instead of applying the 2SLS method, one can make use of the fact that the endogenous
treatment variable is binary. Thus the probit model can be used for the first stage. Let
us call this the “Linear Endogenous Treatment (LET)” model and it is obvious that it is a
special case of LFES model with the restriction that β1 = β0 and u1 = u0 in equation (2.3).

41

2.3

Estimation

Various estimation methods for NFES models are presented in this section. Based on the
estimating equations in Terza (1998), the estimation methods for NFES are discussed below.

2.3.1

Full Information Maximum Likelihood

When a binary choice model is used for regime switching, the treatment equation can be
regarded as another structural equation. Specifically we assume that the treatment variable
follows probit model and the outcome variables do Poisson distribution. Let the one-step ML
estimation method under these assumptions be called Full Information Maximum Likelihood
(FIML).
One of the advantages of FIML is that it is the most efficient estimator achieving the
Cram´r-Rao bound with correctly specified likelihood function. Also unlike other estimators
e
that will be discussed below, it estimates all the parameters separately in a single step. (In
other estimators the structural intercept and covariance between ϵ and v are not separately
identified or, even if they are, it requires further steps to do that.) Despite those advantages
it is still computationally burdensome; it might not numerically converge to any meaningful
solution depending on specific data set, and, if ever, it usually takes a lot of time to make
the bootstrapping very awkward. In the case when it needs numerical integration by using
Gauss-Hermite quadrature, the error from approximation can be substantial depending on
the population. The likelihood of FIML estimator is as below. The assumption used is again
that ϵ and v follow trivariate normal.

Proposition 2.1. The joint density function of y and w conditional on the exogenous

42

variables is
[
f (y, w|z) =

]w
∫
√
√
1
2
√
f (y1 |z, w = 1, 2σ1 ζ1 )Φ∗ ( 2σ1 ζ1 ) exp(−ζ1 )dζ1
π R
[
]1−w
∫
(
)
√
√
1
∗ ( 2σ ζ ) exp(−ζ 2 )dζ
· √
,
f (y0 |z, w = 0, 2σ0 ζ0 ) 1 − Φ
0
0 0
0
π R

where

[
2 ]
√
zγ
∗ ( 2σ ζ ) = Φ∗ (ϵ) = F √ + (σga /σg )ϵ
Φ
g g
1 − (σga /σg )2

Proof The basic proof was given in Terza (1998) and its extention to the NFES model is
given in Appendix A.

The function f (·) denotes the used conditional distribution; it is usually either Poisson or
NegBinII. The above equation is a function of ζ that is being integrated out; the integration
will be computed by Gauss-Hermite quadrature method in actual estimation.

2.3.2

Quai-Maximum Likelihood Estimator

Unless the distributional assumption used in FIML are correct, the FIML estimator might
not be consistent; this is a cost of FIML in exchange for efficiency. By the way there
is another method called Quasi-Maximum Likelihood Estimator(QMLE) that trades the
efficiency with robustness by using weaker condition that only the conditional expectation
function (CEF) is correctly specified. As long as the used likelihood is in the class of linear
exponential family, and the CEF is correctly specified, the estimator is consistent even if
the whole likelihood function is not correctly specified (Gourieroux, Monfort and Trognon,
1984). Given the model in equation (2.1), a natural way to estimate might be running QMLE
43

or Nonlinear Least Squares (NLS) by using the E(yg |x, ϵg ). However, it does not give an
estimable equation due to the ignorance of ϵg ; the unobserved variable needs to be removed
by integrating out from the conditioning set of that CEF. By using the fact that ϵ and v are
correlated, one can construct E(y|z, v).
(
)
1 2
2 ) + xβ + ρ σ v
E(yg |z, v) = exp αg + σg (1 − ρg
g
g g
2

Conditional on z, v determines the value of w. Since z, w makes a sparser σ-field than z, v
does, by law of iterated expectation,
(
)
1 2
2 ) + xβ E[exp(ρ σ v)|z, w]
E(yg |z, w) = exp αg + σg (1 − ρg
g
g g
2

Thus E(y|z, w) can be expressed by using only the observable variables z, w. Then the
estimating equation is obtained as
[

]
) Φ(zδ + ρ σ )
2
σ1
1 1
E(y|z, w) = w · exp α1 +
+ xβ1
2
Φ(zδ)
[
]
(
) Φ(−(zδ + ρ σ ))
2
σ0
0 0
+(1 − w) · exp α0 +
+ xβ0
,
2
Φ(−zδ)
(

(2.4)

where the composite intercepts and βg are identified. As was already seen in equation (2.2)
these parameters are sufficient for identifying ATE. A detailed derivation of the above estimating equation can be found in Appendix B. One can run a QML estimation using the
above CEF. A distributional assumption on y is needed as in FIML; the difference is that
FIML models yg to follow certain distribution with E(yg |z, ϵg ) as CEF, whereas QMLE does
it with E(y|z, w). The integration does not appear in Poisson likelihood based on E(y|z, w)

44

because the unobservable was already got rid of and the correction term does that role instead. Both FIML and QMLE relies on correctly specified conditional mean for consistent
estimation of parameters. However, the conditional mean in QMLE, i.e., E(y|z, w), is expressed by all observable variables that makes the QMLE likelihood simpler than FIML. One
can run a QMLE by using a conditional distribution with the mean E(y|z, w). Specifically
two step method can be employed where the first stage probit estimates are substituted in
the correction terms. It does not, however, have to be carried out sequentially by two steps;
they can be estimated by a single step procedure where all the necessary parameters for
ATE are separately identified (Wooldridge, 2011). However, as Hellstr¨m and Nordstr¨m
o
o
(2008) and Chapter 1 have shown, the single step ML method for estimating ATE in linear
endogenous switching model is relatively less efficient in finite sample; it will be examined
in the sequel whether that is still the case in this nonlinear model with count dependent
variable.

2.3.3

Nonlinear Least Squares Estimator

The above QML method is run by using a likelihood in linear exponential family based on
the condition that the conditional mean function is correctly specified. By the way given
the correctly specified conditional mean function it is also possible to use Nonlinear Least
Squares (NLS) method. This NLS can also be viewed as a method of moment estimator. Let
us write the equation in additive form with the correctly specified conditional expectation
as
y = E[y|z, w] + e,

45

where E(e|z, w) = 0 by definition. The true parameter values are such that it minimizes
E[y − E(y|z, w)]2 and the estimation can be performed by sample analogue. Since the
conditional mean contains the correction terms, it should be estimated through the first
stage probit. Under some regularity conditions this approach gives a consistent estimator.
See Wooldridge (2010, Chapter 12) for detailed discussion.
On the other hand this NLS estimator can also be interpreted as method of moment
estimator. From the minimization problem the true parameters satisfy the first order condition E(dE(y|z, w)/dθ × e) = 0 which can be a moment condition with the instrument vector
dE(y|z, w)/dθ. By the fact that E(e|z, w) = 0 along with law of iterated expectation, it can
be easily shown that any function of instruments can also be a good instrument. Therefore we have infinitely many possible instruments that can serve to increase the asymptotic
efficiency of parameter estimators.

2.3.4

Weighted Nonlinear Least Squares Estimator

We have seen above that NLS with correctly specified conditional mean gives consistent
estimator under some regularity conditions. Also it became clear that the NLS estimator
can also be viewed as a method of moment estimator, from which one can construct infinitely many valid instruments and moment conditions. In order to avoid poor finite sample
performances due to overidentifying restrictions (Altonji and Segal, 1996; Ziliak, 1997), one
can resort to the optimal IV approach (See Wooldridge (2010, Chapter 8)). This can be
done in this case by dividing the instrument by conditional variance; the estimator using
E(dE(y|z, w)/dθ × var[y|z, w]−1 × e) = 0 as moment condition is more efficient than NLS
in the previous section.
One can also draw almost same conclusion by using the weighted nonlinear least squares
46

(WNLS) approach, in which both sides of the error form equation is divided by the square
root of an arbitrary function v(z, w, γ) with nuisance parameter γ. Then the first order
condition becomes E(dE(y|z, w)/dθ × v(z, w, γ)−1 × e) = 0. This WNLS estimator is consistent under correctly specified conditional mean and identification condition; see Wooldridge
(2010, p. 411). Note that the difference between moment condition for optimal IV approach
and first order condition in WNLS is whether the original instrument is divided by var[y|z, w]
or by an arbitrary function v(z, w, γ). Under the condition σ 2 v(z, w, γ) = var[y|z, w] for some
constant σ 2 , the generalized information matrix equality (GIME) holds and the WNLS estimator becomes efficient; if σ 2 = 1 then this essentially restates the efficiency result from
optimal IV. What if σ 2 v(z, w, γ) ̸= var[y|z, w]? Then the inference has to be made robust
due to the failure of GIME. Having said that, the consistency result still holds under the
assumption of identification condition and correctly specified conditional mean. Under the
assumption that y|z, w, ϵ follows Poisson distribution, Terza (1998) found correctly specified
conditional variance var[y|z, w]. If should also be noted that if the Poisson assumption fails,
then the conditional variance will be misspecified and robust inference is called for.
Estimation will be carried out by three steps. The correction term is estimated in the
first step and the structural parameters are estimated in the second step from which the conditional variance is estimated. The last third step again estimates the structural parameters
by using the conditional variance estimated in the earlier step. Terza (1998) has proposed
two approaches to estimating the conditional variance. Among those, the regression based
method will be used in this paper since it is computationally easier to implement. The
derivation for NFES model under the assumption that y|z, w, ϵ follows Poisson is given in

47

the Appendix C that turns out to be
(
)
(
)
2
2
var[y|z, w] = wδ1 δ1 (exp(σ1 )L1,2 −L2 )+L1 +(1−w)δ0 δ0 (exp(σ0 )L0,2 −L2 )+L0 , (2.5)
1
0

2
where δg = exp(αg + σg /2 + xβg ), L1,2 = Φ(zδ + 2ρ1 σ1 )/Φ(zδ), L1 = Φ(zδ + ρ1 σ1 )/Φ(zδ),

L0,2 = Φ(−zδ−2ρ0 σ0 )/Φ(−zδ), and L0 = Φ(−zδ−ρ0 σ0 )/Φ(−zδ). Regression based method
2
estimates the σg that will be used to compute the conditional variance for WNLS. If the

Poisson assumption is true, then the GIME is applicable. Otherwise we need robust inference.
In any case, the parameter estimators are consistent under the regularity conditions given
above.

2.4

Asymptotic Distributions

The asymptotic distribution of FIML estimator is straightforward. Given the likelihood function in proposition 2.1, the score and hessian will be constructed as usual. If the multivariate
normal assumption is correct and so is the likelihood function, then the asymptotic variance
will be simplified. The disadvantage of FIML is that the parameters are not consistent any
more when the likelihood function is misspecified.
Now consider the WNLS estimator. The objective function is (y−E[y|z, w])2 /2·var[y|z, w],
where E[y|z, w] and var[y|z, w] are from equation (2.4) and (2.5). Ignoring the first stage error, the asymptotic distribution can be written under the condition var(y|z, w) = v(z, w, γ)
as (See Wooldridge, 1997)
√

(
) )
[E h(z, w, y, θ) ]−1 ,

(
n(θ − θ0 ) −→ N 0,
d

48

where

[

]
∇θ m(z, w, θ)∇θ m(z, w, θ)′
E[h(z, w, y, θ)] = E
,
v(z, w, γ)

(2.6)

and m(z, w, θ) = E[y|z, w].
Now consider the asymptotic distribution of PQMLE. The likelihood function is constructed using the Poisson distribution with the conditional mean in equation (2.4). Then
the asymptotic distribution is
√

(
n(θ − θ0 ) −→ N 0,
d

)
E[h(y|z, w, θ0 )]−1 E[s(y|z, w, θ0 )s(y|z, w, θ0 )′ ]E[h(y|z, w, θ0 )]−1 ,

where
[

∇θ m(z, w, θ)(yi − m(z, w, θ))
qvar(yi )
]
(yi − m(z, w, θ))∇θ m(z, w, θ)′
×
qvar(yi )
[
′]
∇θ m(z, w, θ)∇θ m(z, w, θ)
E[h(y|z, w, θ0 )] = −E
qvar(yi )

E[s(y|z, w, θ0 )s(y|z, w, θ0 )′ ] =

E

The denominator qvar is the variance implied by the used distribution function in QML.
For WNLS the denominator of the expected Hessian was the conditional variance of y,
whereas qvar, that of expected Hessian and score for PQML, is the variance implied from
the distribution used for quasi-likelihood, i.e. the conditional mean for Poisson QMLE. The
asymptotic variance of PQMLE can be simplified under the condition

V ar[y|z, w] = σ 2 · qvar,

(2.7)

This condition says that the true conditional variance is proportional to the variance implied
49

in the quasi-likelihood. Generalized Conditional Information Matrix Equality (GCIME)
holds under this condition that gives
(
√
n(θ − θ0 ) −→ N 0,
d

(
) )
−σ 2 [E h(y|z, w, θ) ]−1 .

By plugging (2.7) in (2.6), it is obvious that the two asymptotic distribution for WNLS
and PQML are equivalent. Having said that, without the condition (2.7), PQML might be
less efficient than the WNLS. Of course this conclusion is true in as much as the first stage
estimation error is ignored.
Now consider our model with the estimating equation as in (2.7). Although the dependent variable yg conditional on x and ϵg follows the Poisson distribution with the
mean E(yg |x, ϵg ) = exp(αg + xβg + ϵg ), it does not necessarily mean that yg conditional
2
on x and w follows Poisson distribution with the mean E(yg |z, w) = exp(αg + σg /2 +

xβg )Φ(f (zδ))/Φ(zδ). To see this point, mean and variance conditional on z, w are

E[y|z, w] = wδ1 L1 + (1 − w)δ0 L0
(
)
(
)
2
2
var[y|z, w] = wδ1 δ1 (exp(σ1 )L1,2 − L2 ) + L1 + (1 − w)δ0 δ0 (exp(σ0 )L0,2 − L2 ) + L0
1
0

It is obvious that they are neither same nor proportional by a constant. Therefore the
condition for GCIME is not satisfied and the PQML is asymptotically less efficient than
the WNLS. Nevertheless, it should also be noted that the first stage estimation error is
ignored in the asymptotic distribution above, that the small sample behavior can be different.
The asymptotic distribution of the estimators for structural parameters that accounts for
first stage error is straightforward with additional terms on the score functions. Then this

50

adjustment make it impossible to use GCIME and creates a sandwich form variance matrices
both for WNLS and PQMLE.
The discussion so far has been about the structural coefficients inside the exponential
function. When our quantity of interest is ATE, which is a nonlinear function of structural
parameters, the asymptotic approximation of the variance matrix can be obtained by delta
method. Recall that the ATE is estimated as in equation (2.2). Terza (2009) derived the
following result.

Proposition 2.2. Along with the regularity conditions for Uniform Weak Law of Large
Numbers and asymptotic normality, suppose that the nonlinear functions g1 (x, θ) and g0 (x, θ)
are continuous and differentiable at θ0 and that their first derivative with respect to θ satisfies all the conditions in Uniform Weak Law of Large Numbers for objective function. Let
the expectation of their derivatives with respect to θ be denoted by G1 and G0 . Then the
asymptotic distribution of ATE estimator in equation (2.2) is
√
N (AT E − AT E) →d N (0, V ),

where
V = E[T ]2 + (G1 − G0 )A−1 B0 A−1 (G1 − G0 )′
0
0
and
g1 (x, θ0 ) ≡ exp(x, β1 ) , g0 (x, θ0 ) ≡ exp(x, β0 )
(
)
T ≡ g1 (x, θ0 ) − g0 (x, θ0 ) − E[g1 (x, θ0 )] − E[g0 (x, θ0 )]
Proof See Terza (2009).
51

In the above, T is the demeaned “ATE conditional on x”, which is a population property
that is not related to particular estimator being used. Incidentally if the structural parameters are estimated by two-step method, the terms B0 and si (θ0 ) can be easily adjusted by
using the result from two-step M-estimator (See Wooldridge, 2010, Chapter 12).
Under this setting if the two exponential terms, i.e. g1 (x, θ0 ) and g1 (x, θ1 ), are substantially different along the values of covariates x, then the variance of conditional ATE will
also be large. Therefore one can have a large variance of ATE estimator to the extent that
the values of β1 and β0 are different. A lesson from this argument is that ATE estimator
by NFES might become less accurate when there are substantial inequality of the treatment
effect from person to person. On the other hand, there also exist the set of parameters with
which the variance of ATE estimator by NFES can be arbitrarily small. The extreme case
is when the conditional mean for each regime are identical, where the variance of T as well
as the covariance term becomes zero.
The above discussion shows that the consistent NFES method can have larger or smaller
variance depending on the coefficient values. As the conditional mean of two regimes are
similar, then one can expect that ATE estimator by NFES might be very efficient, but when
they are very much different, then it might have larger variance.

2.5

Monte Carlo Simulation

As it was already discussed in earlier sections, the NFES and LFES models are thought to
be consistent for ATE. Therefore the purpose of Monte Carlo simulation is to compare the
linear and nonlinear models for estimating ATE. Incidentally other alternative estimators

52

including LET, estimated by 2SLS and Heckit, and NET model that has acquired wide
popularity needs to be examined in terms of consistency.
Specifically the main objectives of simulation study are: (1) to present the possibility that
NFES estimators can be more efficient than LFES under small variance of T , (2) to find out
how severe the bias in one-regime model is compared to that in two-regime model, and (3) to
check how robust the NFES model is to various distributional misspecifications. In addition
to that, it might also be interesting (4) to clarify the advantage of one-step estimations over
two-step ones in nonlinear models, (5) to compare Poisson QMLE and WNLS in terms of
efficiency, and (6) to find out the advantage of FIML estimator.

2.5.1

Data Generating Processes

For each simulation session, the number of replication is 1000, and the sample sizes are 1000,
3000 and 5000. There will be five Data Generating Processes (DGP). For all the DGPs the
following setup will commonly be used.

x ∼ uniform[−5, 5]
z ∼ Binomial(1, 1/2)
w = 1[.15 − 0.05x − 0.3z + v ≥ 0]
E(y1 |x, ϵ1 ) = exp(β10 + β11 x + ϵ1 )
E(y0 |x, ϵ0 ) = exp(0.1 + β01 x + ϵ0 ),

where the errors follow trivariate normal distribution. The treatment equation is designed
so that the numbers of observation for each regime are approximately same. The population

53

ATE is set equal to one. For the errors, the covariance between ϵ1 and ϵ0 was set equal to 0.5
and the variances of all the errors are set equal to one. What is important is the covariances
between ϵg and v, for which 0.4, 0.5 and 0.6 were used. Below are the descriptions of data
generating processes used for each session.
DGP 0 : β10 = 0.564,

β11 = 0.01,

β01 = 0.1,

yg ∼ Poisson(E[yg |x, ϵg ])

ϵg , v ∼ trivariate normal
This is the ideal case for the NFES estimators. Since the data generating process is nonlinear,
the value of population ATE cannot be computed in closed form; rather the claimed true
R
value is computed numerically. Using Stata⃝ , I tried to find the value of the intercept in

regime one such that the difference between one and the numerically computed

∑

(y1 −y0 )/N

is less than 0.001 with a million observation from the above data generating process. The
value of intercept in regime one, i.e. 0.564 was found in this way. The dependent variables
y1 and y0 follow Poisson distribution for given conditional mean. The other DGP’s for
misspecified distribution will be slight modifications of this basic [DGP 0].
There are three distributional assumptions in the above data generating process: 1)
the dependent variable follows Poisson, 2)the unobservables in conditional mean function
follow the normal distribution, and 3)the selection error follows standard normal, i.e. the
treatment is probit. It might not be very surprising even if the NFES estimator performs
well under [DGP 0] on which the model is based. To be fully reliable and useful the nonlinear
models have to show their validity under misspecified distributions also. This is particularly
important because it has been a major source of critics in favor of simpler linear models that
the nonlinear models rely on strong distributional assumption (Angrist and Pischke, 2010).
There have been some papers addressing this issue: Romeu and Vera-Hern´ndez (2005)
a

54

use flexible functional form for the conditional probability of counts by using polynomial
Poisson expansions. Masuhara (2007) models the joint distribution of errors by Hermite
polynomials. Choi and Min (2009) use Johnson’s SU -normal distribution that is shown to
outperform the normal model in terms of consistency. And Deb and Trivedi (2004) use latent
factor structure and simulated likelihood methods to deal with nonnormality. Although those
methods provide very nice alternatives to the models based on normal assumption, it will
be shown below that the nonlinear model with normality still in many cases performs better
than the linear IV methods.
To check the robustness of NFES, I designed some other data generating processes such
that the above distributional assumptions are violated one by one.
DGP 1 : β10 = 0.564,

β11 = 0.01,

β01 = 0.1,

yg ∼ uniform[0, 2E[yg |x, ϵg ]]

ϵg , v ∼ trivariate normal
In [DGP 1], discrete uniform distribution was employed in place of the Poisson. It is set
such that the left and right end points of the support are zero and two times the conditional
mean.
DGP 2 : β10 = 0.564,

yg ∼ Poisson(E[yg |x, ϵg ])
√
v ∼ (χ2 (5) − 5)/ 10

β11 = 0.01,

ϵg ∼ bivariate normal,

β01 = 0.1,

√
In [DGP 2], for the standard normal selection error, a skewed distribution (χ2 (5) − 5)/ 10
was used2 . The mean and standard deviation are the same as the standard normal. This
2 A correlated trivariate distribution in which two of the errors follow standard normal
and the other follows centered χ2 distribution can be properly modeled by copula density

functions. Nevertheless the purpose of present simulation is not finding a multivariate distribution with parameters that uniquely determines a specific multivariate distribution. In
DGP used above, each random numbers were generated first by creating many standard
normal random numbers as “basis” variables. The dependence structure is then created

55

probit assumption was important since it was this very assumption that made it possible to
write the correction terms as the fractions of two normal cumulative density function. The
conditional mean E[y|x, z] is misspecified without this.
DGP 3 : β10 = 0.516,

β11 = 0.01, β01 = 0.1,
√
ϵg ∼ (χ2 (20) − 20)/ 40, v ∼ normal

yg ∼ Poisson(E[yg |x, ϵg ])

√
In [DGP 3], the case where the structural errors follow (χ2 (20)−20)/ 40 instead of standard
normal is considered. When the structural errors are not normally distributed as in [DGP
0], it directly changes the distribution of dependent variables and thus the population ATE.
To ensure that the ATE is equal to one, the constant term in regime 1 is changed to 0.516.
Normality of ϵg is used in the process of getting rid of ϵg from the conditioning set. Thus
a violation of this assumption causes the composite intercept misspecified. Only under
this assumption, we could obtain a nice result that the composite intercepts in estimating
equation E[yg |x, w] and that in E[yg |x] are identical. Therefore a violation of normality
will make ATE estimator inconsistent. The degrees of freedom in chi-square were chosen so
that the comparison between various estimators easy and meaningful to the extent that they
are shown to be consistent converging to the true value. It was found that the results for
structural errors were more sensitive to distributional misspecification than it is for selection
error. Also when the degree of freedom is as high as that used in selection, a lot of cases
the PQML estimator fails to converge to give nice estimates. One can see from the results
below that this choice of degrees of freedom is indeed good for comparison purpose without
too many extreme outcomes.
by using some common “basis” such that the linear combination gives intended correlation
structure. The errors in [DGP2] and [DGP3] are generated in this way. Stata codes will be
provided upon request.

56

In [DGP 0], the variance of individual treatment effect, i.e. T ≡ g1 (x, θ0 ) − g0 (x, θ0 ) −
(
)
E[g1 (x, θ0 )] − E[g0 (x, θ0 )] was approximately 18.9. The asymptotic distribution in earlier
section says that the variance of ATE estimator is an increasing function of E[T 2 ]. [DGP 4]
is designed to show this.
DGP 4 : β10 = 0.865,

β11 = 0.001,

β01 = 0.35,

yg ∼ Poisson(E[yg |x, ϵg ])

ϵg , v ∼ trivariate normal
In the above process, E[T 2 ] is approximately 52.5. In order to create higher variance of
individual treatment, the vertical distance between two conditional mean function at a fixed
x has been adjusted without changing the true ATE value.

2.5.2

Main Results

The results for [DGP 0] is tabulated for all the discussed estimators except for FIML in
Tables E.1-E.3 for each correlation. The FIML results are separately given in Table E.4. For
other DGPs, only 2SLS, LFES (Heckit), NFES (2PQMLE) and NFES (WNLS), which are
the focus of our discussion, are tabulated in Tables E.5-E.7.
First, compare LFES with NFES under various estimation methods with small T . Tables
E.1-E.3 present the [DGP 0] results with E[T 2 ] = 18.9 that are relevent for this purpose. For
any correlation, it can be seen that NFES are extremely inefficient and their RMSEs are very
large compared to LFES under the sample size 1000. This is predicted by the asymptotic
distribution; the variance of T in nonlinear estimator is still prevalent in small sample.
However, such constant term rapidly disappears as the sample size increases. Also the results
in terms of median and mean absolute deviation (MAD) is not too bad; the medians of NFES
are closer to one than those of linear models. In addition to that, although the Monte Carlo

57

standard deviations of NFES were very large, the degrees of dispersion measured by MAD
are smaller than those of linear models, from which one can conclude that the large amount of
RMSE was caused by some outlying estimates with extreme values. Therefore the measures
that are less affected by the extreme values give more favorable figures for NFES. At any
rate, such extreme estimates disappear as the sample sizes grow larger. With n = 5000,
the mean approaches to the true parameter value sufficiently close, and their Monte Carlo
standard deviations are smaller than those of linear models let alone median and MAD. In
sum, ruling out seemingly absurd estimates, when the variance of T is small, NFES model
does better than the linear models in terms of both consistency and efficiency particularly
as the sample size grows larger. On the contrary, the results under E[T 2 ] = 52.5 in Table
E.8 looks very different. Even under large sample size, NFES estimators are less efficient
than LFES. As seen in Proposition 2.2, the asymptotic variance of NFES estimators can
be arbitrarily large as the variance of T diverges to infinity. However, the results in Tables
E.1-E.3 imply that there can be the cases where NFES becomes more efficient than linear
estimators when the individual treatment effect is not substantially different from person to
person.
Incidentally it has been claimed that LATE and ATE obtained by linear IV and nonlinear
methods respectively might not be substantially different (Angrist and Evans, 1998; Angrist,
2001; Angrist and Pischke, 2009). Nevertheless the simulation results indicate that their
finite sample distributions and the estimates can be very different with each other. This
issue will also be revisited in next section.
Second, we have seen in the above that the NFES estimator is a very useful alternative
to LFES when the variance of T is small. However, it must be shown further that the performances of NFES estimators are not very sensitive to the assumptions on which it is based.
58

If the said nice properties of NFES are valid only under the ideal assumptions, its usefulness
will be much limited. The results of additional simulations under wrong distributions mentioned in previous subsection is tabulated in Tables E.5-E.7. In those tables, only PQML,
WNLS, and linear estimators that are still thought to be consistent were considered.
To begin with, see the case where the discrete uniform distribution was used in the
place of Poisson, i.e., under given E[yg |x, ϵg ], the support is [0, 2E[yg |x, ϵg ]]. Since the
distributional misspecification on dependent variable does not affect the consistency results
of PQML and WNLS, the point of interest should be the efficiency of NFES compared to
linear models. The results in Table E.5 show that the performances of linear and nonlinear
models are not very different from the ones in correct distribution. Regardless of ρ, NFES
model is better than the linear models in terms of efficiency as well as consistency. As in
the correct distribution case, despite some extreme values of estimates in small samples, the
performance of NFES measured by median and MAD is still better than the linear models.
Ruling out the seemingly unacceptable estimates, it can be said that NFES model is still
more efficient than linear models even in small samples. Therefore NFES is robust about
the distributional assumption on dependent variable.
Now consider the case where the true treatment equation is not probit. To that end,
a skewed random variable using χ2 (5) was used. This is important because the correction
terms that we used for PQML and WNLS were derived under the probit assumption. When
violated, then the conditional mean function becomes misspecified and they lose the consistency property. However, the results in Table E.6 show that their behaviors are not very
different from the cases under correct distribution. Although the performances of NFES in
terms of mean and variance are worse than the linear estimators in small samples, they are
better in terms of median and MAD. In larger samples they are a lot better by all criteria.
59

Although a departure from probit misspecifies the conditional mean, the impact is almost
negligible.
Lastly, consider the cases where the distributions for unobservables are skewed. The
advantage of normal distribution assumption was that it leads to same intercept both in
estimating equation (2.7) and in equation (2.2) and thus the estimates from switching regression could directly be used for ATE estimation. Therefore under misspecification, the
switching regression estimates do not give correct information for ATE, which leads to inconsistency. The results for misspecified unobservables are given in Table E.7, where the Monte
Carlo mean of NFES model is farther away than the other cases as expected. However, it
should be noted that it is not only the nonlinear model but also the linear models that are
under strain with misspecified errors. Monte Carlo means for both linear and nonlinear models are also farther away from the true value and their deviation is also noticeable. In other
words, skewed error moves the means away from the true value and makes the estimates less
accurate by increasing the variance. However, the NFES model tends to have smaller RMSE
in larger samples mainly due to its advantage in efficiency. The efficiency gain is particularly
notable when it is measured by MAD; in larger samples the MAD value is merely about half
as much as the ones of linear models. The WNLS is even more efficient than the PQMLE.
Third, from the results in Tables E.1-E.3 and E.5-E.6, the performances of 2SLS and
NFES estimators are more or less the same. As the correlation gets higher, their finite
sample biases grow bigger. However, 2SLS is diverging faster than the NFES, which implies
that the NFES estimators are not only robust to distributional misspecification on y and
v, but also suffering less to the higher degree of endogeneity than 2SLS is. The only case
of concern is when the structural error ϵ is misspecified, where both 2SLS and NFES are
diverging even faster than in the previous cases. However, unlike the above cases, NFES
60

are affected more than 2SLS under higher correlation. Nevertheless due to the advantage
in efficiency the RMSE of NFES in larger sample is still smaller until the correlation is up
to 0.5. Even with 0.6, the RMSE of WNLS is still smaller. In sum, NFES performs better
than the 2SLS unless the structural errors are both misspecified and highly correlated with
the selection error at the same time.
To better understand the simulation results, the sampling distributions for some selected
estimators and DGP are graphically shown in figures. Figure G.1 displays the results under
ideal cases [DGP 0] where the normality conditions hold. Among many estimators with
different sample sizes, only 2SLS, NET(WNLS) and NFES(WNLS) with 5000 sample sizes
for different correlation values are listed. The blue line juxtaposed with the histogram is
nonparametric PDF estimate generated by Epanechnikov kernel with an optimal half-width.
It can be seen from the figure that the NET estimator3 behaves poorly in terms of both
consistency and efficiency as anticipated in earlier sections. Even under large observation
of 5000, sampling mean is very different from those of other estimators. Under multivariate
normality, NFES is clearly consistent and efficient in large samples. For misspecified cases
[DGP 1] and [DGP 2], it is clearly noticeable, in Figure G.2 and Figure G.3, that the NFES
estimator is consistent and more efficient than 2SLS at large sample sizes. Figure G.4 depicts
the sampling distribution with skewed structural error [DGP 3]. Although it creates bigger
finite sample biases for those two estimators, the efficiency loss of 2SLS is more pronounced
than in other misspecified cases. An important lesson from these figures is that the violation
of distributional assumptions on which nonlinear models rely can also cause disadvantage
for 2SLS that is not explicitly seen in traditional asymptotic analysis.
3 The estimators based on NET model will simply be called NET estimators. Similarly

for NFES estimators.

61

2.5.3

Some Other Results

In addition to the main results, here are some interesting, but off the main topic results.
First, examine how large the biases of NET estimators are when the true model is of NFES.
It can be seen in Tables E.1-E.3 that neither by mean nor by median do the NET estimators
converge to the true value. It seems that the asymptotic biases of 1PQML and WNLS
estimators are considerable; it is clear that a single-regime estimation is worse than simple
linear models when the true model is of two-regime. Moreover, the seemingly large difference
of Monte Carlo means of 2PQMLE and NLS reflects that those two estimators are not even
estimating same agreed upon parameters. Gourieroux, Monfort and Trognon (1984) show
that the true parameters are consistently estimated when the conditional mean is correctly
specified. Also under the same condition, the NLS estimator also does the same thing since
it is the sample analogue of appropriately defined moment conditions. By those argument,
correctly specified conditional mean guarantees the convergence of those two estimators to
a same quantity. In other words, a substantial difference even with large sample size is
indicative of conditional mean misspecification. Therefore any parameter restriction in Mestimator should be used with caution.
Second, it can be seen in Tables E.1-E.3 for nonlinear estimators, the one-step QuasiLIML, here NFES(1PQML) and NET(1PQML), is less efficient than the two-step methods,
here NFES(2PQML) and NET(2PQML), in small samples. Such result was already reported in Chapter 1 for linear models where the dependent variable in structural equation
was continuous; present simulation also shows that the finding is still valid when the dependent variable is nonnegative count variable. The results support the claim that two-step
estimation procedure gives more efficient estimator than one-step.

62

Third, compare the 2PQML and WNLS for NFES model in Tables E.1-E.3. As it was
already mentioned above, the asymptotic distribution of QMLE and WNLS are very similar. Under the condition var(u|x, w) = σ 2 qvar, their asymptotic distributions are the same.
Simulation results show that the performances of those two estimators are indeed very similar. Nevertheless it is also clearly noticeable that WNLS is slightly more efficient than the
2PQML, which may be due to the fact that the GCIME does not hold for the 2PQML
estimator.
Forth, according to the results for FIML that were tabulated in Table E.4, the FIML
estimator is considerably more efficient than the others. Nevertheless it does not seem to
converge well to the true parameter value, which may have resulted from the quadrature approximation error. To make this point clearer, another set of simulations using two different
numbers of abscissas, i.e. 8 and 16 were run; more abscissas clearly help the mean approach
to the true value. Incidentally the variances are not affected by number of abscissas. One
can have more accurate FIML estimator by increasing number of abscissas, but it would take
considerable amount of time in actual applications. One other feature that makes the FIML
less attractive is that the PQML and WNLS quickly catches up FIML in terms of efficiency
as the sample size grows bigger. Although FIML is the most efficient among all other estimators, the RMSE for n = 5000 is larger than PQML and WNLS due to the inaccuracy.
On the other hand under small sample sizes the linear estimators do as nicely as FIML.
Also using FIML becomes even less attractive when the standard error has to be found by
bootstrapping; it may take too much time to complete a single session of bootstrap. There
are other disadvantages too; since FIML heavily relies on the distributional assumption, any
violation of them will cause the estimator inconsistent. In sum, FIML does not have any
clear advantage over PQML and WNLS.
63

2.6

Empirical Application

Primary education may increase the human capital and lifetime wage and thereby increase
the opportunity cost of having a child (Becker and Barro, 1988; Barro and Becker, 1989),
and it may help reduce the child’s mortality rate and hence let mothers have fewer children
to reach a desired level of family size (Lam and Duryea, 1999; Schultz, 1994a,b). Other than
that an enhanced literacy can help them use contraceptive method more effectively (Rosenzweig and Schultz, 1985, 1989). Based on those theoretical background, we are interested on
how much the primary education reduces the number of children in Botswana. The sign of
the effect is certainly presumed to be negative. Moreover, those who got primary education
may have better health information for their children, which may possibly reduce the child
mortality. Then we can also expect that the difference between ceb and children might be
smaller for those who got the primary education. In this case the treatment effect would be
greater when ceb was used as the dependent variable.
The data used in this empirical analysis is from Wooldridge (2010, Chapter 21). The
variables description and descriptive statistics are given in Tables E.9 and E.10. There was
a huge increase of enrollment rate in Botswana during 1970s. The female enrollment rate
in early 1970s were roughly 60% and kept increasing for the whole decade until it reached
nearly 100% in 1980(UNESCO, 2011). Due to that increase in enrollment, in 1989, the year
this data set was collected, more than half the total female population had at least seven
years of primary education. Thus this data set captures the ideal time point where there
were even amount of control and treatment groups.
The dependent variables under analysis are children (number of living children), ceb
(number of total children born) and mort (number of dead children) and the covariates are

64

age, agesq, evermarr (ever married), urban (living in urban area), electric (has electricity),
tv (has a TV) and radio (has a radio). The variable of interest, i.e. the treatment variable is
educ7 (finished primary education) and the instrument variable is f rsthalf (born in first half
of year). The correlation between educ7 and f rsthalf is -.106. We are interested in the effect
of women’s primary education on the number of children that she ever has(ceb) and that
of living children(children). Although we are trying all the linear and nonlinear methods
for estimating the ATE of education on fertility, the nonlinear estimators are expected to
perform better in two reason: First, the outcome variable is typical count variable with
small natural numbers and thus modeling the conditional mean as exponential function is
well justified. Second, the ATE conditional on covariates might not be substantially different.
In other words, we would not assume neither substantial difference of causal effects across
different age groups nor any particular time trend.
Table E.12 presents regression results for various models and estimation methods with
children as the dependent variable. In what follows the regime with primary education will
be called regime one with a subscript 1 and the regime without it will be regime zero with a
subscript 0. In Table E.12 first four columns present the estimation results for linear models.
The ATE estimates of LFES(Heckit) is −1.552 but not statistically significant. Although
LET(Heckit) and 2SLS differ only in the first stage regression, the estimates of LET(Heckit)
is almost twice as large as the 2SLS estimate. The LFES(Heckit), LET(Heckit) and 2SLS
give AT E with a lot larger magnitude OLS does, which might be an evidence of endogeneity.
It is, however, very hard to get any meaningful conclusion just by seeing the linear regression
results: the only consistent estimator LFES(Heckit) fails to give significant result, and other
estimators of which estimates are significant do not seem to agree with one another.
The next three columns present the results of NET estimators. We already know that
65

the NET model does not identity the true ATE unless the single regime restriction is true.
Indeed the AT EN ET estimates are substantially smaller than the ones from other estimators.
It was also pointed out in Section 3 that each estimator does not even agree with each other
under wrong restriction, which is well demonstrated here; the magnitude of PQMLE and
NLS estimates are very different and they seem to head to different places. The results show
that the AT EN ET estimate by PQMLE is close to zero and not significant. Although only
NLS gives an estimate weakly significant at 10% level, the magnitude is relatively smaller
than those of linear models; it estimates that the primary education reduces the number of
children by no more than 0.68. Also for semi-elasticity, the PQML estimate does not give
any evidence of effectiveness of primary education. Again only the NLS estimate is weakly
significant reporting roughly 30% decrease of living children. These results seem to mimic
the behavior shown in simulation of last section and it can be an evidence that the NET
model is inappropriate.
The last three (double) columns in Table E.12 list the results of NFES estimators. The
NFES estimates report that the primary education reduces 0.8(PQML) or 1.2(NLS) children.
It is worth mentioning that standard error of NFES estimates are a lot smaller than those of
other estimators, due to which all the three NFES estimates are significant at 1% level. What
is particularly interesting is the fact that the NFES estimates support the validity of 2SLS
estimate by providing similar values. LFES(Heckit) being consistent under probit selection
assumption, it can have higher finite sample bias than the 2SLS when the assumption is
violated as shown in Table E.5. Now can we say with greater certainty that ATE estimates
cannot be substantially different from the LATE by 2SLS as was claimed by Angrist and
Evans (1998) and Angrist and Pischke (2009) for bivariate probit case? As it was already
seen in the simulation results, being similar under weak endogeneity, they start to diverge as
66

the degree of endogeneity becomes higher. Therefore the fact that 2SLS and NFES estimates
are similar would be indicative of relatively weak correlation between selection error v and
structural error ϵ, rather than the validity of the above claim.
The estimated regime one (with primary education) averages

∑

children1 /N for three

estimators are 1.264(NLS), 1.499(2PQML), 1.482(1PQML) and those of regime zero (without
primary education)

∑

children0 /N are 2.488(NLS), 2.340(2PQML), 2.312(1PQML); from

those values one can compute the semi-elasticities, i.e. -0.49(NLS), -0.36(2PQML), and 0.36(1PQML). All those estimates are greater in absolute value than the ones from NET
model. From these, it becomes more obvious that the NET estimators give us information
that looks very much different from what was provided by other estimators. Lastly we
can directly test the restriction put on the NET model. One may use the Wald test of
H0 : β1 = β0 . The p-values for 2PQMLE and NLS are 0.000 and that of 1PQMLE is 0.001
implying that there actually exist two regimes4 . All the above results unequivocally show
that the NET model is not an appropriate model to be used to describe this data set. We
can also test the endogeneity by checking the covariance between v and ϵg . Ignoring NET
model, all the two regime estimators show that the regime one covariance is significantly
positive, whereas the one at regime zero, slightly negative, is not statistically different from
zero. Overall the use of two regime endogenous switching model is well justified.
The above discussion was about treatment effect on the number of living children that
reveals the difference in the desired number of children for each education group. Another
interesting aspect can be the treatment effect on child mortality and those educated mothers
4 Since 2QMLE and NLS use two-step procedure, the asymptotic variance approximation

has to account for the first stage error. One of the advantages of single-step 1PQMLE is that
such first stage error is not present and the inference is straightforward. Although there is
slight difference in the p-values, such trivial difference is not thought to be of any practical
importance.
67

are expected to have fewer dead children(Lam and Duryea, 1999; Schultz, 1994a,b). Thus
the treatment effect might be negative. Since a direct estimation yields extreme outlying
estimates for NFES model5 , an alternative indirect way of estimation procedure is used.
To that end, another regression with the dependent variable ceb is run and presented in
Table E.13. Then the expected child death at regime g for each individual is computed as
mortig = cebig − childrenig . The treatment effect for each observation is then computed
as morti1 − morti0 and their average through the whole population becomes the ATE on
mortality presented in Table E.14. For example, the estimated regime one averages, i.e.
∑
∑

cebi1 /N are 1.402(NLS), 1.697(2PQML) and 1.683(1PQML), whereas those of regime zero
cebi0 /N are 2.890(NLS), 2.680(2PQML) and 2.657(1PQML). The differences in estimated

regime zero averages, i.e.

∑

cebi0 /N −

∑

childreni0 /N are 0.401(NLS), 0.340(2PQML) and

0.345(1PQML) implying that the mothers without primary education lose on average 0.4
children. For those with primary education the quantity is

∑

cebi1 /N −

∑

childreni1 /N of

which numerical values are 0.138(NLS), 0.198(2PQML), and 0.201(1PQML). The implication
is that the child mortality rate is reduced roughly by 0.14 to 0.26 per mother by primary
education. Although the ATE’s on mortality for nonlinear estimators are not significant,
they are still more efficient than the linear ones with an exception of LET(Heckit); the
bootstrap standard deviations of nonlinear estimators are smaller than the linear ones.
5 The 2SLS estimator gives insignificant -.002(.208). Unlike the general results, the NFES

estimators do not give a reasonable estimate; two-step QML estimate is 25.821 with bootstrap standard deviation 784296.5 by 50 replications. It turns out that predicted values for
dependent variable in regime 1, i.e. morti1 are very volatile; about 10% of the observations
have more than 10 predicted child’s deaths and about 1% have more than 100. Strangely
such phenomenon does not occur in regime zero.

68

2.7

Conclusion

The main contribution of this study is to clarify the asymptotic distribution of the ATE
estimator based on NFES model. Unlike other structural parameters, the ATE estimates
are computed by a nonlinear function of the parameter estimates. The estimation error
therefore comes both from the error in parameter estimation and also from the computation
of ATE by the parameter estimates. The asymptotic distribution reveals that each factor
can be written additive separably with a covariance term. The theory predicts that the
efficiency of nonlinear ATE estimator is not taken for granted as in many other nonlinear
cases. However, when the nonlinear estimators are actually more efficient, simulations show
not only that it tends to have smaller finite sample bias, but also that its performance
under misspecified model is not too bad compared with the linear IV or LFES models. The
application shows an example in which this nonlinear methodology can be successfully used.
A nonlinear method is expected to be perform better if the variance of conditional ATE are
not substantial as in the Botswana fertility example.

69

Chapter 3
A Two-Regime CRC Model with
Nonnegative Dependent Variable

3.1

Introduction

The Correlated Random Coefficient (CRC) Models that were first introduced by Heckman
and Vytlacil (1998) provided an alternative way to model the individual heterogeneity. The
main focus having been on clarifying the conditions for identifying the parameters of interest
under the presence of CRC since its inception (Wooldridge 1997, 2003, 2005; Card 2001),
more recent development is focusing on the extension to the cases where the support of
dependent variables are of limited nature. Wooldridge (2007) has suggested a method for
estimating Average Partial Effects (APE) where the count dependent variable as well as
random coefficients correlated with variable of interest are present. Cases of interest in this
article are very similar to it except for the fact that a binary variable with counterfactual
causal model is considered as a starting point. Specifically the goal of this paper is to provide
an estimation method for Average Treatment Effects (ATE) when the dependent variable is
count variable and all the covariates have CRC’s that are correlated with the binary variable.
This paper is organized as follows: In Section 2, previous discussion on CRC model will be
reviewed. Section 3 discusses various CRC models proposed so far in the context of ATE.

70

Section 4 is the core of this article where the ATE estimation method for Nonlinear Tworegime CRC model will be provided. Section 5 discusses two specification tests; the test for
endogeneity is designed for detecting the correlatedness of the random coefficients and the
model selection test is for choosing the model between the ones with and without random
coefficients. The method in this article is dependent on some distributional assumptions and
their sensitivity on the estimation performance will be examined by Monte Carlo simulations
in Section 6. Section 7 is an empirical application of this method and the performance of
other competing estimator will be compared. Lastly, Section 8 is concluding remarks.

3.2

Previous Literature

Random coefficients typically arise from a model where an unobserved variable interacts with
one or more observed variables. It then calls for different approaches due to the random
nature of the coefficients on the observed variables. The randomness of the coefficients
makes the conventional partial effect be random, and the quantities of interest are usually
the means of the random coefficients. In other words, we would like to estimate not the
partial effect but the average partial effect. If a regressor is mean independent from the
unobserved variables, then the APE of the regressor can be consistently estimated by using
OLS (See Amemiya, 1985).
Let the unobserved heterogeneity denoted q and the 1 × K vector of exogenous variables
x . Then for the case where E[q|x] ̸= 0, even if E[q|x] can be appropriately modeled as a
function of x, the APE of a particular variable in x cannot be consistently estimated as long
as the function contains the x with degree one. In such cases, q needs to be expressed by some
proxy variables that does not contain x to consistently estimate the APE of a variable in x.

71

Now let us consider a CRC model where E(q|x) ̸= 0 and proxy variables are not available
as follows. In CRC, no correlation means mean independence, not the usual orthogonality.

y = α0 + (α1 + q1 )x + u

Assume for now that the regressor of interest is exogenous, i.e. orthogonal to the structural
error. Then the model will have the regressor itself inside the composite error, i.e. q1 x + u.
Unlike from the mean independent case, this composite error does not vanish by using
conditional expectation operation and renders the regressor effectively endogenous. Heckman
and Vytlacil (1998) and Wooldridge (2003) have shown that under some assumptions, the
instrumental variables method can be used to consistently estimate α1 .
In sum, when there are uncorrelated random coefficients, OLS gives a consistent estimator
under the mean independence of q conditional on regressor. For CRC models, under the
violation of mean independence, even a regressor that is independent of the structural error
u in the population effectively becomes endogenous due to the random part of the coefficient.
In what follows, our assumption is that the regressor is independent with the structural error
for the sake of simplicity. Too restrictive the independent regressor as above might seem to be,
the solutions for the CRC usually work for the nonindependent cases also (See Wooldridge,
2010).
It must be noted that given a particular regressor x, the coefficients on other regressors
can be both random and correlated with x. The first case to consider is the model where the
CRC with respect to a specific regressor is only on that regressor, i.e. the coefficients on all
other covariates are assumed to be constants. The problem with this case is that a usual IV
conditions are inadequate; even if the mean independence between IV and structural error

72

and between IV and CRC are established, the orthogonality between that IV and composite
error is not guaranteed. In order for the IV to be effective, it has to satisfy an additional
condition that E[q1 x|z] is constant (Wooldridge, 2003).1 Although this condition is not very
strong, its validity may be put in question when x is not continuous(Card, 2001; Wooldridge,
2005).
For the cases where the conditions for IV are not met, there is still another method
that uses control function approaches with stronger assumptions. Heckman (1976) provides
a solution for binary regressor, whereas Garen (1984) does it for continuous one. See the
equation below.
y1 = η1 + z1 δ1 + a1 y2 + y2 z1 γ1 + u1 ,

(3.1)

where z1 is 1 × L1 vector of exogenous variables and δ1 and γ1 are L1 × 1 coefficients vectors.
The common convention is that Greeks are constant and Romans random. Regressors may
or may not have random coefficients which may or may not be correlated with the structural
error. The variables z1 that have constant coefficients without any correlation with u1 will
be called exogenous variable throughout this article. And any variable that is correlated with
structural error or at least has correlated random coefficient will be called endogenous variable; for instance in the above equation the endogenous variable y2 not only has correlation
with u1 , but also has CRC.
One can also enrich the model by introducing some interaction terms between exogenous
and endogenous variables such as y2 z1 in equation (2.1). Let us put some restrictions here;
suppose that y2 is binary endogenous variable and that the coefficient of the interaction
1 Although this condition is for the IV estimation in CRC models, it is still needed

for a random coefficient that is uncorrelated with the regressor. Even if E[q1 x] = 0, it is
not necessarily that E[q1 x|z] is constant. Again no correlation in CRC is not about the
orthogonality but mean independence.
73

term γ1 is constant. Then this model simply becomes Linear Full Endogenous Switching
Regression or LFES model (Keay, 2011) with two regimes between which an individual can
switch. In this model all the coefficients in each regime are nonrandom. For this LFES it
is also possible to introduce CRC to the exogenous variables for each equation (Wooldridge,
2007). Or can one equivalently construct such two-regime CRC model, i.e. a two regime
regression model where the covariates in each equation have random coefficients correlated
with y2 , by directly introducing CRC not only to the endogenous variable but also to the
exogenous variables as well as the interaction. Here is an equivalence result: two-regime CRC
model can be constructed by putting CRC to the all regressors in the equation including the
interaction terms.
The discussion so far was about the models with continuous structural error and hence
continuous dependent variable. In this paper I will provide a model where the dependent
variable takes nonnegative or more specifically natural numbers. Such cases arise very often
in real life such as when the dependent variable is commuting frequency (Terza, 1998) or
number of child (Keay, 2011). In addition to the nonlinearity in the first stage binary choice,
another nonlinearity to the dependent variable of the structural equation will be taken care
of. Although Terza (2009) has already considered such nonlinear model with two regimes
without CRC, its extension with CRC on all exogenous variables has not been proposed so
far. In the following, the former model will be called Nonlinear Full Endogenous Switching
Regression or NFES model and the later be called Nonlinear Two-regime CRC or NTCRC
model. The linear version of Two-regime CRC model was already proposed by Wooldridge
(2007) that will be called Linear Two-regime CRC or LTCRC model.
In what follows Poisson distribution will be used to model the nonnegative response in the
structural model. Incidentally an ensuing natural question is whether all the results in the
74

Linear with Interaction
k
CRC on y kkkkkkk
k
kkk
ukkkk
LFES SS
CRC on all
SSSS
SSSS
SSSS
S)
CRC on z


Nonlinear with Interaction
jj
CRC on y jjjjjj
j
jjjj
tjjjj
NFES TT
CRC on all
TTTT
TTTT
TTTT
TT*
CRC on z


LTCRC

NTCRC

Figure 3.1: Analogy Structure
linear model is valid to the same extent. Of course the model itself does not undergo much
changes although some quantities of interest will not be identified, which will be discussed
in the following sections.
The problem of NFES approach is that the intercept term inside the exponential function
in the structural equation is not identified by itself alone; instead Terza (2009) estimates the
average treatment effect for nonlinear model and such approach will be employed also for
the two-regime CRC model in this paper. As it was already mentioned above, the CRC
on binary endogenous variable makes two-regime switching regression model. The objective
of this paper will be to set out a NTCRC model by bringing once again the CRC to the
two-regime nonlinear switching model, where the average treatment effect will be identified.
Incidentally this is equivalent to the model where all the variables including the interaction
have CRC. The above discussion is summarized in the analogy scheme in Figure 3.1.

3.3
3.3.1

Various Models of CRC
Continuous Endogenous Variable

Let the variable with which random coefficient is correlated be denoted w. By the unobserved
heterogeneity argument the variable w is usually regarded as endogenous, which is also the
75

case here. The CRC can be either only on w or on all other variables. In population regression
model, the variable w and all other covariates can be interacted, which may create many
possible combinations of situations: CRC only on w with no interaction, CRC only on w
with interaction, CRC on all variables with no interaction and finally CRC on all variables
with interaction. Consider a regression model as follows.

y = xβ + τ w + γq + qxδ + u,

where x is a 1×K vector of exogeneous variables and β and δ are K ×1 vectors. The variables
q and w are assumed to be scalar. The former is unobserved variable that is correlated with
the latter, but interacts with x. One of the characteristics of this model is that the variable
w will be correlated with the random coefficients of x, i.e. β + qδ and, at the same time, will
be an endogenous variable in traditional sense because it is correlated with the composite
error γq + u. If an interaction term between x and w is included in this model and also if
that interaction term is again interacted with the unobserved variable q, then, given that
w is binary, the resulting model will be the two-regime CRC model that will be discussed
below.
y = x(β + δ0 q) + w(τ + δ1 q) + wx(κ + δ2 q) + γq + u
A random coefficient model with an endogenous variable with which the coefficients on an
exogenous variable is correlated is created in such manner. This model turns out to be a very
good description of education-fertility behavior which will be discussed in later sections. In
addition to that, this model is very general in that many other previous models can be treated
as its special cases. In case w is endogenous binary variable, a restriction δ0 = δ2 = 0 makes
the model as an ordinary two-regime switching model and also an additional restriction
76

δ1 = κ = 0 will make it as Endogenous Treatment Model as in Terza (1998, 2009). There
are also many other possibilities that are nonetheless special cases of the model given above.
Let us focus on a continuous w here deferring discussion of binary case in subsequent sections. With count dependent variable, suppose the conditional expectation can be modeled
by an exponential function as link function. This approach explicitly considers the dependent variable of the structural model as count variable that does not take negative value. As
the simplest case consider a model of CRC only on w without interaction as below.

E(y1 |z, w, a1 , r1 ) = exp(xδ1 + a1 w + r1 )
w = zδ2 + v2 ,

where x and z are 1 × L1 and 1 × L vectors of exogenous variables with x ⊂ z that are
independent with all the random variables generated in model. The coefficient vectors δ1
and δ2 are L1 × 1 and L × 1 respectively. Also assume that

a1 = α1 + d1
d1 = ψv2 + ed

v2 ⊥ ed
⊥

r1 = θv2 + er

v2 ⊥ er
⊥

In the first equation, α1 = E[a1 ] and d1 = a1 − α1 . The second and third equations
are basically the linear projections of each random variable on v2 . However, we need a
stronger condition that the orthogonal decompositions are not only uncorrelated but also
independent in order to facilitate the identification. The regressor w suffers from endogeneity
unless ψ = θ = 0. Along with those assumptions, we put multivariate normal assumption on

77

the random variables (d1 , r1 , v2 ). Under these assumptions, Wooldridge (2007) has derived
a conditional mean function as below.
(
)
2
2
σr + 2σdr w + σd w2
E(y1 |z, v2 ) = exp xδ1 + α1 w + ψv2 w + θv2 +
2

Although the above equation is estimable once v2 is estimated in the first stage regression,
the semi-elasticity of w, i.e. α1 is not identified due to the fraction term, and this is still the
case even if w is exogenous, where ψ = 0, θ = 0. In order to identify α1 , it is required that
the random coefficient on w is uncorrelated not only with w but also with r1 . Although the
semi-elasticity is not identified, one can still identify the APE of w over z, w and v2 . From
the above equation, the partial effect of w is
(
)
2
2
σr + 2σdr w + σd w2
∂E(y1 |z, w, v2 )
2
= exp xδ1 +α1 w+ψv2 w+θv2 +
·(α1 +σdr +ψv2 +σd w).
∂w
2

By taking the average over the whole population, APE of w is identified. Wooldridge (2007)
also extends to discuss the case where the coefficients of other covariates are also correlated
with w. The results are basically the same; APE is identified although the semi-elasticity is
not.

3.3.2

Binary Endogenous Variable

Recall Figure 3.1. It should be noted that we are already familiar with the models with
CRC only on y, which are equivalent to the two-regime endogenous switching regression
(See Maddala, 1986). For the linear and nonlinear cases, one can use the correction methods
proposed by Heckman (1978) and Terza (1998) respectively. As an extension in linear model,

78

the one with CRC not only on endogenous but also on all other variables, i.e. Linear Tworegime CRC model (hereafter LTCRC), was explored by Wooldridge (2007). Consider a
regression model as follows.

y1 = xd1 + a1 y2 + xy2 g1 + u,

where x is 1 × K vector of exogenous covariates and d1 and g1 are K × 1 vectors of CRC’s.
Also we have a vector of all exogenous variables z such that x ⊂ z. As in Chapter 2, it is
assumed that x is demeaned without loss of generality. Wooldridge (2007) shows that under
the assumptions

E(a1 |z, v2 ) = α1 + φ1 v2
E(d1 |z, v2 ) = δ1 + ψ1 v2
E(g1 |z, v2 ) = ξ1 + ω1 v2 ,

the estimating equation is

E(y1 |z, y2 ) = xδ1 +α1 y2 +y2 xξ1 +ρ1 h2 (y2 , zδ2 )+φ1 h2 (y2 , zδ2 )y2 +h2 (y2 , zδ2 )xψ1 +h2 (y2 , zδ2 )y2 xω1 ,

where
h2 (y2 , zδ2 ) = y2 λ(zδ2 ) − (1 − y2 )λ(−zδ2 ),
where λ(·) is inverse Mill’s ratio.
From this model and estimating equation, one can identify the ATE of y2 on y1 , i.e. α1 .
In the next section, the main contribution of this article, the Nonlinear Two-regime CRC

79

(NTCRC) model and its estimation method will be discussed.

3.4
3.4.1

Nonlinear Two-Regime CRC Model
Model

In what follows we allow for a binary case of the variable of interest w, which calls for an
analysis of ATE under counterfactual framework. Under this setting, the first stage binary
selection equation is assumed to follow probit model, which makes the estimation more
efficient. For the continuous w, the average of the semi-elasticity of w, i.e. α1 was not
identified, while the APE was. Analogously, although the average semi-elasticity of w is
not identified, ATE will be. Consider a model in terms of conditional expectation functions
given below.

E(y0 |z, w, a0 , b0 , a1 , b1 ) = E(y0 |x, a0 , b0 ) = exp(a0 + xb0 )
E(y1 |z, w, a0 , b0 , a1 , b1 ) = E(y1 |x, a1 , b1 ) = exp(a1 + xb1 )

(3.2)

w = 1[zγ + v > 0],

where ag are scalar errors, x is 1×K vector of covariates, and bg are K ×1 random coefficient
vectors. The z is the vector of all the available exogenous variables such that x ⊂ z. For
those random coefficients, our quantities of interest will be the means of random variables.
In what follows, we will use notation var(x) ≡ σ 2 (x) and cov(x, y) ≡ σ(x, y). Also the
regime subscript will be suppressed whenever obvious. Here are the assumptions for further
discussion.

80

Assumptions
1. Unconfoundedness: As was already stated in equation (3.2), w is redundant conditional on a and b that summarize all the information about the determined regime,
i.e.
E(y1 |z, w, a0 , b0 , a1 , b1 ) = E(y1 |x, a1 , b1 ),
and same for Regime 0. Although the main focus is on w, this equation also assumes
the same thing for the unobserved heterogeneity for the other regime.
2. Multivariate Normality: For each regime in equation (3.2), we have

a = α+e
b = β + d,

where α and β are the means of a and b. The vector (e0 , d0 , e1 , d1 , v) of which dimension is (2K + 3) × 1 follows multivariate normal, i.e.

(e0 , d0 , e1 , d1 , v)′ ∼ N (0, V),
where V is the appropriate variance-covariance matrix.
Let the linear projections of e and d on v as

e = σ(e, v)v + ϵ
d = σ(d, v)v + δ.

Note that b, d, δ and σ(d, v) are K × 1 vectors. Specifically, σ(d, v) =
81

[σ(d1 , v), · · · , σ(dK , v)]′ , where bj = βj +dj is the j-th element in b. Since v and ϵ, δ are
orthogonal, they are independent under multivariate normality. Another implication
of the above assumption is that the selection equation follows probit model.
3. Conditional Independence: v and ϵ + xδ are independent conditional on z, w, i.e.,

v ⊥ ϵ + xδ | z, w.
⊥

Under these assumptions, one can obtain the following lemma.

Lemma 3.1. Under the assumptions 2 and 3,

E[exp(e1 + xd1 )|z, w = 1]
[(
= exp

σ 2 (e1 ) +

K
∑

K
∑

σ 2 (d1j )x2 + 2
j

j=1

σ(e1 , d1j )xj +

j=1

×

K
∑∑

)/ ]
σ(d1j , d1r )xj xr 2

j=1 r̸=j

)
(
∑
Φ zγ + σ(e1 , v) + K σ(d1j , v)xj
j=1
Φ(zγ)

for Regime 1, and

E[exp(e0 + xd0 )|z, w = 0]
[(
= exp

σ 2 (e0 ) +

K
∑

K
∑

σ 2 (d0j )x2 + 2
j

j=1

j=1

×

σ(e0 , d0j )xj +

K
∑∑

)/ ]
σ(d0j , d0r )xj xr 2

j=1 r̸=j

(
)
∑K
Φ − zγ − σ(e0 , v) − j=1 σ(d0j , v)xj

82

Φ(−zγ)

for Regime 0.

Proof Consider the Regime 1 only. For notational simplicity the regime subscripts are
suppressed. Derivation for Regime 0 is almost identical. Note that
[

(

)

]

E[exp(e + xd)|z, w = 1] = E exp σ(e, v)v + ϵ + xσ(d, v)v + xδ z, w = 1
]
[
(
)
= E exp (σ(e, v) + xσ(d, v))v exp(ϵ + xδ) z, w = 1
[
(
)
]
= E exp (σ(e, v) + xσ(d, v))v z, w = 1
× E[exp(ϵ + xδ)|z, w = 1]
[
(
)
]
= E exp (σ(e, v) + xσ(d, v))v z, w = 1
× E[exp(ϵ + xδ)|z]
)
(
)
Φ zγ + σ(e, v) + xσ(d, v)
(σ(e, v) + xσ(d, v))2
=
exp
Φ(zγ)
2
(

× E[exp(ϵ + xδ)|z]

For the second term on RHS,
(

K
)2
(
)2
∑
σ(e, v) + xσ(d, v)
= σ(e, v) +
σ(dj , v)xj
j=1

=

σ 2 (e, v) + 2

K
∑

σ(e, v) · σ(dj , v)xj +

j=1

+

K
∑∑

σ(dj , v) · σ(dr , v)xj xr .

j=1 r̸=j

83

K
∑
j=1

σ 2 (dj , v)x2
j

Now consider the third term on RHS. From the normality of ϵ + xδ that is guaranteed by
the multivariate normal assumption,
(
/ )
E[exp(ϵ + xδ)|z] = exp var[ϵ + xδ | x] 2 .

Now that

var[ϵ + xδ | x] = σ 2 (ϵ) + σ 2 (xδ) + 2σ(ϵ, xδ)
=

σ 2 (ϵ) +

K
∑

σ 2 (δj )x2 +
j

j=1

K
∑∑

σ(δj , δr )xj xr + 2

j=1 r̸=j

K
∑

σ(ϵ, δj )xj .

j=1

Collecting those terms we have the stated result.

Remark Terza (1998) directly used the multivariate normal property in order to solve the
(
)
1 2
2 ) was derived under
conditional expectation, i.e. E[exp(ϵ)|z, v] = exp ρσv + σ (1 − ρ
2
the multivariate normal assumption between ϵ and v. Although such approach is also applicable here, it does not give a convenient expression for estimating the average treatment
effect. In the above derivation I used the linear projections of e and d on v that give un
estimating equation of which the coefficients can be directly used to find the ATE. As will
be discussed below, the structural parameters such as β is not independently identified, but
the coefficients on x and x2 from the estimating equation derived by the linear projection
coincides with those coefficients of the estimating equation for ATE.

84

Now let us derive an estimating equation of the model in (3.2). Note that by assumptions
1 and 2,

E[y|z, w, a0 , b0 , a1 , b1 ] = (1 − w)E[y0 |z, w, a0 , b0 , a1 , b1 ] + wE[y1 |z, w, a0 , b0 , a1 , b1 ]
= (1 − w)E[y0 |x, a0 , b0 ] + wE[y1 |x, a1 , b1 ]
(
)
= (1 − w) exp α0 + e0 + x(β0 + d0 )
(
)
+ w exp α1 + e1 + x(β1 + d1 )
E[y|z, w] = (1 − w) exp(α0 + xβ0 )E[exp(e0 + xd0 )|z, w = 0]
+ w exp(α1 + xβ1 )E[exp(e1 + xd1 )|z, w = 1]

By Lemma 3.1.
[(
K
K
∑
∑
2 (e ) +
2 (d )x2 + 2
σ 1j j
[β1 + σ(e1 , d1j )]xj
= w · exp 2α1 + σ 1
j=1

+

K
∑∑
j=1 r̸=j

+(1 − w) exp

[(

j=1

)
(
] Φ zγ + σ(e , v) + ∑K σ(d , v)x
)/
1
1j
j
j=1
σ(d1j , d1r )xj xr 2 ×
Φ(zγ)

2α0 + σ 2 (e0 ) +

K
∑
j=1

+

K
∑∑
j=1 r̸=j

σ 2 (d0j )x2 + 2
j

K
∑

[β0 + σ(e0 , d0j )]xj

j=1

(

)
∑
)/ ] Φ − zγ − σ(e0 , v) − K σ(d0j , v)xj
j=1
σ(d0j , d0r )xj xr 2 ×
Φ(−zγ)

(3.3)

The above equation identifies only σ 2 (dj ), σ(e, v) and σ(dj , v) separately, and the average
of semi-elasticity, i.e. β is not identified due to the nuisance parameters σ(e, dj )’s. It
can be estimated by two step by using the probit model for the selection equation; given
the estimated index zγ, the above equation can be estimated either by NLS or Quasi-ML
85

under Poisson distribution. Or can they still be estimated simultaneously by a single-step
method. See Chapter 2 for detailed discussion of estimation procedure and their asymptotic
distribution. Since the structural parameters α and β are not separately identified, our
interest should lie in average treatment effects.

3.4.2

Identification of ATE

Note that the ATE is E[y1 − y0 ]. One of the easiest way to identify this is by using the law of
iterated expectation, i.e. E[y1 − y0 ] = EE[y1 − y0 |x] as long as the expectation conditional
on x can be derived. To that end, the following lemma is derived.

Lemma 3.2. Under the assumptions given above, the following result holds.
[(
E[exp(e+xd)|x] = exp

σ 2 (e)+

K
∑

σ 2 (dj )x2 +2
j

j=1

K
∑

σ(e, dj )xj +

j=1

K
∑∑

)/ ]
σ(dj , dr )xj xr 2

j=1 r̸=j

Proof Note that

E(e + xd|x) = E(e|x) + E(xd|x) = 0
var(e + xd|x) = var(e|x) + var(xd|x) + 2cov(e, xd|x)
=

σ 2 (e) +

K
∑

σ 2 (dj )x2 +
j

j=1

K
∑∑
j=1 r̸=j

σ(dj , dr )xj xr + 2

K
∑

σ(e, dj )xj .

j=1

For any fixed value of x, e + xd is normally distributed since it is a combination of multivariate normal random variables as was mentioned in assumption 2. Since the mean and
variance of normal random variables are already obtained, the mean of its log-normal variable is trivially found.

86

The ATE conditional on x is

E[y1 − y0 |x] = exp(α1 + xβ1 )E[exp(e1 + xd1 )|x] − exp(α0 + xβ0 )E[exp(e0 + xd0 )|x].

Thus from the above lemma,

E[y1 −y0 |x]
[(
= exp

2α1 +σ 2 (e1 )+

K
∑

σ 2 (d

2
1j )xj +2

j=1

−exp

[(

2α0 +σ 2 (e0 )+

K
∑
j=1

K
∑

[β1 +σ(e1 , d1j )]xj +

K
∑∑

j=1

σ 2 (d0j )x2 +2
j

j=1 r̸=j

K
∑

K
∑∑

)/ ]
σ(d1j , d1r )xj xr 2

[β0 +σ(e0 , d0j )]xj +

j=1

)/ ]
σ(d0j , d0r )xj xr 2 .

j=1 r̸=j

(3.4)
Note the similarity of this equation with the estimating equation in (3); except for the correction functions, the expressions inside the exponential function are identical which makes
the identification of ATE possible. The estimator will be the sample average of equation
(3.4) over the values of x. The fact that each parameter is not identified does not make
any problems for ATE identification. The asymptotic distribution of this ATE estimator is
essentially same as was presented in Terza (2008).

87

3.5
3.5.1

Specification Test
Tests for Endogeneity

In the previous section, we have derived the estimating equation (3.3) and ATE conditional
on covariates (3.4). In these equations no restrictions were imposed and an estimation of
the model without restriction, where the number of identifiable composite parameters is no
less than (K + 4)(K + 1)/2, might cause difficulty in numerical optimization. One can apply
the Lagrange Multiplier (LM) test in order to test for the presence of correlated random
coefficients. Another test called Variable Addition Test (VAT) is also available which is
asymptotically equivalent to LM but easier to apply. This test constructs a conditional
mean by adding appropriately defined variables to create the likelihood function of which
the score under restriction is same as the one used in LM test (See Wooldridge, 2011). The
actual test is performed by Wald test on the significance of coefficients of the added variables.
Revisit the estimating equation (3.3). Inside the exponential function we have each covariate,
the square of each covariate and their cross products. Let the vector of these functions of
covariates denoted x and rewrite the estimating equation for Regime 1 as follows.
(
)
Φ zγ + θ2 + xθ3
E[y|z, w = 1] = exp(xθ1 ) ×
,
Φ(zγ)

(3.5)

where θ2 = σ(e1 , v) and θ3 = σ(d1 , v). As was already mentioned, θ2 is scalar, θ3 is K × 1
vector and θ1 is (K + 2)(K + 1)/2 × 1 vector. The estimating equation for Regime 0 is the
same as above except for the minus sign inside the Φ(·) in correction function.
In order to perform VAT, we need to construct a conditional mean function of which
score is identical to the equation (5) under restriction. Let the restriction or equivalently

88

the null hypothesis
H0 : θ2 = θ3 = 0.
In other words, there is no correlation between selection error v and any other errors in the
structural equations. An LM test can be applied that does not require an estimation of
complicated model without restriction. The VAT is a device that facilitates this LM test by
using an auxiliary regression for Regime 1
)
(
exp xθ1 + θ2 λ(zγ) + θ3 λ(zγ)x ,

(3.6)

where λ(·) is inverse Mill’s ratio. The auxiliary regression for Regime 0 will have −zγ for
zγ. It can be easily verified that the likelihood and score functions derived from (3.6) under
restrictions are same as the one from (3.5). Therefore the score tests on the significance of
coefficients from the above auxiliary regression is equivalent to the LM tests on (3.5). Thus
the Wald tests for (3.6) will give a simple and asymptotically equivalent way to test the null
without estimating the model without restriction. In actual test, λ(zγ) has to be estimated
through the first stage regression.

3.5.2

Model Selection Test

What we have considered above is whether the already given random coefficients are correlated with the selection error or not. Even when the null hypothesis is not rejected, it does
not warrants our coming back to the Nonlinear Full Endogenous Switching (NFES) Regression provided by Terza (1998). In current model, presence of CRC is assumed both in a and
b in equation (3.2). As a natural consequence one also assumes the multivariate normality

89

between those errors and v. On the contrary, the coefficient b is assumed to be constant in
NFES and the multivariate normal joint error distribution is assumed just between a and v.
It should be emphasized that the VAT tests for the endogeneity under the assumption that
b is random, and thus a failure to reject null hypothesis does not imply an acceptance of
NFES. A practical consequence of this observation is that one has to include the square and
cross products of the covariates, which were not included in NFES, even when there turns
out to be no correlation between those random coefficients. Then how can one perform a
test of which the alternative is NFES model? The NFES model assumes that the coefficient
b in equation (3.2) is constant and thus d = 0 as well as σ 2 (dj ) = σ( · , dj ) = 0. From
equation (3.3), the null hypothesis will be the zero restriction on the coefficients of square
and cross product terms inside the exponential functions and the coefficients on xj inside
the correction functions.

3.6

Monte Carlo Simulation

The purpose of the simulation is two-fold: First, it will compare the performances of this
nonlinear ATE estimator with the linear method proposed by Wooldridge (2007). This will
show the advantage of using the nonlinear model for estimating ATE. Second, it will see
the impact of violations of the assumption that were used in order to derive the estimating
equation in the previous section.

3.6.1

Data Generating Processes

The derivation of the estimating equations depends on the three assumptions listed in the
above section. In this section, we examine the performances of the ATE estimator under

90

violations of the first and second assumptions only. In order to check the robustness of those
assumption, a series of simulations are run by using the following data generating processes
(DGPs).

x ∼ uniform[15, 50]
z ∼ Binomial(1, 1/2)
w = 1[1.4 − 0.05x − 0.3z + v ≥ 0]
(
)
E(y1 |x, z, e1 , d1 ) = exp 0.15 + e0 + (0.02 + d0 )x
(
)
E(y0 |x, z, e0 , d0 ) = exp 0.1 + e1 + (0.0059 + d1 )x ,

where y1 and y0 follow Poisson distribution with the mean specified above. This DGP was
designed to make the population ATE approximately equal to unity.
There are three sessions of simulations; DGP 1 deals with the ideal case where the errors
follow multivariate normal distribution and v ⊥ ϵ + xδ. The joint distribution of errors for
⊥
DGP 1 is





∗
 e0 















d∗
0
e∗
1
d∗
1
v
















 ∼ N 













 
0   1 ρ 0 0 ρ
 
 
0  
1 0 0 ρ
 
 
 
0 ,
1 ρ ρ
 
 
 
0  
1 ρ
 
 
0
1









 .







This variance-covariance matrix is in fact correlation matrix since the variances are all equal
to unity. For the simulation, e0 = e∗ /100, e1 = e∗ /100, d0 = d∗ /100, and d1 = d∗ /100 were
0
1
0
1
used. Also for the correlation values ρ, 0.3 and 0.5 were used. This is the case where the
91

above estimator might perform the best, since the actual DGP conforms the assumptions on
which the estimating equation is based.
DGP 2 generates a process where the linear projection errors ϵ and δ are uncorrelated
but dependent with v. For a given v, ϵ is designed to be either the same value of v or −v
by a half probability for each. Then the scattergram will look like a X letter, where the
covariance of the two variables becomes zero, but very strongly dependent with each other.
Another important assumption for Lemma 3.1 is that v follows normal distribution, which
brought the normal cumulative distribution function Φ(·) in the equation. In order to check
the robustness of the estimator on that assumption, DGP 3 uses χ2 (5) distribution to create
skewed errors. The variance-covariance matrix looks just the same as in the above equation.
However, we here drop the multivariate normal assumption; all the marginal distributions
of the errors follow some adjusted χ2 distribution with the predetermined correlation2 .

3.6.2

Simulation Results

Tables F.1 and F.2 list the simulation results for ρ = 0.3 and 0.5 respectively. For each table,
each column for I, II and III lists the results for the DGP I, II and III both for linear and
nonlinear estimator. Let us first consider the performances of nonlinear estimator. In either
Table F.1 or F.2, the column I for DGP 1 shows that the ATE estimator is performing well
as expected. The results show that there are nontrivial amount of outlying values under
the sample size 1000. It is usually the case that the numerical optimization does not work
2 There might be infinitely many such joint distribution and the multivariate χ2 distribution, which is not used here, is just one of them. The reason why multivariate χ2 distribution

is not used is that 1) it is hard to generate the DGP 3 by Stata, and that 2) what matters
is just the failure of the normality for each marginal distribution and its impact on the estimation. In this DGP 3, I generated χ2 distributions by first generating many basis standard
normal distributions. The correlations were created by using some common basis standard
normal distribution. The Stata code will be provided upon request.
92

well in smaller sample size and gives many dubious estimates. In the Table F.1 and F.2, the
outlying values that are greater or smaller than the maximum and minimum values for the
sample size of 3000 were discarded from the generated data. Roughly about 8 to 10% of
the total data is removed in this way. However, whole data are kept for median and MAD
calculation which are not sensitive to the extreme values.
There seems to be no discernible difference between ρ = 0.3 and 0.5 The column for DGP
2 shows that for those two tables the results do not show any clear sign of the adverse impact
due to the violation of the independence assumption. Therefore we can conclude that the
independence assumption is more or less practically negligible.
The last column is for DGP 3, where the multivariate normal assumption is violated. In
this case, it is clearly visible that the estimator suffers from larger variance and bias even
under 5000 observations. It happens because we used the fact that E[exp(u)] = exp(σ 2 /2)
under normality in Lemma 3.1. If the normality fails, then the whole estimating equation
will be misspecified causing inconsistency.
A comparison between linear and nonlinear estimators shows that the nonlinear estimator
has larger variance particularly when the sample size is small. However, it is also clear that
the linear estimators do not show any sign of consistency. The nonlinear estimator keeps
approaching the true value, i.e. one, from sample size of 3000 to 5000, the linear estimator
does not show any convergence as the sample sizes grow. Another interesting aspect is that
the skewed distribution of errors not only affects the nonlinear estimator, but also the linear
one too. This is particularly true in Table F.2; in column III, the linear estimator is actually
moving away from the true value one. In sum, although having larger variance, the nonlinear
estimator is better than the linear one in terms of consistency. It is preferable to use the
nonlinear ATE estimator when the sample size is sufficiently large.
93

3.7

An application: the effect of elementary school education on fertility in Botswana

In this section, the ATE estimator derived above will be applied to the Botswana fertility
data set from Wooldridge (2010, Chapter 21). Detailed descriptions of the data set are
already given in Chapter 2 and thus are omitted here. Before the regression results are
presented, it warrants some justification at this point why the data can be analyzed by the
CRC model framework. Consider a regression equation as below.

children = β0 + b1 age + β2 agesq + u

For the time being let us focus on the motivation of CRC modeling and ignore other covariates
that might also be present in the structural equation. The interpretation of β1 is the marginal
birth per year that is determined by the opportunity cost of childbearing (Becker and Barro,
1988; Barro and Becker, 1989). Since education increases human capital, higher education
would lead to lower b1 . For now just assume β2 is constant. Then b1 is a negative function
of years of education. Thus the above equation becomes

children = β0 + b1 (educ)age + β2 agesq + u.

It is assumed in the above model that agesq does not have the CRC. In other words, all
the heterogeneity of age are assumed to be absorbed into age only. This assumption helps
us dispense with age4 that might possibly cause much trouble in exponential regression.
Now suppose educ is continuous. Then for each value of educ, there are uncountably many

94

regression equations with constant coefficient. Let us divide the support of educ by using
threshold 7, i.e. the duration of primary education in Botswana to have a model below.

children0 = β00 + b01 (educ)age + β02 agesq + u0
children1 = β10 + b11 (educ)age + β12 agesq + u1

The upper equation is defined on {ω|educ(ω) < 7} whereas the lower one on the complement
set.
Next suppose educ is not available but only educ7 is. The information reduction is done
by the following rule.

educ7 = 1 if educ > 7
educ7 = 0 if educ < 7,

where educ = zδ + v. Then the above model can be written as

children0 = β00 + b01 (v)age + β02 agesq + u0
children1 = β10 + b11 (v)age + β12 agesq + u1
educ7 = 1[zδ + v > 0]

95

Considering the nonnegativity of children,

E[children0 |age, u0 ] = exp(β00 + b01 (v)age + β02 agesq + u0 )
E[children1 |age, u1 ] = exp(β10 + b11 (v)age + β12 agesq + u1 )
educ7 = 1[zδ + v > 0]

It is expected that there are negative correlations between bg1 and v; in other words σdg v s in
equation (3.3) might be negative. Higher education can reduce fertility either by increasing
opportunity cost or by providing people with more information on, say, contraception. Although omitted for simplicity so far, other variables such as evermarr, urban, and electric,
which might have CRC’s too, will also be included in the following regression. Therefore all
the variables but agesq have the CRC’s and their square and cross product will be included
in the estimating equations.
Before running regression using NTCRC model, it might be a good idea to test for
endogeneity by VAT. The test statistic for both the regime that follows χ2 (10) is 46.76 with
p-value .0000. The test statistics for Regime 1 and 0 that follow χ2 (5) are 26.13 with p-value
.0001 and 20.64 with p-value .0009. Thus the null hypothesis of no endogeneity is rejected
and we will run a regression with full-blown model.
The regression results for LFES and NFES models are summarized in Tables F.3 and F.4.
First and second columns in Table F.3 show the results by OLS and 2SLS. Next columns are
the ones by LFES estimated by Heckman’s correction method. NFES model is estimated
by using both Poisson Quasi-ML and NLS methods. The results for NTCRC model are
provided separately in Table F.4. Although both NLS and Poisson QMLE is available also
for NTCRC, the estimate for NLS does not give a reasonable value; the estimated ATE for
96

NLS is −5.28 and thus only the QMLE results are listed. All the standard errors in Tables
F.3 and F.4 are bootstrap standard errors. The bootstrap ran 500 replications. Among those
the bootstrap standard error for ATE in NTCRC model was particularly large. Since the
maximum number of children in the data set was 13, it might be highly unlikely that the
absolute value of ATE is greater than 13. By that reasoning, all the data with estimated
ATE over 13 were discarded to get the bootstrap standard error presented in the table. In
Table F.4, each basic covariate is numerically labeled such as age (1), evermarr (2) and so
forth. Using that numbers cov(1)=cov(δ1 , v), cov(2)=cov(δ2 , v), and so on.
Let us now see the estimation results for NTCRC model. As it was already expected in
theory, there are indeed, although insignificant, negative correlations between the selection
error and the coefficients of age for both the regimes. The estimated ATE is −1.020 which
is very close to the estimates from NFES model in Table F.3. Let us now see the LTCRC
estimates in Table F.4. The ATE estimate is substantially large and highly significant,
which is indicative of inconsistency of the estimator. Remember that both LTCRC and
NTCRC estimating equations are based on the conditional expectation and thus these two
estimators cannot be consistent at the same time. As long as we are convinced that the
effect of education on fertility is negative, the lesson is that the inconsistency of LTCRC can
be substantial, which supports the usefulness of NTCRC model provided in this article.
Finally one can perform the model selection test discussed in Section 5.2. The null
hypothesis that this model is in fact NFES is that the coefficients of all the bilateral products
as well as the covariances between δ and v are equal to zero. The Wald statistics of this null
hypothesis for Regime 1, Regime 0 and both the regimes are 106.74 (df=10), 339.63 (df=10)
and 602.32 (df=20) with p-values all equal to .0000, which justifies the model specification
as NTCRC.
97

3.8

Conclusion

We have so far considered the endogenous switching regression where there is a random
coefficient correlated with the switching variable under the count dependent variable. The
count dependent variable requires a nonlinear modeling using the exponential conditional
mean function. Although it is impossible to identify the means of each CRC, we have seen
that the ATE, which might be more interesting, can be identified. Simulations show that
this ATE estimator by nonlinear two-regime CRC model performs well with large sample
size.

98

APPENDICES

99

Appendix A
Simulation Results for Series
Estimator

100

Note to Tables A.1-12:

Each table presents the series estimator results for specified correlation and selection error.
Simulations were run for ρ = .0, .4, .5 and .6. For each correlation, three types of selection
error were considered. For a table, each column shows the monotone functions, i.e. identity,
inverse Mill’s ratio, and normal CDF used for binary choice index. Subcolumns p = 1, 2 and
3 specify the degree of polynomial of monotone function. Each row shows the distribution
used for u for each sample size.

101

Table A.1: Simulation results for series estimators with ρ = 0.0, ξ: normal
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
2 (5)
u: χ
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
0.905
2.497
2.499
1.009
0.177
0.177
1.002
0.070
0.070
0.890
2.048
2.050
1.000
0.188
0.188
1.000
0.074
0.074
1.047
1.665
1.665
1.002
0.108
0.108
0.997
0.071
0.071

identity
p=2
p=3
0.672
0.184
5.768
18.913
5.777
18.931
1.010
1.018
0.180
0.292
0.180
0.292
1.001
1.002
0.070
0.072
0.070
0.072
0.668
-0.235
4.759
28.807
4.771
28.833
1.001
1.009
0.186
0.309
0.186
0.310
1.000
1.000
0.075
0.077
0.075
0.077
1.066
66.547
2.042 1592.114
2.043 1593.462
1.000
0.918
0.141
2.064
0.141
2.066
0.997
0.998
0.072
0.073
0.072
0.073

p=1
1.238
13.086
13.088
1.104
1.760
1.763
1.030
0.732
0.733
1.673
22.520
22.530
1.121
2.398
2.401
1.013
0.739
0.739
0.664
10.238
10.244
0.940
1.753
1.754
1.008
0.997
0.997

102

IMR
p=2
-4.003
160.863
160.940
1.470
9.016
9.028
0.870
2.567
2.570
-2.652
139.004
139.052
1.694
15.246
15.262
0.891
2.419
2.421
-0.076
30.095
30.114
1.242
4.146
4.153
1.062
2.668
2.669

p=3
-4.536
2144.130
2144.137
2.085
40.975
40.989
0.758
18.714
18.715
-66.012
1954.183
1955.331
1.946
38.925
38.936
0.751
17.300
17.302
-16.218
415.140
415.497
-2.349
48.550
48.666
0.387
16.232
16.243

p=1
0.002
15.011
15.044
0.956
2.212
2.213
1.006
1.101
1.101
-0.380
26.305
26.342
0.941
3.741
3.742
0.988
0.951
0.951
1.412
12.484
12.490
1.040
2.220
2.220
0.973
1.403
1.403

normal CDF
p=2
p=3
-9.141
-2.676
207.018 2245.681
207.266 2245.684
0.997
0.117
8.300
46.724
8.300
46.732
0.861
0.650
2.712
24.053
2.716
24.056
-8.688
47.569
161.743 1967.634
162.033 1968.185
0.932
0.402
11.778
44.746
11.778
44.750
0.821
0.650
2.682
22.353
2.688
22.356
1.304
23.753
26.002 518.243
26.004 518.742
-4.611
1.482
142.931
67.768
143.041
67.770
1.011
1.909
2.848
21.341
2.848
21.361

Table A.2: Results for ρ = 0.0, ξ: t(5)
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
1.462
11.395
11.404
1.000
0.095
0.095
1.001
0.064
0.064
1.251
3.150
3.160
0.959
2.136
2.136
0.995
0.069
0.069
1.727
12.393
12.415
1.002
0.121
0.121
0.996
0.069
0.069

identity
p=2
1.470
11.423
11.433
1.001
0.095
0.095
1.000
0.065
0.065
1.118
2.484
2.487
0.962
2.109
2.109
0.995
0.070
0.070
1.624
10.564
10.582
1.001
0.125
0.125
0.996
0.069
0.069

p=3
1.215
13.376
13.378
1.000
0.102
0.102
1.001
0.067
0.067
2.949
39.619
39.667
0.997
1.384
1.384
0.995
0.072
0.073
1.502
10.555
10.567
1.000
0.128
0.128
0.997
0.072
0.072

p=1
0.537
5.228
5.248
1.021
0.996
0.996
0.966
0.563
0.564
0.711
4.924
4.933
0.991
1.261
1.261
0.999
0.530
0.530
0.530
10.995
11.005
1.010
0.990
0.990
1.011
0.543
0.543

103

IMR
p=2
p=3
8.813
18.942
122.882 580.978
123.130 581.255
1.183
3.112
2.973
31.698
2.979
31.769
0.872
0.903
1.473
9.677
1.478
9.678
0.669
-0.795
33.028 1066.294
33.030 1066.296
1.054
2.382
3.918
28.998
3.918
29.031
1.050
0.995
1.426
9.878
1.427
9.878
-6.811
-15.234
173.140 262.276
173.316 262.778
0.891
2.720
3.382
27.108
3.384
27.162
0.957
0.937
1.475
9.421
1.476
9.421

p=1
-1.648
74.963
75.010
0.969
1.305
1.306
1.043
0.586
0.588
0.975
8.052
8.052
1.036
1.589
1.589
0.986
0.566
0.566
2.146
34.055
34.075
1.014
1.669
1.669
0.983
0.562
0.562

normal CDF
p=2
p=3
1.447
-4.180
83.649 524.571
83.650 524.597
1.150
-0.667
3.011
36.946
3.015
36.984
0.942
0.613
1.870
13.924
1.871
13.930
-0.022
-24.906
33.815 2086.952
33.830 2087.113
1.192
2.860
3.125 106.597
3.131 106.614
1.041
0.938
1.843
14.212
1.843
14.213
-2.217
19.497
154.943 392.817
154.977 393.252
0.863
-1.823
3.076
33.335
3.079
33.455
0.908
0.815
1.926
13.655
1.928
13.656

Table A.3: Results for ρ = 0.0, ξ: χ2 (5)
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
-0.190
23.632
23.662
1.027
2.031
2.031
0.789
5.320
5.325
0.753
6.102
6.107
1.027
1.071
1.071
0.990
0.155
0.155
1.502
14.174
14.183
0.846
2.177
2.182
0.994
0.207
0.207

identity
p=2
-0.139
25.422
25.447
1.032
2.024
2.024
0.808
4.891
4.894
0.535
7.119
7.134
1.022
1.021
1.022
-1.301
56.188
56.235
1.480
14.044
14.052
0.860
2.259
2.264
0.998
0.199
0.199

p=3
4.089
114.077
114.119
1.025
2.064
2.064
0.790
5.353
5.357
0.048
18.836
18.860
1.028
1.048
1.048
-1.298
56.188
56.235
1.448
16.133
16.139
1.818
23.846
23.860
1.000
0.211
0.211

p=1
0.819
6.847
6.849
1.048
1.285
1.286
0.963
0.629
0.630
1.108
6.519
6.520
0.984
1.848
1.848
0.978
0.721
0.721
1.201
4.734
4.738
0.880
1.861
1.865
0.979
0.915
0.915

104

IMR
p=2
p=3
2.004
5.072
42.215 831.961
42.227 831.971
1.289
2.570
5.439
35.175
5.446
35.210
0.900
0.704
2.695
18.370
2.696
18.373
-9.782
-7.016
175.169 1084.230
175.501 1084.260
1.196
-1.794
5.267 102.992
5.271 103.030
1.148
1.030
2.230
18.927
2.235
18.927
0.357
-26.260
53.792 469.023
53.796 469.815
2.901
-0.413
49.073
72.928
49.110
72.941
1.057
0.765
3.777
17.255
3.777
17.257

normal CDF
p=1
p=2
p=3
1.272
3.022 -14.460
7.037
56.951 903.143
7.042
56.987 903.275
0.956
1.175
-0.249
1.964
4.368
45.661
1.965
4.371
45.678
1.044
1.072
0.845
0.655
2.589
23.248
0.656
2.590
23.249
1.195
-9.550
34.038
8.013 186.925 976.187
8.016 187.222 976.746
1.075
1.094
4.281
1.914
4.474 120.464
1.915
4.475 120.508
0.970
1.126
0.977
0.757
2.648
25.814
0.758
2.651
25.814
0.628
-1.474
14.801
8.254
51.984 753.007
8.262
52.043 753.133
1.084
3.467
7.182
2.138
53.878 107.651
2.140
53.934 107.828
0.965
1.162
1.735
0.880
3.305
23.844
0.880
3.309
23.855

Table A.4: Results for ρ = 0.4, ξ: normal
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
1.585
1.435
1.550
1.656
0.211
0.689
1.653
0.063
0.656
1.602
1.279
1.414
1.654
0.186
0.680
1.654
0.067
0.657
1.630
0.669
0.919
1.554
1.465
1.566
1.617
0.069
0.621

identity
p=2
1.550
1.571
1.665
1.656
0.205
0.688
1.653
0.064
0.657
1.577
1.441
1.553
1.655
0.178
0.679
1.655
0.068
0.658
1.647
0.709
0.960
1.555
1.469
1.570
1.618
0.069
0.622

p=3
1.850
10.140
10.175
1.657
0.204
0.688
1.653
0.067
0.657
1.800
7.871
7.911
1.655
0.181
0.680
1.655
0.072
0.659
1.643
0.846
1.063
1.552
1.528
1.625
1.618
0.071
0.622

p=1
1.307
5.927
5.935
0.957
1.796
1.796
0.945
1.440
1.441
1.335
5.523
5.534
0.990
1.695
1.695
0.935
1.120
1.122
0.557
13.044
13.051
0.954
1.930
1.931
0.888
0.943
0.950

105

IMR
p=2
-1.812
63.999
64.061
0.705
6.261
6.268
0.949
5.583
5.583
-1.399
58.693
58.742
0.830
5.922
5.925
0.943
5.478
5.478
2.528
32.373
32.409
0.681
11.416
11.421
1.013
2.861
2.861

p=3
-244.475
4349.198
4356.120
-0.264
32.845
32.870
1.195
17.378
17.379
-218.822
3561.923
3568.700
0.255
35.032
35.040
1.534
18.083
18.091
54.997
1091.634
1092.969
-0.374
56.784
56.800
-0.488
20.391
20.445

p=1
1.860
8.261
8.306
2.376
2.813
3.131
2.296
3.409
3.647
1.455
9.688
9.699
2.342
2.826
3.129
2.301
3.116
3.377
1.883
17.769
17.791
2.471
5.579
5.769
2.290
1.436
1.931

normal CDF
p=2
p=3
-0.862
98.108
66.486 3190.363
66.512 3191.841
2.823
4.548
4.549
40.649
4.900
40.803
2.512
2.830
4.274
22.629
4.534
22.703
-1.487
78.668
62.218 3133.514
62.268 3134.476
2.798
4.030
4.555
42.870
4.897
42.977
2.504
2.436
3.948
23.089
4.225
23.134
2.775
-51.650
32.364 1110.114
32.412 1111.362
2.601
7.852
6.516 152.745
6.710 152.899
2.712
5.325
2.837
25.032
3.313
25.402

Table A.5: Results for ρ = 0.4, ξ: t(5)
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
2 (5)
u: χ
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
1.409
2.607
2.639
1.598
0.086
0.604
1.604
0.054
0.607
1.751
15.885
15.903
1.650
0.891
1.102
1.602
0.067
0.605
1.774
1.966
2.113
1.696
0.115
0.705
1.690
0.061
0.693

identity
p=2
2.225
18.243
18.284
1.600
0.087
0.607
1.606
0.055
0.609
1.782
16.054
16.073
1.652
0.873
1.090
1.604
0.068
0.608
1.799
2.097
2.244
1.698
0.119
0.708
1.691
0.062
0.693

p=3
2.288
19.057
19.100
1.600
0.092
0.607
1.605
0.057
0.608
1.720
15.538
15.555
1.653
0.856
1.076
1.604
0.071
0.608
2.166
7.903
7.988
1.695
0.119
0.705
1.690
0.063
0.693

p=1
0.560
5.776
5.793
0.875
1.128
1.135
0.965
0.504
0.505
1.101
5.884
5.884
0.875
1.206
1.213
0.945
0.565
0.568
0.825
4.384
4.387
0.893
1.105
1.111
0.942
0.520
0.523

106

IMR
p=2
0.223
18.087
18.104
0.830
2.996
3.000
1.015
1.392
1.392
7.428
80.305
80.562
1.210
5.961
5.965
1.136
1.612
1.617
2.110
50.278
50.291
0.766
3.736
3.744
0.807
1.468
1.481

p=3
6.892
711.156
711.180
0.825
14.825
14.826
1.373
8.595
8.603
-10.197
691.627
691.718
1.918
26.496
26.512
1.697
10.620
10.642
-40.554
767.518
768.642
0.666
15.999
16.003
1.360
8.368
8.376

p=1
2.466
8.557
8.682
2.348
1.330
1.894
2.244
0.523
1.349
1.515
9.981
9.995
2.238
3.160
3.394
2.249
0.583
1.379
2.742
6.341
6.576
2.476
2.016
2.499
2.446
0.529
1.540

normal CDF
p=2
p=3
3.017
-12.984
16.688 1016.387
16.809 1016.483
2.796
3.758
2.956
20.996
3.459
21.176
2.647
2.821
1.852
12.566
2.478
12.697
5.061
11.073
61.300 747.279
61.435 747.347
2.803
3.455
4.053
30.221
4.436
30.321
2.835
3.044
2.011
15.660
2.722
15.793
5.056
72.784
58.716 1266.234
58.856 1268.267
2.818
3.713
3.012
21.298
3.518
21.470
2.653
2.657
1.848
12.220
2.479
12.332

Table A.6: Results for ρ = 0.4, ξ: χ2 (5)
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
1.726
3.445
3.521
0.807
19.300
19.301
1.602
0.251
0.652
0.036
34.923
34.937
1.227
9.042
9.045
0.855
18.134
18.135
1.454
3.380
3.411
1.233
9.881
9.884
1.615
0.129
0.628

identity
p=2
1.225
12.631
12.633
0.825
19.152
19.153
1.606
0.248
0.654
0.220
33.714
33.723
1.252
8.805
8.808
0.872
17.918
17.918
1.521
3.689
3.726
1.248
9.540
9.544
1.621
0.137
0.636

p=3
4.124
48.106
48.208
0.697
22.329
22.331
1.606
0.251
0.656
0.271
52.038
52.043
1.051
10.061
10.061
0.830
18.865
18.866
1.707
7.703
7.736
1.243
9.832
9.835
1.627
0.150
0.645

p=1
1.235
5.758
5.763
0.860
1.501
1.507
0.904
0.856
0.861
1.478
5.905
5.924
0.837
1.600
1.608
0.874
0.866
0.875
1.160
6.246
6.248
0.783
2.428
2.437
0.902
0.757
0.764

107

IMR
p=2
0.229
26.793
26.804
1.041
7.222
7.222
0.390
6.148
6.178
0.467
65.448
65.451
0.913
5.490
5.491
0.648
5.677
5.688
1.187
21.007
21.008
0.877
9.586
9.587
0.860
2.054
2.059

p=3
30.498
1313.152
1313.483
-1.822
61.044
61.109
0.960
17.520
17.520
7.669
690.188
690.220
-0.252
38.116
38.137
-0.647
39.131
39.166
-6.888
1121.906
1121.934
-1.680
50.114
50.185
-0.128
16.384
16.423

p=1
2.139
5.254
5.376
2.357
1.903
2.338
2.366
0.875
1.622
2.376
9.930
10.025
2.428
1.710
2.228
2.351
0.972
1.664
2.169
6.961
7.059
2.413
2.987
3.304
2.341
0.681
1.504

normal CDF
p=2
p=3
0.775
24.472
30.090 1332.741
30.091 1332.948
2.733
2.109
4.446
69.006
4.772
69.015
2.528
2.314
3.334
24.728
3.667
24.763
4.986
-12.510
59.551 814.498
59.684 814.610
3.008
5.698
5.476
45.458
5.832
45.700
2.774
2.880
2.808
33.121
3.322
33.174
2.538
-40.546
19.943 1167.326
20.002 1168.065
2.818
10.402
5.788 135.818
6.067 136.143
2.694
5.406
2.356
22.334
2.902
22.765

Table A.7: Results for ρ = 0.5, ξ: normal
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

identity
p=1
p=2
p=3
1.751
1.715
1.997
1.401
1.517
9.293
1.590
1.677
9.347
1.821
1.822
1.823
0.222
0.216
0.214
0.851
0.850
0.850
1.816
1.817
1.817
0.060
0.060
0.064
0.818
0.819
0.819
1.771
1.746
1.898
1.184
1.338
5.350
1.413
1.532
5.425
1.821
1.822
1.822
0.196
0.186
0.189
0.844
0.842
0.843
1.817
1.818
1.818
0.065
0.066
0.070
0.820
0.821
0.821
3.642 -152.907 -152.886
54.878 3751.113 3751.115
54.941 3754.269 3754.270
1.772
1.772
1.772
0.104
0.105
0.113
0.779
0.779
0.781
1.765
1.767
1.771
0.066
0.071
0.108
0.768
0.770
0.779

108

p=1
1.308
5.795
5.803
0.940
1.869
1.870
0.925
1.420
1.422
1.353
5.525
5.537
0.974
1.750
1.751
0.918
1.106
1.109
0.779
5.864
5.868
0.906
2.089
2.091
0.933
0.981
0.983

IMR
p=2
p=3
-1.748 -223.241
66.855 3905.942
66.911 3912.374
0.681
-0.167
6.120
32.301
6.128
32.322
0.938
1.166
5.778
17.438
5.778
17.439
-1.332 -211.368
60.928 3223.614
60.973 3230.602
0.812
0.415
5.686
33.790
5.689
33.795
0.939
1.506
5.655
18.523
5.655
18.530
3.737
41.465
64.437 888.309
64.495 889.230
0.239
-61.222
7.406 1422.404
7.445 1423.764
1.817
-72.234
16.911 1492.984
16.931 1494.779

p=1
2.225
8.544
8.631
2.715
2.960
3.421
2.640
3.503
3.867
1.807
9.346
9.381
2.673
3.034
3.464
2.645
3.158
3.561
2.791
7.768
7.972
2.842
3.171
3.668
2.636
1.243
2.055

normal CDF
p=2
p=3
-0.054
71.763
69.193 2887.926
69.201 2888.793
3.226
4.965
4.529
40.405
5.046
40.599
2.944
3.502
4.298
22.576
4.717
22.714
-0.759
61.653
64.255 2838.222
64.279 2838.870
3.180
4.326
4.551
41.661
5.046
41.794
2.941
3.133
3.929
23.448
4.382
23.544
7.904
-19.124
78.898 1295.865
79.199 1296.021
2.999
29.286
5.429 527.206
5.785 527.964
3.944
-73.574
15.817 1969.699
16.088 1971.110

Table A.8: Results for ρ = 0.5, ξ: t(5)
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
1.306
7.878
7.883
1.753
0.082
0.757
1.759
0.050
0.761
2.265
11.937
12.004
1.755
0.100
0.761
1.755
0.065
0.758
2.110
6.737
6.828
1.859
0.159
0.874
1.862
0.058
0.864

identity
p=2
2.090
11.556
11.607
1.756
0.083
0.761
1.762
0.051
0.764
2.313
12.426
12.495
1.759
0.104
0.766
1.758
0.067
0.761
2.119
6.601
6.695
1.861
0.167
0.877
1.863
0.058
0.865

p=3
2.152
12.338
12.392
1.757
0.088
0.762
1.761
0.052
0.763
2.238
10.230
10.305
1.764
0.145
0.777
1.759
0.069
0.762
5.009
70.993
71.106
1.859
0.172
0.876
1.863
0.060
0.865

p=1
0.556
5.503
5.521
0.851
1.132
1.142
0.949
0.511
0.513
1.085
5.171
5.172
0.943
1.011
1.013
0.913
0.639
0.645
0.961
4.437
4.437
0.871
1.420
1.426
0.944
0.558
0.561

109

IMR
p=2
0.455
21.766
21.773
0.789
3.022
3.029
1.016
1.368
1.368
0.694
26.726
26.728
1.104
2.692
2.694
1.067
2.300
2.300
2.997
40.355
40.405
0.419
5.204
5.236
0.771
1.432
1.450

p=3
-3.359
879.925
879.936
0.739
14.768
14.770
1.397
8.331
8.341
8.374
407.675
407.741
12.207
268.626
268.860
14.594
342.987
343.256
-32.207
887.756
888.377
-0.556
21.634
21.690
0.537
7.954
7.967

normal CDF
p=1
p=2
p=3
2.801
3.397
-0.139
8.941
19.344 1156.982
9.121
19.492 1156.983
2.697
3.249
4.499
1.336
2.888
20.764
2.160
3.661
21.057
2.569
3.082
3.437
0.529
1.809
12.175
1.656
2.758
12.417
2.778
3.693
1.471
7.736
20.710 410.407
7.937
20.885 410.407
2.575
3.202
-7.942
1.018
2.993 292.329
1.876
3.716 292.466
2.587
3.175
-8.899
0.738
2.671 337.136
1.751
3.445 337.281
2.458
60.007
20.749
5.877 1359.200 476.175
6.055 1360.480 476.584
2.917
3.209
5.544
1.956
3.350
29.005
2.739
4.013
29.359
2.800
3.063
4.313
0.603
1.703
11.422
1.898
2.675
11.892

Table A.9: Results for ρ = 0.5, ξ: χ2 (5)
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
1.820
2.751
2.871
1.555
4.775
4.807
1.758
0.258
0.800
0.254
33.037
33.046
1.335
10.238
10.243
0.997
18.615
18.615
2.585
19.918
19.981
1.758
0.244
0.797
1.755
0.296
0.811

identity
p=2
1.379
11.212
11.218
1.577
4.544
4.580
1.763
0.255
0.804
0.447
31.418
31.423
1.364
9.983
9.990
1.015
18.424
18.424
2.533
19.806
19.865
1.760
0.295
0.816
1.764
0.300
0.820

p=3
3.770
41.059
41.153
1.439
7.980
7.992
1.763
0.258
0.806
1.164
47.442
47.442
1.227
10.567
10.569
0.966
19.523
19.524
4.956
45.317
45.490
1.748
0.476
0.886
1.763
0.309
0.823

p=1
1.261
5.494
5.500
0.831
1.605
1.614
0.877
0.849
0.858
1.498
5.905
5.926
0.790
1.641
1.655
0.845
0.862
0.876
1.197
6.037
6.040
0.711
1.832
1.855
0.845
1.157
1.168

110

IMR
p=2
0.199
26.939
26.951
0.947
7.436
7.436
0.320
6.540
6.575
0.300
60.848
60.852
0.855
5.729
5.731
0.541
5.734
5.752
3.327
52.412
52.463
-0.487
21.893
21.943
0.883
4.033
4.035

p=3
20.936
1211.318
1211.482
-1.624
54.468
54.532
0.833
17.694
17.695
0.825
648.799
648.799
-0.637
35.654
35.692
-0.616
36.839
36.874
5.477
1014.042
1014.052
-0.813
27.023
27.084
0.263
17.755
17.771

p=1
2.421
5.171
5.363
2.706
2.089
2.697
2.713
0.884
1.928
2.665
9.067
9.218
2.802
1.796
2.544
2.723
0.918
1.952
2.480
6.409
6.578
2.871
2.506
3.128
2.656
1.302
2.107

normal CDF
p=2
p=3
1.174
34.170
29.922 1349.311
29.923 1349.719
3.133
3.035
4.489
60.785
4.970
60.819
2.962
3.072
3.347
24.876
3.879
24.962
5.009
-8.262
55.814 760.331
55.958 760.387
3.505
6.911
5.696
43.801
6.223
44.198
3.206
3.572
2.803
32.939
3.567
33.039
4.877
4.803
46.194 1058.208
46.356 1058.215
2.357
4.825
21.745
36.775
21.787
36.973
3.040
4.997
3.414
24.746
3.977
25.066

Table A.10: Results for ρ = 0.6, ξ: normal
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
1.915
1.342
1.624
1.982
0.238
1.011
1.974
0.055
0.975
1.943
1.052
1.412
1.985
0.225
1.010
1.976
0.063
0.978
1.955
0.764
1.223
1.917
0.217
0.942
1.917
0.066
0.919

identity
p=2
1.879
1.436
1.684
1.983
0.232
1.010
1.975
0.056
0.977
1.918
1.185
1.499
1.986
0.214
1.009
1.977
0.064
0.980
1.922
1.123
1.453
1.923
0.250
0.956
1.918
0.066
0.920

p=3
2.143
8.169
8.249
1.983
0.231
1.010
1.975
0.059
0.976
2.008
3.545
3.686
1.987
0.215
1.010
1.977
0.067
0.979
2.132
4.645
4.781
1.923
0.253
0.957
1.918
0.069
0.920

p=1
1.300
5.694
5.702
0.922
1.948
1.949
0.906
1.366
1.369
1.363
5.512
5.524
0.958
1.815
1.816
0.902
1.058
1.062
1.121
5.532
5.534
0.840
2.353
2.358
0.861
0.819
0.831

111

IMR
p=2
-1.608
69.991
70.040
0.673
6.047
6.055
0.927
5.835
5.836
-1.144
64.164
64.200
0.819
5.594
5.597
0.933
5.529
5.529
-4.785
152.666
152.775
0.659
9.886
9.892
0.827
3.603
3.607

p=3
-193.518
3321.385
3327.076
0.004
31.575
31.591
1.122
17.307
17.308
-192.521
2704.632
2711.547
0.686
31.594
31.596
1.423
18.686
18.691
4.184
782.850
782.856
-8.181
348.054
348.175
-0.399
32.080
32.111

p=1
2.605
9.142
9.282
3.038
3.186
3.782
2.973
3.507
4.025
2.197
9.076
9.154
2.987
3.362
3.905
2.984
3.014
3.608
3.445
14.623
14.826
3.224
3.657
4.280
3.029
1.385
2.457

normal CDF
p=2
p=3
0.895
38.808
72.123 2551.367
72.123 2551.647
3.603
5.272
4.544
39.956
5.237
40.184
3.367
4.184
4.235
22.248
4.851
22.475
0.241
35.246
66.997 2508.761
67.002 2508.995
3.526
4.452
4.624
39.343
5.269
39.494
3.378
3.910
3.744
23.460
4.435
23.639
0.366
17.115
152.234 1219.060
152.236 1219.167
3.797
23.476
8.948 401.758
9.375 402.387
3.584
9.358
3.346
87.408
4.228
87.807

Table A.11: Results for ρ = 0.6, ξ: t(5)
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
1.446
7.859
7.871
1.912
0.077
0.915
1.918
0.045
0.919
1.143
19.862
19.863
1.913
0.121
0.921
1.900
0.065
0.902
1.994
0.596
1.159
2.033
0.092
1.037
2.035
0.056
1.036

identity
p=2
2.157
9.976
10.043
1.916
0.078
0.920
1.921
0.045
0.922
1.088
21.975
21.976
1.916
0.121
0.924
1.903
0.065
0.905
2.570
13.838
13.926
2.035
0.095
1.039
2.053
0.410
1.130

p=3
2.219
10.750
10.819
1.917
0.083
0.921
1.920
0.047
0.921
1.189
19.205
19.206
1.916
0.126
0.925
1.903
0.068
0.905
-14.227
396.369
396.661
2.036
0.098
1.040
2.052
0.410
1.129

p=1
0.562
5.293
5.311
0.828
1.160
1.172
0.931
0.524
0.529
0.728
4.783
4.791
0.772
1.499
1.516
0.903
0.739
0.746
0.941
4.994
4.994
0.898
1.498
1.501
0.925
0.563
0.568

112

IMR
p=2
p=3
0.498
-11.457
23.983 1027.405
23.988 1027.481
0.735
0.709
3.163
14.463
3.174
14.466
1.013
1.409
1.344
8.027
1.344
8.038
2.240
3.905
18.476 358.296
18.518 358.307
0.933
1.523
4.518
25.017
4.518
25.022
0.862
0.986
5.396
20.919
5.398
20.919
1.218
-17.108
16.735 495.433
16.736 495.763
0.822
0.082
3.789
15.135
3.793
15.163
0.960
0.085
4.975
25.166
4.975
25.183

p=1
3.171
9.272
9.523
3.057
1.432
2.507
2.904
0.541
1.980
2.845
6.649
6.901
3.125
2.401
3.207
2.893
0.767
2.042
3.017
6.849
7.140
3.162
2.345
3.190
3.161
0.584
2.238

normal CDF
p=2
p=3
3.872
11.510
20.461 1310.067
20.662 1310.109
3.702
5.147
2.830
20.154
3.912
20.576
3.528
4.083
1.754
11.680
3.077
12.080
4.253
51.089
16.360 1135.995
16.680 1137.099
3.963
4.964
4.036
31.489
5.007
31.737
3.412
4.494
5.591
21.474
6.089
21.756
20.454
20.701
414.109 535.525
414.566 535.888
3.603
5.496
3.285
21.537
4.191
22.001
3.725
6.038
5.150
44.941
5.827
45.223

Table A.12: Results for ρ = 0.6, ξ: χ2 (5)
monotone functions
degrees of polynomial
obs=100
mean
s. d.
RMSE
obs=500
mean
u: normal
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: t(5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE
obs=100
mean
s. d.
RMSE
obs=500
mean
u: χ2 (5)
s. d.
RMSE
obs=1000
mean
s. d.
RMSE

p=1
1.978
3.035
3.189
2.358
10.869
10.954
1.923
0.248
0.956
0.571
27.921
27.925
1.463
11.088
11.098
1.182
18.243
18.244
-7.626
237.699
237.856
1.856
0.886
1.232
1.895
0.163
0.910

identity
p=2
1.641
8.456
8.480
2.384
11.195
11.280
1.930
0.244
0.962
0.746
26.659
26.660
1.494
10.818
10.830
1.200
18.064
18.065
-2.993
251.249
251.281
1.885
0.909
1.269
1.923
0.191
0.942

p=3
3.899
40.539
40.642
2.252
7.897
7.996
1.930
0.247
0.962
2.299
39.932
39.953
1.460
10.984
10.993
1.145
19.343
19.344
-5.618
264.834
264.917
1.881
0.893
1.254
1.922
0.211
0.946

p=1
1.297
4.568
4.578
0.803
1.721
1.732
0.849
0.818
0.832
1.476
5.926
5.946
0.745
1.689
1.708
0.819
0.861
0.880
1.078
10.765
10.765
0.700
2.492
2.510
0.789
0.928
0.951

113

IMR
p=2
p=3
0.203
6.196
25.212 1097.418
25.224 1097.430
0.832
-1.308
7.599
46.393
7.601
46.450
0.265
0.721
6.557
17.481
6.598
17.483
0.254
-5.936
55.876 643.289
55.881 643.326
0.801
-1.061
6.117
32.627
6.120
32.692
0.442
-0.535
5.515
32.703
5.543
32.739
0.476
-17.813
109.312 826.419
109.313 826.633
0.331
0.582
8.895
31.604
8.920
31.606
0.782
1.053
2.266
15.294
2.277
15.294

p=1
2.702
4.901
5.188
3.067
2.277
3.076
3.080
0.911
2.270
2.959
8.291
8.519
3.182
1.899
2.893
3.106
0.908
2.293
2.942
13.383
13.523
3.173
2.064
2.997
3.073
0.850
2.241

normal CDF
p=2
p=3
1.417
45.284
27.891 1357.101
27.894 1357.823
3.521
4.062
4.513
48.607
5.170
48.704
3.415
3.858
3.255
24.531
4.052
24.697
5.113
-3.693
50.636 729.581
50.803 729.596
4.001
8.114
6.071
41.818
6.772
42.419
3.633
4.312
2.765
32.058
3.817
32.229
6.034
18.191
120.664 608.739
120.769 608.982
4.010
6.456
6.382
48.811
7.057
49.115
3.790
5.538
2.507
25.215
3.751
25.620

Appendix B
Simulation Results for All Other
Estimators

114

Note to Tables B.1-18:

Each table presents the all the results but series estimator for specified correlation and IV.
Simulations were run for ρ = .0, .4, .5 and .6. For a table, each column specifies the estimator
categorized under different types of distributions for u. Each row shows the distribution used
for selection error ξ for each sample size. M, S, R, Md denote mean, Monte Carlo standard
deviation, RMSE and median respectively.

115

Table B.1: Simulation Results: Strong IV and ρ = 0.0
obs
100

M
S
R
Md
500
M
ξ:
S
normal
R
Md
1000
M
S
R
Md
100
M
S
R
Md
500
M
ξ:
S
2 (5)
χ
R
Md
1000
M
S
R
Md

IV Heckit1
0.803
0.976
18.556
7.140
18.557
7.140
1.022
1.006
1.040
1.046
1.327
1.528
1.328
1.529
1.004
1.010
1.012
1.015
0.681
0.662
0.681
0.662
1.024
1.045
0.732
1.955
14.543
4.979
14.546
5.069
1.005
1.396
1.036
1.603
1.398
1.394
1.399
1.519
1.078
1.399
1.052
1.411
0.631
0.665
0.633
0.781
1.048
1.350

u: normal
Heckit2 QLIML QFIML
0.809
0.135
0.975
7.539
18.502
1.431
7.542
18.522
1.431
0.935
0.921
0.926
1.041
1.019
1.030
1.264
3.349
0.601
1.265
3.349
0.602
1.002
1.022
1.017
1.011
0.974
0.995
0.649
1.367
0.497
0.650
1.367
0.497
1.032
0.995
1.006
0.909
0.032
1.027
6.455
23.840
1.449
6.455
23.860
1.450
1.003
0.960
1.007
0.985
0.909
0.991
1.302
3.765
0.573
1.303
3.766
0.573
1.008
0.957
1.011
1.003
0.975
1.021
0.618
1.788
0.499
0.618
1.788
0.499
1.033
1.019
1.032

116

IV Heckit1
4.440
1.126
108.561
6.028
108.616
6.030
0.988
1.004
1.102
1.034
2.413
1.092
2.415
1.092
1.064
1.023
1.002
1.015
0.752
0.669
0.752
0.669
1.011
1.012
-2.554
2.144
99.600
5.654
99.663
5.769
1.346
1.422
0.893
1.555
1.455
1.386
1.459
1.493
1.104
1.363
1.017
1.390
0.708
0.686
0.708
0.789
1.116
1.310

u: χ2 (5)
Heckit2 QLIML QFIML
1.124
0.060
1.006
6.146
26.560
1.546
6.148
26.577
1.546
0.972
0.938
0.985
1.026
1.024
1.032
1.213
3.539
0.877
1.214
3.539
0.877
1.035
0.991
1.014
1.010
0.993
1.017
0.664
1.198
0.911
0.664
1.198
0.911
1.020
1.002
0.995
1.092
1.623
0.986
5.975
24.003
1.537
5.975
24.011
1.537
1.056
1.096
0.835
0.966
0.936
0.732
1.085
2.806
0.981
1.086
2.807
1.017
0.936
1.012
0.759
0.965
0.892
0.689
0.624
1.756
0.925
0.625
1.759
0.976
0.932
0.985
0.631

Table B.2: Results for Strong IV and ρ = 0.4
obs
100

500
ξ:
normal
1000

100

500
ξ:
χ2 (5)
1000

M
S
R
Md
M
S
R
Md
M
S
R
Md
M
S
R
Md
M
S
R
Md
M
S
R
Md

IV Heckit1
1.259
1.361
24.316
9.359
24.318
9.366
1.203
1.218
0.816
0.895
1.742
1.583
1.751
1.586
1.051
1.060
0.951
0.967
0.690
0.663
0.691
0.663
1.032
1.018
5.185
2.227
71.770
4.895
71.892
5.046
1.229
1.432
0.687
1.502
6.310
1.580
6.318
1.658
1.102
1.393
1.046
1.367
0.639
0.616
0.640
0.717
1.090
1.333

u: normal
Heckit2 QLIML QFIML
1.195
-1.082
1.294
8.948
25.431
1.307
8.950
25.516
1.340
1.194
1.136
1.280
0.881
0.626
1.245
1.353
3.221
0.548
1.358
3.243
0.600
1.080
1.031
1.237
0.960
0.900
1.059
0.648
0.928
0.619
0.649
0.933
0.621
1.028
0.980
1.088
1.156
-1.879
1.325
5.504
24.664
1.381
5.506
24.832
1.418
1.113
0.969
1.255
0.854
0.740
1.249
1.430
2.620
0.529
1.437
2.633
0.585
1.022
1.008
1.195
0.971
0.902
1.139
0.646
1.067
0.539
0.647
1.071
0.557
0.996
0.944
1.114

117

u: χ2 (5)
IV Heckit1 Heckit2 QLIML QFIML
0.320
1.171
0.847
-1.217
1.205
13.649
4.706
6.192
23.647
1.444
13.666
4.709
6.194
23.751
1.458
1.202
1.250
1.211
1.192
1.208
0.745
0.936
0.881
0.671
0.925
3.649
1.729
1.310
2.687
0.822
3.658
1.730
1.315
2.707
0.825
1.000
1.026
1.007
1.031
1.048
0.930
0.943
0.918
0.816
0.602
0.815
0.678
0.695
1.557
0.827
0.818
0.680
0.700
1.568
0.918
0.975
0.998
0.978
1.011
0.211
1.384
2.491
1.365
-0.404
1.175
43.334
8.577
9.176
25.317
1.555
43.336
8.705
9.183
25.356
1.565
1.239
1.566
1.196
1.046
1.105
0.918
1.525
0.899
0.667
0.805
2.085
1.318
1.405
3.649
0.866
2.086
1.419
1.408
3.664
0.887
1.129
1.409
1.062
0.996
0.757
1.021
1.323
0.937
0.761
0.482
0.631
0.601
0.671
1.858
0.842
0.631
0.683
0.674
1.874
0.988
1.058
1.317
0.997
0.960
0.075

Table B.3: Results for Strong IV and ρ = 0.5
obs
100

500
ξ:
normal
1000

100

500
ξ:
χ2 (5)
1000

M
S
R
Md
M
S
R
Md
M
S
R
Md
M
S
R
Md
M
S
R
Md
M
S
R
Md

IV Heckit1
0.979
1.357
24.720
9.161
24.720
9.168
1.251
1.233
0.775
0.869
1.864
1.589
1.878
1.595
1.052
1.075
0.935
0.952
0.715
0.670
0.718
0.671
1.030
1.025
4.799
2.193
59.319
4.692
59.440
4.841
1.265
1.483
0.648
1.464
7.227
1.549
7.236
1.617
1.093
1.390
1.031
1.342
0.644
0.591
0.645
0.683
1.098
1.324

u: normal
Heckit2 QLIML QFIML
1.158
-1.361
1.374
8.891
28.617
1.273
8.892
28.714
1.327
1.254
1.128
1.365
0.853
0.621
1.280
1.413
2.870
0.512
1.421
2.895
0.583
1.087
1.024
1.243
0.945
0.877
1.154
0.663
0.900
0.393
0.665
0.909
0.422
1.028
0.975
1.117
1.128
-0.612
1.395
5.354
33.681
1.303
5.355
33.720
1.362
1.165
1.153
1.306
0.819
0.909
1.290
1.501
3.030
0.490
1.512
3.031
0.570
1.025
1.009
1.232
0.950
0.850
1.190
0.651
1.291
0.388
0.653
1.300
0.432
0.992
0.966
1.121

118

u: χ2 (5)
IV Heckit1 Heckit2 QLIML QFIML
0.695
1.498
0.608
-0.633
1.361
37.964
9.773
15.743
23.151
1.361
37.965
9.786
15.748
23.208
1.408
1.214
1.269
1.162
1.312
1.274
0.681
0.901
0.829
0.567
0.998
3.592
2.078
2.201
3.691
0.664
3.606
2.081
2.207
3.717
0.664
1.016
1.062
1.012
0.999
0.652
0.864
0.921
0.877
0.808
0.707
1.030
0.704
0.778
1.624
0.469
1.039
0.708
0.787
1.635
0.553
1.011
1.019
0.993
1.004
0.518
0.647
1.756
0.866
-1.120
1.169
23.916
6.539
8.393
24.404
1.459
23.918
6.583
8.394
24.496
1.468
1.215
1.569
1.227
1.263
1.017
0.853
1.459
0.875
0.826
0.769
3.762
1.598
1.431
4.111
0.799
3.765
1.663
1.436
4.115
0.832
1.053
1.334
0.949
0.982
0.396
0.968
1.317
0.898
0.835
0.509
0.759
0.695
0.747
1.346
0.628
0.760
0.764
0.754
1.356
0.797
1.044
1.310
0.982
0.962
0.221

Table B.4: Results for Strong IV and ρ = 0.6
obs
100

500
ξ:
normal
1000

100

500
ξ:
χ2 (5)
1000

M
S
R
Md
M
S
R
Md
M
S
R
Md
M
S
R
Md
M
S
R
Md
M
S
R
Md

IV Heckit1
0.646
1.340
25.625
8.788
25.628
8.794
1.325
1.329
0.736
0.844
1.988
1.600
2.005
1.608
1.068
1.073
0.920
0.938
0.749
0.686
0.754
0.689
1.027
1.021
4.452
2.216
48.623
4.660
48.746
4.816
1.324
1.550
0.628
1.435
7.991
1.548
8.000
1.608
1.085
1.357
1.014
1.319
0.652
0.566
0.652
0.649
1.088
1.326

u: normal
Heckit2 QLIML QFIML
1.101
-0.581
1.407
8.717
20.334
1.251
8.717
20.395
1.316
1.292
1.166
1.368
0.827
0.595
1.295
1.475
2.543
0.476
1.485
2.575
0.560
1.065
1.028
1.267
0.931
0.897
1.137
0.688
1.090
0.360
0.691
1.095
0.385
1.016
0.988
1.084
1.145
-0.533
1.484
4.814
26.714
1.288
4.817
26.758
1.375
1.199
1.295
1.383
0.791
0.899
1.323
1.562
2.946
0.463
1.576
2.948
0.565
1.018
1.036
1.255
0.930
0.827
1.189
0.659
1.155
0.389
0.663
1.168
0.433
0.987
0.937
1.100

119

IV Heckit1
1.536
1.481
31.611
10.474
31.615
10.485
1.235
1.251
0.819
0.918
2.794
1.623
2.800
1.625
1.017
1.068
0.894
0.941
0.903
0.740
0.909
0.742
1.021
1.032
-2.554
2.671
99.600
9.893
99.663
10.033
1.346
1.790
0.893
1.501
1.455
0.999
1.459
1.118
1.104
1.489
1.017
1.435
0.708
0.606
0.708
0.746
1.116
1.431

u: χ2 (5)
Heckit2 QLIML QFIML
0.884
0.024
1.361
6.833
20.848
1.361
6.834
20.871
1.983
1.217
1.244
1.274
0.799
0.573
0.998
1.607
4.106
0.664
1.620
4.128
0.440
1.017
1.044
0.652
0.841
0.844
0.555
0.902
1.573
0.767
0.916
1.580
0.786
0.972
0.991
0.506
0.619
0.710
1.243
17.531
27.404
1.626
17.535
27.406
1.644
1.232
1.231
1.030
0.740
0.653
0.769
1.624
4.086
0.753
1.644
4.101
0.620
0.945
0.942
0.476
0.887
0.747
0.504
0.744
1.709
0.575
0.752
1.728
0.576
0.978
0.894
0.335

Table B.5: Results for Weak IV and ρ = 0.0
obs
100

M
S
R
Md
500
M
ξ:
S
normal
R
Md
1000
M
S
R
Md
100
M
S
R
Md
500
M
ξ:
S
2 (5)
χ
R
Md
1000
M
S
R
Md

IV Heckit1
0.980
10.114
0.638 172.074
0.638 172.315
0.976
0.981
0.995
0.857
0.545
6.247
0.545
6.249
0.990
0.979
0.976
0.923
0.453
1.501
0.453
1.503
0.967
0.939
1.035 -10.261
0.613 265.733
0.614 265.971
1.031
1.047
1.013
0.934
0.547
8.995
0.547
8.995
0.996
0.954
1.046
1.113
0.452
5.328
0.454
5.329
1.037
1.065

u: normal
Heckit2 QLIML QFIML
0.983
0.691
0.963
0.636
32.420
1.096
0.637
32.421
1.096
1.002
1.021
0.964
0.994
1.253
0.992
0.537
6.483
0.735
0.537
6.488
0.735
0.993
0.974
0.956
0.973
0.906
0.963
0.450
2.831
0.569
0.451
2.832
0.571
0.973
0.972
0.968
1.069
0.005
0.995
0.614
24.815
1.094
0.618
24.835
1.094
1.068
0.946
0.976
1.150
1.232
0.987
0.548
5.896
0.707
0.568
5.900
0.707
1.128
0.958
0.992
1.231
1.156
1.012
0.460
3.455
0.553
0.515
3.459
0.553
1.196
1.109
1.004

120

IV Heckit1
1.018
-1.436
0.640
93.693
0.640
93.724
0.999
1.048
1.017
2.092
0.524
77.712
0.524
77.720
1.005
1.016
1.014
0.762
0.434
6.072
0.434
6.077
1.001
0.995
0.978
-3.914
0.630 223.518
0.631 223.572
0.991
0.842
1.033
0.716
0.537
13.047
0.538
13.050
1.009
1.006
1.031
1.052
0.457
5.238
0.458
5.238
1.029
1.032

u: χ2 (5)
Heckit2
1.014
0.630
0.630
0.984
1.018
0.518
0.518
1.004
1.010
0.429
0.429
1.005
1.009
0.623
0.624
1.012
1.164
0.535
0.559
1.143
1.222
0.453
0.505
1.202

QLIML QFIML
2.490
0.998
24.057
1.007
24.103
1.007
0.853
0.961
0.988
0.981
5.271
0.582
5.271
0.583
1.190
0.993
1.053
1.057
2.668
0.803
2.669
0.805
1.038
1.008
0.948
0.910
30.667
0.983
30.667
0.987
1.211
0.874
1.076
0.938
6.180
0.664
6.180
0.667
1.101
1.034
0.791
0.728
3.327
0.825
3.333
0.869
0.874
0.748

Table B.6: Results for Weak IV and ρ = 0.4
obs
100

M
S
R
Md
500
M
ξ:
S
normal
R
Md
1000
M
S
R
Md
100
M
S
R
Md
500
M
ξ:
S
2 (5)
χ
R
Md
1000
M
S
R
Md

IV Heckit1
1.586
2.836
0.633
35.243
0.863
35.291
1.578
1.438
1.400
0.037
0.526
24.103
0.661
24.123
1.407
1.138
1.268
1.130
0.433
6.115
0.509
6.117
1.282
1.014
1.587
2.641
0.594
34.063
0.835
34.103
1.588
1.471
1.432
1.638
0.530
23.301
0.684
23.310
1.417
1.135
1.310
0.994
0.428
8.085
0.529
8.085
1.310
1.092

u: normal
Heckit2 QLIML QFIML
1.588
-3.068
1.586
0.606
20.161
1.022
0.844
20.568
1.178
1.587
0.224
1.575
1.398
-0.585
1.330
0.521
5.314
0.677
0.655
5.546
0.753
1.404
0.250
1.307
1.264
-0.057
1.206
0.428
2.832
0.527
0.503
3.023
0.566
1.274
0.566
1.167
1.617
-3.805
1.549
0.579
23.632
1.033
0.846
24.116
1.170
1.615
0.115
1.491
1.540
-0.501
1.398
0.532
5.135
0.675
0.758
5.350
0.783
1.524
0.315
1.321
1.460
0.279
1.230
0.426
2.615
0.501
0.627
2.713
0.551
1.444
0.578
1.152

121

IV Heckit1
1.536
-2.592
0.597 240.707
0.802 240.734
1.508
1.283
1.383
0.670
0.515
9.581
0.642
9.586
1.401
1.116
1.277
1.006
0.440
3.613
0.520
3.613
1.287
1.019
1.586
1.557
0.633
51.806
0.863
51.809
1.578
1.350
1.399
-0.065
0.528
25.444
0.662
25.467
1.433
1.117
1.295
0.700
0.435
9.323
0.525
9.328
1.320
1.063

u: χ2 (5)
Heckit2
1.539
0.592
0.800
1.523
1.387
0.513
0.643
1.407
1.276
0.436
0.516
1.280
1.610
0.638
0.883
1.612
1.508
0.529
0.733
1.519
1.455
0.435
0.629
1.453

QLIML QFIML
-2.606
1.433
25.049
0.941
25.307
1.035
-0.266
1.391
-1.625
1.192
6.592
0.778
7.095
0.801
-0.122
1.394
-0.315
0.899
3.314
0.767
3.565
0.774
0.286
1.103
-1.096
1.523
40.577
1.066
40.631
1.187
0.331
1.510
-0.406
0.957
7.720
0.871
7.847
0.872
-0.074
1.102
-0.084
0.635
3.917
0.793
4.065
0.873
0.400
0.483

Table B.7: Results for Weak IV and ρ = 0.5
obs
100

M
S
R
Md
500
M
ξ:
S
normal
R
Md
1000
M
S
R
Md
100
M
S
R
Md
500
M
ξ:
S
2 (5)
χ
R
Md
1000
M
S
R
Md

IV Heckit1
1.729
3.130
0.592
62.997
0.939
63.033
1.720
1.555
1.497
0.508
0.514
10.178
0.715
10.190
1.498
1.152
1.335
0.838
0.428
3.494
0.544
3.498
1.353
1.009
1.726
3.822
0.573 169.952
0.925 169.976
1.731
1.506
1.501
4.126
0.506
77.381
0.712
77.444
1.494
1.124
1.366
1.141
0.410
4.313
0.550
4.315
1.369
1.063

u: normal
Heckit2 QLIML QFIML
1.734
-3.201
1.721
0.588
21.170
0.998
0.941
21.583
1.231
1.741
0.232
1.623
1.495
-0.280
1.388
0.510
5.184
0.649
0.710
5.339
0.756
1.494
0.463
1.333
1.332
0.350
1.229
0.423
2.425
0.492
0.537
2.510
0.543
1.350
0.683
1.165
1.754
-2.987
1.692
0.566
27.520
1.022
0.942
27.807
1.234
1.753
0.771
1.623
1.608
0.403
1.417
0.493
5.296
0.652
0.782
5.329
0.774
1.598
0.572
1.332
1.509
0.702
1.244
0.402
2.793
0.472
0.648
2.809
0.532
1.499
0.707
1.172

122

IV Heckit1
1.686
-2.573
0.600 182.319
0.911 182.354
1.704
1.564
1.488
0.541
0.520
7.570
0.713
7.584
1.505
1.132
1.349
0.900
0.427
5.431
0.552
5.432
1.354
1.038
1.747
1.056
0.593
17.854
0.954
17.854
1.740
1.615
1.492
0.421
0.545
8.400
0.734
8.420
1.515
1.099
1.354
1.039
0.417
2.638
0.547
2.639
1.364
1.075

u: χ2 (5)
Heckit2
1.685
0.589
0.904
1.713
1.496
0.518
0.717
1.515
1.358
0.423
0.554
1.361
1.771
0.592
0.972
1.747
1.599
0.546
0.811
1.615
1.500
0.415
0.650
1.501

QLIML QFIML
-0.896
1.595
28.523
0.930
28.586
1.104
0.460
1.571
-0.625
1.209
8.002
0.771
8.166
0.799
0.151
1.350
0.042
0.894
3.894
0.708
4.010
0.716
0.425
0.463
1.409
1.639
37.053
1.009
37.055
1.194
0.973
1.594
0.790
0.972
8.326
0.828
8.328
0.829
0.501
0.860
0.905
0.714
4.161
0.726
4.162
0.780
0.567
0.336

Table B.8: Results for Weak IV and ρ = 0.6
obs
100

M
S
R
Md
500
M
ξ:
S
normal
R
Md
1000
M
S
R
Md
100
M
S
R
Md
500
M
ξ:
S
2 (5)
χ
R
Md
1000
M
S
R
Md

IV Heckit1
1.870
2.304
0.571
13.195
1.041
13.259
1.879
1.702
1.590
1.115
0.498
6.704
0.773
6.705
1.602
1.155
1.401
0.892
0.421
2.709
0.581
2.711
1.423
1.008
1.890
0.524
0.557 120.541
1.050 120.542
1.894
1.771
1.593
1.180
0.501
12.399
0.777
12.401
1.612
1.144
1.431
0.902
0.404
1.596
0.591
1.599
1.441
1.066

u: normal
Heckit2 QLIML QFIML
1.837
-0.322
1.848
0.612
16.511
0.969
1.037
16.563
1.288
1.826
0.885
1.716
1.589
0.222
1.421
0.495
5.268
0.625
0.769
5.325
0.754
1.590
0.724
1.388
1.397
0.529
1.231
0.416
2.149
0.458
0.575
2.200
0.513
1.420
0.776
1.136
1.910
1.180
1.804
0.545
15.472
0.942
1.060
15.473
1.238
1.902
1.495
1.724
1.696
1.464
1.472
0.492
4.972
0.636
0.852
4.994
0.792
1.703
1.079
1.357
1.570
0.895
1.244
0.389
2.999
0.422
0.690
3.000
0.487
1.566
0.797
1.152

123

IV Heckit1
1.836
1.918
0.617
44.461
1.039
44.470
1.837
1.665
1.618
0.601
0.513
21.976
0.803
21.979
1.621
1.231
1.441
1.013
0.431
4.355
0.617
4.355
1.448
1.057
1.922
2.613
0.617
86.358
1.110
86.373
1.936
1.728
1.610
0.280
0.491
14.177
0.783
14.196
1.612
1.118
1.437
0.645
0.426
3.557
0.610
3.574
1.472
1.031

u: χ2 (5)
Heckit2
1.837
0.612
1.037
1.826
1.628
0.511
0.810
1.633
1.452
0.429
0.623
1.461
1.963
0.618
1.145
1.993
1.805
0.574
0.989
1.813
1.645
0.422
0.771
1.652

QLIML QFIML
1.730
1.695
27.031
0.948
27.041
1.176
0.965
1.635
1.247
1.251
7.572
0.750
7.576
0.791
0.939
1.279
0.990
0.979
3.972
0.668
3.972
0.668
0.718
0.587
1.934
1.755
37.442
1.112
37.454
1.344
2.310
1.700
3.106
1.000
8.526
0.871
8.783
0.871
1.150
0.624
1.035
0.674
5.245
0.683
5.246
0.756
0.564
0.371

Appendix C
Other Tables in Chapter 1

124

Table C.1: Relative Efficiencies (Ratio of MSEs)
ρ
0

ξ
normal

χ2 (5)

0.4 normal

χ2 (5)

0.5 normal

χ2 (5)

0.6 normal

χ2 (5)

Heckit1 vs LIV
obs. u: normal u: χ2 (5)
100
6.03
18.69
500
7.01
8.51
1000
4.43
3.26
100
13.66
16.15
500
8.36
6.69
1000
8.36
7.92
100
8.13
14.70
500
5.70
4.24
1000
2.07
5.02
100
20.34
7.62
500
3.36
6.77
1000
2.74
7.72
100
10.43
2.17
500
4.15
2.83
1000
1.87
4.31
100
39.64
8.52
500
4.02
8.20
1000
3.96
3.23
100
5.47
9.33
500
3.01
6.49
1000
2.51
2.98
100
30.86
2.44
500
3.50
6.22
1000
3.11
5.28

Heckit2 vs QLIML
Heckit2 vs Series
Heckit2 vs QFIML
2 (5) u: normal u: χ2 (5) u: normal u: χ2 (5)
u: normal u: χ
3.01
2.78
0.04
0.06
6.75
324.48
1.94
2.09
0.23
0.52
0.75
4.89
1.27
2.26
0.59
1.89
1.06
1.26
1.13
0.63
0.05
0.07
8.23
298.49
0.97
2.95
0.19
0.88
0.85
0.95
1.04
2.14
0.65
2.44
0.66
0.81
0.44
4.44
0.02
0.06
6.74
8.42
1.75
2.16
0.20
0.49
1.22
4.47
4.93
1.84
0.92
1.72
1.09
1.44
1.10
0.46
0.07
0.03
202.95
24.78
1.10
3.00
0.17
0.40
14.52
2.16
1.77
1.28
0.74
2.15
0.80
0.86
0.43
0.14
0.02
0.01
7.27
15.05
1.73
0.90
0.17
0.09
1.39
3.00
4.57
1.56
0.40
0.49
1.14
2.15
1.05
0.52
0.06
0.03
150.73
13.20
1.14
1.67
0.14
0.34
20.02
5.13
1.72
2.40
0.44
1.12
0.89
0.99
0.43
0.66
0.02
0.08
8.49
9.09
1.72
2.12
0.14
0.07
1.56
2.97
3.92
0.82
0.31
0.74
1.20
1.50
0.90
0.38
0.08
0.01
102.44
98.67
1.21
2.33
0.13
0.14
24.76
1.70
1.57
1.60
0.43
0.59
1.01
0.90

125

Table C.2: Ratio of estimations with AvarQLIM L < AvarT wo−step
ρ

ξ
normal

0.0 ξ: t(5)

χ2 (5)

normal

0.4 ξ: t(5)

χ2 (5)

normal

0.5 ξ: t(5)

χ2 (5)

normal

0.6 ξ: t(5)

χ2 (5)

obs. u: normal
100
0.54
500
0.54
1000
0.52
100
0.53
500
0.57
1000
0.56
100
0.55
500
0.58
1000
0.55
100
0.48
500
0.53
1000
0.53
100
0.53
500
0.52
1000
0.53
100
0.54
500
0.56
1000
0.55
100
0.48
500
0.53
1000
0.54
100
0.52
500
0.57
1000
0.56
100
0.53
500
0.58
1000
0.56
100
0.48
500
0.54
1000
0.56
100
0.52
500
0.56
1000
0.57
100
0.53
500
0.59
1000
0.60

126

u: t(5)
0.56
0.52
0.52
0.52
0.55
0.55
0.56
0.58
0.56
0.48
0.53
0.54
0.53
0.54
0.54
0.52
0.58
0.58
0.48
0.55
0.55
0.52
0.54
0.56
0.56
0.58
0.58
0.48
0.55
0.56
0.54
0.54
0.57
0.55
0.59
0.60

u: χ2 (5)
0.53
0.53
0.54
0.53
0.56
0.53
0.56
0.57
0.54
0.51
0.56
0.54
0.51
0.51
0.58
0.55
0.59
0.55
0.51
0.54
0.56
0.51
0.54
0.54
0.51
0.56
0.61
0.54
0.54
0.57
0.53
0.56
0.58
0.55
0.57
0.59

Appendix D

.

.

Figures in Chapter 1

.

.

.

.

.

.

Figure D.1: Sample Cumulative Density Functions under Strong IV with ρ = 0.6 and
obs=1000.
Note: The upper row is for normal ξ and lower row is for chi-squared ξ. The left column is
for normal u and right column is for chi-squared u. The dotted, short-dashed and
long-dashed line represent FIML, LIML and Heckit estimators respectively.

127

.

.

.

.

.

.

.

.

Figure D.2: Sample Cumulative Density Functions under Weak IV with ρ = 0.6 and
obs=1000.
Note: Graphs are placed by same way as in Figure D.1.

128

Appendix E
Proofs in Chapter 2
E.1

Proof of Proposition 1

Although the basic proof is presented in Terza (1998), the following generalizes it to the two
regime setting.
Lemma 1

∫ ∞

[

2
zγ + (σ1a /σ1 )ϵ
g(v|ϵ)dv = F √
1 − (σ1a /σ1 )2
−zγ

]

Proof First imagine the joint density function of v and ϵ. Then the expression is the
probability of v ∈ [−zγ, ∞), i.e. d = 1 conditional on ϵ.
A linear projection can be written as
(
( σ )2 )
σ1a
with e ∼ N 0, 1 − 1a
v = 2 ϵ+e
σ1
σ1
Then
2
w = 1[zγ + (σ1a /σ1 )ϵ + e > 0]
2
= 1[e > −zγ − (σ1a /σ1 )ϵ]
[
2 ]
−zγ − (σ1a /σ1 )ϵ
e
= 1 √
> √
1 − (σ1a /σ1 )2
1 − (σ1a /σ1 )2

Therefore

2 ]
zγ + (σ1a /σ1 )ϵ
P(w = 1|ϵ) =
≡ Φ∗ (ϵ)
g(v|ϵ)dv = F √
2
1 − (σ1a /σ1 )
−zγ

[

∫ ∞

as was to be shown.

Proposition The joint density function of y and d conditional on the exogenous variables

129

is
[
f (y, w|z) =

]w
∫
√
√
1
2
√
f (y1 |z, w = 1, 2σ1 ζ1 )Φ∗ ( 2σ1 ζ1 ) exp(−ζ1 )dζ1
π R
[
]1−w
∫
(
)
√
√
1
∗ ( 2σ ζ ) exp(−ζ 2 )dζ
· √
f (y0 |z, w = 0, 2σ0 ζ0 ) 1 − Φ
0 0
0
0
π R

Proof Note that

∫ ∞
f (ϵ1 , w = 1) =
f (ϵ1 , w = 0) =
f (ϵ0 , w = 1) =

f (ϵ1 , v)dv

−zγ
∫ −zγ

f (ϵ1 , v)dv

∫−∞
∞

f (ϵ0 , v)dv

−zγ
∫ −zγ

f (ϵ0 , w = 0) =

f (ϵ0 , v)dv

−∞

Let f (·|z) ≡ g(·).
)
( ∫ −zγ
∫
∫
g(ϵ0 , v) dv dϵ0 =
g(y0 |w = 0, ϵ0 ) · g(ϵ0 , w = 0) dϵ0
g(y0 |w = 0, ϵ0 )
−∞
R
R
∫
=
g(y0 , w = 0, ϵ0 ) dϵ0
R

= g(y0 , w = 0)
)
(∫ ∞
∫
g(ϵ1 , v) dv dϵ1 =
g(y1 |w = 1, ϵ1 ) · g(ϵ1 , w = 1) dϵ1
g(y1 |w = 1, ϵ1 )
−zγ
R
R
∫
=
g(y1 , w = 1, ϵ1 ) dϵ1

∫

R

= g(y1 , w = 1)
Thus we have
g(y, w) ≡ g(y1 , w = 1)w · g(y0 , w = 0)1−w
(∫
)w
(∫ ∞
)
g(y1 |w = 1, ϵ1 )
=
g(ϵ1 , v) dv dϵ1
R

(∫
×

R

−zγ

g(y0 |w = 0, ϵ0 )

130

( ∫ −zγ
−∞

)1−w

)
g(ϵ0 , v) dv

dϵ0

Recovering the original notation
[∫
f (y, w|z) =

R

]w
(∫ ∞
)
f (y1 |z, w = 1, ϵ1 )
f (ϵ1 , v|z) dv dϵ1
−zγ

[∫
·

f (y0 |z, w = 0, ϵ0 )

( ∫ −zγ

]1−w

)

f (ϵ0 , v|z) dv dϵ0
[∫
]w
(∫ ∞
)
=
f (y1 |z, w = 1, ϵ1 )
f (ϵ1 |z)f (v|ϵ1 , z) dv dϵ1
R

R

−zγ

[∫
·

−∞

)

]1−w

f (ϵ0 |z)f (v|ϵ0 , z)dv dϵ0
]w
[∫
(∫ ∞
)
f (y1 |z, w = 1, ϵ1 )f (ϵ1 |z)
f (v|ϵ1 , z) dv dϵ1
=
R

=

R

−∞

−zγ

[∫
·

[∫

R

f (y0 |z, w = 0, ϵ0 )

( ∫ −zγ

R

]1−w
( ∫ −zγ
)
f (y0 |z, w = 0, ϵ0 )f (ϵ0 |z)
f (v|ϵ0 , z)dv dϵ0
−∞

]w

f (y1 |z, w = 1, ϵ1 )f (ϵ1 |z)Φ∗ (ϵ1 ) dϵ1
[∫
·

Let

R

]1−w
(
)
f (y0 |z, w = 0, ϵ0 )f (ϵ0 |z) 1 − Φ∗ (ϵ0 ) dϵ0

ϵ
ζ≡√ ,
2σ

ϵ ∼ N (0, σ 2 )

(E.1)

Then
[
f (y, w|z) =

∫

1
√
π R
[

√
√
2
f (y1 |z, w = 1, 2σ1 ζ1 )Φ∗ ( 2σ1 ζ1 ) exp(−ζ1 )dζ1

]w

]1−w
∫
(
)
√
√
1
2
× √
f (y0 |z, w = 0, 2σ0 ζ0 ) 1 − Φ∗ ( 2σ0 ζ0 ) exp(−ζ0 )dζ0
π R

as was to be shown.

131

E.2

Derivation of Estimating Equation for NFES Model

In what follows the regime subscripts were omitted for simplicity. For each regime
E(y|z, v) = exp(α + xβ)E[exp(ϵ)|z, v]
(
)
1 2
2)
= exp(α + xβ) exp ρσv + σ (1 − ρ
2
)
(
1 2
2 ) + xβ + ρσv
= exp α + σ (1 − ρ
2
The second equation derived by the result in Appendix A in Terza (1998).
Note that for regime 1,
∫ ∞
exp(ρσv)p(v|v > −zδ)dv
E[exp(ρσv)|v > −zδ] =
−zδ
∫ ∞
p(v)
=
exp(ρσv)
dv
P(v > −zδ)
−zδ
∫ ∞
1
=
exp(ρσv)p(v)dv
Φ(zδ) −zδ
∫ ∞
( v2 )
1
1
√ exp −
exp(ρσv)
=
dv
Φ(zδ) −zδ
2
2π
∫ ∞
( 2ρσv − v 2 )
1
1
√ exp
dv
=
Φ(zδ) −zδ 2π
2
∫ ∞
( (ρσ)2 )
( 2ρσv − v 2 )
( (ρσ)2 )
1
1
√ exp
=
exp −
exp
dv
Φ(zδ) −zδ 2π
2
2
2
( (ρσ)2 ) ∫ ∞ 1
( (v − ρσ)2 )
1
√ exp −
=
exp
dv
Φ(zδ)
2
2
−zδ 2π
( (ρσ)2 ) ∫ ∞
( v2 )
1
1
√ exp −
dv
=
exp
Φ(zδ)
2
2
−(zδ+ρσ) 2π
=

( (ρσ)2 )
1
Φ(zδ + ρσ)
exp
Φ(zδ)
2

By the same reasoning it can be shown that for regime 0,
( (ρσ)2 )
1
E[exp(ρσv)|v < −zδ] =
exp
Φ(−(zδ + ρσ))
Φ(−zδ)
2
Since G(z, v) ⊂ G(z, w), by law of iterated expectation,
(
)
1 2
2 ) + xβ E[exp(ρ σ v)|z, w]
E(y1 |z, w) = exp α1 + σ1 (1 − ρ1
1
1 1
2
)
(
1 2
E(y1 |z, w = 1) = exp α1 + σ1 (1 − ρ2 ) + xβ1 E[exp(ρ1 σ1 v)|v > −zδ]
1
2
) Φ(zδ + ρ σ )
(
σ2
1 1
= exp α1 + 1 + xβ1
2
Φ(zδ)
132

Similarly
(
)
1 2
2 ) + xβ E[exp(ρ σ v)|v < −zδ]
E(y0 |z, w = 0) = exp α0 + σ0 (1 − ρ0
0
0 0
2
(
) Φ(−(zδ + ρ σ ))
σ2
0 0
= exp α0 + 0 + xβ0
2
Φ(−zδ)
Therefore the estimating equation is,
[
]
(
) Φ(zδ + ρ σ )
2
σ1
1 1
E(y|z, w) = w · exp α1 +
+ xβ1
2
Φ(zδ)
[
]
(
) Φ(−(zδ + ρ σ ))
2
σ0
0 0
+ (1 − w) · exp α0 +
,
+ xβ0
2
Φ(−zδ)
where y = wy1 + (1 − w)y0 .

E.3

Derivation of Conditional Variance for WNLS Estimator

It was shown in Appendix A. in Terza (1998) that
(
)
1 2
E[exp(cg )|v] = exp ρg σg v + σg (1 − ρ2 )
g
2
(
)
(
)
2
2
var[exp(cg )|v] = exp 2ρg σg v + 2σg (1 − ρ2 ) − exp 2ρg σg v + σg (1 − ρ2 )
g
g
Let all the expectations below are conditional on z. Then
[
]
E[exp(cg )2 |w] = E E[exp(cg )2 |v] w
[
]
[
]
= E var[exp(cg )|v] w + E E[exp(cg )|v]2 w
Thus we need to find out the expressions for the two terms on the RHS.
[
(
)
(
) ]
2 (1 − ρ2 ) − exp 2ρ σ v + σ 2 (1 − ρ2 ) w
E[var(exp(cg )|v)|w] = E exp 2ρg σg v + 2σg
g g
g
g
g
[ 2
]
[
(
) ]
= exp 2σg (1 − ρ2 ) · E exp 2ρg σg v w
g
]
[
(
) ]
[ 2
− exp σg (1 − ρ2 ) · E exp 2ρg σg v w
g
)
(
[ 2
2 )] − exp [σ 2 (1 − ρ2 )]
=
exp 2σg (1 − ρg
g
g
[
(
) ]
× E exp 2ρg σg v w
(E.2)
133

Also

[
E[E(exp(cg )|v)2 |w] =

(

]

)

2
E exp 2ρg σg v + σg (1 − ρ2 )
g

w

(
)
[
]
2
= exp σg (1 − ρ2 ) · E exp(2ρg σg v)|w
g
Therefore
By the way

[ 2
]
[
]
E[exp(cg )2 |w] = exp 2σg (1 − ρ2 ) · E exp(2ρg σg v)|w
g
[
]
E[exp(cg )|w] = E E(exp(cg )|v) w
) ]
[
(
1 2
= E exp ρg σg v + σg (1 − ρ2 ) w
g
2
(1
)
[
]
2
= exp σg (1 − ρ2 ) · E exp(ρg σg v)|w
g
2

Therefore
var[exp(cg )|w] = E[exp(cg )2 |w] − E[exp(cg )|w]2
[ 2
]
[
]
= exp 2σg (1 − ρ2 ) · E exp(2ρg σg v)|w
g
(
)
2 (1 − ρ2 ) · E [ exp(ρ σ v)|w ]2
− exp σg
g g
g
From the results in Appendix A,

{

}
Φ(zδ + 2ρ1 σ1 ) ( Φ(zδ + ρ1 σ1 ) )2
2
2
var[exp(ϵ1 )|w = 1] = exp(σ1 ) exp(σ1 )
−
Φ(zδ)
Φ(zδ)
}
{
Φ(−zδ − 2ρ0 σ0 ) ( Φ(−zδ − ρ0 σ0 ) )2
2
2
−
var[exp(ϵ0 )|w = 0] = exp(σ0 ) exp(σ0 )
Φ(−zδ)
Φ(−zδ)

Also
[
]
[
]
var E[y|z, w, ϵ1 , ϵ0 ] z, w = var w exp(α1 + xβ1 + ϵ1 ) + (1 − w) exp(α0 + xβ0 + ϵ0 ) z, w
= w exp(α1 + xβ1 )2 var[exp(ϵ1 )|z, w]
+(1 − w) exp(α0 + xβ0 )2 var[exp(ϵ0 )|z, w]
+2w(1 − w) exp(α1 + xβ1 ) exp(α0 + xβ0 )
× cov[exp(ϵ1 ), exp(ϵ0 )|z, w]
2 var[exp(ϵ )|z, w = 1]
= w exp(α1 + xβ1 )
1
+ (1 − w) exp(α0 + xβ0 )2 var[exp(ϵ0 )|z, w = 0]
{

}
Φ(zδ + 2ρ1 σ1 ) ( Φ(zδ + ρ1 σ1 ) )2
2
2
= w exp(α1 + σ1 /2 + xβ1 )2 exp(σ1 )
−
Φ(zδ)
Φ(zδ)
}
{
Φ(−zδ − 2ρ0 σ0 ) ( Φ(−zδ − ρ0 σ0 ) )2
2
2
+(1 − w) exp(α0 + σ0 /2 + xβ0 )2 exp(σ0 )
−
Φ(−zδ)
Φ(−zδ)
134

2
Let δg = exp(αg + σg /2 + xβg ), L1,2 =

Φ(zδ + 2ρ1 σ1 )
Φ(zδ + ρ1 σ1 )
, L1 =
, L0,2 =
Φ(zδ)
Φ(zδ)

Φ(−zδ − 2ρ0 σ0 )
Φ(−zδ − ρ0 σ0 )
, and L0 =
. Then the last equation can be simply writΦ(−zδ)
Φ(−zδ)
ten as
)
)
(
[
]
(
2
2 exp(σ 2 )L
2 + (1 − w)δ 2 exp(σ 2 )L
var E[y|z, w, ϵ1 , ϵ0 ] z, w = wδ1
0 0,2 − L0
1 1,2 − L1
0
And
var[y|z, w, ϵ1 , ϵ0 ] = var[wy1 + (1 − w)y0 |z, w, ϵ1 , ϵ0 ]
= wvar[y1 |z, w, ϵ1 , ϵ0 ] + (1 − w)wvar[y0 |z, w, ϵ1 , ϵ0 ]
+ 2w(1 − w)cov[y1 , y0 |z, w, ϵ1 , ϵ0 ]
= wE[y1 |z, w, ϵ1 ] + (1 − w)E[y0 |z, w, ϵ0 ]
Taking E[·|z, w] at both sides,
[
]
E var[y|z, w, ϵ1 , ϵ0 ] z, w = wE[y1 |z, w = 1] + (1 − w)E[y0 |z, w = 0]
= wδ1 L1 + (1 − w)δ0 L0
Finally,
[
]
[
]
var[y|z, w] = var E[y|z, w, ϵ1 , ϵ0 ] z, w + E var[y|z, w, ϵ1 , ϵ0 ] z, w
(
)
(
)
2
2
= wδ1 δ1 (exp(σ1 )L1,2 − L2 ) + L1 + (1 − w)δ0 δ0 (exp(σ0 )L0,2 − L2 ) + L0
1
0

135

Appendix F
Tables in Chapter 2

136

Table F.1: Simulation Results for rho = 0.4
Linear Models
NET (1 Regime Nonlinear)
2SLS LET(Hkt) LFES(Hkt)
1PQML
2PQML
NLS
WNLS
n=1000
mean
0.726
0.742
0.747
-18813768 -1037.586
-6220.699 7.289E+13
mc. st. dev
2.463
2.448
2.404
378300000 30871.585 186544.680 2.055E+15
RMSE
2.478
2.462
2.417
378800000 30889.051 186648.410 2.056E+15
median
0.825
0.845
0.855
-91.134
-0.296
0.747
1.177
MAD
1.453
1.446
1.444
94.588
3.387
2.257
1.757
n=3000
mean
0.804
0.820
0.818
-2389.450
-4.619
-2.516
-0.347
mc. st. dev
1.255
1.254
1.256
19778.908
16.242
17.923
18.548
RMSE
1.270
1.267
1.269
19922.838
17.187
18.264
18.597
median
0.814
0.804
0.809
-187.915
-0.239
0.616
0.270
MAD
0.809
0.798
0.806
188.648
1.881
1.407
1.208
n=5000
mean
0.892
0.904
0.904
-904.755
-1.712
-0.857
-0.340
mc. st. dev
0.980
0.982
0.982
5179.628
6.637
10.024
3.833
RMSE
0.986
0.987
0.986
5258.226
7.170
10.194
4.061
median
0.907
0.913
0.906
-220.859
0.033
0.495
0.065
MAD
0.647
0.630
0.628
192.470
1.388
1.034
0.981
Note: The true ATE=1, The number of Monte Carlo repetition is 1000. LET(Hkt): LET model estimated by twostep Heckit, 1PQML: one step Poisson Quasi-Maximum Likelihood estimator, RMSE=root mean squared error,
MAD=mean absolute deviation

137

Table F.1 (cont’d)

n=1000

n=3000

n=5000

mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

NFES (2 Regime Nonlinear)
1PQML 2 PQML
NLS WNLS
-0.528
-0.084
-3.555 -0.100
17.753
14.587 105.187 18.089
17.818
14.628 105.286 18.122
0.909
0.984
1.096
0.971
1.132
1.160
1.198
1.138
0.856
0.885
0.879
0.926
1.487
1.322
1.444
1.298
1.494
1.327
1.449
1.300
0.934
0.937
1.004
0.958
0.679
0.674
0.706
0.649
0.983
0.980
0.973
0.996
0.933
0.935
0.987
0.912
0.933
0.935
0.988
0.912
1.027
1.022
1.001
1.044
0.539
0.540
0.594
0.528

138

Table F.2: Simulation Results for rho = 0.5

n=1000

n=3000

n=5000

mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

Linear Models
2SLS LET(Hkt) LFES(Hkt)
0.620
0.634
0.635
2.512
2.517
2.522
2.541
2.544
2.549
0.784
0.820
0.789
1.467
1.457
1.461
0.748
0.763
0.761
1.254
1.255
1.256
1.279
1.277
1.279
0.735
0.770
0.772
0.809
0.803
0.803
0.835
0.847
0.847
0.986
0.988
0.988
0.999
1.000
1.000
0.881
0.888
0.888
0.638
0.640
0.631

139

NET (1 Regime Nonlinear)
1PQML 2PQML
NLS WNLS
-2.097E+11 -107.190
-70.525 1.E+07
6.902E+12 1678.496 1008.897 3.E+08
6.905E+12 1681.979 1011.429 3.E+08
-90.775
-0.391
0.779
0.878
93.716
3.289
2.149
1.821
-4252.613
-7.376
-3.185
-1.604
56506.153
37.336
18.650 17.555
56666.026
38.264
19.113 17.747
-151.285
-0.702
0.292
0.028
133.619
2.212
1.452
1.330
-622.799
-2.529
-1.083
0.221
2763.448
8.182
10.301 26.549
2832.979
8.910
10.510 26.561
-191.688
-0.260
0.517
-0.121
131.812
1.558
1.084
1.112

Table F.2 (cont’d)

n=1000

n=3000

n=5000

mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

NFES (2 Regime Nonlinear)
1PQML 2 PQML
NLS WNLS
-4.675
-4.115 -19.700 -1.291
101.905
79.327 383.617 28.446
102.063
79.491 384.175 28.538
0.942
1.002
1.141
1.010
1.071
1.128
1.189
1.121
0.811
0.820
0.777
0.852
1.364
1.337
1.494
1.320
1.377
1.349
1.510
1.328
0.932
0.936
0.950
0.954
0.651
0.646
0.721
0.625
0.935
0.940
0.937
0.954
0.952
0.946
1.004
0.920
0.954
0.948
1.006
0.921
1.016
1.017
1.017
1.029
0.550
0.546
0.559
0.536

140

Table F.3: Simulation Results for rho = 0.6

n=1000

n=3000

n=5000

mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

Linear Models
2SLS LET(Hkt) LFES(Hkt)
0.589
0.610
0.615
2.444
2.424
2.417
2.478
2.455
2.447
0.743
0.725
0.761
1.500
1.500
1.483
0.690
0.703
0.703
1.269
1.269
1.267
1.306
1.304
1.301
0.724
0.723
0.727
0.830
0.843
0.846
0.777
0.787
0.788
1.012
1.015
1.015
1.036
1.037
1.037
0.830
0.835
0.855
0.655
0.658
0.656

141

NET (1
1PQML
-3178.000
10050.000
10050.000
-73.208
76.071
-1579.442
17617.457
17688.204
-134.971
112.208
-411.791
1144.600
1216.761
-158.018
104.547

Regime Nonlinear)
2PQML
NLS
WNLS
-83.600
-430.3381 2933.343
828.368 7701.3678 80317.211
832.677 7713.4375 80370.723
-0.760 0.50252408
0.818
3.636 2.3827269
1.897
-9.535 -5.6418819
326.938
43.824
29.77737 9798.644
45.073 30.509119 9804.063
-1.165 0.33116622
-0.149
2.613 1.4962865
1.456
-3.600 -1.1125125
-1.055
8.520 7.3739959
5.262
9.683 7.6706274
5.649
-0.705 0.53132847
-0.315
1.930 1.0969538
1.192

Table F.3 (cont’d)

n=1000

n=3000

n=5000

mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

NFES (2 Regime Nonlinear)
1PQML 2 PQML
NLS
WNLS
-2.296
-362.847
-4313.352 -143.539
39.603 11411.335 126080.480 4209.908
39.740 11417.134 126154.270 4212.388
0.983
0.998
1.170
1.188
1.113
1.165
1.176
1.150
0.717
0.725
0.670
0.782
1.650
1.595
1.842
1.517
1.674
1.618
1.871
1.533
0.945
0.943
0.989
0.974
0.685
0.688
0.706
0.664
0.901
0.907
0.893
0.923
0.976
0.973
1.054
0.954
0.981
0.978
1.059
0.957
0.975
0.986
0.993
1.007
0.556
0.547
0.596
0.539

142

Table F.4: Simulation Results for FIML estimator

n=1000

n=3000

n=5000

correlation
# of abscissas
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

0.4
8
2.143
1.342
1.763
2.014
0.737
1.888
1.048
1.374
1.880
0.605
1.837
0.916
1.241
1.822
0.507

16
1.336
1.525
1.561
1.430
0.725
1.348
0.975
1.035
1.394
0.488
1.460
0.967
1.071
1.471
0.486

143

0.5
8
2.365
1.325
1.902
2.246
0.726
2.234
1.121
1.667
2.141
0.547
2.215
0.903
1.514
2.195
0.552

16
1.395
1.524
1.574
1.523
0.747
1.557
1.032
1.173
1.586
0.513
1.650
0.853
1.073
1.641
0.403

0.6
8
2.511
1.630
2.222
2.376
0.749
2.302
1.074
1.687
2.271
0.550
2.254
0.920
1.556
2.197
0.511

16
1.422
1.606
1.661
1.593
0.843
1.670
0.904
1.125
1.633
0.462
1.656
0.906
1.118
1.617
0.544

Table F.5: Simulation Results for DGP 1

ρ = 0.4 n=1000
mc.

n=3000
mc.

n=5000
mc.

ρ = 0.5 n=1000
mc.

n=3000
mc.

n=5000
mc.

mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD

Linear Model
2SLS LFES(Hkt)
0.622
0.644
3.195
3.200
3.218
3.220
0.858
0.864
1.581
1.539
0.892
0.904
1.439
1.436
1.444
1.439
0.995
0.979
0.953
0.955
0.891
0.904
1.104
1.101
1.110
1.105
0.894
0.916
0.728
0.714
0.509
0.540
3.336
3.309
3.371
3.340
0.759
0.826
1.614
1.590
0.827
0.839
1.463
1.464
1.473
1.472
0.921
0.931
0.933
0.930
0.821
0.834
1.131
1.128
1.145
1.141
0.853
0.860
0.737
0.734

144

NFES
2PQML
WNLS
149.774
584.489
11454.700
18081.155
11455.667
18090.568
0.979
1.027
1.280
1.246
0.946
0.983
1.604
1.521
1.605
1.521
1.024
1.065
0.778
0.774
0.994
1.027
1.014
0.980
1.014
0.980
1.023
1.028
0.585
0.564
-6.616E+05 -1.324E+15
2.069E+07 4.103E+16
2.070E+07 4.106E+16
1.030
1.032
1.248
1.182
0.922
0.971
1.497
1.427
1.499
1.428
1.045
1.067
0.727
0.712
0.960
0.997
1.026
0.979
1.027
0.979
1.016
1.037
0.582
0.559
See next page.

Table F.5 (cont’d)

ρ = 0.6 n=1000

n=3000

n=5000

mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

Linear Model
2SLS LFES(Hkt)
0.402
0.428
3.387
3.364
3.440
3.412
0.849
0.864
1.620
1.614
0.771
0.784
1.503
1.500
1.520
1.516
0.852
0.874
0.958
0.959
0.768
0.779
1.149
1.147
1.172
1.168
0.800
0.811
0.766
0.768

145

NFES
2PQML WNLS
-9.205 -12.746
173.719 301.270
174.019 301.584
1.101
1.012
1.244
1.213
0.796
0.959
1.888
1.465
1.899
1.466
0.983
1.121
0.722
0.689
0.950
1.013
1.009
0.945
1.011
0.945
0.992
1.070
0.541
0.550

Table F.6: Simulation Results for DGP 2

ρ = 0.4 n=1000
mc.

n=3000
mc.

n=5000
mc.

ρ = 0.5 n=1000
mc.

n=3000
mc.

n=5000
mc.

mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD

Linear Model
2SLS LFES(Hkt)
0.791
0.503
2.329
2.788
2.338
2.832
1.062
0.823
1.278
1.517
0.918
0.619
1.260
1.520
1.263
1.567
0.930
0.653
0.816
0.999
0.857
0.553
0.956
1.148
0.966
1.232
0.898
0.580
0.639
0.778
0.523
0.100
2.581
3.067
2.625
3.197
0.824
0.449
1.447
1.786
0.905
0.512
1.279
1.559
1.283
1.634
1.013
0.621
0.821
0.988
0.810
0.411
0.942
1.148
0.961
1.291
0.847
0.466
0.610
0.731

146

NFES
2PQML WNLS
-9.268
-3.302
321.882 135.961
322.046 136.029
1.085
1.121
0.965
0.926
1.085
1.130
1.212
1.157
1.215
1.164
1.016
1.041
0.606
0.597
0.998
1.015
0.827
0.829
0.827
0.829
0.974
0.987
0.484
0.483
0.422
0.578
6.241 13.511
6.268 13.517
0.975
1.203
0.944
0.913
1.145
1.177
1.129
1.093
1.139
1.107
1.122
1.148
0.603
0.570
1.038
1.048
0.744
0.726
0.745
0.727
1.029
1.035
0.446
0.427
See next page.

Table F.6 (cont’d)

ρ = 0.6

n=1000

n=3000

n=5000

mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

Linear Model
2SLS LFES(Hkt)
0.540
0.035
2.549
3.185
2.590
3.328
0.884
0.423
1.321
1.646
0.822
0.345
1.275
1.569
1.287
1.700
0.904
0.475
0.841
1.024
0.739
0.253
0.953
1.173
0.988
1.391
0.750
0.269
0.629
0.789

147

NFES
2PQML WNLS
-40.132 -2.731
1267.348 77.542
1268.015 77.632
1.069
1.064
0.849
0.871
1.137
1.163
1.120
1.068
1.128
1.080
1.153
1.173
0.541
0.549
1.069
1.075
0.719
0.712
0.722
0.716
1.080
1.077
0.426
0.424

Table F.7: Simulation Results for DGP 3

ρ = 0.4 n=1000
mc.

n=3000
mc.

n=5000
mc.

ρ = 0.5 n=1000
mc.

n=3000
mc.

n=5000
mc.

mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD

Linear Model
2SLS LFES(Hkt)
0.364
0.361
6.723
6.285
6.753
6.317
0.676
0.646
2.681
2.697
0.750
0.749
3.039
3.059
3.049
3.070
0.816
0.839
1.599
1.622
0.713
0.713
2.198
2.219
2.217
2.237
0.761
0.769
1.239
1.260
0.537
0.492
6.434
6.505
6.451
6.525
0.745
0.735
2.611
2.651
0.419
0.405
3.132
3.145
3.185
3.201
0.605
0.589
1.719
1.704
0.702
0.673
2.253
2.304
2.273
2.327
0.787
0.731
1.297
1.338

148

NFES
2PQML
WNLS
-1097.543
34.431
34992.925
874.882
35010.165
875.520
1.241
1.235
1.836
1.826
1.861
1.370
14.839
3.718
14.864
3.736
1.158
1.222
0.989
0.953
1.331
1.367
1.959
1.693
1.987
1.733
1.105
1.201
0.769
0.772
6.409E+05 2.076E+05
1.969E+07 5.937E+06
1.970E+07 5.940E+06
1.425
1.409
1.571
1.356
1.420
1.497
2.629
2.370
2.663
2.422
1.273
1.335
0.888
0.852
1.566
1.591
2.160
1.563
2.233
1.671
1.328
1.419
0.746
0.789
See next page.

Table F.7 (cont’d)

ρ = 0.6 n=1000

n=3000

n=5000

mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

Linear Model
2SLS LFES(Hkt)
0.278
0.231
6.323
6.510
6.365
6.556
0.744
0.715
2.634
2.705
0.575
0.541
3.067
3.122
3.096
3.156
0.768
0.729
1.586
1.612
0.638
0.599
2.278
2.337
2.307
2.371
0.673
0.610
1.371
1.411

149

NFES
2PQML
WNLS
169.761
4308.951
4842.930 131368.520
4845.870 131439.140
1.466
1.605
1.402
1.328
2.485
1.755
24.906
4.314
24.950
4.379
1.506
1.526
0.830
0.805
1.703
1.630
2.836
2.129
2.921
2.221
1.473
1.416
0.646
0.612

Table F.8: Simulation Results for DGP 4

ρ = 0.4

n=1000
mc.

n=3000
mc.

n=5000
mc.

ρ = 0.5

n=1000
mc.

n=3000
mc.

n=5000
mc.

mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD
mean
st. dev
RMSE
median
MAD

Linear Model
2SLS LFES(Hkt)
0.646
0.888
3.689
3.854
3.706
3.855
0.819
0.988
2.099
2.069
0.725
0.910
1.881
1.834
1.901
1.836
0.721
0.877
1.217
1.187
0.732
0.914
1.389
1.354
1.415
1.356
0.762
0.935
0.917
0.841
0.505
0.724
3.577
3.512
3.611
3.523
0.751
0.925
2.017
1.979
0.663
0.833
1.847
1.806
1.878
1.814
0.653
0.821
1.232
1.230
0.655
0.831
1.394
1.366
1.436
1.376
0.685
0.874
0.895
0.834

150

NFES
2PQML
WNLS
-18.308 -156.597
205.225 1979.583
206.131 1985.846
1.213
1.026
1.764
2.137
0.864
0.535
1.998
3.068
2.003
3.103
0.947
0.900
1.058
1.232
0.935
0.730
1.439
1.937
1.440
1.955
1.064
1.078
0.770
1.004
-8.922
-13.194
93.284 107.502
93.810 108.435
1.164
1.360
1.468
1.609
0.911
0.641
1.713
2.531
1.715
2.557
0.919
0.957
0.903
1.059
0.850
0.671
1.393
1.792
1.401
1.822
0.909
0.896
0.835
1.113
See next page.

Table F.8 (cont’d)

ρ = 0.6 n=1000

n=3000

n=5000

mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD
mean
mc. st. dev
RMSE
median
MAD

Linear Model
2SLS LFES(Hkt)
0.321
0.584
4.308
3.733
4.361
3.756
0.692
0.849
2.039
1.954
0.567
0.729
1.822
1.794
1.873
1.814
0.611
0.758
1.215
1.164
0.595
0.759
1.380
1.353
1.438
1.374
0.624
0.798
0.885
0.852

Table F.9: Variables Description
children
ceb
mort
educ7
age
agesq
evermarr
urban
electric
tv
radio
frsthalf

number of living children
children ever born
number of dead children
= 1 if educ ≥ 7
age in years
age2
= 1 if ever married
= 1 if live in urban area
= 1 if has electricity
= 1 if has tv
= 1 if has radio
= 1 if mnthborn ≤ 6

151

NFES
2PQML WNLS
-7.682 -12.635
64.476 117.247
65.058 118.038
1.483
1.342
1.463
1.403
0.862
0.920
1.816
1.853
1.821
1.854
0.971
1.114
0.912
0.869
0.858
0.850
1.480
1.547
1.487
1.554
1.087
1.223
0.822
0.690

Table F.10: Descriptive Statistics
Variable
children
ceb
mort
educ7
age
agesq
evermarr
urban
electric
tv
radio
frsthalf

Mean
2.267
2.441
.173
.555
27.405
826.460
.476
.516
.140
.092
.701
.540

Std. Dev.
2.222
2.406
.511
.496
8.685
526.923
.499
.499
.347
.290
.457
.498

Min
0
0
0
0
15
225
0
0
0
0
0
0

Max
13
13
7
1
49
2401
1
1
1
1
1
1

Table F.11: Regression Results: selection equation
Variable
age

1 Regime
1PQML
-0.013
(0.019)
-0.001**
(0.000)
-0.306***
(0.049)
0.256***
(0.044)
0.413***
(0.079)
0.831***
(0.111)
0.491***
(0.046)
-0.211***
(0.044)
0.269***
(0.032)

2 Regime
1PQML
-0.007
(0.018)
-0.001***
(0.000)
-0.301***
(0.050)
0.258***
(0.044)
0.418***
(0.073)
0.811***
(0.089)
0.490 ***
(0.048)
-0.231***
(0.042)
0.278***
(0.031)

probit
-0.014
(0.018)
agesq
-0.001**
(0.000)
evermarr
-0.306***
(0.049)
urban
0.257***
(0.043)
electric
0.412***
(0.079)
tv
0.828***
(0.111)
radio
0.492***
(0.045)
frsthalf
-0.215***
(0.042)
constant
0.271***
(0.032)
L-likelihood
-2371.668
Note: The second and third columns report the results
from single step estimation. All the figures in the parenthesis
are bootstrap standard errors. *: significant at 10%,
**: 5%, ***: 1%

152

Table F.12: Regression Results: dependent variable children
Variable
OLS

Linear Models
2SLS LET(Hkt)

LFES(Hkt)

ATE
educ7

age
agesq
evermarr
urban
electric
tv
radio
constant

-0.398 ***
(0.046)

-1.185 *
(0.691)

0.272 *** 0.262 ***
(0.019)
(0.021)
-0.002 *** -0.002 ***
(0.000)
(0.000)
0.694 *** 0.610 ***
(0.054)
(0.096)
-0.246 *** -0.178 **
(0.047)
(0.078)
-0.337 *** -0.233 **
(0.074)
(0.114)
-0.330 ***
-0.155
(0.085)
(0.182)
0.027
0.153
(0.053)
(0.126)
-3.540 *** -2.880 ***
(0.035)
(0.385)

cov(epsilon, v)

L-likelihood
R2
sigma

-2.232 ***
(0.432)

-1.552
(0.979)

0.249 ***
(0.021)
-0.002 ***
(0.000)
0.499 ***
(0.080)
-0.088
(0.066)
-0.094
(0.098)
0.078
(0.124)
0.322 ***
(0.099)
3.508 ***
(0.246)
1.108 ***
(0.257)

R1
R0
0.251 *** 0.384 ***
(0.030)
(0.049)
-0.003
-0.003
(0.001)
(0.001)
0.194 ** 0.930 ***
(0.098)
(0.198)
0.206 ** -0.478 ***
(0.088)
(0.172)
0.197
-0.512
(0.126)
(0.347)
0.467 ***
-0.563
(0.142)
(0.698)
0.620 ***
-0.052
(0.129)
(0.294)
1.878 **
(0.943)
2.550 ***
-0.524
(0.454)
(0.905)

NET (1 Regime)
1PQML
2PQML
-0.103
-0.120
(0.363)
(0.324)
-0.046
-0.053
(0.158)
(0.142)

0.340 ***
(0.009)
-0.004 ***
(0.000)
0.326 ***
(0.030)
-0.101 ***
(0.024)
-0.162 ***
(0.044)
-0.203 ***
(0.061)
-0.015
(0.032)
-5.514 ***
(0.085)
-0.060
(0.094)
-1088.243

0.586
1.431

0.563
1.471

0.589
1.427

0.595
1.417
153

NLS
-0.681 *
(0.394)
-0.303 *
(0.172)

0.340 *** 0.265 ***
(0.009)
(0.012)
-0.004 *** -0.003 ***
(0.000)
(0.000)
0.325 *** 0.291 ***
(0.029)
(0.033)
-0.101 *** -0.085 ***
(0.023)
(0.027)
-0.161 *** -0.135 ***
(0.042)
(0.051)
-0.201 ***
-0.108 *
(0.058)
(0.070)
-0.014
0.035
(0.030)
(0.037)
-5.507 *** -4.059 ***
(0.077)
(0.240)
-0.056
0.099
(0.085)
(0.104)
1283.413

8597.940

Table F.12 (cont’d)
Variable
ATE

1PQML
-0.830 ***
(0.318)

NFES (2 Regime)
2PQML
-0.841 ***
(0.322)

NLS
-1.224 ***
(0.375)

R1
R0
0.412 *** 0.294 ***
(0.019)
(0.015)
-0.006 *** -0.003 ***
(0.000)
(0.000)
0.268 *** 0.312 ***
(0.045)
(0.042)
0.006 -0.129 ***
(0.041)
(0.034)
-0.070
-0.155 *
(0.057)
(0.079)
-0.060
-0.121
(0.079)
(0.155)
0.067
0.000
(0.061)
(0.046)
-6.843 *** -4.728 ***
(0.101)
(0.153)
0.390 ***
-0.081
(0.178)
(0.169)

R1
R0
0.410 *** 0.293 ***
(0.018)
(0.015)
-0.005 *** -0.003 ***
(0.000)
(0.000)
0.270 *** 0.309 ***
(0.044)
(0.042)
0.003 -0.127 ***
(0.040)
(0.033)
-0.074
-0.150 *
(0.057)
(0.077)
-0.063
-0.111
(0.078)
(0.153)
0.063
0.004
(0.060)
(0.045)
-6.813 *** -4.697 ***
(0.098)
(0.146)
0.373 **
-0.062
(0.169)
(0.159)

R1
R0
0.333 *** 0.239 ***
(0.022)
(0.017)
-0.004 *** -0.003 ***
(0.000)
(0.000)
0.219 *** 0.293 ***
(0.050)
(0.045)
0.031 -0.111 ***
(0.045)
(0.036)
-0.028
-0.142 *
(0.065)
(0.085)
0.075
-0.062
(0.089)
(0.174)
0.169 **
0.021
(0.069)
(0.049)
-5.671 *** -3.677 ***
(0.092)
(0.169)
0.656 ***
-0.009
(0.202)
(0.178)

educ7

age
agesq
evermarr
urban
electric
tv
radio
constant
cov(epsilon, v)

L-likelihood
-1067.756
1303.732
8521.373
2
R
sigma
Note: All the covariates are demeaned. All the figures in the parenthesis are bootstrap standard errors.
154

Table F.13: Regression Results: dependent variable ced
Variable
OLS

Linear Models
2SLS LET(Hkt)

LFES(Hkt)

ATE
educ7

age
agesq
evermarr
urban
electric
tv
radio
constant

-0.462 ***
(0.048)

-1.187
(0.725)

0.269 *** 0.260 ***
(0.021)
(0.023)
-0.002 *** -0.002 ***
(0.000)
(0.000)
0.734 *** 0.657 ***
(0.058)
(0.101)
-0.248 *** -0.186 **
(0.049)
(0.082)
-0.389 *** -0.293 **
(0.078)
(0.119)
-0.389 ***
-0.228
(0.089)
(0.190)
0.001
0.118
(0.057)
(0.134)
2.698 *** 3.101 ***
(0.037)
(0.405)

cov(epsilon, v)

L-likelihood
R2
sigma

-2.496 ***
(0.483)

-2.118 *
(1.084)

0.243 ***
(0.023)
-0.002 ***
(0.000)
0.517 ***
(0.087)
-0.073
(0.072)
-0.119
(0.105)
0.063
(0.136)
0.329 ***
(0.109)
3.828 ***
(0.276)
1.229 ***
(0.287)

R1
R0
0.251 *** 0.388 ***
(0.033)
(0.053)
-0.003 *** -0.003 ***
(0.001)
(0.001)
0.174 0.936 ***
(0.107)
(0.221)
0.233 ** -0.419 **
(0.097)
(0.190)
0.229 *
-0.507
(0.138)
(0.374)
0.537 ***
-0.426
(0.153)
(0.769)
0.733 ***
-0.048
(0.142)
(0.325)
2.308 **
(1.046)
2.944 ***
-0.296
(0.493)
(1.003)

NET (1 Regime)
1PQML
2PQML
NLS
-0.179
-0.189 -0.897 **
(0.373)
(0.343)
(0.419)
-0.074
-0.078 -0.372 **
(0.152)
(0.141)
(0.172)

0.339 ***
(0.009)
-0.004 ***
(0.000)
0.321 ***
(0.029)
-0.093 ***
(0.023)
-0.172 ***
(0.043)
-0.215 ***
(0.079)
-0.023
(0.114)
0.518 ***
(0.083)
-0.052
(0.091)
-84.416

0.605
1.513

0.588
1.545

0.608
1.509

0.615
1.495
155

0.339 *** 0.264 ***
(0.009)
(0.012)
-0.004 *** -0.003 ***
(0.000)
(0.000)
0.321 *** 0.283 ***
(0.029)
(0.034)
-0.093 *** -0.071 ***
(0.023)
(0.027)
-0.171 *** -0.140 ***
(0.042)
(0.050)
-0.214 ***
-0.106
(0.057)
(0.070)
-0.023
0.033
(0.030)
(0.037)
0.520 *** 0.783 ***
(0.077)
(0.104)
-0.049
0.131
(0.084)
(0.103)
2287.245

-9545.770

Table F.13 (cont’d)
Variable
ATE

1PQML
-0.974 *
(0.591)

NFES (2 Regime)
2PQML
-0.983 **
(0.400)

NLS
-1.488 ***
(0.511)

R1
R0
0.405 *** 0.293 ***
(0.018)
(0.015)
-0.005 *** -0.003 ***
(0.000)
(0.000)
0.287 *** 0.291 ***
(0.047)
(0.044)
-0.016 -0.101 ***
(0.041)
(0.035)
-0.103 *
-0.139 *
(0.060)
(0.080)
-0.106
-0.069
(0.083)
(0.158)
0.053
0.001
(0.063)
(0.051)
0.164 0.645 ***
(0.111)
(0.179)
0.300
0.011
(0.183)
(0.179)

R1
R0
0.404 *** 0.292 ***
(0.018)
(0.014)
-0.005 *** -0.003 ***
(0.000)
(0.000)
0.289 *** 0.289 ***
(0.045)
(0.043)
-0.018 -0.099 ***
(0.040)
(0.033)
-0.106 *
-0.136 *
(0.057)
(0.076)
-0.109
-0.061
(0.081)
(0.151)
0.051
0.004
(0.061)
(0.046)
0.171 0.656 ***
(0.105)
(0.154)
0.289 *
0.023
(0.171)
(0.159)

R1
R0
0.318 *** 0.239 ***
(0.021)
(0.018)
-0.004 *** -0.003 ***
(0.000)
(0.000)
0.240 *** 0.273 ***
(0.052)
(0.047)
0.006 -0.081 **
(0.045)
(0.037)
-0.056
-0.128
(0.067)
(0.085)
0.033
0.007
(0.096)
(0.174)
0.165 **
0.025
(0.073)
(0.053)
0.098 0.809 ***
(0.102)
(0.190)
0.579 ***
0.091
(0.212)
(0.185)

-66.489

2305.122

-9480.499

educ7

age
agesq
evermarr
urban
electric
tv
radio
constant
cov(epsilon, v)

L-likelihood
R2
sigma

156

Table F.14: Average Treatment Effects
children
2SLS

E(y1 )
E(y0 )
AT E

ceb

mort

1.741
1.914
0.173
2.926
3.101
0.175
-1.185 *
-1.187
-0.002
(0.691)
(0.725)
(0.220)
LET(Heckit)
E(y1 )
1.276
1.332
0.056
E(y0 )
3.508
3.828
0.32
AT E
-2.232 ***
-2.496 ***
-0.264 **
(0.432)
(0.483)
(0.123)
LFES(Heckit)
E(y1 )
0.326
0.19
-0.136
E(y0 )
1.878
2.308
0.43
AT E
-1.552
-2.118 *
-0.566 *
(0.979)
(1.084)
(0.291)
1PQML
E(y1 )
1.482
1.683
0.201
E(y0 )
2.312
2.657
0.345
AT E
-0.83 **
-0.974 **
-0.144
(0.337)
(0.438)
(0.152)
2PQML
E(y1 )
1.499
1.697
0.198
E(y0 )
2.34
2.68
0.34
AT E
-0.841 ***
-0.983 ***
-0.142
(0.310)
(0.373)
(0.123)
NLS
E(y1 )
1.264
1.402
0.138
E(y0 )
2.488
2.89
0.402
AT E
-1.224 ***
-1.488 ***
-0.264
(0.357)
(0.454)
(0.166)
Note: All the figures in the parenthesis are bootstrap standard errors.
The three nonlinear estimators on the bottom are all from NFES model.

157

Appendix G
Figures in Chapter 2

Figure G.1: Selected Monte Carlo Simulation Results for DGP 0.
Note: Each column represents the sampling distribution of 2SLS, NET (WNLS) and NFES
(WNLS) estimator from left to right. Each row represents that of ρ = .4, .5 and .6 from top
to bottom. The sample sizes are 5000 for all. Among these five grids, the middle one
represents the population ATE, i.e. 1.

158

Figure G.2: Selected Monte Carlo Simulation Results for DGP 1.
Note: Each column represents the sampling distribution of 2SLS and NFES (WNLS)
estimator from left to right. Each row represents that of ρ = .4, .5 and .6 from top to
bottom. The sample sizes are 5000 for all. Among these five grids, the middle one
represents the population ATE, i.e. 1.

159

Figure G.3: Selected Monte Carlo Simulation Results for DGP 2.

160

Figure G.4: Selected Monte Carlo Simulation Results for DGP 3.

161

Appendix H
Tables in Chapter 3

162

Table H.1: Simulation Results for ρ = 0.3
DGP

I
II
III
linear nonlinear
linear nonlinear
linear nonlinear
n=1000
mean
0.5640
2.192*
1.2664
1.584*
1.2505
5.366*
mc. st. dev
1.4758
3.703*
1.7732
2.326*
2.4280
15.545*
RMSE
1.5389
3.890*
1.7931
2.399*
2.4409
16.146*
median
0.5930
1.031
1.1954
0.961
1.0098
1.041
MAD
0.9266
1.023
1.0931
0.939
1.2917
1.322
3.5610
7.286
4.3419
5.947
5.0139
15.939
IR
n=3000
mean
0.5781
1.361
1.3657
1.233
1.2210
2.430
mc. st. dev
0.7665
1.613
0.9766
1.423
1.1970
5.627
RMSE
0.8750
1.653
1.0429
1.442
1.2172
5.806
0.5971
1.064
1.3409
0.913
1.1089
1.074
median
MAD
0.5106
0.547
0.6304
0.488
0.7878
0.809
1.9470
2.398
2.4461
2.403
3.1001
5.169
IR
n=5000
mean
0.5356
1.158
1.3692
1.035
1.2210
1.606
mc. st. dev
0.6042
0.807
0.7265
0.766
0.9759
2.919
RMSE
0.7620
0.822
0.8150
0.767
1.0006
2.981
median
0.5220
1.032
1.3442
0.890
1.1648
1.046
MAD
0.4223
0.394
0.5115
0.367
0.6017
0.589
IR
1.5592
1.750
1.8633
1.705
2.3029
3.032
Note: The Monte Carlo standard deviation is substantially large for the sample
size of 1000. The * indicates that the outlying values that are greater or smaller
than the maximum and minimum values of for the sample size of 3000 were
discarded from the generated data.

163

Table H.2: Simulation Results for ρ = 0.5
DGP
n=1000

mean
mc. st. dev
RMSE
median
MAD
IR
n=3000
mean
mc. st. dev
RMSE
median
MAD
IR
n=5000
mean
mc. st. dev
RMSE
median
MAD
IR

linear
0.7653
1.4605
1.4792
0.7267
0.9558
3.6600
0.7548
0.7953
0.8323
0.7494
0.5424
2.0694
0.7155
0.6110
0.6740
0.7179
0.4144
1.5459

I
nonlinear
1.358*
1.843*
1.877*
0.984
0.927
5.074
1.147
1.044
1.055
0.975
0.478
2.085
1.139
0.749
0.762
1.031
0.413
1.594

164

linear
1.3439
1.7058
1.7401
1.3196
1.0801
4.3459
1.3890
0.9522
1.0286
1.3690
0.6060
2.4681
1.3440
0.7306
0.8076
1.3386
0.5071
1.9085

II
nonlinear
1.634*
2.590*
2.667*
0.931
0.990
6.482
1.182
1.421
1.432
0.872
0.505
2.331
1.024
0.788
0.789
0.865
0.388
1.657

linear
1.6385
2.3073
2.3940
1.4693
1.3670
5.5827
1.5936
1.3847
1.5066
1.4946
0.8624
3.2039
1.7004
1.0403
1.2541
1.6007
0.6847
2.5199

III
nonlinear
3.457*
9.524*
9.836*
1.045
1.200
11.164
1.937
4.738
4.830
1.028
0.764
3.935
1.418
1.757
1.806
1.006
0.574
2.697

Table H.3: Regression Results for Linear and NFES models
Variables
ATE

OLS

educ7

-0.419***
(0.049)

2SLS

LFES

-1.152*
(0.596)

NFES (PQML)
-0.956***
(0.278)

NFES (NLS)
-1.103***
(0.315)

-1.168**
(0.528)
R0
0.431***
(0.043)

R1
0.419***
(0.020)

R0
0.300***
(0.016)

R1
0.339***
(0.022)

R0
0.250***
(0.018)

age

0.273***
(0.017)

0.263***
(0.019)

R1
0.271***
(0.020)

agesq

-0.002***
(0.000)

-0.002***
(0.000)

-0.004***
(0.000)

-0.004***
(0.001)

-0.006***
(0.000)

-0.004***
(0.000)

-0.005***
(0.000)

-0.003***
(0.000)

evermarr

0.685***
(0.052)

0.612***
(0.080)

0.135*
(0.078)

1.088***
(0.138)

0.212***
(0.047)

0.331***
(0.043)

0.174***
(0.055)

0.321***
(0.046)

urban

-0.259***
(0.046)

-0.172**
(0.084)

0.187***
(0.064)

-0.596***
(0.120)

0.063
(0.051)

-0.152***
(0.040)

0.103*
(0.059)

-0.141***
(0.044)

electric

-0.468***
(0.066)

-0.287*
(0.161)

0.117*
(0.093)

-0.781***
(0.224)

0.010
(0.079)

-0.242**
(0.113)

0.105
-0.244*
(0.090)
(0.127)
See next page.

165

Table H.3 (cont’d)
Variables
OLS
constant
-3.514***
(0.245)
cov(ϵ, v)

2SLS
-2.815***
(0.620)

LFES
0.085
1.253***
(0.268)
(0.478)

NFES (PQML)
-0.100
0.397**
(0.097)
(0.154)

NFES (NLS)
-5.668*** -4.077***
(0.327)
(0.469)

2.202***
(0.304)

0.714
(1.398)

0.963***
(0.295)

-1.484**
(0.605)

-0.204
(0.192)
1299.311

-0.213
(0.218)
8534.8903

L-likelihood
R2
0.585
0.564
0.593
σ
1.433
1.469
1.418
Note: All the standard errors presented above are bootstrap standard errors. *, ** and *** indicates the significance at 10%, 5% and 1% levels respectively.

166

Table H.4: Regression Results for LTCRC and NTCRC models
Variable
ATE
educ7

age (1)
agesq

evermarr (2)
urban (3)

electric (4)

LTCRC

13.376***
(2.347)
R1
R0
0.104*** 0.343***
(0.030)
(0.040)
-0.003** (0.001)
0.009***
(0.002)
-0.047
1.377***
(0.164)
(0.299)
0.132
-0.178
(0.155)
(0.287)
0.396**
(0.175)

0.686
(0.919)

age*evermarr
age*urban
age*electric
evermarr*urban
evermarr*electric
urban*electric
constant

0.999*** 1.440***
(0.345)
(0.502)

167

NTCRC
-1.020
(1.633)

R1
0.405***
(0.040)
0.005***
(0.001)
2.180***
(0.844)
-0.265
(0.382)

R0
0.106
(0.088)
-0.001
(0.001)

0.690**
(0.321)
0.826***
(0.319)
-0.675** -0.480
(0.335)
(2.471)
-0.097** -0.009
(0.041)
(0.008)
0.013
0.012
(0.025)
(0.008)
0.009
-0.010
(0.010)
(0.054)
0.060
0.132**
(0.427)
(0.066)
0.479*
0.202
(0.272)
(0.178)
0.295
0.088
(0.287)
(0.248)
-1.666
7.010*** (1.842)
(0.749)
See next page.

Table H.4 (cont’d)
Variable
LTCRC
NTCRC
R1
R0
R1
R0
cov(ϵ, v)
1.208*** -0.677
2.438** 1.353
(0.460)
(0.553)
(1.068)
(2.435)
cov(1)
0.022
0.135***
-0.108
-0.041
(0.033)
(0.043)
(0.074)
(0.121)
cov(2)
0.675** 0.621**
0.290
6.954**
(0.287)
(0.313)
(0.414)
(3.374)
cov(3)
-0.085
0.431
-0.312*
-0.225
(0.263)
(0.279)
(0.182)
(2.127)
-0.869
0.953*
-0.094
-0.255
cov(4)
(0.583)
(0.563)
(0.680)
(1.240)
L-likelihood
1341.602
2
R
0.6036
σ
1.4026
Note: The bootstrap error for NTCRC ATE estimator is calculated
by excluding all the replication with ATE values over 13. cov(1) is
the covariance between v and the coefficient of variable (1), i.e. age.
Variable number for each basic covariate is indicated in the variable
list. cov(2) and other numbers are defined similarly.

168

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