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2/05 p:ICIRC/Date0ue.indd-p.1

ESTIMATION AND TESTING IN DYNAMIC, NONLINEAR PANEL
DATA MODELS

By

Margaret Susan Loudermilk

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Economics

2006

ABSTRACT
ESTIMATION AND TESTING IN DYNAMIC, NONLINEAR PANEL DATA MODELS
By

Margaret Susan Loudermilk

This dissertation consists of three chapters that address issues of estimation and testing
in dynamic, nonlinear panel data models. Chapter 1 deals with an example of the peculiar
difficulties that can arise in estimation of nonlinear models. l\-*Iany economic variables occur
as fractions and percentages. In these cases, the fractions instead of the level values are the
variables of interest. Estimating models with fractional response variables can present chal-
lenges due to the presence of corner solution outcomes at 0 and 1 and continuous outcomes
in the interval (0, 1). Most standard estimation techniques are inappropriate in this setting
because they are designed for variables that are either entirely continuous or take on only
a discrete number of values. This chapter demonstrates an easily implemented method for
estimating fractional response variables and presents an application of the technique to the
determination of firm dividend policy.

Chapter 2 studies the sensitivity and relative performance of average partial effect es-
timates. Typically, partial effects are the quantities of interest. for policy analysis. For
linear models. these are often simply the parameter estimates. However, obtaining partial
effects is more complicated for nonlinear models because these estimates will depend on all
of the model’s explanatory variables in a way that is not separable, except in special cases.
Therefore, when some important individual specific explanatmy variables are unobserved,
consistent estimates of the partial effects may not be available. Instead, estimates of the
partial effects averaged over the distribution of the unobservables. average partial effects,

may be used as the variables of interest for policy analysis. Current estimation techniques

for dynamic, nonlinear panel data models require strong assumptions on economic models.
\Vhich assumptions are maintained affects generality, ease of computation, and even which
quantities can be estimated, but little evidence exists on the relative performance of dif-
ferent estimation techniques for nonlinear panel data models. Since few economic models
conform to such restrictive assumptions, it is important to know how sensitive estimates in
these models are to econometric specifications. This chapter includes both simulations and
empirical analysis.

Chapter 3 addresses a more general problem of testing the assumption of homoskedas-
ticity in nonlinear models with unobserved effects. As a practical matter, heteroskedasticity
is of little concern in linear models since it. does not affect consistency or unbiasedness of
estimators, and standard errors can easily be corrected to perform inference. However, in
many nonlinear models the presence of heteroskedasticity is of greater consequence because
it. changes the functional form of the estimator. The class of tests known as score tests is
ideal for cases in which the alternative hypothesis is complicated or computationally difIicult
because it only requires estimation under the null for implementation and is invariant to a
many alternative hypotheses. Thus, such a test can be formulated for the null hypothesis of
homoskedasticity against a general alternative that encompasses many prevalent specifica-
tions of the variance as special cases or locally invariant alternatives. In this chapter, a test.
for heteroskedasticity is proposed for two dynamic latent variable models, namely the panel

probit and fractional response models, and applications of the test are presented for each.

Copyright by
Margaret Susan Loudermilk
2006

DEDICATION

To my family

ACKNOWLEDGEMENTS

The journey to a PhD is a long and often difficult one. Many people have helped me
on mine, and I want to take this opportunity to thank them. First, I would like to thank
the members of my guidance committee, Leslie Papke, Ana Herrera, and Valentina Bali, for
their contributions and support. I especially want to thank my advisor Jeff VVooldridge for
educating me not only about econometrics but also about being a researcher, a teacher, and a
scientist. Jeff 5 insight, willingness to talk about his own experiences, and his understanding
of the need for balance between career and personal life has made my experience here much
more rewarding.

I also want to thank Thomas Jeitschko, Deborah Foster, Mary Hoffman, Nicole Furnari,
and Patrice \Vhitely for their efforts, particularly during my time on the job market. There
are too many other people at Michigan State that have provided support and guidance during
my years here to name them all. However, they all have my gratitude.

I could never have made it through graduate school without the love and support of
my family. My parents were the first scientists in my life and much of the inspiration and
motivation for my achievements has come from them. My two brothers have also contributed
to my doctorate, each in their own way, and they have my thanks as well.

Without a doubt the person who has contributed the most to my doctorate is my fiance,
Alan Bester. Alan has been there for me through all the ups and downs of graduate school
and has been unwaivering in his love, support, and belief in me. Following each other through
the dissertation process hasn’t always been easy, but I have gained trememdously from his
insight and advice, whether he knows it or not.

Finally, I would like to acknowledge financial support for this research from the American

Association of Univeristy Women Educational Foundation’s American Fellowship.

vi

TABLE OF CONTENTS

List of Tables x
1 Estimation of Fractional Dependent Variables in Dynamic Panel Data
Models with an Application to Firm Dividend Policy 1
1.1 Dynamic Fractional Response Model With Unobserved Effects ........ 5
1.1.1 Model Specification ............................ 5
1.1.2 Estimation ................................. 6
1.1.3 Computational Issues ........................... 9
1.2 Empirical Application .............................. 11
1.2.1 Dividend Policy Theory ......................... 11
1.2.2 Data .................................... 14
1.2.3 Estimation and Results .......................... 16
1.2.3.1 Ordinary Least Squares .................... 16
1.2.3.2 Linear Dynamic Panel Data Model .............. 18
1.2.3.3 Fractional Response Panel Data Model ............ 20
1.2.4 Specification Testing ........................... 24
1.3 Discussion ..................................... 30
2 An Examination of the Sensitivity of Average Partial Effects in Panel
Probit Models 31
2.1 Model Specifications ............................... 34
2.2 Simulation ..................................... 38
2.3 Application .................................... 63
2.3.1 Data .................................... 64
2.3.2 Estimation ................................. 66
2.4 Discussion ..................................... 69
3 A Score Test for Heteroskedasticity in Dynamic Latent Variable Models 74
3.1 Introduction .................................... 74
3.2 Score Test for Heteroskedasticity in the Dynamic Probit Model ........ 77
3.2.1 Panel Probit with Exponential Heteroskedasticity ........... 80
3.2.2 Panel Probit with Quadratic Heteroskedasticity ............ 81
3.3 Score Test for Heteroskedasticity in the Dynamic Fractional Response Model 82
3.4 Empirical Applications .............................. 84
3.4.1 Dynamic Probit Model: Persistence of Union
Membership ................................ 84
3.4.2 Dynamic Fractional Response Model: Determination of Firm Dividend
Policy ................................... 85
3.5 Discussion ..................................... 88
A Chapter 1 Summary Statistics & Variable Definitions 89
B Chapter 2 Summary Statistics & Parameter Estimates 91
C Chapter 3 Summary Statistics 110

vii

Bibliography 1 14

viii

1.1

1.2

1.3

1.4

1.6

1.7

2.1

2.2

2.3

2.4

2.6

2.7

3.1

3.2

Al

A2

[3.1

B2

B3

LIST OF TABLES

Pooled Ordinary Least Squares Regression with Industry and Yearly Dummies 17

FD 2SLS Estimates Including Lead Selection Indicator ............ 21
First-Diffcrenced Two—Stage Least. Squares Estimates ............. 21
Dynamic, Two-Limit, Random Effects Tobit Estimates ............ 23
Average Partial Effect Estimates ........................ 25
Reset-type Specification Test ............................ 27
Dynamic, Two-Limit, Pooled Tobit Estimates ................. 29
Static Simulation, p20 .............................. 45
Dynamic Simulation, p = .25 ........................... 47
Dynamic Simulation, p = .75 ........................... 53
Dynamic Simulation, Cile? y“) .......................... 59
Static Brand Choice ............................... 67
Dynamic Brand Choice, c,|;r:.,j .......................... 70
Dynamic Brand Choice, c,|:r,:. ‘3/2'0 ........................ 72
Restricted Estimates for the Dynamic Probit Model .............. 85
Restricted Estimates for the Dynamic Fractional Response Model ...... 87
Compustat Quarterly & Annual Data Summary Statistics .......... 89
Industry Dummy Variable Definitions ...................... 90
AC. Nielsen Yogurt Data Summary Statistics ................. 91
Parameter estimates for Table 2.1 ........................ 93
Parameter estimates for Table 2.2 ........................ 99

ix

B.4 Parameter estimates for Table 2.3 ........................ 105

8.5 Parameter estimates for Table 2.4 ........................ 109

C.1 Union Membership Data Summary Statistics ................. 110

CHAPTER 1

Estimation of Fractional Dependent Variables in Dynamic Panel

Data Models with an Application to Firm Dividend Policy

Many economic variables occur as fractions and percentages. Examples include firm market
share, employer 401(k) contribution match rates, and TV Nielsen ratings. In these and
similar cases, the fractions, instead of the level values, are the variables of interest. For
example, market size varies across industries, but market share remains a meaningful measure
of concentration and market power regardless of the absolute market size. What differentiates
these variables from an econometric standpoint is that they are not probabilistic outcomes,
yet they possess both two-corner solution outcomes and continuous outcomes in the interval
(0. 1). Consequently, most standard models are inappropriate for estimation. The approach
presented in this paper provides a consistent estimator for fractional dependent variables with
panel data in the presence of both lagged dependent variables and unobserved heterogeneity.
The technique is then applied to firm dividend policy, which demonstrates the potential
effects of ignoring the unique nature of fractional response variables.

In cross-sectional settings, fractional response models have used the standard logit frame-
work under quasi-maximum likelihood estimation (QMLE) to address misspecification issues
as proposed by Papke and Wooldridge (1996) or a linear model using the log-odds trans-
formation. However, the log—odds framework is not appropriate for estimating models that
have a fractional dependent variable with a. substantial number of observations at either zero
or one, as is the case in the application undertaken in this paper. In panel data settings with

unobserved effects, the argument that provides consistency of the “fixed effects" logit model

estimated by conditional maximum likelihood does not carry through when the dependent
variable is fractional, evidently making the logit QMLE approach inconsistent for estimation
with fixed effects. The consistency of this estimator appears to be entirely a consequence of
the logistic distribution’s functional form and depends on the binary nature of the dependent
variable. (See Wooldridge (2002), p.491 for an illustration of the logit case.)

Estimation of dynamic, nonlinear panel data models is complex even without the addition
of a fractional dependent variable. Using logit or probit models in a panel data setting
introduces an “incidental parameters” problem since in nonlinear models with unobserved
effects it is not possible to separate the unobserved effects from the maximum likelihood
estimates (MLE) of the explanatory variables’ parameters, and, except in special cases,
no known transformations will eliminate the unobserved effects. Chamberlain (1992) and
Wooldridge (1997), for example, present transformations for some multiplicative models. For
fixed effect. models, the number of parameters for the unobserved effects will increase. with
the number of cross-sectional observations, N, making consistent estimation impossible with
a fixed number of time periods, T. MLE is still consistent as the number of time periods
approaches infinity. but typically in panel settings the. number of cross-sectional observations
tends to infinity while the number of time periods is small. In random effects settings, the
joint likelihood of the dependent variable (ylt- yNt) can no longer be written solely as the
product of the marginal likelihoods of the dependent variables, Hit» and will require bivariate
integration (Baltagi 2001, Hsiao 1986). Thus, to deal with unobserved heterogeneity in
nonlinear models, typically either a distribution for the unobserved effect must be specified
or a semiparametric approach will be necessary.

Semiparametric approaches allow consistent estimation of model parameters without as—
sumptions on the distribution of the unobserved effects, but a major limitation of these

approaches is their inability to estimate partial effects or average partial effects (APEs).

Even if one is only interested in parameters, currently, semiparametric approaches have
some limitations. Honore and Kyriazidou (2000a) consider semiparametric estimation of
different types of tobit models with individual specific effects, but the approach requires all
of the regressors to be strictly exogenous. This prevents inclusion of lags of the dependent
variable. Honore and Kyriazidou (2000b) extend the estimation to a logit framework that
allows for lagged dependent variables but with assumptions on strictly exogenous covariates
that eliminate the use of time dummy variables. Honore and Lewbel (2002) provide an al-
ternative semiparametric estimator that allows for general predetermined regressors instead
of only lagged dependent variables that achieves x/N consistency by assuming that at least
one of the regressors is independent of both the errors and the unobserved effect. In ad-
dition, semiparametric methods are subject to the usual bias-variance trade-off present in
nonparametric methods.

If a lagged dependent variable is included as a regressor when unobserved heterogeneity
is present, ordinary least squares estimation (OLS) will be inconsistent since the lagng
dependent variable will be correlated with the time invariant unobserved effects. Fixed effects
estimation will also be inconsistent with fixed T. Monte Carlo evidence from Heckman (1981)
shows that. the bias created by incidental parameters is quite significant in dynamic models.
In these cases, consistent estimation will depend on the treatment of the initial condition, 3/2‘0-
There are three prevalent parametric methods for dealing with initial conditions in nonlinear
models. One approach treats the initial conditions as nonrandom. Another method is to

specify a distribution for the initial condition given the unobserved effect so that the joint

 

density of the {ll-it} can be written as f(yg. ...,yT z,c) = f(;/1, ....”Tl-go.z,c)f(;r/0|z:,c). In
addition, Heckman (1981) suggests approximating the conditional distribution of the initial

condition to avoid having to find it.

This paper follows the methodology proposed by Wooldridge (2005a) to deal with the

initial conditions problem in dynamic, nonlinear panel data models and will use a tobit
specification with corner solutions at both zero and one to estimate fractional dependent
variables. While the approach addresses some problems present in other models, it retains
many of the usual drawbacks of parametric models including the need to specify a conditional
distribution for the unobserved heterogeneity, referred to herein as the auxiliary distribution.
In addition, the model requires the explanatory variables other than the lagged dependent
variable to be. strictly exogenous; however, it does allow for the use of time dummies.

An application to firm dividend policy is presented to provide a concrete example where
the fractional response panel data estimator is appropriate and careful treatment of fractional
response variables has potentially important policy implications. The application considers
the determination of the share of payouts to firms' shareholders made as share repurchases
versus traditional cash dividends. This is an ideal example since 1) share repurchases as a
fraction of total payouts is defined on the interval [0, 1]; 2) dividend policy theory suggests
that both unobserved heterogeneity and state dependence are relevant in determining share
repurchases; 3) the variables of interest are not considered to be endogenous in the current
literature; and 4) a substantial fraction of observations are observed at both corner solution
outcomes. The estimation demonstrates that conclusions drawn when neglecting the dual
corner solutions can be misleading.

This chapter proceeds as follows: Section 2 presents the econometric model for dynamic
panel data with fractional dependent variables in the presence of unobserved effects. Model
specification. calculation of quantities of interest, estimation, and computational issues are
all addressed. Section 3 presents an application of the technique to firm dividend policy

theory, and section 4 contains conclusions.

1.1 Dynamic Fractional Response Model With Unob-

served Effects

1.1.1 Model Specification

Specifimtion of the model begins with a latent variable setup that allows for two corner

solution outcomes, zero and one.

313} = tan + 9(yi,t—1)P + (a: + Na (1)
'uitl(3z‘~yi,t—1~---".Ui0~Ci) “IMO-0121) (2)
0 if 3);", g 0
flit = yr, if 0 < y}; < 1

1 if 31,321

This is sometimes referred to as the “Two-Limit” or “Doubly-Censored” Tobit Model. zit
is comprised of strictly exogenous regressors, (7,: represents the unobserved effect. and u.“
is a normally distributed error term. For notational simplicity, let flit—1 E .(]("I/i.t_1). The
function g(.) allows the effect of gnu] to differ depending on its realization in the previous
period. For example, the lagged dependent variable might have. a different effect if in the
previous period there was a corner solution instead of an intermediate value. Such a. case
is easy to imagine for market share applications in which a firm engages in some degree
of competition when it has a positive share but not. all of the market. the firm acts as a
monopolist at one corner, and has exited the market or is considering entry at the other.
Using the latent variable setup above, the density of y“ given Sits 927.t— 1, and c,- can be

derived as follows

 

-2' “r - .‘I'J—ll’ — (~-
[)(yitzolzitagiJ—l-Ci) = ‘I’( It I 2)

(Tu

 

34H + {Int—1P + c- — 1
PU/it = 1l32’t‘9m—1-(7i) ‘1’ ( I l l > (4)

 

 

 

(7U
flit _ 3H7 " git—ll) " Ci
/’(!/it S ;'/l3it~!li.t—1-('il : ‘1’ ( ) (5)
On
8170/21 E 3/l32ft-9t.t—1~(‘t) _ _1_ (ya — Zit’t - 9i,t—1P — (72') (6)
By an On

To simplify notation further, define U’zfit = (zit.g,-.t_1) and i3 = (y, p) which, together with

(3) - (6), specify the density of I/z‘t given (2.,t,g,:‘t_1. (1,3) as

 

—'l!',‘t/3 — (‘1')I[yit:0] (I) (“Vt/3 + ('2' — 1)I[y’:t:l]

(711

ftfyitlu’itJ-‘fim : <I>(

(Tu

_ . I[()<yr <1]

1 . __ v 3_ ’7. If

X l: 65 (Mt 7’ (t1 (1)] (7)
(Tu 0U-

 

where 9 represents the vector of parameters.

1.1.2 Estimation

Wooldridge (2005a) proposes estimation of dynamic, nonlinear panel data models with 1111-
observed heterogeneity by modeling the distribution of the unobserved effect conditional on
the initial value and any exogenous explanatory variables. There are several advantages
to specifying the distribution of c, conditional on 31,0. These include the ability to choose
a flexible auxiliary distrilmtion, the ability to specify the auxiliary distribution such that
standard software packages can be used for estimation, and the fact that average partial
effects are identified and can be estimated with little difficulty. These features are described
in greater detail below.

Under the assumptions that the dynamics of the conditional distribution are. correctly
specified and 2,1 : (:i1,...,.:,-T) is strictly exogenous conditional on ci, the joint density of

(31,1.yy1’) given (gm. 2,, (5,1) is given by

T
f(.1/i1~3/iTl3/i0~ 32%?) = H ftiyitl'U-'it~ (It: 60 (8)
t=1

The idea behind the methodology is that since the density of (ya, yiT) given (ya), 2.1-4:2)
is already available under these assumptions, only the density of c,- given (ym, 2,1) needs to
be specified in order to proceed with estimation. In addition, this density is not restricted
by the assumptions used above to derive the density of (31,31, y”) given (m0, 2,1,(31'), so it
can be chosen based on convenience, flexibility, or any other criteria.

In order to construct the likelihood function, it is necessary to integrate over the distribu-
tion of the unobserved effect (1,. This requires specification of the density of 1' given Luz-0,2,1)
with parameter vector (5, denoted h(c|y,:0. 2,1,5). Given h(c|yi0, 3,3; (5) is a correctly specified

model for the density of 0,: given (317:0. 22-), the log likelihood function is

T
1i(9-5)=109 / H ftfllitl'witaciigl INCH/50,2215)“ (9)
t=1
N T
L = 2109 f H ftfyitl'witacii 0) thlyta 227: (5)0”? (10)
2'21 t=1

In general, the quantities of interest in Tobit models will be E (yla) and E (ylw, O < y < 1),
as well as the partial effects of the explanatory variables. Using properties of the normal

distribution, the conditional expectations are given by

B + — 1 1 — ," {3 — -? l' .13 — c:
Efyzftlwzim'zil = ‘1’ (_u I" P" > + [(1) (————u't' ”) — (b (————l "" ’)]
an 0n 0U

 

 

X E(yitlwit~cis0 < yit<1) (11)
where
1— ,9- 3— —'.v-13—
P(0<y<1|:r) = [q>( “l” “)—<1>(—"—"—”)] (12)
a" all
and
— wit 8— c!- 1 — nil-t 3 _Ci
, 0M ) — (p( ”u )
E(y,it|w,;t. (32-. (l < yit < 1) = wit/3 + (1,: + an (13)

(I) (l—witfl—ci) _ (I? (—witU—Ci)

an 0n

. . . — t- 3— - —
C‘ombmmg these terms and definmg (D1 = (I) (id—£1) (P2 = (1) (_11—1) 691 =

On

0a 011

— !- 3—3 . — r- :3— -
(f) (id—Ll). and (.62 = (b (w) produces

witd + c,- —

 

E(;l/2'(|U’itJ-'il = ‘1’ (

1 ,
a ) + (use + v.) [e2 4151] + 0.. [<01 — (>21 (14)
U
Notice that. the expression in (13) is clearly analogous to that of E (yitlivit. (3:. y“ > 0) in the
standard Tobit. model where the typical Mills Ratio is replaced by a similar quantity for the
interval between the two corners.

Then, by taking derivatives of the conditional mean equation with respect. to the ex-

planatory variables, the partial effects are given by

 

 

”u

3E -- wr ( , w: {3 + c- — 1 ' , . ,
(311" "t l) — 1(0( I" 7 ) + ’7 [‘1’2 — q>1]+ 0101321!er Ci + Sultan - 02) (15)
“-

(hit (7“

 

 

 

 

 

 

 

2
35(1/2'tl'W-itmi, 0 < ya < 1) z _ ”fair 6’2 - 651 A, $2 — (b1 (16)
0:“ a“, (D2 — «p1 ’ <1>2 — <I>1
I
()E l' u" .(f‘ /’.‘7' t— t!" H + (r- —1 I
(.lztl It I) 2 1. 195 It 2 + ng t—l [(1)2 _ (1)1] . ..
ayat~l on an i
I
”gut—1 , , .
+ :7 (wit/3 + (7:: + 9i.t—1)(¢’1 - (>2) (17)
u
’ 2
(9E('l/,jtlw,'t, (1i, 0 < ”if < I) _ I p-(Iit—l-qu—l (3)2 — e51 I (62 — (1)1 18
(9' . — Wat—1 - (I) _ (D + Min—1 —q, _ q, ( )
Uzi—1 Ga 2 ‘2 1

However. the. partial effects cannot be estimated due to the presence of the unobserved effect.
Instead. the partial effect. averaged across the distribution of the unobserved effect. the APE.

can be estimated in the following way

7’l(3t~yt—1aC§6) E Elyitliit~!/i.t—1~Cil (19)
#(3t~3/t—1)= E(~,-l""(3t-yt—1~C2f¥9ll = E{Elmfltlit—l‘ciiell'f/MhSill (‘20)

= E [f711(2),yt_1.('i;/))h(('|yi(),:i;(f)(fc (21)

This provides a \/ N consistent estimator

1

N
[1.(.:t,,I/t_1) = N Z Efmfzt-f/t—1~“i§0)l;’/i013il (22)
i=1

from which the APEs can be obtained by taking derivatives with respect to 3t and yt_1.

£2

. N .
allfavyt—l) 1 8El7'7'(3t~yt—lsci?6)lyi()s

(‘):- = I»? a:-
It i=1 It

] (23)

 

3

 

- N *

0/!(3p m— 1) 1 3El'mtzt (It—1 . ('2'; 0) hm» ~

__ 24

N; ( >
2:

33111—1 — affix—1

1 . 1 . 3 Computational Issues

Estimation of the model can be carried out using standard software that allows for two-limit

random effects Tobit specifications if the density of c,- is specified in the following way
I .
(72"in 32' = 00 + (113/20 + 0231+ 0i. (25)

with a,- ~ N (0, 03). Including the entire vector 2,- along with Mo allows the unobserved
heterogeneity to be correlated with both the initial condition and the exogenous variables.
This distribution for the unobserved effect is similar to that. employed in the random effects
probit model proposed by Chamberlain (1980) but uses the full vector :2- instead of the time
averages of the exogenous variables to allow for more general correlation. Substitution for

C, produces

 

0'21.

2217+ 9:14P + 00 + a’1'3/2'.()+ 0231+ at - 1
Ple/it = llfliita Sngzioaail = ‘1’< a (27)
U

BP(_y.,-t g yl'l1};f.Z,’.gi0.a,‘) _ id) (ya - zm — 92'.t—1/7 — 00 - 0131a) — (12:1 - at) (28)
L

—z"‘ — -_ -—a —(_r r- —a z-—(z-
my“ =0l‘U’itazisgioaai) :<I>( zt/ 92.1 1P 0 1.110 2 1 2.) (26)

 

 

 

8y (71 I 0U.
The log-likelihood function is then obtained by integrating the density of (31,71, 3;”) given

(tel-t, 3,, gm, a.,;) against the distribution of a,-_

N T
1 . (I
L = E :109 / I I ftfyitlwits3i~fli0aaiigl E—(b (0—) (1(1- (29)
i=1 ‘1

t=1 0'

This log-likelihood function has the same form as that for the random effects Tobit model

with the explanatory variables u,',1t.zi.y,-0. By iterated expectations and defining (1)1 =

 

(I) (-—2vit.’)’—OO—altho—(1221-)’ ($2 : (I) (I—wB—fl)—01yio—(tgzi)a ([31 = (f) (_“litB—nQ—alyiO—GQH),

an an (in

652 = 95 (1—witB—oo—myzjo‘aft

a“ ) and 01} = (Tu + (Ta, the conditional mean function, defined

as [1.(2t, yt_1) in (19), can be written as

"lit/f + (10 + (IN/m + ”22,2: — 1

0e

 

("(lt'aa 215 yzio; 0) Z (I) ( ) + (wit/3 + 00 + mm + 0222?) ° "
X [$2 _ in] + 0?! [(2)1 ‘_ $2] (30)

and the APEs are given by

 

ar(u’itazi~yit) 9) ll 1 231

0:0' 2N.

a [021

Since the distribution of 0,; is now fully specified in terms of observables and a normally

u: .
HI]

 

 

0v] (ml-I3 + Oz0 + Qlyin + (1222’) (032 — 9:51) + [$2 — (1)1] } (31)

distributed error, partial effects on P(yit = Olwitmi) and P(yit = llwitmi) can also be
computed.

W'hile standard software can be used to implement the estimator outlined above under
these assumptions on the auxiliary distribution, special programming is required to use more
general specifications of the unobserved effect. In addition, since estimation of the fractional
response variable employs the truncated normal distribution, better convergence properties
should be attainable by exploiting the trimming of the normal distribution’s tails in the

optimization routine.

10

1.2 Empirical Application

1.2.1 Dividend Policy Theory

There is a substantial literature, both theoretical and empirical, that investigates the motiva-
tions behind firm payouts to shareholders. A small but growing literature has also developed
addressing the choice of paying dividends either through cash distributions or through share
repurchases. The determination of what fraction of payouts will be made to shareholders
in the form of share repurchases is an ideal application of the dynamic fractional response
model for several reasons. First, a substantial fraction of dividend paying companies make
either 0% or 100% of their payouts in the form of share repurchases in any given year. In
the sample used in this application, over 20% of the observations occur at each corner. In
addition, there are very few concerns raised in the existing empirical literature related to
problems of feedback or endogeneity in the explanatory variables. Also, the concept of state
dependence or persistence in dividend policy is central to the theoretical literature. The
importance of dynamics is based on the work of Lintner (1956) who showed that firms are
reluctant to reduce cash dividend payments since it may be viewed as a negative signal of
future performance. Therefore, share repurchase programs may be used to distribute changes
in earnings that. are expected to be transitory while changes in cash dividends may reflect
permanent changes in earnings. Finally. unobserved firm characteristics are recognized as
potentially important in explaining firms‘ dividend policy decisions, but there are few empir-
ical studies that have attempted to use panel data to correct for unobserved heterogeneity.
The most common types of analyses performed in the past have been univariate comparisons
and least squares estimation on averages of firm level annual data. However, recently some
advances have been made in the application of panel data techniques and in addressing other

econometric issues.

11

Jagannathan, Stephans, and Weisbach (2000) propose a number of testable implications
of the hypothesis that firms use share repurchases to distribute temporary cash flows. First,
they expect that firms with greater uncertainty about future cash flows, as measured by the
volatility of operating income, will have a larger percentage of repurchases. Since operating
cash flows tend to be more. permanent than nonoperating cash flows, they also predict a
negative relationship between the ratio of payouts from share repurchases and operating
income, and a positive relationship with nonoperating income. In addition, Jagannathan,
et a1. argue that share repurchases may be used by management when they believe that
the stock is undervalued, causing the proportion of repurchases to be negatively related to
the market-to—book value of the stock. They provide an analysis of descriptive statistics of
firm characteristics by payout method and perform a multinomial logit estimation to predict
the choice of payout. method. In the univariate comparisons, they find that repurchasing
firms have lower operating incomes, higher nonoperating incomes, higher volatility, and
poor stock performance as predicted by theory. The multinomial logit model produces
the same conclusions with the exception that nonoperating income is not significant in the
multivariate analysis. However, it is difficult to interpret how these results generalize to
firms’ choice between dividends and share repurchases unconditionally since a firm’s choice
of whether or not to increase payouts and by what method may be related to unobserved
characteristics.

Fenn and Liang (2001) investigate the relationship between firms’ payout policy and man-
agers’ stock incentives. In general, dividend policy theory suggests that insider ownership
of stock aligns the incentives of management and shareholders, reducing agency problems
and leading to higher payouts of firm cash flows. However, managerial stock incentives may
also influence the composition of payouts to shareholders. Since the value of insiders’ stock

options is negatively related to future dividend payments, stock options create incentives for

12

managers to make payments in the form of share repurchases instead of dividends. Penn and
Liang estimate the ratio of repurchases using a two-limit Tobit model on firm-level averages
of annual data for 1993-1997. They use management shares and stock options, net operat—
ing cash flow, market-to-book ratio, log of assets, debt-to—assets (leverage), and volatility of
operating income as explanatory variables. Of these, only management options, market-to-
book ratio, and volatility are significant, and all three exhibit. a positive relationship with
the percentage of payments made through repurchases. Their finding of a positive and sig-
nificant coefficient for volatility agrees with that of J agannathan, et a1. (2000), but they find
the opposite relationship between market-to—book ratio and share repurchases. In addition,
Penn and Liang find a negative sign on operating income, which is also consistent with the
results of J agannathan, et al., but the coefficient estimate is not significant. Penn and Liang
also provide an alternative explanation for the role of options in share repurchase decisions.
They suggest that insider stock options could act as a proxy for unobserved characteristics
since firms with substantial growth opportunities may rely more heavily on stock options
in providing executive compensation or may be more uncertain about the timing of invest-
ment. opportunities. However, they believe that they have adequately controlled for growth
opportunities through their selection of explanatory variables.

Moh’d, et a1. (1995) also study the hypothesis that. paying out cash dividends may
reduce agency costs by providing outside monitoring of managers. They use an 18 year
balanced panel of firms and include both industry effects and dynamics in their analysis.
However, since they apply weighted least squares to the panel without instrumenting or using
a transformation such as fixed effects or first differencing, their estimates will be inconsistent
if unobserved effects are present.

Manos (2002) is perhaps the first paper in this literature to deal with sample selection

directly. The study analyzes dividend payouts for a panel of firms from the Bombay Stock

13

Exchange using a Tobit model. Manes finds evidence of sample selection and applies a
Heckman correction. It is uncertain to what degree Manos’ results may generalize to firms
on the US stock exchanges.

The application presented in this paper looks at some of the most. commonly cited de-
terminants of firm dividend policy to assess how estimates of these factors, as well as policy
analysis, may be impacted by correcting for dual corner solution outcomes, dynamics, and
unobserved heterogeneity. Since OLS estimation has been a prevalent method of analysis of
firm dividend policy in the past, the estimation performed in this paper will begin with OLS
as a basis of comparison for the results obtained after addressing these issues. Subsequently,
the model will be augmented to take into account unobserved effects and dynamics within a
linear model, as suggested by theory. Finally, the fractional response panel data model will
be implemented to address the econometric problem arising from the presence of two corner

solution outcomes.

1.2.2 Data

The data come from Compustat’s Industrial Annual and Industrial Quarterly databases
for 1992-2002 and includes all companies active during this period without missing data.
This data set exhibits a large number of share repurchase outcomes at each corner with
approximately 37% at zero and 25% at one. However, the number of observations per firm
varies due to a combination of entry, exit, and missing data. Therefore, a sample was
created by dropping all firms that were not present at the beginning of the sample and
treating attrition as an absorbing state, meaning any observations following a period in
which the firm was not observed were dropped. Here t = 0 refers to 1992 so firms that did

not appear in both 1992 and 1993 were dropped. The underlying assumption when treating

14

attrition in this manner is that at t = 1 the data represents a random sample and that
any sample selection occurs through exit after this point. Implications of the unbalanced
panel for estimation are addressed along with other estimation issues in section 1.2.3. Four
observations were also dropped with share repurchase ratios that were either negative or
greater than one. These appear to be data entry errors. This leads to a final sample with
11.628 observations on 1,800 firms.

The variables collected include: volatility of earnings, market-to—book ratio, operating
income, non—operating income, dividends, and share repurchases. Volatility of earnings is
defined as the standard deviation of the ratio of quarterly operating income to assets and is
calculated from Compustat Industrial Quarterly File data items 21 and 44. The market-to-
book ratio is defined as the ratio of market value of equity multiplied by shares outstanding
to the book value of equity from Compustat. Industrial Annual File data items 24, 25, and 60.
Operating income is the ratio of operating income to assets, Compustat Industrial Annual
File data items 13 and 6. Non-operating income is the ratio of non—operating income to assets,
Compustat Industrial Annual File data items 61 and 6. The share repurchase and dividend
data come from Compustat Industrial Annual data items 115 and 127, respectively. The
share repurchase ratio is then computed as repurchases divided by the sum of repurchases and
cash dividends. A detailed discussion of share repurchase measurement issues can be found
in .Iagannathan. et. a1. (2000). Volatility, operating income, and nonoperating income are all
expressed relative to assets to control for firm size. All Compustat data items are measured
in millions of dollars. In addition, firm SIC codes were used to construct industry dummy

variables. These definitions as well as summary statistics are provided in the appendix.

15

1.2.3 Estimation and Results

1.2.3.1 Ordinary Least Squares

Ordinary least squares estimation if a linear model is often a benchmark due to its theoretical
and computational simplicity as well as the small number of assumptions required for con-
sistency of the estimates. With a panel of data, OLS can be performed on the observations

pooled across the cross-sectional units. 2', and the time periods, t. Given the model,

ya = .lli,t— 1P + 226:7 + (a (32)

pooled ordinary least squares (POLS) coefficient estimates will be consistent under a zero

 

conditional mean assumption, E l‘z’t yi.t_1, Sal = 0, and a rank condition. No other restric-
tions on the distribution of the error term are required. Use of the OLS model will ignore
functional form issues arising from the doubly-censorrxl nature of the dependent variable.
Since the share repurchase ratio is a fraction, predicted values. especially those for response
probabilities, should always lie in the unit interval in order to be sensible. Predicted values
from POLS can occur outside this interval though because a one unit increase in an explana-
tory variable will always have. the same effect. on the response probability, regardless of the
starting value. In addition. this specification does not take into account potential firm level
unobserved effects. Ignoring such unobserved heterogeneity, if present, would lead the POLS
estimates to be inconsistent. due to omitted variable bias. It is also important to remem-
ber that the zero conditional mean assumption will be violated when a. lagged dependent
variable is present if there are time invariant unoljiserved effects. The violation arises from
the correlation between the unobserved effect, which is a component of qt along with the
i(‘liosyn(;tratic error, and the lagged dependent. variable.

In the case of this application, flit is the share repurchase ratio and 3,} is made up of the

exogenous covariates (i.e, market-to—book ratio, operating income, non-operating income,

16

Table 1.1: Pooled Ordinary Least Squares Regression with Industry and Yearly Dummies

 

 

 

Share Repurchase Ratio Coefficient Estimate
Lagged Repurchase Ratio 0.675
(0.009)
lVIarket-to—book Ratio -0.0001
(0.00003)
Operating Income Ratio 0.063
(0.018)
N onoperating Income Ratio —0.093
(0.085)
Volatility of Earnings 0.011
(0.114)

 

Number of observations = 9828
Number of Firms = 1800

R—squared = 0.68

Note: Bold type indicates significance at the 1% level.
Quantity in parentheses is standard error.

and volatility). In order to control for industry level differences, the exogenous variables are
augmented with 10 industry dummy variables: agriculture, mining & construction, manufac-
turing, transportation, retail, wholesale, communications, financial, services. and utility. In
addition, a set of year dummies are included to control for changes occurring over time that
are common to all firms. POLS estimates with heteroskedasticity robust. standard errors
are computed for the full sample. The estimates are provided in Table 1.1. The estimated
coefficients for industry and year dummies are excluded here and in subsequent tables for
brevity. The quantities given in parentheses are standard errors. hi’Iarket—to-book ratio and
non-operating income both have negative coefficient estimates, and operating income and
volatility both have positive signs. However, only the lagged share repurchase ratio, market-
to—book ratio, and operating income are statistically significant. The result for operating

income is contrary to that obtained by both Jagannathan, et a1. and Fenn and Liang, as

17

well as that predicted by theory. Overall, the POLS estimation only shows empirical support
for the ideas that low stock prices contribute to a policy of increased share repurchases and

that there is persistence in share repurchase ratios.

1.2.3.2 Linear Dynamic Panel Data Model

Firm dividend theory suggests that. both unobserved firm level effects and past dividend
policy are determinants of share repurchase ratios. Therefore, the econometric model should

be modified to include an unobserved effect such that
ya = that—1p + 3m + Ci + Hit (33)

where c,- is a firm specific unobserved effect and ”it is an idiosyncratic error term. Since c,; is
not observed, estimatation of this model requires a transformation to remove its effect. The
treatment of attrition as an absorbing state makes first-differencing (FD) a logical choice,

and applying this transformation the model becomes
Alla = Ayi,t—1P + Aim + Aua (34)

where Arit 2 (Bit — 31:11-1. This removes the unobserved effect, ci, as well as any time
invariant regressors, such as industry dummies, from the estimation equation. Since the FD
transformation removes the unobserved effect. before estimation, no assumptions need to be
made about the form of (32'. However, inclusion of Aqu violates the zero conditional mean
assumption required for consistency of the FD estimator, E [A'ttitlALI/igt_1,AZit] = 0. Thus,
estimation of this model will require implementation of an instrumental variables procedure.
W'ith first-differencing, (z,.y.,‘t_2.y,,t_3. y“) are all available as instruments at time t,
but, in order to limit the number of overidentifying restrictions, only (A2”, ;I/i,t_2,yist_3)
will be used as instruments in the first stage regression. The first-differenced two-stage least

squares (FD 2SLS) estimator corrects the problems inherent in the OLS estimation caused by

18

ignoring the effects of unobserved heterogeneity. The estimator shares the desirable features
of the OLS model that no assumptions about the form of the error distribution are needed
for consistency beyond those on the conditional mean and that no assumptions about the
form of the unobserved effects are required. However, it also shares the negative feature
of ignoring functional form problems that can lead to predicted values outside of the unit
interval.

In this application, there may also be concern over using a balanced subpanel for estima-
tion since firm entry and exit are likely to be related to financial performance measures. In
other words. firms that appear in every period may systematically differ from those which do
not. Typically, in the existing literature on firm dividend policy, unbalanced panels are dealt
with either by performing estimation on a balanced subpanel from the data or on firm level
averages of annual data. Use of the latter precludes dynamics in the econometric model.
Even when selection is random or ignorable, estimation on the balanced panel is inefficient
since it is in effect throwing away data. Therefore, it is important to determine if sample
selection is present. Testing for sample selection was not performed in the previous section
because, in addition to the problems with OLS estimation noted above. if unobserved effects
are omitted in the econometric model but are correlated with selection, inference on the
significance of sample selection may be misleading. In order to produce consistent estimates
on an unbalanced panel, selection may be related to 32' or (r, but may not. be correlated with
the error term. To test for selection in the panel with attrition, variable addition tests can
be performed like those outlined by Wooldridge (1995, 2002). The most straightforward
method is to include a lead of the selection indicator, 32'..t+ls as a regressor and test for
significance. using a t-test. (See, for example, Papke 1994.) The results of this estimation
on the full sample are shown in Table 1.2. The estimates show that the lead variable is

not. significant. in explaining share repurchases after conditioning on the other regressors and

19

unobserved effects. It is now reasonable to conclude that sample selection is ignorable in the
econometric model. The FD estimator is, therefore, consistent for the unbalanced panel or
a balanced subpanel. After dropping the lead of the selection indicator from the regression,
the FD QSLS estimates are given in Table 1.3.

The lagged dependent variable is positive and highly significant with a magnitude sim-
ilar to that obtained from POLS, indicating that there is substantial persistence in the
percentage of payouts made through share repurchases even after accounting for unobserved
heterogeneity. The contrast between the other estimates obtained here and under POLS is
striking. The market-to-book ratio. which was significant at the 1% confidence level under
POLS, is no longer significant after controlling for unolmerved effects. The coefficient on
non-operating income has changed in sign from negative to positive and is now highly sig-
nificant. This is the relationship predicted by theory and obtained by .Iagannathan, et al.,
but is opposite that obtained by POLS. Volatility has also become highly significant and
has increased substantially in magnitude, making the results under FD QSLS consistent with
previous results, studies, and theory. Consequently, the results of the dynamic, linear model
suggest. that companies use share repurchases to pay out transitory changes in earnings but
do not support a role for stock performance in dividend policy. In addition, the difference
between these results and those of the OLS estimation suggests a potentially important role

for unobserved firm heterogeneity in determining dividend policy.

1.2.3.3 Fractional Response Panel Data Model

Since the OLS estimates and the estimates from the dynamic, linear model support different
but not mutually exclusive theories of the determination of firm payout methods, the natural
question is whether one or the other, both, or neither of these theories will be supported once

the dual corner solutions are accounted for in the estimation. However, in order to address

20

Table 1.2: FD 2SLS Estimates Including Lead Selection Indicator

 

Share Repurchase Ratio

Coefficient Estimate

 

 

Legged Repurchase Ratio 0.726
(0.201)
Market-to—book Value —0.00002
(0.00001)

Operating Income Ratio 0.371
(0.094)

N on-operating Income Ratio 0.385
(0.155)

Volatility of Earnings 0.875
(0.397)

Lead Selection Indicator -0.026
(0.018)

 

Number of observations 2 4736
Number of Firms = 1129

Note: Bold type indicates significance at the 1% level, italics 5%.
Quantity in parentheses is standard error.

Table 1.3: First-Differenced Two-Stage Least Squares Estimates

 

Share Repurchase Ratio

Coefficient Estimate

 

Lagged Repurchase Ratio
Market-to—book Value
Operating Income Ratio
Non-operating Income Ratio

Volatility of Earnings

0.606
(0.169)

0.00002
(0.00001)
0.352
(0.082)
0. 333
(0.138)

1.020
(0.352)

 

Number of observations = 5189
Number of Firms = 1129

Note: Bold type indicates significance at the. 1% level, italics 5%.
Quantity in parentheses is standard error.

21

this functional form problem in the context of the model from section 1.1, more restrictions
on the error distribution will be required as well as assumptions about the distribution
of the unobserved effect. For estimation of the dynamic, fractional response model with
unobserved effects. equation (33) is now the latent variable equation in the two-limit Tobit
model specified in (1). The auxiliary distribution is specified as described in section 1.1.3

such that
c,- = (.10 + 0131,70 + (1’22,- + ai (35)

where 22' includes the values of the time varying exogenous regressors (i.e., market-to—book
ratio, operating income ratio, non-operating income ratio, and volatility of earnings) in
every time period. The inclusion of the exogenous variables from all time periods will limit
the number of observations to only those for firms with data available in all time periods,
creating a balanced panel for estimation of the fractional response model. Consequently,
testing the ignorability of selection will not be possible for the fractional response model
under this specification of the unobserved effect. However, tests of selection measures such
as the number of periods a firm appears in the sample with alternate specifications failed to
show a significant selection effect.

The results of the estimation are displayed in Table 1.4. While the parameter estimates for
non-operating income and volatility become insignificant once the nonlinearity is accounted
for, a more significant difference between this model and the previous specification is in
the size of the coefficient estimates. The parameter estimate for operating income is now
almost twice as large, and the coefficient estimate for market-to-book ratio is 100 times
larger. However, all of the coefficient estimates maintain the same signs. Looking at the
larger implications of the model, it is clear that results that were highly significant under

both of the previous specifications continue to be highly significant here (e.g., a positive

22

Table 1.4-: Dynamic, Two-Limit, Random Effects Tobit Estimates

 

 

 

Share Repurchase Ratio Coefficient Estimate
Lagged Repurchase Ratio 0.484
(0.025)
lVIarket-to-book Value -0.002
(0.001)
Operating Income Ratio 0.687
(0.125)
Non-operating Income Ratio 0.192
(0.245)
Volatility of Earnings 1.107
(0.689)

 

Number of observations 2 4530
Number of Firms = 453

Note: Bold type indicates significance at the 1% level.
Quantity in parentheses is standard error.

relationship with (merating income) while those that were significant in only one now appear
as marginally significant. This is the case for both market-to—book value and volatility of
earnings with coefficient estimates just below the 10% confidence level. Thus, what appeared
to be strong empirical support for different theories in the previous models is now far less
conclusive.

Comparing these parameter estimates still provides an incomplete. picture of the different.
implications of these estimators empirically and, more importantly, for policy analysis since
the quantities of interest for (.lctermining the effects of the explanatory variables are the
partial and average partial effects. L'ndcr POLS and FD 2SLS. the coefficient estimates
are also the partial effects. However, in the fractional response. model this is not the case.
Instead, the APEs are computed using the parameter estimates from the fractional response
model and equation (31). The APE estimates are shown in Table 5. The directions of the

effects are. the same as those for the parameter estimates, but the size of the effects has

23

increased by 50-100‘X. Such a large difference in magnitude could be of great importance in
policy analysis.

Standard errors for the average partial effects can be obtained by the delta method or
bootstrapping. Due to the complicated form of the average partial effects in the case of
the doubly-censored Tobit model, bootstrap standard errors are more practical. Bootstrap
sampling can also provide asymptotic refinements when the sampling scheme appropriately
recreates the dependence structure of the data. This is possible given a correctly specified
parametric model under certain regularity conditions. (See Andrews 2001.) Bootstrap stan-
dard errors for the APEs, computed with 500 replications, are also included in Table 5. In
addition to the difference. in magnitudes, the standard errors show that, in contrast to the

parameter estimates, all of the APE estimates are highly statistically significant.

1.2.4 Specification Testing

A major drawback of using the Tobit model for estimation of a fractional response variable
is that it is only consistent under the assumption of normality of the error distribution.
The empirical relevance of departures from normality, however, depends on the extent of
their effects on the estimation results. This section attempts to address this concern as well
as other issues of misspecification. Another concern raised by this estimation method is
that misspecification of the error distribution may also cause inconsistency. The fractional
response panel data estimator described above can be implemented under more general
distributional assumptions for the unobserved effect, but this specification has two features
to recommend it. First, as previously discussed, the adoption of this distribution of the
unobserved effect allows estimation to be performed with standard software. In addition,

this class of models is prevalent in current empirical work. To evaluate the impact of such

24

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specification problems, a functional form test can be performed.
By extending Ramsey’s Reset test to index models, Papke and Wooldridge (1996) derive
a test that can be performed as a general functional form diagnostic for fractional response

models. In this “Reset-type” test, the null model is given by
y}; = 113116 + “2' + ”it (35)
with :17, = (3,3, yiy_1, gm). The alternative model is then
11;} = Inf + 711(1)“th + 172072103 +01? + “it (37)

and a Lagrange h‘lnlt.i1i)lier (LM) statistic is computed for the null hypothesis m = 0, 7,2 =
0. The idea is that. if the model has been correctly specified, nonlinear functions of the
explanatory variables will have no additional explanatory power. Table 6 presents the results
of the Reset-type specification test described above. Column 1 shows the estimation results
and Ramsey’s Reset test statistic. for the POLS model estimated in section 1.2.3.1. Here
the variables included in :1: are the lagged repurchase ratio, market-to-book value, operating
income ratio, nonoperating income ratio, and volatility of earnings. The Reset test statistic
clearly indicates rejection of this model specification. This is not surprising given that the
model ignores firm specific unobserved effects and nonlinearities. Column 2 shows estimation
results and the Reset test statistic for the fractional response model from the previous section
using the same set of explanatory variables. The fractional response model takes into account
the effects of the nonlinear response probability, lagged dependent variable, and unobserved
heterogeneity, but it is subject to the normality assumption discussed above and specific
assumptions about the distribution of c.,-. The test statistic for this model is substantially
smaller than that for the POLS model, but it still clearly indicates misspecification.

What is the cause. of the misspecification problem detected though? Unfortunately, the

Reset test is not helpful in answering this question. One possiblity is that instead of having

26

Table 1.6: Reset-type Specification Test

 

 

 

 

Share Repurchase Ratio POLS RE Tobit RE Tobit
Lagged Repurchase Ratio 0.675 0.484 0.571
(0.009) (0.025) (0.076)
Lagged Repurchase Ratio2 -0.146
(0.081)
l\»farket—to-book Value -0.0001 -0.002 -0.003
(0.00003) (0.001) (0.002)
Market-to-book Value2 0.0000004
(0.0000002)
Lagged Repurchasesa=Market-to—book 0.004
(0.003)
Operating Income Ratio 0.063 0.687 1.080
(0.018) (0.125) (0.231)
Operating Income Ratio2 42.994
(0.437)
Legged Repurchases*Operating Income 0.103
(0.221)
Non-operating Income Ratio -0.093 0.192 0.547
(0.085) (0.245) (0.528)
Non-operating Income Ratio2 -0.415
(0.555)
Lagged Repurchases*Non-operating Income -0.535
(0.847)
Volatility of Earnings 0.011 1.107 1.288
(0.114) (0.689) (1.248)
Volatility of Earnings2 —2.708
(9.135)
Lagged Repurchases*Volatilit. y 1. 105
(1.469)
Reset statistic 241.77 147.27 4.26
(if 2 2 2
5% critical value. 5.99 5.99 5.99

 

Note: Bold type indicates significance at the 1% level, italics 5%.
Quantity in parentheses is standard error.

27

misspeeified the error distribution, there may be nonlinear functions of the explanatory
variables which are significant and have been omitted from the model. To test this possibility,
column 3 estimates the fractional response model again including in :1: quadratic functions of
the explanatory variables and interactions of the lagged dependent variable with the other
explanatory variables. The coefficient estimates for the explanatory variables that were
included in both two-limit Tobit. estimations are qualitatively the same in terms of sign and
significance. and they are quantitatively similar in terms of the size of the coefficient estimates
when the quadratic and interaction terms are added. However, now the model passes the
Reset test. Due to the similarity of the estimation results across both of the linear models,
which are unaffected by assumptions about the form of the errors or unobserved effects, and
the fractional response model as well as the results of the Reset-type functional form tests,
it seems reasonable to believe that inconsistency due to the lack of robustness of the Tobit
model to non-normality of the error distribution is not of great concern in this application.

To test whether it is appropriate to include a firm specific effect, a pooled two-limit
Tobit model was estimated and a likelihood ratio test of the significance of the within-panel
variance was performed. Results of the pooled two-limit Tobit estimation and likelihood
ratio test statistic are provided in Table 1.7. The test clearly rejects the hypothesis of no
unobserved firm effects. This is further supported by the differences between the coefficient
estimates for the pooled Tobit and those obtained from models that account for unobserved
heterogeneity (i.e., RE Tobit and FD 2SLS), as well as by the similarity of the pooled tobit
estimates to those of POLS, which ignores these. effects.

Comparing results across all four models estimated provides additional support for the
appropriateness of the fractional response panel data model in this application. Parameter
estimates for the lagged share repurchases are positive and significant across all models and

are similar in magnitude. The parameter estimates are somewhat higher for the POLS

28

Table 1.7: Dynamic, Two-Limit, Pooled Tobit Estimates

 

 

 

 

Share Repurchase Ratio Coefficient Estimate
Lagged Repurchase Ratio 0.962
(0.020)
l\-’Iarket-to—book Value -0.002
(0.001)
Operating Income Ratio 0.138
(0.060)
Non-operating Income Ratio —0.081
(0.203)
Volatility of Earnings -0.471
(0.404)
Likelihood Ratio Statistic 279.49

 

Number of observations = 4530
Number of Firms = 453

Note: Bold type indicates significance at the 1% level, italics at 5%.
Quantity in parentheses is standard error.

and pooled Tobit. models though, possibly indicating an upward bias created by neglecting
unobserved heterogeneity. Estimates for market-to-book value are consistently negative and
small in magnitude across models but are substantially smaller under POLS and FD 2SLS in
which the n(.)nlinearity in the dependent variable is not taken into account. The relationship
between nonoperating income and share repurchases predicted by theory is supported by
the results of the FD 2SLS and fractional response panel data model, but under POLS and
pooled Tobit the opposite relationship is obtained. Again, the change in the sign of the
parameter estimate is likely caused by neglecting unobserved heterogeneity. Volatility also
appears to suffer from a substantial downward bias when unobserved effects are ignored. In
summary, estimates are consistent across both the models that are and are not affected by
the assumptions of I‘iorinality of the error terms and of a specific form of the unobserved

effect when all factors are taken into account.

29

1 .3 Discussion

This chapter develops a method for estimating fractional dependent variables with panel
data in the presence of unobserved effects and lagged dependent variables. The estimator
allows for a wide variety of specifications for the density of unobserved heterogeneity. Aver-
age. partial effects are identified and easy to computable. In addition, a special case of the
estimator can be implen'iented by standard software that includes routines for random effects
T obit models. An appli ‘ation of the technique to firm dividend policy also demonstrates the
potential effects of neglecting the doubly-censored nature of fractional responses. Primarily,
this application shows that it. is important to recognize that average partial effects are the
relevant quantity for empirical analysis in dynamic, nonlinear panel data models with unob-
served effects. In chapter 2, the robustness of APE estimates to specifications of unobserved
effects and intial conditions is examined further using both simulations and an appli ‘ation
to household brand choice.

Finally, there are several potential extensions to the results presented here. First, in—
corporation of heteroskedasticity into both the structural and auxilliary densities could be
considered. Also, Wooldridge (2000) presents an approach that could extend the methodol-
ogy employed in this paper to allow for feedback to future explanatory variables. it might
also be interesting to investigate how the proportion of observations occurring at corner
solution outcomes affects the differences between the ordinary least squares and dynamic
linear and nonlinear model estimates. In addition, programming the general model to allow
for any auxiliary distribution and to take advantage of potential computational gains would

be desirable.

30

CHAPTER 2

An Examination of the Sensitivity of Average Partial Effects in

Panel Probit Models

In nonlinear panel data models, the usual estimates of interest from regression analysis,
partial effects, are not parameter estimates as they typically are in linear models. Instead,
partial effects depend on all of the model explanatory variables, including unobserved effects,
through a nonlinear function. Consequently, nonlinear panel data models are difficult to es—
timate because unobserved effects can not be separated from maximum likelihood estimates
except. in special cases. When lagged dependent variables are present, dynamics add another
layer of complexity due to the introduction of initial conditions. Three methods are generally
used for estimation of dynamic, nonlinear panel data models that produce consistent param-
eter estimates: random effects. bias corrected fixed effects. and semiparametrics.1 However,
not all of these methods are able to identify partial effects and average partial effects (APEs),
partial effects averaged over the distribution of the unobservables, which are more relevant
empirically and for policy analysis.

Wooldridge (2005b) argues that. in the literature too much attention has been given to
identification of parameters and not enough to partial effects. As discussed by W’ooldridge,
this criticism is particularly important in the context of latent variable models since only
the sign and relative magnitude of the parameters from these models and the total effect of
covariates on the response probabilities have quantitative meaning. However, there has been

a recent focus in the econometrics literature on estimating limited dependent variable models

 

1For details on bias correction methods, see Hahn and Newcy (2004).

31

using semiparametric methods such as those devised by Honore and Kyriazidou (2000), Hon-
ore and Lewbel (2002), and Arellano and Carrasco (2003). Semiparametric methods allow
for agnosticism about. the form of the unobserved effect and initial conditions in estimation,
but current methods do not allow for identification of APEs. In comparison, random effects
(RE) models can indentify APEs but require specification of a parametric model for the
densities of both the unobserved effect and the initial condition. While it is known that
under misspecification of these densities, RE parameter estimates are inconsistent, the be-
havior of the associated average partial effect estimates is generally unknown. Since partial
and average partial effects are typically the true quantities of interest for answering empirical
questions. if average partial effects are relatively insensitive to the specification of unobserved
effects and initial conditions, the focus on more complicated semiparametric methods may
be unnecessary.

This chapter presents an examination of the behavior of average partial effects through
simulation analysis and an empirical application as a step in filling this gap in the literature.
In addition to addressing an econometric question, understanding the behavior of APEs has
important. practical implications. This is illustrated in a statement made by Heckman in his
2001 Nobel Lecture: “Different assumptions about the sources of unobserved heterogeneity
have a profound effect on the estimation and economic interpretation of empirical evidence,
in evaluating programs in place, and in using the data to forecast new policies and assess
the effect of transporting existing policies to new environments.”2 The quote addresses the
need for care in the choice of assumptions employed in econometric modeling of unobserved
heterogeneity in order to obtain valid empirical results. Since the true functional forms of the
distributions of the initial condition and unobserved heterogeneity are generally unknown, if

RE methods are to be used for policy analysis, it is important to know about the sensitivity

 

2Heckman, 2001, p.686.

32

of quantities of interest to such assumptions and econometric specifications.

Currently, little evidence on the sensitivity of dynamic, nonlinear panel data estimators
to functional form assumptions exists. However, a few studies compare parameter estimates
across methods and parametric specifications for the binary response model. Chintagunta,
et al (2001) compare traditional logit and probit estimation methods for a discrete choice
model of household brand choices with the semiparametric method developed by Honore and
Kyriazidou (2000). They compare the parameter estimates and standard errors produced
by the various models and also perform Monte Carlo experiments to examine the sensitivity
of the estimators to the assumptions made concerning unobserved effects. They conclude
that: 1) All methods lead to significant. estimates but that the size of the estimates varies
greatly across specifications: 2) Conditional logit models produce more robust. estimates of
the coefficients on the exogenous variables but produce poor estimates of the coefficient
on the lagged dependent variable; and 3) Estimates of state dependence greatly depend
on the specification of unobserved heterogeneity. However. Chintgunta, et a1 assume that
gm is exogenous in all specifications where a lagged dependent variable is included, which
limits the applicability of their results to a broader class of models. Chay and Hyslop
(2000) also compare estimation methods in dynamic, binary response panel data models.
They perform both Monte Carlo experiments and estimate a model of female labor force
participation to assess the performance of random effects, fixed effects, and the Honore and
Kyriazidou methods. In contrast with Chintagunta, et al, Chay and Hyslop include multiple
specifications for the initial conditions in their study. They find that when initial conditions
are misspecified the degree of state dependence tends to be substantially overstated and the
effects of the exogenous covariates are underestimated.

Hyslop (1999) performs a similar comparsion of the linear probability model, a static ran-

dom effects probit model, and a dynamic random effects probit specification in an empirical

33

application of female labor force participation and by comparing their prediction capabili-
ties. Hyslop finds that the differences between the parameter estimates depend upon the
specification of the errors and unobserved heterogeneity but are often comparable. Ham
and Lalonde (1996) also conduct a limited comparison of estimates from a duration model
application to job training programs when different assumptions are made about the form
of unobserved heterogeneity. They do not. draw any substantive conclusions about the effect
of varying these assumptions though since the focus of their paper is on sample selection
issues.

Thus, all previous studies focus on parameter estimates even though the quantities of
interest in binary response models are typically the conditional response probabilities and
partial effects. Instead, this chapter proposes an examination of the ability of random effects
models with flexible specifcations for the conditional distributions of unobserved heterogene-
ity and intial conditions to produce good estimates of quantities of interest even if the model
is incorrectly specified. This will be accomplished through both Monte Carlo experiments
and an empirical application. The results of the simulation study are abstract and difficult
to put into the proper context alone. Thus, the empirical application helps to clarify the
findings of the simulation. The outline of the chapter is as follows: Section 2 describes the
model specifications to be considered. Section 3 presents the Monte Carlo study. and section
4 examines APE sensitivity in an empirical application. A discussion of the results and

conclusions is presented in section 5.

2. 1 Model Specifications

The paper restricts attention to the latent variable model since this is the leading case in

which parameters have no meaningful interpretation. This is also the case most commonly

34

considered in previous studies on parameter estimates. Among latent variable models the
simplest case is that of the binary response model. The dynamic, binary response panel data

model is specified as

yit = 1(1‘2113 + Pym—1 + Ci +1121 > 0) (1)

I’ll/a = llJ‘it--yi.t—1~Ci) = FfIitfi + Pym—1 + Ci) (2)
POM = Ulla tat—rat) = 1 — F0413 + Pym—1 + Ci) (3)
”W = llTitw-yiJ—L met/,0) = Pfyit = 1l£it~~3lzi,t—1) (4)

where F(.) is some well-behaved cumulative distribution function, 1' = 1, ..., N indexes cross-
sectional units, t = 1, T indexes time periods, and 1(1‘213 +py,;t_1 + c,- + Hit > 0) is an
indicator function equal to 1 when the expression inside the parentheses is greater than zero
and is equal to zero otherwise. y", is the dependent variable, :L‘it represents the exogenous
variables, and 0,- represents unobserved time invariant individual characteristics. ”it is an
error term that has zero mean, variance 03, and is normally distributed in the probit case,
F(.) = <I>(.,) or has a logistic distribution, F(.) = A(.), in the logit case. In empirical
applications, the choice between the logit and probit models is largely based on convenience
since typically there is nothing from economic theory to recommend one form over the other.

However, parameter estimates generally differ by a sulmtantial factor between these models.

The conditional density of y“ is then given by

, , - 1— ,-
ftf3/itl4l7it- sync—1.0:) = F (In/3 + Pym—1 + Gill/7" [1 — F (1172113 + prim-1 + 02)] y" (5)

and the log-likelihood function for the traditional dynamic random effects model is given by

N T
I. = Z(”II/l./'(;Uz'0|-'7i~"1)Hf(;I/1t|!/2'.t—1~We(‘i)l"((?il4l'i~(VII/"(63) (5)
i=1

t=1

with h((:.,-|1:,-,(5) representing the density of 0,- given :13,- with parameters (5. Average partial
effects are then obtained as the partial effect after “integrating out" the unobserved effect.
Standard errors for the average partial effects can be obtained by the delta method or
bootstrapping.

From the equation for the log-likelihood function, it is clear that specification of f (yiolri. (3,)
and h(c,:|;r,-, 5) is required in order to proceed with estimation. Since the. data generating pro-
cesses are unknown and consistency of maximum likelihood depends on correct specification,
it is difficult to determine the best. way to proceed. Ideally, specifications for unobserved
effects and initial conditions that are very simple to estimate would lead to good estimates
of quantities of interest even if the true underlying model was quite complicated. To in-
vestigate the feasiblity of such an approach for obtaining estimates of APE’s, several fairly
simple but. flexible models for the condtional distributions of c,- and 3110 have been chosen.
Three different specifications of the intial condition are studied. First, the initial condition

is treated as exogenous in the sense that

PfyiOlTiaCi) = P(yi0)' (7)

The degree to which this assumption is a concern in practice depends on the application.
For example, this assumption may not be distasteful in the context of brand choice, as in
Chintagunta, et a1, since the initial brand choice may, in fact, be unrelated to most individual
specific characteristics. This is in contrast to settings such as estimating a wage equation
or labor force participation where this assumption would be difficult. to justify at best. Two
specifications also approximate the distribution of the initial condition. Approximation of the
distribution of the initial condition was proposed by Heckman ( 1981). The approximations

used here are

P(yi()=1|1‘z'~ci) = “172013) (8)

36

P0120 = 01723021) = @070 + 77,1351 + 77202:) (9)

which allow for varying degrees of correlation between the initial condition and individual
characteristics. The. second specification is more flexible because it allows for a separate
intercept and does not impose the same parameter values on the initial period that are
estimated for t 2 1. Also, four specifications for the distribution of unobserved heterogeneity

are considered. These are given by 3

Cilxi ~ Nf‘U”()+‘¢’/‘ifi~0(21,) (10)
01113 ~ N(Ui’0+'¢ifzt+w2fi2+¢3fi3,03) (11)
Glitz: N Mil/’0+‘t’iifz'sagefPfAiJ-fil) (12)
Cilfi ~ Nth/'50 + elf.- + 1.1925512 + $31713» 036$!)(Aifill (13)

The distribution of the unobserved effect in (10) gives rise to what is sometimes called
Chamberlain’s Random Effects Probit Model or the Mundlak-Chamberlain Device and cre-

4. The second specification allows for

ates correlation between c and a: through the mean
additional flexibility without attempting to incorporate heteroskedasticity, which compli-
cates estimation, by using a polynomial in the exogenous regressors for the mean function.
The last two specifications contain the same mean functions as previosly described but allow
for exponential heteroskedasticity as well.

In addition to the traditional RE method, models specifying the distribution of the unob—
served effect conditional on the initial condition will be estimated. This approach is proposed

by Wooldridge (2005a). Wooldridge points out that, since the density of (gm, 31,7) given

(31,0, 1,7, 0,3) is already available without assumptions on the initial condition, only the den-

 

2

3For notational convenience, when :17, is a vector .177. or 3:? indicates that each element of the

vector is squared or cubed.
4Mundlak (1978), Chamberlain (1980)

37

sity of C, given (yi(),:i:,:), denoted h(cly,-0,17, 6). needs to be specified in order proceed with
estimation. Therefore. given }2((i'|yi0,1:,-; (5) the log likelihood function for the random effects

model is

N T
L = :10!) j H f(;l/itl!/i.t—1217it-Ci) h(<‘i|ym,fi;(”(1510) (14)
1:1 t=1

For this method. the distributions of the unoberserved effect are similar to those used in
the traditional RE setting but include an additional term taking into account the initial

condition

(fil-I'i- I/z'e N N(“() + n1M0 + ”211723 ”3) (15)
('z'l-‘I'zw. .1110 N N010 + ”ll/it) + 02-13 + "ail—'12 + “411713-03 (16)
«m me ~ NW) +0111“) + M23. aim-potrfl) <17)
"ii-7'27» U20 ~ NW) + (113/10 + 024th + ”3:132 + (14-733, figs-'IIPOi-‘Ei2ll (18)

By specifying the distribution of the unobserved effect in this way, correlation is allowed

between the exogenous variables. unobserved effect, and the initial condition.

2.2 Simulation

Attention in the sinmlation study is restricted to the probit. model with the variance of
Hit» 03 normalized to 1 to simplify estimation and comparison of the estimates. A l\.-'Ionte
Carlo study was performed for panels of multiple lengths in both the time and cross-section

dimensions such that T = 5. 25 and N = 100. 500.5 The true data. generating process (DGP)

 

"In every case the magnitudes of N and T specified maintain the standard asymptotic properties for

panels since '1‘ is small relative to N.

38

in the simulation is given by

y” = (cud + pg/i,t_1 + (7,: +11“, 1 = 1, ...,T (19)
Mo = Lin/3 + ('2' + “2‘0 (20)
Hit N N(0.1) (21)
Cilil‘i ~ Ntcie+'¢5’ifi-03(A0+/\i4512)) (22)

while estimation is performed using (7)-(9) and (10)-(13) for traditional RE or (15)-(18) for
the method proposed by Wooldridge. Thus, the variance of CilI,‘ in the DC? is heteroskedas-
tic but of a different form than in the estimated specifications, and the form of the initial
condition in the DGP is a combination of the estimated specifications. The simulations
present a comparison of the parameter and average partial effect estimates produced when
the correct distribution of the unobserved effect is specified. namely normally distributed
unobserved effects, but the moments of the distribution may be incorrectly specified.6 In
addition. the. degree of st ate dependence varies while the coefficients on the exogenous vari-
ables remain fixed in order to examine whether the relative size. of the parameters influences
the senstivity of the estimates. The parameter values used are ,131 = .25, 132 = 1.5, and
p = 0,025. or 0.75. Note that for p=0 no lagged dependent variable is included in the
data generating process (DGP). Estimating this static model, provides a basis of compari-
son for determining the. effects of misspecification of the unobserved effect alone. All other
model parameters (i.e., 03. x\, w, a and 7]) will be fixed across designs. The “true“ values of
the APEs discussed below were obtained by simulating the partial effects for each ‘ase and

evaluating the average at. the mean value of the covariates.

 

6Research is ongoing to check the robustness of the results obtained here by assuming a non-normal
distribution for the data generating process of the unobserved effects but performing the estimation using a

normal (,listribution. Thus. the distribution itself, in addition to the mean and variance. is misspecified.

39

APE results for the static case (i.e. p = 0) are shown in Table 2.1, which includes 1,000
replications for each of the four panels (A, B, C, and D). Parameter estimates significant
at the 10% confidence level or above are obtained in all cases. and the magnitudes of the
biases are similar across models and panel sizes, as shown in Table 8.2 in the appendix. On
average, the parameter estimate for $1 differs from its true value by .075 and that for :52
differs by .449. This is approximately 30% of their true values. However, all tests of the
hypothesis xi? = 1.5 are rejected at the 10% level. Similarly, 9 of the 16 cases reject the
hypothesis that .31 = 0.25 at the same level. In Table 2.1, estimates are shown in bold print
if they are both significant at greater than or equal to the 5% level and fail to reject the
hypothesis that they are equal to their true value. Estimates shown in italics are similarly
significant at. 10%. In contrast, only 2 of 16 APE estimates for $1 reject the hypothesis that
they are equal to their true value and approximately 1 / 3 of the APE estimates for 2:2 fail to
reject this hypothesis. All of the estimated APEs are statistically significant at the 5% level
with one exception that is significant at 10%. In addition. the bias of all these estimates
is less than the average bias of the parameter estimates as a percentage of their true value.
No particular specification of the unobserved effect appears to provide better estimates than
any other overall. This could be concerning since statistically significant APE estimates are
obtained that are statistically different from their true values in some cases. However, the
differences in magnitude are sufficiently small in all cases that the deviation from the true
value may not. be important from an empirical standpoint, particularly since the size of the
bias is not growing either with the sample size or the number of time periods. Also, it is
interesting to note that the APE estimates are most robust to misspecification of c,- when
the panel is small. As the panel size grows, correct specification of the conditional mean of (3,:
given 1:,- appears to become more important as shown in panel D. However, misspecification

of the conditional variance does not seem to have a great effect on the estimates even as the

40

panel size changes.

Tables 2.2 and 2.3 present simulation results for the dynamic case using the traditional
random effects approach where p = 0.25 and 0.75, respectively. Again, 1,000 replications were
performed in each case. The results for the dynamic models are more complicated than those
for the static case and differ slightly according to both the degree of state dependence and
the way in which the density of the initial condition is estimated. The parameter estimates
associated with Tables 2.2 and 2.3 are given in Tables 8.3 and B.4 in the appendix.

For ,0 = 0.25, all parameter estimates for $2 are statistically significant and nearly all of
those for 1:1 are as well. p is not well estimated by any model in panel A, which has the
smallest sample size, but the estin'iatcs are statisitically significant in most other cases. It is
interesting to note that the bias of the parameter estimates tends to be smallest when the
unobserved effect is specified as in (13), especially for initial conditions specified in columns
(i) and (ii). This corresponds to the. most flexible. specification for (1,; and the more simple
specifications for 3120» but this effect is smaller in larger panels. Further, no combination
of panel length, intial condition specification, and unobserved effect specification leads to
a set of parameter estimates that are all significant and fail to reject the hypothesis that
they are equal to their true values. The results of the APE estimation are more encouraging
though. Insignificant APE estimates only occur when parameter estimates are statistically
insignificant. Among the statistically significant APEs all fail to reject the hypothesis that.
they are equal to their simulated values with one exception, which is for $2 in panel B under
specification (13) for c,- and (i) or (iii) for "gt-0. In general, the most simple specification for
the initial condition, (i), appears to lead to the “worst” APE estimates since no combination
of unobserved effect specification and panel size leads to all three APEs being simutaneously
well estimated. In addition, no specification is able to accurately estimate all three APEs

when p = 0.25 for the. smallest sample size as the average parital effect of p is only consistenly

41

well estimated when T = 25. However, when T = 25 using (10) and (ii) or (iii) all APEs are
consistenly estimated for panel C and with ( 11) or (12) and (iii) all APEs are consistenly
estimated in panel D.

For {2 = 0.75, which has a higher degree of state dependence, if the parameter estimates
are statistically significant then the APE estimates are significant and not statistically dif-
ferent from their simulated values at the 10% confidence level or higher, except for 1‘02 in
panel B with (12) and (i). However, many fewer of the corresponding parameter estimates
are statistically significant and not significantly different from their true values as indicated
by the bold and italic print. Overall, for T=5 homoskedastic models seem to produce the
best APE estimates. This is seen by observing that for (i)-(iii) and (10)-(12) all three APEs
are consistenly estimated in panels A and B by using (10) or (11) to specify the distribution
of the unobserved effect. Interestingly. for p = 0.75 and T = 25, all three APEs are only
well estimated when (12) and (iii) are specified. This may indicate that. the importance of
allowing for heteroskedasticity in the variance of the unobserved effect increases as either
the degree of state dependence, the number of time periods, or both increase. In contrast,
when the number of time periods is small, all specifications for the initial condition and the
conditional mean are able to produce a well estimated set of APEs.

Overall. Tables 2.2 and 2.3 share. many similarities. Specifically, almost all APEs are
statistically significant and close to their true values, except. those with corresponding in-
significant parameter estimates. In addition, the relative insensitivity of the APE estimates
to misspecification is further emphasized by comparison with the. findings of earlier studies on
the sensitivity of parameter estimates. Chintagunta, et al found that parameter estimates for
state dependence varied substantially depending on the specification of unobserved effects.
The results in Tables 2.2. 83,23, and B.4 show variation in the parameter estimates across

the specifications of CilTi~ which seems to depend primarily on the specification of the. initial

42

condition. More importantly, even when parameter estimates vary the most across the distri-
butions of unobserved heterogeneity, the changes in the magnitudes of the APE estimates are
very small. Chay and Hyslop found that when initial conditions are misspecified the effect of
state dependence is overstated and the effects of the exogenous regressors are understated.
Instead, the simulations presented here show that in larger panels all parameter estimates
were underestimated, regardless of the degree of state dependence. In smaller panels, how-
ever, when the initial condition was specified as l"(y,0 = 1|;Iti,c,j) : <I>(-I)0 + 77,117.,- + 712(5)
parameter estimates were all understated when the unobserved effects were estimated as
homoskedastic and all overstated when they were heteroskedastic. However, the degree of
variation in the APE estimates was much less, and APEs were more sensitive to the specifi-
cation of the. intial condition than that of the unobserved effect.

Table 2.4 shifts to consider the case when the distribution of c,- depends on the initial
condition in addition to the exogenous covariates. This simplifies estimation and reduces
the number of assumptions required by removing the need to model the distribution of 31230
separately. This is of even greater importance in light of the results above, which show that.
APEs are more sensitive to the specification of fliO than ci. The results of these simulations,
using the specifications of unobserved heterogeneity described in (15)-(18), immediately show
two other reasons to prefer this method to that of specifying the distributions of Cil$i and
Iliul-‘I'is’3i separately. First, there is never a case where if is statistically significant and
equal to its true value and the associated APE estimate is not. as well. Second, using this
method, a larger number of the estimated APEs are consistently well estimated than under
any single specification for ”min-,1, r:,-. Further, when the degree of state dependence is high,
for every panel size there is a specification for the unobserved effect that leads to all three
APEs being simultaneously consistently estimated. As under the previous method, with

T =2 5 homoskedastic specifications for the unobserved effect seem to produce the best APE

43

estimates and allowing for heteroskedasticity appears to become more important as the
sample size increases.

The results obtained in this section are fairly abstract. Taken together, they suggest that
if APE estimates are statistically significant, they tend to provide good estimates of their
true expected values despite potential misspecification of the distributions of unobserved
effects and initial conditions. In cases where dynamics are present and a distribution for the
initial condition is specified. there is also a benefit to increased flexibility in the specification
of the initial condition and to allowing for heteroskedasticity in the unobserved effect in
larger panels. Whereas, in smaller panels, simplicity in estimation is more important. In
addition. while. the specification of the unobserved effect seems to have only a small effect on
APE estimates, the specification of initial conditions has a large effect on APEs. This may
be a good argument for using the approach in which the distribution of unobserved effects
are. specified conditional on the initial condition and exogenous regressors, as proposed by
Wooldridge. In the following section, a similar approach is used in an empirical application.
The results of the application support the findings described here and make the implications

of the simulation study more concrete.

44

Table 2.1: Static Simulation, p=0

DGP c,-|.r,j ~ NW’O + ‘d’ifz’flngO + M5132»

 

 

Estimated Process

APE Estimates

 

(A) T=5, N=1()()

 

 

 

 

 

 

 

 

 

2:1 .132
o-I-r. ~ Mn, + t/y’lr, 03) 0.041 0.247
(0.018) (0.020)
mm ~ Mm) + e’lf, + me? + e333. 03) 0.041 0.245
(0.018) (0.020)
(3).]; ~ .‘\"(t“() + W117i.03(6.1'p(/\’1.f,')) 0.041 0.244
(0.018) (0.020)
(It-tr,- ~ N (on + viii, + (£52132 + 'lg')3f.,-3, 03erp(/\'1;IT,-)) 0.041 0.246
(0. 021) (0.022)
Simulated APE 0.037 0.221
(B) T25, N2500 :171 3:2
an, ~ was” + @313. 03) 0.041 0.247
(0.008) (0.009)
c,;|.r,- ~ Mao + ‘t/vifz- + use? + «pm-3, 02,) 0.041 0.244
(0.008) (0.008)
olzm ~ N(r’vo + em. agca‘p(/\'1f,')) 0.041 0.244
(0.011) (0.010)
(film,- ~ .I'\"(t(’() + *0/11}; + 052.1372 + 0231723.03(31‘])(/\'1;17,j)) 0.037 0.229
(0.011) (0.033)
Simulated APE 0.033 0.199
(C) T225, N=100 2‘1 (1:2
«in, ~ lye-'0 + t""’1.f,j.r73) 0.043 0.254
(0.008) (0.015)
c,- |.r,j ~ /'\"(‘I.-"(‘) + LV'lzfj + 09.132 + "023233, 03) 0.042 0.255
(0.008) (0.015)

confirmed on negrt page

Table 2.1 (continued)

 

 

 

 

 

 

 

 

Estimated Process APE Estimates
(‘l'l'l‘i ~ ./\‘r(t‘l"() + ‘lf-‘Ilfi. 0391740313» 0.042 0.253
(0.008) (0.015)
(film, ~ N(c:0 + eta,- + 02.5,? + 2.9333, agexpo’lan 0.037 0.229
(0.011) (0.033)
Simulated APE 0.051 0.304
(D) T225, N=500 1‘1 172
a, |;r,j ~ New“ + am. 03) 0. 042 0.255
( 0. 004 ) (0.007)
qty, ~ Nova +015, + 52.27,? + 213.133. 03) 0.042 0.255
(0.003) (0.007)
(311-,- ~ Meg + 2.125,. firepo’flm 0. 042 0.254
( 0. 004 ) (0.007)
(_.‘,'l.l',' ~ 1)."(1/9'0 + til/1:17,- + 11”ng + 0,3223. ag(%.r[)(/\’1fi)) 0.041 0.249
(0.004) (0.011)
Simulated APE 0.049 0.292

 

 

Quantity in jiiarcnthcses is standard error.

f‘istiinates in bold are significant at the 5% level X1 simulated equals estimated APE (italics 10%).

46

Table 2.2: Dynamic Simulation, p 2 .25

DGP (.‘jlflfi ~ N(L/}0 + 'tf-’,1fi,0’3()\0 + /\

"’20

117i

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

 

 

 

 

 

(i) /’(!/2'0l:'7i~'=-z‘) = P0170)
(A) T25, N=IOO $1 1’2 yt—i
cilzri ~ New.) + 44.27,. 03) 0.041 0.244 0.037
(0.019) (0.020) (0.037)
(ii-737: ~ N('U"() + ‘c'l‘i 11.} + 1;".7211712 + 1493:1743 0.039 0.235 0.034
(0.018) (0.021) (0.036)
(3.1-lat,- ~ N(t/’() + 05,133, 034.271(/\’1f,~)) 0.042 0.246 0.046
(0.026) (0.032) (0.070)
c,j}:r,- ~ New.) + 44:17, + 02132 + 0""3fi3,0(2,c:r})()\’1f;)) 0.048 0.251 0.057
(0.032) (0.041) (0.088)
Simulated APE 0.037 0.219 0.037
(B) T25, N2500 3‘1 1‘2 yt_1
(3)1,- ~ New, + 4413. 53) 0.041 0.246 0.042
(0.009) (0.010) (0.017)
('4,le ~ N(I.T’() + gift?- + 12ng + 093.1713 0.039 0.233 0.035
(0.018) (0.020) (0.038)
(Till‘i ~ NW!” + 'z/rifi, 03(«:.1:p()\'1f,-)) 0.047 0.249 0.050
(0.025) (0. 027) (0.068)
mm, ~ Nm, + 1.2123- + «4.9.7,? + 0’33133,U(2,erp()\’1f))) 0.047 0.246 0.050
(0.027) (0.023) (0.095)
Simulated APE 0.033 0.198 0.033
(C) T225, N2100 11 1‘2 ytfil
75-13:,- ~ Nam + 44.17,. 03) 0.045 0.273 0.045
(0.008) (0.015) (0.018)
alas, ~ NM) + 4212:- + 49223:? + 03133.03 0.040 0.0241 0.039
(0.008) (0.016) (0.017)

47

continm—rd on next page

Table 2.2 (continued)

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

(3,417,: ~ N(t'0 + ‘g";"117,.0;2,e1'1)(A’11‘,)) 0.042 0.254 0.042
(0.008) (0.015) (0.018)
cilzri ~ N(0’*() + viii, + 09-21742 + 42333-3, o§crrp(/\’1:E,')) 0.041 0.243 0.039
(0.009) (0.016) (0.019)
Simulated APE 0.050 0.302 0.050
(D) T225, N2500 131 1‘2 yt_1
(,-,,-|a.~,- ~ Ntu’o + 1.2137,. 03) 0.045 0.272 0.045
(0.004) (0.006) (0.008)
0,-1.4, ~ NW0 + 03.17,: + 02.5,? + 0327,1453) 0.041 0.242 0.040
(0.006) (0.009) (0.010)
mg,- ~ Naso + 44.7,. 03.2.2:p(x,.r,)) 0.042 0.254 0.042
(0.004) (0.007) (0.008)
(5)1", ~ N(t"’0 + 0117.,- + 0”ng + 093.7713, ogexpO’lfI-D 0.043 0.254 0.043
(0.009) (0.010) (0.017)
Simulated APE 0.048 0.290 0.048

 

 

48

continued on next page

Table 2. 2 (continued)

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

 

 

 

 

 

(ii) P (3170 = 1|$i~cil (“$20.13)
(A) T=5, N=100 171 :1’2 l’lt—l
my. ~ N040 + 41.17,: , 03) 0.039 0.239 0.038
(0.016) (0.018) (0.035)
Cil-Tz‘ N N (1110 + @053: + 01213-2 + 053.133, (73) 0.038 0.228 0.033
(0.016) (0.019) (0.035)
cilxl- ~ N('0"’0 + 11"),11fi03631‘p()\,1fi)) 0.041 0.239 0.049
(0.022) (0.032) (0.071)
(:ilzlri N N((/'() + ‘(iiifi -+- 1.0211712 + '¢I‘3Lfi3, 03(’..’IT])(/\’1:I—li)) 0.042 0.238 0.051
(0.028) (0.037) (0.096)
Simulated APE 0.037 0.219 0.037
(B) T=5, N=500 1'1 :172 yt_1
aim ~ N(0’ro + 11.43.1383) 0.040 0.241 0.040
(0.008) (0.009) (0.017)
7:03;,- ~ N050 +0117.- +412sz + 0.23.1713, 03) 0.038 0.227 0.033
(0.016) (0.020) (0.036)
(3)4,- ~ N010 +'0’Jifi.oge1'p()\'1f,j)) 0.039 0.230 0.045
(0.018) (0.022) (0.075)
aim ~ NM + 03.5,; + 2.425)? + 4317.3, ogexp(/\-'lzfi)) 0.043 0.242 0.055
(0.027) (0.031) (0.088)
Simulated APE 0.037 0.219 0.037
(C) T=25, N=100 .731 T2 yt_1
cz-lafz- ~ N(0”0 + 0,-5.5). 03) 0.045 0.272 0.045
(0.008) (0.015) (0.018)
(rilzri ~ N(0’10 + 0.3.5,: + 092.5)? + 023.133.7731) 0.046 0.271 0.044
(0.009) (0.015) (0.018)
cm), ~ N(11’1()+'¢?'1fi.oge.cp(/\'1fi)) 0.041 0.241 0.040
(0.007) (0.015) (0.016)

49

continued on next page

Table 2.2 (continued)

 

 

Estimated Process

APE Estimates

 

.3

 

 

 

 

 

cilari ~ N(‘ti’5'() + 0911?,- + 1122:1742 + 053271‘ ,oge$p(/\'1fi)) 0.043 0.252 0.043
(0.011) (0.017) (0.022)
Simulated APE 0.050 0.302 0.050
(D) T=25, N=500 1‘1 (1‘2 yt_1
7.1.1:,- ~ M010 + 41.11.15,.03) 0.045 0.272 0.045
(0.004) (0.006) (0.008)
(isz‘ N N ('U'I'U + @5413 + $217.22 + 1423.17.13.03) 0.046 0.271 0.045
(0.007) (0.010) (0.011)
(3)2") ~ N(t"’0 + t’i'irfi. ogea'p(/\’1:f,j)) 0.042 0.252 0.042
(0.003) (0.007) (0.008)
7,-1.1. ~ NM) + 0,-3.5,- + 1.1217,? + 4:53.133, ogezrpO/lffi) 0.042 0.253 0.044
(0.006) (0.008) (0.015)
Simulated APE 0.048 0.290 0.048

 

 

50

continued on next page

Table 2. 2 (continued)

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

 

 

 

 

 

(iii) I’M/1:0 = WIN-'1?) = (1)020 + 0,1171”. + 02(3)
(A) T=5, N=100 .771 :1'2 gt_1
CAT". N NM) + iii” 03) ((05118) ((0204219) (0&1)
“ll-'14 N N (d0 + 'l/"i-‘F i + $.02:th + ills-133» 0'3) ((000429) 81203233) (%%%%)
. O . . '
villi ~ Next, + up). 03c1.'p(,\’1~f.i)) (0).?)421” (%.2)%26) (%%%20)
(il- lil'i ~ /\/v(-(;',»0 + (lift + 022172.? + ((33179, (7301.74 Alli”) ((3%5358) ((333155) ((513%)
Simulated APE 0.037 0.219 0.037
(B) T=5. N=500 :171 .‘l'2 l/t-l
(fl-[13 ~ N010 + 011.13. 03) 0.042 0.242 0.051
(0.011) (0.016) (0.031)
(’il-l'i N N050 + (lift tel/(2:132 + ‘4’}31713J73) (%%42(i) ((23259) ((0%?)
. . . O
(3)17. N N050 + 5933:). 03.415035)» Egg) ((3.35%) ((50331; )
Oil-Ti N N (1,30 + villi—72: + E92132 + 'lz>:337i3.03€$l?()\'13"71)) ((39%?) ((3%4252) ((0%7891)
Simulated APE 0.037 0.219 0.037
(C) T225, N=100 21 2‘2 yt_1
(71-11,- ~ Mayo + 01.13, 03) 0.046 0.272 0.047
(0.009) (0.016) (0.020)
ml ~ W0 +4417.- + ””2""? + ”"317"; ”3) ((0%19) ((2314146) (($352)
7.7)..) ~ My,“ +.),/1.5).03.41:,)(/\’113)) ((30385) (0)2020) (%%4226)
. . O .

continued on next page

Table 2.2 (continued)

 

 

Estimated Process

APE Estimates

 

3

 

 

 

 

 

cilrzrzj ~ N(z."10 + 9’1 f) + 059122 + '0'2313' ,ogexpfl’lffi) 0.044 0.255 0.049
(0.013) (0.019) (0.035)
Simulated APE 0.050 0.302 0.050
(D) T=25. N=5()0 271 2‘2 yt_1
(fl-Ia?) ~ .Nwo + 't'L/‘iflfi. 03) 0.046 0.273 0.047
(0.005) (0.008) (0.014)
(3)4,- ~ N(t~~0 +4113 + 02:5)? +-¢317.,;3,ag) 0.046 0.272 0.049
(0.007) (0.011) (0.017)
(-,j|:1.‘l- ~ N(t"() + 13117). 031%.17])(,\’1 171)) 0.039 0.230 0.044
(0.010) (0.053) (0.022)
Cilri ~ N(0’"0 +4531?) + 17,213? + t/33372j3,oge23p()\’1fi)) 0.043 0.255 0.029
(0.007) (0.012) (0.029)
Simulated APE 0.048 0.290 0.048

 

 

Quantity in parentheses is standard error.

Estimates in bold are significant. at the 57 level & simulated equals estimated APE (italics 10%).

52

Table 2.3: Dynamic Simulation, p = .75

DGP c.,-|;1‘.i~ 07(po + (finch/m + Man-2))

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

 

 

 

 

 

(i) /’(!/2fi(’1|-‘I*i~'fi) = ”((170)
(A) T=5, N=100 171 £132 yt_1
mm,- ~ N(t/’0 + (vi/1.13.53) 0.035) (0.211) (0.100)
(0.017 0.026 0.028
cm) ~ NM) + 0313- + 02.5)? + 03333, 03) (0.035) (0.208) (0.099)
0.016 0.021 0.034
cilz‘ri ~ N(t="10 + (4211",;ogeafp()\'1f,;)) (00 214213) ((212)4205 ) (($16,262)
Cil-Ti N N (‘00 + ‘L’V'ifz‘ +1152??? + U333. 03833190311170) ((8%?) ((326%) ((3102932)
Simulated APE 0.037 0.219 0.109
(B) T25, N=500 2:1 162 yt_1
cum-z- ~ NM) +'t/«‘/1.’17,j.03) (0.035) (0.212) (0.106)
0.007 0.012 0.011
(Jilly: ~ No.10 + 0’14?) + 2.21233? + 023333, 03) (0.035) (0.208) (0.102)
0.008 0.010 0.017
an) ~ M00 +1ng5,-.03....1.-;)(A'1.L7.)) (00%3290) ((324119) $102673)
cilrrz- ~ N(-t”() + wig?)- + 71523:? + t?3f,;3, 03721-120425») ((300429) ((2).?)(3) ((310052)
Simulated APE 0.033 0.196 0.098
(C) T=25. N=100 2:1 12 gm
0,: (4;,- ~ Nam + 04.51.03) 0.039 0.228 0.113
(0.008) (0.019) (0.013)
l<> (2.35 ta. (2.31..

continued on next page

Table 2. 3 (continued)

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

c.,j|a:,_j ~ N(t""0 + '0'L"1f,;.o(21e:rp(A’1f,-)) 0.044 0.263 0.131
(0.009) (0.015) (0.018)

(‘il-Tz'. ~ N(t"0 + @5711?) + @2232 + 0'32?) ,UgeIp(/\,127,j)) 0.043 0.253 0.125
(0.009) (0.017) (0.023)

Simulated APE 0.050 0.299 0.149

(D) T=25, N=500 .771 1‘2 yt-1
alas,- ~ N(t’10+«t:~(.f.1.02) (0.038) (0.228) (0.114)

0.003 0.009 0.006

cilzri ~ NM) + 0,1117.- + 02:5,? + 1.93.5.- 0.038 0.225 0.113
(0.007) (0.011) (0.006)

cila‘,’ ~ Nam + t/:§5..03(axp(1’115,:)) 0.044 0.263 0.131
(0.004) (0.007) (0. 008)

(film; ~ N(-t/*() + 01:17. + 0221712 + 1123:171- oge$p(/\'1:f,t)) 0.043 0.253 0.126
(0.008) (0.009) (0.014)

Simulated APE 0.048 0.288 0.144

 

 

continued on next page

Table 2.3 (continued)

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

 

 

 

 

 

(ii) [’(3/70 = lllzw Ci) = W140”)
(A) T=5, N=100 :1'1 .772 gt_1
cm,- ~ N(t"vo+-0’1’1f,-.o(21) 0.034 0.206 0.101
(0.015) (0.024) (0.027)
(Tiliri ~ N(t/’0 + ((21.73 + '09ng + 653.133. 63) 0.033 0.202 0.098
(0.015) (0.020) (0.034)
(Bil-Ti ~ N(z.’"’() + 05123. alga/203113)) 0.039 0.233 0.114
(0.020) (0.022) (0. 060)
(:zjlzrz- ~ N(t”() + "(lift + '0’523742 + 0’23371-3,oge:l'])(/\'1;f,j)) (0.042) (0.236) (0.128)
(0.023) (0.037) (0.098)
Simulated APE 0.037 0.217 0.109
(B) T=5, N=500 3:1 3:2 yt_1
cjlaf- ~ M00 + (9'56?) 0.034 0.207 0.105
I I 1 1 (I
(0.006) (0.011) (0.012)
7:27).” ~ N040 + 'Ulfifzj + 02.132 + 43.13-13.63) 0.034 0.202 0.101
(0.007) (0.009) (0.017)
cilzri ~ N(t”() + tirifi. 03e2.'p()\’1f,j)) 0.041 0.235 0.125
(0.016) (0.018) (0.058)
cilzrl- ~ N090 + ti'ifi + 0237.)? + -u';3f,j3, oge:vp()\'lf,-)) 0.042 0.242 0.143
(0.018) (0.032) (0.102)
Simulated APE 0.033 0.196 0.098
(C) T=25, NZIOO (131 $2 yt—l
c241,- ~ N(t’¥'() + 't-L'fifzj. 03) 0.038 0.227 0.113
(0.008) (0.020) (0.013)
al.,-,7 ~ N(t-"’() + 04.13 + ([12:13? + 723.133, 63) 0.038 0.226 0.111
(0.008) (0.020) (0.013)
('l'll‘i ~ N(1.«"'() + 133.174. oge:1:p(/\’1fi)) 0.044 0.262 0.130
(0.008) (0.015) (0.018)

continued on next page

Table 2.3 (continued)

 

 

 

 

 

 

 

Estimated Process APE Estimates
e,- LT?" ~ N(L"() + 0’11}: + tirgfl-Q + 113133» 03e$p()\'1f;)) 0.043 0.252 0.126
(0.009) (0.016) (0.025)
Simulated APE 0.050 0.299 0.149
(D) T=25, N=500 1‘1 (1‘2 yt_-1
(71m) ~ NM) + 4.4.5,, 03) 0.038 0.227 0.114
(0.003) (0.009) (0.006)
Cilm,’ ~ N(t"'0 + ukifi + 1.22.??? + 11":3233. 03) 0.037 0.225 0.113
(0.005) (0.010) (0.007)
(fl-[271: ~ Nwo + '0’.-'(:17,j.ogerp()\'1:17i)) 0.044 0.262 0.132
(0.004) (0.007) (0.008)
eilzri ~ N(2.«’!0 + U41") + 2.521312 + #:31330361190’11170) 0.042 0.252 0.126
(0.006) (0.008) (0.015)
Simulated APE 0.048 0.288 0.144

 

 

continued on next, page

Table 2. 3 (continued)

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

 

 

 

 

 

(iii) P0110 = 1|:z:,-,c,~) = (“00 + T2313 + 02%)
(A) T=5. N=100 .171 .172 gt_1
ez-lzrz- ~ Moo + 0113-. 63) 0.035 0.207 0.102
(0.017) (0.030) (0.029)
al.,-z- ~ N(</v() + 0’16,- + (02.13? + 03.133. 63) 0.033 0.209 0.094
(0.022) (0.027) (0.070)
eila‘i ~ N000 + #91172:- 036:1‘])()\,1f?j)) 0.038 0.192 0.125
(0.028) (0.053) (0.072)
(:zjlafl- ~ N(u"'0 + ((1.171 + ’U')2;F,j2 + 033.133, 03(’IIT])(/\,11fi)) 0.052 0.257 0.152
(0.036) (0.043) (0.103)
Simulated APE 0.037 0.217 0.109
(B) T25, N=500 1171 1'2 lit—1
pip-l- ~ M00 + 01.23. 03) 0.035 0.208 0.108
(0.008) (0.021) (0.017)
(aim ~ M00 + 0,-1.1:- + 0:203? + 03.5.3, 63) 0. 0.34 0.206 0.107
(0.019) (0.020) (0.064)
cl-lx,‘ ~ Nm, + 01.5,. 03(i.77[)()\’1f,j)) 0.041 0.196 0.129
(0.020) (0.055) (0.070)
eilrrl- ~ N(L7'() + 'c'z'ifi + 0922,7212 + 01:3172-3, ogemp()\'1:f,-)) 0.045 0.226 0.109
(0.026) (0.017) (0.074)
Simulated APE 0.033 0.196 0.098
(C) T=25. N=100 171 1‘2 yt_1
(rim ~ Nun + 01113:. 63) 0.039 0.228 0.113
(0.008) (0.021) (0.014)
nip-i ~ Nana + 0.11.13- + «02.132 + 03.79, 63) 0.037 0.225 0.113
(0.008) (0.022) (0.013)
eilrti ~ N(l('() + 't-f'r’lfi. agerpbvlffl) 0.036 0.210 0.109
(0.012) (0.055) (0.035)

continued on next page

57

Table 2.3 (continued)

 

 

 

 

 

 

 

 

Estimated Process APE Estimates
n<<>> 333 as 033
Simulated APE 0.050 0.299 0.1498
(D) T=25. N=500 £131 £172 yt_1
elzm ~ N(‘U"0 + 493.1%. 03) 0.038 0.227 0.114
(0.005) (0.012) (0.009)
Glut. ~ N(0’v()+-e’1.f,¢ “02.1372 + 03133. 03) 0.038 0.225 0.114
(0.005) (0.011) (0.006)
'(W 6-35 6-33 6-33
elm,- N N (““0 + Vii? + 34:22.2 + #3133 0361100137”) (%'%0%) (005152) ((00331)
Simulated APE 0.048 0.288 0.144

 

 

Quantity in parentheses is standard error.
Estimates in bold are significant at the 5% level & sin‘ullated equals estimated APE (italics 10%).

Table 2.4: Dynamic Simulation. e.,-|.r.,-, y“)

DGP ('Ijlil‘z' N lV(L’:’() + $313,

03(A0 + All 1712))

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

 

 

p = .25
(A) T=5. N=100 1‘1 12 .Ut—l
Cil‘vb 3M) N N010 + 013410 + (12172303) 0.040 0.233 0.036
(0.018) (0.021) (0.039)
('«ill'i» the ~ N010 + (113/20 + 02173: + 031712 + 041373. 0'3)
0.049 0.259 0.048
(0. 027) (0.032) (0.069)
_ " 2
(‘il417isl/i0 "’ N010 + all/i0 + (121% Ug€TP(/\I117i )) 0050 0.253 0.064
(0.038) (0.052) (0.111)
r _ _ _ ' - 2
(7.1.1:), 3M) ~ /\I (00 + (Ilyl‘o + (121,: + (1313.12 + (14:133. ogexpfl’lasi ))
0.056 0.262 0.065
(0.041) (0.042) (0.096)
Simulated APE 0.037 0.219 0.037
(B) T=5. N=500 1’1 T2 yt—l
(311124120 N N(00 + all/2'0 + 02ft~ 03) 0.041 0.235 0.038
(0.007) (0.009) (0.017)
(ilIi 1110’” N(”0 + all/i0 + 0211'. + (13171.2 + 041713103.)
0.049 0.279 0.046
(0.011) (0.017) (0.030)
, _ ‘ 2
Gliri. .1120 "V W (00 + 0’19“) + 02sz 036179931131“ )) 0.041 0235 0.038
(0. 007) 2.(0 009) (0.017)
(32111211100 N N((!() + ”IMO + (12.17," + (13.732 + (.14.!"3 .30 (331])(Ailfi2))
0.041 0.235 0.038
(0.007) (0.009) (0.017)
Simulated APE 0.033 0.198 0.033

 

 

59

continued on next page

Table 2.4 (continued)

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

 

(C) T225, N=100 171 .172 yt_1
“ilJ’ia t/iO N N(”-0 + ”1WD + ”21.723 ”221) 0.041 0243 0.040
(0.008) (0.015) (0.019)
Cill‘i- 3110 N N(00 + 013/10 + 021727 + 031742 + 045513, 03)
0.049 0.279 0.046
(0.011) (0.017) (0.030)
_ - 2
(“zilfz'a yiO ~ MOO + alt/i0 + 02117-5 0361‘P()\'1172' )) 0.043 0.244 0042
(0.017) (0.038) (0.048)
_ _ _ " 2
Gil-731'. y2'0 N N ((310 + 013/20 + 02174 + 035812 + 04-75273, 065$P(/\’133i ))
0.046 0.272 0.047
(0.014) (0.026) (0.049)
Simulated APE 0.050 0.302 0.050
(D) T=25, N=500 $1 12 yt—l
Ciliris 31130 N N010 ‘1' 013/270 + 0247723 (73) 0.039 0.240 0.040
(0.006) (0.011) (0.014)
("2'le 3110 ~ N(00 + alt/2'0 + 02172“ + 0317-42 + 041103903)
0.049 0.276 0.045
(0.009) (0.015) (0.030)
. _ — 2
CilT‘i» 3110 N 0’ (“'0 + alt/i0 + 0212‘: 038119031174 )) 0.047 0.248 0_040
(0.010) (0.033) (0.019)
. - 2
(’il-Ti» ”to ~ N((10 + “ll/i0 + (12.13 + (1311722 + (14:133. (Igearpbvlari ))
0.043 0.280 0. ()4 7
(0.010) (0.019) (0.025)
Simulated APE 0.048 0.290 0.048

 

 

continued on next page

60

Table 2.4 (continued)

 

 

 

 

 

Estimated Process APE Estimates

p = .75
(A) T=5, N=100 :171 9:2 lit—1
Cilfzh yiO N N(00 + 013/10 + 021513 03) 0.036 0.208 0.100

(0.017) (0.021) (0.035)
(1)1723!)sz ~ N((10 + (11)/)0 + (12.17) + (13117232 + (14461320121)

0.046 0.243 0.117
{0.025) (0.030) (0.061)

_ — 2
01173» 910 ~ N(00 + 0191.0 + 02171: 0121611701172“ )) 0.042 0.211 0,103
(0.031) (0.036) (0.085)
_ _ _ - 2
fill—Pi» 920 ~ N010 + 013/10 + 0211' + 0311312 + (141331 03141120310.- ))

0.054 0.248 0.115
(0.035) (0.041) (0.086)

 

 

 

 

Simulated APE 0.037 0.217 0.109
(B) T=5, N=500 1'1 2:2 yt_1
Cilf'i» 31130 N N010 ‘1' (113110 ‘1' 02171110121.) 0035 0.210 0.106

(0.007) (0.009) (0.017)
('i|;1:,i.gl-0 ~ N((10 + ”ll/10 + (12:13 + (13.1712 + (14:17.13. (73)

0. 048 0.249 0.120
(0.009) (0.006) (0.016)

. _ — 2
CilfITj. 3111) N N (00 ‘1' 013/10 + 024172? 01216-111904“ )) 0.035 0.208 0.105
(0.008) (0.016) (0.019)
_ _ _. - 2
Cill'i» 3110 N N010 + a13/10 + 0242' + 031312 + 0'441310121835190'1331 ))

0.035 0.208 0.105
(0.008) (0.016) (0.019)

 

Simulated APE 0.033 0.196 0.098

 

 

continued on next page

61

Table 2.4 (continued)

 

 

Estimated Process

APE Estimates

 

 

 

 

 

 

 

(C) T=25, N=100 1‘1 172 yt-1
"il-Tis ”10 N N(”() + ”ll/2'0 + ”24170 0121.) 0.030 0213 0.106
(0. 008) (0.016) (0.017)
Cilin. yin ~ lV((')() + (113/10 + (.1in + 031122 + (1.4133. 03)
0.046 0. 205 0.129
(0.012) (0.018) (0.026)
_ - 2
Gil-Ta .910 N N(00 + (113/10 + 02100391190'1171 )) 0.037 0.210 0.101
(0.013) (0.025) (0.039)
_ _ _ - 2
(i'-1|ilfi~y1() ~ N010 + 013/10 + 02171 + 031712 + 04113, 039171003131 ))
0.044 0.262 0.126
(0.012) (0.023) (0.041)
Simulated APE 0.050 0.299 0.149
(D) T=25. N=500 :1-1 2‘2 91-1
(2)"le ”11) N N(”() + ”211/210 + ”24172.1 0(2).) 0036 0.214 0103
(0.002) (0.007) (0.006)
('1|$1~.U10 N NW) + a1.1/10 + 021:1 + 031712 + 04-7713~ 03)
0.044 0.267 0.131
(0.003) (0.010) (0.020)
_ — 2
GIT-1, .010 N N010 + (11910 + 0211.0121FTPW1371 )) 0.033 0.225 0109
(0.009) (0.036) (0.020)
, _ _ _ - 2
"ilJ'ie .Uz'0 ~ ”(”0 + ”1.1/10+ ”2411' + "34112 + (14-1’13103"J'I'(/\'1372fi ))
0.045 0.273 0.124
(0.006) (0.015) (0.022)
Simulated APE 0.048 0.288 0.144

 

 

Quantity in 1.)arentheses is standard error.

Estimates in hold are significant at the 5% level & simulated equals estimated APE (italics 10%).

62

2.3 Application

The empirical application undertaken is a model of household brand choice. In the context
of household brand choice 2‘ = l, N represents the household observed, t = 1, ...,‘T is the
purchase occasion observed, c,- are unobserved time invariant household characteristics, 2,,
are observed explanatory variables, and Mt is an indicator variable for whether or not a
specific brand was purchased. Only two brands can be considered with the binary model
previously specified, but it can easily be, extended to encompass all brands with a multinomial
specification.

In the dynamic model of brand choice, households are assumed to maximize lifetime
expected utility over two goods: the product under study and a composite good. The product
studied is comprised of multiple brands. Under standard assumptions, microeconomic theory
shows that households will choose the brand, j, that provides both the lowest quality adjusted
price and the highest expected utility. Quality is assumed to depend on observed product
and individual characteristics, whether the brand was consumed in the previous period (i.e.,
habit effects), unobserved time invariant characteristics, and random shocks, fit- With only
two brands, if utility is quasilinear in the composite good, the utility of the good studied
is logarithmic, and a household chooses to consume only brand j in period t, then utility
maximization subject to a household budget constraint with prices, (Ijitv leads to a brand

choice decision rule given by

ylit = 1 {31.151 - Z2it52 + "riym—i - 723/2i,t—1 — 17101121) + whim) + Ci + €1it- 622:: 2 0}

Redefining the variables such that (Lit = zlit“22ita Pit = l'n,(q2,jt)—ln(q1,-t), and (fit 2 Clit”€2it

the decision rule can be rewritten as
ya = 1{/31“z't + [1.1/uh + £2921. + Ci. + 8a 2 0} (23)

63

the decision rule can be rewritten as
ya = 1 {/3131 + wept—1 +5791: + (11‘ + eit Z 0} (23)

This equation now mirrors that of (1). Thus, the underlying microeconomic theory leads to
a. simple econometric model of brand choice by assuming that the probability of purchasing
a given brand follows some well-behaved cumulative distribution function as discussed in

Section 2.7

2.3.1 Data

AC. Nielsen data on yogurt purchases for a sample of households in Sioux Falls, South
Dakota from 1986 to 1988 are used in this investigation.8 A.C. Nielsen chose Sioux Falls as
a representative market because its population demographics resemble those of the United
States as a whole and because they were able to monitor purchasing at. all major grocery
stores in that area. Sample households were issued magnetized cards that were presented
at the grocery store when checking out. All of the households’ purchases were then scanned
and the information provided to the data collection agency. In addition, the agency collected
household demographic information and information on weekly marketing from grocery lo-
cations.

The full data set consists of 1318 households and 17,679 purchase occasions. Each ob—
servation represents a purchasing occasion and contains a household identifier, the date of
purchase, the brand of yogurt purchased (Yoplait, Dannon, Nordica, or other), value per
ounce of store or mamifacturer coupons (if used), price per ounce before applying coupons,

indicators for whether the brand had a special display or advertising feature at the time of

 

7See Chintagunta. et al (2001) for a detailed discussion of the theoretical model of dynamic, discrete

brand choice.

”I would like to thank Ekaterini Kyriazidou. Department of Economics. UCLA for providing the. data.

64

purchase, and data on household characteristics. Nordica has the largest market share for a
single brand with 4,739 purchases followed by YOplait with 3,813 and Dannon at 1,834. How-
ever, the majority of yogurt purchases are made of “other” brands, including store brands,
which comprise 7,293 purchases together. In order to obtain the largest possible sample,
Nordica and Yoplait will be the brands considered for the analysis. Using this data, vari-
ables for price per ounce in log differences and variables for special displays and features were
created to correspond with the variables pit and 11’it described at the beginning of section
2.3. The variables for special displays and features are defined as the difference in indica—
tor variables for whether Yoplait or Nordica were featured or on special at the time of the
purchase occasion.

Since some of the specifications to be considered include a lagged dependent variable,
households without at least 3 purchases were dropped from the data set. This creates an
unbalanced panel in which the minimum number of household observations is 3 and the
maximum is 305. The household with 305 observations appears to be an outlier since the
number of purchase occasions is roughly continuous between 3 and 45 per household with
no other observations between 65 and 305. Here t = 0 refers to a household’s first purchase
occasion, and the number of time periods in which a household is observed depends on
the number of purchase occasions. In addition, if a household purchases a brand other
than Yoplait or Nordica, all subsequent purchases are treated as a new household with the
next purchase occasion in which Yoplait or Nordica is purchased defined as t = 0. These

modifications lead to a final sample with 1,398 households and 5,618 observations for t _>_ 1.

65

2.3.2 Estimation

Standard software, such as Stata, can be used for estimation of some models described
in section 2.1, but estimation of these models under most assumptions on the conditional
density of the initial condition, f (y()|z.c), or unobserved heterogeneity, h(c|o), generally
requires special programming to allow for specification of the densities, integration over
the distribution of the unobserved effect, and calculation of the average partial effects with
standard errors. Results for the static brand choice model are shown in Table 2.3.2. All
of the estimates obtained are. statistically significant at the 5% level. The magnitude of
the differences between parameter estimates across unobserved effects specifications is 0.02
to 0.06 and 0.0001 to 0.006 for APE estimates. The APE estimates indicate that a. 1%
change in the relative price of N ordica or Yoplait leads to a 30% change in the probability
of purchasing a. particular brand on average. Also, being on a special display or having an
advertising feature when the other brand does not increases the probability of purchasing that
brand by 8 or 9%. respectively. The estimated effect of a price change on brand choice seems
extremely high unless consumers view different brands of yogurt. as very close substitutes.
As in the simulations, the results obtained in the dynamic case show more variance
across different specifications. However, as with the static. case, APE estimates vary by
a smaller amount than parameter estimates. The results for P(y,~0 = 1|;r,,c,;) : (Perl-OB)
and P(y,~0 = 1|;r.,;,c,-) : (Mm) + 7/133,- + T]2C‘,f) are shown in Table 2.6. Estimates from the
dynamic model indicate a large variance in consumer price sensitivity. After controlling for
past purchase behavior, a 1% change in the relative price between brands leads to a change
of 8%-18% when P(y,-0 = lll‘i, c1: ) = (I)(;r,:0,d) and of approximately 30% in the probability
of purchase when P(y.,-0 = 1|:1:,-_. (.),) = (1)070 + 7/133,- + 772e,). Habit effects are shown to be very

strong in brand choice decisions. Unlike the effect of price, there is little variation across

66

Table 2.5: Static. Brand Choice

 

 

Estimated Process Coeff. Estimates

APE Estimates

 

(1) (fill? N Altai/‘0 + 117113903)

 

 

 

 

 

 

 

 

 

 

Price -0.998 -0.305
(0.138) (0.009)
Display 0.255 0.078
(0.098) (0.032)
Feature 0.300 0.092
(0.052) (0.022)
(2) C‘zilJT-z' N N (We + v’JIf-zi + $2132 + ¢"’:3fzf3- 03)
Price -1.060 -0.310
(0.151) (0.011)
Display 0.285 0.084
(0.112) (0.035)
Feature 0.314 0.092
(0.057) (0.025)
(3) C'il-Ti ~ N090 + till-2030313179150)
Price —0.997 -0.304
(0.138) (0.009)
Display 0.256 0.078
(0.098) (0.032)
Feature 0.300 0.092
(0.052) (0.022)
(4) (all) ~ Ntlé’e + vi‘j'fi + Ul’2fi2 + #:3133- Grimm/V1450)
Price -1.056 —0.309
(0.151) (0.010)
Display 0.280 0.082
(0.112) (0.035)
Feature 0.316 0.092
(0.057) (0.025)

 

Note: All estimates are significant. at the 5% level.

Quantity in parentheses is standard error.

67

specifications in the effect of previous brand choice on current brand choice. Purchasing
a particular brand in the previous period leads to an estimated increase in the probability
of purchasing the same brand of 36-38%. The effect of a special display on brand choice
is essentially unchanged from the static case, varying only between 8 and 10%, and the
probability of purchasing a brand when it has an advertising feature is estimated to be 12-
16% in the dynamic case. The results from Table 2.6 show that there is substantial variation
in the APE estimates across specifications for 312-0 for some variables. In addition, there
is some variation across specifications for unobserved heterogeneity. The variation is small
across different specifications of the mean and larger across specifications of the variance.
Table 2.7 shows the estimates obtained by specifing the distribution of the unobserved
effects conditional on the initial condition and the exogenous regressors. The APE estimates
for lagged brand choice are not statistically significant, and the other estimates are similar
to those obtained by the other models. However, the variance of the parameter estimates
is larger than in Tables 2.5 or 2.6. The APE estimates indicate that a 1% change in the
relative price leads to a 29-37% change in the probability of purchasing a particular brand.
Also, being on a special display or having an advertising feature when the other brand does
not increases the probability of purchasing that brand by 6—9% or 9—11%, respectively.
Thus, the results of the empirical application mirror those obtained in simulations. Pa-
rameter estimates tend to vary more across specifications of the unobserved effects than
APE’s for both the static and dynamic models. APE’s are relatively insensitive to the spec—
ification of unobserved effects but tend to vary by a larger amount with the specification of

initial conditions.

68

2.4 Discussion

The results of the simulations and empirical analysis suggest that average partial effects are
relatively insensitive to the specification of unobserved heterogeneity, and simple, flexible
specifications do a good job of estimating quantities of interest even when the true distribu-
tion of the unobserved effects differs from that estimated. However, average partial effects
appear to be considerably more sensitive to the specification of initial conditions, and this
problem does not seem to be easily solved by choosing a more flexible specification for the
distribution of the initial condition. This concern can be mitigated to an extent by using
the approach of Wooldridge (2000a) though.

Ongoing research on this topic includes simulation evidence on the performance of APE
estimates, specified as previously described, when the unobserved heterogeneity is actually
non-normally distributed. For example, the data generating process for the conditional
distribution of the unobserved effect could be specified as a mixture of normals where both the
mean of the conditional distribution and the mixing probabilities depend on the exogenous
regressors. In addition, efforts are being made to improve the computation time of the
models described in the paper. Substantial reductions in the computation time are likely to

require a change to estimation by simulation methods.

69

Table 2.6: Dynamic Brand Choice, (3,- 1r,

 

 

 

 

 

 

 

 

 

 

 

Estimated Process Coeff. Est. APE Est. Coeff. Est. APE Est.
(1) Gil-Ti N Nttf’o + and)
Lagged Brand Choice 1.792 0.378 1.830 0.377
(0.047) (0.054) (0.048) (0.095)
Price —0.842 -0.177 -1.539 -0.317
(0.152) (0.022) (0.157) (0.056)
Display 0.392 0.082 0.447 0.092
(0.108) (0.025) (0.109) (0.032)
Feature 0.695 0.146 0.588 0.121
(0.058) (0.026) (0.058) (0.036)
(2) Willi-'2' N N(‘¢”0 + '¢"’i-’I_’-i + $211712 + #31713. 03)
Lagged Brand Choice 1.750 0.362 1.783 0.363
(0.055) (0.060) (0.056) (0.105)
Price -0.831 -0.172 -1.520 -0.309
(0.176) (0.025) (0.181) (0.064)
Display 0.406 0.084 0.493 0.100
(0.124) (0.028) (0.125) (0.038)
Feature 0.706 0.146 0.563 0.115
(0.066) (0.030) (0.066) (0.040)
(3) CiISF-i N Nfd'o + ’v5’ifi~0361‘19()\'1fi))
Lagged Brand Choice 1.772 0.375 1.791 0.372
(0.047) (0.037) (0.045) (0.077)
Price ~0.392 -0.083 -1.243 -0.108
(0.151) (0.030) (0.162) (0.054)
Display 0.466 0.099 0.476 0.111
(0.108) (0.022) (0.128) (0.036)
Feature 0.771 0.163 0.579 0.149
(0.058) (0.020) (0.062) (0.035)

 

 

 

70

continued on next page

Table 2. 6 (continued)

 

 

 

 

 

 

Estimated Process Coeff. Est. APE Est. Coeff. Est. APE Est.

(4) ‘72Tl51'i ~ N(U”() + £9,117.) + ‘u”21f,j2 + 11,3133, (7(2,c:17p(/\’1:fi))

Lagged Brand Choice 1.731 0.360 1.748 0.369
(0.055) (0.041) (0.051) (0.081)

Price -0.379 —0.079 -1.133 -0.102
(0.173) (0.034) (0.144) (0.057)

Display 0.482 0.100 0.451 0.103
(0.124) (0.025) (0.116) (0.040)

Feature 0.783 0.163 0.557 0.141
(0.066) (0.023) (0.062) (0.038)

 

 

 

Note: All estimates are significant at the 5% level.

Quantity in parentheses is standard error.

Column (1) P(yz:0 = 1w.) = (crew/3)

Colunm (iii) [)(312'0 = 1‘17), (3,“) = <I>(r)0 + 0,117.) + 172(1))

71

Table 2.7: Dynamic Brand Choice, c,|:r,-. y“)

 

 

Estimated Process Coeff. Estimates APE Estimates

 

(1) (Hill‘s .1170 N Nfee + 011/270 + 0211723 0'3)

 

 

 

 

 

 

 

Legged Brand Choice 0.439 0.090
(0.760) (0.126)
Price -l.801 -0.371
(0.158) (0.073)
Display 0.411 0.085
(0.109) (0.019)
Feature 0.532 0.110
(0.058) (0.009)
(2) (filial/10 N N(00 + 013/110 + 02ft + 03152 + 01415-13. 03)
Laggcd Brand Choice 0.421 0.085
(0.858) (0.143)
Price -l.775 -0.360
(0.183) (0.080)
Display 0.442 0.090
(0.126) (0.021)
Feature 0.548 0.111
(0.066) (0.009)
. _ - 2
(3) Gila/'1': 3120 N Nfflo + (113/70 + 02% (736171203132: ))
Lagged Brand Choice 1.339 0.092
(0.664) (0.196)
Price -4.215 -0.289
(0.128) (0.055)
Display 0.910 0.062
(0.086) (0.031)
Feature 1.271 0.087
(0.048) (0.026)

 

72

continued on nezrt page

Table 2. 7, continued

 

 

 

 

 

Estimated Process Coeff. Estimates APE Estimates
- _ _ “ 2

(4) Gill’s the N N010 + 013110 + 02312”. + 03132 + a4$i3~036$P(/V1$i ))

Lagged Brand Choice 1.390 0.102
(0.691) (0.194)

Price -4.034 -0.297

' (0.138) (0.067)

Display 0.989 0.073
(0.096) (0.035)

Feature 1.239 0.091
(0.051) (0.030)

 

 

Note: Bold type indicates significance at the 5% level.

Quantity in parentheses is standard error.

73

CHAPTER 3

A Score Test for Heteroskedasticity in Dynamic Latent Variable

Models

3. 1 Introduction

In linear models heteroskedasticity does not affect consistency or unbiasedness of estimators
and can easily be corrected for in standard errors in order to perform inference. However, in
many nonlinear models the presence of heteroskedasticity is of greater consquence because
it changes the functional form of the estimator. For this reason, the ability to test for
heteroskedasticity in nonlinear models is important. The difference between the two cases

can be seen by first observing that given a linear panel data model
Eftlitlfzt, 0i) = aTit/3 + (’2' (1)

with the partial effect of :1: on [3 given by

3Efyitl17i~ 02:) =
(91‘,-

 

a (2)

does does not depend on any features of the distribution of the unobserved effect, c.,-. In
addition, it is simple to consistenly estimate )3 by fixed effects, as discussed in chapter 1. In
some. cases, such as a raiidem-coefficient model in which there are individual specific slope.
coefficients

Etyalirz', I32:- C-i) = Stair-132‘ + Ci. (3)

it is not possible to estimate the individual parameters, 13,-, in the traditional panel setting

with small T. However, under the assumption that E (ass-t) = E (23,) for .r'jt = 13a — 17,, it

74

is still possible to consistently estimate average partial effects using FE. When a nonlinear
model is considered though, such simple solutions are no longer available. The simplest case

is that of a panel probit model, given by
Pfl/a = Hie-ft) = (pfilfitfi + 0i) (4)

Again. the partial effects conditional on c,- will not depend on any features of the dis-
tribution of the unobserved effect, but, without knowing a value of the individual specific
effect at which to evaluate the partial effect, these quantities can not be obtained. The APE,
however, will depend on the entire distribution of c,- given 1‘,- in general. The importance of
the assumptions made about the form of this distribution for consistently estimating APEs
was shown in chapter 2. For cases in which ignoring potential heteroskedasticity in the un-
observed effect may affect estimation of the APEs having a test for heteroskedasticity which
is easily implemented could be important.

General test. procedures are difficult to construct for dynamic, nonlinear panel data mod-
els with unobserved heterogeneity though. As explained by Wooldridge (2005b), it is not
possible to identify the effect of covariates on the variance, 8Var(y|;r. c) / (9:13, averaged over
the distribution of the unobserved effect. as it is the average partial effect. Consequently, to
learn anything about the conditional variance it must be modeled directly.1 Thus, tests for
heteroskedasticity in limited dependent variable models are generally obtained on a case-by-
case basis under specific assumptions about the structure of the unobserved effects. This
paper will focus on testing the null hypothesis of homoskedasticity through exclusion restric-
tions using a score test. This simplifies estimation by avoiding the need to estimate the model

as heteroskedastic, which would be necessary using Wald or likelihood ratio statistics. In or-

 

1See W'ooldridgc (2005b) for further explanation and examples using Poisson regression and duration

models.

75

der to accomplish this. a parametric form for the unobserved heterogeneity will be specified.
Throughout the paper. normality of the distribution of the unobserved effect and a given
specification for its conditional mean are maintained. Clearly, one can question the validity
of these assumptions and their effect on the proceedure. As many authors have noted, the

2 How-

validity of standard test statistics will depend on the auxiliary assumptions made.
ever. the benefit of this approach is that, by assuming normality and formulating the test.
through exclusion restrictions, estimation of the restricted model is possible via traditional
random effects methods for both the probit and tobit. In addition, the specification of the
conditional mean can be made more flexible without any substantive changes to the test
proceedure as long as it is a function of the exogenous covariates and the initial condition
and is linear in parameters. In theory, the assumption of normality of the unobserved effect
could also be tested either before or after testing for heteroskedasticity, but this is beyond
the scope of this paper.

The remainder of the chapter proceeds as follows: Section 1 constructs the score test
for the dynamic, panel probit model. Section 2 extends the test to the case of a fractional
dependent variable. and section 3 presents applications of the test to both the dynamic

probit and fractional response models. Section 4 concludes.

 

2800 White (1981, 1989). Bera and Jarque (1982), Pagan and Hall (1983), Godfrey (1987), ct al.

76

3.2 Score Test for Heteroskedasticity in the Dynamic

Probit Model

The dynamic, panel probit model will be maintained as the model of interest throughout

this section. Consider the dynamic, panel probit model of chapter 2 the following form
ya = 1(1‘it13+ Pym—1 + Ci + an > 0) (5)
Uitffl'i~ JIM—1s y2:0~ (‘2?) N NU)» 1) (6)

Using the latent variable setup above, the density of Slit given In» y,,t_1, and c,- can be

derived as follows
PM: = 1lilfit~~,;I/i.t—1~sz) = (130151.13 + Putt—1 + '32:) (7)
Pfilit = 0|4'7it~»;'/i,t—1~‘3zi) = 1 — ‘1’(:17it{3+/’:Ui,t—1 + (’27) (8)
This leads to a conditional density for l/z't given (In, yiy_1 . (ii) of
fr(.I/z't|-ra~ yet—1ft: 6’) = ‘1’ (Tit/3 + pint—1 + My” [1 — ‘1’($a.<'3+ P!/zi.t—1 + 01)] by” (9)

with 6 = (d, p).
Following the method proposed by Wooldridge (2005a) and given Mom-0.113; (5) is a cor-

rectly specified model for the density of c, given (Lt/21% :Ifl'), the log likelihood function is

h(cly,~_(), 17,-; (”dc (10)

 

T
“(0, d) = [0]] f H j~t(.’/itf4"it~ ”Lt-1i ('; (I);
‘ t=1

Defining \II = f [Hill ffyitlTit» y,.t_1,c;b’)] h(c|:r,j.y,i();(5)dc, the score is given by s,(d. (5) =

01,705) mime)
( ()0 ‘ ()6

 

) where

 

 

 

 

- V — T
(”2160) 1 3f(yit|$it.3/i.t—1~C':9) _ ' .
(90 — 2 E ./ H 86 [1(le1, 3110.0)(16
7:1 _t=1
- N — T ‘
(VI-((1.0) 1 (’)}]((f).l".,(/'():(5)
’86 — E; E / tl'll ffyitll'it~yi.t- 1.62.0) 50, ’ dc
7: ,:

 

77

Then a test statistic can be constructed by evaluating the score at the resticted parameter

values suggested by the null hypothesis and evaluating

N N ‘1 N
1 ~ ~ ~I .. 2
S = E 3i, E 3233i E 3i N Kg (11)

where q is the number of constraints imposed and s7,- = s]:((), (5) denotes the score evaluated
at the restricted estimates. Notice that when the statistic is computed, the 1/\II terms will
cancel so that only the portion of the derivates appearing within the brackets needs to be
evaluated in order to obtain it. The outer product form of the asymptotic variance shown in
(11) has the advantage of not requiring additional calculations for estimation of the variance
matrix and is invariant to reparameterizations. However, the nominal size of the test can be
distorted in many cases such that it over rejects the null hypothesis. The expected Hessian
form is often a better alternative as an estimate of the asymptotic variance when it is easy
to obtain since it is always positive definite if it exists, is invariant to reparameterizations,
and has smaller size distortions in many cases. Thus, the estimate of the variance matrix
used may affect computational simpicity, finite sample properties, and the power of the test.

To make the test operational, more structure needs to be imposed on the unobserved
effect. As explained above, a correctly specified model for the density of c,- given (350,171) is
needed. For this test proceedure, normality of the conditional distribution of the unobserved
effect and a specification for the conditional mean are maintained. The conditional mean
is only restricted to be a function of the exogenous covariates and the initial condition
and to be linear in parameters. Assuming normality of the unobserved effect is neccessary
in order to obtain estimates of the restricted model parameters through standard random
effects maximum likelihood estimation. A test of the distributional assumption could also
be implemented either before or after testing for heteroskedasticity but is not addressed in

this paper.

78

Under the assumption that Gilt/2'0, :13,- = cro+cr’1f,j +02y2-0 +a,-, where f,- is the time average
of the exogenous variables and a, is an individual specific, zero mean, normally distributed

error term,

P0111 = llritiyi.t-—lvai) = ‘1’ (fit/3 + Pym—1 + 00 + Gift + 023/20 + 0i) E ‘1’f (12)

ft

(1,2)” [1_ q);—yit] (13)

This specification for the unobserved effect is often referred to as the Mundlak-Chamberlain
device and allows for correlation between the exogenous variables and unobserved heterogene-
ity through the conditional mean. In addition, the conditional variance of the unobserved
effect is assumed to be given by a function g(1:,, MO; '7') such that homoskedasticity is obtained,
V or(a,~) = 03, when exclusion restrictions are imposed on the parameters, 7. This implies
that. under the null the combined error term will have constant variance, u + a N N (0, (72),
as well. These restrictions on the variance of the unobserved effect encompass several com-
monly used structures for heteroskedasticity. This includes exponential, linear, and quadratic
heteroskedasticity as special cases as given by 9(1), MOW) = e;zrp(’yo + 711:, + 723/0)2 and
g(:r,-. gm; 7) = (70 + 7117.,- + 723,0)‘7, = 1. 2, respectively. For all three cases, the exclusion
restrictions 71. 72 = 0 imply 9(70) = 03,. Note that now in h(c|:r,-,y,:0;6), (5 = (o, 7) where
o are the parameters of the conditional mean and 7 are the parameters of the conditional
variance and that a, N N(0, 9(1), yi0;e/)). Under these assumptions the log-likelihood can

be writ ten as

T 1 a
1, 6,6 :10 j ' d 14
( ) g / gft ( 9(xi-3/i0i7))0( Millet/10:7) a ( )

Now the null hypothesis of heteroskedasticity can be tested by applying the exclusion re-

 

 

strictions. Under these restrictions the log likelihood function becomes
T 1 a
l,(0,6) = log / H ft (—) (b (-—) da (15)
t=1 a a

79

While this is not equivalent to the random effects probit maximum likelihood estimator,
consistent parameter estimates can be obtained from the random effects probit with I“,
g/,,,_1, 1. f,, and 31,0 as explanatory variables.

After some algebra, the score for the unrestricted model, s,(6, 6), with variance function
first/ion) E 92'» 3a = (frail/21th), “'2' = (list/i0), 6’ = (tip), a = (00:01.02), and

9'9 (a/\/(T,) E (pa, can be written as

 

 

 

 

 

 

r- T -
8146.6) 1 / (yt‘Qf)¢th ( 1 )
_, = _ d 16
()6 \II E (1)}‘9t [1 _ (pflyt [—91 49a a ( )
8/,(0.6) 1 / (lit—(pf)¢fwi ( 1 )
_f = _ _ <25 da (17)
(90 ‘1’ £11 c} gt [1 — <1) fly‘, 69.- a
.. ' T
3mm) 1 / y, H, ( a2) (v,,/‘g,)
= _ <I>z 1—<I> 4t 1+— —— d 18
37 ‘1’ E f [ f] 92' 92' $0 a ( )

Details for the test under exponential and quadractic heteroskedasticity are given in the

following sections.

3.2.1 Panel Probit with Exponential Heteroskedasticity

Assume that g(;r,, 31201.7) 2 erp(7()+71f,+’72y,0)2. Under this specification of the conditional

variance

1 a
1,6(, =log / ft (( )q')( _ )da (19)
1: earp( )0 + 712?, + 723/20) ”PUG + ”7351i + 72610)

Then heteroskedasticity in the unobserved effect can be tested as H0 : 71 = 0, 72 = 0. Under

 

 

the null, the restricted model estimates can be obtained from the usual random effects probit
estimator with variables .17“, ,i/,,,_1, 1. 37,, ”,0 as shown in (15). For 7 = (70. 71, 72), the score

for the unrestricted model can be shown to be

M 2 l / fi (Lat-q’fozt (_1_) cada (20)

 

06 \II

 

 

 

 

 

 

 

 

 

 

 

_ . f T
alga») 1 / (gt/t — cf) aft“, ( 1 )
- — __ — cf) da (21)
80 \II{ 121—119)} yt[1—q>f]yt w; a
81'1“) l / ficyit [1 — <I> 11:11.1 (1+ 93) (— wt )¢ da (22)
57 ‘1’ t=1 f f 92‘ M97 0
(23)
Imposing the exclusion restrictions 71,72 = 0, this reduces to
T 1 '
61,(6,(5) 1 / ¢f2t ( 1 ) ( a )
,. z —— —- ()5 ——- da (24)
()6 \Il { _t1;[1(1yy¢.f_yt_[1—¢f]yt (Ia (7a
- T '
.- (1))
algae) = i / (wy f MW; (1) ,5 (5:) da (25)
C1Jt=1¢ t[1—Qf] t 00, 0a
. . T
i)l,(6,b) _ _1_ (1)3/1't[ f]_1 31—1 (12 _Lt', _(_1_
('97 — ‘1' {/ ,1}qu 3 1+ if 0a. (15 ”a da (26)

 

3.2.2 Panel Probit with Quadratic Heteroskedasticity

Now consider g(T,, gm; '7') = (“/0 + 71f, + e/2;1/,_0)2. The response probability and conditional

density of y,t will be the same as those given in ( 12) and (13) above. Then

I (H (5) _10q {/ [H ft] (( (70 + ”til; + 72%)) it) ((70 + vii-11+ 72310)) dd} (27)

Again, the restricted parameter estimates can be obtained from random effects probit esti-

 

 

mation on T,,, y,,t_1, 1, f,, y,0, and the null hypothesis of homoskedasticity can be expressed

through the exclusion restrictions '71 = 0, 72 = 0.

31,-(9. a)
66

 

61.,(6. 6)
30

 

31,;(0. a)
(97

 

{firs

t 1

{/g.
{/i

:Is

is

 

9..
ll
H

 

 

(yt-‘PfoZt ( 1 )
—- (bad
‘1’th [1 - ‘I’flyt_ W” a
(fl—ty— (pf) ¢fwi (_1_) (Dada
2
(pl/it 1_(1, 1"-‘hit (1 (L) (_
f [ fl ] + 92'

 

81

u,

(28)

(29)

6912“} (30)

Imposing the exclusion restrictions, this reduces to

81,(0.6) 1 ”T (yt—q))¢zt 1
__ = _ [fl f f (_

 

 

 

 

 

 

(90 \Il _t=l ®}_yt [1_ ¢f1yt_ 0a) '1
- ' T '
61,-(6. 0) _ 1 (y: — ‘1’f) (bfw, 1 a
an _ 117 f H 1-yt _ lit ((7) q) (a—) do. (32)
1:1 (bf [1 q’f] — (I a
. ' T
01,0111) _ 1 / yit 171/11 (1.2 w,- o. ,
8‘7 _ ‘1] £11 of [1 cf] 1+ 0,2, 03 a a“ da (33)

3.3 Score Test for Heteroskedasticity in the Dynamic

Fractional Response Model

In this section, the test statistic is extended to the case of fractional dependent variables.

Recall from chapter 1 the dynamic fractional response model given by
:1:
11,; = Tit," +!/i.t—1/) + («‘2' + ”it
-|(-- - -)~N<0 2>
11,, :1:,.y,q,_1, ...,y,0,c, ,0“

0 if 31;} g 0
L’lit = 3);", if 0 < yft < 1

1 if 31/521

W ith the conditional mean of c, defined as

I _ ..
"111110-11 = 00 + 0117 + 021/10 + Oi (35)

This leads to the response probabilities

 

 

I ._
. —3—”2'n — yarn”) — 00 - 0137-1— 021/10— 02‘ _ . .
[’(3/11 =01y271—1111/‘i-y1001) = ‘1’ ( a ) = (1’0 ('36)
u
Jr,” + ,1/,‘,_1/) + (10 + (1113+ (12)/,0 + (1,,- —— 1 _
1’f3/11= 1|3/1.t—1»11~3/10»azt) = ‘1’< a = 4’1 (37)
U.

82

 

 

3P0!“ S Elli/114.171.3110. (lit) _ Lo (ya — 93117 — yet—110 - O0 - Ot'lfi - 02.1110 — at)

1
(9y 0,, 0,, : —(Z>,, (38)

an

From which the density of y,,)y,,,_1..r,. 3,1,0, 0, can be written as

:_ (1)0[yzt lQIff/zt l—(Dz/[ <y1t< l
011 ‘

ftff/itff/ii-LT‘ryiOa(1296) (39)

By using this density in place of (13), the test outlined above can be implemented. Under
the restricted model the log-likelihood function is obtained by integrating the density of
(I’lil~ 11,71) given (;1/,y_1.:1:,,31,011,) against the distribution of a,

1 a

-——d> (—) da
0a 0a

Thus, the log-likelihood function has the same general form as in ( 15) above, and the re-

T
lif9~5)=109 / Hft(yit|yi,t~1axia31101096) (40)

t=l

stricted model parameter estimates can be obtained from the random effects two—limit. tobit.
The score for the unrestricted model follows similarly to that of the dynamic probit and is

given by

 

 

 

al.- 1 2.. <11 <12 22:19 1
(96 ,1, {/ f1I=I1F( au)(1lyt 01¢: +1111“ 1],,2 +1l0 < y: < 110..) Ada ( )
T _
Bl, 1 11), (1‘51 (fig 117,0
_ 2 _ __ . : _ . : _ 1. 42
aa ‘1, {/ L1F( au)(1lyu 0l¢1+1lyn 1ng +1l0 < yt<1l 0,, ) Ada} ( )
i_l / |T|p(_1_ _1[,. _Olfl _1[ . _1]¢_2§ +1[0< .<1]£_§ - Ada (4.3)
{)0}; — ‘1’ (:1 012‘ Jlt _ @161 3!!! — Q2 2 ylt U.

 

—=;{/[fir (1+§)(V"fl)1ada}

 

(=1
where
1" = $0 I (1,1 2 0—(23/ z
u
I _.
E1 = (_1-117 — Hut-19 — 00 — 01172:— 023/10 - 02'.)
I _
E2 = (1117+ yzt,t—1/) + a0 + 01171 + 023/10 + 01' — 1)
I _

E3 = (3111 — 93117 — yet—10 — a0 — 01331 — 023/10 — 02:)

1 .

91'.

83

The score test for heteroskedasticity can then be computed for any form of the conditional
variance such that 0, ~ N(0, g(1:,,y,0;'y)) where the variance is constant under the null
hypothesis. In this case, the variance is assumed to exhibit exponential heteroskedasticity

as specified in section 3.2.1.

3.4 Empirical Applications

3.4.1 Dynamic Probit Model: Persistence of Union

Membership

Wooldridge (2005a) estimates a simple model of union membership to illustate the method
used above for simultaneously dealing with initial conditions and unobserved heterogeneity
in nonlinear, dynamic panel data models. Here, the same application is used to demonstate
the test for heteroskedasticity in this setting derived above. The data consists of observations
on 545 working men over an 8 year period from 1980 to 1987 with indicator variables for
union status, y,, = 0,1, and marital status, 23,, = 0,1. Applying the dynamic probit model

to union membership,

”(I/zit = lfiflii—li ---,1/z:0»-’I'2:~Cz') = ‘Pf/tI/m—i +1’3-‘Ifz't + (12') (45)

In addition, a set of year dummies are included to control for changes occurring over time
that are common to all workers. Estimation of the restricted model, can be carriedout using
standard software that allows for random effects probit specifications, and the density of c,
is specified as c, = (10 + 0111:, + (123/71) + a, and (1, ~ N(0, 03).

The parameter estimates under the restricted model are given in Table 3.1. Only the
estimated coefficients for the primary variables of interest are included for brevity. The

quantities given in parentheses are standard errors. Given these parameter estimates, the

84

Table 3.1: Restricted Estimates for the Dynamic Probit Model

 

 

 

 

Union Membership Coefficient Estimate
Lagged Union Membership 0.875
(0.094)
l\v-"farital Status 0.168
(0.111)
Test Statistic 10.31
(if 7
5% critical value 14.07

 

Number of observations = 4360
Number of workers = 545

Note: Bold type indicates significance. at the 1% level, italics 5%.
Quantity in parentheses is standard error.

scores of the unrestricted model are evaluated at the restricted estimates and the score
statistic calculated. The statistic is also shown in the table and is distributed X3, making
the critical value for the test approximately 14 at the 5% level. Thus, the test fails to reject

the null hypothesis of homoskedasticity.

3.4.2 Dynamic Fractional Response Model: Determination of Firm

Dividend Policy

The dividend policy application from chapter 1 will be used to demonstrate the test as applied
to the fractional response model. Recall that in this application y,t is the share repurchase
ratio, 1“,, is made up of the exogenous covariates (i.e, market-to-book ratio, operating income,
non-operating income, and volatility), and c, is a firm specific unobserved effect. In order to
control for industry level differences, the exogenous variables are augmented with 10 industry

dummy variables. In addition, a set of year dummies are included to control for changes

85

occurring over time that are common to all firms.3 As before, estimation of the restricted
model, (40), can be carried out using standard software that allows for two-limit random
effects tobit specifications if the density of c, is specified as in (35) and a, ~ N(0. 03). Here f,
includes the time averages of the exogenous regressors (i.e., market-to—book ratio, operating
income ratio, non-operating income ratio, and volatility of earnings).

The parameter estimates under the restricted model are given in Table 3.2. Only the
estimated coefficients for the primary variables of interest are included for brevity. The
quantities given in parentheses are standard errors. Given these parameter estimates, the
scores of the unrestricted model are evaluated at the restricted estimates and the score
statistic calculated. The statistic is also shown in the table and is distributed X3: making
the critical value for the test approximately 11 at the 5% level. Therefore, the test clearly
rejects the null hypothesis of homoskedasticity with a value above 24. This may imply that
the model is heteroskedastic.

However, rejecting homoskedasticity is also potentially compatible with inclusion of a
regressor that is correlated with the error due to omitted variables or measurement error.

Consider the following probit model in which the i and t subscripts have been supressed

.7/1 = 1(Z1771 + 7.7/2 + 11.1 > 0) (46)

:1/2 217721 + 227722 +12 (47)

where yg is continuous and (111,112) has a zero mean, bivariate normal distribution and is
independent of z. Thus, yg is endogenous if 111 and 112 are correlated. In this case, if only
(46) is estimated the test statistic could reject the null hypothesis of constant variance even
if 111 and 1.12 are homoskedastic. It is possible to estimate this model and test for exogeneity

of yg by using a two-step procedure such as that of Rivers and Vuong (1988). Given the

 

3For additional details on firm dividend policy theory and data set construction see sections 1.2 and A.

86

Table 3.2: Restricted Estimates for the Dynamic Fractional Response Model

 

 

 

 

Share Repurchase Ratio Coefficient Estimate
Legged Repurchase Ratio 0.483
(0.025)
l\v~'Iarket-to-book Value —0.002
(0.001)
Operating Income Ratio 0.644
(0.120)
Non—operating Income Ratio 0.185
(0.243)
Volatility of Earnings 1.032
(0.666)
Test Statistic 24.33
df 5
5% critical value 11.07

 

Number of observations = 4530
Number of firms 2 453

Note: Bold type indicates significance at the 1% level, italics 5%.
Quantity in parentheses is standard error.

joint normality of (111,12) and Va'r(u1) = 1

u1 = (4211,12 + 61 (48)

where (~21 is normally distributed and E (Q) = 0. Now

311 = 1(31m + T312 + curl/’2 + 6’1 > 0) (49)

and e1|z. yg. [1.12 is normally distributed with zero mean and constant variance. Since v2 is
not observed, this model is estimated in two steps: 1) Perform an OLS regression of 312
on z: and save the residuals, 1.72; and 2) Estimate the probit model of y1 on :1, 312, and fig.
The exogeneity of 312 is then tested with the usual t-statistic for the hypothesis w] = 0.

However, if 312 is endogenous the usual standard errors and t-statistics will not be valid, and

87

the asymptotic variance of the two-step estimator can be dervied using standard results for

maxumum likelihood estimation.

3.5 Discussion

This paper presents a score test for heteroskedasticity in the dynamic probit and tobit panel
models with unobserved heterogeneity through the use of exclusion restrictions. While the
test is derived under the assumption of normally distributed unobserved heterogeneity and
requires specification of the conditional mean. it leads to a computationally simple test that
is straightforward to implement. This is demonstrated through an application of the test to
a model of firm dividend policy using the dynamic. fractional response model.

There are several avenues open for additional research. The power properties of the test
need to be studied and an evaluatation of its performance against different types of het-
eroskedasticity should be performed. In addition, a test for non-normality of the conditional
distribution of the unobserved effect might also be developed along similar lines, which would

allow testing for this underlying assumption.

88

APPENDIX A

Chapter 1 Summary Statistics & Variable Definitions

Table A.1: Compustat Quarterly & Annual Data Summary Statistics

 

Variable Mean Std. Dev. Min Max
Share Repurchase Ratio 0.281 0.362 0.000 1.000
hrv'larket-to-book Value 6.191 154.095 —533.895 10,473.960
Operating Income Ratio 0.148 0.173 -1.419 4.864
Non-operating Income Ratio 0.001 0.026 -0.388 1.080
Volatility of Earnings 0.013 0.021 0.000 0.541

 

89

Table A2: Industry Dummy Variable Definitions

Industry Dummy 4-digit SIC Code

 

 

Agriculture 0000-1000

Mining & Construction 1000-1099, 1200-1799

Manufacturing 2000-3999
Transportation 4000-4799
Communications 4800—4899
Utility 4900-4999
Wholesale 5000-5199
Retail 5200-5999
Financial 6000—6799
Services 7000—8999

 

90

APPENDIX B

Chapter 2 Summary Statistics & Parameter Estimates

Table. 8.1: AC. Nielsen Yogurt Data Summary Statistics
Variable Mean Std. Dev. Min Max

 

 

Brand 1.23 1.24 0 3
Yoplait. price per ()7. 0.10 0.01 0.04 0.13
Yoplait special 0.01 0.11 0 1
Yoplait featured 0.03 0.18 0 1
Dannon price per oz 0.07 0.01 0.04 0.08
Dannon special 0.04 0.19 0 1
Dannon featured 0.17 0.38 0 1
Nordica price per oz 0.08 0.01 0.06 0.10
Nordica special 0.00 0.00 0 0

Nordica featured 0.03 0.18 0 1

 

91

Table 8.2: Parameter estimates for Table 2.1

 

 

Estimated Process

Coefficient Estimates

 

 

 

 

 

 

 

 

(A) T=5, NZIOO 1‘1 11:2
«2qu ~ We + warm» 03> (3'33) (11%?)
Cililiz' ~ N (1190 + ‘wifz' + $21512 + 11135513, 0%) ((2)3725) (10%;; )
6.117.: ~ Nwo + viii. ogexpe’lfa) (13)-$7) (10%22)
Cil'Iz' N NW0 +'¢"1fz' + $21712 + '¢'3fi3a 01216331001150) ((510%) ((1)9217)
(B) T=5, N=500 2:1 2:2
elm.- ~ W + x 03) (fig) (313315 )
0211131 “J NW0 + til/"1152' + $23312 + $3551.31 012;) ((610314) (%)%14§1)
car.- ~ M2120 + viii, 01218231903950) ((3.32%) ((0)131)
. . Q
Cill‘z' N NW0 + U’ifz‘ + $2132 + 1.4355213, 0363319039170) (21035) ((1)8121)
(C) T225, N=100 x1 :02
cilri ~ N0!) + 13;, 02) 0.178 1.064
a (0 032) (0 042)
til-“'72: ~ NW0 +w'11'7i + $211712 + “112311713303 ((3107581) (1()%§154)
our. ~ Neva + 1.52: ozexpe’lzf.» (3113?; ) (Wilt)
Cilil‘z: N N ('1/3‘0 + viii + $211722 + $311713. 03611019031170) (3.10:3) (lo-(2)5431)

 

92

continued on next page

Table B. 2 (continued)

 

 

 

 

 

Estimated Process Coefficient Estimates
(D) T225, N=500 $1 $2
an, ~ N(0~ + a, 03) 0.177 1.061

(0.014) (0.020)

c,|1:.,1 ~ N(1/;0 + Vii-z? + $2322 + 1113123, 03) 0.176 1.062
(0.014) (0.019)

(312:,- ~ N(e’;0+tr'1fzj,03exp()\'lfi)) 0.175 1.051
(0.015) (0.020)

cilari ~ N(t/)0 + 0131?,- + #321712 + '1/13117'113. ageIpOx'lfi» 0.175 1.051
(0.016) (0.020)

 

 

Quantity in parentheses is standard error.
Estimates in bold are significant at the 5% level & 8 equals its true value (italics 10%).

93

 

Table 8.3: Parameter estimates for Table 2.2

 

 

Estimated Process Coefficient Estimates

 

 

 

 

 

 

 

 

 

(i) ”Mollie 01') = 130/10)
(A) T=5, N=100 2:1 2:2 111—1
ci|xi ~ N(u”0 + 7.0325,. 53) 0.172 1.028 0.159
(0.080) (0.100) (0.164)
an, ~ N(v"0 + 7/117, + «(12:17,? + 1713.153, 03) 0.174 1.046 0.152
(0.081) (0.104) (0.162)
cilai ~ N(1f’() + wifi,ogexp()\'1fi)) 0.187 1.106 0.187
(0.026) (0.214) (0.319)
(Iilifii ~ N(1/’0 + 10(ij + 7/}2-‘172’2 + 013.7713, age:r[)(/\’1fi)) 0.245 1.284 0.253
(0.174) (0.327) (0.448)
(B) T=5, N=500 131 1'2 yt_1
qty,- ~ N(’d’0 + 1171317,, 53) 0.172 1.018 0.176
(0.056) (0.044) (0.083)
(.),-lat,- ~ N(ui10 +1135, + (02:5,? + 335,3, 03,) 0.174 1.033 0.157
(0.079) (0.105) (0.167)
cila‘i ~ Mao + (7’15), 0362312013170) 0.215 1.123 0.205
(0.032) (0. .207) (0.316)
6,-1.8,- ~ N(1Z.’0 + 2.1325,- + 1712.75,? +w3fz-3,agearp()\’1fi)) 0.261 1.348 0.252
(0.160) (0.309) (0.520)
(C) T=25, N=100 .131 1‘2 yt_1
8,15,- ~ N(1/10 + 7:35,. 03,) 0.176 1.054 0.174
(0.032) (0.042) x (0.071)
cilxi ~ Nam +4133,- + ((12:17,? + 11.327.15.03) 0.175 1.056 0.173
(0.033) (0.047) (0.075)
all» ~ Ive/,0 + w’ :5. 0265;9(1’ 5)) 0.176 1.056 0.175
7 l 1 Z a 1 I (0.031) (0.044) (0.070)
qty,- ~ M1110 +7122: + «52:5,? + 33133, agexp(1’,;5,)) 0.179 1.069 0.173
(0.039) (0.061) (0.085)

 

94

continued on next page

Table B. 3 { continued )

 

 

 

 

 

Estimated Process Coefficient Estimates
(D) T=25, N=500 1:1 12 yt_1
6,11, ~ N(21’ro + 91:51.03) 0.174 1.041 0.174
(0.014) (0.019) (0.030)
6,-1.1, ~ N(:.00 + .1137, + 111215,? + 1193133, 03,) 0.177 1.052 0.173
(0. ()4 1) (0.025) (0.054)
mm, ~ N(l./’() +'t/”1fi.a(2,exp(x\’1fi)) 0.174 1.052 0.175
(0.015) (0.020) (0.032)
Cil’Iz’ ~ N(1/’0 + u’rifl- + 1152132 + 113-133, 0381p(/\,1fi)) 0.183 1.070 0.179
(0.052) (0.045) (0.079)

 

 

continued on next page

95

Table B. 3 ( continued)

 

 

Estimated Process

Coefficient Estimates

 

(ii)

P0110 = 111F561) = 90610.13)

 

 

 

 

 

 

 

 

(A) T=5, N=100 271 :02 yt_1
6,15,- ~ M2110 + 11113.03) 0.161 0.977 0.156
(0.067) (0.091) (0.150)
(slur,- ~ N(‘I,/’() + 1011f,- + 1112:1712 + "435171310120 0.166 0.990 0.143
(0.072) (0.090) (0.155)
ale,- ~ M100 +'tb'1f,j,ogexp(/\'lfi)) 0.180 1.040 0.194
(0.101) (0.196) (0.315)
1,11,. N N((_/10 + 0.5.1:,- + 012.77,? +"1/13272-3,031::B])(A’1:Ei)) 0.204 1.161 0.205
(0.144) (0.245) (0.466)
(B) T=5, N=500 x1 1’2 311—1
5,11,. N 111(0),, +1/;’1;1‘-,-,ag) 0.160 0.970 0.163
(0.039) (0.040) (0.074)
an, ~ M00 + «055, + 0225,? + 10315353,) 0.163 0.985 0.142
(0.068) (0.097) (0.161)
an, ~ N000 + 4015,.03125190’15,» 0.190 1.099 0.217
(0.103) (0.245) (0.406)
aye,- ~ NW0 + 055,- + 02:15,? + 1:13:59, agexpo’lf,» 0.212 1.196 0.242
(0.139) (0.241) (0.426)
(C) T225, N=100 171 $2 yt_1
c,|:1:,- ~ Nam + 01.5,. 03) 0.175 1.045 0.172
(0.031) (0.043) (0.067)
cm, ~ N(.,1,0 + 0317,: + 02:6,? +-135,-3,ag) 0.176 1.046 0.171
(0.033) (0.045) (0.073)
am ~ N000 + 21:15,. 0361119315,» 0.176 1.043 0.172
(0.031) (0.043) (0.070)
cm, ~ M00 + 031?,- + 02:15,? + 0:31.33, ogexp(/\’1fi)) 0.181 1.058 0.180
(0.058) (0.057) (0.097)

 

96

continued on next page

Table B. 3 (continued)

 

 

Estimated Process

Coefficient Estimates

 

 

 

(D) T=25, N=500 1‘1 1‘2 yt_1
alas, ~ N000 + 03.6,.03) 0.173 1.040 0.173
(0.014) (0.019) (0.031)
me,- ~ N(00 + 0.1117,- + 40217,? + 035,3, 03) 0.176 1.041 0.173
(0.048) (0.024) (0.059)
5,11,- ~ N070 + 0131340341,)0'15») 0.173 1.040 0.173
(0.014) (0.019) (0.030)
5,14,- ~ N010 + 0:117, +0213? + 01313-3, agexpo’lf,» 0.172 1.059 0.182
(0.026) (0.046) (0.061)

 

 

continued on next page

97

Table 8.3 (continued)

 

 

Estimated Process Coefficient Estimates

 

 

 

 

 

 

 

 

 

(iii) 130110 = lll‘z'» Ci) = @070 + 721271 + 77261)
(A) T=5, N=100 $1 272 01—1
6,-1.1, N M110 + 11115,. 53) (0.176) (1.033) (0.206)
0.080 0.110 0.205
(1)1,- ~ NW0 +1317,- + 1.1217,? + 13133113) @1311) $1135) ((5210):)
412:.- ~ N016 1.151.311.1031,» ((321313 ) (63313) ((3158112)
CiITz‘ ~ N(1/10 + ((21:17,: + #1211712 + “((13:33, ogewpbvlffl) (%22%;) (k4???) (0)4522»
(B) T=5, N=500 $1 $2 yt_1
(alrm N N ("1’0 411401.03) ((0.11%) (0%1575) ((021256)
(Til-731 N N000 + $1371 + 11721512 + 133313103) $11905” (1012133)) ((32335)
4111 ~ M10 + 1111131110310) ((32139)) (323375) (31543871 )
Cz'le-z' N N (10 +'¢'1f1 + 1.101712 + 1433313103855190'133» 321915494) (b12273) $131312)
(C) T=25, N=100 I1 1‘2 yt_1
I<> .113. .1113 318.5
1.12:.- ~ M10 + 111. + 12:52 113113.13) {3210157) ((9)7128) ((3)1317)
'<<” 139 as 171)
cilia: N N (10 + 112?}- + 122.7712 + "13f13,03610(/\’1f1)) (13)-g?” ($871) $219127)

 

98

continued on next page

Table B. 3 (continued)

 

 

Estimated Process

Coefficient Estimates

 

 

 

(D) T=25, N=500 :171 1‘2 yt_1
6,11,- ~ M10 + 1’11}. 03) 0.178 1.052 0.182
(0.020) (0.026) (0.057)
all, N N010 + (1’11?) + 11217,? + 13013.02) 0.182 1.066 0.196
(0.042) (0.031) (0.080)
(Ll-)2, ~ N(t="'() + urifi, agexpfliffi) 0.182 1.062 0.202
(0.024) (0.037) (0.091)
6,-(1, ~ N(z/’0 +1117, + (1215,? + .1135,3,03exp(1\'1;5,)) 0.184 1.086 0.194
(0.030) (0.058) (0.118)

 

 

Quantity in parentheses is standard error.

Estimatts in bold are significant at the 571 level & 3 equals its true value (italics 10%).

99

Table B.4:

Parameter estimates for Table 2.3

 

 

Estimated Process

Coefficient Estimates

 

 

 

 

 

 

 

 

 

(0 100101331101) = 130/10)
(A) T=5, N=100 11:1 11:2 yt—l
0,);1, ~ N050 +135153) 0.174 1.310 0.503
(0.085) (0.109) (0.178)
(5)2,- ~ N(*t/’() + {(91.27) + 10211712 + 103513310121) 0.174 1.044 0.499
(0.082) (0.107) (0.171)
cilcrz; ~ N(u’ 010 + 101.23,, 0 36227120112727» 0.193 1.118 0.562
(0.114) (0.212) (0.333)
(r.,-|:I.',j ~ N(1/’0 + (0127,: + '1/1221-2 + ((3.253 ,0 2e2p(z\’1:fi)) 0.238 1.269 0.593
(0.172) (0.300) (0.452)
(B) T25, N=500 1'1 1‘2 yt_1
6,11,: ~ M10 + 11313.53) 0.169 1.014 0.509
(0.035) (0.048) (0.075)
cl-Iatz- ~ N00 + 1117,: + 121,211 1313-3, 53) 0.174 1.023 0.504
(0. 040) (0.059) (0.087)
Gil-Ti ~ N(tl’0 +1le 0(2,e:0p(/\'1:r,;)) 0.187 1.148 0.671
(0.113) (0.220) (0.364)
Cilm-i N N000 + lb‘ifj + ”(192113-12 + '(b'3fi3,0’ 028$p()\1f2j)) 0.259 1.323 0.640
(0.148) (0.291) (0.429)
(C) T=25, N=100 .231 1:2 yt_1
5,11, ~ M110 + 11517,. 03) 0.178 1.054 0.524
(0.035) (0.046) (0.077)
Ci|zrzj N NW2) + 11417) + ’11)ng + $3179, 02) 0.174 1.056 0.523
(0.035) (0.048) (0.078)
c,-|a:, ~ N010 + 11117,.agexp(1’113)) 0.177 1.054 0.526
(0.035) (0.047) (0.074)
cilxi ~ N000 + ufif, + 102212+ (11322-3 ,00262p()\’1fi)) 0.179 1.061 0.522
(0.044) (0.049) (0.084)

 

100

continued on next page

Table B.4 (continued)

 

 

Estimated Process

Coefficient Estimates

 

 

 

(D) T=25, N=500 $1 172 yt—l
c1|1~1~ N(1’10+12’1f1-,03) 0.176 1.051 0.525
(0.015) (0.021) (0.034)
(~111- ~ NW0 +1151 + 12512 +1-313-313) 0.177 1.052 0.531
(0.049) (0.026) (0.051)
0111“,: ~ N(l£'() + '1’.r'1f1-.0121€:rp()\’1f1-)) 0.175 1.051 0.525
(0.015) (0.021) (0.036)
al.11- ~ N(1/10+1/r1271- +1251? + '1/23231310361p()\’1f1j)) 0.179 1.058 0.526
(0.049) (0.027) (0.058)

 

 

continued on next page

101

Table B.4 (continued)

 

 

Estimated Process

Coefficient Estimates

 

 

 

 

 

 

 

 

 

(ii) P0110 = llirzfacz') = (1)0103)
(A) T25, N=100 2:1 2:2 yt_1
“I“ N NW) + 11:51.03) ((111%) ((011951) ((01%)
(Til-'11 ~ (Wt/’0 + 'U"'141—’2' + “$241712 + 11311713» (73) ((01022) ((1131) ((012%)
(2.11.: ~ Mum 11117113611111» (00.113152) (119973) (1315111113)
(71".271- ~ N(1/10 + 'd’i‘fi + [11:22:12 + 111931713, ag(%fr])(/\’1:I_1j)) ((00. 210148)) ((1012112)) ((%Z7551))
(B) T=5, N=500 1‘1 1‘2 yt_1
(.111,- ~ M10 + 14.51, 0121) 0.159 0.963 0.489
(0.030) (0.040) (0.074)
C’ill‘i N N (110 +9913? + 1.1925512 + 93-513. 0.21) ((00026) ((2)1321) ((1088?!)
Cilirzi ”V N (9”0 + 9911171. 038177401 171)) $119030) (1)0313” ((213942?)
Glitz“ N N (”"0 + $133 + 1112132 + $3173» 038100351) ) ((00958) (11.12%) ((10005)
(C) T=25, N=100 1‘1 1‘2 311—1
c1|231 ~ N(10 + 11:13. 03) 0.174 (1.040 (0.521)
(0.035) 0.046) 0.076
.1> 119.. 11%) 11.2.0.)
c.1150,- ~ N(l_/’() +1/r1f1-.agerp()\’1f1~)) ((31:33 (1)0343) ((0501701)
(>111- ~ NW0 + 12125.,- + 12151-2 +'1/)3f13,0121€:rp()\'1f1)) (0117315) (111355) (011381))

 

102

continued on next page

Table B.4 (continued)

 

 

Estimated Process

Coefficient Estimates

 

 

 

(D) T=25, N=500 :51 2:2 yt—l
cilzrl- ~ M00 + 01:13.03) 0.172 1.038 0.521
(0.014) (0.020) (0.034)
0111:,- ~ NW0 +0313 + 1192f? + 03333, 03) 0.173 1.039 0.523
(0.031) (0.024) (0.045)
0,117;- ~ NW0 +'drifii,ogexp()\’1fi)) 0.173 1.038 0.522
(0.015) (0.020) (0.034)
Cilil‘,‘ N Wyn/’0 + 'l/I'ifl‘ + $21712 + #2311713, 0'361‘p()\,1f,j)) 0.176 1.044 0.523
(0.031) (0.029) (0.056)

 

 

103

continued on next page

Table 3.4 (continued)

 

 

Estimated Process

Coefficient Estimates

 

(iii)

P(yz'0 =1|23i,c2j)= @070 + 7111*? +0201)

 

(A) T=5, N=100

 

 

 

 

 

 

 

 

1131 £172 yt—l

aim ~ NW0 + 03:13.53) 0.175 1.029 0.528
(0.086) (0.114) (0.199)

oily) ~ N030 +17;;;z-:,,-+1/22;132 + 03.53, 03) 0.181 1.144 0.508
(0.124) (0.220) (0.380)

(Lila?) ~ N(-u"0 + 0113.03.22119’150) 0.255 1.283 0.827
(0.187) (0.335) (0.495)
nil-'17.- N NW) 1. 9,353- + (52.5.2 + (1.3.133, ogcmpfl’lfifl (%22%81) (big) (%75%18)

(B) T25, N=500 231 1‘2 yt_1
(2);» ~ NW0 + 0hr. (72) 0.174 1.015 0.537
1 l 1 2 a (0.041) (0.056) (0.133)

c115.- ~ M1170 + 114.5) + 02:5)? + (03:533. 03) 0.186 1.131 0.581
(0.120) (0.219) (0.375)

(fit-Ia?) ~ No.90 + 8:55.. oge$p(/\'1fi)) 0.277 1.343 0.874
(0.190) (0.342) (0.488)
Cilia: N N(’L/’0 + 1.01171 + #923332 + $33713, 036391293171» ((631%) $532)) ((3756531)

(C) T=25, N=100 1'1 :52 yt_1

Cz-Iaji ~ NW0 + 41/155). 03) 0.179 1.055 0.528
(0.036) (0.049) (0.087)

aim ~ N050 + 5:115.- + 27:12:15.2 + 03513.03) 0.170 1.059 0.536
(0.036) (0.049) (0.091)

0133- ~ No.50 + 'I,/2';1?t.o2_ea:p(/\'f1)) 0.181 1.003 0. 555
I . 1 z a 1 I (0.036) (0.056) (0.115)
Cilmi ~ NW0 + 97112 + 1021122 + #931713, 03815.0()?» £010?” (10%?1) $515385)

 

104

continued on next page

Table B.4 (continued)

 

 

Estimated Process

Coefficient Estimates

 

 

 

(D) T=25, N=500 x1 2:2 91—1
(film): ~ M990 + 023:5). 03) 0.176 1.052 0.531
(0.020) (0.020) (0.000)
cilati ~ Nwo +0315.- + 025,-? + 03:133. 03) 0.178 1.054 0.534
(0.031) (0.025) (0.048)
Cilil‘i ~ N(Lz"-'0 + ibifi.oge:z'p(x\’1fi)) 0.179 1.056 0.558
(0.025) (0.038) (0.103)
cl-lxz- ~ N010 + (1:55,- + 0227.2 + 1/1'32713,ogexp(/\’1:f,;)) 0.185 1.072 0. 557
(0.052) (0.048) (0.116)

 

 

Quantity in parentheses is standard error.

Estimates in bold are significant. at. the 5% level 83 ,3 equals its true value (italics 10%).

105

 

Table 13.5: Parameter Estimates for Table 2.4

 

 

Estimated Process Coefficient Estimates

 

p=.25

 

(A) T=5, N=100 :1'1 .172 y,_1

 

(Yd-Ti» ya) ~ N010 + a13/10 + 021303)
0.176 1.037 0.163
(0.084) (0.119) (0.184)

('1I'Iz', .1110 ~ N010 + 013/10 + “2171+ 031712 + 045171310121)

0.241 1.237 0.245
(0.100) (0.311) (0.375)

(f.l'l;l‘2f, gig ~ N((10 + (twin + (12.17). oge:l'])()\'1-;I:i2))
0.299 1.469 0.307
(0.240) (0.449) (0.628)

__ _ _ - ' 2
(.'1j|:1:,j.y,-0 ~ N((10 + 01y“) + (.121),- + (13:17,:2 + (1.1;r.)'3.oge:tp(x\'11:i ))
0.384 1.593 0.423
(0.269) (0.445) (0.611)

 

 

(B) T=5, N=500 I1 2:2 91—1

 

011417253110 ~ .N((l() + 013/10 + (12:5), 03)
0.178 1.031 0.164
(0.029) (0.044) (0.075)

(Til-131'. 910 ~ N010 + 0131260 + 02172: + 033512 + 0'4f13~ 03)

0.190 1.090 0.181
(0.046) (0.082) (0.133)

_ - 2

(71le 3070 ~ N010 + 01.1110 + 0215012161‘1’0'1171 ))
0.178 1.028 0.164
(0.029) (0.045) (0.074)

.. . , ,— —2 —. 2... f.2
(Ari-.31“) ~ N((.1() + (113/;()+ (12:12,: + (13.1?) + a4$i3.oamp()\11i ))

0.178 1.028 0.104
(0.029) (0.045) (0.074)

 

 

continued on next page

106

Table B. 5 (continued)

 

 

Estimated Process

Coefficient Estimates

 

 

 

 

 

(C) T=25, N=100 271 2:2 yt_1
(fill-”172'. 311.0 N N010 + ”ll/2'0 + (12171303)
0.180 1.060 0.174
(0.039) (0.058) (0.089)
01151711070 ~ N010 + 01.1110 + 02172: + 03171.2 + 0441:3303)
0.190 1.090 0.181
(0.046) (0.082) (0.133)
_ — 2
5112.. 1.0 ~ Mao + 011100 + 424193410934.- 0
0.194 1.102 0.181
(0.076) (0.140) (0.206)
. ~N‘ .. ,-. -.2 -.3 2. , 17-2
c, |.r,, gm ((10 + mg“) + (12.1:2 + (1,327; + (14cm ,Ua€$P( 11172. ))
0.189 1.114 0.197
(0.064) (0.113) (0.217)
(D) T=25, N=500 1131 $2 Kit—1
Gil-Ti» 1910 ~ N(0'0 + (1111/10 + 0213'» 03)
0.178 1.058 0.170
(0.031) (0.041) (0.082)
all)?» 910 ~ N010 + 01920 + 0213' + 031712 + (14523: ‘73)
0.191 1.09.? 0.183
(0.053) (0.086) (0.127)
. _ - 2
(72711723 .1110 ~ N010 + a191:0 + 021171,, 01216173031132? ))
0.196 1.115 0.189
(0.078) (0.155) (0.101)
....,.~N. . ,.—. 7.2 .—.3 2 Ar?
(«1 In. .910 (00 + (113/10 + 02171 + 0312. + 041?: flaw?“ 1172 ))
0.187 1.106 0.196
(0.063) (0.106) (0.088)

 

 

continued on next page

107

Table B. 5 (continued)

 

 

Estimated Process Coefficient Estimates

 

p=.75

 

(A) T=5. N=100 21:1 11:2 yt_1

 

(’ilri» 3/2'0 N N010 1' all/i0 + 02171.03)
0.178 1.035 0.497
(0.088) (0.118) (0.181)

— — — 2
Cilia 910 ~ N010 + 01010 + 02374 + 031512 + (14113439011)

0.232 1.207 0.590
(0.142) (0.258) (0.361)

- 2

(Til-Ti» ”7.0 N N((10 + (113/20 + (1211723 03"1’7’0‘132 ))
0.289 1.403 0.691

(0.244) (0.411) (0.005)

r — T 2 — 3 2 . ‘. 2
(Bil-T1» 910 N (V (00 + 013/10 + 0212: + 0312: + 041% aUaEIPWfi/i ))

0.344 1.549 0.749
(0.237) (0.390) (0.594)

 

 

(B) T=5, N=500 x1 2:2 01—1

 

015131.910 ~ N010 + 013/10 + 02:55 03)
0.173 1.029 0.519
(0.035) (0.051) (0.085)

. ' — ,7 2 . ,? 3 2
(fill/1,3110 N (V (00 + 013/10 + 02sz + 0312' + (1411 10a)

0.204 1.058 0.511
(0.041) (0.061) (0.099)

. _ — 2
(3017143110 ~ M00 + 011910+ 021175 0381‘PW11‘2' ))
0.168 1.029 0.516
(0.035) (0.054) (0.092)

,T T - I 2 . , I— 2
Gil-Ta 3110 N N010 + 01910 + (121-1 + 03112 + (1417213: Ua€$P()\1fi ))

0.168 1.029 0.516
(0.035) (0.054) (0.092)

 

 

continued on next page

108

Table 8.5 (continued)

 

 

Estimated Process

Coefficient Estimates

 

 

 

 

 

(C) T=25, N=100 1:1 (122 yt—l
(‘z'lél‘zw .710 ~ NW) + "11110 + ”2413503)
0.177 1.058 0.527
(0.38) (0.051) (0.084)
..I. . N [V , ‘ .7. ‘ :12 -_3 2
c, 1.1.1/)0 1 ((10 + 013/10 + (12.1, + 031.1 + (141:. .00)
0.191 1.095 0. 535
(0.059) (0.090) (0.117)
_ - 2
Cil$i~ 1110 ~ N010 + 011/10 + 02175 0361190412,: ))
0.198 1.128 0.540
(0.079) (0.170) (0.208)
._ ~N . .. .—.2 .—.3 2.. Art?
(11121 3110 (“0 + (113/70 + 0212 + 0312 + (1411 saa‘ J‘p( 11'? ))
0.185 1.102 0.534
(0.061) (0.098) (0.195)
(D) T=25, N=500 2‘] 1'2 yt_1
CilIi~ 3110 ~ N010 + 011910 + 02% 03)
0.175 1.056 0.509
(0.011) (0.023) (0.027)
all)?» 3110 ~ N010 + 013110 + 02351 + (#131712 + 045513~ (73)
0.190 1.090 0.537
(0.048) (0.081 ) (0.107)
r _ - 2
(32'le 3110 "V (V (00 + 01910+ (I211~0121€1‘P(/\'1171 ))
0.196 1.200 0.551
(0.063) (0.044) (0.099)
.va .-. ,.-.2 ,-,3 22'. AI—“2
(31.4511 y,0 ((10 + 013/10 + (121} + (1.3172 + (141?; ,(7“(.I‘p( 11., ))
0.187 1.117 0.544
(0.044) (0.036) (0.100)

 

 

Quantity in parentheses is standard error.

Estimates in bold are significant at the 571- level 81.. 5’ equals its true value (italics 10%).

109

APPENDIX C

Chapter 3 Summary Statistics

Table C.1: Union Membership Data. Summary Statistics
Variable Mean Std. Dev. Min Max

 

 

Union Membership 0.244 0.430 0 1

l\"Iarital Staus 0.439 0.496 0 1

 

110

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