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This is to certify that the

dissertation entitled
PANEL DATA WITH CROSS-SECTION VARIATION IN THE

SLOPES AS WELL AS THE INTERCEPT:
THE EFFECTS OF UNIONS ON WAGES

presented by

CHRISTOPHER MARK CORNWELL

has been accepted towards fulfillment
of the requirements for

Ph . D . degree in Economics

 

 

m 3&9?

Major professor

 

Date Noveniber 13, 1985

MS U is an Affirmative Action/Equal Opportunity Institution 0-12771

 

MSU

 

 

 

BEIURNING MATERIALS:
Place in book drop to
remove this checkout from

 

LIBRARIES
gnu—cunnin- your record. FINES will
be charged if book is
returned after the date
~- '“" stamped below.
‘ m1 8 1 1994

SE? 2 1.2008

 

 

 

A 443‘?

PANEL DKTA HIT“ CROSS-SECTION VARIATION IN THE SLOPES
AS HELL AS THE INTERCEPT:
THE EFFECTS OF UNIONS 0N "AGES

by

Christopher Mark Cornwell

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Economics

1985

ABSTRACT

Combining time-series and cross-section data is useful in
controlling for omitted or unobservable individual Specific attributes
which may be correlated with the explanatory variables in a
regression. A regression function that does not condition on the
individual specific effects will not identify the parameters of the
model. Econometric models that assume the availability of panel data
are usually of the constant slopes and variable intercept form. This
study considers a panel data model with cross-sectional variation in
some of the slopes as well as the intercept.

An established literature exists on the estimation of the simple
model. The choice of estimation procedures depends on the assumptions
about the individual effects. We distinguish three sets of
assumptions: (1) fixed effects, (2) random effects uncorrelated with
the regressors, and (3) random effects correlated with the regressors.
The fixed effects model is estimated by analysis of covariance, or
within. Generalized least squares is the standard procedure when the
effects are random and uncorrelated with the regressors. When the
effects are random and correlated with the regressors, the instrumental
variables estimator introduced by Hausman and Taylor is appropriate.
Each of these estimators is asyhptotically well-behaved in the case of
inany individuals and few time periods. For the general model we derive

the analogous within, GLS, and Hausman-Taylor instrumental variables

estimators. Furthermore, we prove that these estimators possess the
same properties in the general model that they have in the simple model.
Then, we apply some of our theoretical results to an attempt to
measure the impact of unions on wages. Conventional wisdom suggests
that cross-section estimates are upwardly biased due to the positive
correlation of unobserved individual specific attributes, or ”ability",
with union status. Most often this bias is addressed through a fixed
effects specification of the simple model. However, this approach is
criticized for ignoring the sectoral dependence of the individual
effects. We consider a special case of our model in an attempt to deal
with this criticism. We conclude the conventional wisdom is confirmed

in our empirical investigation.

ACKNOWLEGHENTS

The completion of this dissertation leaves me indebted to many
people. To a few people the debt I owe is particularly large.

My dissertation chairman, Peter Schmidt, has consistently provided
careful guidance and patient instruction. From him I have learned most
of what I know about econometrics. The debt I owe to him is certainly
beyond my capacity to repay.

Dan Hamermesh has similarly provided often needed guidance and
direction. My interest in labor economics is due, in large part, to
him.

I also owe much to Harry Holzer. I have benefitted greatly from
his comments and advice.

Perhaps the greatest debt is that which I owe to my wife. Without
her love, encouragement, and support this research would still be
incomplete.

In addition, I am very grateful for the constant support of my

parents and the excellent typing provided by Terie Snyder.

11

TABLE OF CONTENTS

CHAPTER ONE:
INTRODUCIION..OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO0.0...

CHAPTER TWO:
FIXED EFFECI‘SOOOOOOOOOOOOOOOOOOOOOO...0.0.0.0....OOOOOOOOOOOOOO

CHAPTER THREE:
RANDOM EFFECTS UNCORRELATED WITH THE REGRESSORS................

Appendi-XOOCOOOOOOOOOOOOOIO....0..0.OOOOOOOOOCOOOOOOOO0......

CHAPTER FOUR:
RANDOM EFFEch mRRELATED WITH TI‘IE REGRESSORS O O O C O O O O O O O O O I O O O 0

Appendix AOOOOOOOOOCOOOO00.00....0.0COO-OOOOOOOOOOOOOOOOOOOO

Appendix BOOOOOOOOOOOOOCOO0.0......0......OOOOOCOOOOOOOOOCOO

CHAPTER FIVE:
ESTIMATION OF UNION WAGE DIFFERENTIALS I O O O O O O O O O O O O O O O O O O O O O O O .

CHAPTER SIX:
SUMMARY AND mNCIIUSIONSOOOOOO0.0..0...OOOOOOOOOOOOOOOOOO0......

FOOTNOTES

Chapter TWOOOOOOOOOOOOOOOOOO...OIOOOOOOOOOOOOOOOOOOOOOO00......

Chapter ThreeOOOOOCOOOIOOOOOOO00.0.0.0...OOOOOOOOOOOOOOOOOOOOO.
Chapter FourOOOOOOOOOOCIOOOOOOII.00......OOOOOOOOOOOOOOOOOOOOOO

Chapter FiveOOOOOOIOOOOOOOOOOOOO0.0...OOOOOOOCOOOOOOOOO00......

REFERENCESOCOOOOOOOO0.0.0.000...00.0.00...0..OOOOOOOOOOOOOOOOOOOOOO

iii

Page

15
24

25
38

43

47

68

74
75

77
80

82

LIST OF TABLES

TABLE 1
Means (and Standard Deviations)................................

TABLE 2
Cross-Section Estimates Dependent Variable: LOG Wage..........

TABLE 3
Panel Estimates: Intercept Varying Dependent Variable:

Lm wage...O...0.0.0....00......OOOOCCOCOOOOOOOOOO0.0.0....

TABLE 4 _
Estimates: Slopes and Panel Intercept Varying Dependent

variable: ch; wageOOOOOOOOOOOOOOOOOOOOOOOOOO000......0....

TABLE 5
Percent Union Wage Effects.....................................

iv

52

54

56

63

66

CHAPTER ONE

INTRODUCTION

Panel or longitudinal data are simply time-series observations on a
cross-section. The applied economist may have several years of data on
individuals, households, or firms. Typically the number of time
observations (T) is small and the number of cross-sectional units (N) is
large. Combining time-series and cross-section data is particularly
useful in controlling for omitted or unobservable attributes specific to
the cross-sectional unit (henceforth taken to be an individual) which
are correlated with the explanatory variables. A regression function
which does not condition on these individual specific effects will not
identify the parameters of the model.

Econometric models that assume the availability of panel data most

often take the form

(101) Yit =3 Xit'B + 01 + cit , i=1,ooo’N, F190009T ,

where X1: is a vector of explanatory variables and sit is an iid
S

error. The a1 are individual specific parameters, or effects. 80, in

(1.1) each individual has a unique intercept.

How we estimate (1.1) depends on our assumptions about the 01.
From the literature we can identify three distinct cases: (1) fixed
effects, (2) random effects uncorrelated with the regressors, and (3)
random effects correlated with the regressors. Case (1) is the weakest
set of assumptions. Here the individual effects are taken to be
constant over time (no other assumptions about them are necessary). In
(2) the at are assumed to be iid random variables that are uncorrelated
with all of X. This specification is sometimes referred to as the error
components model. The last case drops the independence assumption and
allows the individual effects to be correlated with some of X.

This study has as its focus the estimation of an obvious
generalization of (1.1) - a panel data model which allows cross-

sectional variation in some of the slopes as well as the intercept.

Such a model can be written as

a ' ' =
(1.2) Y1: X1: 8 + Wit 61 + Sit, 1 1,ooo,N, F1’000,T ,
where ”it is a vector of explanatory variables associated with
coefficients that depend on 1. (Alternatively, we could partition X and
a ' ' .1 \ a

B and write Yit xlit 811+ XZithZF + €1t)' Clearly, if “it constant
then (1.2) reduces to the simple, intercept varying model.

Our investigation proceeds as follows. The next three chapters
present theoretical results. Chapter Two considers the fixed effects
case; Chapter Three takes up the case of random effects uncorrelated

"PC\,
with the regressors; then, the case of random effecté uncbrrelated with

\w

.r

the regressors is covered in Chapter Four. For each set of assumptions,

we first review the results on estimation established for the simple

model. Then, we extend these results to the general model. In each
case we are interested in estimators which have good asymptotic
properties as Nrn while T is fixed.

In Chapter Two this means deriving an analog to the within (or
analysis of covariance) estimator of the simple model. We also show
that under normality, the within estimator for the general model is the
conditional MLE. The error components model is traditionally estimated
by generalized least squares. So, in Chapter Three, we derive the GLS
estimator for our model and prove that the properties of GLS in the
simple model carry over to the general model. The groundwork for
Chapter Four is laid by Hausman and Taylor (1981) (hereafter referred to
as H-T). They develop an instrumental variables procedure for the
simple model in which the individual effects are random and correlated
with some of the regressors. We derive a similar instrumental variables
estimator for our model, and following H—T, detail conditions under
which it differs from the fixed effects estimator.

In Chapter Five, we apply some of our theoretical results in an
empirical exercise where we attempt to measure the impact of unions on
earnings. Using data from the years 1978-1981 of the Michigan Panel
Study of Income Dynamics (PSID), we estimate: (1) the simple cross-
sectional earnings equations for the four years of our sample; (2) the
usual panel data model in which only the intercept varies across
individuals; and (3) a special case of our general panel data model.
Conventional wisdom states that the cross-section estimates are upwardly
biased due to the positive correlation of unobserved individual specific

attributes - collectively referred to as ”ability” - with union

status. Most often this bias is addressed through a fixed effects
specification of the simple model. We note some criticisms of this
approach and examine a Special case of the fixed effects version of our
model that attempts to deal with the criticisms. For the sake of
comparison, we also estimate (2) and (3) under both sets of random
effects assumptions. In general, we are able to confirm the
conventional wisdom that cross-section estimates of the union wage
effect are upwardly biased.

In Chapter Six, we present a summary of our results and offer some

final remarks on panel data models in which some slopes as well as the

intercept vary cross-sectionally.

CHAPTER THO

FIXED EFFECTS

2.1 Introduction

 

In this Chapter, we consider the estimation of (1-1) and (1-2)
under the weakest set of assumptions; i.e., fixed effects. Since the
number of individual Specific parameters increase with sample size, we
focus our analysis on the estimation of B. In particular, we seek
estimators of B that are consistent in the common panel case of large N
and small T.

First, we review the estimation of the simple model. We derive the
”within" estimator of covariance analysis, which possesses the above 1
consistency Property. This is equivalent to maximum likelihood. “:1
Chamberlain (1980) demonstrates the incidental parameters problem can
also be circumvented through a conditional likelihood approach. He
derives the conditional MLE of B in (1.1) which is also equivalent to
the within estimator.‘

Secondly, we extend the results of the standard model to the more
general model which allows cross-sectional variation in some of the

slopes as well as the intercept. There exists a substantial literature

on the case in which all coefficients vary across 1 (see, for example,

‘\
.1

Judge et a1 (1985, sectioé 13.5). In this case, the model can be
1

1

considered as N seemingly unrelated regressions; and if not all the
coefficients vary across 1, then cross-equation restrictions are

implied. Since we are concerned with an asymptotic theory in which N+00
.TT“

and T is fixed, this treatment is unsatisfactory. Mundlak (1978) has ;
\‘ed~'

1 \ \\»fi

investigated this case, and notes (given standard assumptions about the
errors) the cross-sectionally constant regression coefficients can be
estimated by a version of least squares.

So, we follow Chamberlain, and derive the conditional MLE of B in
(1.1). This is shown to be equivalent to the obvious least squares
estimator, which is a comforting result. Our conclusions are summarized

in section four.

2.2 The Standard Model

Recall the usual representation of a linear regression model with

panel data. This is described in (1.1) as

Y axit's +0 +6 131,0009N; talgooogTo

it i it

where git is assumed to be iid N(o,oz). The n1 are incidental 1

parameters and B is a K-dimensional vector of cross-sectionally constant

coefficients.

As outlined in the previous section, we seek an estimator

of B which is consistent for the usual panel case of large N and small

T. It is well known that the “within” estimator of covariance analysis

~

possesses this consistency property. Let us review this estimation

procedure.

As a matter of notation, define

(2.2.1) Y = Y .

 

 

This allows us to write

(2.2.2) Y1 = X18 + a1 + 81 .

 

Then, we may consider all NT observations as

(202.3) Y :3 X3 + m* + e D

where
Y1
(2.2.4) Y ' Y2 ’
YR

 

 

and

 

>4

N
a.” H

z".

 

 

 

 

Z 000M H

 

 

er 0 ...0 “I

D = I s e = 0 e : , a = a ,
(2.2.5) N T : T. : * ?
0 00.. e1. ’ 1 0N

 

 

 

 

where GT is a T-dimensional vector of ones. Notice D is simply a matrix
of individual indicator (dummy) variables.

The within estimator of B is derived as follows. Let

a _ I '1 a
(2.2.6) MD INT D(D D) D .
Then transform the data by premultiplying (2.2.3) by MD, thereby

obtaining

"DY a Mst + MDDn* +_MDF ,

which reduces to

(2.2.7) MDY a MDXB + Mus ,

since HDD=O.. This transformation changes a vector of observations into
deviations from individual means.l Least squares applied to (2.2.7)

yields the within estimator of 8, defined as

- (x'an)'lx'u Y .1;

(2.2.8) 5 n

W

‘

A

Under the condition that xit varies over time, 8W is consistent as

N*” for fixed T.

It should be clear that the within estimation does not depend on
normality of the errors. When we invoke normality, we see that maximum-
likelihood is, in fact, analysis of covariance. Chamberlain (1980)

derives the conditional MLE of B, which he shows is also equivalent to

O

8 The conditional likelihood approach employs a set of sufficient

“0
statistics for the Q1, removing any incidental parameters problem. The

consistency of Bw is confirmed by the coincidence of the conditional and

joint MLE's.

2.3 A Generalization

 

As described in (1.2), a straightforward generalization of the

standard panel data model is

6 +6 i=1’000’N; tal,...’T .

a c t
x1e B + "it 1 1c

Yit

(’1 ‘1 \\
\

The "it and\6 are L-dimensional vectors of explanatory variables and

i
coefficients, respectively. The remaining variables and parameters are
defined as in the simple model.

This distinguishing feature of this model is that we allow for
cross-sectional variation in some of the slapes as well as the
intercept.‘ Obviously, if ”it is a constant, (1.2) reduces to the simple
model. Again, we seek a consistent (as N+co for fixed T) estimator

OfB. a

Let “1 a ("11’ "12,000, “1'9'. This allows 118 to write

10

(2.3.1) Y1 = x13 + ”151 + 51 .

Then, considering all NT observations, we obtain

(2.3.2) Y = xs + Q 6* + e .

where
W1 61
(20303) Q " ”2 , 6* a 62 o
wN 6N

 

 

 

 

This general model can be estimated by least squares. By analogy to the
within transformation, we premultiply (2.3.2) by the idempotent matrix

Mo, which is defined as

o -1!
u - INT“Q(QQ> Q

Q
M1
(2.3.4) _ u
2
MN
with
, -r~
(2.3.5) M1 =- IT - u1(v1'w1) ”‘1' .

11

Then, we may apply least squares to the transformed model,

. . M = M M ,
(2 3 6) QY QXB + QC

which yields the following estimator of 8:

(2.3.7) B 1 1 1 1 1 1

i i

-1 -1
= ' ' a 2 ' Z ' o
w (x MQX) x MQY ( x M x ) x M Y

The estimatorgw is consistent for fixed T if (x'qu)‘l+o as
N*”. Essentially, this is a condition which requires sufficient

temporal variation in X1 not explained by W1. When W1 is only a
constant term, we have the familiar condition that xit must vary over
time.

As in the standard model, the above is straightforward and does not
depend on normality. We now invoke normality to prove that'é'w is also
the conditional MLE of 8.

Following Chamberlain, consider the 61 as incidental parameters for
which we need to find sufficient statistics. The likelihood of Y,
conditional on the sufficient statistics, will not depend on the 51.
Maximizing this conditional likelihood should provide a consistent
estimator of B.

To prove this, we first show that Wi'Y1 is sufficient for 61.

Consider the (T+L)Xl vector

Ill
04

(2.3.8) y -

12

The vector y has a (singular) multivariate normal distribution with

mean u and covariance matrix X, which with the above partitioning gives

u1 = X18 + "151
= t I
"2 "1 X15 + "1 "151
2
(2.3.9) 211 a IT
2 == 02W 'W
22 1 1
2
212 = 0 W1
2 l
221 0 W1

Standard results on normal distributions imply the distribution of Y1

conditional on Wi'Y1 is (singular) normal with mean

0 a "l _
E<Y1 w1 Y1) "1 + 212222(y2 "2)

(2.3.10)

‘1
u t _
W1(W1 W1) W1 Y1 M1X18 p

and covariance matrix

a ' =3 — 1 = 2
cov(Y1 W1 Y1) z r z z a M

(2.3.11) 9 11 12 22 21 1 .

1

Neither (2.3.10) nor (2.3.11) depends on 61. Hence, Wi'Y1 is indeed

sufficient for 61.

To construct the conditional likelihood function, we need an
O

explicit formula for the conditional density. Here we employ the

standard result [see, e.g., Rho (1973, p. 528)] that if y~N(u.Z) with

13

dim(Y)=P and rank (2) = k<P, then

- k/ZE- 1/2exp [-1/2(y-u)'2+(Y‘U)] 9

(2.3.12) f(y) = (2n)
where 2+ is any generalized inverse of 2, and E is the product of the k

positive eigenvalues of 2. Since M1 is idempotent (with rank T-L),

.1.
(2.3.13) 91 = _§.u1

and
(2.3.14) 6 = (02)(T-L) .
Given the conditional mean of the distribution,

(2.3.15) Y E(Y V 'Y ) = M1(Y1-X18) .

1 " 1 1 1

Therefore, we obtain the following expression for the conditional

density:

(2.3.16) f“‘1’"1"1’ = (zn)‘(T'L)/zo“T‘L>exp[- -1—2-(Y1-X18)'M1(Y1-X18)] .
20

Since observations are assumed independent across i, we may

multiply over i to obtain the conditional likelihood function,

Q
3

(2.3.17) : =- (21v)-N(T-L)/zo-N(T-L)exp[- Z(II—221: (vi-xieriui-xisn.

14

This likelihood function is indeed maximized by 3W as given in (2.3.7).

2. Conclusion

 

We have considered an extension of the usual fixed effects panel
data model in which some of the explanatory variables may have
coefficients which vary across 1.

The results we obtain are essentially the same as those obtained in
the standard, simpler case. Our model may be estimated by least
squares, and the resulting estimates (of the cross-sectionally constant
coefficients) are consistent given a reasonable condition on the
variability of the regressors. Under normality this estimator is in
fact the conditional MLE. However, we cannot claim asymptotic
efficiency. Unlike the direct MLE, the conditional MLE will not, in
general, be efficient in the sense that its asymptotic variance equals
the Cramer-Rao lower bound (see Andersen (1970)). On the other hand, we

do not know of any estimators with superior asymptotic behavior.

CHAPTER THREE

RANDOM EFFECTS

UNCORRELATED WITH THE REGRESSORS

3.1 Introduction

 

Having demonstrated the consistency of the within estimator in the
usual (fixed T) panel case, we must now acknowledge two drawbacks of the
fixed effects specification. First, for small T, the within estimator
is not fully efficient since it ignores variation between individuals.
Secondly, time-invariant explanatory variables are orthogonal to the
within transformation and therefore cannot be incorporated into a fixed
effects model. This is potentially a serious problem, since in many
applications, attention is focused on the coefficients of such variables
(e.g. on the coefficients of race or education in an earnings equation).

As a remedy to the problems of fixed—effects models, a random
effects specification is sometimes proposed. Random effects models take
the individual effects to be iid random variables independent Of the
explanatory variables and the disturbance. Estimation of the simple

a

model, also referred to as the error components model, is well

documented. The basic results are reviewed in the next section.

15

16

Then, in section three, we extend the results of the error
components model to our more general model where some of the slope
coefficients are allowed to vary cross—sectionally. Our model, under
the assumption of random effects, is essentially a Swamy random
coefficient model where some of the coefficients do not vary across i.

Section four summarizes our results.

3.2 Error Components

 

Assume the a in (1.2) are iid N(0,aa2) variables. Furthermore,

i

let 011 be uncorrelated with the columns of (x,e). Under these

assumptions, the usual panel data model has an error components

structure, where

(3.201) vit a a1 + cit 9

and therefore

a 0
(3.2.2) Y t Kit 3 + v

1 it'

Then, considering all NT observations,
(302.3) Y a XB + v o
with v = Dn* + s .

Traditionally, models 11k.‘<3.2.3) are estimated by generalized

least squares. The GLS estimator of B is defined as

l7

‘ = . -l -l , -1
(3.2.4) BGLS (x n x) x n Y ,
where
= g 2 2 .
(3.2.5) o cov(v) 0 INT + 00 DD
and
a 2
-l a l 2 , 2 a a
o +Ta
a
Now, like BW’ BGLS is consistent as N+w for fixed T. But, BGLS is

asymptotically more efficient than B". This efficiency gain is a result
of the exploitation by GLS of both within and "between” (across

individuals) variation. Within estimation only uses variation within

1. However, this efficiency gain disappears and BGLS + 8W as Tr”. 1

An additional advantage of the random effects specification is that
time-invariant explanatory variables can be incorporated into the model
(Recall the within transformation annihilates time-invariant
regressors). So, in the classical panel case (large N, small T), the
error components model may be preferred.

One caveat is in order. The consistency of aGLS depends crucially
on the assumption that the a1 are uncorrelated with the columns of X.
This is often an unreasonable assumption; and, if violated, gGLS is no
longer consistent.2 Ironically, it is inconsistency in the presence of

such correlation that led to thbsoriginal fixed effects model.

N

18

3.3 A Generalization

 

Generalizing the error components model to include cross-

sectionally varying slope coefficients is straightforward. Let
(3.3.1) 5 = 6 + u .

Assume the “1 are iid N(O,A) random variables, with A E COV(u1). As in
the previous case, take the “1 to be uncorrelated with the explanatory

variables and 8. Given (3.3.1), the full model may be defined as

=- I 3
(3.3.2) Y xits+wt6 +v ,

it i 0 it

where

(3.3.3) Vit 3 "it “1 + Cit o

More conveniently, we have

(3.3.4) Y=XB +W60+v
and

(3.3.5) v = Qu* + e ,.

where s

19

w1 u1
W = W2 and u* = u2 .
"N "N

 

 

 

 

Applying GLS to (3.3.4) we obtain the following estimator of 8:

(3.3.6) chs = (x'n' 1/2Mw*n' llzx)”1MWn" l/ZY ,

where

(3.3.7) MW* = 1_w*(u*ou*)‘1w*-=I_Q- 1/2w[(9- 1/2w).9— l’zwl’lw'n‘ 1/2’
and

2

(3.3.8) 9 = COV(v) = 0 INT + QAQ' .

with A 2 IN a A. Note that

(3.3.9) 9‘1 - 11,. HQ + 0(Q'Q)‘1r‘l(o'o)’lo' .
0

where T = 02(Q'Q)-l + A. (For the derivation of n‘l, see the appendix

which concludes this chapter).

~

Now, as in the usual error components model, BGLS is consistent as

S
Nfir for fixed T. Then, do the efficiency results of GLS carry through

’4’”

to this more general model? Indeed, we would be surprised if they did

20

not. It is straightforward to prove that EGLS is efficient relative to

~

EW for fixed T and that'fi B as T+m.

+
GLS

If BGLS is efficient relative to BW’ then [COV(Bw)-COV(BGLS)]

must be a positive semidefinite (PSD) matrix. It is well known that for
any two nonsingular matrices A and B, (A-B) is PSD if and only if (B.1 -
A‘l) is PSD. Therefore, our problem is to show that

1/2 1

x --—— x' M x
02 Q

- 1/2 -
(3.3.10) x'n Mugs

is PSD.

Rewriting (3.3.10), we obtain

1/2 ___g .
X 2 X MQX

X'Q- l/ZII'Q- l/ZW(W'Q-IW)-IW'Q- 1’2] Q-

0

(3.3.11)
x - x'n'IW(w'n'lw)‘lw'n‘lx - —%-x' MQX .
O

= x'n‘l

Now, given (3.3.9),

(3.3.12) x'n‘lx =.—% X'MQX + x'Q(Q'Q)-IP'I(Q'Q)'IQ'X
a A

and

Xin‘lw(w1n‘1w)‘1w'n‘1x

(3.3.13) .
= x'Q(Q'Q)'1r‘IENL(nNL'r‘lzNL)‘IENL'r‘l<Q'Q)‘lQ'x .

 

21

e a N—dimensional vector of one's. So we need

h a
w ere E eN D IL’ N

NL
only show that

X'Q(Q'Q)“’r‘1(Q'Q)‘IQ'x
(3.3.14)

1 'r“1<Q'Q)"IQ'x

- X'Q(Q'Q)_1F-IENL(ENL'F- ENL)’1ENL
is PSD.

For simplicity define

s = x'Q(Q'Q)‘l
(3.3.15) R a 1""1

_ a -1 '
8 SR ENL(ENLRENL) ENL .

Note that R is PD. Then, consider the PSD matrix,
(3.3.16) (s-§) R (s-E)‘ .

Expanding (3.3.16), we obtain

SR§'-§RS'
I -1 I I I I
SRS + 35E NéE NLRB Ni ENL RENL(ENL RENL)ENL (SR)

3
_ I I I I
SRENL(§NL RENL) ENI. (SR)

533' + §R§'

-l
- I I
SRENL(ENL RENL) ENL(SR) ’

22

which is equivalent to (3.3.14). Hence (3.3.10) is psn, and EGLS is

efficient relative to aGLS for fixed T.

However, as T*” this efficiency gain disappears. That is to say,

as T¥~,

llzflnlfl‘ 1/2X - ——%— X'M X]+0 I

1 . -
(3.3.17) T‘ [x n a Q

or
-i.- [X'Q(Q'Q)-II‘_I(Q'Q)-1Q'X
(3.3.18)
I I ‘1 -1 I ‘1 ‘1 I ‘1 I ‘1 I
‘ X Q(Q Q) P ENL(ENL P ENL) ENL P (Q Q) Q X]+O .
To see this, assume %-Q'Q,-%~X'Q, and-%~Q'X have finite nonsingular
limits.

In particular, let
(3.3.19) -l,fQ'Q+U , 711-. Q'x+v, and 7:,— x'Q+v'

Note that (Q'Q)-l+0, This implies r - 02(Q'Q)-1+A+A and r"l+A-l. Then,

1 l l

1. 1.-1-11.-1_1_. .-
033m TXQ(§Q® r %QQ) Tvau
‘.

A” u’v

\

.and

 

23
1 , 1 , -1 -1 , —1 -1 , -1 1 , -1 1 ,
:r-XQ(.-r—QQ) r Ream.“ Em) Em.“ (Too) Tax

-1 , —1 -1
ENL A u v..

1 1

A_ E (NA-l)

+ v'u" ”L

Since -%»A-l+0, (3.3.18) is confirmed, implying EGLS is equivalent
to?w for large T.

Now, the caveat concerning the assumption of independence of the
individual effects with the regressors also carries through to this
model. It remains that GLS has little appeal if purchased from

unreasonable assumptions. In the next Chapter, we consider a random

effects model in which the independence assumption is dropped.

3.4 Summary

In this Chapter, we have considered the estimation of our general
model in a random effects context. Taking the 61 to be iid random
variables independent of the regressors and the disturbance, we sought
to extend the results of the usual error components model to this more
general model.

As in the fixed effects case, the results we obtain for the general
model are essentially the same as those established for the simpler
model. Estimation is by GLS, and the GLS estimator of 8 is consistent

for fixed T. Moreover, for fixed T, E is efficient relative to?w

GLS
GLS¥§ 88 T*”). And, the random effects specification does

allow the inclusion of time-invariant explanatory variables.

is
However, as in the standard'error components model, the

\

unreasonableness of the independence assumption reduces the appeal of

(however,.§

the random effects specification.

QiAPTER THREE

APPENDIX

 

CHAPTER THREE

APPENDIX

We make use of the following fact:
Fact: Provided the relevant inverses exist,
(A+BDB')‘1 = A‘lBEB'A“1BEB'A‘1+A‘IBE(E+D)‘IEB'A‘1,

where E = (B'A‘lB)”1.

The covariance matrix of V is given by

(11.1) n = 021m. + my .

In applying the above fact,

2

let A .. a I, B=Q, D=A, and 1: =- (B'A-IB)-l =- on’Q . Then,

’n'1 = —;. 1 - é q(Q-Q)‘IQ' + <1(<1'<1>‘1 [62(0'0)’1+A1(Q'Q)‘IQ'
0'
(A.2) ' s

1 .-1‘-1 . -1.
--a—2-HQ+Q(QQ)‘I' (00) Q '.

where r - 02(Q'Q)-1+A .

24

CHAPTER FOUR

RANDOM EFFECTS

CORRELATED WITH THE REGRESSORS

The conventional random effects specification allows us to include
time-invariant explanatory variables which cannot be incorporated into a
fixed effects model. In the usual fixed T case, GLS estimation of this
Specification is more efficient than within estimation of the fixed
effects model. However, these improvements usually come at the expense
of an untenable assumption: that the individual effects are
uncorrelated with all the regressors.

In this Chapter we drop this assumption. We investigate random
effects panel data models in which the individual effects are assumed
correlated with some of the explanatory variables. As in the previous
two chapters, we focus on the fixed T case and begin by discussing the
results, established by Hausman and Taylor (1981), for a model with only
an intercept that varies across 1. Briefly, they use prior information
to construct exogeniety restrictions that are then employed to derive a
consistent and asymptotically efficient instrumental variables
estimator. They also derive the conditions under which it differs from
the fixed effects estimator. ~

Next, we generalize the H—T analysis to include 810pes that vary

25

26

across 1. Although the derivation of our estimator is more complicated,
we show that the results of H-T carry through to the general model. In

the last section we offer a brief summary.

6.2 The Hausman—Taylor Analysis

In addressing the problems associated with both the fixed effects

and error components Specifications, H-T consider the following model:

'Y + a + s ,

=- I
(4.2.1) Y x“: s + z 1 it

it i

where 21 is a J—dimensional vector of time-invariant explanatory
variables (notice it is not indexed by t) and Y is a conformably
dimensioned parameter vector. The oi, as in the error components model,
are assumed to be iid N(0,oa2) random variables. However, unlike the
usual random effects specification, H—T take the at to be correlated
with some of the columns of X and Z.

According to H-T, consistent and asymptotically efficient
estimation of all of the parameters in (4.2.1) hinges on our ability to
distinguish columns of x and 2 which are not correlated with the a1.

To examine this, let us adopt a more convenient form of the

model. Let

EN

N
oON
I

(4.202) 2 - 2* n-6,]: ’ 2* '

 

 

2‘"-

27

So, we may write

(4.2.3) Y=XB+2y+V,

where V = Du*+e (as before). Then, suppose we have prior information on

which of the columns of X and Z are correlated with the at. Let

(4.2.4) x = [x1,x2] , z = [21.22] .

where X1 is NTXkl, X is NTsz, 2 is NTle, and 22 is NTij (and

2 l
kl+k2=K, j1+jz=J). For fixed T, assume

1

'D l D
(4.2.5)
.1. ' l ’
N X2 Dn*+ hx¢0 fi-Zz Dn*+hz¢0 .

Now, it should be noted that although the condition E(a1|X1t,21) a 0
fails, consistent, though inefficient estimates of B and Y may still be
obtained from the within regression.l First, we estimate 8 by within,
obtaining a“ defined in (2.2.8). Secondly, we compute the within

residuals, (Y-XBw). From the within residuals we estimate the individual

means, defined as
(4.2.6) .d = PD(Y-X8w) - ZY + Da* + PD(e+error).

__'- s
where PD=D(D'D) 1D' and “error” denotes estimation error from the within

‘

regression. Treating PD(e+error) as an unobservable zero mean

28

disturbance, we attempt to estimate Y from (4.2.6). We know OLS and GLS

are inconsistent for Y since the a are not independent of 22. However,

i
if the columns of X1 (which are uncorrelated with Dn*) provide
sufficient instruments for the columns of Z2 (which are correlated with
Du*), consistent estimation of‘y from (4.2.6) is possible. A necessary
condition for this is that the model must include at least as many time-
varying exogenous variables as time—invariant endogenous variables;
i.e., it must be that k1>jz.

If this condition is fulfilled, instrumental variables applied to

(4.2.6), using as instruments
(412.7) B = [x1,zl] ,
yields the following estimator for Y (denoted Y"):

A a ' -1 ' A
(4.2.8) '7w (2 PBZ) z de ,

where PB = B(B'B)-1B', the projection onto the column space of B. This

estimator is consistent for fixed T, but not fully efficient since it is
calculated from the within-residuals. (Recall a" is not fully efficient
since it ignores between variation.)

. Now, consistent and asymptotically efficient estimates of B and Y
can be derived if these parameters can be identified using prior

information like that given in (4.2.5). Even without (4.2.5) all of the

elements of B are identifiable as is clear from the within regression

fl
(1.2.. X'HDX is nonsingular). waever, without this information, no

‘

elements of‘Y are identifiable. But, given (4.2.5) we have the set of

29

instruments.2

(4.2.9) A = [MD,X1,21]
and the corresponding projection PA, suggesting the following

proposition:

Proposition 1(H-T): A necessary and sufficient condition for the

 

identification of (B,Y) in (4.2.3) is that

x l
PA [xl,zl]

be nonsingular.
And, associated with this rank condition is the order condition to which
we referred earlier:

Proposition 2 (H—T): A necessary condition for the identification

 

of (B,Y) in (4.2.3) is that kl>jz.

SuPpose the parameters of (4.2.3) are identified by the information

in (402-5)o3 Let
(4.2.10) 9- 1/2 a MD + OPD 3 INT-(l-G)PD ,

where 6 = (l-TBIZ) 1,2 = [oz/(02+Toza)]1/2. 4 Then, perform

instrumental variables on the transformed equation,

Q
3

(4.2.11) 9" 1’21: = n" UZXB + 12" 1’22)! + 9' l”V .

30

using the set of instruments given by A in (4.2.9). This procedure
yields consistent and asymptotically efficient estimates of B and Y.

Equivalently, and more computationally convenient,5 we may apply OLS to

(4.2.12) FAQ‘ 1’21 = pAn‘ 1’sz + FAQ“ ”221 + PAS?- 1le .

where PA is the projection onto the column space of A. We denote the
estimators of (B,Y) obtained from (4.2.11) or (4.2.12) as (6*,Y*) .

Now, in evaluating the information given in (4.2.5), three cases
are possible: under-identification, exact-identification, and over-
identification. First, in the under-identified case (kl<j2), §*=§w and
Y* does not exist. Secondly, in the case of exact-identification

(klajz), 3*;EW and Y*=Yw, where Y" is defined in (4.2.8). Finally, if

the model is over-identified (kl)jz), (3*,Y*)¢(EH,YH) and (5*,Y*) is

Q

6
",Yu)'

more efficient than (a
4.3 H-T: An Extension

This section extends the H-T analysis to our more general model.
In doing so, we are able to incorporate time-invariant explanatory
variables and allow for endogeneity of some regressors based on

correlation with the individual effects. The version of our model we

consider here is

I \sI
tY +W1t61+e

,.

- 0
(4.3.1) in x“ B + 21 1t .

31

where zit is a JXl vector of explanatory variables. We treat the zit as
time-invariant in the sense that any time variation in the zit must be
fully explained by the time variation in the wit'7 Again,

let 61:60+u1, where "i are taken to be iid N(O,A) random variables.
However, we now consider the case where the “i are correlated with some

of the regressors.

Consider all NT observations and write our model as
(4.3.2) Y=XB+ZY+W5°+VI

where

(4.3.3) z - z , z = z '

oON

 

 

 

 

zfi
N

pane

r-J

and V = QU* + e. Suppose, for fixed T,

l
N

1 1
— x ' QU*+O _ Zl'QU*+O

I
N 1 N W QU*+O

(4.3.4)

1_ . 1 .
N X2 QU*+gx¢0 'fi 22 QUf+gzto ,

where X and Z are partitioned in the same way described by (4.2.4).

4.
Note that we have assumed the effects to be uncorrelated with some of

‘

the columns of X and Z, and all of the columns of W.

32

Using prior information like (4.3.4), H—T derived two different
estimation procedures for their model. One combined the prior
information with the fixed effects regression to obtain consistent
though inefficient estimates of the parameters. The other applied
identifying restrictions to an n' 1’2 transformation of the equation to
obtain consistent and asymptotically efficient parameter estimates.
Next, we construct analogous procedures for our model.

First, consider estimation based on the fixed effects regression.
The translation of the initial steps of the H-T procedure to our model
is direct and straightforward. We simply estimate 8 by applying OLS to
(2.3.6), thereby obtaining?w in (2.3.7), and then form the least
squares residuals, (Y-Xfiw). At this point the translation becomes more
subtle. The next step in H-T is to calculate d in (4.2.6) by
premultiplying the residuals by PD. This would suggest the d analog in
our model should be constructed by premultiplying our residuals by
PQ - Q(Q'Q)-1Q'. But this is not optimal.

The correct procedure in our case is to calculate8

‘a a a“ l/2(Y_x§") a 9" l/ZZY +9" l/Zwso

(4.3.5)

1/2

+ 9- (QU*+e+error) .

where

(4.3.6) 11" 1’2 -%.u + F

33

and

1/2 1/2 /2]- 1/2(Q,Q)- 1/2

- 2

(4.3.7) F = Q(Q'Q) [o INL+(Q'Q) A(Q'Q)1 Q'.
Then, if k1>j2, 9 we perform instrumental variables on (4.3.5), using
the set of instruments10

_ - 1/2 _ - 1/2
(4.3.8) 3* — n B — n (xl,zl,w) .

This yields

‘Yw
(4.3.9)
= [(le)'n- l/ZPB 9' l/2(Z.W)]'l(z,w)"1$1" l/ZPB E
as. * *
ow '

 

 

-l
with P3* = 3*(B*'B*) B*' .
To understand this, recall the definition of the H-T 9- 1,2 in

(4.2.10). Suppose we substitute (4.2.10) for PD in the calculation

of d. The, instrumental variables using 3* a n‘ 1’2(xl,zl) yields

1/2 - 1/2 -1 . - 1/2 - 1/2 _ *
PB n z) z 9 PB n (Y xsw) .

(4.3.10) 1" = (242'
* 1:

which is generally different from‘Yw in (4.2.8).11 However,

when kl-j2 (the exact identification case), both (4.2.8) and (4.3.10)

‘4
3

are equivalent to12

‘

34

(4.3.11) 'Yw - (B'z)"l B'(Y-x§w) .

So, in the case in which this procedure is appropriate (in the
sense it is equivalent to "efficient” estimation when kl=j2), it makes
no difference in the H—T model whether we use pD(Y-x§w) or
9- l[2(‘17-Xg ) for d or whether we em loy (X z ) n- 1/2()( Z )

w 9 p 1, l or 1, l as
instruments. In our model, it is never the case that substitution of Pb

for 9- 1,2 and B for 3* results in estimates equivalent to the correct

Y" andlgow.given in (4.3.9).

In sum, consistent but inefficient estimation, based on the fixed
effects regression, of all the parameters in (4.3.2) is possible if the
columns of X1 provide sufficient instruments for the columns of 22. As
before, the inefficiency is grounded in the use of the fixed effects
residuals.

Now we turn to efficient estimation of the model. Specifically, we
seek identifying restrictions from which instruments may be formed to

estimate our model consistently and asymptotically efficiently.

Following H-T, consider the set of instruments

(4.3.12) A = Inq,xl,zl,w],

and the projection onto the column space of A, PA. Then, given
Propositions l and 2, rank and order conditions for identification are
easily derived. The order condition, which is mentioned above, is the
same as in H-T; namely that kl>j2 is a necessary but not sufficient
condition for the identificatioh'of B,Y, and 60 in (4.3.2). The rank

\

condition is almost the same as in H-T: a necessary and sufficient

35

condition for the identification of all the parameters in (4.3.2) is
that G'PAG be nonsingular, where G = (X,Z,W).

Suppose the rank condition is fulfilled by the information in
(4.3.4).13 Then, similarly to (4.2.11), we transform (4.3.2) by

- 1/2

our 9 and perform instrumental variables using the set of

instruments

= - 1/2
(4.3.13) A* n (MQ,x zl,w) .

19

Equivalently, we may apply OLS to

(4.3.14) * A* A*
~ 1/2 - 1/
+ PA*Q W60 + PA*Q 2v ,

where PA* is the projection onto the column space of A*. This yields

~*
~* — .— — — -
(4.3.15) Y 8 (6'9 1/2P 9 1IZG) IG'Q 1/2P 9 lle .
* 4* 4*
60

 

 

These estimates are consistent and asymptotically efficient.
Returning to the information given in (4.3.4), we (again)
distinguish three separate cases. Appendix A formally derives the

characteristics of §*; Y*, and‘30* when the model is under-identified,

exactly-identified, or over—identified. Although the derivations

\

differ,14 the characteristics of‘these estimators when kéjz are

.‘

essentially the same as in the H-T model. To summarize,

36

if k1<jz, 8 =8" and (Y ,60 ) does not exist. If k=j2,

~

*av *.~ * ~ ~ ~
8 =8” and (Y ,60 ) = (YW’KOW)’ when (YW’GOW) is defined in (4.3.9).
And, if kl>32, (s ,y '30 ) s (BW’YW’KOW) with the former being more
efficient than the latter.

Finally, there is the following computational note. While it is

possible to calculate 9-1/2 (or F) and estimate the transformed model

(see Appendix B), it is not necessary. Instead, we may directly

calculate
~*
;W ~* -1 2 -1
~ and Y , using 3 (or F ), Recall a is defined in
6014 30 *
0

 

 

(3.3.9) as [—%-NQ+Q(Q'Q)_lF-1(Q'Q)-IQ']. However, this may be expressed
a

88

-l l 2
(4.3.16) :2 =- M + F .
37' Q

So, in the consistent but inefficient procedure,

(4.3.17) 9"1’298 a“ 1’2 = 1241303'n"lli)'113'fi"1 .
*

Simplifying (4.3.9) to

7 _
w I.
(4.3.18) - [(2.11)'F213(B'n'ls)‘ls'1?2(z,w)]'l
6ow (z,w)'FZB(B'n'ls)’lB'Fzfi .

37

And since p = “Q is a projection onto p 9- 1/23, the efficient
A* Q

estimator in (4.3.15) can be computed as

~*

~* ... .. .-
(4.3.19) 1 =- {c'[-% MQ+FZB(B'FZB) ls'rzlc} 1G,[_%MQ+F23(B.F23) lB'leY

~ * O O

5

O

 

 

4.4 Summary

We have considered a random effects specification of our model in
which the unreasonable independence assumption of Chapter Three is
dropped and time-invariant explanatory variables are added.

Following Hausman and Taylor (1981), we derived a consistent and
asymptotically efficient estimator of our model using identifying
restrictions constructed from prior information about which explanatory
variables are uncorrelated with the individual effects. We also derive
conditions under which this estimator differs from the within estimator
discussed in section three of Chapter Two.

This represents a significant improvement over the fixed effects
specification since we now can estimate coefficients of time-invariant

variables and gain efficiency without requiring that all the regressors

\
3

be exogenous.

CHAPTER FOUR

APPENDICES

CHAPTER FOUR

APPENDIX A

To derive the characteristics of §*, Y*, and 3 * when kl-é-jz, we

0
will make use of the following lemma, due to Trevor Breusch (personal
communication):

Lemma: Let H and C be nxm and nxp matrices respectively, such that

H and H'C both have rank m. Then H and H(H'H)"1H'C have the same column

Space. ~*
Case I (under-identification): If kl < j2 , B = 8W and ~ *
6
0

does not exist.

First, consider 8. The ”efficient" estimator of 8, §*, is obtained

(separately) by OLS of PA 9. lle on the part of PA 9- 1/2x orthogonal
* s
- 1/2 - 1/2
to PAJR (Z,W). Now, since A* a n ‘ (M0,xl,zl,W) a (HQ’B*),

p . M + projection onto the part of B* orthogonal to HQ; i.e.,

A* Q

P a h + p B*(B*'PQB*)-IB*'P

(11.1) A. Q Q Q
- 1/2 . — 1/2 - 1/2 -1 , - 1/2
HQ + PQQ B(B fl PQQ B) B Q PQ

But, 9 - n + FB(B'FZB)-1B'F

A* Q

38

39

and
- 1/2 _ 1 , 2 -1 . 2
(4.3) PA*n — a-MQ + FB(B F B) B F .
Therefore,
(A.4) PA n’ 1”x =-l-M x + FB(B'F2B)-1B'F2X
* 0 Q
(4.5) Plan“ 1/2(z,w) = FB(B'FZB)-1B'F2(Z,W)
(A.6) PA 9- 1’22 = l.u Y + FB(B'FZB)‘IB'F2Y .
* a Q

Given the above lemma (with H = PB and C = F(Z,W)). when k1 < 12, the
rank of B determines the rank of 9A n‘ 1,2(Z,W). Thus, both
*
PA 9- l/2(Z,W) and EB share the same column space and null space.
*

Hence, the part of PA 9- 1”X orthogonal to P n. l/2(Z,W) must also be
*

orthogonal to F8. This part of PA 9’ 1/2(Z,W) must also be orthogonal

*
- l 2
to PB. This part of PAH I X is %-MQX. SO. when k1 < 32:

—v* a 1 . 1 -1 1 . 1 . 2 -l . 2
s [(3.th) 6.14Qx] (E-MQX) [a-MQ+FB(B F B) B F ]Y

I -1!
(x MQX) x M Y

(A.7) Q

:38”.
Now, consider Y and 60. Since the column space of PA 9" l[2(Z,W)
‘ *
fl
equals the column space of EB, when k1<jz, PA*D- 1”(LED is not of full

column rank. So, a (J+L) ~dimensiona1 vector 5 such that

4O

_ Y
P Q 1/2(Z,W)E = O and cannot be distinguished from
A* 6
0
~*
Y Y
+ g. Therefore, * does not exist in the under-

60 60
identified
case.

~*~
Case II (exact-identification): If k1=j2, B =8w and

~* «I

Y - Y N

N * ~ .
60 6CW

Again, rank [F(Z,W)] = rank (FB). 80, following the argument in
~* ..
case I, 8 = 8W when k1 = jz.

~* ..
Since 8 = B in this case,

w
«0*
Y .. ~ ..
* - OLS of P Q llzd on P D l/2(Z,W)
'50 As A.»
(A.8)
7w - 1/2~ - 1/2
~ = OLS of PB 9 d on PB 9 (Z,W)
50w * *
Now,
in
(14.9) phn" “221' a 61‘ n0? + Fs(s'rzs)‘ls'rzfi

41

and

(4.10) PB 9" ”23 = n" 1/28(B'n'ls)'ls'n‘12i = 9" l/213(13's1'113)’113'1922i,
*
since B'Mdd = 0. Then note
(4.11) PB 11" 1/2(Z,W) = 11" 1/ZB(B'Q_IB)-IB'F2(Z,W) .
*
So, the regressions of (A.8) yield
?*
(A.12) ~ * =- [(Z,W)'F28(B'F28)’18'F2(z,w)]’1(z,w)'F28(B'FZB)”IB'FZE
60
?w
‘ 2 —1 -1 2 2 -1 2~
(11.13) ,. ~ =- [(Z,W)‘F B(B'n B) B'F (Z,W)‘F B(B'n B)B'F d ,
6
ow

l

which are not generally equivalent since B'FZB t s'n’ B (specifically,

X'F2X=t Kin-1X). However, when k = j2 , B'F2(Z,W) is nonsingular and

1
therefore
3* 3'
(1.14) ~,, - 3" ‘- [B'F2(Z,W)]-IB'F23.

42

8”
~* ~

Case III (over—identification): If k1>j2 , t Yw
~ * ~

60 60”

 

 

 

 

and the former are more efficient.
If kl>j2, rank (FB) ) rank F(W,Z). Then, the column space of

P Q— l”(Z,W). Intuitively, this means that there are parts

A*
of P 9“ 1/2x orthogonal to PA 9‘ l/2(Z,W) even though they are not

A* *

~* ~
orthogonal to F3. Hence 8 ¢ 8".

..* ~ ~ ~ ~
Since 8 t 8W in this case, (Y-XB*) i d = (Y-XBW)' Additionally,

I

there is the general nonequivalence of B'fl- B and B'FZB (which is not

~* ~
Y Y"
mentioned by H-T). So, for two reasons, ~ * t ~ . And,
60 6OW
~* ~
Y Y“
because ‘~ * is asymptotically efficient, ~ is not in this
60 6011

case.

(BAPTER FOUR

APPENDIX B

We now consider the computation of the consistent and
asymptotically efficient estimates defined in (4.3.15). First, we need
a consistent.n— 1,2. More precisely, we need consistent (as N+¢)

estimation of 02 and A, since

- 1/2 1 l
9 = M +F = M
:7 Q 37 Q
(13.1)
+Q<Q'Q)‘ ”2 [azimm'm”2(1NaA)<0'0”21‘ “(on)" ”20'.

As in the H-T model, 02 is derived from the within residuals. Let

3'1 =- th, E = MQe, and hi =- I—X(X'X)-1X'. Then, the SSE from the within

regression in (2.3.6) may be written as
(13.2) "12'”? = Z'E' - E'§(i'i)

and therefore

(3.3) a - m?‘"i? . W Y Q ")0

43

This estimator is consistent:

~~ 2

(B.4) plim o = plim N T-2

r<2

Y "2
since 3'? = s 'e - e'Q(Q'Q)_1Q'e .

A consistent estimator of A may be constructed as follows. Perform

instrumental variables on

(13.5) (Y-xsw) =- 2), + ms + 011* + e ,

0

using B = (X1,ZI,W) as instruments. From the IV residuals, we can form

(8.6) 5 = ee' - . ,

where e1 denotes a TXl vector of IV residuals.. Obviously, a is not a

consistent estimator of Q. But,
(8.7) 050 =- ozo'o + Q'QUNDMQ'Q .
and

(3.8) (Q'oflo'fiom'ofl - 02(0'0)’1 - 11,84.

\
C

This suggests

45

(8.9) 2 = I(wi'wi)-lwi'eiei'wi(wi'wi) -32(w1'w1)-1]

1
_.):
Ni

Since the IV estimates of B, Y, and do are consistent,

A: 1 I -1 I I
plim A plim fi-i [(W1 N1) W1(W1u1+€1)(wiu1+€1) W1(W1 W1)]
1 2 , -l
filo ("1"1)
1
(8.10)
= plim-£2 uu '
N i i
1
3A.
Hence, A is consistent.
2 ‘ - 1/2

Given a and A, Q can be easily calculated once the
troublesome terms of F are decomposed into matrices of their eigenvalues

and eigenvectors. For example, (Q'Q)- 1’2 = PD- lIZP', where P is an
orthogonal matrix of the eigenvectors of Q'Q, and D is a diagonal matrix '
of the eigenvalues of Q'Q.

To implement the efficient estimation procedure defined by

(4.3.14), note that (4.3.2) and (4.3.4) may be expressed as a system of

one structural equation and two reduced form equations:

1/2

(8.11) 9" 1’2)! =- n‘ 1’sz + n” 1,227 + n’ ”21:60 + n' v
- 1/2 - - 1/2 - 1/2 - 1/2 - 1/2

(8.12) n x:z fl 9 xlnllm 2111 12m 1111le MQIIM-l-nl
- 1/2 .. - 1/2 ‘ - 1/2 - 1/2 - 1/2

(8.13) n 22 n xlnnm zlnzzm 171123441 11011244412.

46

1/2X and 9- 1,22 are calculated from their

2 2

reduced forms. However, estimation of (8.12) and (8.13) is made

The fitted values of Q-

cumbersome due to the presence of HQ. Alternatively, we may "parse out

"Q in both reduced form equations by premultiplying by PQ. This allows

us to calculate Q. 1,222 from a regression of F22 on FB. To obtain

9- l/ZXZ, we must replace MQX2 ”parsed out“ by Pb. 80, we form

fl_ 1’2 = M + P Q. l/ZX = M + FX where PX are the fitted
x2 0x2 Q 2 0x2 2' 2

values obtained from a regression of FXZ on PB.

Finally, 0. 1IZXZ and n. 1/222 can be combined with X1 and Zr; in a

least squares regression to obtain consistent and asymptotically

efficient estimates of B, Y, and 60.

CHAPTER FIVE

ESTIMATION OF UNION

WAGE DIFFERENTIALS

The previous three chapters have presented some theoretical results
on the estimation of panel data models with cross-sectional variation in
both slopes and intercept. In this chapter we will apply these results
to the problem of measuring the impact of unions on wages.1 For the
purpose of exposition let us begin with a brief methodological review.

Prior to the existence of large longitudinal data sets, attempts to
measure the union wage differential were characterized by the estimation
of various forms of a standard cross-sectional earnings equation. To
consider a specific form, let

13 1,0...“ ’

(5.1.1) Y = x1'8 + U 6 + e

i i i ’

where Y1 is the natural logarithm of the wage, X1 is a K—dimensional
vector of explanatory variables (e.g., education, experience, sex), 01
is a binary variable indicating whether individual i's job is covered by
a union contract, and 61 is an iid disturbance. Ordinary least squares
estimation of (5.1.1) provides an estimate of 6 which is interpreted as

the union relative wage differential.

47

48

One of the usual objections to estimates based on (5.1.1) is that,
given high union wages, firms will hire workers of commensurate quality,

resulting in

(5.1.2) 8(6101) > 0 .

In other words, there exists an omitted variables problem, since in the
simple cross-section we are unable to control for those individual wage
determining attributes - collectively referred to as ”ability" - which
are correlated with union status. Consequently, 81 is positively
correlated with U1 and the least squares estimate of 6 is biased upward.

With the availability of panel data, these individual attributes,
or effects, can be taken into account. Typically this has been done

within a fixed-effects framework,2 where

1=l’.0.,N’ FI’OOO,T O

a O

+ +
it t5 a e

i

and n1 is the individual effect, assumed constant over time. The

inclusion of the<11's can be viewed as a form of differencing. Indeed,
this is the essence of the within transformation, to difference away the
correlation between the error and union status. Then, applying least
squares to the within-transformed equation yields an unbiased and
consistent estimate of the union wage differential.

_The notion of omitted variable bias is supported by those studies
in which models like (5.1.3) are estimated. Invariably these analyses
report lower union wage effectsfiihan do cross-section studies.3

\

Now, models like (5.1.3) are also subject to criticism. First,

49

there is the issue of measurement error with respect to the union status
variable. This problem is not addressed in this paper, but Freeman
(1984) and Chowdhury and Nickell (1985) correctly point out that errors
in the reporting of union status are accentuated in longitudinal studies
since the estimation of panel data models usually depends on a small
number of union status changes.

A second criticism, which we do consider, is offered by Stewart
(1983). He notes that the standard fixed effects model (varying
intercept only) ignores the possibility that the individual effects may
be sector dependent. In recognizing that the processes which determine
wages are different in the union and nonunion sectors, Stewart
constructs a model which allows the union wage differential to vary
across people. As we will see below, his model is just a special case
of the fixed effects version of our model presented in section 3 of
Chapter Two.

We may express Stewart's model as
a 0

where

(5.1.5) ¢it

S

Clearly, if A 8 1 (5.1.4) reduce; to the standard fixed effects model.

If’l ¢ 1, then the individual effects are sector dependent. Stewart

50

suggests A<l since union contracts, in their standardization of wage
rates, ”may alter the link between pay and productivity." That is to
say, the individual effects may have greater impact in the nonunion

sec tor.4

The Stewart model is seen as a Special case of our own when we

write

a I
(5.1.6) Yit x” B + out;1 + a1 + an ,

where 61 =‘Y + (X-l)011 is the union wage differential for individual
i. For simplicity, in our empirical application of Stewart's model, we
consider an unrestricted version of (5.1.6), one in which 61 is not
constrained to be a linear function of the individual effect. This
allows us to investigate, with little computational difficulty, fixed
and random effects estimation of Stewart's model.5

In the next two sections, we present our empirical results on the
union wage effect. We compare estimates from the three structural
frameworks described in (5.1.1), (5.1.3), and (5.1.6). In addition, we
contrast the fixed and random effects estimates of the simple and

general panel data models. Our conclusions are presented in section

four.

5.2 Data

 

Our data set consists of 1706 heads of household taken from the

k
Michigan Panel Study of Income Dynamics (PSID). We consider the four

‘

years 1978-1981, and restrict the sample to include only those

51

individuals who report receiving a nonzero wage (each year), are not in
the military, and were between the ages of 18 and 65 in 1978. The
variables used are those commonly found in an earnings equation. They
are described in some detail below. Table 1 gives the means and
standard deviations of these variables for each of the four years and
for the pooled sample.

The dependent variable is the natural logarithm of the wage (LW).
The set of regressors includes the following continuous explanatory
variables. Education (ED) is measured by years (grades) of schooling
completed. Notice this variable is time-invariant. Experience (X) is
recorded as the number of years that an individual reports he/she has
worked full time. Now, in the PSID, the values of experience in 1979-
1981 are the same as the value of experience recorded in 1978.
Therefore, we construct experience responses for the last three years.
We do this by simply adding one year of experience to the 1978 value for
each of the following three years. The tenure (TEN) variable is not the
preferred "tenure with the employer.” For 1978-1980, the PSID only has
observations for ”tenure on the job.” So, this is what we use, measured
in months. As is standard procedure, we also include a quadratic in
experience (X2) and tenure (TENZ).

Other individual characteristics are defined by a set of binary
variables. If an individual lives in a standard metropolitan
statistical area (i.e., a city with population > 50,000), city size
(SMSA) is given the value 1. .Similarly, we use a redneck (REG) dummy,
set equal to 1, if an individual lives in the South. Marital status
(MARR) is recorded with a valuefibf 1 if married. The gender variable

(SEX) is equal to 1 if an individual is male. An individual's race

52

'LAILEI

Means (and Standard Deviations)

 

 

Variables 1978 1979 1980 1981 8001.81)
1w 6.3706 6.4779 6.5773 6.6616 6.5218
(.4843) (.4825) (.4707) (.4688) (.4902)
8) 12.1219 12.1237 12.1237 12.1237 12.1232
(2.8715) (2.8687) (2.8687) (2.8687) (2.8687)
x 13.9461 14.9461 15.9461 16.9461 15.4461
(11.0018) (11.0018) (11.0018) (11.0018) (11.0561)
x2 315.4619 344.3540 375.2462 408.1383 360.8001
(412.2647) (433.4538) (454.7202) (476.0537) (446.0018)
TEN 70.0305 77.7907 88.4877 119.2339 88.8882
(77.6660) (81 .0625) (90.0519) (103.6219) (90.6037)
Tm? 10932.7456 12618.6829 15936.4297 24947.4209 16108.9448
(23348.9671) (25332.7136) (30691.7007) (40331.7051) (31108.8640)
5184 .6940 .6858 .6899 .6858 .6889
(.4610) (.4643) (.4627) (.4643) (.4630)
RE: .4302 .4314 .4308 .4326 .4313
(.4953) (.4954) (.4853) (.4956) (.4953)
MAR .7145 .7216 .7315 .7292 .7242
(.4518) (.4484) (.4433) (.4445) (.4469)
sr-x .8224 .8224 .8224 .8224 .8224
(.3823) (.3823) (.3823) (.3823) (.3822)
sacs .6676 .6676 .6676 .6676 .6676
(.4712 (.4712) (.4712) (.4712) (.4712)
CB .3458 .3646 .3581 .3628 .3579
(.4893) (.4825) (.4707) (.4688) (.4902)

N= 1706

53

(RACE) is defined as 1 if he/she is white (the nonwhite category only
consists of blacks). Union coverage (i.e., whether a person's job is
covered by a union contract), rather than union membership, is chosen as
the union status variable. It (CB) is set to 1 if a person's job is
unionized. Finally, we include a series of one—digit occupation dummies
to control for the effect of occupation on wages.

In the next section we combine the data with the structural models

described in 5.1 to examine the impact of unions on wages.

5.3 Estimation and Results

 

Here we present the results of estimation. Our primary concern is
to gain a better understanding of the union wage effect. Of related
interest is a comparison of the results obtained from the simple and
general panel data models (in particular, the within and H-T
estimators). We proceed by considering, in turn, the models described
by (5.1.1), (5.1.3). and (5.1.6).

As mentioned in section one of this Chapter, the first attempts to
measure the union wage differential were simple cross-section studies
where ordinary least squares was applied to an equation like (5.1.1).
we replicate this procedure on each of the four years of our sample.
These cross-section results are given in Table 2. In general, our
estimates are very similar to those obtained in other cross-section
studies. Two exceptions are the coefficients of tenure and tenure-
squared. In no year are these coefficients estimated with much

\

precision. This is due, at least.in part, to the fact that our tenure

‘

variable is defined as "tenure on the job” rather than the preferred

54

TAILEZ

(toss—Section Estimates

 

 

 

Dependent Variable: 100 Wage
Emlanatory

Variables 1978 1979 1980 1981

n) .0472 .0481 .0505 .0547
(.0039) (.0038) (.0037) (.0037)

x .0238 .0235 .0214 .0167
(.0028) (.0029) ( .0029) (.0032)
x2 -.00044 -.00047 -.00038 -.00032
(.00007) (.00007) (.00007) (.00007)

M om372 om336 -0m264 omo
(.00360) (.00336) (.00300) (.00288)
TENZ -.00004 .0000006 .00014 -.00012
(.00013) (.0001253) (.00010) (.00009)

3154 .1173 .1164 .1170 .1193
(.0187) (.0183) (.0177) (.0179)

RE; -0048]. _00547 “00520 -00509
(.0185) (.0183) (.0177) (.0181)

MARR .0700 .0827 .1102 .1325
(.0263) (.0264) (.0264) (.0272)

$81 .2585 .2299 .2035 .1865
(.0334) (.0331) (.0327) (.0331)

Ram .1071 .1389 .1204 .1305
(.0202) (.0201) (.0196) (.0199)

CB .1758 .1573 .1915 .1809
(.0188) (.0182) ( .0179) ( .0182)

i2 . _ .550 .551 .556 .535

N = 1706 . Standard errors in parentheses

55

"tenure with the employer.” The coefficient on union status (8) ranges
from .157 to .191, which implies a union wage differential of 17 to 21
percent.6 Like our parameter estimates in general, this range of
estimated union wage differentials is in agreement with earlier cross-
section results. Perhaps also not surprising is the decline and rise in
the union wage effect over the four year period. This may be explained
by the incompleteness of union cost-of-living adjustments (COLA's)
during the inflationary period of the late 1970's, and the
countercyclical nature of the union wage differential (demonstrated by
the effects of the 1980 recession). .

Since we strongly suspect the selectivity of union workers causes
an upward bias in the cross-section estimates of the union wage effect,
we turn to the panel data model of (5.1.3), where each individual has a
unique intercept (oi, the intercept effect). For the sake of
comparison, we first estimate this model by OLS. These results are
presented in the first column of Table 3. Notice they are vary much
like those obtained by the simple cross-section regressions. This is
because OLS ignores the longitudinal nature of the data. In other
words, OLS assumes no correlation between the explanatory variables and
the individual effects. So, the 8 calculated by OLS is upwardly biased
for the same reason as the union status coefficients given in Table 2.

The claim of this bias in the cross-section and OLS/panel estimates
of 6 is supported by the fixed effects results given in the third column
of Table 3. Performing OLS on a within transformation of the data

(i.e., deviations from individual means) yields an estimate of the union

\
3

Panel Estimates:

56

TAILE3

Dependent Variable:

 

Intercept Varying

 

 

 

Explanatory
Variables OLS GLS Within HT
83 .0514 .0728 .1573
(.0020) (.0152) (.0677)
x .0233 .0960 .1164 .1155
(.00151) (.0017) (.0027) ( .0027)
X2 -0m5 “.le -om3 -0m2
(.00004) (.0001) (.00007) (.00007)
TEN .00648 .00360 .00096 .00180
(.00156) (.00003) (.00003) (.00003)
Tm? -.0000003 -.0000011 -.0000003 -.0000005
(.0000004) (.0000002) (.0000002) (.0000002)
(.0094) (.0156) (.0157) ( .0157)
RE -00519 ".0427 ’00416 “.0308
(.0094) (.0285) (.0298) (.0291)
“ARR o lm3 omsa "' om27 "omds
(.0138) (.0132) (.0131) ( .0131)
381 .2298 .3393 .2912
(.0171) (.1065) (.1154)
(.0104) (.0917) (.1761)
(:8 .1803 .0975 .0945 .0941
(.0095) (.0096) (.0096) (.0096)
E2 .536 ‘. .397 .405 .403
N - 17%. Standard errors in parentheses.

57

status coefficient of .094, a 65 percent decrease from the cross-section
estimate.7 With the exceptions of experience and experience-squared,
the impacts of the other explanatory variables are reduced by allowing
for the individual effects. The within estimates are unbiased and
consistent even if the effects are correlated with the regressors.
However, they are not fully efficient and the within transformation
removes all time-invariant variables (ED, SEX, RACE) from the model. As
we stated at the outset of Chapter Three, this is a potentially serious
problem if one is interested in, for example, the return to schooling.

An alternative specification of (5.1.3) is the error components
model, which was reviewed in section two of Chapter Three. In this
case, the a1 are taken to be iid random variables uncorrelated with the
regressors. If this assumption is true, generalized least squares based
on consistent estimates of the variance components will produce
consistent and asymptotically efficient estimates of all the parameters,
including the coefficients of the time-invariant variables; and GLS is
simply computed. It is equivalent to OLS on a (1-0) differencing of the
data, where 6 - [oz/(02+Toza)] 1’2 (see Note 4 of Chapter Four).

2 and a2 can be derived from the within

Consistent estimates of o a

residuals (see H—T, p. 1384). In our case,
(5.3.1) 0 = .025, on = 3.418, 9 B .043 .

The results of GLS estimation are listed in the second column of

Table 3. Now we have argued that there exists an omitted variables

is
problem in the cross—section. Since GLS also assumes no correlation

'D

between the effects and the explanatory variables, the error-components

58

specification is vulnerable to the same criticism leveled at the cross-
section estimates. The GLS estimate of 6 (.098) is higher than the
within estimate (.094). However, it is surprisingly low given the
argument of omitted variables bias. In any case, if the effects are
correlated with union status, the GLS estimate is biased and
inconsistent. The coefficient on education, which we also expect to be
correlated with the effects, is closer to the OLS estimate. In sum,
unless we are prepared to reject the story of selectivity of union
workers, the fixed effects specification should be preferred over the
error components model for measuring the union wage differential.
However, we need not rely solely on the fixed effects version of
(5.1.3). Instead, we may let the n1 be random variables correlated with
the regressors. Then, following the H—T coefficient procedure described
in Chapter Four, we are able to include time-invariant explanatory
variables, and obtain consistent and asymptotically efficient estimates,
thereby meeting the objections to the OLS, within, and GLS estimators.
Implementation of this procedure requires that we be able to
distinguish those regressors that are correlated with the effects from
those that are not. Since the effects presumably control for ability,
obvious choices as endogenous variables are union status and
education. In addition, applications of this technique by H-T and
Chowdhury and Nickell lead us to include experience and tenure (and
their quadratic terms) as endogenous variables.8 Recalling the H-T
order condition for identification of the model, we know that we need at
least as many time varying explanatory variables (SMSA, REG, MARR)
uncorrelated with the effects ab‘we have time invariant explanatory

\

variables correlated with the effects (ED). Clearly this condition is

59

fulfilled here.9 In fact, the model is over-identified. Hence, the H-T
estimator should yield an efficiency improvement over the within
estimator for this model.10

Beyond determining which explanatory variables are endogenous, the
only major difficulty is computational. However, the computational
difficulty can be reduced. Following H-T (Appendix B), the fitted
values of the endogenous variables (both time-varying and time-
invariant) can be calculated from their reduced forms in a manner that
reduces the size of the estimation problem from sample size NT
regressions to sample size N regressions. The predicted values of the
time-invariant endogenous variables are obtained from a regression of
these variables on the time-invariant exogenous variables and the

individual means of the time varying exogenous variables. The

 

calculation of the predicted values of the time-varying endogenous
variables is almost as simple. They are derived from a regression of

the individual means of the time-varying endogenous variables on the

 

time-invariant exogenous variables and the individual means of the time-
varying exogenous variables. To these predicted individual means-we
must add the true deviations from means since the correct prediction of
the time-varying endogenous variables is calculated with the set of
instruments that includes the projection used to transform the data into
deviations from means. Then, fitted values are combined with the
variables that are uncorrelated with the effects to obtain consistent
and asymptotically efficient estimates of 8 and d.

The results from the H-T efficient estimation of (5.1.3) are
presented in the last column of\Table 3. First, note that the estimated

‘

coefficient on union status is essentially the same as that calculated

60

by within. This should not be surprising since correlation between the
effects and union status is assumed by this specification. Education is
also taken as endogenous and the effect of this is a marked increase in
its coefficient, to .157, over both the OLS (.051) and GLS (.073)
estimates. This rise in the returns to schooling is not in accordance
with a story of positive correlation between education and ability
leading to an upwardly biased OLS (or GLS) estimate. But, H-T point out
that when the amount of education is endogenous, there may be a negative
correlation between ability and the amount of education chosen.ll Their
application of this procedure to an investigation of the returns to
schooling reveals a similar rise over OLS and GLS estimates.

The coefficients of the other time-varying explanatory variables
are reasonably close to the within estimates. The sex and race
parameters, which cannot be estimated by within, are both considerably
different from either the OLS or GLS estimates. In particular, notice
the effect of race has essentially vanished.

Now, the H-T procedure produces parameter estimates which support
the omitted-variables bias argument, and are consistent and fully
efficient. However, within the framework of (5.1.3) we are still
vulnerable to the criticism of Stewart. Thus, we next consider the
unrestricted version of (5.1.6), where the union wage differential is
allowed to vary cross-sectionally. We examine (as we did for the simple
model), the fixed-effects and two random-effects specifications of this
special case of our general panel data model.

First, we take 6 and mi in (5.1.6) to be fixed over time. The

i
\
fixed effects version of the genbral model is estimated by performing

\

OLS on the within (deviations from means) transformation of the simple

61

model (see chapter two). These results are presented in the second

column of Table 4. Since T=4, the individual 61's cannot be estimated

equals .093, which is only

A

consistently. However, their mean, fi-i 61
slightly less than the usual fixed effects estimate. In general, the
coefficients of all the explanatory variables are not too different from
those calculated by within on the simple model, even though the effects
are now composed of both 61 and n1. Finally, notice the time-invariant
variables are (again) eliminated by the transformation.

We have consistently argued that any estimation method which
assumes no correlation between the regressors and the effects is
inappropriate for measuring the union wage effect. However, for the
sake of comparison, let 61 and a1 be iid random variables uncorrelated
with the explanatory variables. With this specification we estimate by
GLS, but we first need consistent estimates of the variance of sit and

the covariance matrix of (01,01). Using the estimates defined in

Appendix B to Chapter Four, we obtain

“2

o = .032
(5.3.2)
. .774 .003
A a o

.003 .043

Given (5.3.2) we can calculate a consistent estimator of n" 1/2

(again, see chapter four, appendix B). Then, GLS is obtained by
performing an.a- 1’2 transformation on the data and running OLS.

The resulting estimates, which are only consistent if the
uncorrelatedness assumption is‘true, are given in the first column of

‘

Table 4. As in the simple model, the GLS estimate of the union wage

62

effect (.095) remains higher than, though close to, the fixed effects
estimate. The estimates of the other parameters, with the exception of
the coefficients of the experience and tenure variables, are very
different from those calculated by GLS on the simple model. One reason
for this must be the inclusion of the 61 as part of the random effects,
which are assumed to be uncorrelated with the regressors. These results
should not be too upsetting, however, since they are based on an
unrealistic assumption and are therefore biased and inconsistent.

Finally, we address the drawbacks of within and GLS. Now we take
the a1 and 61's to be random variables correlated with the explanatory
variables. To this version of (5.1.6) we apply our extension of the H-T
analysis. The variables taken to be endogenous in the simple model are
also assumed to be correlated with the effects here. The only exception
is union status, which is now exogenous.12 Like the simple model, our
model is over-identified, and therefore the H-T procedure should yield
consistent and asymptotically efficient estimates of the general model.

These estimates are presented in the last column of Table 4. After
distinguishing those variables that are uncorrelated with the effects,
as in the simple model, the main difficulty is computational. In an
analogous fashion to the procedure outlined for the simple model, we
calculate the fitted values of the endogenous variables from their
reduced forms. For the general model, however, efficient estimation
requires the reduced forms be transformed by

9 - 1,2 (constructed from 02 and A given in (5.3.2)). (General reduced

form expressions for those variables correlated with the effects are

62

63

TAILE4

Estimates: Slopes and Panel Intercept Varying
Dependent Variable: 1m thge

 

 

Ebcplana toxy
Variables GLS Within 111‘
n) .3364 .3743
(.0045) (.0227)
x .1236 .1141 .1677
(.0031) (.0026) (.0123)
x2 -.00142 -.00061 -.00095
(.00008)- (.00007) (.00033)
TEN omzm OM72 -0m168
(.00120) (.00096) (.00480)
Tm? -.00004 -.00003 -.00001
(.00004) (.00003) (.00016)
(.0175) (.0153) (.0328)
R .1272 -00220 -o%77
(.0297) (.0290) (.0556)
MARR -.0212 -.0091 -.0208
(.0151) (.0126) (.0237)
SEX 1.1048 .3498
(.0573) (.1658)
RACE -.0610 -.4510
(.0522) (.1135)
, (13 .0955 .0930 .0854
' (.0074) (.0128)
E’ .911 . .403 .791

 

N = 1706. Standard errors in parentheses.

64

defined in chapter four, appendix B).

1/2 transformed) reduced

Direct application of OLS to the (0 -
forms is cumbersome since the set of instruments includes not only the
exogenous variables, but also the projection used to transform the
fixed-effects model. This projection can be ”parsed out” in both
reduced forms before performing OLS. However, to obtain the correct
predicted values of the time-varying endogenous variables, the ”within”
transformed exogenous variables must be added back to the fitted values
calculated from the least-squares regression with the ”within"
projection ”parsed out”. (This is formally described in appendix B to
chapter four). The predicted values of the endogenous variables are
then combined with the (0 - 1’2 transformed) exogenous variables in an
OLS regression to obtain consistent and asymptotically efficient
estimates of the parameters in the general model. The union status
coefficient is estimated to be .085, implying a union wage differential
of 8.9 percent. This is very close to the mean of the 61's derived from
the within regression. As in the simple model, the H-T estimate of the
return to schooling is much higher than that calculated by the original
OLS regression. This lends further support to the story offered by H—T
of a negative correlation between ability and education when the amount
of schooling is made endogenous. The coefficients on the other time-
varying explanatory variables are closer to the within estimates of the
general model than those obtained by GLS. However, one particularly
peculiar result is the estimate of the race parameter. In the simple
model the race effect essentially disappears. In their return to
schooling exercise, HJT report b‘similarly reduced effect of race on

‘

earnings from their efficient procedure; but, a negative race effect of

65

such magnitude is completely unexpected. Why the inclusion of 61 as

part of the individual effect would lead to this result is unclear.

5.4 Conclusions

 

Table 5 contains a tabular summary of the union wage differentials
obtained from the different models and estimation techniques we have
considered. Because of the selectivity of union workers it is likely
that ability is correlated with union status. If union status is
observed without error, then we conclude that there is no real
justification for measuring the union wage differential from the simple
cross-section, or from a panel estimation method which does not allow
for correlation between the regressors and the individual effects. The
simplest means of estimating the union wage effect is within estimation
of the usual fixed effects model. These estimates are unbiased and
consistent. Similar, and likewise consistent results may be obtained by
estimating a fixed effects version of our general model (e.g., Stewart's
Specification). In this way we can let the union wage differential vary
cross-sectionally (or allow for sectoral dependence of the effects).
Finally, if we are also concerned about the coefficients of the time
invariant explanatory variables, we can apply H-T to the simple
(intercept varying) model and obtain consistent and asymptotically
efficient estimates of all the parameters in the model (provided the
parameters are identified). In the general model, the H-T analysis
provides somewhat peculiar results for the coefficients of the time-
invariant variables. This may be due to misspecification, since union

‘

status is not endogenous in the general model. In sum, all of the

()6

TABLELS

Percent Union Wage Effects

 

 

 

Cross-Section Panel OLS WITHIN GLS HT
1978: 19.22 Intercept Varying: 19.76 9/91 10.24 9.87
1979: 17.03 Slopes and
1980: 21.11 Intercept Varying: 9.75 10.02 8.92

1981:

19.83

 

67

estimation methods which allow for correlation between the individual
effects and the explanatory variables yield union wage differentials

that are much smaller than those obtained in simple cross-section or

OLS/panel estimation.

CHAPTER SIX

SUMMARY AND CONCLUSIONS

A regression function which does not control for omitted or
unobservable variables that are correlated with the explanatory
variables will not identify the parameters of the model. Conventional
estimation of such a regression function will produce biased and
inconsistent results. However, the availability of panel data allows us
to control for these omitted or unobserved characteristics through the
inclusion of individual specific parameters or effects.

The focus of this study is on the estimation of panel data models
in which there is cross-sectional variation in some of the slopes as
well as the intercept. A well established literature exists on the
estimation of the simpler case in which only the intercept varies cross—
sectionally. The results for the simple model are a function of the
assumptions about the individual effects. We identify three different
cases: (1) fixed effects, (2) random effects uncorrelated with the
regressors, and (3) random effects correlated with the regressors.

For each set of assumptions we review the appropriate method of
estimation for the simple model_and then extend these results to our
more general model. In both the review and the extension, our primary

interest is in estimation techniques that behave well when we have a

68

69

small number of time observations on a large cross-section, a typical
case with longitudinal data.

First, we consider the weakest set of assumptions: fixed
effects. In this case, the individual effects, while differing across
people, are assumed to be constant for each person. Traditionally, the
simple model is estimated by OLS on the within transformed data. The
within estimator is consistent for fixed T. We develop the analogous
transformation for the general model and show that OLS on our
transformed model is also consistent (given a reasonable assumption
about the variability of the regressors) in the case of fixed T. In
addition, we prove that under normality, the within estimator of the
general model is also the conditional MLE.

Two drawbacks of the fixed effects specification are noted. The
first, and perhaps less serious, is that (for both models) the within
estimator is not fully efficient when T is small. Secondly, time-
invariant explanatory variables are orthogonal to, and therefore
eliminated by, the within transformation.

One solution to these two problems is to adopt a random-effects
specification where one assumes the individual effects are iid random
variables uncorrelated with the regressors. In the model with only a
variable intercept, estimation is by GLS. The GLS estimator is
consistent and efficient relative to within when T is small. We derive
the GLS estimator for our model and show that the results from the
simple model carry over to the general model. However, the consistency
of GLS in both models hinges on the assumption that the effects be
uncorrelated with the explanatoTy variables. This assumption is not

justified in most empirical applications.

70

Finally, we consider the case of random effects which are
correlated with the regressors. Under this set of assumptions, Hausman
and Taylor (1981) derive an instrumental variables procedure for the
simple model that allows the inclusion of time invariant variables, and
yields consistent and asymptotically efficient estimates. Their IV
estimator is unique in that it uses the included exogenous time-varying
variables as instruments for the endogenous time-invariant variables.
Specifically, their procedure requires that we have at least as many
time-varying explanatory variables uncorrelated with the effects as we
have time invariant explanatory variables correlated with the effects.
This is essentially an order condition for the identification of the
parameters in the model. We apply the HJT analysis to the case in which
slopes and intercepts are allowed to vary. An analogous order condition
for the identification of our model is obtained, and a consistent and
asymptotically efficient IV estimator is derived. Then, following H-T,
we detail conditions under which the efficient IV estimator of our model
differs from within.

After our theoretical examination of panel data models in which
slopes and intercepts are allowed to vary, as an empirical exercise we
consider the estimation of unions' impact on earnings. The issue of the
union wage effect offers an appropriate empirical question for the
application of a special case of our model.

The first attempts to measure the union wage effect were conducted
within the framework of a simple cross-sectional earnings equation
containing a union status dummy. These studies ignored the selectivity
of union workers. Given high union wages, firms tend to select more

5

able workers, producing a positive correlation between union status and

71

the disturbance. Put differently, there is an omitted variables problem
since ”ability” is ignored in the cross-section regression. The result
is a measure of the union wage differential which is upwardly biased.

With the availability of panel data, the typical response to the
biased cross-section results has been through the standard fixed effects
panel data model. Here individual specific intercepts (effects) are
included to control for ability. This can be viewed as a form of
differencing, which is the essence of the within transformation.
Estimation by within yields an unbiased and consistent estimate of the
union wage differential.

However, the usual fixed effects model is not without its
critics. Stewart (1983), noting that the processes which determine
wages are different in the union and nonunion sectors, constructs a
model that allows for the individual effects to be sector dependent.
This is equivalent to letting the union wage differential vary across
the individuals in the sample. Thus Stewart's model is just a special
case of the fixed effects version of our general panel-data model. In
his model, the union wage differential is constrained to be a linear
function of the individual effects. Estimation of his model is by
nonlinear least squares. We consider an unrestricted version of
Stewart's model which is estimated by within. In either case, the
individual union wage effects cannot be estimated consistently as long
as T is fixed. However, we can calculate the average union wage
differential for the sample.

Using data from the years 1978-1981 of the PSID we estimate: (1)
the simple cross-sectional earnings equation, (2) the usual fixed

effects model, and (3) the unconstrained Stewart model. Since within

72

estimation of (2) and (3) eliminates time invariant variables (e.g.,
education), and because the resulting estimates are not fully efficient
in the case of small T (here, T=4), we also estimate the random-effects
(correlated and uncorrelated with regressors) specifications of each.

From (1) we obtain estimates of the union wage differential in the
range of 17 to 21 percent for the years considered. These results are
in agreement with other cross-section studies. Estimation of the
standard fixed-effects model yields a much smaller union wage
differential of 9.9 percent. Averaging the individual union wage
effects obtained from within estimation of (3) produces a measure of the
union wage differential of about 9.7 percent. Similar results are
obtained when we take the effects to be random and correlated with the
regressors. In general, we find that in every case where we allow
correlation between the regressors and the individual effects, the story
of upward bias in the cross-section is confirmed with substantially
reduced estimates of the impact of unions on wages.

We conclude from this empirical exercise that unless there is
interest in the coefficients of the time invariant explanatory
variables, the fixed effects specification of either the simple or
general model provides a satisfactory framework for measuring the union
wage effect. And, since within estimates of each are always consistent,
they can serve as a basis of comparison for results from more
restrictive models.

Our final remarks concern what remains to be done. Aside from
other applications of the theoretical results presented here, there are
further extensions of our modelfifio be considered. One is to allow some

\

of the variables associated with the cross-sectionally varying

 

73

coefficients to be correlated with the individual effects. Another
extension might explore the limits of conditional likelihood analysis in
a system of simultaneous equations with panel data, where some of the
slope coefficients vary across individuals. Such considerations are

tOpics for future research.

 

FOOTNOTES

CHAPTER TWO

NOTES

1 To see this, notice MD may also be expressed as

l I
MD — INT - [IN 8 T eTeT ] .

So, prelmultiplication of the data by MD yields

 

 

 

Y11 ’ Y1
Y - Y
12. l - _1
MDY a : ' Yi ‘ ‘T’ E Yit
Y _-
1T Y1
F
Y21 ’ 2
Y22 T Y2
Y2T ‘ Y2
YN1 ' YN
Ynz ' YN
YNT ' YN

 

 

and MDX similarly.

74

CHAPTER THREE
NOTES

1 Recall the within estimator,

A

_. '1.
8w — (x MDX) x MDY .

where MD = I - D(D'D)-1D'. Let PD = D(D'D)-lD'. The "between"

estimator is defined as

“ _l
= ' .
BB (x PDX) x PDY.

The projection PD transforms the data into individual means. So, BB is

derived from a regression of

 

 

 

 

Y1 x1
2 on X2 ,
YN *8

and therefore uses variation across individuals.

may be expressed‘as a matrix weighted average

A

Now, BGLS

A

of 8W and BB:

75

76

fl

2 -1 2
= I I I I
BGLS (x (MDX + e x PDX) (x MD + 0 x PD) Y.
2 2 02
where 6 = 1 - T8 =.—___—_— . For fixed T, the use of between
1 2 2
o +To a

variation by GLS results in an efficiency gain over within. But,

A

02 + O as T + w implying 8 = B

l .
GLS W for arge T

2 Except where the individual effects are correlated with all of the

columns of X. In this case GLS = within (see Mundlak (1978)).

CHAPTER FOUR
NOTES

1Specification tests are outlined on pp. 1382-3 of H-T.

2Any vector orthogonal to a time-invariant vector can be used as an
instrument. Since the time—invariant n1 are the only components of the
disturbance which are correlated with an explanatory variable, MD may be
included as an instrument. As H-T note, the time-invariance of the (11

provides N(T-l) linearly dependent instruments for (4.2.3). (See H-T p.

3These identifying restrictions can be tested. See H-T, pp. 1388-9 for

details.

1/2

éThe matrix Q - transform 9 into a scalar matrix; i.e.

fl - 1,299- 1/2 = 021 When used to transform (4.2.3), 0 — 1/2

NT.
yields a simple (1-0) differencing of the data,

Y - (149))?1 = [xit-(l—efiils + 621)! + 0:11 + (ct-(l-efiil.

it

3

OLS applied to this transformation is GLS (see note 2). This is

computationally convenient.

77

78

5This procedure combines the computational convenience of the Q _ 1/2

transformation with the simplifications provided by PA° First, PA

applied to the exogenous variables yields the variables themselves.
Secondly, the projection of the endogenous variables onto the column

space of A can be derived by using only individual means (see H—T

appendix B).

6The proofs of these results are given in Appendix A of H-T.

7Formally we assume MQZ = O (i.e., Mil1 = 0, V1).

8Derivation of Q- 1’2 is a straightforward application of Wansbeek and

Kapteyn (1982).

9Or, equivalently if k1 + J1 + L > J + L.

10Since (9‘ lIZXI, 9- 1,221, Q- lIZW) is more highly correlated than
(x1, zl w) with 9‘ 1’20:, z, u).
- 1/2 1/2

11Since 9 z = 02 and n" (Y-XBw) = 0(Y-X8w) .

y" a [2;8(8'n"18)"lB'Z]-1 Z'B (B'fl-IB)-IB'(Y‘XBW)-

This is generally different from Yw in (4.2.8); i.e., (4.2.8) uses

79

while (4.3.10) uses

2
I I
B'n'ls _ xl I(MD+O 1>D)x1 8 x1 21
I I
0 21 x1 0 Z1 21

Incidently, B'B t B'Q-IB is another reason (one H-T do not mention) for

A

5*
Yw t Y in the over-identified case.
12Since B'Z is nonsingular when k1 = jz.

13Again, the identifying restrictions may be tested. See note 6.

14Compare with H-T Appendix A.

CHAPTER FIVE
NOTES

1An extensive literature exists on the estimation of union wage
differentials. Two excellent surveys, discussing methodological issues

as well as empirical findings, are Freeman and Niedoff (1981) and Lewis

(1983).

2A random effects/GLS approach to this problem would not make sense for
obvious reasons. However, at least one study, Chowdhury and Nickell
(1985), has applied the H-T analysis to the question of unions' effect

on wages.
3See Lewis (1983) for a critique of these studies.

4See Freeman (1980).

5The restricted version of Stewart's model can be estimated consistently

by nonlinear least squares (i.e., searching over the values of 1).
6Percentage differentials are calculated by (exp(5)-1)100.

7Percentage changes are calculated by differences in natural logarithm.

80

81

8The occupation dummies are also treated as endogenous.

9The rank condition is also fulfilled.

10Recall the H—T efficient estimator is equivalent to within in the

exactlky identified case.

11See also Criliches (1977) and Criliches, Hall, and Hausman (1978).

12Union status is now a part of W in our general model (see (4.3.4)).
While this may be intuitively unsatifying, allowing parts of W to be

correlated with the efffects makes estimation of the model overly

complicated.

REFERENCES

Chamberlain, G. (1980), "Analysis of Covariance with Qualitative Data,”
Review of Economic Studies, 47, 225-238.

Chowdhury, G. and Nickell, S. (1985), "Hourly Earnings in the United
States: Another Look at Unionization, Schooling, Sickness, and
Unemployment Using PSID Data," Journal of Labor Economics, 3, 38-

69.

Freeman, B. (1984), "Longitudinal Analysis of the Effects of Trade
Unions,” Journal of Labor Economics, 2, 1-2.

Criliches, 2., Hall, 8., and Hausman, J. (1978), "Missing Data and Self—
Selection in Large Panels," Annales de L'Insee, 30-31, 137-176.

Criliches, Z. (1977), "Estimating the Returns to Schooling: Some
Econometric Problems," Econometrica, 95, 1-22.

Hausman, J.A. and Taylor, W. (1981), "Panel Data and Unobservable
Individual Effects," Econometrica, 14, 1377-1399.

Judge, G.G. (1985), The Theory and Practice of Econometrics, New York:
Wiley.

Lewis, H.G. (1983), Union Relative Wage Effects: A Survey, Chapters 5
and 7, presented at the NBER Conference on the Economics of Trade

Unions.

Mundlak, Y. (1978), "Models with Variable Coefficients: Integration and
Extension,” Annales de L'Insee, 30-31, 483-504.

Mundlak, Y. (1978), ”On the Pooling of Time—Series and Cross-Section
Data," Econometrica, 46, 69-86.

 

Rao, C.R. (1973), Linear Statistical Inference and Its Applications, New
York: Wiley.

Stewart, M.B. (1983), "The Estimation of Union-Wage Differentials from
Panel Data: The Problem of Not-So-Fixed Effects," Mimeo,
Princeton University Industrial Relations Section.

Wansbeek, T. and Kapteyn, A. (1982), ”A Class of Decompositions of the
Variance-Covariance Matrix of a General Error Components Model,”

Econometrica, 50, 713-724.

 

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