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THREE ESSAYS ON NONLINEAR MODELS FOR
FRACTIONAL RESPONSE VARIABLES WITH TIME-
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YOUNG GUI KIM

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THREE ESSAYS ON NONLINEAR MODELS FOR FRACTIONAL RESPONSE
VARIABLES WITH TIME-VARYING INDIVIDUAL HETEROGENEITY

By

Young gui Kim

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Economics

2009

ABSTRACT
THREE ESSAYS ON NONLINEAR MODELS FOR FRACTIONAL RESPONSE
VARIABLES WITH TIME-VARYING INDIVIDUAL HETEROGENEIT Y
By
Young gui Kim

This dissertation consists of three chapters on nonlinear panel models for fractional
response variables. The first chapter extends the framework of Papke and Wooldridge
(2008) by allowing unobserved individual heterogeneity to vary over time. An interaction
term between a time effect and an individual effect of Kiefer (1980) and Lee (1991),
known as an interactive effect, is added to the model. This allows each cross-sectional
individual to respond differently to time effects common to all individuals. Based on
Papke and Wooldridge (2008), I discuss the pooled Quasi maximum likelihood estimator
(QMLE) and the multivariate weighted nonlinear least squares (MWNLS) estimator to
estimate the model in strictly exogenous case. The two-step procedure of Rivers and
Vuong (1988) and the single-step estimator of Wooldridge (2007) are discussed in the
endogenous case. The methods are applied to analyze the effect of school spending on
math test pass rates of 4th graders in Michigan. According to the Monte-Carlo
simulations, the fractional time-varying model gives the least root mean squared errors
among all three models in a reasonable range of correlation between the regressors and
individual heterogeneity.
The second chapter continues studying fractional response models and considers binary
endogenous explanatory variables (EEVs). Because EEVs are not continuous, I modify

the bivariate probit model to derive a conditional mean fimction for a fractional response

variable. The pooled quasi-limited information maximum likelihood estimator (QLIMLE)
is proposed based on this mean function. The conditional mean function for a fractional
response variable and the reduced form for a binary endogenous variable are estimated
together to identify all parameters of interest [Wooldridge (2007)]. Also the average
treatment effects (ATEs) of a binary EEV are discussed. The estimation method is applied
to study the effect of fertility on women's fractions of working hours. The simulations
show the root mean squared errors of the ATEs of the fractional bivariate model are less
than both the linear and the fractional probit model.

In the last chapter, I discuss the hurdle model for a fractional dependent variable.
Binary endogenous explanatory variables and time-varying individual heterogeneity are
considered as before. The hurdle model allows us to separate the determination of comer
solutions (y=0 or y>0) from the decision of the amount of dependent variables
conditional on y>0. This hurdle model allows us to use different sets of EEVs for the
comer solution equation and the equation for the amount and to obtain the ATEs
conditional on y>0. I study the fertility effect of the second chapter again by using the

hurdle model.

ACKNOWLEDGMENTS

I cannot express enough thanks to my advisor, Professor Jeffrey M. Wooldridge for
his enthusiastic advice and support. Without his advice and discussion, I could not
have written this dissertation. He also provided me the data for the first chapter. I am
very gratefill to Professor Timothy Vogelsang. His generous support made me possible
to successfully finish the doctoral program. I would like to thank my committee
members, Professor Leslie E. Papke and Professor Yuehua Cui for their valuable
comments and constructive discussions. I would like to express graditute to my family

for their understanding and encouragement.

iv

LIST OF TABLES

1

1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8

2.1
2.2
2.3
2.4
2.5
2.6

3.1
3.2
3.3
3.4
3.5
3.6

TABLE OF CONTENTS

FRACTIONAL RESPONSE MODELS WITH TIME-VARYING

vi

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

INDIVIDUAL HETEROGENEITY 1
Introduction 1
Basic Model 3
Estimation methods with strictly exogenous variables 7
Estimation methods with endogenous explanatory variables 12
Test statistics for time-varying individual effects 15
Empirical application: math test pass rates 17
Monte-Carlo simulations 24
Conclusion 27
THE FRACTIONAL MODELS WITH BINARY EN DOGEN OUS

EXPLANATORY VARIABLES AND TIME-VARYING

INDIVIDUAL HETEROGENEITY 30
Introduction 30
The Model with binary endogenous explanatory variables 33
Estimation methods . 36
Application: fertility effect on working hours of women 40
Monte-Carlo simulations 45
Conclusion 48
THE HURDLE MODEL FOR FRACTIONAL RESPONSE

VARIABLES WITH BINARY ENDOGENOUS EXPLANA-

TORY VARIABLES AND TIME—VARYING INDIVIDUAL

HETEROGENEITY 50
Introduction 50
The basic fractional hurdle model 51
Estimation methods under strict exogeneity 54
Estimation methods with binary endogenous explanatory variables ................. 56
Application: fertility effect on working hours of women 61
Conclusion 65

Appendix: A Tables for Chapter 1 66
Appendix: B Tables for Chapter 2 81

 

Appendix: C Tables for Chapter 3

BIBLIOGRAPHY

 

 

vi

88

93

Table A.1

Table A2

Table A.3

Table A.4

Table A.5

Table A6

Table A7

Table A8

Table A.9

Table A.10

Table All

Table A. 12

Table A.13

Table A.14

Table B. 1

Table 32

Table B.3

LIST OF TABLES

Descriptive Statistics ......................................... 67
The Linear and the Fractional Model: Exogenous Spending ........ 68
The Fractional Time-varying Model: Exogenous Spending ......... 69
The Linear and the Fractional Model: Endogenous Spending ....... 70

The Fractional Time—varying Model: Endogenoussspending,
Two-step Estimator .......................................... 71

The Fractional Time-varying Model: Endogenous Spending,

Limited Information MLE ..................................... 72
The Scale Factors at Different Spending Levels (in 1995 and

2001) ...................................................... 73
Mean Squared Errors and Biases of the APEs ( Increasing 7),

Pro = 0.1 — 0.5) ............................................. 74
Mean Squared Errors and Biases of the APEs ( Increasing 77,

Pm = 0.6 — 0.9) ............................................. 75
Mean Squared Errors and Biases of the APEs ( Decreasing 77,

Pm = 0.1 — 0.5 ) ............................................. 76
Mean Squared Errors and Biases of the APEs ( Decreasing 71,

Pm = 0.6 — 0.9) ............................................. 77
Mean Squared Errors and Biases of the APES (pm 2 0.2) .......... 78
Mean Squared Errors and Biases of the APEs (pxc = 0.3) .......... 79
Mean Squared Errors and Biases of the APEs (ch = 0.4) .......... 80
Descriptive Statistics ......................................... 82
The Linear and the Fractional Model: Exogenous Fertility ......... 83
The Hactional Time-varying Model: Exogenous Fertility .......... 84

vii

Table B.4

Table 35

Table B.6

Table C. 1

Table C .2

Table C.3

Table C4

The Linear and Fractional Model: Endogenous Fertility ........... 85

The Fractional Time-varying Model: Endogenous Fertility ......... 86
Mean Squared Errors and Biases of the ATEs .................... 87
The Fractional Hurdle Model: Exogenous Fertility ................ 89
The Fractional Hurdle Time-varying Model: Exogenous

Fertility .................................................... 90
The Fractional Hurdle Model: Endogenous Fertility ............... 91

The Fractional Hurdle Time-varying Model: Endogenous
Fertility .................................................... 92

viii

Chapter 1 F RACTIONAL RESPONSE MODELS WITH TIME-VARYING
INDIVIDUAL HETEROGENEITY

1.1 Introduction

Some economic variables are measured as fractions, percentages, or proportions.
Examples include market shares, 401(k) participation rates, test pass rates, expenditure
shares, and cost shares. Many models and estimation methods have been proposed for
fractional dependent variables. Fractional variables always takes values between 0 and
1 including two extreme values. A simple linear model is easy to be estimated and
interpreted, but the fitted values could be greater than 1 or negative even though the
dependent variable is defined in the unit interval. One might use the log-odd transformation
to avoid this problem, but we may not recover the original dependent variable without
further assumptions.

Papke and Wooldridge (1996, 2008) proposed a new approach to modeling fi'actional
response variables based on a correctly specified conditional mean function. I focus on
their two estimation methods. The first one is the pooled quasi-maximum likelihood
estimator (QMLE) based on the Bernoulli distribution. With large enough samples, models
can consistently estimated without distributional assumption by their estimation methods
[For detailed discussion, see Papke and Wooldridge (1996), Gourieroux, Monfort, and
Trognon (1984)]. The second method is the weighted non-linear least squares (WNLS)
estimator for a non-linear mean function. A multivariate WNLS (MWNLS) estimator is
proposed to address serial correlations in the cross sectional dimensions of their panel

data.

In this chapter, I combine this framework with the time-varying individual effects
of Kiefer (1980) and Lee (1991). In most panel analyses, unobserved individual
heterogeneity has been assumed as a time-invariant factor. Kiefer (1980) and Lee (1991)
introduced time-varying individual heterogeneity in a linear model by adding interaction
terms between individual effects and time effects. I use this interaction term (sometimes
called an interactive effect) to relax the constant individual heterogeneity assumption in
our non-linear modell. This allows each individual to respond differently to the common
time effects. One example where this can be useful is in a model of income where the
productivity of an individual (unobserved ability) varies over the business cycle [Ahn,
Lee and Schmidt (2007)]. In our application, the effect of school spendings on math test
pass rates in Michigan is studied by using school district level data. The model with
time-varying individual heterogeneity allows each school district to respond differently to
the difficulty levels of the math tests.

The remainder of the chapter is organized as follows: In section 2, I introduce the basic
model. Section 3 and section 4 contain estimation methods under the assumption of strictly
exogenous explanatory variables and endogenous explanatory variables, respectively. I
provide three test statistics for the null hypothesis of no time-varying individual effect
in section 5. I apply the estimation methods to the test pass rates for fourth graders in
Michigan and conduct Monte-Carlo simulations in sections 6 and section 7. The last

section concludes.

1.2 Basic Model

Let us assume N randomly drawn cross-sectional units from the population and T

 

While Lee (1991), and Ahn, Lee, Schmidt (2007) considered mulitple time-varying factors in a linear model,
one time-varying factor in a nonlinear model is considered because of an identification problem.

2

observations for each cross-sectional observation. The number of cross-sectional units is
assumed to be substantially larger than the number of time series observations (large N
fixed T asymptotics).

Let’s assume a correctly specified conditional mean function:

E (yitixitvci) = 9 (33215 + 772%), (H)
where i indexes a cross-sectional unit and t indexes time (2' = 1,2, ..., N, t = 1, 2, ..., T).
yit denotes a fractional dependent variable for individual 2' and time t that is continuous
between 0 and 1 (also Mr: is allowed to take 0 or 1 with positive probability), and wit is a
1 x K vector of explanatory variables. 0,- indicates unobserved individual heterogeneity
(an individual effect) and 7it is its corresponding coefficient that is allowed to vary
over time. 722: can be regarded as a time effect in that this coefficient is common to all
cross-sectional units at time t. Therefore, 77tCz' is sometimes called an interactive effect
[Bai (2005)]. This term is assumed to be linear additive in the model. To ensure that the
fitted values take values between 0 and l, cumulative distribution functions (cdf) can be
used for the mean function of yit- In particular, the standard normal cdf, <I>(-), is used
in this chapter because this gives a computationally simple estimator when the model
contains unobserved heterogeneity and endogenous regressors.

The main contribution of this paper is to allow the effect of unobserved individual
heterogeneity to vary over time in the linear index. One might attempt to consider the
most flexible individual efiect which varies freely over time (Cit)- However, there are two
problems: First, it is impossible to distinguish a time-varying individual effect (Cit) from

an idiosyncratic error (Hit) without further assumptions. Second, it seems unreasonable

3

to assume that individual heterogeneity, such as innate ability, changes over time. An
alternative is an interaction term between a time effect (in) and a constant individual
effect (Ci). We can interpret this interactive effects in two ways: the effect of individual
heterogeneity is varying over time or time dummies have individual specific intercepts.
This interactive effect model includes the constant individual effect model as a special
case and still retains a manageable structure. I set 771 = 1 as a normalization.

Because this is a nonlinear model, we are interested in the partial effects not the

coeflicients. Dropping subscript i, we can obtain the effect of the kth

regressor,
as“, on yt from 6E(yt|:ct,c)/6:ct,k = fikqbcrtfi + UtC) if $t,k is continuous, or
<I>(:r:§1)fl + UtC) — @(rémfi + me) where mil) and 10:0) denote two different values
of mt, k if 931:, k is discrete. As we can see, the partial efi‘ect depends on set and c. It

is a good idea to evaluate this effect at interesting values of x’s or average the partial
effects across individuals. However, we may not observe c. Therefore, a popular
method is to calculate the average partial effects (APEs) by averaging the partial effects
across the distribution of c. The APEs can be obtained from Ec[flk¢(a:tfi + ntc)] or
Ermine + m6) —- mime + men.

In order to identify both fl and APEs, we use two assumptions:

E (yitlxia 02') = E (yrtll‘ita 02'), (12)
where 3,; E (in, ..., xiT) is a set of explanatory variables in all time periods. The
first assumption is that xit conditional on c,- is strictly exogenous. This assumption
rules out dynamic models and feedback models. Also this implies that we do not

consider traditional simultaneous models and the omitted variable problem. If appropriate

4

instrumental variables are available, this assumption can be relaxed so that explanatory
variables are allowed to be endogenous.

The second assumption is

c,- = tb+§3,-€+a,,a,-|:1:,- ~ N(o,a?,), (1.3)
where T,- (E T‘1 231:1 :r:,-t) is a 1 x K vector of time-averaged explanatory variables
and a,- denotes the part of unobserved individual heterogeneity which is not correlated with
33,. 0,2, is the variance of c,- given 13,-. This Mundlak (1978)-Chamberlain (1980) device
is adopted to restrict the distribution of c,- conditional on 33,-, which enables us to get the
APEs by using the law of iterated expectations (LIE). This implies that constant individual
heterogeneity is correlated with time-invariant parts of strictly exogenous explanatory
variables. For the purpose of flexibility, one can consider more general functional forms
for D(c,- |x,-) by adding squared and/or interaction terms. Also parametric modelling for
Var(a,-|:r:,) or relaxing the normality assumption of a,|:c,~ can be considered, but I focus
on (1.3) in this chapter. [For detailed discussion about more general specifications for
D(c,-|:r:,~), see Papke and Wooldridge (2008).]

Under assumptions (1.2) and (1.3),

E (yrtlfcz', at) = ‘1) (WP + $715 + Ut’fif + mar) (14)
nta,|:r:, ~ N (0, 77%02) .
53,-, must contain only time-varying elements so that there is no perfect multicollinearity

between 23,-, and 5,. Adding time dummies to the model is desirable in order to allow

different intercept for each time. Even though nttb is absorbed into the coefficients of time
dummies, I do not consider time dummies explicitly for notational simplicity.

By the properties of the normal distribution, we can rewrite the equation (1.4) as

<I> 77M + 113213 + 772317326

 

Err/mm.) 2 2 (1.5)
1 + 77, 0a
5 9 Watt + witflat + iiéat),
where 't/Jat E 471% , fiat = 2 2 , and Eat E _’71_§2_2 denote
1+7], 0a 1+1], 0a 1+7], 0a

scaled parameters. The subscript a means that all parameters are scaled by 0,2, and
the subscript it means those parameters depend on time due to 77t- Because we can
observe (1, :r:,-t,E,-), the scaled parameters Wat, fiat, Eat) are identified. Papke and
Wooldridge (2008) proposed methods to estimate parameters and APEs consistently
without distributional assumptions about D(y,-, |:c,, 0,).

By the LIE, the average structural function is obtained as follows.

Er, [PU/lat + Svtfiat + fitter” - (1-6)
The APEs for each t can be obtained by differentiating equation (1.6) with respect to
CL‘t,k. 513,- is redundent in the conditional mean firnction and c,- and :r:,-t are independent
conditional on 52,-. With these two assumptions, we can use the arguments in Wooldridge
(2002, sectional 2.2.5). Therefore, the APEs of a: k at time t can be consistently estimated
by N-1 2,121 Bat,k¢ (that + 1,13,“; + sing“) for each t. A scale factor also can
be obtained from (NT)"'1 2;:1 ELI ¢ (12201, + mitBat + iiéat) We can compare

coefficients of the nonlinear models with those of linear models after multiplying the

coefficients of the nonlinear model by this scale factor.

1.3 Estimation methods with strictly exogenous variables

Let us define wit E (1, z,t,E,) and 7rt E (t/Jgt, 5:1,, €;t)’ for notational simplicity.
Then, the conditional mean function is <I>(w,t1rt). Before proceeding, I need to discuss
two practical issues: normalization of 0,2, and recovering parameters of interest. For
identification, 0a is set to l as normalization, but this does not affect the results because
we have a nonlinear model. From now on, I will drop the subscript a. Our interest lies
in 6 E (10’, 77’, 5’, 5’)' not 7r E (rt/1, 7rI2, ""7rf’1‘) , and there are specific relationships
between 7t and 0 (For example, wt = 7,0/ 1 + 77?). Therefore, two approaches can be
used: the classical minimum distance (CMD) estimator and estimation with constraints.
First, the CMD estimator recovers 6 from 7r by minimizing the weighted differences in
the relationships between 7r and 6. This CMD estimator uses the variance-covariance
matrix for it as weight. The detailed explanation for the CMD estimator will be provided
later. Second, one can estimate the model after imposing the relationships directly on the
objective functions. Estimation methods with constraints are straightforward.

Now focus on the ways to estimate 7r consistently. Given the conditional mean fimction,
there are many possible estimators. One estimator is the pooled nonlinear least squares
(PNLS) estimator. This estimator is consistent and x/N-asymptotically normal (T is fixed).
However, the pooled NLS estimator is ineflicient because this assumes homoskedasticity,

but this assumption does not hold. A reasonable model for the variance, V(y,-t |:r:,-), has the

following form:

V (yit I 3373) = 790021:th [1 - ‘1’ (witfltlL (1-7)

where r is a constant. [Readers are referred to Papke and Wooldridge (1996) for detailed
discussion] Based on (1.7), the weighted nonlinear least squares (WNLS) estimator can
be an alternative. In this chapter, the pooled quasi-maximum likelihood estimator (QMLE)
based on Bernoulli distribution is used instead of the WNLS estimator. The pooled QMLE
is equivalent to the WNLS when the inverse of (1.7) is used as a weighting matrix , but
the pooled QMLE is simpler because the NLS estimator needs preliminary results to
obtain the weighting matrix. Also this estimator is strongly consistent even if the true
distribution of y is not Bernoulli once the first moment is assumed to be correctly specified
because the Bernoulli distribution is belong to the linear exponential family (LEF) [Papke
and Wooldridge (1996), Gourieroux, Monfort and Trognon (1984)]. The pooled QMLE
is obtained by maximizing the sum of the following log-likelihood function. I call this
estimator the pooled fractional probit estimator (FPE) as Papke and Wooldridge (2008)
did.

git (“121;”) = yit10g[<P(t/Jt+ mttflt + irétll (1-8)

+(1- yit)108 [1 — PM: + xitfit + 551623)]
E y,t10g[<b(w,t7rt)] + (1 - 3m) log I1 — (PIWitthl-

Another possible estimator to enhance the efliciency is the multivariate weighted
nonlinear least squares (MWNLS) estimator which considers serial correlations among
errors within a cross sectional unit 2'. Let define w,- E (D, Dr“, 03,) which is a
T x (T + 2TK) matrix of explanatory variables. D E ITxT denotes a set of all time
dummies. Data denotes T x TK matrix which has 10,-, for its diagonal elements and

0 for all off-diagonal elements. In the same way, D5,- is defined as T x TK matrix

with its diagonal elements of 35,-. This matrix contains all interaction terms between
E, and D. Suppose the conditional mean function for y,- E (31,1, y,2, ..., yiT)’ be
m,(w,-, 7r) E <I>(w,-7r). Applying MWNLS estimator requires the variance-covariance
matrix of 3),, Var(y,-|:r:,-). Along with (1.7), we need to form Cov(y,-t, y,3|:r:t),t # 3.
Instead of using a parametric model for Var(y,- [33,), its working version proposed in the
generalized estimating equations (GEE) literature is used. Also the standardized errors
are assumed to have the same correlation (exchangeable correlation assumption). The

common correlation is calculated by f) = [N T(T —— 1)]_1 Z,N___1 ELI 2,3753 é,té,-S,

 

where éit = flit/«@(witfit) [I —- <I>(w,t‘7l't)] and {fit E yit — (I)(wit7i't). ‘~’ means that all
are evaluated at the preliminary consistent estimates from the pooled FPE. The MWNLS

estimator solves

N
ngnZIyi‘mi(wiv7r)iII7i—1Iyi‘mi(wiv7I)I (1-9)

i=1

V,- = Diag [6,(1—6,)]1/2C

,

(away [in-(1 — in] 1/ 2

where I7,- is the T x T matrix where its diagonal elements are from (1.7) and its off-diagonal
elements from the ‘working correlation matrix’. C(23) is the T x T matrix where its
diagonal elements are l and its off-diagonal elements are p [Papke and Wooldridge (2008),
Liang and Zeger (1986)]. This MWNLS estimator is equivalent to the GEE when the same
weighting matrix is used. Note that we cannot consider the serial correlations when we
recover the parameters by the CMD estimator because pure cross-section data is available
for each t. Therefore only constrained MWNLS (or constrained GEE) can be used to

consider possible serial correlations.

For testing hypotheses and using the CMD estimator, we need a consistent estimator of
asymptotic variance matrix. Hypotheses include independence between the unobserved
heterogeneity and regressors (H0 : g = 0) and time-invariant individual heterogeneity
(H0 : 77t = 1). Because we do not assume that the conditional variance of y is correctly
specified, a consistent estimator of asymptotic variance robust to heteroskedasticity and
serial correlation has so-called sandwich form [Arellano (1987), Wooldridge (2002)].

With regularity conditions, the pooled FPE is x/N-asymptotically normal with the

variance of

REE (erMLE) = A-IBA—l/N, (1.10)
where A = N-1 iii. 23;. (mic-m), B = N-1 2.1212; Eases, and
is}, = Vw€,t(fr‘) [Wooldridge (2002, Theorem 12.3)].

In the case of the MWNLS estimator has the following asymptotic variance.

—1
N
AvarnMWNLs) = vamgvflvm. (1.11)

i=1

N A A
x zvflmgt/i—la,agig—1vnm,

i=1

N —1
X(ZV7rfn;I/i—1V7rrh, /N,

i=1

where Vwrh, is the Jacobian of the mean firnction, m,- is m,(w,-, 7r) evaluated at 7?, and
u,(E y,- - m,(w,, 7%)) is a residual vector.
With scaled estimates (it) and their asymptotic variances (E), the CMD estimator is

used to estimate (9 from fr. We have (T + 2K) parameters of interest (6), but 7? is a

10

(T + 2TK) x 1 vector. Let’s suppose h(-) be a continuously differentiable function to
represent relationships between 6 and 1t (7r = 12(6)). The CMD estimator minimizes the

A

distance between it and h(6).

H31; [7} — h(0)]’§—1 [7r — h(6)]. (1.12)
From the first order condition for 6 and a standard mean value expansion around 6, the

following asymptotic distribution of 6 is obtained.

W (a — 9) fl: N (0, ways—111(9)) , (1.13)
where H (6) (E V9h(6)) is the (T + 2TK) x (T + 2K) Jacobian of h(6). The appropriate

estimator of asymptotic variance of 6 is

A

A A I A A
Avar(6) = (H E‘lH)-1/N, (1.14)
where H is H evaluated at 6. When we use the estimators with constraints to get 6 directly,

the robust asymptotic variance of 6 has the same form with (1.14).

1.4 Estimation methods with endogenous explanatory variables

Now let’s relax the strict exogeneity assumption given appropriate instrumental
variables. Time-varying individual heterogeneity is considered as above. One endogenous
variable case is considered in this chapter, but considering multiple endogenous variables
case is straightforward. The basic model with an endogenous explanatory variable is as

follows:

11

E (Mull/£132: Zrt, Ci1,vit1) = E (yitliyith 261,011,1511) (1-15)
= ‘1’ (0113/62 + 26151 + 771611 + Um)
where yitg denotes an endogenous explanatory variable which is assumed to be correlated
not only with individual heterogeneity (0,1) but also with unobserved omitted variable
(1),“). 2,, (E (2,,1, z,t2)) denotes a set of exogenous explanatory variables and
instrumental variables. Instrumental variables (2,,2) should not have significant effects on

ym and must be at least partly correlated with 3,1,,2.

The Mundlak(1978) -Chamberlain(l980) device is applied again (cillz, ~

N ($1 + E2'51"" ai11031))-

E(yrt1|yit2, Zr. Tm) = (1’07th + alyit2 + 221151 + 77155161 + 7"211) (1-16)
where rm denotes a composite error of man and 22,131. In this structural equation, mtg is
assumed to be correlated with 11",“.

The reduced form for y,t2 is assumed as

mm = Wt + 22152 + 5162 + 73:52 E 101127 + Trt2~ (1-17)
This equation can be consistently estimated by the pooled OLS estimator which gives
the same estimates with the fixed effects estimator because the averages of explanatory
variables are added. Endogeneity of y,t2 implies that mtg is correlated with both

ail and 22,“. I assume that rm given rm is conditionally normal and this leads

m1 = Pth‘tZ + 612:, eitIZz' ~ N(0, T?)-

12

Substituting r,t1 with Pth't2 + Git,

E(yrt1|yz't2. 22', TM) = @(ntwt + alyitZ + Zit151 + with (1-18)

+Pt7‘z't2 + err)

 

<I> mm + 0113622 + 3213161 + 61:26 1 + ptt‘z'tz
1 + 7%
Because joint normality of (7,”, 73,2) is assumed, the final model (1.18) comes from

properties of normal distribution. T1 is set to 1 as normalization.

With a continuous endogenous variable, two possible approaches can be used to
estimate the equation (1.18) consistently: the two-step estimation procedure and the
limited information maximum likelihood estimator(LIMLE). The LIMLE is called the
single step estimator compared to the two-step estimation procedure. The two-step
procedure proposed by Rivers and Vuong (1988) is as follows.

1. Estimate the reduced form for yg (1.17) and get the residuals, 63,2.

2. Replace 7,152 with 53,2 as an additional regressor and estimate the structural model
(1.18).

Now all explanatory variables are observable and the omitted variable problem can be
avoided by adding 72,2. Even though fag is added to the model, mg is still allowed not
to be strictly exogenous. Therefore, I use the FPE instead of the MWNLS estimator. The
MWNLS estimator requires the strict exogeneity as the generalized least squares (GLS)
estimator in linear models does. The endogeneity (H0 : (0,; = 0) can be tested by using the
Wald statistics. If the null hypothesis is rejected, we should adjust the asymptotic variance

for inference because the final model (1.18) contains a generated regressor (Fug) from the

13

first step estimation. In other words, the asymptotic variance of W (6 — 6) depends on
the asymptotic variance of x/N (7i — 7) [For detailed discussion, Wooldridge (2002) ch.
12.4 and Papke and Wooldridge (2008) Appendix 1].

This two-step procedure has some advantages. Along with computationally simplicity,
the endogeneity can be easily tested because we do not have to adjust asymptotic variances
under the null hypothesis. However, the asymptotic variances should consider generated
regressors if the exogeneity is rejected, and this approach cannot be used if endogenous
explanatory variables are not continuous. .

Wooldridge (2007) proposed the quasi limited information MLE (QLIMLE) of
nonlinear models without additional assumptions of the joint distribution of y1 and yg.

Substituting mg with (ym — 212,127), we can rewrite the equation (1.18) as

 

 

77112 +ay- +z- 6 +17'z'f
E(yit1I3/it2azi1wit2) = ‘I’I t 1 1 “2 m 1 t z 1 (1-19)
1+7?
”1+7?
E <I>(w,-t17r).

Because I assume a correctly specified mean firnction and the Bernoulli is the member
of the linear exponential family, the Bernoulli QMLE estimates the equation (1.19)
consistently. To identify all parameters, equation (1.19) is estimated along with the
equation (1.17). The the quasi-log-likelihood function is as follows:
121(9) = yit 108 Iq)(wz't17rt)I + (1 — yit)108 [1 - @(wz'tlfitll (121)
—(1/2) loge?» — mm [(m + waif/032] ,

where 032 E Var(r,t2|w,t2). Compared to the two-step procedure, the QLIMLE

14

might be more efficient because both the structural equation and the reduced form for
the endogenous variable are estimated together, and estimating valid standard errors
is straightforward. Also this approach can be used when the models contain discrete
or limited endogenous variables, while the two-stage estimator is invalid [Wooldridge

(2007)].

1.5 Test statistics for time-varying individual effects

In this section, I discuss test statistics for testing constant individual heterogeneity
(H0 : 17, = 1). Let us start from the basic estimating equation of Papke and
Wooldridge (2008), E(y,,|:r,) = (I) (7,!) + 510,,6 + 53,0. To consider time-varying
individual heterogeneity, consider a more general estimating equation, E (y,,|:r:,) =
(130/1 + $63 + M + 5::(52'5 X Ttl) E 4’ (wz'fl + Mité >< T0) E mu, Where 716 X Tr
indicates interaction terms between 33,7; and all time dummies. The degree of freedom
does not depend on the number of regressors in this case.

The first statistic is a regression-based test which is also known as a variable addition
test. This is a special case of Ramsey’s RESET in that fitted values are used as additional
explanatory variables. The procedure is as follows:

1. Estimate the null model and get the coefficients of E,- (E).

2. Add interaction terms (3,? x T,) between 5,? and time dummies.

3. Estimate the model of 37,, on 1, a set of all time dummies, :r,,, E,, (25,75 x T,).

4. Test whether all coefficients of the interaction terms are 0 (H0 : (it = 0).

This test can be implemented by using usual statistical packages such as E-views or
STATA. Also it is easy to make the test statistics robust to possible misspecified second

moments.

15

The score or Lagrangian multiplier (LM) statistic is ideally suited for specification
testing. Only restricted estimates are needed to implement the score test. In most
cases, restricted models are quite easier to estimate. Let the restriction be c(6) = O,
(6 E (7r’, 69’) and its Jacobian matrix be C'. With consistent estimates, each element
of the statistic is evaluated at the restricted parameters, 6 = (71’, 0’)’. Because I do not

assume correctly specified variance, the general form of the statistics should be used.

I
LM :11, £545) 547—15” [EAT—lééi—lé’] ‘1 Sis-(‘6') 2. X%T_1),
i=1 i=1

(1.22)
where S, (6) is the score function evaluated at the restricted estimates. A and B are defined
in (1.10), respectively. the LM statistic should be 0 evaluated at the unrestricted estimates
(6). If the restriction is valid, the LM statistic evaluated at the restricted estimates (6) also
should be close to 0. The following procedure can used to produce the same test statistic
with (1.22). Divide 6 into the (K — T + 1)><1 vector of 7r and the (T — 1)x1 vector of 6,.
Let Vnm,, and V, t m,, denote the gradients with respect to 7r and 6,.

1. Run a multivariate regression thm,, on Vnm,, and obtain the 1 x (T — 1)
residuals, f,,. Then make 11,,f,, by multiplying 11,, by each elements of i‘,,, where 11,, is
the residuals from the restricted model.

2. Run the regression 1 on fr,,f,, and construct LM = NT — SSR, where SSR is the
usual sum of squared residuals. This statistic has a limiting XCZZLI distribution.

The last one is the minimum distance statistic which comes from the CMD estimator.

Going back to the time-varying model, let’s suppose a vector of the restricted parameters

be a E (w’, 1, 6’, 5')’, where 77, = 1 is imposed and g(-) shows the relationship between rt
16

and (1 [7r = 9(a)]. The minimum distance statistic is

N [(5% — Nana—10; — 9(a))j — N [(5% — h(6))’§"1 (a — 14%)] 3 X%_,. (1.23)
This statistic is based on the distance between two minimized objective firnctions. The
' intuition of the statistics is the same with that of likelihood ratio (LR) test. If the restriction
is true, the distance must be close to 0. However, we need to get 6 and 62 to obtain this

statistic like the LR test.

1.6 Empirical application: math test pass rates

In 1994, Michigan reformed K-12 school funding system in order to equalize
educational opportunities in terms of school spending. From a policy prospective, it is
important to estimate the effect of school spending on student performance consistently.
Papke (2005, 2008) found that increase in per-student spending has significantly positive
effect on student performance measured as pass rates on fourth grade math test (the
Michigan Educational Assessment Program, MEAP, test). In the paper, she allowed the
endogeneity of spending and used foundation grant as an instrument variable for school
spending. While Papke (2008) adopted the fixed effects estimator to control school
heterogeneity in the linear model, Papke and Wooldridge (2008) provided a new approach
to consider a nonlinear conditional mean function with unobserved heterogeneity. They
also found a significantly positive effect of school spending on math test pass rates. I use
the same data with Papke and Wooldridge (2008). Table (A. 1) provides simple descriptive
statistics for key variables over the years of 1995 through 2001 used in this chapter. Papke
and Wooldridge (2008) used 501 school districts, while 503 districts are used here.

Equation (1.24) shows the mean firnction for math test pass rates with time-varying

17

unobserved school district heterogeneity.

E(math4,,|-) = <I>[T,6, + [31 log(avgre:rpp),, + figlunch,, (1.24)

 

+53 log(enr0ll),, + €110g(avgrexpp), + 52W,-
+€3Wi + 77M].
where math4,, denotes the fraction of 4th graders who pass the MEAP fourth grade math
test in district 2' for year t. The sample includes 503 school districts and 7 years. T, is a
set of time dummies to allow a different intercept for each year. Papke and Wooldridge
(2008) used three explanatory variables: the 4-year averaged value of spending per pupil
in real dollar (log(avgrexpp)), the fraction of students who are eligible for free or reduced
lunch program (lunch), and the number of enrollment (log(enroll)). Because Papke
(2005) found that not only the current spending but lagged spendings have significant
effects on the performance, averaged real spending per pupil in first, second, third, and
fourth grade (avgrerpp) is considered. Logarithm transformation implies a diminishing
effect of spending and enrollment on the test pass rates. The purpose of this model is to
find the effect of spending on the test pass rates, so all variables which might be correlated
with spending and have significant effect on the test pass rates should be controlled. Two
control variables are added into the model; lunch is used as a proxy for the poverty rate or
economic well-being, and log(enroll) is for controlling school size.
Our estimation procedure is as follows: (1) estimate equation (1.24) for each year
by using the pooled FPE and get total 49 parameters (it). There are 7 parameters and

each school district has 7 years of observations. (2) Recover the parameters of interest

18

6 = (6’,17’,B’,§’) by the CMD estimator from if. Also the model with the restriction of
7r = h(6) is estimated by both the pooled constrained FPE and the constrained MWNLS
estimator.

Table (A2) and (A.3) contain the estimates of Papke and Wooldridge (2008)’s and a
linear model. Because all models are nonlinear, the APEs of all three explanatory variables
are provided to compare these results with those of the linear model. The first column of
the table (A.2) contains the results of the fixed effects estimator of the linear model, and
the next columns are the replicated results of Papke and Wooldridge (2008). The table
(A.3) shows the estimation results of the time-varying model (the model with time-varying
individual heterogeneity). Three estimation methods are used: the CMD estimator after
the pooled F PE for each time, the constrained FPE (the constrained fractional probit
estimator), and the constrained MWNLS estimator. The constrained estimators indicate
the estimators after imposing the relationships between 7r and 6 on the conditional mean
function.

The results show two things. First, the effect of school spending on the performance
is still significant and positive after considering time-varying individual heterogeneity.
The coefficients of log(avgre:cpp) of the time-varying model (0.949 ~1.103) are slightly
bigger than those of Papke and Wooldridge (2008) (0.883 ~0.886), but it is meaningless to
compare the coefficients directly. Comparing the APEs, I find the effect of the time-varying
model (0.214 ~0.228) is smaller than those of Papke and Wooldridge (2008) (0.298).
Under the time-varying heterogeneity assumption, increase in spending per student by 10
percentage points leads increase in test pass rates by 2.1 to 2.2 percentage points. These

APEs of the nonlinear models are smaller than the coefficient of log(avgre:r:pp) in the

19

linear model.

Second, f7, varies fi'om 0.84 to 1.24. Table (A2) contains the results of the LM test and
the variable addition test. The constant heterogeneity hypothesis is reject at 5% significant
level. Also the minimum distance statistic in the table (A.3) is 35 and its p-value is zero
to four decimal places. Because the LM test and the variable addition test are to test
a necessary condition for the time-varying heterogeneity, minimum distance statistics
is greater. These three test statistics show that the assumption of constant individual
heterogeneity is too restrictive. a .

Under proposal A, each school district was given the foundation grant based on its
spending level per student in 1994. As a result, all districts received per-student spending
at least as much as a basic grant. For example, in 1995, school districts spent under $4200
in 1994 were given $4200 per student or an additional $250 per student. High revenue
districts were held harmless in that they could receive at least as much amount as before.
As discussed in Papke (2005) and Papke and Wooldridge (2008), the foundation grant
variable is a non-smooth function of spending in 1994. After controlling for spending
in 1994, it seems reasonable to assume that the foundation grant is not correlated with
unobserved shocks that affect the test pass rate. On the other hand, it is easily show that
the foundation allowance is partially correlated with the average expenditure per student
because a district’s revenue is largely determined by the foundation amount.

As discussed above, the estimating equation for math test pass rates has the contrl

fiinction form,

20

E(math4,,|-) = <I>[T,6,1 + alog(avgrexpp),, + dlllunch,, (1.25)
+51210s(em‘oll)z't + €11Wz‘ + 513t10sIT€$pp94li
+£12W + ptt‘z‘tz + eitia

where log(re:z:pp94) denotes logarithm of real expenditure per pupil in 1994. Adding this
variable allows us to assume tha the foundation allowance is unrelated to the error, e,,.

The variable r,,2 is the reduced form error from the equation for log(avgre:r:pp) (1.26).

The reduced form equation is

log(avgr‘e:r:pp),, = T,6,2 + dlglunch,, + 621 log(enr0ll),, (1.26)

+522t10s(T6$W94)r + 5231: 103(9mntlrt + éztmz'
+£2210g(enroll), + 7,,2,

where log(grant),, denotes logarithm of real foundation grant. I have two estimation

strategies. First, estimate equation (1.26) and get residuals (73,2). Substitute r,,2 in (1.25)

with the residuals and estimate the model by the pooled FPE (and the CMD estimator) or

the constrained FPE. Second, equation (1.26) and equation (1.25) can be estimated at the

same time by the pooled Quasi-LIMLE (the single step estimator).

Table (A.4) contains three results: the first column shows the result of the fixed effects
instrumental variable estimator, and the rest columns are the replicated results of Papke
and Wooldridge (2008) using the two-step FPE and the quasi-LIMLE, respectively. School
spending still has a significantly positive effect on the test pass rates, and the effects are

much greater than those in the exogenous case. This implies that the effect of spending

21

 

has downward bias under the exogeneity assumption. The QLIMLE has smaller standard
errors and bigger APEs than the two-step procedure. The coefficient of spending in the
linear model is between them. In both methods, t-statistics reject the null hypothesis of the
exogeneity of spending. The hypothesis of constant individual heterogeneity is rejected at
5% significant level.

Table (A5) and table (A.6) show the results of the fractional model with time-varying
individual heterogeneity. Table (A5) contains the results of the two-step procedure, while
table (A.6) reports the results of the single-step estimator (QLIME). In both tables, the
spending has a significantly positive effect on the math test pass rates. In the two-step
procedure, both the CMD estimator and the constrained F PE report slightly different
coefficients of the spending and the constrained F PE seems to be less efficient than the
CMD estimator. In the single step estimator, both estimation methods give quite similar
results and the constrained methods (CQLIMLE) looks more eflicient. The APEs of the
fiactional model with time-varying individual heterogeneity are from 0.507 to 0.592,
and are similar with the coefficient of the linear model. Under endogenous spending
assumption, increase in school spending per student by 10 percent points results in
increase in the math test pass rates by 5.6 percentage points. The exogeneity of spending
is rejected at 5% level, and the constant individual heterogeneity is also rejected.

Table (A.7) shows the scale factors at difierent spending levels. Using linear models,
Papke (2005) found different effects of spending at different spending levels and different
initial performance levels, by estimating separate models for various subgroups. Papke
and Wooldridge (2008) used a more parsimonious approach to allowing nonconstant

partial effects by applying the fractional probit model, but with constant coefficients on
22

the heterogeneity. The table contains the scale factors for nine cases: strictly exogenous
and endogenous spending and the two-step procedure and the single-step estimator for
endogenous spending. For each case, three results are reported: the fractional probit model
of Papke and Wooldridge (2008), the CMD estimator and the FPE (or the CMWNLS
estimator in exogenous case) of the time-varying model, for the earliest and latest years
in the data, 1995 and 2001. From the results, we can see the effect of school spending is
greater at school districts with lower spendings. Let us focus on the results of the QLIMLE
for the time-varying model with endogenous spending because we already reject the time
constant heterogeneity and exogeneity of spending. The APEs are 0.668 in 1995 and 0.526
in 2001 at 5th the percentile of spending, while the APEs are 0.456 and 0.327 in 1995 and
2001 at 95th percentile. We can find diminishing APEs of school spending as spending
levels go up. The difference in the APEs in 1995 and 2001 of Papke and Wooldridge
(2008) is 0.03 (0827-0797). When we allow time-varying heterogeneity, the difference
becomes 0.142 (0.668-0.526). Generally, there are nontrivial differences in the APEs
across different parts of the spending distribution when time-varying factors are allowed

on the unobserved heterogeneity.

1.7 Monte-Carlo simulations

We can study the small-sample properties of the various estimation methods, when the
models are and are not correctly specified, via a simulation study. Because some of the
models are time consuming to estimate, I use 500 Monte Carlo replications. Nine different
sample size combinations are used: the number of cross-sectional units are 100, 300, and
500 and the number of time periods are 5, 7, and 10.

Equation (1.27) shows the data generating process of an explanatory variable.

23

at = men/ham (1.27)

Ci weft N N(011)r

where £13,, denotes an explanatory variable and is allowed to be correlated with unobserved
heterogeneity (c,). The variance of 27,, is set to 1.

The data generating process of a fractional response variable is as follows. H Beroulli
outcomes, 10,, h are generated with probabilities that depend on the covariate, the unknown
heterogeneity, and unobservables u,,. The observed fractional response is the fraction of

successes.

pit = PIwz'th = 1) = @(VytTt + Exit + "tci + “it)auit ~ NIO. 1) (1-28)
1 H
ya: = ‘1; 2: with, with N Bernoulli(p,,),

where T, is time dumrfirizels. The data generating mechanism is the same as creating the
fractional variable, y,,, from the Binomial distribution in which the probability of a
success is defined as equation (1.28) and there are H trials. I use H = 1000.

The normalization I use on the time-varying coefficients is 171 = 1. I set up two
different populations: 17, is increasing by 0.1 and by —0.1 over time. 7y, is T x 1 vector
of known numbers and I set 6 = 1. Let the correlation between a: and c be ch- Because
variances of :r and c are set to 1, the correlation of :2: and c is the same with the coefficient,
7,36. Nine different correlations between :1: and c from 0.1 to 0.9 are considered. The
larger p3,C is, the better the APEs of the fractional model with time-varying heterogeneity

fit intuitively.

24

Only exogenous case is considered because both the endogeneity effect and time-
varying individual heterogeneity might be mixed up so that we have difficulty in
interpreting results. Six estimators are compared. The first one is the fixed effects
estimator for the purpose of comparison. The next two are the pooled FPE and the
MWNLS estimator proposed by Papke and Wooldridge (2008). The last three are the
CMD estimator based on estimates of the pooled FPE for each time, the constrained F PE
(CFPE), and the constrained MWNLS (CMWNLS) estimator.

Because the model in the population is nonlinear, a closed form for the average paratial

effects is difficult to obtain. In stead, the APEs in the population are approximated as

APEt,pop = Em) [fies ("lytTt + 5n + 771:0 + 1%)] (1.29)

22

'l

[3i ‘2]: d> VytTt + Exit
N . 2
that is, I use a sample average across the distribution of 22,, but with the true values of the

parameters inserted.

Equation (1.30) shows the conditional mean function and the estimated APE:

E(y,,|:c,) = ‘I’ITtTt + Eth’t + 6,573,) (1'30)
- 1 N - -
APE, = fitIV— Z ¢(EtTt + Erma + 5%),
i=1

where, of course, these will differ by estimation method. To indicate different estimation
methods, I write APE,,,.

By comparing the root mean squared errors (RMSE) and biases between the APEs
of the population and those of all estimators, I attempt to find the effect of considering

time-varying individual heterogeneity on the APEs.

25

 

RMSE, = \J % iMPEu — APE,,p0p)2 (1.31)
Table (A8) through table (A.14) cofrfalin the simulation results. Two cases are

considered as explained above: an increasing individual effect and a decreasing individual
effect. Table (A8) through (A.11) show the RMSEs and biases from 1022c =0.1 to 0.9 when
N is 500. The results shows the RMSEs become bigger as pzc increases. When pm is
large (ch > 0.5), all estimators have quite similar RMSEs. In large pm, the time-varying
model gives slightly larger RMSEs than even a linear model. Note that high correlations
between a: and c mean there is not enough variations in sc, which results in imprecise
estimates and large RMSEs. So I restrict Pro to reasonable ranges (from 0.2 to 0.4). Table
(A.12) through (A.14) report the results of different N and different T. In most cases,
the time-varying model gives better results than not only the linear model but the model
of Papke and Wooldridge (2008). The constrained MWNLS estimator, which accounts,
albeit in a misspecified way, for serial correlation, generally shows the best results. For
examples, when p5,, = 0.4 and 17, is increasing, the RMSEs of the time-varying model are
0.0208~0.0203 with N = 500 and T = 10. The RMSE of the linear model is 0.0270, and
those of Papke and Wooldridge (2008) are 0.0270~0.0271. If we calculate the APEs at

different percentiles, Papke and Wooldridge (2008) model must show lower RMSEs than

the linear model.

1.8 Conclusion
I allow unobserved individual heterogeneity to vary over time in a fractional response
model. Based on the framework of Papke and Wooldridge (2008), the model is extended

by adding an interaction term (an interactive effect) between a time effect and an individual

26

effect proposed by Kiefer (1980) and Lee (1991). This interaction term can be interpreted
as a time-varying coefficient of individual heterogeneity or an individual specific intercept
at each time. Therefore, each individual is allowed to respond differently to the common
effects (the time effect). This setting contains the constant individual effect model as

a special case and we still have a manageable form. The Mundlak-Chamberlain device
is used to restrict of the distribution of individual effect conditional on explanatory
variables. Based on a conditional mean function for fractional dependent variables, the
pooled fractional probit estimator and the multivariate non linear least squares estimator
are discussed as in Papke and Wooldridge (2008). Two cases are considered: strictly
exogenous explanatory variables and continuous endogenous variables with appropriate
instrumental variables. Especially, I adopt the single-step estimator of Wooldridge
(2007) along with the traditional two-step estimator when the model contains endogenous
explanatory variables. Also I mention three test statistics for time-varying individual
heterogeneity. In particular, the variable addition test can be conducted by using usual
statistical software programs such as the STATA even though the time-varying model
cannot be estimated by those kinds of software.

I apply the estimation methods to analyze the effects of school spending per student
on students’ performance in terms of the math test pass rates in school district level.
The time-varying individual heterogeneity model allows each school district to respond
difl'erently to common time effects such as the difficulty level of the test. I find that
the effect of school spending on test pass rates is still statistically significant after
considering time varying individual heterogeneity and the APEs are quite similar in both

the time-varying model and the time-constant heterogeneity model. However, when the

27

APEs are evaluated at different percentiles of spending and in different years, the APEs of
the time-varying model are smaller that those of the time-constant model. The hypothesis
of a constant individual effect is rejected by all three test statistics.

The Monte-Carlo simulations are also conducted to find finite sample properties of
estimators. Two cases are considered: an increasing and a decreasing individual effect
using nine different samples with N=100, 300, 500 and T=5, 7, 10. Also different
correlations between the regressor and individual heterogeneity from 0.1 to 0.9 are
considered. When our interest is restricted to the reasonable ranges of the correlations
between an explanatory variable and individual heterogeneity (0.2~0.4), the time varying
model gives less RMSEs than both the linear model and the model with constant individual
effects. Especially, constrained MWNLS estimator which considers a possible serial

correlation shows the best results among all estimator.

28

Chapter 2 THE FRACTIONAL MODELS WITH BINARY ENDOGENOUS
EXPLANATORY VARIABLES AND TIME-VARYING INDIVIDUAL
HETEROGENEITY

2.1 Introduction

In this chapter, I continue to study the fractional response model, but now I allow for
the presence of a binary endogenous explanatory variable. I combine the problem of an
endogenous binary explanatory variable with unobserved heterogeneity. As in the first
chaper, I allow time-varying coefficients on the individual heterogeneity.

In the first chapter, continuous EEVs are considered and an extension of the two-step
control function approach proposed by Rivers and Voung (1998) is used. However, it is
well known that there is no consistent two-step estimator when EEVs are not continuous
in nonlinear models. Instead, the bivariate probit model is modified to derive a conditional
mean function and the single step estimator. Recently, Wooldridge (2007) has shown
that in some cases — fractional responses being a leading one — quas-limited information
maximum likelihood can be used. Therefore, the main contribution of this chapter is to
derive the conditional mean function for fiactional dependent variables with binary EEVs
and to extend Wooldridge’s approach to the panel data case.

The model here has many applications. For example, y,,1 could be the proportion of
questions correctly answered on a test, and y,,2 an intervention, such as getting individual
tutoring help. Or, 37,,1 can be the fraction of pension assets invested in the stock market
and y,,2 and indicator for taking a “financial awareness” class. The empirical example

I use here is to analyze the effect of fertility on female labor supply. Women are still

29

primarily responsible for providing child care and doing housework. Jacobsen (1994)
pointed out that women spend four times as much time for child care as men. Therefore,
it is not surprising that many studies has found a negative relationship between fertility
and women’s labor supply. However, to interpret this causal relationship clearly, we have
to consider the endogeneity of fertility in female labor supply equation. Using my model
and estimation method, I can address two kinds of endogeneity: fertility and female labor
supply are likely to be jointly determined and unobserved individual heterogeneity might
affect both variables.

To consider this endogeneity, simultaneous equation systems may be used to consider
joint determination of fertility and labor supply. However, if there is a misspecification in
any equation, this misspecification transmits its inconsistency to all system. In other words,
even if our interest'lie in the labor supply equation and this model is correctly specified,
inconsistent estimates might be obtained when the other model has a misspecification
problem. An alternative is to use an instrumental variable estimation method with
appropriate instruments for fertility in reduced form for labor supply. Rosenzweig and
Wolpin (1980) used twins in the first birth as an instrument. Because twining occurs
randomly, this would produce consistent estimates, but we have a sample size problem.
There are only 87 twin mothers in Rosenzweig and Wolpin (1980). Angrist and Evans
(1998) proposed a new instrumental variable, an indicator for whether the sexes of the
first two children are the same, for fertility based on the parental preference on the mixed

sex sibling in developed country2. Angrist and Evans (1998) compared two instrumental

 

Park and Cho (1995), and Arnold and Zhaoxiang (1981) use parental preference for boys in developing
countries.

30

variable results: twins at the first birth and the same sex. This allows them to find the
effect of children of different ages. Carrasco(2001) extended Angrist and Evans (1998) to
panel data, but he focuses on the same sex. Some studies use lagged dependent variables
as proxies to control unobserved heterogeneity, while panel data allow us to control
individual heterogeneity explicitly. Instead adding lagged dependent variables, I use
Mundlak(1978) - Chamberlain (1980) device to restrict the distribution of unobserved
heterogeneity.

Another issue is definitions of labor supply and children variables. There are two major
measures of labor supply variables: job participation (extensive margin) and working
hours (intensive margin). Angrist and Evans (1998) used participation, weeks worked, and
hours per a week, while Carrasco’s (2001) dependent variable is labor force participation.
The effect of fertility on labor supply is not clear a prior because an income effect and a
substitution effect coexist. However, in most cases mothers are likely to withdraw entirely
from labor market or at least reduce their working hours to take care of their new born
babies. The binary dependent models might underestimate the effect of the fertility in that
the binary indicator does not change as long as mothers take part in the labor market even
if they reduce working hours. In this paper, fraction of working hours to total available
hours is used as a dependent variable. Many researchers use linear models for working
hours, but working hours have two limits: maximum time limit and 0. Therefore, linear
models cannot ensure the fitted values exist between these limits. Also constant effects of
independent variables might be too restrictive. In this paper, I focus on the effect of new
born babies because most of the effect of fertility depends on that more than, for example,

on the number of children [Carrasco (2001)].
31

3

2.2 The Model with binary endogenous explanatory variables

As in the first chapter, I assume N randomly drawn cross-sectional units from the
population and T observations for each cross-sectional observation. Again, large N and
fixed T asymptotics is used.

Let us assume a correctly specified conditional mean function as

E(l/it1I3/z't2a Zitla Ci1> Uitl) = War/m + 26161 + mm + Urn), (2-1)
where 2' indexes a cross-sectional unit and t indexes time (2' = 1,2, ..., N ,t = 1,2, ..., T).
y,,1 denotes a fractional dependent variable which can take on any value between zero and
one, including zero and one. The variable 37,,2 indicates a binary endogenous explanatory
variable (EEV). In this chapter, I focus on one EEV case; in principle, it one can extend
the approach to more than one EEV. 2,,1 is a set of strictly exogenous regressors. The
interactive effect, 17,c,, is added to consider time-varying individual heterogeneity again.
v,,1 is an unobserved omitted variable and is thought to be correlated with y,,2. Also
the standard normal cumulative distribution function (cdf) is used for the mean function
of y,,1 to make the expected value between zero and one. This particular choice also
provides some computational advantages.

Our main interest lies in the partial effect of a binary EEV. We can obtain this from
©(a + z,161 + 77,c1 + 27,1) — <I>(z,161 + 771501 + 11,1) dropping the subscript z'. The partial
effect depends on 2,1, cl, and 11,13. We can evaluate the effect at interesting values of 2’s
or average the effects across individuals, but we may not observe c1 and 11,1. Therefore,

a popular method is to calculate the average treatment effects (ATEs) by averaging the

 

This partial effect is called treatment effect when the variable of interst is binary or discrete.

32

treatment effect across the distribution of c1 and v1. Therefore,

ATEt = E(c1,v1)l‘1’(0t + Ztrfit + 772:61 + vtl) - ‘I’Iztlfit + 77:01 + ”all (22)
To identify the parameters and the ATEs, the Mundlak(1978)-Chamberlain(l980)

device is adopted to restrict the distribution of individual heterogeneity (c1).

— ' 2
C,1|Z, v‘ N061 +z,C1+a,1,aa1), (2.3)
_ T
where z, E (2,1, ..., 2,17), Z, E l/T :tzl Z,,, and 031 E Var(a,1|z,). Z,, E (2,,1, Z,,2)
is a set of all exogenous variables and z,,2 indicates a set of instrumental variables.

Rewrite the mean function with the device as follows.

Eiyitllyit2aziarit1) = @(W/Jr + ayrtz + 22321.31 + mirCr + Tm). (2-4)
where r,,1 E (77,a,1 + v,,1) denotes a composite error. The endogeneity of y,,2 implies
that r,,1 is correlated with y,,2.

To consider the nonlinearity of yg, I use the probit model. There is no theoretical
difference between the probit and logit model. However, because plugging probit fitted
values for y,,2 into the structural model is not valid, we exploit the properties of the
normal distribution. (1 refer to Wooldridge (2002) in the cross section case for probit.)

The equation (2.5) shows the reduced form model for y,,2:

E (yit2lzitv 012) = (“27162 + 622 + 22,-,2) (2-5)

_ 2
Cr2|2r m N(¢2+Zr€1+ai2,0a2),

33

where c,2 indicates individual heterogeneity but is assumed invariant over time for
simplicity. v,,2 is unobservable but assumed independent of 2,, and 0,2 for identification.

021 is the conditional variance of a,2. With the Mundlak (1978) - Chamberlain (1980)

device,

Eillz’tQIZz'» 73252) = (PU/12 + zitB2 + 5&2 + 7‘62) (2-6)
= ‘I’Iwit27T2 + 711:2), 703152 E (lazz'tiz'),
where 7,,2 E (a,2 + 2),,2) is defined to represent a composite error and is independent
of z,,. 112,,2 and 7r2 are defined for notational simplicity. Because the endogeneity of
y,,2 implies that (r,,1, r,,2) are correlated, a linear projection of r,,1 on 7,,2 and joint
normality are assumed as: r,,1 = p,r,,2 + s,,, s,,| (2,, r,,2) ~ N (0, ”r? ) p, is allowed
to vary over time because r,,1 contains a time-varying factor, the coeflicient. From joint

normality of (r,,1, r,,2) and independency of 5,, on y,,2 and 2,,

Uri/’1 + ayit2 + 221151 + UtEiCI + Writ? (2 7)

E(yz‘t1Iyz't2,Zz'ta7‘rt2) = ‘13
(n+7,2

‘1)(7W17 + avyirz + 2211617 + UtZiCIT + P11773112)

 

‘I>(wrt17rlt—r + Purim),

where subscript 7 means all parameters are scaled by 7'. For notational simplicity,
w,,1 E (1, y,,2, z,,1, 2,) and its corresponding coefficient vector, 7r”, are defined and I
drop subscript 7' from now on.

z, and r2 conditional on c1 and 121 are redundant in equation (2.1) and c1, TI, and y2

are independent conditional on (2,, r2). By the law of iterated expectations,

34

ATEt = E07137?) [‘1’ (mi/11 + a + Zt151 + 77225241 + PtTit2) (2-8)

41’ (mt/)1 + Ztlfit + with + pt'l‘z’tzll-

From the properties of the normal distribution, we can rewrite (2.8) as

 

“Ll-p?

4, nt¢1+zt151+nt5iC1 ]
1+p§

 

2.3 Estimation methods

The models with binary EEVs may not be estimated by the traditional two-step
estimation, where generated regressors in the first step estimation are added as additional
explanatory variables or used proxy variables for EEVs. In the nonlinear settings, this
two-step estimator produces inconsistent estimates because we cannot pass the expectation
through nonlinear functions. An alternative is to use the full information maximum
likelihood estimator (FIMLE) afier assuming the joint distribution of (371,372) given 2.
However, the F IMLE produces inconsistent estimates if the assumed joint distribution is
not true in the population or there is misspecification in any equation.

In this chapter, the bivariate probit model is modified to obtain a conditional mean
function for yl and the Bernoulli quasi maximum likelihood estimator (QMLE) is used
based on the conditional mean function. So, I call this the fractional bivariate probit

estimator (FBPE). To get the conditional mean function, take expectation with respect to
7‘2.

35

E(yrt1|yit2,zz‘) = (2.10)
yitZEIq)(wit17I1t+Pt7‘it2)l7‘it2 > —wz'1:27r 2]
+(1—yit2)EI(p(wit17I1t+pt7'it2)l7"z't2 < —wz't27T21

Because Q is assumed normally distributed,

wz't27r2
y' ,
E(yit1Iyrt2»Zr) E ——q,(wi:22,,2) ‘I’Iwz't17r1t+/Jt7‘2)¢(7‘2)d7‘ (2-11)
2
—OO
‘wit27r2
/ @(witlfllt + Pt72)¢(7"2)dt‘

“—00

 

l-ww
(Di-wit27r2)

 

+

(1’2Iwz't1771t, (23/7172 - llwz't27r2t, (2.1/11:2 - 1).0ti
‘I’IIZyz'tz — 1)wz’t27r2ti
To identify all parameters, the structural equation (2.11) and the reduced form (2.6) can

E 777.2,.

be estimated together. Wooldridge (2007) argues that, because the Bernoulli log likelihood
is in the linear exponential family, only the mean E (y,,1 |y,,2, 2,) needs to be correctly
specified along with the probit model for 37,,2 conditional on 2,. Here I am just applying
a pooled method, and so the argument is essentially identical. The quasi-log-likelihood

firnction is

ln€,, = yitl ln(mit) + (1 - yit1) 111(1 — mit) (2-12)
+yit21niq)(wit27r2)i + (1 — 312732) lnll — @(wit27r2)i

Where mit = E (yitllyit27 Zil-

With consistent estimates, the ATEs of y,,2 can be obtained from equation (2.13).

36

 

A l N A A A A A _'A
N 3:1
‘/1 + p,
4, firth + ZnEl + 7%»?le ].
1 + p?
The endogeneity of 3,7,,2 can be easily tested by the two step estimator of the

 

Rivers-Vuoung (1988). If some of the p, are different fi'om zero, we have to use the F BP
estimator to account for endogeneity of y,,2. If p, is zero, the conditional mean firnction

can be reduced to the simple mean function without endogeneity, that is

‘1’2Iwit1711t.(2%'t2 '— 1)wit27r2ta 0] : ‘I’IIwrtflltI‘I’lIQyz‘w “‘ 1)wit27r2ti

 

 

_ E (D (217', 7f], .
etch-,2 ‘1)wz‘t27I2ti <I>1<2y32 —1>w.t2w2t1 I z 1 I
For inference, we need a consistent estimator for the asymptotic variance.
Avar(7r)— — A—IBA_1/N,N (2.14)
N T
Where A = N4 2 Z —EIV%1D€itI )I B=N_1 Z Z 21 SrsITF (”lsz'tIW)’,

i=1 t=1 i=1t1 3:1
and 5,,(7r) E V7, 1n 6,,(7r) [Wooldridge (2002) Theorem 12.3]. The gradient of the

quasi-log-likelihood function for each (i, t), V7, 1n 6,,(7r), can be written as

 

 

51111321 _ yitl — mit (15Iwzt19’57r 104504131) (2 15)
a ‘ -(1— MW —1) 1""1 '
7T1, mzt mzt yit2 10712772
E yitl *- mz‘t g“
mitIl — mit) z

 

(23/2152 - 1)wit27r 2 — (21/222 — llpt’wz‘tflr 1

l—pg

where (b() indicates the standard normal probability density fimction (pdf) and I define

”11731 =

9,,1 to simplify notations for the Hessian matrix.

37

In the same way,

am (it = Ilitl — mz‘t (291:2 - 1)¢Iwit27r2)¢(vit2)
(97f 2 mitIl - mit) ‘I’II2yz't2 - 1)wit27r2i
yit? - ‘I’I’wz'1127r 2)

®Iwit27r2) [1 - ‘thitzfiz
_ yitl ‘- mit g_ + 36132 — <I>(w,,27r2)

mitIl - mu) ”2 ¢Iwit27r2l [1 - @(wz'tzflz
71212 = witl” 1 - (23/62 - 1)Pt(2yit2 - 1)wit27r2'

(/1 — pi?

Because 7r2 appears in both the structural equation and the reduced form for 372, there is an

 

513222 (2-16)

 

)] [yin - (PIwitmll $212

 

 

)] uithitQ

 

additional term.

The score function for p, is

 

 

 

 

 

 

3111522 = 93131 — mit (2.17)
apt mitIl — mit)
X (231272 — 1)¢2Iwit17r1t3(2yit2 — 1)w,,27r 2» (23/222 — llptl
“(23/222 - llwz't27r2l
E yitl — mit 9'2:
mitIl — mit) 2 '0’
where ¢2(-) denotes the bivariate standard normal pdf.
Some components of V72r 1n 6,,(7r) are
621M —1 2 ,
—— = g- at, :13. , (2.18)
Orlan’l 2:21; m,,(l —— m,,) ”1 z 1 “1
_ = 't 't 'tlr
6316p; i=1,=1 mitIl — mit) 2 1 z p z
T
6211.16 —1 I
_ = 9't19't2x't1x' -
6216762 1;; m'z'tI1— mit) z z 2 “2

The remaining elements of the Hessian matrix can be obtained in the similar way.

The test for time-varying individual heterogeneity is conducted based on the more

38

 

general function for yl as in the first chapter. See the first chapter for detailed explanation
about the more general function. Also the LM (score) test, the variable addition test, and

the minimum distance test are conducted as before.

2.4 Application: fertility effect on working hours of women

I apply the method to study the effect of fertility on working hours of women with two
or more children. As I mentioned above, the same sex indicator is used as an instrumental
variable. Therefore, the interpretation of the average treatment effects of a new-bom is the
effect of fertility moving from the second child to the third one on fractions of working
hours of women with two or more children.

Panel Study of Income Dynamics (PSID) for 1986-1989 is used as in Carrasco (2001).
I refer readers to Carrasco (2001) for detailed explanation about the data. The PSID is
longitudinal survey over 5000 households since 1968 and consists of two independent
subsamples: equal probability sample of US household and below 150% of the poverty
line. The data in this paper includes both subsamples to increase sample size. The data
contains 1442 married or cohabitating women with two or more children between the
ages of 18 and 55 in 1986. Table (B.1) shows descriptive statistics for key variables. The
women spent about 19% of their total hours to work and there is only one observation of
1. 14.3% of women had a new born baby and 37% of them had children aged between 2
and 6 years in 1986. Their average age is 31.302 and about 77.3% of them are black. As
the ratio of women have new born babies decreases, the fraction of working hours to total
hours of women increases.

The conditional mean function for labor supply is

39

tht + afe'rt,, + kid26,t51 + hinc,,,Bg + 010,619,363

”1+7?

+ 0961354 + ntdsefrz'C 1 + fltkid262'42 + UthinciC 3 + 10:73:12]

EIf’UJhitI') E (Di (2-19)

 

 

2
1+7,

where T, indicates a set of year dummies. f wh is the ratio of working hours to total
available hours and f ert is a fertility variable which equals 1 if the mother has a child aged
1 at t+ 1, because it is not clear that a child who is a year old at the time of interview means
a mother gives birth to the child at that year or previous year [Carrasco (2001)]. Two more
exogenous variables are added: kid26 takes 1 if a woman has children aged between 2
and 6 years and hinc is a logarithm form of husband’s income. Mothers’ race (black)
and age (age) are also controlled. The time averages of exogenous variables are added to
control individual heterogeneity. dsea: is a same sex indicator, which equals 1 if the first
two children have the same sex. This variable is used as an instrument variable for the
fertility based on parental preference on mixed siblings. Because of indistinguishability
and exogeneity of black and age, their time averages do not appear in the conditional
mean function for f wh. To consider the endogeneity of f art, a linear projection of
r,,1 = p,r,,2 + e,,, 5,,I2, ~ N (0, 7?.) is assumed. r,,1 denotes an unobserved omitted
variable in the model for f wh and is assumed to be correlated with f ert. 7,,2 appears in
the reduced form for f ert.

Equation (2.20) shows the reduced form for f ert.

40

E ( fert,,|-) = <I>(w,T, + 6dsex,, + kid26,,71 + hinc,,72 + blue/9,73 (2.20)
+age,,”y4 + d—se-Eptl + 73326,).2 + Wflg + 7",,2).

To obtain the conditional mean function for f wh, the bivariate probit model is used and
the result looks like equation (2.11). The conditional mean function for f wh along with
the reduced form for f art is estimated by Bernoulli QMLE.

For the purpose of comparison, a linear model is also estimated and reported. Let us
explain how to estimate the linear model before looking at estimation results. In strictly
exogenous case, the fixed effects estimator is used. In binary EEVs case, I use the two-step

estimator.

yin = wz‘t17r1 + Um

312752 = 1011222712 + ”62 > 0),
where y,,1 and y,,2 denote the fractional dependent variable and a binary EEV,
respectively. Because the endogeneity of 37,,2 implies v,,1 = pv,,2 + 6,,1. I define
a new regressor, yit2¢Iw¢t27r2)/<P(wz't27rz) - (1 - yrt2)¢(wrt27r2)/ [1 - Q’Iwazmll,
add it to control the endogeneity. The motivation of this approach is as follows.
The expectation of v,,1 conditional on m,,l is pE(v,-,2 | w,,1) + E (6,,1 | m,,l),
where E(e,,1 | 112,,1) = 0 by independency of 6,,1 on m,,l. E(v,,2 | w,,1) can
be Obtained from yitzEIvitz I ’Uz‘t2 > —wit27T2) + (1 — yit2)EIvit2 I Uitz <
—w,,27r2). If we assume the probit model for y,,2, we can prove E(v,,1 | w,,1) =

yit2¢Iwit27I2)/(I)Iwit27r2) - (1 - yit2l¢Iwit27I2V I1 - ‘I’Iwz't27r2ll- Therefore, we

41

estimate the probit model for 37,,2 at the first step, and the fixed effects estimator is used
after adding the generated regressor [Freedman and Sekhon (2008)]. As Wooldridge
pointed out, the fixed effects instrumental variable (FEIV) estimator can be used under
weaker assumptions in that the FEIV does not require yg follows a probit. Because the
linear model is used to compare the coefficients of the linear model with the ATEs of the
fractional model and the fractional model assumes the probit reduced form for yg, I use
this Heckrnan approach.

One can use the plug-in method, where the fitted values for y,,2 at the first step,
(I) E <I>(w,,27r‘2), is used an IV for y,,2. Because E(y,,2 | w,,2) = <I>(w,,27r2) holds and
we have a linear model, this plug-in method also produces consistent estimates4.

Table (3.2) and Table (33) show the estimates of various models for strictly exogenous
fertility case. Table (B.2) reports the results of the linear and the fractional model. All three
methods show significantly negative effect of fertility on fractions of working hours. The
coefficient of fertility in the linear model is -0.021 and the ATEs of fertility in the fractional
model are -0.023 in both the FPE and MWNLS estimator. Compared to women without
new born babies, mothers with the new born reduce their fractions of working hours by
2.3% points. Also table (32) contains two test statistics for time-varying heterogeneity.
While the LM test statistics accept the null hypothesis of constant individual heterogeneity
(H0 : r7, = 1), the variable addition test rejects the hypothesis.

Table (B.3) contains the results of the fraction model with time-varying individual

heterogeneity. The first column and the third column are the estimates of the constrained

 

According to the simulation results in this chapter, both two-step estimator and the plug-in estimator show
quite similar results in term of the root mean squared erros, compared to the average treatment effects in the
population.

42

pooled F PE and the constrained MWNLS where the restriction of 7r = h(6) is imposed
directly on the model. The second column contains the estimates of the CMD, where the
set of parameters of interest 6 is recovered from 7r by the classical minimum distance
estimator. The ATEs are -0.022 ~-0.023 and these number are quite similar with those of
the linear and the time-constant heterogeneity model. Also the GMM test statistic cannot
reject the null hypothesis of r7, = 1.

Table (B4) and table (B.5) contain the results of the linear, the fractional model, and
the fractional time-varying model for endogenous fertility case. According to table (B.4),
the effect of fertility on the fi'action of working hours is -0.142 in the linear model, while
the ATEs of fertility in the fractional model is -0. 129. Also table (B.5) shows quite similar
results with the ATEs of the fractional model in table (B4). The ATEs in table (B5) are
-0. 122 ~ -0.126 depending on estimation methods. In the same way, we can interpret these
results as follows: mothers with new born babies reduce their fiactions of working hours
by 12.9 ~ 12.2 % points compared to women without. According to table (B.1), the mean
values of f wh are around 0.2. Therefore, the ATEs of fertility seem not only statistically
but also economically significant and meaningfirl. The exogeneity of fertility is rejected in
both the fractional model (the estimate of the correlation between TI and r2 is 0.214 and
its standard error is 0.058) and the fractional time-varying model (the p-value of the Wald
test for endogeneity is 0.0293), but the time-constant individual heterogeneity cannot be

rejected.

2.5 Monte-Carlo simulations
Simulations are conducted to find finite sample properties of a linear model, inconsistent

two-step procedure, and the single-step estimator. For the purpose of comparison, the

43

ATEs of a binary EEV are computed and their mean squared errors and biases are reported.
The number of replication is 500 and I generate eight samples with N=300, 500, 700, 1000
and T=5, 7.

The equation (2.21) shows the data generating process for a binary EEV.

xit = II'YastTt + 7.22271 + 73663 + ”it > Oil (221)
c,,u,, ~ N(0,1),
where 3,, and 2,, denote a binary EEV and its instrumental variable, respectively. c,

indicates unobserved individual heterogeneity and is assumed to be correlated with 2,,.

Also T, (time dummies) are added to the model to allow different intercepts for each time.

2., = 72cc.- + (p — 730..., we ~ N(0,1). (2.22)

Equation (2.22) is the data generating process for 2,,. Also 2,, is set to be correlated with

C,.

p = (DITytTt + A/yxxit + Ci + vit): (2-23)
”it E Putt + eita ez’t N N(0,1),

H
1 .
yz‘t E E Z yithiyith ~ Bernoulldp),
h=1
where 1),, denotes an unobserved omitted variable and is correlated with 2,, through u,,.
The fractional dependent variable, y,,, is generated based on the Bernoulli distribution

with probability of 77. As before, I use H = 1000 Bernoulli draws. The average treatment

effect is approximated as

N
1
ATEt,p0p z N Z [(FITytTt + 73,73: + Ci + vit) _ @IVytTt + Ci + ”20] - (2-24)

i=1
The equation (2.24) shows the ATEs for time t in population.
As mentioned above, the root mean squared errors (RMSEs) and biases are compared

based on the ATEs of three estimation methods.

1. The fixed effects estimator of a linear model

yit E TytTt + Vyxfliz‘t + 53/in + pvz'tz- (2.25)
PIxit = 1|2,) E @IaxtTt + axzzit + 5225i)-
(1) Estimate the second equation of (2.25) by the probit estimator and generate
a new regressor, 6,,2 E x,,63,,/<I>,, -— (1 — fL'it)$it/(1 — <I>,,), where 65;, =
¢I5¥xtTt + (3222:2711 + Emit) and ‘i’z‘t = ‘PIflxtTt + 63332,, + 3x223)- (2) Replace vitz
with 6,,2 and estimate the model by the fixed effects estimator. (3) get ‘y‘yx.

2. The Two-step fractional probit estimator (FPE)

E (yitlxr‘ta 27;) E ‘I’IvytTt + 7372:3327. + (lg/zit + P712112) (2-26)
(1) Estimate the same equation of (2.26) by the probit estimator and get 0,,2. (2)

Substitute 22,,2 with ’17,,2 and estimate the first equation by the fractional probit estimator

based on the conditional mean function. (3) Compute the ATEs from (2.27).

A 1 N . A e _ M

‘(I’I’lytTt + (lg/252' + BEEN]-

45

3. Single-step fractional bivariate probit estimator (FBPE)

(D2[wit17r1, (2:13“ — ”102127er (23311 _ Up]. (2.28)
‘I’Kzaiz‘t — 1)wz't27T2]

(1) Estimate the equation (2.28) by the pooled QMLE. (2) Calculate the ATEs from

 

E(yit|$itv 21') =

(2.29).

'iytTt + ‘nyivz't + 53/in <I> ‘rytTt + 5yz5i

N
A 1
ATEt,FBPE = 7V- 2 ‘1’ -
(/1+ 2)?

i=1 1+2)?

 

 

(2.29)
Table (B.6) contains simulation results. The numbers reported are the RMSEs and
biases of three estimation methods. As N and T increase, the RMSEs decrease. The
RMSES of the FBPE gives the best fits among three methods. For examples, the RMSE
of the FBPE is 0.0238 and the RMSEs of the linear and the FBE are 0.0295 and 0.0294,
respectively when N = 1000 and T = 7. The FPE and FE estimators are inconsistent in
this setting, but the FPE performs somewhat better. The nonlinearity in the population

seems to lead to this result.

2.6 Conclusion

In this chapter, I extend the first chapter to a binary endogenous explanatory variable
(EEV) case. Because the EEV is discrete, the traditional two-stage procedure of Rivers
and Vuong (1988) produces inconsistent estimates. Alternatively, the full-information
maximum likelihood estimator (FIMLE) can be used. However, if distributional
assumption is not true in population or there is misspecification in any equation of the
system, the FIMLE also is an inconsistent estimator. I modify the bivariate probit model

to derive a conditional mean fimction for a fractional dependent variable with a binary

46

endogenous variable. Based on this mean function, the pooled quasi-limited information
maximum likelihood estimator is proposed. For identification, the conditional mean
function along with a reduced form for a binary endogenous variable are estimated
[Wooldridge (2007)]. This single-step estimator gives simple asymptotic variances and
possibly enhances efficiency of estimates. The estimation methods are applied to the effect
of fertility on women’s fraction of working hours. Based on the parental preference on the
mixed siblings, the same sex indicator of the first two children is used as an instrumental
variable to consider the endogeneity of fertility. The ATEs of fertility are 0023 under
exogenous fertility assumption and -0.129 under endogenous fertility assumption. The
exogeneity of fertility assumption is rejected, but time-constant individual heterogeneity
cannot be rejected. The ATEs of fertility imply that mothers with a new born reduce
their working hours by 12.9% points compared to women without new born babies. Also
simulations are conducted and the ATEs of a binary endogenous regressor are compared
in a linear model, the inconsistent two-step fractional probit estimator (FPE), and the
fractional bivariate probit estimator (FBPE). The FBPE gives the best fit among the

estimation models and methods.

47

Chapter 3 THE HURDLE MODEL FOR F RACTIONAL RESPONSE VARIABLES
WITH BINARY EN DOGENOUS EXPLANATORY VARIABLES AND
TIME-VARYING INDIVIDUAL HETEROGENEITY

3.1 Introduction

This chapter studies the hurdle model (or, the generalized Tobit model) for fractional
variables. Also time-varying individual heterogeneity and binary endogenous variables
(EEVs) are considered as in the second chapter. The fractional variables can take 0 or 1
with positive probabilities. This stack of 0’s or 1’s occurs because fractional variables
cannot be defined outside the unit interval or are comer solutions (optimal choices). Our
two applications in the first and second chapter show two different examples. In the first
chapter, nontrivial proportion of school district reports 100% of test pass rates because test
pass rates cannot exceed 1. In the second chapter, some mothers report 0 working hours
and this stack of 0’s seems to be the result of optimal choices.

The fractional hurdle model allows us to separate the determination of comer solutions
(y = 0 or y > 0) from the decision of the amount of y conditional on y > 0, where y
denotes a fractional dependent variable. By using the fractional hurdle model, we can
allow different sets of explanatory variables for the equation for comer solutions and the
equation for positive fractional dependent variables. Also, we can obtain both E (y | :13)
and E (y | ac, y > 0), and their corresponding average treatment effects (ATEs) can be
calculated. The motivation of the second chapter is that the effect of fertility might be
underestimated when the binary model for job market participation is used because the

binary model ignores women who are working but reduce their working hours after having

48

new babies. The ATEs of fertility in E (y, | x, y > 0) shows how much the binary model
possibly underestimate the effect of new born babies.

After deriving a conditional mean function, the single step estimator of Wooldridge
(2007) is used. To obtain a conditional mean function the bivariate probit model is
modified as before. The estimation methods are applied to analyze the fertility effect
on women’s labor supply by using the same data as in the second chapter. Two average
treatment effects (ATEs) of the fertility are obtained and compared with those in chapter 2.
The first ATEs are obtained using the whole sample and the second ATEs are calculated

conditional on the fractions of working hours are greater than 0.

3.2 The basic fractional hurdle model

I assume N randomly drawn cross-sectional units and T observations for each cross-
sectional unit. Also N is assumed to be greater than T (large N fixed T asymptotics). The
basic hurdle model for fractional response variables consists of two equations. The first
equation is (3.1), which indicates the probability that the fractional dependent variable,

yit, takes 0 or positive values.

P(dit = llivz't,cz'1)= 20(3/z‘t > lez‘tacil) = @(xit51+ Ci1)- (3-1)
Cillxi ~ N(1/11 + Iii/3’2 + ai1)avar(aillxi) = 021,
where 2' indicate a cross sectional unit and t indicate time (i = 1,2, ..., N ,t = 1,2, ...T).
dit equals 1 when yitl takes positive values. $it and Gil denote a set of explanatory
variables and individual heterogeneity, respectively. The individual heterogeneity is

assumed invariant over time. One can easily consider time-varying heterogeneity by using

49

interactive effects model. The distribution of c1 is restricted by using the Mundlak(1978) -
Chamberlain(1980) device.

From the properties of the normal distribution,

P(dit = llfliz') = ‘1’ (2/11 + aSit/31 + fir/32 + a2:1) (32)
$1 + 3321.51 + 532'52

2
‘/1+aa1

E @(witfll)2wit5(lixit1ji)i

(I)

 

where 33,; E (5132-1, ..., xiT) is a set of explanatory variables in all times and i E
N “1 Sill xit is a set of time averages of suit. Because only scaled parameters are
identified, I drop subscript a from now on and define wit and 7r1 for notational simplicity.

The second equation is (3.3), which shows the mean function conditional on y > 0.

E(yitlxita 0224121: > 0) = “5132251 + fltCz'Ql (3-3)
Cizlivz' ~ NW2 + 5352 + 012, 0,212),
where ntcig is added to consider time varying individual heterogeneity. Applying the

Mundlak (1978) - Chamberlain (1980) device leads the equation (3.4).

 

E(yit|5’3ia yit > 0) = (I) (mt/12 + 1172151 + 77221—5152 + 77:02'2) (3-4)
77th +1“ 51 +0535 _
— Q) 2t 2 2 z 2 : ©(wit7r2t)'
1+ "t aa2

Therefore, the conditional mean function for y is

50

E(yit|$z') = P(yit = Olmz‘) x 0 +p(yz‘t > Olivi) X E(yit|$iayit > 0) (3-5)

¢(wit7rl)<1>(wz't7r2t) = (1)2(wz't7711 witmta 0) E m(wz‘ta 7ft),

where (1)2 indicates the bivariate standard normal cumulative density function (cdf). The
conditional mean function turns out to be the product of two standard normal cdfs and
we can rewrite the mean function by using the bivariate standard normal cdf with the
correlation of 0. I will use this property later to obtain the average treatment effects (ATEs)
of a binary explanatory variable.

The partial effects of the kth explanatory variable, :1: k, can be obtained either fi'om
taking derivatives of the conditional mean fimction with respect to :1: k if x k is continuous,
or from the differences of the mean firnction evaluated at two values of 53k if £12k is discrete.
As before, these partial effects depend not only observables but unobserved c1 and c2.
Therefore, it is a good idea to calculate the average partial effects by averaging partial

effects across the distributions of cl and 02.

APEt = E(g;,c1,02)[13k¢($t51 + Ci1)‘1’($t51 + Utcz'2) (3—6)
+Bk‘b($t51 + Ci1)¢($t51 + ntcz‘2)]
ATEt = E(x,01,c2)[<1>(a:§1)51 + Ci1)<1>($,(g1)51 + 7725622)
-‘I’(fv,(;0)51 + Ci1)®($§0)51 + 7mm],
where APEt and ATEt stands for the average partial effects and the average treatment
effects of 33k at t, respectively and 5125:) and 239) denote sets of explanatory variables for

two different values of witk- Exploiting the Mundlak (1978) - Chamberlain (1980) device

51

and assuming the independency of (ail: Gig) conditional on 33,-, we can rewrite (3.6) as

 

E
ME = E5,- ——’“—2 ¢(wt771)q)(wt772t) (3.7)
1+0a1
E
+ ’3 2 <I><wur1>¢<wmt>
1+77t0a2

ATE. = E— [E(w§1’vr1)<1><w§”vr2t) - <I><w§°)«1>¢><w§0)vr2t)] ,

where 102(3) and 102(3) denote sets of explanatory variables for two different values of wit-
3.3 Estimation methods under strict exogeneity

Based on the equation (3.5), we can always use the pooled nonlinear least squares
(NLS) estimator. This pooled NLS estimator gives consistent estimates, but might be
inefficient because of possible heteroskedasticity and/or serial correlation in errors. In
this paper, the pooled quasi-maximum likelihood estimator (QMLE) and the multivariate
nonlinear least squares (MWNLS) estimator to consider the serial correlation in errors
are used. Also the conditional mean function for y along with the probit model for d is

estimated because of the identification problem. The log likelihood function for the pooled

Bernoulli QMLE is

lit = W ln'm(wz'ta 7ft) + (1 — yit) 11111 — m(wita 7%)] (3-8)
+dit1n q)(wz't7r1) + (1 — dit)1n[1 — @(wz'tfilfl-
The MWNLS estimator might enhance the efficiency by using the possible correlation

in errors within a cross—sectional unit. The MWNLS estimator solves

52

N

main: [di — 9(wz'7r1)]' I751 [012' — @(wz'filfl (3-9)
i=1
+ [112' — m(wia ”)1, {7,51 [92; — m(wz', 70]

~ ~ ~ ]1/2

Vi1 = Diag[<1>,-(1—<I>i) ~ ]1/2

Conway [FE-(1 — <I>.->
v.2 = May W1 — Emil/2 C(fiiwiag W1 — Eva->11”,

where m,- E m(w,-, 7r) and (I),- E <I>(w,-7r1). ‘~’ indicates that all elements are evaluated at

the preliminary estimates. After estimating (3.2) and (3.5) by the nonlinear least squares

(NLS) estimator, one can calculate the correlations (76f, 772’ ) by using the standardized

residuals from the NLS estimates. See Papke and Wooldridge (2008) and the first chapter

for detailed explanation.

With consistent estimates, the ATEs can be obtained from (3.10).

N

A 1

ATEt = N E [@(w§1)ir1)<r>(w§1)ir2t) — c(w§0)fr1)<r>(w§0)fr2,) . (3.10)
i=1

3.4 Estimation methods with binary endogenous explanatory variables

Now relax the strict exogeneity assumption. I consider a binary endogenous explanatory
variable (EEV) case with appropriate instrument variables. Estimation methods with
continuous EEVs can be found in the first chapter. In continuous EEVs case, the two-step
estimator by Rivers and Voung (1988) can be used and this estimator produces consistent
estimates. However, there is no convenient two-step estimators but inconsistent plug-in
estimator with binary EEVs (Wooldridge, 2007).

The equation for corner solutions is defined as follows.

53

dit = 1(alyz't2 + zit1fi1+ 611 + Um > 0)adit 5 (31m > 0)- (3-11)
Cillzz' ~ N(l/Jl + 52732 + 021,021),
where z, E l/T 2:1 Zita zit = (2,11,11,12). zit contains both explanatory variables
(Zitl) and instrumental variables (Zit2)- yit2 is a binary EEV and Um denotes an
unobserved omitted variable that is assumed to be correlated with yitg. The general index
fimction, 1(-), is used, but I assume the probit model for d to exploit the properties of the

standard normal cumulative density fimction (cdf).

From the Mundlak (1978) - Chamberlain (1980) device,

dit = 1(1/11 + 0113/2752 + 2123151 + 2732 + a11+ vitl > 0) (3-12)
= 1(101 + alyit2 + Zinfil + izfiz + Tm > 0)

1(wit17r1'i' Titl > 0)awit1 5(1azit1a2i)-

The following equation shows the mean function conditional on y > 0.

E(yit1|yz't2, Zitla 07:2, 11212, yitl > 0) = (1901231212 + 221151 + 77tcz'2 + Uit2)(3-13)
- 2
Ci2|~Ti ~ N(I/Jz + 22:52 + W2, 0,12),
where vitg denotes an unobserved omitted variable and is assumed to be correlated with

yit1°

In the same way, we can rewrite the mean function as

54

E(yit1|yit2a Ziarit2w3/z't1 > 0) = ‘I’lflth + 023/212 + 221151 (3-14)
+77t32'52 + Tm]

©(wz’tl7r2 + 732:2), 731:2 5 W112 + vit2-

Now let’s define the reduced form for yg. Because the reduced form is also a nonlinear

function, the Mundlak(l978)-Chamberlain(1980) device is applied.

yit2 = 1(Zz't7 + 613 + “213 > 0)- (3.15)
c- lz- N(Y/J ‘- - 2
23 2t V‘ 3+ZZ<+aZ3i0a3)a

We can rewrite (3.15) as

yitz = 1(¢3 + Zit'Y + 52C + 731:3 > 0) (3-16)
5 1(wz‘t27r 3 + T213 > 0), W3 5 023 + vit3-
The endogeneity of yit2 implies that r213 is correlated with both Titl but 73,52.

Therefore, linear projections of TM and Titg on 73,53 are assumed.

2
Titl = P173153 + eit1,V(eit1|Zit) = T1- (3-17)

2
”Fitz = p2trit3 + Baal/(Brawn) = T2t-
After assuming joint normality (rm, mtg, 727.3), I modify the multivariate probit model
to obtain the conditional mean function for y,“ by taking expectation.

Equation (3.12) can be rewritten as follows.

55

llyit2,ziirit1) = ©(wz't17T1 + Tm) (3-13)
wz’t17T1 + P1T1t3
1 + 7%

 

llyz'tzzzwriw) = <1)

@(wz'tlfll + 1017313)-

Because only scaled parameters are identified, I drop subscript for simplicity.

From the results of the second chapter,

Er3(ditlyit2,zi) =

Er3[q)(wit17r 1 + P1Tz't3)] (3-19)
wit27r3
——yit2 / <I>(w- 7r + -
P 7‘ t )QW )dT

 

1 —wit27r3
- 31212 f
+ (1’0”th + Plr't )¢(7“3)d7"3
(I)(_wit27r3) 00 z 7' 3

‘1’2lwz't17r1, (23/212 - 1)wz't27T3, (231m — 1)p1]
‘I’l(2yz‘t2 - 1)’wz't27r 2]

 

m21it(wz'tla wit2, 7T)

In the same way, we can rewrite the equation (3.14) as

E(yit1|yit2aZitafit21yitl > 0) = 9(wz't17r1 + TM) (320)

= <I> witifli + P2trit3
2
1 + Tu

 

Taking expectation with respect to 73 leads

56

Er3(yit1lyit2,ziayit1 >

0) = Er3[@(wit17T 2t + Pztmsfl (3-21)
wit27r3

 

yit2 / @(w
—— 't17r2t + 92th: )¢(T3)d7‘3
@(wz't27r3) z z 3

—OO
1 "wit27r3
_ yit2
+ ‘1)(w't17r2t+ p2t"'t3)¢(7‘3)d7“3
(I’(“wit27r3) z 1

—OO
©2lwz't17r2t,(2yit2 - 1)wz't27r3, (2.71212 - Um]
(91(23/2'232 '— 1)wit27T2]

 

m22it(wit1a 10212, 7f)-

To obtain the conditional mean function for yl,

E(yit1|yit2,zi) =

I I I

10017;: = llyit2,zi) X E (yitlyz‘t2,zivyitl > 0) (3-22)

®2[wit1“11(2yit2 - 1)wit27r3i (231212 - 1)!)1]
‘I’[(29it2 — 1)wz‘t27r2]
X ©2Iwz’t17r2ta (23/212 - 1)wz't2773a (23/212 — 1)P2t]
(Plat/212 - 1)wit27r2]

 

 

m21it(wit1’ 10212, 7r) X m22it(wit1, wit2a 7r)

m2it(witlv wit2a 7T),

where 7r E (alt, ”2t? 7r3)’. After estimating the model, we can test the endogeneity of

y2 based on p1 and pgt. To identify all parameters and possibly enhance the efficiency

of estimates, the equation (3.22) is estimated along with (3.19) and (3.16) by the pooled

QMLE.

57

111321 = dit 111(m21it) + (1 - dit) 1n(1 — 7712121) (3.23)
+yit1 111(m2z't) + (1 — yit1)1n(1 ‘ "1221)
Han 1nl‘1’2(w7;127r3)] + (1 - 21212) 11111 - (I)(wit27r3)]-

Now let us derive the ATEs of the binary EEV, 312. Two assumptions hold: yg is
independent of (c1, 211,02, v2) conditional on (2,73) and (2., T3) are redundant in the
structural mean function. From Wooldridge (2002 section 2.2.5), the ATEs can be obtained
from differences in the following function evaluated at yitg = 0 and yitg = 1. Dropping

the subscript z',

E(cl,v1,c2,v2)[q)(a1yt2 + Ztlfil + 01 + 'Utl) (3-24)
x 130123112 + 21151 + 77102 + ”Ut2)]

= E(z,r1,r2)[q)(wt17r1 + Tt1)(p(wt17r2t + 712)]

= E(z,r3) [©(wt1711 + P1Tt3)‘1’(wu7rzt + 9215713)]

By exploiting the properties of normal cdfs after transforming the errors, we can rewrite

(3.24) as follows and consistent estimator for the ATEs is

 

 

 

 

N (1)- w(1) . .
1 111 7r 7r
AT Et _N 21% t1 1 wtl 2t P1P2t (3.25)
(0)
4921(me 7r12 wil 7r2t P1P2t

where wall) and w(t1)den°te sets of explanatory variables with yztg = 1 and yth— — 0,

respectively.
58

Testing endogeneity of y2 (H0 : p1 = pgt = 0) and time-constant individual

heterogeneity (H0 : 77t = 1) are conducted in the same way in the chapter 1 and chapter 2.

3.5 Application: fertility effect on working hours of women

I apply the estimation methods to study the effect of fertility on working hours of
women with two or more children using the same data as in the second chapter. The main
contribution of this chapter is to separate the determination of labor market participation
from the determination of fractions of working hours of women. According to the
results from the second chapter, the effect of a new born on mother’s working hours
is significantly negative after controlling the endogeneity of fertility. This implies that
women outside job market would not take part in the market after delivering babies.
Therefore, focusing on women who remain in the market produces more meaningful
implication.

The estimation model for comer solutions is

E(‘d,-, = 1|.) = chm/Jun + 61 f «271,-, + kid26,t,32 + hing-1553 (3.26)
+black,54 + amt/35 + Mia,- + REE/37 + Wm; + rm),
where f wh is the fraction of working hours to total hours and f ert is a fertility dummy,
which is 1 if a woman has the new born at t+l. Two time-varying explanatory variables
are added: kid26 is 1 if a woman has children aged between 2 and 6 and hinc is logarithm
of husband’s income. dsea: is 1 if a woman has the same sex children and is used as
an instrument variable (IV) for the fertility. This IV is based on parental preference on

mixed siblings. Also black and age are controlled. However, because they are likely to be

59

independent on individual heterogeneity and we may not identify the coefficients, their
time averages do not appear in the equation. Tt indicates a set of all time dummies to
allow different intercepts.

The mean function conditional on positive fractional variable is

E(fwhit|fwhit > 0, ) = @(l/JZtTt + dlfertit + kid26it62 + hincz-t63 (3.27)
+blackz-64 + ageit65 + ESE—£13,156 + E32621}; + 523752-68 + mtg).

In this chapter, I use the same explanatory variables for both the determination equation
for a corner solution and the mean fitnction conditional on a positive dependent variable.
However, it is straightforward and sometimes desirable to allow different explanatory
variables for each equation. In our application, some regressors might affect the decision
of whether mothers participate the labor market, but might not affect the amount of hours
worked.

The probit model is assumed for fertility.

E ( f ertitl) = <I>(w3tTt + Vldsexit + kid26it'yg + hincing (3.28)
+blackn4 + 0962175 + 33—65176 + miw + W178 + T113)-
Table (C.1) contains the estimates of the fractional hurdle model under strictly
exogeneity assumption. The first 3 rows reports the estimates of equation (3.26) and the
next 3 rows are estimates of equation (3.27). The reported ATEs are fi'om equation (3. 10).
I report two ATEs: the ATEs conditional on f wh > 0 and the ATEs. The ATEs conditional

on f wh > 0 are obtained based on estimates of equation (3.27). The ATEs of fertility

60'

is -0.022 in both the QMLE and the MWNLS estimator and the ATEs conditional on

f wh > 0 are -0.014~-0.013. The ATEs are quite similar with those of the simple fraction
model, but the ATEs conditional on f wh > 0 are about half of the ATEs. This difference
occurs because some women drop out of the labor market after delivering babies and the
ATEs conditional on f wh > 0 do not count of them. Both the LM test and the variable
addition test accept the hypothesis of constant individual heterogeneity (H0 : 771: = 1).

Table (C.2) reports the estimates of the hurdle fractional model with time-varying
individual heterogeneity. As we can see, the GM test statistic also accepts the hypothesis.
Therefore, it is not surprising that the ATEs are quite similar with those of the fractional
model with constant heterogeneity.

The next two tables (table (C3) and table (C.4)) contain the results under endogenous
fertility assumption. The same sex indicator is used as an instrumental variable. In table
(C3), the first 4 rows report estimates of equation (3.26) and the next 4 rows contain
estimates of equation (3.27). In table (C3) and table (C4), the ATEs are obtained by using
formula (3.25) and the ATEs conditional on f wh > 0 are obtained by using the formula in
the second chapter based on estimates of equation (3.27). The ATEs are -0.129 and quite
similar with those of the simple fractional model. Time-varying individual heterogeneity
hypothesis is rejected and the endogeneity of fertility is accepted marginally. In table
(C.4), we can interpret the results similarly in table (C3). The ATEs are -0.124 ~-0.117
depending on the estimation methods. the ATEs conditional on f wh > 0 are -0.07 ~-0.09.

Even though the point estimates of 77,; are 1.131 ~0.037, their standard errors are too
big. The time-varying heterogeneity is rejected by all test statistics. This might imply that

we cannot estimate the models precisely because of their nonlinearity and the number of

61

parameters estimated. Therefore, let’s focus on the results of the model with time-constant
individual heterogeneity. As a result, fertility is marginally endogenous in the model and
individual heterogeneity is time-constant. The ATEs of fertility are significantly negative,
but the values are quite similar in both the fractional model and the hurdle fractional
model. the ATEs conditional on f wh > 0 imply that the women who are still working
after delivering babies reduce their fraction of working hours by 9 % points. This fertility
effect on working women is usually ignored in the binary model, because their binary

indicators for labor market participation after having babies do not change.

3.6 Conclusion

In this chapter, I consider the hurdle model for fractional dependent variables. Binary
EEVs and time-varying individual heterogeneity are considered as before. With the hurdle
model, we can separate the determination of comer solutions from the decision of the
amount of dependent variables. In the application, some women report 0 working hour
and these might be their optimal choices.

By using the fractional hurdle model, we can allow different sets of explanatory
variables for each equation and calculate the average treatment effects (ATEs) conditional
on the positive dependent variable. These ATEs are ignored in the binary model.

To derive a conditional mean fimction, the bivariate probit model is modified. Based on
the derived conditional mean function, the pooled QMLE is used and the reduced form for
the binary endogenous variable and the mean fimction for comer solutions are estimated
together for identification. I propose how to compute the average treatment effects (ATEs)
of binary endogenous variables. The estimation methods are applied to analyze the effect

of fertility on women’s fraction of worked hours. The ATEs are -0.129 and the ATEs

62

conditional on f wh > 0 are -0.09 under endogenous fertility assumption. The endogeneity
of fertility is accepted marginally, but the time-varying heterogeneity is rejected by all

three test statistics.

63

Appendix A Tables for Chapter 1

64

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90

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