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SPECIFICATION TESTS IN PANEL DATA MODELS WITH
UNOBSERVED EFFECTS AND ENDOGENOUS
EXPLANATORY VARIABLES

presented by

Carrie A. Falls

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SPECIFICATION TESTS IN PANEL DATA MODELS WITH
UNOBSERVED EFFECTS AND ENDOGENOUS EXPLANATORY
VARIABLES
By

Carrie A. Falls

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY

Department of Economics

2007

ABSTRACT

SPECIFICATION TESTS IN PANEL DATA MODELS WITH UNOBSERVED
EFFECTS AND ENDOGENOUS EXPLANATORY VARIABLES

By

Carrie A. Falls

My dissertation consists of three chapters on specification testing in panel data
models containing unobserved heterogeneity and endogenous variables. The first
chapter, "Regression Based Specification Testing for Panel Data Models Estimated
by Fixed Effects ZSLS," derives specification tests for linear panel data models
estimated by fixed effects or fixed effects instrumental variables methods. I propose
convenient, regression-based tests for endogeneity, over-indentification, and
non-linearities. Importantly, some versions of the tests are robust to
heteroskedasticity and serial correlation. As a illustration, I test for non-linearities
and for endogenous Spending in analyzing the afi‘ect of spending on student
performance.

The second chapter, "Testing for Correlated Random Effects in Panel Data
Models Estimated using Instrumental Variables," proposes variable addition tests for
correlation between the unobserved heterogeneity and the instrumental variables. In
addition to the standard model with a single additive effect, I consider the extension
to models with unit-specific trends. As an illustration, I test for correlated district
effects and trending district effects using panel data on student performance and
educational spending.

The third chapter, ’Testing the Conditional Variance and Unconditional Variance

in Panel Data Models Estimated by Fixed Effects ZSLS," develops a robust
regression based test for heteroskedasticity. In addition, I derive a test for the
unconditional variance. I apply the test for heteroskedasticity to a panel data model
that explains student performance in terms of spending, poverty rates, and

enrollment.

For my mom, Kelsey Gibson

iv

ACKNOWLEDGMENTS

I would like to thank Jeff Wooldridge for encouraging my interest in

econometrics. I am very fortunate to have had his excellent training and assistance.

TABLE OF CONTENTS

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

CHAPTER 1
Regression Based Specification Testing for Panel Data Models
Estimated by Fixed Effects ZSLS ............................................................... 1

CHAPTER 2
Testing for Correlated Random Effects in Panel Data Models
Estimated using Instrumental Variables ................................................... 35

CHAPTER 3
Testing the Conditional and Unconditional Variance Matrix
in Panel Data Models Estimated by Fixed Effects ZSLS .......................... 57

APPENDIX
Stata Commands for Implementing the Tests in Chapter One ................ 85

REFERENCES ....................................................................................... 91

vi

LIST OF TABLES

TABLE 1
Estimates of the Percentage of Students with a Satisfactory Score
on the 4’” Grade Math Test .................................................................... 31

TABLE 2
Fixed Effects IV, Random Effects IV, and Random Effects IV
including time averages of the instruments ........................................... 54

vii

Regression Based Specification Testing for Panel Data Models
Estimated by Fixed Effects ZSLS

1. Introduction

Panel data methods have become very important in a broad range of empirical
studies. The most convincing studies allow the unobserved effect, or unobserved
heterogeneity, to be correlated with the time-varying explanatory variables.
Therefore, fixed effects (FE) methods are most often used for policy analysis and for
estimating equations where omitted heterogeneity is viewed as being an important
source of bias. The standard fixed effects estimator has been used in numerous
studies.

While estimation by fixed effects allows for arbitrary correlation between the
unobserved effect and the explanatory variables (which are necessarily time varying),
it is not without cost. Estimation afier the within transformation requires a form of
strict exogeneity imposed on the explanatory variables. That is, for consistent
estimation of the parameters, the time-varying explanatory variables must be
uncorrelated with the idiosyncratic error in all time periods. Therefore, while much
attention is given to the relationship between the unobserved heterogeneity and the
observed explanatory variables, it is always possible that the explanatory variables
are correlated with the idiosyncratic errors resulting in asymptotically biased
estimation. When dealing with panel data we need to examine the relationship
between the explanatory variables and the idiosyncratic error contemporaneously as

well as in other time periods.

Wooldridge (2002, Section 10.7) proposes a simple variable-addition test of the
assumption that the idiosyncratic error at time t is correlated with the explanatory
variables in the next time period. This test has power to detect feedback from
unexpected changes in the dependent variable this period to future explanatory
variables. But often we are interested in testing for contemporaneous correlation
between the explanatory variables and idiosyncratic errors, as this kind of
endogeneity occurs when there is measurement error, omitted variables, or
simultaneity. I Show, given the availability of instruments, simple tests for
contemporaneous endogeneity are fairly straightforward to compute.

When one or more explanatory variables are thought to be contemporaneously
endogenous, it is popular to combine FE methods with instrumental variables (IV)
methods. Wooldridge (2002, Chapter 11) contains a discussion and references.
Although fixed effects-instrumental variables (FE-IV) methods are becoming more
widely used, little work has been done on extending standard specification tests to
the FE-IV framework. When we decide explanatory variables are endogenous and
apply FE-IV, it is important to test any overidentification conditions — just as in a
pure cross-sectional or time series framework.

If we maintain standard assumptions, such as homoskedasticity and serial
independence of the idiosyncratic errors, applying tests of endogeneity and
overidentification are fairly straightforward. But there is much interest recently in
making inference fully robust to any second moment matrix of the idiosyncratic
errors. In this chapter I propose a framework for regression-based tests of standard

assumptions afier estimation by FE-IV. The framework can be used to test for

endogeneity of some explanatory variables, for overidentification restrictions in the
context of FE-IV, and non-linearities. I cover tests that are valid only if the
idiosyncratic errors satisfy the “ideal” assumptions, as well as tests that are fully
robust to serial correlation and heteroskedasticity.

Davidson and MacKinnon (1993) and Wooldridge (1995a) propose
regression-based tests for endogeneity that can be applied to single-equation
cross-sectional or time series equations, but they do not apply directly to panel data
models estimated by fixed effects. The framework here follows most closely that of
Wooldridge (1995a), but I explicitly treat the presence of the unobserved effect. The
asymptotic analysis is with a fixed number of time periods and an increasing
cross-sectional sample size.

The remainder of this chapter is organized as follows. Section 2 contains a brief
literature review. In Section 3 I present the basic model and estimation method. In
Section 4 I obtain a general specification test afier F E-ZSLS estimation. Section 5
presents examples for executing tests of endogeneity, over-identification, and
non-linearities. Section 6 provides conditions for applying the derived tests to
unbalanced panels. In Section 7 I apply the tests to a panel data model that explains
test pass rates in terms of spending, poverty rates, and enrollment. I test for
endogeneity of spending — afier controlling for a school fixed effect. The data set,
used by Papke (2007), comes from Michigan schools from 1992 through 2004.
Section 7 contains a brief conclusion. Stata commands for implementing the test for

endogeneity, over-identification, and non-linearities are provided in the appendix.

2. Background

The literature on Specification testing in panel data models is surprisingly thin,
and has mostly concentrated on comparing different estimators via a Hausman (l 978)
test. Wooldridge (2002, Chapter 10) covers a variety of tests in the context of
standard fixed effects estimation, but he does not consider tests of contemporaneous
correlation between the regressors and the idiosyncratic errors. Such tests require the
availability of instrumental variables, and I assume that I have suitable instruments.

Testing for endogeneity and testing over-identifying restrictions is pretty well
established in pure cross-sectional and pure time series contexts. Hausman (1978)
has the seminal paper, and Davidson and MacKinnon (I993, Chapter 7) includes a
general treatment. Davidson and MacKinnon offer regression-based versions of the
standard Hausman test. [Ruud (1984) was an early proponent of regression-based
versions of Hausman’s tests] But these and other references cover only non-robust
tests — effectively, they maintain that one of the estimators in a Hausman comparison
is relatively efficient. Wooldridge (1990, l995a) proposed a framework for robust,
regression-based statistics. But Wooldridge’s framework is for time series
applications and does not easily cover panel data models with unobserved effects.

Below, I effectively extend Wooldridge’s (l995a) framework to the case of
unobserved effects panel data models estimated by fixed effects-instrumental
variables procedures. In addition to deriving tests under a full set of ideal
assumptions, 1 derive fully robust versions, too. The asymptotic analysis is for a
fixed time series dimension with the cross-sectional sample size growing. Thus, the

asymptotics is standard random sampling asymptotics.

3. Model and Assumptions

1 consider the standard unobserved effects panel data model
y;,=x,~,fi+c,-+u,-,, t= I,2,...,T, (3.1)

where i denotes a random draw from the cross section. The vector x i, is 1 x K of
explanatory variables, c,- is the time-constant unobserved effect, and u,-, is the
idiosyncratic error. To allow estimation by instrumental variables, let 2;, be a l x L
vector of instruments, where L 2 K. Thus, the set of observations for unit 1' is
{(xi,,y,-,,z,-,) : t = l, . . . , 7}. I assume random sampling in the cross-section
dimension with a sample of size N. The time series dimension, T, is fixed in the
asymptotic arguments. Typically, 2;, and xi, would overlap — for example, each
would include a full set of time dummies to allow unrestricted secular effects.

In this chapter, I make no assumption regarding possible correlation between x,-,
and 0;. I also allow the instruments to be arbitrarily correlated with c;. Therefore, to
consistently estimate B, the unobserved effect is removed via the within
transformation. For each (i,t), let)?“ = y" — )7,- be the time-demeaned dependent

variable, where j) = T‘1 2;] y,-, is the time average. Similar notations hold for 52,-,
and an. After applying the within transformation to equation (3.1) the model
becomes
fitt=55m3+iiiu t= 1,.-.,T, (3.2)
If (xi, : t = l,...,T} is strictly exogenous — that is, E(x:-su,-,) = 0, all s,t = l,...,T,

then 5051-51711) = 0. Thus under strict exogeneity of the explanatory variables

estimation of equation (3.2) by pooled OLS is a consistent procedure for the
estimation of B. However, to ensure identification, the rank condition must be
satisfied. That implies, at a minimum, the explanatory variables must vary across
time. This leads to the so-called within or fixed-effects estimator. Since we are
interested in the case where the explanatory variables cannot be treated as strictly
exogenous we estimate (3.2) by pooled ZSLS, using time-demeaned instruments 2,7.
This produces the fixed effects-two stage least squares (F E-ZSLS) estimator.

A useful expression for the estimator is obtained by defining X,- and 2i as the
T x K and T x L matrices of time-demeaned regressors and instruments, respectively,

and j},- as the T x 1 vector of time-demeaned dependent variables. Then

- (EXHIBI-

Inserting equation (3.2) gives

-l

N N -1 N ’1
= 13 + (Eff-Z) (Z 2’2) (Z '29?) (3.3)
i=1 i=1 i=1

where the last equality uses Z;u',- = Zfi-ui.
For each i, we can write
T T T
"I" ..I .. " " ..I .. "I ..I
ZiZi = 22,1212, XIZI' = 2x112”, and Ziu; = Ell-tuft.
t=l t=l t=l
We can easily state sufficient conditions for consistent estimation of B, with fixed T

asN-+ oo.

ASSUMPTION A.l: For all s,t = 1,..., T,
E(z§su,-,) = 0. (3.4)
Assumption A.l is a strict exogeneity condition on the instrumental variables —
after eliminating the unobserved effect, c,~ — and implies E(§;-,uiz) = 0,1 = 1,. . ., T. It
follows then that N“1 2:1 Zui fl 0. Assumption A.l is sufficient for tests of

endogeneity and overidentification. However, testing for non-Iinearities requires a

stronger zero conditional mean assumption. Specifically, we will require
E(uiz|Zi) = 0- (35)

Equation (3.4) implies that the instruments are not linearly related to the
idiosyncratic error whereas the zero conditional mean rules out nonlinear

relationships as well. Hence, maintaining the zero conditional mean is much more

restrictive, although often intended in structural models.

The second assumption we need for consistency of FE-ZSLS is a rank condition.
First, define H as an L x K parameter matrix from the population linear projection of
56,-, onto 2,,,t = 1,...,T, that is,

—l

T T
Ewan) Efrem). (3.6)
1:]

t=l
o o o ..I .- 0 o
For this matrix to eXIst, we need to assume T E z- 2- IS nonsm ular, an
1:] u 1‘

assumption that rules out time-constant instruments and instruments that are perfectly

collinear. Given I'I, define the population projection x ,,,
Note that this is not the same as projecting 56,-, onto 2,, for each I; in effect, because
the F E-ZSLS estimator is a pooled estimator, the projection in (3.7) pools across

time.

The FE-ZSLS estimator effectively uses 35:, = 2,,fI in place of 56,-, in the

estimation of B whereI'I= [Zr—l 2:12 z,-,'z,-,]_l Zfi, Z_ _, ,,x,, Isthe obvious,

consistent estimator of TI. That is,

NT "1

N T
Z 5‘53]??? ZZXE'PII (3.8)

i=1 (=1 i=1 (=1

E)
II

QTIG’.‘ NA!“
XX: X.- XX, y.-
i=1

Expression (3.8) makes it clear that Z_ ,E(5c'-‘"”'-" ,) must have full rank, for which we

need 2;, E(z z'-,52,,) to have full rank. Thus, we have the following assumption:

ASSUMPTION A.2: (i) rank 2;, E(2§,2,-,) = L and (ii) rank 23;, E(2§,52,-,) =

Of course, necessary for the rank condition is the order condition, L 2 K. That is,
there must be at least as many instruments as explanatory variables.

THEOREM 2.1: Under assumptions Al and A2, the pooled ZSLS estimator
obtained from a random sample in the cross-section is consistent for B.

Proof: The asymptotic first order properties are derived by rewriting equation

(3.8) as

A*’

. N Mm: -' N
p = fi+ N-1 25?, X,- N-1 ZX, u,- (3.9)
i=1 i=1

Add/vi!

Under A.2, N‘l 23:156- 2?,- is non-singular (with probability approaching one) with

A*’A*

plim (N4 2:, X,- X,) = A4, where A .=_ E(X}"'}f}‘). Further more, under
Assumption A.l, the
N <.\*’ ..
plim N“l 2X,- u, = E(X}'"u,-) = 0.
i=1

Therefore, by Slusky’s theorem (for example, Wooldridge (2002, Chapter 3),plim

B=B+A‘l o0=B.

The asymptotic normality of W (B — B) results from the asymptotic nomality of

N"”2 2:, X’f'ub which follows from the central limit theorem under A.1, A2 and

finite second moment conditions. Thus, the asymptotic distribution of

_ N -- 1 .
N “2 Zi=1 Xf u,- 15
N d
N-W- ZXf’u, _. Nonnal(0,B),
i=1
where
B —-_= E()"(}"u,-u;-)"(f)
But we also need to use the important result that having to estimate II in the first

stage does not affect the asymptotic distribution. Formally,

N M, N
N—l/ZZX,‘ u,- = N—l/ZZEIU,‘+OP(I).

i=1 i=1

It follows that
,/N(i3 — 13) 5’» Normal(0,A‘lBA‘1 ).

Importantly, the asymptotic variance A‘lBA“l places no restrictions on

Var(u,-|z,,c,-), and so it is fully robust. It is the basis for robust inference using the

FE-ZSLS estimator (which is the specific FE-IV estimator we cover in this chapter).

The asymptotic variance can be simplified by ruling out heteroskedasticity and/or

serial correlation. However, the interest in this chapter is in deriving robust tests,

such assumptions are considered later as a special case.

4. A General Framework for Specification Tests

We now propose a framework, which extends Wooldridge (1995a), that allows
various specification tests after FE or FE-IV estimation. For example, if interest lies

in testing whether IV methods are necessary, we want to test whether x,-, is correlated

10

(3.10)

(3.11)

(3.12)

(3.12)

with the idiosyncratic error, u ,,. In deriving the test statistic and its properties I
generalize the methodology by defining a set of misspecification indicators that are
then tested for correlation with the idiosyncratic error.

Let v,-,(0) be a l x Q vector of misspecification indicators. I allow v,-,(0) to
depend on some G x 1 vector of parameters (0) as well as elements of {z,,,x,-,,y,-,}.
The parameter vector 0 can include B and other parameters. Sufficient for our testing
purposes is that we have a W -consistent estimator of 0, a very weak requirement
that follows generally by the central limit theorem.

For notational convenience, let v,, = v,,(0). Technically, it would be more
appropriate to distinguish the “true” value of 0 from a generic value, but in the
interests of keeping the notation simple, I let 9 denote the true value.

Ideally, we would state the null hypothesis as Ho : E(v:-,uiz) = 0, t = 1, . . . , T.
However, because we are removing the unobserved effect via time-demeaning (the

within transformation), the null is effectively
T
H0: 2 Hagan) = 0, t= 1,...,T. (4.1)
(=1

Taking the null as stated in (4.1) has a couple of consequences. First, the test
maintains that the selection indicators are actually strictly exogenous. Second, the
time-demeaning complicates the asymptotic analysis because of the degeneracy it
introduces into the limiting distribution. But I take care of that in my analysis.
Because the basis for testing is F E-IV (with FE as a special case), the tests are

actually based on

11

T T
Z 2131,12,.) = Z E(i3:-,u,-,) = 0. (4.2)
t=l t=l

That is, it does not matter whether the idiosyncratic errors have been demeaned. This
equivalency is a result of the idempotent matrix identity. However, if we took away
the summations (4.2) would not hold. This observation simplifies the asymptotic
results, just as with analyzing the usual FE estimator (see Wooldridge (2002, Chapter
10)).

Clearly, the fact that the null is effectively (4.2) limits the scope of the tests. For
example, a test for AR(I) serial correlation would take 12,-, = u ,-,,_1 , a scalar
However, under the null hypothesis,of no serial correlation, equation (4.2) is violated
due to the negative serial correlation that is induced by the within transformation.
Therefore, the test statistic proposed in this chapter is limited to tests where the
misspecification indicator is strictly exogenous under the null. In Chapter 3 I
explicitly consider the problem of testing for serial correlation after FE-ZSLS
estimation.

Tests based on (4.2) are derived under assumptions that are extensions of those
made in section 3. Each assumption is assumed to hold under the null.

ASSUMPTION B.l: Assumption A.1 holds and, in addition, (i)

2;, E[ii,,(9)'u,,] = 0 and (ii) 2;, E[Vgii,-,(0)'u,-,] = 0, where V913,,(0)' is the
Q x G Jacobian of 13,-,(0)'.

Part (ii) of Assumption BI is an auxiliary assumption that extends Wooldridge

(1995a). It ensures that the estimation of 0 does not affect the limiting distribution of

the test statistic so long as [N (0 - 9) = 0,,(1). In most cases, requiring Assumption

12

B. 1 (ii) is innocuous. It is satisfied in the test for endogeneity and tests of
overidentification under A.l , and it holds for testing for non-linearities when we
impose E(u,-,|Z,-) = 0. As a general rule, Assumption B.l(ii) holds when the null
hypothesis is defined in terms of zero conditional means. If the misspecification
indicator is linear, then the assumption generally holds under zero correlation
assumptions.

Equation (4.2) suggests using the sample analog,

N T A,
N-1 2 2 3,2,, (4.3)

as the basis for a test, where 22,, = y,, — x,,B are the FE-ZSLS residuals and ‘13,, is
evaluated at B. To use (4.3) as the basis for a test, we must derive the limiting
distribution of
N T I
N-“ZZZWI (4.4)
i=1 #1
However, it is usually the case that the limiting distribution of (4.4) is different from
the limiting distribution of
N T
N-“2 Z Z 3;,” (4.5)
i=1 1:1
In other words, the asymptotic distributions of JN (B — B) and JN (B - 0) typically
affect the limiting distribution of the moment conditions. While deriving a quadratic
form of (4.4) that has an asymptotic chi-squared distribution is not difficult, the
resulting statistic would not be easy to compute. A Simpler approach is to transform

(4.5) so that replacing u,, with IL, does not affect the limiting distribution of the test

13

statistic. This results in robust test statistics that are easy to compute using a series of
regressions, and are asymptotically equivalent so long as a N -consistent estimation
procedure is used in the first-stage estimation, as Shown by Wooldridge (1990). This
brings us to our second assumption, which extends Wooldridge (1990, l995a) to
allow exlicitly for time—demeaning in a panel data context.

ASSUMPTION B.2: Assumption A.2 holds. In addition, letting 134,-, = (2,,, i3”),

define the linear projection
x; = fig-,1“, (4.6)
where

T -1 T
F = [ZEMIMIDI ZE(w§,5c,-,). (4.7)
#1 (=1

Further, define the projection residuals

r,-, a 9,, —5&;!;A, (4.8)
where
T '4 T
A = [Ziggy/52;; :1 2150533,). (4.9)
t=1 t=l
Assume
T
rank T“ ZE(r§,r,-,) = Q. (4.10)

t=l

As discussed in Wooldridge (1 995a) in the case with a single data dimension,

redefining 3;; to be the projection onto the extended vector (2,,, 1%,), and then netting

l4

out 52,-,, leads to computationally simple tests. The same is true here. Of course, we

have to operationalize the procedure by plugging in first-stage estimators. Therefore,

let
N T A -1 N T
=2: 2A Witwit z 20 Witxit,
i=1 t=l i=1 t=1

.. 4) “ ’..\ .. 4:
xit = WIIF- Where W1: = (21:,vz'1) -

N
ZZ§IH§I Zin-tvu.

i=1 t=l i=1 t=l

and 2",, = i3}, - 55,-,A. In other words, 35,-, are the fitted values from the pooled

regression
51,,on2,,,$,,,t= 1,...,T,i= 1,...,N (4.11)
and then the 7",, are the residuals from the pooled regression
P,,on§,-,,t=1,...,T,i=1,...,N. (4.12)

Now, we obtain a test statistic by adjusting equation (4.3) by replacing ii}, with 7",,
Thus, we are partialling out the portion of the instruments and misspecification
indicators that are linearly related to the explanatory variables in (3.2). Equivalent to
the parialling out in (4.12) is to simply add ‘13,, to the fixed effects estimating
equation (3.2). That is, adding the misspecification indicators to (3.2), then
estimating the augmented equation by pooled-OLS using time-demeaned variables, is
equivalent to, first estimating f7, then regressing 12,-, on F,,. However, regardless of
how the partialling out is achieved it is this partialling out of i3,,, the misspecification

indicators, that allows us to ignore the first-stage estimation. That is, we can ignore

15

the estimation of B when deriving the limiting distribution of the test statistic. The
proposed test statistic is based on
N‘1 22mm, (4.13)
i=1 (=1
Our goal is to Show that, under H0 and Assumptions 3.] and 8.2,
N T N T
N-'/2 2 2 21,4, = N‘“2 2 2 r;,u,-, + 0,,(1), (4.14)
i=1 (=1 i=1 (:1
where r,, = 13,-, — 33A are population residuals; then, the central limit theorem can be
applied to the second piece. To establish (4.14), we first Show
N T
N-1/222r,, u,,= N-1/222r;,a,, +o,,(1). (4.15)
i=1 (-1 i=1 #1
To Show (4.15) we can apply a mean-value expansion and use Assumption B.l(ii).

Then,

N-l/2 2 2’ v,,u, = N-l/2 2 23' v,,u,, + (2 E(v,,v;,u,,)> 7M9 - 6) + 0,,(1)

i=1 (=1 i=1 (=1

= film”2 2 Z I'd-(1217+ 0p(1).

i=1 (=1

Next,

N4” 2 2(XIIA) “it: A(N-l/2N 2 EN xi( “(7)

i=1 (=1 i=1 (=1

N T
+ JN(A — A)'(N‘l Z: 2:171”)

i=1 (=1

16

N T
= NOV-”2 22353221,) + 0p(1)~op(1)

i=1 I‘=l

N T
= A'(N‘l/2 22§;'a,-,) +o,,(1)

i=1 t=l
because WM — A) = 0p(1) and N_12f:iz¢:1§;’fiit = op(l). Finally, it can be
shown that
N T N T
N-“2 2 2 fi’fii, = 1w“2 2 2x3’ai, + 0,,(1).
i=1 1:1 i=1 z=1

Collecting terms we have established (4.15).
We still need to show
N T N T
N'”2 2 Z r2112“ = N‘”2 Z Z rI-tui, + 0p(l).
i=1 t=l i=1 t=1

But

N T N T N T

2W“2 Z 2 4,12,, = N‘“2 Z Z rauo - (N-1 Z Z rim-x) ml? — 13).
i=1 t=1 i=1 (=1 i=1 (=1

Now, 2;] rz-txi, = 2’: l rip?“ (because ri, is a linear combination of vectors that

have been time demeaned). Further, 56,-, = 5:; + g), where, by construction,

21:] E(2:-,g,-,) = 0 and 2;] E(i'2:-,g,-,) = 0. Therefore, because r), is a linear function

of conga”), it follows that 2;, E(r§,g,-,) = 0. So

2;] E(r§~,5%,-,) = 21:] E01152?!) = 0, where the last equality follows because ri, is

the population residual fi'om the regression in, on 56;}, t = 1,. . ., T. We have shown

that N‘1 2:1 2;] 7'17in = 013(1) (by the law of large numbers); combined with

17

JN—(fi — fl) = 0p(l) we have
N‘”:2 2:] H rituit = N‘“2 2).: 21H ruui, + op(l). When we combine the
previous results we arrive at the important relationship in (4.14).

Now obtaining a test statistic is easy. Define

T T T
C E Var<Z film) = ZZE(uitu,-sr;sri,), (4.16)

#1 (=1 s=l

so that, by (4. 14),
N T d
or“2 2221,22,, 4 Normal(0, C) (4.17)
i=1 t=1

under Ho. It follows immediately that, if C a C then the statistic

i=1 t=l I'-1 t—l

T ’ T
= (22424:) (NC) ‘_N( 2%)
.=1 (:1 i=1 [:1

has a limiting XZQ distribution under H0. Of course, the actual form of the test

N T ’
f = (N‘l/ZZZ ram-JC- '(N-1/222r3,a,-,) (4.18)

~

statistic depends on the estimator used for C. We first derive the fully robust version
of the test. That is, a test statistic that is robust to heteroskedasticity of unknown form
and within cluster serial correlation. Then, with large N and small T, we can choose

C‘ to be completely general:

T T
C‘ = 111-1 (222 a,,t2,~s;}s;,-,). (4.19)

Using equation (4.19) the test statistic in (4.18) becomes

N T N N T
f, = 2 2 21,12), 2 2 2 ”aka-ST), 2 2 iii-(12;, (4.20)

This fully robust statistic cannot, in general, be computed via regressions. However,
there exists a test statistic that is asymptotically equivalent to (4.20) under Ho, and
local alternatives, that can be easily computed using simple regression. The

alternative statistic isjust the fully robust Wald statistic from the regression of

fi;,on;‘it; i= 1,...,N;t=1,...,T. (4.21)
In other words, let i be the Q x 1 vector of coefficients on the 1 x Q vector Pit. Then
we just compute the robust Wald test for significance of 1. Many software packages,

such as Stata, have such a feature for pooled OLS regressions.

The test statistic based on (4.21) can be written as

N T T 1 N T
I A A A I" A A, A
aw: ZZeiteisrisrit 22mm: , (4.22)
i=1 t=l

where, the é it are the residuals from the regression in (4.21). The regression form of

the test replaces ( 12,-, 12,3) with ( é i, e‘is) in the estimation of C. Because 3. E» 0 under

H0, this alteration does not affect the limiting distribution of the test statistic. In fact.

1" and i” can be shown to be asymptotically equivalent in the sense that fr — fr 5 0
under local alternatives such that ,1 N = 0(N‘1/2). The argument is straightforward
and similar to Wooldridge (1990). In fact, any estimator of C that is consistent when
A = 0 will lead to a test asymptotically equivalent to fr under N‘“2 local
alternatives.

If we impose certain homoskedasticity and serial independence assumptions, we

19

can compute a simpler form of the statistic.

ASSUMPTION B.3. The idiosyncratic errors are homoskedastic in the sense that
E(u§?-,|z,-,v,-) = 024 = 1,...,T, (4.23)

where z; = (2,-1,...,z,-T) and V,- = (v;1,...,v,-T).

Note that the homoskedasticity assumption is stated in terms of the squared
errors. Because we have not assumed E(u,-,|z,-, vi) = O -— we used the weaker
assumption of zero correlation, in addition to Assumption B.l(ii) - Var(u,-,|z,-, vi)
does not generally equal E(u,2,|z,-, vi). In most cases, we would expect (4.23) to hold
only if E(ui,|z,-,v,-) = 0 and Var(u,-,|z,~,v,') = 02, and so we view Assumption 3.3 as
being an assumption about the conditional variances. The homoskedasticity
assumption rules out variances that depend on the instruments or misspecification in
any time periods. In addition, the unconditional variance must be constant across I.

ASSUMPTION BA: The errors are serially uncorrelated in the sense that
E(u,-,u,-S|z,-,vi) = 0, all s i t. (4.24)
Again, the no serial correlation assumption is stated in terms of E(u;,u,-s|z,-, vi)
rather than Cov(u,-,, uislzi,v,~); the two are equivalent under E(u,~,|z,-,v,-) = O for all t.
It is possible that E(u,~,u,-S) = 0 but the conditional covariances are not zero. For the
simplified form of the test to be true, we need to make Assumption 3.4.
We can simplify the expression for C in (4.16) under Assumptions 3.3 and 3.4.

First consider the terms when t = s. Then, by iterated expectations,

20

E(u;2¢r:'trit) = E[E(u,z;ritrizlzi,vi)l = E[E(u)21lzi,vi)r:yrit]
= 02E(r;,r,-,), t = 1,...,T.

Next, for! rat s,

I I
E(uituisrisrit) = E[E(uituisrisritlziavi)]

= E[E(uituis|2i,Vi)r;-srizl = 0-
Combining these results gives
T
C = a2 2E(r§,r,-,). (4.25)
(=1
Therefore, a consistent estimator of C under the additional assumptions 3.3 and 3.4

is

N T
C = 62N-1 2 2 21,2”, (4.26)
i=1 r=l

where 62 fl 02 as N —» 00. As is well known, the sum of the FE-IV residuals across I
for each i is identically zero because of the within transformation. In effect, one
degree of freedom is lost for each i. As in the case with exogenous variables -— see,
for example, Wooldridge (2002, Chapter 10), the following estimator is consistent for
02 with Tfixed:
N T
62 =(N(T—1))-122zz,?,. (4.27)
i=1 t=l

Using (4.26) as C in (4.20) gives the nonrobust statistic

21

’ N T *1
r-pm) [((MT- 1)>—' EN: 2 m.) (2 2 22.2..) :l
[:1 t=l i=1 t=l i=1 (=1

N T
° 22421317) (4.28)

=~(r—1>(ZZ£M) (2321.) (232) (212.4)

i—I t-l i=l t-l i=1 t=l

gr)

II
A
.Ffl2
M’s

where R27; is the usual R-squared from the pooled regression in (4.21). Under the

null hypothesis and the extra assumptions B.3 and 3.4,
N(T— I)Ru~ 1Q. (4.29)

Importantly, failure of either B.3 or B4 causes the test statistic in equation (4.29) to
no longer have an asymptotic chi-squared distribution. Further, as discussed by
Wooldridge (1990, 1991), the nonrobust version of the statistic has no systematic
power for detecting violations of B.3 or B4: the limiting distribution of i'nr is a
quadratic form an a random normal vector, and the distribution may lead to
asymptotic rejection frequencies either higher or lower than the nominal size.

Because the robust version of the test has the same limiting distribution as Tm
under local alternatives when B.3 and B4 are true, nothing is lost in terms of
asymptotic power against local alternatives by using a robust test even when the ideal
assumptions hold. Because the fully robust test is, these days, fairly easy to compute,
there is a strong case to be made for computing the robust version, even if the

nonrobust version is also computed.

22

5. Examples

I now provide several examples of how the testing framework in Section 4 can be
applied after FE or FE-IV estimation. The appendix contains a list of commands that

can be used to implement the tests in a popular software package, Stata.

5.1. Testing for Endogeneity
To test whether a subset of the time varying explanatory variables is endogenous,

we write down a partitioned model,

yiz =Xi1131+xitzl32+6i+um t= 1,2,...,T, (5-1)
where xm is 1 x K1 and xuz is l x K2. The vector x512 is assumed to be strictly
exogenous (with respect to the idiosyncratic error); both xm and xitz are allowed to

be correlated with of. Interest lies in testing if xm is correlated with the idiosyncratic

errors. In an FE environment, the null hypothesis is
Ho :E(5e;.,,u,-,) = 0, t= 1,...,T (5.2)

Under the null hypothesis that xm is exogenous, the estimating procedure is standard
fixed effects, which means that the instruments 2;, used to estimate the null model are
just x”.

In order to test the null hypothesis that xm is exogenous, we need at least K 1
time-varying instruments. Let hm be a 1 x L1 vector of strictly exogenous
instruments with L] 2 K1. Then the full set of instruments for estimation under the
alternative is q i, a (h m ,xi12)- The regression-based Hausman test that compares FE
and F E-ZSLS can be obtained as follows. Estimate the reduced form equations (K 1 of

them)

23

xix] = (1211—1 ‘1' 01+ Vii (5-3)
by fixed effects, and obtain the FE residuals, 13,-, = x,” — q ,-,FI. This 1 x K, vector is
the vector of misspecification indicators. Because x,” is exogenous under Ho,
2,, = x,, in this case. So, regress
fi,,onx,-,,i= 1,...,N,t= 1,...,T (5.4)

and obtain the residuals, 2",, Let 12,-, be the FE residuals from estimating (5.1). Then
the fully robust statistic is to testjoint significance of 2. in the artificial equation
12,-, = 2,,22 + error,, (5.5)

using pooled OLS and a fully robust variance matrix. Notice that the dimension of 2,
as always, is the degrees of freedom in the limiting chi-square distribution: K1 in this
case.

Because the first-stage FE estimations should be done anyway to determine the
strength of the instruments, the only additional step is in the regression (5.4) — a
trivial set of pooled OLS regressions. (If x,,1 is a scalar, it is a single regression.) The

nonrobust test is N(T— I)R,2, from the regression (5.5).

5.2. Testing Overidentifying Restrictions
We can also test the validity of overidentifying restrictions in a fully robust
manner. Let 2,, be the 1 x L vector of instruments with Q = L — K > 0. As discussed
in Wooldridge (I995a), the misspecification indicator, in this case for panel data, can
be any I x Q subset of 2,, that is not also in x,-,. To compute the statistic, we obtain

the FE-ZSLS residuals, 12,-,, the first-stage fitted values, 35,-, from the regression 56,-, on

24

211, and let 9,, be the demeaned misspecification indicators (which do not depend on
estimated parameters in this case). Then 1",, are the residuals from the pooled OLS

regressions 9,, on §,,, and then the test is carried out as in equation (5.5).

5.3. Testing for Non-linearities

Testing for non-linearities in unobserved effects models with possibly
endogenous variables poses some challenges because the nature of the alternative
model is not clear. Of course, we can always add nonlinear functions of exogenous
and endogenous variables and reestimate a more general equation by F E-ZSLS. But it
is also useful to have parsimonious tests, such as Ramsey’s (1969) RESET. The tests
in Section 4 can be used to obtain RESET-like tests in the current framework.

With large N and small T, we cannot add nonlinear functions of x,, 6 + c, to the
model and estimate these because c, is unobserved. One approach is to first eliminate
c, through the within transformation, as usual, where we again partition the equation

into endogenous and exogenous explanatory variables:
)71: = 55121131 +55iz2fi2 + 171:, t= 1,...,T. (5-6)
Applying RESET to this equation directly would mean choosing misspecification
indicators such as (56,” [31 + 5?,QBZ)2 and (52,,1 m + 56,,232)3 (and possibly higher
order powers). With exogenous explanatory variables this choice produces a valid
test, and then 9,, = [(5E,,fi)2, (56,,B)3] is a sensible choice for a two
degrees-of-freedom test. Because the x,, are exogenous, 317'} ,, = 52,,. The l x 2 vector
5“,, is obtained by regressing each element of 3,, on 3%,,

When it,” is thought to be endogenous, (55,,fi)P would be generally correlated

25

with 12,, even if there are no non-linearities in the model. One possibility, which
extends Wooldridge (1 995a), is to replace 56,,1 with the first-stage fitted values, 3? ,,1 ,
from 5%,,1 on 2,,. Then, the misspecification indicators become, say,

(jig-Iii, + i,,2f32)2 and ((32,131 + 56,,232)3, where [3 is the FE-ZSLS estimator. Such
a test should have power against general forms of nonlinearity. But, like with RESET
in pure cross section or pure time series contexts, it is not clear how to modify the

model in case of rejecting the null, linear model.

6. Applying the Tests to Unbalanced Panels

The previous treatment assumed that a balanced panel data set is available. In
practice, one often has to deal with unbalanced panels. As discussed by Wooldridge
(l995b) and Semykina and Wooldridge (2005), accounting for unbalanced panels
when selection is endogenous can be very difficult. However, as shown formally by
Semykina and Wooldridge (2005), FE-IV estimation has some robustness to
endogenous selection: selection in each time period can be arbitrarily correlated with
the unobserved effect and the instruments. Therefore, we can easily allow that kind
of selection in the current framework, too.

The selection indicators, which we call s,,, appear in formulas -— even sample

averages — in a nonlinear fashion. For example, the time average of the dependent

variable can be written as )7, = 77‘ 21:, s,,y,,, where 3,, = 1 means that we
observed the data point, and T, = 21:15” Of course, the FE-ZSLS estimator is a

nonlinear function of sample averages. This is important because zero correlation

assumptions are now difficult to work with. So, if the selection indicator, v,,(0) is a

26

function of observables q ,, — observable at least when the other variables are — then

the simplest statement of H0 is

E(uitlzila--”2179411a---1‘11'T1Sila---aSiT) '=' E(u11|21,qi,51) = 0, (6-1)
where s, is the T -vector of selection indicators. In other words, like the instruments
and misspecification indicators, selection must be strictly exogenous under Ho.
Assumption (6.1) essentially rules out correlation between the idiosyncratic
errors and selection in every time period. But 3,, may be a function of 2,, q,, and the
unobserved effect, c,. In cases where data are missing in some time periods in an
essentially random way — even if the rate of missingness is higher for some units —

then the tests in the previous sections apply directly to the unbalanced panel.

7. Application

Estimating the causal relationship between spending and student performance is
complicated by unobserved factors. Generally, a model for student performance
would include family demographic and economic variables such as income, size, and
attitudes towards schooling. While, it is often the case that such family attributes are
unobservable they are likely to be correlated with district level spending. For
example, prior to the passing of Proposal A in 1994 increases in spending were
obtained by the passage of local millages. Since communities with higher incomes
and perhaps higher values on education were more likely to pass such millage,
changes in spending were likely to be correlated with family demographic and
economic variables.

In addition, it is plausible to think that the district has its own attributes affecting

27

students’ performance. For example, districts may sytmatically differ in how they
allocate funds. If a district allocates more to schools that are not preforrning as well
we would expect the effect of spending to vary across districts. Thus, it is plausible
that unobserved heterogeneity exists across districts as well as unobserved family
demographic and economic variables. Since both are thought to be correlated with
spending ignoring them leads to inconsistent estimation of spending. However, if
the unobserved district effect is constant across time, estimation by fixed effects
eliminates the district effect. This leaves us with the issue of omitting family
demographic and economic variables which can cause spending to be correlated with
the idiosycratic errors, as pointed out by Papke (2005, 2007).

Papke (2007), examines the effect of spending on the pass rates of the
fourth-grade MEAP math test using district-level data from 1992 through 2004.
Noting the likely endogeneity of spending, Papke uses the dramatic changes in
funding of Michigan public schools brought about with the passage of Proposal A in
1994. Proposal A imposed spending floors and phased in spending ceilings. Prior to
Proposal A, a large fraction of per pupil spending was determined by the school
district through local property taxes. Proposal A revoked the districts’ ability to
increase the millage rate and replaced it with a grant system financed through the
state School Aid Fund.

For our purposes, the significance of Proposal A is that it provides an instrument
for spending. Specifically, the level of the foundation grant is a suitable IV candidate
provided fixed effects are allowed in the pass rate equation. Papke (2007) uses the

foundation grant as an instrument for spending. In an earlier analysis using school

28

level data Papke (2005) finds evidence of edogenous spending by comparing OLS
and IV estimates using the standard Hausman test. There are probably two reasons
for her findings. First, the presence of unobserved school effect which is likely to be
correlated with spending. Second, the inability to control for family demographic and
economic variables as well as parental influence may lead to omitted variable bias.
That is, without controlling for the fixed effect, it is hard to tell where the
endogeneity comes from. Papke notes the likelihood of unobserved heterogeneity
across schools and applies both fixed effects and fixed effects ZSLS in estimating the
effect of spending. She finds substantially larger spending effects for the FE-2SLS
estimates, but does not formally obtain a Hausman test after controlling for the
unobserved heterogeneity.

Using Papke’s (2007) district level data, I reconstruct her estimation of the
spending-perfonnance model and test for the endogeneity of spending and
non-linearities using the fiilly robust test derived in section 4.

Following the example in (5.1), the pass-rate equation is partitioned as
M = xiz1fi1 + 16112132 + 01+ “it, (7-1)
where y,, is the percentage of students, within the district, who pass the
fourth-grade MEAP math test. The observed time varying explanatory variables are
partioned in that x,,1 ,the variable being tested for endogeneity, is average real
spending in the current and previous three school years. Using a four year average
allows spending in the first, second, third, and fourth grade to affect performance on

the standardized tests given in the fourth grade. In addition, she uses the log of

average real spending noting that for a given increase in spending the affect on

29

performance may vary depending on the level of spending. That is, using the log of
spending states that the percentage change is constant as compared to the nominal
change in spending having a constant effect on pass rates. The strictly exogenous
explanatory variables, x,,2, include, the log of enrollment (lenrol) and the percentage
of students eligible for the federally subsidized school lunch program (lunch) were
both appear in quadratic form allowing for diminishing or increasing effects. The
unobserved district effect, c,, is allowed to be arbitrarily related to both x,“ and x,,2.
The instrument,(lf0und), for spending is the log of the district foundation grant
beginning in the 1994/1995 school year.

Table 1 contains fixed effects and fixed effects-instrumental variables estimates
of equation (7.1). The fixed effects-instrumental variables estimate for the log of
average real spending is 11.99 points higher than the fixed effects estimate. That is,
for a 10% increase in average spending the FE-IV estimate results in a 1.2% higher
pass rate than the FE estimate. However, Papke (2007) argues that allowing for
heterogeneity across districts may not be enough, it may be that the effect of
spending varies with performance. For example, a district that has low pass rates
prior to Proposal A may benefit more from an increase in spending than districts that
were already experiencing high pass rates. Thus, she separates the regression analysis
into districts of high verses low performance by using the average pass rate for the
first three years of the sample. This partitioning of the data allows the effect of
spending to vary with high or low initial performance. The estimates for lunch and
enrollment are parcitcally insignificant and only lunch is significant at the 10% level.

However, that does not mean that we haven’t missed important non-linearities. For

30

example, it could be that the affect of enrollment depends on spending.

Table 1
Estimates of the Percentage of Students with a Satisfactory Score on the 4’” Grade
Math Test (math4)
fl 1

LExplanatory Variables (1) FE (2) FE-IV l

 

 

 

 

'log(average real spending) “24.73 36.77
: (3.20) ,(5.44)

 

 

 

 

 

 

 

 

 

 

 

 

 

. l
llunch ,-.066 l-.073
l _ 1 (.037) (.038) “l
: log(enrollment) l —3.37 —.934 I
t 2 1 (1.97) (2.16)
1 year dummies? E yes yes j
[Within R2 __ V_ _ .394 1.392 , _ ,
(observations 15’000 [5,000 l

 

 

I test for endogenous spending using the procedure outlined in section 5.1. The
first step is to obtain the residuals, 12,,. That is, I estimate equation (7.1) under the
null hypothesis saving the residuals as uhat. Recall that under the null hypothesis
spending is strictly exogenous. Thus, the estimation method is standard fixed effects.

Next, I obtain the residuals, 12,,, from the FE regression,
log(avg. real spending) on log(grant), lunch, log(enroll), and year dummies (7.2)
To obtain 2,, I need to partial the explanatory variables out of 12,,. That is, I regress,
9,, on lunch, log(enroll), log(average real spending), and year dummies (7.3)
and save the residuals as f,,. Lastly, to compute the fully robust test statistic 1 regress,
12,, on 1",, (7.4)
using a fully robust variance matrix estimator. The robust t-statistic for 1",, is -l .84.

Thus, at the 10% level we reject the null hypothesis that spending is exogenous.

However, the p-value is .066, so the evidence that spending is endogenous is

31

minimal. The non-robust version of the test is N(T— 1)R2 from the regression in
(7.4). There are 500 school districts each with 10 years of observations and the

R2 =. 0017. Thus, the non-robust test statistic is 7.65. Thus, we reject the null
hypothesis that spending is exogenous at the 1% level. However, we must maintain
assumptions B.3 and B4 to justify the non-robust version.

Next, I test for nonlinearities. Given that we have evidence supporting the
endogeneity of spending, the first step is to ontain the residuals, 12,,, from estimating
equation (7.1) by FE-IV. Secondly, we need to estimate the reduced form of
spending. That is, obtain the fitted values, say spendinghat, from the pooled-OLS

regression of
dlog(avg. real spending) on dlog(grant), dlunch, dlog(enroll), and year dummies, (7.5)

where the d before each variable indicates that it has been demeaned. Next, we need
to obtain the misspecification indicators. Using demeaned variables obtain the fitted

values, yhat, from the pooled-OLS regression of

dmath4 on spendinghat, dlunch, dlenroll, and year dummies. (7.6)

A . A
The misspecification indicators are the squares and cubes of dmath4. To obta1n r,, we

regress each misspecification indicator, 1751211742 and W3, on spendinghat,
dlunch, dlog(enroll) and year dummies, saving the residuals, f,” and Pm. Lastly, we
regress 12,, on f,” and 1“,,2 using the robust cluster command and test for their joint
significance. The resulting Wald statistic is 7.54 which is significant at the 5% level.
However, the test does not indicated what non-linearity is missed. In our model, we

could try adding interactions between spending and enrollment and the same for

32

lunch.

8. Conclusion

I have derived robust regression-based tests after estimation by fixed effects
two-stage least squares. The tests offer simple methods to detect endogenous
explanatory variables, over-identifying restrictions, and functional form
misspecification. An important contribution is that the tests are easily adjusted to
account for arbitrary forms of heteroskedasticity and within cluster serial correlation.

1 apply the test for endogenous explanatory variables to a panel data model
explaining fourth grade pass rates on Michigan Education Assessment Program math
test in terms of spending, poverty rates, and enrollment. Using data fi'om Michigan
public school districts from 1992 through 2004, I test the null hypothesis that upon
eliminating the unobserved heterogeneity of the school district, spending is
exogenous. Applying both the fiilly robust and non-robust versions, I find mixed
statistical evidence that spending is endogenous. The robust version of the test
provides evidence that spending is endogenous at the 10% level but not at the 5%
level. However, the non-robust version is significant at all levels. This illustrates the
need for developing tests for heteroskedasticity and serial correlation in the context
of FEZSLS.

I have not analyzed the small-sample performance of the tests, as estimation by
FE ZSLS is not suited for data sets with a small number of cross-sectional units.
Generally, instrumental variable methods do not perform well in small samples.

However, a case worth examining would be panel data models that have a growing

33

time dimension and a small fixed cross-sectional size. In such circumstances the test
statistic and assumptions can be adjusted accordingly. For example, one would need
to make assumptions regarding the dependency of the data, as the asymptotic analysis
would focus on the behavior of the test statistic as the time dimension grows
infinitely. The more complex, yet interesting, case would be allowing for grth in
both the time-dimension and the cross-section. Under such circumstances
assumptions need to be made regarding the relative rates of growth.

Furthermore, the tests derived in this chapter focus on the relationship between
the idiosycratic error and the time-varying explanatory variables. It would be of
interest to derive a robust regression- based test about the presence or form of the
unobserved heterogeneity when the observed explanatory variables are endogenous

for other reasons.

34

Testing for Correlated Random Effects in Panel Data Models
Estimated using Instrumental Variables

1. Introduction

In the previous chapter, 1 derived tests concerning the idiosyncratic errors in
linear panel data models containing possibly endogenous explanatory variables. In
the context of fixed effects estimation, 1 showed how to obtain computationally
simple, fully robust tests to detect endogeneity of explanatory variables. I also
showed how to test overidentification restrictions in the context of fixed effects
instrumental variables (FE-1V) estimation.

The fixed effects estimator and its instrumental variables analogue have proven
to be very useful in empirical work, and the tests in the previous chapter should add
to the empirical researcher’s tool kit. But the FE-IV estimators can have large
sampling variances, making inference difficult or at least imprecise. In the context of
the linear model with an additive, time-constant effect, it is very common to compute
a Hausman (1978) statistic that compares the FE and RE estimates — at least for the
coefficients on variables that change in the cross section as well as across time.
Sometimes, the additional assumption under which RE is consistent - essentially,
that the heterogeneity is not correlated with the covariates — is not rejected, and then
one prefers the more precise RE estimates. See Wooldridge (2002, Chapter 10) for

recent discussion and for obtaining simple regression-based, fully robust tests.

35

Of course, it is possible to formally test for differences between random effects
instrumental variables (RE-IV) and FE-IV, and that is the subject of this chapter. I
propose simple variable-addition tests that are easily made robust using standard
software packages. In addition to treating the standard case of a single, additive,
time-constant effect, I also cover the so-called “random growth” model, which allows
for unit-specific linear trends as well as level effects.

My approach is related to, but distinct from, existing work in the literature.
Hausman and Taylor (1981) derive estimation and testing methods in the context of
linear panel data models containing correlated unobserved effects — that is, where the
unobserved effects are correlated with some of the explanatory variables. In their
setup, they maintain that all explanatory variables are uncorrelated with idiosyncratic
errors, but allow some explanatory variables to be correlated with the heterogeneity.
Hausman and Taylor’s main focus is on estimating the parameters associated with
time-invariant explanatory variables (which rules out the use of the usual FE
estimator). Hausman and Taylor impose exogeneity assumptions on some of the time
varying explanatory variables in developing an estimation procedure using
instruments to estimate the parameters of the observed time-invariant variables. The
instruments are the within transformations of the time-varying explanatory variables
that are assumed to have no relationship with the unobserved component. Therefore,
Hausman and Taylor do not rely on instruments from outside the model as they
assume the time varying explanatory variables are strictly exogenous. In addition,
they extend the work of Hausman in the context of testing for correlation between the

unobserved component and the included explanatory variables. However, they too

36

assume conditions simplifying the variance-covariance matrix.

Comwell, Schmidt, and Sickles (1990) propose an efficient instrumental
variables method similar to Hausman and Taylor (1981) for estimating coefficients
that vary over cross-sectional units as well as the intercept. Similar to Hausman and
Taylor, estimation does not rely on instruments from outside the model as they
assume that some of the time varying explanatory variables are uncorrelated with the
unobserved heterogeneity. Therefore, there must be enough variables that are
uncorrelated with the unobserved heterogeneity for the model to be identified.

Probably the closest paper to the current work is Arellano (1993), who develops a
test for correlated effects by decomposing the T time periods into two sets of
estimating equations. The first T — 1 equations comprise the equations that lead to the
usual fixed effects (or within-groups) estimator. The last equation leads to the
between-groups estimator. From these equations, Arellano (1993) shows how the
usual Hausman test comparing RE and FE can be obtained under the full set of
random effects assumptions (which include homoskedasticity and serial
independence of the errors). More importantly, Arellano shows how his approach
leads readily to a fully robust test. .

Arellano (1993) also shows how to extend the test to dynamic models. He covers
the case where the instrumental variables are the strictly exogenous variables from all
time periods. Here I am interested in general cases where a set of strictly exogenous
instruments is available — so, omitted variables or simultaneity can be treated.
Nevertheless, Arellano’s approach expanded to my setting can be used to obtain a

fully robust test in the context of RE-IV and F E-IV estimation. The drawback to

37

Arellano’s approach is computational. It requires forward filtering and then special
software to compute a fully robust variance matrix. The test 1 derive here can be
computed using standard software that computes RE-IV estimates and allows for
fully robust inference. Plus, as I mentioned, I do not restrict attention to a model with
a lagged dependent variable.

Metcalf (1996) extends the procedures of Hausman and Taylor to models
containing endogenous variables in addition to the correlated unobserved effects. The
test statistic developed is then pertaining to possible correlation between the
instruments and the unobserved component. The drawback is that, as with many of
the existing tests, the test requires a full set of random effects assumptions, including
serial independence of the idiosyncratic errors, and homoskedasticity of the variance
of the unobserved effect and the idiosyncratic errors. Plus, Metcalf’s statistic is not
readily computable after estimating models using standard sofiware.

Ahn and Low (1996) further extend the testing literature regarding panel data
models by reformulating the Hausman test in the context of GMM estimation. They
note that the Hausman test statistic for testing correlation between the unobserved
component and the regressors implies that the individual means or time averages of
the regressors are exogenous. (This point was made in the pioneering work by
Mundlak (1978) in the context of the standard linear panel data model). Ahn and
Low’s alternative GMM statistic incorporates a much broader set of moment
conditions that are valid if each of the time-varying explanatory variables is
exogenous. The key feature of Ahn and Low’s alternative GMM test statistic is that,

unlike the Hausman test, it has power in detecting nonstationary coefficients of the

38

regressors.

Unlike many of the previous tests, the tests derived in this chapter do not rely on
assumptions about the structure of the variance-covariance matrix of the composite
error I derive a variable addition test for the key assumption of RE-IV estimation,
namely, that the unobserved effect is uncorrelated with the instruments. In addition, I
extend the usual model to allow for individual specific trends. The tests are easy to
compute while allowing for arbitrary forms of heteroskedasticity and serial
correlation under the null.

In the next section, I develop the variable addition method for testing the random
effects assumption pertaining to the unobserved component. In Section 3, I extend to
test to account for possible trending of the unobserved component. Section 4
discusses the ramifications of applying the tests to unbalanced panels. In Section 5, I
apply the tests to a panel data model that explains test pass rates in terms of spending,
poverty rates, and enrollment. 1 test for correlated unobserved district effects and for
possible trending of the unobserved district effect. The data set, used by Papke
(2007), comes fiom Michigan schools from 1992 through 2004. Section 6 contains a

brief conclusion.

2. The Standard Model and Assumptions

Consider the standard panel data model containing unobserved heterogeneity in

the cross-section
yit=xitp+ci+uits I: 19"‘9T9 (21)

where 1' denotes a random draw from the cross section. The dependent variable is y,,

39

and x,, is the 1 x K vector of observed explanatory variables, some of which are
thought to be correlated with the idiosyncratic errors u,,, or even u,, for r :1: t. In
many applications, some elements of x,, are strictly exogenous in the sense that they
are uncorrelated with the idiosyncratic errors in all time periods. We also allow x,, to
be arbitrarily correlated with unobserved heterogeneity, c,.

If we allow some elements of x,, to be correlated with the idiosyncratic errors
then we need some extra information to estimate ,6. I assume we have a set of
instrumental variables 2,,, a 1 x L vector with L 2 K. Typically, 2,, and x,, would
overlap - for example, each would include a full set of time dummies to allow
unrestricted secular effects. The set of data observed for each cross-sectional unit 1' is
{(x,,,y,,,z,,) : t = 1,. .., T}. I assume random sampling in the cross-section
dimension with a sample of size N. The time series dimension, T, is fixed in the
asymptotic arguments.

The assumption I maintain in the instruments is that they are strictly exogenous

in the sense that
E(z;,u,,) = 0, t,r = 1,...,T. (2.2)

Assumption (2.2) allows arbitrary correlation between 2,, and 0,. But the instruments
may not be correlated with the idiosyncratic errors at any time period. For example,
changes in 11,, cannot cause changes in future values of the instruments.

In this chapter, [propose robust, computationally simple tests for whether the
instruments are correlated with unobserved heterogeneity. In addition to (2.2), the
key assumption, imposed by a random effects using instrumental variables (RE-IV)

approach is

40

5(01121) = 13(61), (23)
so that the heterogeneity is mean independent of 2,. (Actually, for consistency, we
can relax (2.3) to zero correlation, but (2.3) is useful for stating alternative
assumptions and for obtaining simple inference under a full set of random effects
assumptions.) If Assumption (2.3) holds along with (2.2) and a standard rank
condition, then the RE-IV estimator is consistent for 3. (Recall, if we apply RE, we
can allow x,, and 2,, to contain elements that do not change over time.) If (2.3) is
violated, but we have sufficient time-varying instruments for the time—varying
elements of x,,, then we can estimate [3 (or at least the elements on the time-varying
components of x,,) consistently using fixed effects instrumental variables (FE-IV).
But, as is well known, the FE-IV estimator can have large sampling variance because
we are removing time averages and also applying instrumental variables. If (2.3)
holds, it can be much more efficient to use RE-IV. And, if we have an instrument that
is essentially randomized, we can treat it as being orthogonal to the time-constant
heterogeneity as well as the idiosyncratic errors.

As I discussed in the introduction, one can directly compare the RE-IV and
FE-IV estimates via a Hausman (1978), but computing a fully robust test that does
not maintain serial uncorrelatedness and homoskedasticity of the idiosyncratic errors,
along with homoskedasticity of Var(c,|z,), is cumbersome. Instead, I propose a
variable addition test.

To derive the test, it is easiest to think of an alternative to (2.3) as
E(Ci|21) = E(Cil§i) = 50 + 31-5, (2-4)

where 2, is the time average of 2,, :

41

T
2,- = T" 22,-, (2.5)
t=1

In what follows, 2, would not include averages of time-period dummies or other
aggregate time variables, but we use the same notation for simplicity. Assumptions
such as (2.4) date back to Mundlak (1978) for the linear model with strictly
exogenous explanatory variables and Chamberlain (1980) binary response models.
Recently, Papke and Wooldridge (2007) use assumptions like (2.4) to estimate
nonlinear models for fractional response with endogenous explanatory variables.

Under (2.4) we can write
0, = £0 +2,J,‘ + a,, E(a,|z,) = 0. (2.6)
Plugging (2.6) into (2.1) gives the augmented equation
y,,=x,,fi+§0+2,§+a,+u,,, t=1,...,T. (2.7)
Under the maintained assumptions, the composite error in (2.7), say v,, = a, + u,,, is
uncorrelated with 2,, so, in particular, v,, is uncorrelated with (z,,,2,). Therefore, we
can estimate (2.7) by instrumental variables using (in addition to a constant) (z,,,2,)
as the IVs. Interestingly, it turns out that, whether we use pooled IV or RE—IV on
(2.7), the estimator of ,0 is the FE-IV estimator. This is an important result as
generally estimating (2.7) by RE-IV would be inconsistent without assuming (2.4).
However, since it is the FE-IV estimator we do not need to maintain assumptions
pertaining to the relationship between the unobserved heterogeneity and instruments.
To verify that the RE-IV estimator of (2.7) is identical to the FE-IV estimator, we
use the characterization of RE-IV as a pooled IV estimator on quasi-time-demeanded

data. In fact, let 2 be any value in [0,1], and consider the equation

42

y,, — 7151,- = (x,, — 22,),8 +(1— 102,5 + v,, — 217,1: 1,...,T, (2.8)

or
51,,=2,,fi+(1—2)2,§+i'1,,,t= 1,...,T, (2.9)
where the intercept is suppressedjust for notational simplicity. (In fact, we can just
include it in 2,.) When 2 = 0 we have the pooled-IV estimator and when 2 = 1 the
FE-IV estimator.
The instruments are then
51': = [(211 “ 450,512] (2-10)

We can absorb (1 — 2) into C and since (z,,-22,) is a linear combination of 2, and 2,,
we define the set of instrumental variables as (2,2,). The first stage regression of
random effects two stage least squares, RE- ZSLS, regresses 2,, on 2,,, 2, obtaining
the fitted values 2,. As a result of the orthogonality between 2,, and 2, it is
algebraically equivalent to regress 2,, on 2,, and 2,, on 2, then combining linear
projections gives the fitted values 2,.

Regressing 2,, on 2,, gives

@-
(:11:

This is equivalent to

‘1 N T

2,) (2221,11,) (2.11)
1—11=1

(zit) (fiiZMHQH—M 12)
1—11-1

(i iii-1311)-] (2N: 5321-1551) (2.12)

M4 1M4

K
II
~

i=1 (=1 i=1 [=1

43

Therefore regressing 2,, on 2,, is the same as regressing 2,, on 2 ,,. Regressing 2,,

on 2, results in the following:

N T -1 N T N -1 N T
22222,- zzzzseu =74 221,) 22:23,..-»2.)
i=1 I: 1:] t=l i=l 1:] (=1
N *1 N
=7“l 22:2, T(1-A)2(2§2,)
l=l =1
N *1 N
=(1-2) 22:2,- 2(2§5c,), (2.13)

or, (1 — 1.) times the coefficient estimate from regressing 2, on 2,. Combining the two
linear projections results in the fitted values
2, = 2,r‘1, + (1 - 2.)2,fi2 (2.14)

The final step in two-stage random effects is to regress 52,, on 2, and 2,,. Recall
that the estimated slope coefficient for explanatory variable a in multiple regression
analysis is the same as the slope coefficient from simple regression analysis if we;
first regress explanatory variable a on all of the others, then regress the dependant
variable on the residuals fiom the first regression. Therefore, to show that adding the
time averages of the instruments results in the fixed effects estimator for 2,, we split
the regression into two parts. First regress 2,, on 2,, obtain the residuals 2,,, then
regress 57,, on 2,,. The coefficient pertaining to 2,, is the same as the coefficient on 2,,
in the regression of 5),, on 2, and 2,,. This shows that 5",, is equal to 2,, which is equal
to 2,,fi 1. Secondly, I show that the quasi-demeaning of the dependent variable does
not alter the coefficient estimate.

a o _ A o c In} .. o _ c
We can replace x,, With (1 — A)z,1'12 tn the regressmn of x,, on 2, smce z, 15

44

orthogonal to 2,,. Therefore the coefficient on 2, can be written as
N -1 N
(1 - l)< 2:5,) (2 21-2,)1'12 = (l - 101-12
i=1 1':

;it = §it- (l —/1)2,fiz

Then,

Substituting in 2,, = 2,,fI, + (1 — ”2,122 gives
r: = 36:17- --(l 102771212 = 5i1fi1+(1- Diifiz - (1 - Miifiz = 3:11:11 = 12:7

Next I show that the quasi-demeaning does not alter the estimation.

OCI~

x mm

M2
Ms

BRmSLS = (Z: Z x 11x")

i-l (:1

N T A’A
= (2255115511).

-
ll

y—n

u—n

'tO’it—Ayi)

.5“:

M2
Ma
51’

N
11
_
'IT
~

~
|l
#
N
ll
—

11 11

A A

E. [V] 2 M 2

E M ~a M -1
3.22. 32>.
.2 ..

II; M 2 'M 2
TL M ~1 M ~a
3‘9. :1’

E} 3"

11 E
i?)

E
a

Extending Mundlak (1978), a test for correlated effects is a test of H0 : 4‘ = O in

equation (2.7). In most applications, we would include year intercepts in (2.7), and

these, of course, would not be included in 2,. We can estimate (2.7) by RE-IV using

instruments (21:3,) as instruments or by pooled ZSLS using the same instruments. In

the latter case, the test should definitely be made robust to heteroskedasticity and

serial correlation. If we think the standard RE assumption E(v,v,|z,) = E(v,v,) = Q

where Q has the standard random effects form, then the usual nonrobust test based on

RE-IV works. But, we can also use a robust test in the context of RE-IV in order to

45

(2.15)

(2.16)

(2.17)

allow arbitrary heteroskedasticity and serial correlation.

3. Trending Unobserved Heterogeneity

In the previous section we examined the standard unobserved effects model
where the unobserved effect has the same partial effect on y), in all time periods. This
assumption is to strong for some applications. For example, F riedberg (1998)
analyzes the affect of divorce laws on divorce rates using state-level panel data.
When she omits a state-specific trend she finds no evidence that divorce laws affect
divorce rates. However, when she includes a state-specific trend the estimated effect
is large and statistically significant. Therefore, we extend the previous model to
allow for individual-specific slopes or a random trend. Consider the following

extension of equation (2.1):
yi, =Ci+git+xitfi+uits t=1,2,...,T. (3.1)

The difference between equation (2.1) and the above is that the individual-specific
trend is an additional source of heterogeneity, as in Heckman and Hotz (1989). As

before, we allow (c,-, g,) to be arbitrarily correlated with xi! and 2),. If we could

maintain strict exogeneity of the observed explanatory variables we could estimate fl
by first differencing equation (3.1) then estimating the differenced equation by fixed
effects or differencing again and estimate the twice differenced equation by pooled
OLS. Regardless of which method one chooses to use, there must be at least three
time periods. Typically the method of choice is based on the serial correlation of the
idiosycratic error. Recall that if a variable has an autocorrelation equal to one, that is,

it follows a random walk, then differencing that variable results in the differenced

46

variable having an autocorrelation of zero. Therefore, if the idiosycratic error is
believed to follow a random walk then the twice differencing approach leads to
standard errors that are serially correlated.

However, if it is not feasible to maintain strict exogeneity of the explanatory
variables then one needs to estimate the differenced equation using fixed effects two
stage least squares or difference again and estimate the twice differenced equation
using pooled ZSLS. As you can imagine the twice differencing or differencing and
quasi-demeaning, along with using instrumental variables, can result in large
sampling variances, making inference difficult. Thus, if the FE-IV assumption that

the unobserved heterogeneity is time constant holds, that is that

E(g1|21) = E(g1) (3.2)
then one prefers the more precise F E-IV estimates. Or, if an alternative version of the

RE-IV assumption that
E(Ci|Zi) = E(Ci) (3-3)
and
E(gi|Zi) = 5(81') (3-4)
holds then one would prefer the more precise pooled-IV estimates.
Therefore, we are interested in deriving a variable addition method to test
whether such advanced estimation methods are warranted. As in the standard
unobserved effects model, illustrated in the previous section, we assume that any
correlation between the unobserved heterogeneity and the instruments arises through

the mean and now trending behavior of our instruments. Thus, to derive the

appropriate variable to add to equation (3.1) we need to model the trending behavior

47

of the instrumental variables.

Let the following describe the nature of our instruments across-time:

z;,=di+h,-t+v,-,, t= l,2,...,T. (3.5)

This equation states that the behavior of the instruments across time is unique for
each cross-sectional unit. While it may seem odd to think that both the structural
model and instruments contain possible trending unobserved heterogeneity consider
the following example. Papke (2007) estimates the effect of spending on student
performance using data from Michigan public school districts where she uses the
amount of the foundation grant as an instrument for spending, since the foundation
grant varies by school district and across-time it is obvious that the trending behavior
of the foundation grant is unique for each school district. The question is whether the
unobserved district effect trends in the pass-rate equation.

Secondly, to derive the appropriate test, consider the alternative to (3.2), (3.3),

and (3.4) as

E(Cilzi) = E(Cildiahi) = 710 + ”1611+ ”Zhi (3-6)
and

E(gilzi) = E(g1|di,hi) = 10 + lldi + A2hr (3-7)

That is, we assume that the unobserved heterogeneity in equation (3.1) is related to
our instruments through their trending unobserved heterogeneity in equation (3.5).

Under equations (3.6) and (3.7) we can write
0,- = 71'0 + fl1di+ flzh; + a,, E(a,-|z,-) = 0 (3.8)

and

48

g,- = 10 + 11d; + 11211; + bi, E(bilzi) = 0. (3.9)
Substituting equations (3.8) and (3.9) into equation (3.1) gives,

ya = Ito +7tldi+7t2hi+ lot+lldfl+lzhit+xitfl (3.]0)
+a,-+b,-t+ui,,t= 1,...,T.

In addition to a linear time trend, we have the levels of the heterogeneity in (3.5) and
also interacted with a linear time trend.

Under the strict exogeneity of 2),, and assumptions (3.8) and (3.9), estimation of
(3.10) by pooled IV is consistent, as the composite error, vi, = a,- + b it + up, is
uncorrelated with Zi- That is, inserting equations (3.8) and (3.9) into (3.1) controls for
any correlation between the instruments and the unobserved heterogeneity. In
addition, we can test for the significance of the trending unobserved heterogeneity in
equation (3.1) by testing the null hypothesis that 7t1,7t2, A} and 12 are equal to zero.

Using equation (3.10) to test for correlated effects is not directly applicable
because we do not observe d,- and h ,-. Instead, we can use proxies for the unobserved
heterogeneity obtained from unit-specific regressions. Remember, we are obtaining a
test of the null hypothesis that there is no correlated heterogeneity between the
heterogeneity in the structural model and the instruments. It is natural to regress each
instrument on a constant and linear time trend, separately for each cross section unit,
1'. This gives us, for each i, a I x 2L vector (8,5,) (where, of course, we leave out
time period dummies). In effect, we are looking for correlation between (c,-,g,-) and
(at, 51)-

Obtaining the (21,-, 5,) can be obtained unit-by-unit. If N is not too large, the

49

pooled regression
zi,0ndl,-,a2;,...,dNi,d1; °t,...,dN,' 0t, i = 1,...,N; I = 1,...,T (3.11)

can be used to obtain the intercept and slope for each 1'. However we obtain the
individual-specific slopes and trends, we estimate the equation

y,, = 7:0 + 1:121,- + 7121;, + 101+ 11,21,” Azfiit+xitl3+ e,,, t = 1,...,T (3.12)
by some instrumental variables method, where the instruments are
(1,3),5 ,-,t, flit, 5,1,2”). The test for no correlated heterogeneity is thejoint test that
7:1,7r2, A] and AZ are all zero, which is a test of 4L restrictions. In practice, we may
want to be selective about the elements of 2;, we include in equation (3.5). A more
efficient test is to use a generalized method of moments procedure with an
unrestricted weighting matrix. Alternatively, if under the null we maintain that
{an : t = 1, . . . , T} is serially independent and homoskedasticity, conditional on 2;,
then a random effects ZSLS estimator can be used. It is not the standard estimator
because of the presence of b it in the error term. This would be the efficient IV
estimator if the typical random effects assumptions hold, but we can also guard
against violation of these assumptions by making inference robust.

It is important to understand that in (3.12) there is not incidental parameters
problem from “estimating” the intercept and trend terms, :1,- and Iii. First, they are not
parameters they are heterogeneity. Second, for each 1' these are simply fimctions of
{2), : t = 1,. . . , T}, and so the test from (3.12) is looking for whether these functions
of 2,- appear to be correlated with the heterogeneity in the error term. In fact, we

couldjust have stated the alternative as

E(c,-|z,-) = Ito + 7:121,- + £25; (3.13)

50

and
E(g,-|z,-) = 2.0 + 21121,- + 2211‘). (3.14)
It is interesting to note that, if (3.13) and (3.14) are true — so that the unit-specific
intercepts and slopes act as sufficient statistics for the relationship between (ci,g;)
and z,- — then IV estimators using (3.12) are actually consistent (with T fixed, N -» 00)
even when the null is false. In fact, we can replace the conditional expectations with
linear projections, but we are still assuming a specific relationship between (c;,g,-)
and 2;. IV methods that eliminate (ci,g,-) from (3.1) require no such assumptions.

It may be that we strongly suspect that there is unobserved heterogeneity in the
cross-section that is correlated with the instrumental variables, and we want to allow
that while testing whether the individual-specific trend is correlated with the IVs. It is
very common to estimate models using fixed effects IV methods without checking
for individual-specific trends. If we want to test the null hypothesis (3.2) without

restricting E(c,-|z,-), then it is natural to specify the testing equation
)2), = 50 + z"): + 101+ 2121;” Azfiit+xnfi + e", t=1,2,...,T, (3.15)
where 2,- = T‘1 2:12;, is the time average. We can estimate this equation by pooled
ZSLS, using instruments (1,2),1, 21,1, 5,1,2”) or, even better, random effects IV, as the
RE-IV estimator has a chance of being efficient under the null hypothesis. In
particular, if E(v,-v:-|z,-) == E(v,-v:-) = og-jyj'T+ 031;, where jT is the T x 1 vector of
ones and v,- is the vector of composite errors c,- + u,-,, then the RE-IV estimator is the
asymptotically efficient IV estimator under H0. Actually, in most cases it is

preferable to include a full set of year dummies in xi, and then drop the overall linear

time trend, lot. Either way, we test for joint significance of (1] , 22).

51

An alternative to RE-IV estimation with 2,- included as regressors is to estimate
)2), = m+111£15t+ Azfiit+x,-,[3+c,-+ 11,-, (3.16)

by FE-IV, where the instruments are the time dummies, flit, ligt, and 2;, Once £1,- and
Iii have been obtained, the test is easy to cm out. The approach is preferred because
it does not impose restrictions on the relationship between c,- and 2;.

If we fail to reject the null hypothesis that the unobserved heterogeneity in the
cross-section is time-invariant, we would want to estimate [3 by the more efficient
FE-IV. However, if we reject the null hypothesis we must estimate equation (3.1) by
either twice differencing equation (3.1) then estimating the twice differenced
equation by pooled-IV or first difference equation (3.1) then estimate the differenced

equation by FE-IV.

4. Extension to Unbalanced Panels

As is well known, applying standard panel data methods to unbalanced panels
may result in inconsistent estimators if the sample selection mechanism is not
appropriately “exogenous.” As discussed by Wooldridge (2002, Chapter 17), fixed
effects methods are more robust in the presence of missing time periods because
selection is allowed to be arbitrarily correlated with the unobserved heterogeneity.
This holds true in the model (3.1) estimated by IV methods that eliminate (ci, gi). So,
we might first difference and the use fixed effects, or sweep away the
individual-specific trends in the instruments before applying IV. Using the same
arguments in Wooldridge, it is easy to show that a sufficient condition for

consistency on the unbalanced panels is

52

E(u,,|z,,s,1,...,s,T,) = 0, I: 1,...,T, (4.1)

where 3,, is the selection indicator for unit 1' in time t: s,, = 1 if we observe the data,
and zero otherwise.

Extending the test in Section 3 to the case of unbalanced panels is, in principle,
straightforward. Let T, be the number of time periods observed for unit 1'. Then, for
all units with T, 2 3, we can obtain 21,- and H, as before, and use these in either
equation (3.12) or (3.16). (Equation (3.15) is not as attractive because in the
unbalanced case, putting 2, in as regressors is no longer completely general.)
However, we must be cautious in interpreting a rejection, because (£1,,li,) are

functions of (s,, , . . . ,s,7~). One way to see this is that (2155,) are from the OLS
regression
s,,z,, on s,,,s,, - t,t = 1,...,T. (4.2)
Therefore, if we find that (3,,fi,) are statistically significant, it could be because
(d,,h,) is correlated with the trend, g,, but it could also just be that g, is correlated
with selection. Fortunately, the next step would be clear in either case: we would
estimate (3.1) by IV methods that eliminate c, and g,, which would allow selection to

be correlated with g,- and the instruments to be correlated with g,.

5. Empirical Example

In chapter one, I extended the analysis of Papke (2007) by testing for endogenous
spending in the pass rate equation after controlling for the unobserved district effect.
Using the level of the foundation grant as an instrument for spending 1 found mixed

evidence about the exogeneity of spending.

53

In this chapter, I examine the spending-performance model further by applying
the tests for an unobserved district effect and for possible trending of the district
effect. In section two, I showed that adding the time averages of the instruments to
the pass rate equation then estimating the augmented equation by RE-IV results in
the FE-IV estimator for ,8 while supplying a test for the random effects assumption
regarding the unobserved heterogeneity stated in (2.3). In the table below, I provide
the estimation results for FE-IV, RE-IV, and RE-IV with the time averages of the
instruments. Because this is a balanced panel, we do not have to worry about sample
selection issues.

Table 1
Fixed Effects IV, Random Effects IV, and Random Effects IV including time

averages of the instruments:
Dependent Variable is Percentage of Students with a Satisfactory Score on 4* Grade

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

__ ___ -_ ,. _ 1 _ Matthest

1 Variable _ _ 1 FE-IV RE-IV 1 RE-IV with 2,- 1
1 log(average real spending) 1 36. 77 21.38 F 36. 77
1 ; (5.44) (2.77) (5.44) 1

lunch -. 073 —.352 —. 073 l
L (.038) (02) (.038) 1 7
1 log(enrollment) ; —. 934 -. 809 —. 934 1
1 - (2.17) (.401) (2.17) 1
j average lunch ‘ ! -—.433
1 ,2 +7 _ , __# 13-046) fl
1 average log(enrollment) .015

. . (2.25) "k A1

1 average log(real foundation grant) 1 —28.42
1 _ _ (6.59) . #7”;
1numberofdistricts _ :500 1500 500 __ ‘
1R2 1 .392 1 .385 .392 1

 

 

 

 

 

Note: All specifications include a full set of year dummy variables. -

Consistent with the derivation in Section 2, the RE-IV with the time averages of

the instruments yields the exact same estimates as FE-IV. However, there is a rather

large difference between them and the RE-IV estimates. Loosely this is an indicator

54

of correlated unobserved effects. Testing the joint significance of the three time
average variables gives a Wald statistic equal to 104.72 which is clearly significant at
all levels.

Since the data spans 10 years it is possible that the district effect follows a linear
trend. As 1 suggested earlier, this could easily happen if school districts adjust how
they allocate funds across schools in the district. For example, Papke (2007) breaks
the analysis up into districts that have initial low pass rates and districts that have
moderate or high pass rates prior to Proposal A. This results in spending having a
much higher effect on pass rates for the districts with initially low pass rates than
their counterparts. Following the same thought, if some schools within a given
district have much lower pass rates allocating more fimds to those than to schools
with higher pass rates is likely to increase the district’s overall pass rate. As districts
learn the most beneficial way of allocating funds we would expect the district effect
to increase. Therefore, 1 apply the test for trending unobserved district effects by
estimating equation (3.16) and testing the joint significance of flit and liit. After
estimating (3.16) by FE-IV, the Wald statistic for the joint significance of cilit and Iiit
is 1.56 with a p-value of .45. That is, we fail to reject the null hypothesis that the

unobserved heterogeneity is time—invariant.

6. Conclusion

This chapter has developed variable addition tests for correlated unobserved
heterogeneity in panel data models containing endogenous variables. The tests can be

computed through a series of regressions and are robust to arbitrary forms of

55

heteroskedasticity as well as within cluster serial correlation by simply computing
robust standard errors when estimating the augmented equation. The benefit of the
variable addition method is that it not only allows for computational ease but allows
the structure of the tests to adjust to fit various hypotheses. For example, we can tests
the random-effects assumption that the conditional expectation of the unobserved
heterogeneity does not depend on the instruments or test the fixed-effects assumption
that the unobserved heterogeneity is time-invariant.

I have not analyzed the finite sample properties. However, panel data and
instrumental variables methods are perhaps not well-suited for data sets that have a
small number of cross-sectional units. The test developed in this chapter does not
readily extend to panel data models with a small cross-sectional dimension and large
time-dimension. However, as panel data continues to grow and statistical software
becomes more and more sophisticated, the need for tests valid under large N and T
become obvious.

I apply the variable addition test for correlated unobserved effects in analyzing
the effect of spending on student performance. The study uses data on the pass rates
for fourth-grade MEAP math tests from 1992 through 2004 at Michigan Public
school districts. 1 recreate the results of Papke (2007) and apply the test for an
unobserved district effect. Adding the time averages of the instruments to the RE-IV
equation I find evidence supporting an unobserved district effect. I expand this to

test for possible trending of the unobserved district effect.

56

Testing the Conditional and Unconditional Variance Matrix in
Panel Data Models Estimated by Fixed Effects ZSLS

1. Introduction

In the previous chapters, I derived tests concerning the idiosyncratic errors and
unobserved heterogeneity in linear panel data models containing possibly
endogenous explanatory variables. In the first chapter, I developed a robust
regression based test for endogenous explanatory variables in the context of fixed
effects estimation. In addition, I show how to test for overidentification and
non-linearities in models estimated by fixed effects instrumental variables (FE-IV).
In the second chapter, I provide simple variable-addition tests regarding the
unobserved heterogeneity in the context of panel data models estimated by
instrumental variables. The variable addition method allows for test statistics that are
easily made robust to arbitrary forms of heteroskedasticity and serial correlation
using standard software packages. In addition to covering the standard case of a
single, additive, time-constant effect, I show how to test for correlated effects that
follow a linear trend.

While it is well known that the presence of heteroskedasticity or serial correlation
in an otherwise properly specified model does not affect the consistency of the
estimator. It does lead to inefficient parameter estimates and inconsistent estimation
of the variance-covariance matrix. Where inconsistent estimation of the
variance-covariance matrix leads to invalid test statistics. As a result, the most
convincing studies report robust standard errors. While this is primarily due to

advances in software packages as virtually all packages compute robust standard

57

errors with a simple command. Despite the ease in computing robust standard errors
researchers are still interested in testing for serial correlation and heteroskedasticity
as if either are detected more efficient estimation methods such as GMM, or GLS are
available.

Wooldridge (2002, Section 10.5) proposes a simple regression based tests for
serial correlation in panel data models with strictly exogenous regressors. The
popular statistical software, Stata, implements Wooldridges’ test for autoregressive
or moving-average behavior of the idiosyncratic errors in models estimated by
random effects or fixed effects. The test is very attractive because of its’ robustness
and computational ease. Drukker (2003) presents simulation evidence that
Wooldridges’ test has good size and power properties in balanced and unbalanced
panel data models, with or without conditional heteroskedasticity, in reasonably sized
samples. Wooldridge notes that testing for serial correlation is more complicated in
the fixed effects setting, in that, under the null hypothesis of no serial correlation in
the idiosycratic error the time-demeaned errors are serially correlated. This
complicates the analysis since we can only estimate the time-demeaned errors. In
general the correlation induced by the time-demeaning is —l/(T- 1). Therefore, at a
minimum we need to make the test for serial correlation robust to serial correlation.
In addition, while in the case of strictly exogenous regressors one can ignore the
estimation error in ,6 when obtaining the asymptotic distribution of the test statistic,
that is not so in the absence of strictly exogenous regressors. Thus, while fixed
effects-instrumental variables (FE-IV) methods are becoming more common it is not

surprising that there are no readily available tests for serial correlation.

58

Breusch and Pagan (1979) and White (1980) propose regression-based tests for
heteroskedasticity that can be applied in both time-series and cross-sectional studies.
However, both impose auxiliary assumptions regarding the fourth moment for the
tests to be valid. Wooldridge (l 990) derives a partially out method resulting in
regression-based tests without maintaining auxiliary assumptions under the null
hypothesis. In particular, Wooldridge shows that de—meaning all functions of the
explanatory variables used in the test results in a test for heteroskedasticity that is
robust to heterokurtosis. While the method used in this chapter follows most closely
that of Breusch and Pagan (1979), the within transformation of fixed effects results in
a regression-based test that is robust to heterokurtosis. In the standard fixed effects
model with strictly exogenous regressors the Breusch-Pagan (1979) test for
heteroskedasticity can be used with a degrees of freedom adjustment to account for
the time-demeaning. However, as noted in the previous paragraph, it is typically the
case that one cannot ignore the estimation error in ,6 when obtaining the asymptotic
distribution of the test statistic. Generally accounting for the first-stage estimation
results in test statistics that are not computationally simple. Conveniently, in this
chapter I show that the Breusch-Pagan test for heteroskedasticity can be extended to
panel data models estimated by FE-IV.

The remainder of this chapter is organized as follows. Section 2 contains a
review of existing tests for heteroskedasticity and serial correlation. Section 3
provides the basic FE-ZSLS model and assumptions. In section 4, I develop a robust
regression-based test for heteroskedasticity after estimation by FB-ZSLS. In section 5,

I obtain a test for serial correlation. In Section, 6 I apply the test for

59

heteroskedasticity to a panel data model that explains test pass rates in terms of
spending,poverty rates, and enrollment. The data set, use by Papke (2007), comes
from Michigan school districts from 1992 through 2004. Section 7 contains a brief
conclusion. Throughout the chapter I assume random sampling in the cross-section
with a fixed time dimension and growing cross-sectional dimension. That is, the

asymptotics is standard random sampling asymptotics.

2. Background

Testing for heteroskedasticity and serial correlation is well established in the pure
cross-sectional and time series contexts. Of course serial correlation is not considered
in pure cross-sectional analysis, as there is no time dimension. Breush and Pagan
(l 979) and White (I 980) have the seminal papers on regression based tests for
heteroskedasticity. While the tests they derive can easily be extended to panel data
models with strictly exogenous explanatory variables there is no clear extension to
models containing endogenous variables. However, when extending such tests to
panel data models estimated by fixed effects we must make sure the tests are robust
to serial correlation. This changes the standard form of the tests as neither address
robustness to serial correlation since the tests were derived for cross-sectional
models. In addition, they both impose homokurtosis under the null hypothesis.
Conveniently, Wooldridge (l 990) shows that partialling out the averages of the
variance misspecification indicators leads to a version of the Breush and Pagan test
that is robust to heterokurtosis. This implies that we do not have to address issues of

heterokurtosis when deriving a test for heteroskedasticity due to the

60

time-demeaning.

While the literature on testing for serial correlation is vast, most of the existing
tests today stem from Durbin and Watson’s (1951) famous d-test. Durbin (l 970)
extended the seminal work of Durbin and Watson with the more exact h-test and the
regression based m-test. A drawback to the d,h, and m tests are that they use
first-stage residuals in computing the test statistic. This of course is not an issue when
the regressors are exogenous. However, in models containing endogenous
explanatory variables typically, the asymptotic distribution of the test statistic will
depend on the first-stage estimation error. Generally accounting for the first-stage
estimation error is not the problem but rather, developing a test statistic that is easy to
compute is the issue.

Godfrey (1976) developed an instrumental variables analog to Durbin’s h-test by
replacing the log-likelihood function of Durbin’s general theorem with the
autoregressive IV criterion function. Durbin’s variable addition m-test was extended
to models estimated by instrumental variables in the unpublished work of Breush
(I 978). Breush’s procedure replaces the OLS residuals with their IV counterparts,
resulting in a regression based test for serial correlation in time series models
containing endogenous variables. Godfrey (I 994) extends on Breush’s procedure by
redefining the set of instruments used in estimating the augmented equation to
increase the precision of the test. However, such methods do not extend to panel data
models with endogenous variables as the form of exogeneity is more restrictive. In
addition, testing for serial correlation in the fixed effects setting is complicated in

that the within transformation induces serial correlation when the idiosycratic errors

6]

are uncorrelated.

Wooldridge (2002 chapter 10) extends Durbin’s m-test to panel data models
estimated by fixed effects. Wooldridges’ test can be used to test for both
autoregressive or moving-average idiosyncratic errors and is robust to conditional
heteroskedasticity. Another benefit of the Wooldridge test is that it is very easy to
implement, Stata currently implements the test with a simple command. Bhargava,

F ranzini, and Narendranathan (l 982) extend Durbin and Watson’s d-test, and the
Berenblut-Webb (1973) test for the null hypothesis that the idiosyncratic errors
follow a random walk, to panel data models estimated by fixed effects. Arellano
(I990) develops a test for serial correlation in panel data models containing lags of
the dependant variable and unobserved heterogeneity. Arellano derives the limiting
distribution of the covariance matrix under non-normality and presents a Wald test
for testing covariance restrictions after estimation by three-stage least squares

(3 SLS). However, Arellano derives the limiting distribution assuming the unobserved
heterogeneity is uncorrelated with the instrumental variables thereby avoiding
complications resulting from the within transformation. Inoue and Solon (2005)
propose a portmanteau test for serial correlation in panel data models estimated by
fixed effects using the conditional likelihood function. In comparison to the
Wooldridge test, the test proposed by Inoue and Solon is harder to compute. They do
suggest that the test can be used in models estimated by FE-IV. However, it turns out
that the lack of strictly exogenous regressors affects the asymptotic distribution of the
test statistic. Okui (2007) applies double asymptotics and a kernel estimator to extend

the test for serial correlation developed by Ljung and Box (1978) to panel data

62

models estimated by fixed effects. Okui notes that the biasness that results from using

the time-demeaned errors arises from the inability to estimate the long-run variance.

3. General FE-IV Model

Consider the standard unobserved effects panel data model
yi,=x;,,6+ci+u;,, t= l,2,...,T, (3.1)

where i denotes a random draw from the cross section. The vector x,, is 1 x K of
explanatory variables, c; is the time-constant unobserved effect, and “it is the
idiosyncratic error. To allow estimation by instrumental variables, let 2;, be a l x L
vector of instruments, where L 2 K. Thus, the set of observations for unit 1' is
{(xi,,y,-,,z,-,) : t = 1,..., T}. I assume random sampling in the cross-section
dimension with a sample of size N. The time series dimension, T, is fixed in the
asymptotic arguments. Typically, 2i, and x it would overlap — for example, each
would include a full set of time dummies to allow unrestricted secular effects.

In this chapter, I make no assumption regarding possible correlation between x;,
and c,. I also allow the instruments to be arbitrarily correlated with ci. Therefore, to
consistently estimate ,6, the unobserved effect is removed via the within
transformation. For each (i, t), let jig, = y,, — j),- be the time-demeaned dependent
variable, where )7 = T‘1 z 1:] y,, is the time average. Similar notations hold for 56,-,
and it“. After applying the within transformation to equation (3.1) the model

becomes
Yit = 551%“? flit, t: 1,...,T, (32)

If (xi, : t = l, . .., T} is strictly exogenous - that is, E(x:-su,-,) = 0, all

63

s,t = 1,..., T, then Ritz-silly) = 0. Thus under strict exogeneity of the explanatory
variables estimation of equation (3.2) by pooled OLS is a consist procedure for the
estimation of ,6. Where
N ‘1 N
i3 = (xv-I 21%;" 3? ) N-1 23217;;-
i=l i=1

However, to ensure identification, the rank condition must be satisfied. That
implies, at a minimum, that the explanatory variables must vary across time. This
leads to the so-called within or fixed-effects estimator. Since we are interested in the
case where the explanatory variables cannot be treated as strictly exogenous we
estimate (3.2) by pooled ZSLS, using time-demeaned instruments 2,7. This produces
the fixed effects-two stage least squares (FE-ZSLS) estimator. Therefore, we develop
tests for serial correlation and heteroskedasticity under the following conditions

ASSUMPTION A.l: For all s,t = 1,. . ., T,

E(uit lZi) = 0- (3-3)

Assumption A.1 is a strict exogeneity condition on the instrumental variables —
after eliminating the unobserved effect, c,- — and implies 15121-121“) = 0,! = 1,. . ., T.
Of course, to ensure identification, the rank and thus order condition must be
satisfied. The rank condition implies, that the demeaned instruments are sufficiently
linearly related to the demeaned endogenous variables. Necessary for the rank
condition is the order condition, that there are at least as many time-varying
instruments as there are time-varying explanatory variables. That is, L > K.
Specifically we require,

ASSUMPTION A.2: (i) rank 23,; 1:12:12”) = L and (ii) rank 2;, E(2},5e,-,) =

64

THEOREM 3.1: Under assumptions A.1 and A.2, the pooled 2SLS estimator
obtained from a random sample in the cross-section is consistent for 6.

Proof: The asymptotic first order properties are derived by rewriting the equation

for 0 as

. N o*'/..\* “I N N"
p = B+ N-l XX,- X,- N" ZX; u,- _ (3.4)
' i=1

i=1

Aid/>30!

Under A.2, N4 2:] X,- X; is non-singular (with probability approaching one) with

Alt/All!

plim (N‘1 2:] X,- Xi) = A”1 , where A a E(}\"}"')"(f). Further more, under

Assumption A.l, the

i=1

N M,
plim (Iv-12X,- 11,-) = E()"(;-*’u,-) = 0.

Therefore, by Slusky’s theorem (for example, Wooldridge (2002, Chapter 3),plim

fi=fi+A4-0=a
The asymptotic normality of IN ([3 — 6) results fi'om the asymptotic nomality of

N‘”2 2:1 Xf'ui, which follows from the central limit theorem under A.l, A2 and

finite second moment conditions. The asymptotic distribution of N'“ 2 2:] Xf'u; is

N
N-“2 227%,- 3 Normal(0,B), (3.5)
i=1
where
B a Ed’f’uiupt’?) (3.6)

65

But we also need to use the important result that having to estimate II in the first
stage does not affect the asymptotic distribution. Formally,
N I.>*’ N ..
N‘UZZXi u,- = N-l/ZZXy’ui+op(1). (3.7)
i=1 i=1

It follows that

7M} — p) 5’» Normal(0,A‘lBA" ). (3.8)
Importantly, the asymptotic variance A’l BA‘l places no restrictions on
Var(u,-|z,-,c,-), and so it is fully robust. It is the basis for robust inference using the
F E-ZSLS estimator (which is the specific FE-IV estimator we cover in this chapter).

The sample analog of B is

N T T
13 =N—1 22;; 33,53,353; 35:; (3.9)
i=1 t=l s=l
where 12;, = y,, — 523,3 are the FE-ZSLS residuals. The variance estimator in (3.9)
is fully robust to both serial correlation and heteroskedasticity. However, the
asymptotic variance can be simplified by ruling out heteroskedasticity and serial
correlation.

ASSUMPTION A.3. The idiosyncratic errors are homoskedastic in the sense

that

E(u,2,|z,-) = 02,: = 1,...,T, (3.10)

where z; = (2,1,...,z,-T).
Under A.l, E(u,~,|z,-) = 0, A.3 implies Var(u,~,|z,-) = 02, and so we view

Assumption A.3 as being an assumption about the conditional variances.

66

ASSUMPTION A.4: The errors are serially uncorrelated in the sense that
E(u,-,u,-S|z,-) = 0, all 3 ¢ t. (3.11)

Under A.l, (3.11) is equivalent to Cov(uiz.uis|zi) = 0
Under A.3 and A.4 we can simplify the expression for B in (3.6). First consider

the terms when t = 3. Using the law of iterated expectations,
E( 2-*' «an _ E 2»*' "It: , _ EE 2 _»*’ «at:
uitxi! xi!) ‘ [(uitxit xitlz,) — [ (uitlzl)xit xii]
2 “It, "II:
= 0' E(xi’ x”), t = 1,...,T.
Next fort at s,

E(uituisx¥ Mk) = E[E(uituisxit 16323)]

I
- E[E(uizuislzi)55ft 55;] = 0

Combining these results gives
B: 02 Z E(55;!;'x 35*) (3.12)

However, if either assumption A.3 or A.4 fail to hold the variance in (3.12) is
bias. If we find evidence of heteroskedasticity and/or serial correlation we can use the
fully robust variance estimator given in (3.9). Up until recently it was believed that in
the absence of serial correlation we could extend the heteroskedasticity robust (HR)
variance estimator that is consistent in cross-sectional models White (I 980).
However, Stock and Watson (2006) show that with fixed T, the conventional HR
variance estimator, for cross-sectional models, applied to the fixed effects estimator

is inconsistent with or without the degrees of freedom adjustment. This inconsistency

67

is a result of having increasingly many incidental parameters. This issue arises
because the entity means must be estimated, however, with fixed T we cannot
consistently estimate them. As we will see when developing a test for serial
correlation, if replacing the population variables with their estimates changes the
limiting distribution of the test statistic, we must account for the estimation error in
computing the test statistic. Stock and Watson model the asymptotic bias and propose
a bias-adjusted estimator. In addition, their derivation can be used to adjust the

variance matrix under no serial correlation.

4. Testing for Heteroskedasticity

In general, deriving a test for heteroskedasticity involves testing for correlation
between the squared idiosyncratic errors and the instruments after controlling for 02.
Therefore, let hi, 2 h “(2,7) denote a l x q vector of any time-varying, nonredundant
functions of z,-. Ideally, we would state the null hypothesis as
H0 : E[h;-,(u,3, — 02)] = 0, t = 1,. . ., T. However, because we are removing the
unobserved effect via time-demeaning (the within transformation), the null is

effectively
E[i&§,(a,2,-n2)] =0,t= 1,...,T, (4.1)
where
n2 = 02/0 —1/T). (4.2)
This value comes from the fact that, under A.l, A.3, and A.4,

Ema) = E[(uu - 32,92] = E(u;?-,) + 15029) - 2563.373)

68

= 02 + (02/7) — 2(02/7') = 020 — 1H). (4.2)
Equation (4.2) leads to the following estimator for 02
=[N(T- 1)] '22”. (4.3)
i=1 1:]
Using (4.3) the sample analog of (4.1) is,
N-1 Z 2 h §,(u',-,— 37 (4.4)
i=1 t=l
where a,, = y,, - inf} are the FE-ZSLS residuals.

To use (4.4) as a basis for a test, we must derive the limiting distribution of
N“”2 EN: 2 h,,(u,-, — n 2.) (4.5)
i=1 1:]
Typically the first-stage estimation causes the limiting distribution of (4.5) to be
different from the limiting distribution of
1w“2 2 2 h ;,(u-,.2,- (4.6)
i=1 t=l
However, since h i, is a function of the instruments, which are strictly exogenous by
A.l , we can show that the first-stage estimation does not effect the limiting
distribution of (4.5), provided we add an additional assumption. To see what that
assumption is, use a mean value expansion gives

N—l/2 Z Z hit(fi;21_

i=1 (:1

69

N T
___. N—l/Z Z Z h:t(fiizt _ fiZ)

i=1 1:]
N T
+ N-1 Z; :2; 2;,2a,,3ei,,/1—V(i3 — p) + 0,,(1), (4.7)
,= =
where JJV (0 - [3) = 0,;(1) by the central limit theorem. Therefore, we need to show
that N'1 211:1 2 1T: 1 iii-1222,3555, = 0p(l ). This assumption is generally not true under
A.3 and A.4. Therefore, we add

ASSUMPTION A.5: E(fi,~,5&,-,lz,~) = E(z‘i,~,5é;,),t = 1,...,T.

This assumption effectively restricts the conditional covariance between the
demeaned errors and regressors to be constant. In other words, it is a kind of system
homoskedasticity assumption. Wooldridge (1990) discusses similar assumptions in
the context of IV estimation using pure cross section or pure time series data. In the

panel data case, we might think of

xi! = di + Zitn + Vita
E(Vitlzi) = 0,

and then if we assume the covariance matrix of {(uib v,,) : t = 1,..., T} is constant
conditional on 2;, then Assumption A.5 holds. If we drop Assumption A.5, the test is
much harder to compute.

Under A.1, A.3, A.4, and A5, and letting A = E(z‘i;,5i,-,), we have

T T T
2150133235335) = zElh§,E(fiir5iizl'Z‘iz)l 5 2501210
(=1 (=1 t=l

T
= 5(2 I3},)A
(:1

=0

70

because 21:! h;, = 0. By the law of large numbers we have
07-12:] I; iii-(237,663, = op(l) and since op(l) x 0p(l) = 0,;(1) we can rewrite
(4.5) as
N T .. 2 N T
M“2 2 211502,, — 372) = 1w"2 2 2 111,02}, — 372) + 0,,(1). (4.9)
i=1 #1 i=1 t=l
We still have to show that the estimation of n2 can be ignored asymptotically.

But

1W“2 2 Z h;,(u;?,— (4. 10)

i=1 t=l

=N‘“ZZZiii-5<a%— nZ>+N"’ZZZh’-5<n -n>
i=1 t==l i=l t=l

= N-l/zzzhituiit—n
i=1 t=l

because 2:1,“ = 0. Therefore, the asymptotic distribution of (4.5) is equivalent
to (4.6). That is,

N-l/Zfiéh,,(a,,—fi2)=N—1/222h,(u,2,— n2)+op(1). (4.11)

i=1 t=l i=1 r=l
To obtain a test statistic, define
T T T
”I .. .. .. "I "
C a Var Z h,,(u,2, — n2) =22 mu}, — n2)(u;?—s — n2)h,.,h,~,]. (4.12)
’=1 (:1 s=1
It is important to note that the variance-covariance estimator must be robust to serial

correlation. This is true even though we are maintaining A.4, no serial correlation in

the idiosyncratic errors, for two reasons. First, the within transformation induces

7l

serial correlation, and second, the squared errors might actually contain serial
correlation, such as in dynamic models of heteroskedasticity.
Then by (4.11) and (4.12) we have,

N-“2 22h; ,(u,, — W) —+ Normal(0, C). (4.13)
i=1 t=l

Now, if C' 5’, C, obtaining a test statistic that has a chi-squared asymptotic
distribution is easy. The Lagrange Multiplier (LM) test is,

N T
(221719!“2))(NC)1<ZZTI;,(E%—fi2)). (4.14)

i-I t=l
Under the null hypothesis (4.14) has a limiting 7(2Q distribution, where Q is the

degrees of freedom. The degrees of freedom come from the number of restrictions
being tested, which in this case is the dimension of hi). Of course, the actual form of
the test statistic depends on the estimator used for C. As noted previously, it is
important to use a robust variance estimator even if we maintain A.4. Therefore, we
choice C‘ to be completely general:

N T T 2 2

‘ .. ’2 A ’2‘ A ”I "
C = N ' ZZZ (us. — lexuis — 712)h3¢his- (4.15)
i=l l=l s=l

Using the variance estimator in (4.15) the test statistic in (4.14) becomes

’ N T
=<zzh,(uu 32)) ZZGi—fiZXEi-fizflilfiis (4.16)

i=1 t-l i=1 (:1

N T “ A2
- (223w. - 32>)

i=1 t=l

This fully robust statistic in (4.16) cannot, in general, be computed via regressions.

72

However, there exists a test statistic that is asymptotically equivalent to (4.16) under
Ho, and local alternatives, that can be easily computed using simple regression. The

alternative statistic isjust the fully robust Wald statistic from the regression of
Efionniim i=l,...,N;t=1,...,T. (4.17)
In other words, let 2 be the Q x 1 vector of coefficients on 5,7. Then we just compute
the robust Wald test for significance of i, that is, using a variance matrix that is
robust to heteroskedasticity (to all for nonconstant conditional fourth moments in ii ,7)

and serial correlation (to allow serial correlation of arbitrary form in 11%,). Many

software packages, such as Stata, have such a feature for pooled OLS regressions.

The test statistic based on (4.17) replaces 3,27 — 1‘72 in obtain C with the residuals

from (4.17), 1",, s 3,2, - y? — 71,7110 get

. N T ’ N T .. ..
6r: (22:; Muir—n )) 22 rims/32,123.. (4.18)
H

=1i=11=1
N

22 h;,(u?.-—n )).
i=lt=l

Because i f» 0 under Ho, this alteration does not affect the limiting distribution of
the test statistic. In fact, 5‘ and 5 can be shown to be asymptotically equivalent in the
sense that 5‘ -3r 5 0 under local alternatives such that A N = 0(N‘I/2). The argument
is straightforward and similar to Wooldridge (l 990). In fact, any estimator of C that
is consistent when A = 0 will lead to a test asymptotically equivalent to 3r under
N‘“2 local alternatives.

We summarize the steps for the test of H0 : E(z'i;?',|z,-) = 112.

PROCEDURE 4.1. (i) Estimate the model by FE-IV and obtain the squared

73

F E-IV residuals, ’53,.

(ii) Choose Q misspecification indicators 11,7 = h ,-,(z,-), and take away time
averages to form 171,7.

(ii) Run a pooled OLS regression $301111“, t = l,...,T,i = 1,...,Nand use a
fully robust Wald statistic for joint significance of 11,7. Its asymptotic distribution is
16 Q-

F or simplicity, I assumed the indicators 11,-, did not depend on estimated
parameters. By extending the previous approach, it can be shown that the same
asymptotic analysis goes through under A.l to A.5. In particular, we can ignore the
first-stage estimation of any parameters appearing in 11 ,7, provided the estimators are

JN -consistent.

5. Testing the Unconditional Variance Matrix

The test in the previous section can be used for testing whether the conditional
variances and covariances do not depend on the instruments, 2,. Often in panel data
cases we are interested in testing whether the unconditional variance matrix of the
idiosyncratic errors takes on the standard, scalar form. That is, we want to test the
null hypothesis

H0 : E(u,-u:-) = 0217, (5.1)

against the alternative that some of the off-diagonal elements are nonzero or that the
diagonal elements change across-time. Here, u, is the T x 1 vector of idiosyncratic
errors. As we have seen, because we use the within transformation, we cannot test

hypotheses about E(u,-u;-) directly. Instead, we must use the time-demeaned errors.

74

Using the time demeaned errors as a test for serial correlation in the idiosycratic
error leads to complications. This is because under the null hypothesis of no serial
correlation in the idiosycratic errors, the time-demeaned errors will exhibit negative
correlation. However, if (5.1) holds, the negative correlation induced is known.

As we derived in the previous section,
502%,) = 02(1 — 1/7). (5.2)
Further,the covariance between 12 ,7 and a,, is
5(1711il'is) = E[(uiz - 170041.: - 171') = E(uizuis) - E(uizl71) - E(uisfii) + 5071171)
= o - 02/T— 02/T+ az/T = —02/T (5.3)
Thus, the negative correlation induced is
Corr(z'i,7fi,-s) = -1/(T— l) (5.4)
for all t :1: 3.
To derive a test of (5.1) define Q a E(ii,-z‘i:-). It is important to note that, while
E(u,-u:-) is generally has rank T, the time-demeaning causes 0 to be singular with
rank T — I. This does not cause any problems in what follows because we will use a
matrix to select out the elements we wish to test. In fact, let R be a Q x T2 matrix
such that the null hypothesis can be stated as
H, : R'vec(Q) = 0, (5.5)
where vec(-) is the “vectorization” operator. An important property of vec operator is
vec(ABC) = (C ' ® A)vec(B), for any matrices A,B , and C. and we will use this

property repeatedly below. Under the null hypothesis in (5.1), Q has diagonal

elements 02(1 — l/ T) and off diagonal elements —02/ T. An unrestricted version of Q

75

(except that it has rank T— I) has elements (93,, s,t = 1,..., T .. Under (5.1), we have
restrictions on the diagonal elements, a)“ = 0522 == can. If we want to test these
restrictions directly, we can choose R to pick off the appropriate elements.
Depending on what the restrictions are, we need to be careful to not choose
redundant restrictions. For example, if T = 2, we cannot test a) 11 = (022 because the
FE-IV residuals are identifical for the two time periods for each observation. In fact,
we can only test restrictions on Q for T 2 3. Even then, we must be aware of the
reduced rank of 0.

Consider the T = 3 case. Then, under H0,

2 2."L2 ”12

30 3° 3°

_ _L2 2.2 __L2
Q— 30 30 30
.l2 -.1_2 2.2

3“ 3° 3“

We can test the restrictions on 0 using a variety of linear combinations of the
unrestricted variances and covariances. For example, for testing a)“ = (022, it is

easily seen that we can choose R as
R’=(1ooo—10000).

which, of course, leads to a one degree-of-freedom test. We can bring in covariances

by, say, testing a)” = ——a)2] - (023 by adding a row to R':

R, 1000—10000
1—1—1000000

Then

76

Rivec(Q) = aJ11 -" a022
a111 -w21 -w22

and our goal is to test whether this linear combination is zero.

We base the test on a consistent estimator of Q when it is unrestricted.
N 1
Q = N-1 202,327) (5.6)
i=1

where i},- = )7,- - 5&0 are the T x l vectors of FE-ZSLS residuals. Then, the sample
analog of (5.5) is,
N
R'vec<N" ZG‘Tfiib) = R'vec(fl). (5.7)
i=1
In order to use (5.7) as a test of (5.5) we must derive the limiting distribution of
N
R'vec(N’l/2 213,35). (5.8)
i=1
In Section 4, we were able to obtain assumptions under which the estimation of B
by FE-ZSLS did not affect the limiting distribution of the test statistic under H0 (and
local alternatives). Further, in situations where x,, is strictly exogenous, the limiting
distribution of (5.8) would not depend on the limiting distribution of ,/I_V ([3 — 0).
See, for example, Wooldridge (2002, Chapter 7) and Inoue and Solon (2005).
Unfortunately, if some elements of x,, are endogenous, the limiting distribution of
(5.8), even under H0, depends on that of JN (B - ,8). It is my purpose to derive the
appropriate asymptotic variance that can be used to test the general hypothesis in
(5.5).

Write y', = )"(fl + ii, and 3,- = )7, -)"(,-B. Then

77

31= 151-503 = Xifi + iii -XiB,
and so
A A, .. A .. A
41173 = (iii - X103 - 3))(171- X103 - 13)),-
Straightforward algebra gives
A A, a .. .. a .. A A ..
£71171 = 14111: - 171(13 - W'X} - X1113 - [3117:- + Xiw - m3 - 31%» (59)
Using the expression in (5.9) we can write
N I N N A
N-l/2 2 a,,-5,- : M“2 2 31,31;- N*”2 2: 32,03 - (3)01”, (5.10)
i=1 i=1 i=l
N N h ..
- N-V2 Zia-(13 - ma:- + N-“2 253w — (ix/i - 1W,-
i=1 i=1

Applying the vec operator to the third term in (5.10), that is,

111“”2 23:, 5m? — axis — 131% gives,
N A A
N’mvec‘(2(j(i @501) ° veC((B-B)(fl-l3)') (5-11)
i=1
Rearranging terms,
N o A A
(N‘l 2W1 ® }(3)) ° WWCCW - BXH - 13),)
i=l

where N-1 20?, <53 2%,) is 0,,(1) and mu} — fi)(,0 - 13)’ is a,,(1) since N(fi — p)
is 079(1) by the Central Limit Theorem and (3 - [3) is OP( 1 ). Therefore, we can

ignore the term in (5.] l) in deriving the limiting distribution of (5.8) since 0p( l) .

0,,(1) =op(l).

After applying the vec operator to equation (5.10) and noting that equation (5.1 l)

78

is op(l) gives

N N
vec(N'l/2 2 3,31) = vec(N‘l/2 2 52,321.) (5.12)
i=1 i=1

(Iv-12y eu,)>1_ (3 m- (Iv-1 2(u,®X)>J— (13 m+op<1>

Using (5.12) and multiplying through by R' gives the representation

N N
R'vec(N"I/2 2 3,31) = R'vec(N‘1/2 2 32,321.) (5.13)
5 .

—R(N-‘Z(X 5331,0175 13)
—R(N“Z(u,®X)>J—(fi 131+op<1>

To use (5.13) as the basis for a test we replace [N (B - fl) with its first-order

1 1
representation [E(X19‘ X1“)]‘1N‘1/2 21:] X’“ u,- (which holds up to op(l))

i
N

N
R'vec N‘”2 2 15,3; = R'vec N‘”2 Z u',-z'i1- (5.14)
i=1 i=1

N 1
— R’[N“ 2 (X518 a,)]1E(X;-* X*>1"

i=l

N 1
'N_]/2 2X?! ”1

i=l

N ,
— R’[N-' Z (a, ®X,)][E(X}' 1??)1"

N 1
. N‘“2 2X1“ u,- + 0p(l).

i=l

To simplify (5.]4), define the following: A] = E[X,- 81 iii], A2 = E[ii,- ® X,], and

79

I
C = E(X1-“ X1“). Substituting gives

N N
R’vec N-'/2 2 3,31- : R’vec NW2 2 31,311- (5.15)
i i
N 1
-R’(A1+A2)C_IN_“/2ZX1-“ u,-+0p(l).

i=1

Furthermore, letting v,- = R'vec(z‘i,-fi1-) and s,- = R'(Al +A2)C‘1X13" a,, (5.l5) can be

written as

N N
M“2 2 i3,- = N“2 20,- — s,-) + op(1). (5.16)

i=l l=1

Because E(v,-) = 0 and E(s,-) = 0 when E(u,-|z,-) = 0 and (5.1) holds, we have

N
N-l/2 Z 5, 5’» Normal(0, D), (5.17)

i=1
where

D a Var(v, — s,-). (5.18)

Given a consistent estimator D f, D (as N -5 00), we can define the test statistic

N ’ N
8 = (N40 25,-) 15-1 (N‘1/229,), (5.19)
i=1 i=1

which converges in distribution to xZQ under H0 (assuming that D has full rank Q).

Of course, the actual form of the test statistic depends on the estimator used for D.
Since we do not want to rely on additional assumptions, we use a completely general

estimator for D:

80

N
1‘) = N" 2(3, — 5,)(1‘2,— 3,)’. (5.20)
i=1

Using (5.20) the test statistic in (5.19) becomes

N ’ N ‘1 N
3 = (291') Z (91—§i)(9i “57), (291'), (5-21)
i=1 i=1

i=1
whose computation is not too onerous. Still, it cannot be computed simply via
regressions, and so it should be programmed in standard statistical packages, such as

Stata, that allow fixed effects instrumental variables methods.

6. Application

In chapter one, 1 extended the analysis of Papke (2007) by testing for endogenous
spending in the pass rate equation after controlling for the unobserved district effect.
Using the level of the foundation grant as an instrument of spending I found mixed
evidence about the exogeneity of spending. In chapter two, I examined the
spending-performance model further by applying the tests for an unobserved district
effect and for possible trending of the district effect. The results indicated correlated
district effects. In this chapter, I apply the test for heteroskedasticity. It is important
to note that the results of Stock and Watson (2006) do not affect the test for
heteroskedasticity. However, if we find evidence of heteroskedasticity, the Stock and
Watson result shows that even if the errors are uncorrelated across-time the standard
heteroskedasticity robust variance estimator is bias. While this is an important
finding, it is very unlikely that serial correlation would not be a problem and thus

justifying a variance estimator that is only robust to heteroskedasticity. Of course,

8]

nothing rules out heteroskedasticity, serial correlation is generally more of a problem
with fixed T.

The spending-performance model analyzed by Papke (2007) models the pass rate
on the fourth grade MEAP math test as a function of enrollment, poverty rates, and
average real spending per student. Noting the likely endogeneity of spending Papke
uses the level of the foundation grant as an instrument for spending. In addition, she
uses the percentage of students eligible for the federal free lunch program as a proxy
for poverty rates. Therefore, the test for heteroskedasticity developed in Section 4
involves regressing the squared fixed effects residuals on functions of enrollment,
lunch, and the foundation grant. Papke uses the log of enrollment, spending, and
foundation grant to allow for diminishing effects. Therefore, we use the levels,
squares and cross-products of lunch, log(enrollment), and log(foundation grant) to
test for heteroskedasticity. Testing the joint significance of the nine variables, after
time-demeaning, gives a fully robust Wald statistic of 27.27 which gives a p-value of
about .0016. Therefore, the conditional variance matrix of the idiosyncratic errors
does not appear to have the standard form 0217. At a minimum, inference needs to
be made robust. Plus, this suggests that more efficient estimation, particularly,

generalized method of moments, can improve efficiency over FE-ZSLS.

7. Conclusion

The chapter has developed a test for heteroskedasticity and general test regarding
the unconditional variance of the FE-IV estimator. The test for heteroskedasticity can

be computed by regressing the squared residuals, from FE-IV estimation, on various

82

functions of the time demeaned instruments. A key benefit of the regression based
test is the ease in computing a test statistic that is fully robust. This is critical since
the test uses residuals for FE-IV, as the within transformation leads to serially
correlated time demeaned errors if the idiosycratic errors are uncorrelated. I apply the
test to the spending-performance model analyzed by Papke (2007). The study uses
data on the pass rates for the fourth-grade MEAP math test from 1992 though 2004 at
Michigan public school districts. I test for heteroskedasticity using the levels,
squares, and cross-products of the variables used to estimate the FE-IV equation. The
results suggest that the conditional variance matrix of the idiosyncratic errors does
not have the standard form 0217.

While the test for serial correlation cannot be computed via regressions it is
actually a general test regarding the unconditional variance. In addition, the
asymptotic distribution can be used to adjust the variance of the Wooldridge test that
is currently programed in Stata 8.0 and higher. That is, the asymptotic distribution
derived in this chapter can be used to extend the Wooldridge test to panel data
models estimated by instrumental variables methods.

I have not analyzed the finite sample properties of either test. However, Drukker
(2003) provides simulation evidence that the Wooldridge test, for autoregressive or
moving average errors in panel data models, performs well in reasonable sized
samples. Therefore, we would expect that if the Wooldridge test is adjusted using the
asymptotic distribution derived in this chapter it might preform equally well. The
tests derived in this chapter are readily extended to panel data models with fixed

cross-sectional size and growing time dimension. Such a change in asymptotics

83

would necessitate assumptions regarding the stationarity or time dependency of the

variables.

84

Appendix: Stata Commands for Implementing the Tests in Chapter 1

Testing for Endogeneity:

Partition the model as

yit = 1111131 +xiz2fi2 + 01+ ”it, t=1,2,-..,T,
where x,72 is strictly exogenous and we are interested in testing if x,,] is
endogenous.
1) Obtain 12,7 as the residuals from the fixed effects regression.
a) areg y,, x,,, x,72, absorb (fixed effect identifier)
b)predict uhat,resid

The last command saves the residuals from the fixed effects regression as uhat.
Note that the predict command must directly follow the regression command.

In the next step if there is more than one endogenous variable step two must be
carried out for each endogenous variable. In addition, there must be at least as many
instruments as possible endogenous variables.

2) Since under the null hypothesis x ,7, equals v,-, and by the first order conditions
of FE N‘1 21:] Z; x,,, 12,7 = 0, we need to estimate v,, as the residuals from the
following FE regression,

a) areg x,,, x,,2 2,7, absorb (fixed effect identifier)
b)predict vhat,resid
3) Obtain 1",, as the residuals from partially out all explanatory variables from

vhat. Note you do not want to include the instruments in this regression.

85

a) areg vhat x,,l x,,z , absorb (fixed effects
identifier)

b)predict rhat,resid

4) The fully robust test statistic is the robust t or F test of rhat from the

following regression.

a)reg uhat rhat, robust cluster(fixed effects
identifier)

b) test rhat

The non-robust version of the test is simply N(T— l)R2. Where R2 is from the

regression in step 4.

86

Testing Overidentifying Restrictions

When testing for over-identification we do not need to estimate the
misspecification indicators as they are simply a subset of the instruments.

1) Obtain 12,, as the residuals from the FE-ZSLS regression using the full set of

instrumental variables.

a)xtivreg dependent variable strictly exogenous
explanatory variables (endogenous variables = list of
instruments),fe

b)predict uhat,resid

2) In computing the linear projection in equation (4.6) we can omit v,, since the

misspecification indicators are a strict subset of the instruments. However, there will
be a regression for each endogenous variable. That is, perform parts a and b for all

endogenous variables then form xhat as (xhatl ,...,xhatk1) where k1 is the number of

endogenous variables. Thus, it must be that k1+ q = L where q is the number of
over-identifying restrictions and L is the total number of instruments.
a)areg endogenous explanatory variable
instrumental variables, absorb(fixed effects identifier)
b)predict xhat,xb
3) For each over-identifying restriction, 53,-” ,...,f-,-,q is obtained as the residuals
from the q FE regressions

a) areg over-identifying restriction 1 all

87

explanatory variables,absorb(fixed effect identifier)
b)predict rhat1,resid
4) The test statistic is obtain as the fully robust Wald or F-test on f,,1,..., $7“,
from the regression;
a)reg uhat rhatl,...,rhatq, robust cluster(fixed
effects identifier)
b)test rhatl, ...,rhatq
The non-robust version of the test is simply N(T - l)R2. Where R2 is from the

regression in step 4.

88

Testing for Non-linearities

First apply the within transformation to remove the unobserved effect then

partition the equation into endogenous and exogenous explanatory variables:

in = 55121131 +55i12fi2 + 17:1, t = 1,...,T.

1) Under the null hypothesis the estimation procedure is FE-IV. Therefore,

121., can be obtain from the following regression,
a) xtivreg dependent variable strictly exogenous
explanatory variables (endogenous variables = list of

instruments),fe
b)predict uhat,resid
2) Next, we need to estimated the reduced form of the endogenous variables.
That is, for each endogenous variable obtain the fitted values from the following
regression using demeaned variables,
a)reg demeaned endogenous variable demeaned
exogenous variables demeaned outside instruments
b)predict endogenous variable hat
3) Since the misspecification indicators depend on endogenous variables we
replace the endogenous variables with their reduced form estimations obtained in
step 2.. The simplest way to obtain the misspecification indicators is by first
obtaining the fitted values from the following regression using demeaned variables,

a)reg demeaned dependent variable demeaned

89

strictly exogenous explanatory variables fitted values
of endogenous variables.
b)predict yhat
4) Using yhat we can obtain a,,, = [aim/2, + 52,7232? and
9,,2 = [(1,181 + 52,9132? by squaring and cubing yhat. That is,
a) genfim=yhat2
b) gen ha=yhat3
5) Obtain Fm and fig as the residuals from partially out all demeaned
explanatory variables from the misspecification indicators. Note you do not want to
include the instruments in these regressions.
a)reg a”, demeaned xgl demeaned X”2
b)predict rhatl,resid
c)reg 9&2 demeaned xy, demeaned Xflz
d)predict rhat2,resid
6) The fully robust test statistic is the robust t or F test of rhatl and rhat2 from
the following regression.
a)reg uhat rhatl rhat2, robust cluster(fixed
effects identifier)

b)test rhatl rhat2

90

References

Ahn S.C. and S. Low (1996). "A Reforrnulation of the Hausman Test for Regression
Models with Pooled Cross-Section-Time-Series Data," Journal of Econometrics 71,
309-319.

Arellano, M. (1990). "Testing for Autocorrelation in Dynamic Random Effects
Models," Review of Economic Studies 57, 127-134.

Arellano, M. (1993). "On the Testing of Correlated Effects with Panel Data," Journal
of Econometrics 59, 87-97.

Bhargava, A., L. Franzini, and W. Narendranathan, (1982). "Serial Correlation and the
Fixed Effects Model," Review of Economic Studies 49, 533-549.

Breusch, TS. (197 8). "Some Aspects of Statistical Inference for Econometrics,"
Unpublished Ph.D. thesis, Australian National University.

Breusch, T. S., and AR. Pagan (1979). "A Simple Test for Heteroskedasticity and
Random Coefficient Variation," Econometrica 50, 987-1007.

Berenblut, I. I., and G. I. Webb. (1973). "A New Test for Autocorrelated Errors in the
linear regression model," Journal of Royal Statistical Society, Series B, 1, 33-50.

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

Comwell,C., P. Schmidt, and R., Sickles. (1990). "Production Frontiers with
Cross-Sectional and Time-Series Variation in Efficiency Levels," Journal of
Econometrics 46, 185-200.

Davidson, R., and J .G. MacKinnon (1993). Estimation and Inference in Econometrics:
New York: Oxford University Press.

Drukker, D. (1998). "Testing for Serial Correlation in Linear Panel-Data Models,"
Stata Journal 3, 168-177.

Durbin, 1., and G. Watson (1951). "Testing for Serial Correlation in Least Squares
Regression-ll," Biometrika 38, 159-178.

Durbin, J. (1970). "Testing for Serial Correlation in Least Squares Regression When
Some of the Regressors Are Lagged Dependent Variables," Econometrica 38, 410-421.

Friedberg, L. (1998). "Did Unilateral Divorce Rates Rise? Evidence from Panel
Data," American Economic Review 88, 608-627.

Godfrey, L. G. (1994). "Testing for Serial Correlation by Variable Addition in
Dynamic Models Estimated by Instrumental Variables," The Review of Economics and

91

Statistics 76, 550-559.

Godfrey, LG, (1976). "Testing for Serial Correlation in Dynamic Simultaneous
Equation Models," Econometrica 44, 1077-1084.

Hausman, J. (1978). "Specification Tests in Econometrics," Econometrica 46,
1251-1271 .

Hausman, J ., and W. Taylor (1981). "Panel Data and Unobservable Individual
Effects," Econometrica 49, 1377-1398.

Heckman, J. J, and V. J. Hotz (1989). "Choosing among Alternative Nonexperimental
Methods for Estimating the Impact of Social Programs: The Case of Manpower Training,"
Journal of the American Statistical Association 84, 862-875.

1m, K. S., S. C. Ahn, P. Schmidt, and J. M. Wooldridge (1999). "Efficient Estimation
of Panel Data Models with Strictly Exogenous Explanatory Variables," Journal of
Econometrics 93, 177-201

Inoue, A., and G. Solon (2006). "A Portmanteau Test for serial Correlated Errors in
Fixed Effects Models," Econometric Theory 22, 835-851.

Ljung, GM. and G.E.P Box (1978). "On a Measure of Lack of Fit in Time Series
Models," Biometrika 65, 297-303.

Metcalf, G., (1996). "Specification Testing in Panel Data with Instrumental
Vatiables," Journal of Econometrics 71, 291-307.

Mundlak, (1978). "On the Pooling of Time Series and Cross Section Data,"
Econometrica 46, 69-85.

Okui, R. (2007). "Testing Serial Correlation in Fixed Effects Regression Models: The
Ljung-Box Test for Panel Data," Unpublished manuscript.

Papke, LE, (2004). "The Effects of Spending on Test Pass Rates: Evidence from
Michigan," Journal of Public Economics 89, 821-839.

Papke, LE, (2007). "The Effects of Changes in Michigan’s School Finance System,"
working paper.

Papke, L. E. and J. M. Wooldridge (2007). "Panel Data Methods for Fractional
Response Variables with an Application to Test Pass Rates," working paper.

Ramsey, J. B. (1969). "Tests for Specification Errors in Classical Linear Least
Squares Regression Analysis," Journal of the Royal Statistical Society, Series B, 31,
350-371 .

Ruud, P. (1984). "Tests of Specification in Econometrics," Econometric Reviews 3,
21.1-242.

92

Semykina and J .M. Wooldridge (2005) "Estimating Panel Data Models in the
Presence of Endogeneity and Selection: Theory and Application," unpublished working

paper.

Stock, J .H. and M.W. Watson (2006). "Heteroskedasticity-Robust Standard Errors for
Fixed Effects Panel Data regression," National Bureau of Economic Research Technical
Working Paper Number 323.

White, H. (1980). "A Heteroskedasticity-Consistent Covariance Matrix Estimator and
a Direct Test for heteroskedasticity," Econometrica 48, 817-838.

Wooldridge, J .M. (1990). "A Unified Approach to Robust, Regression-Based
Specification Tests," Econometric Theory 6, 17-43.

Wooldridge, J .M. (l995a). "Selection Correction for Panel Data Models under
Conditional Mean Independence Assumptions," Journal of Econometrics 68, 1 15-132.

Wooldridge, J .M. (1995b). "Score Diagnostics for Linear Models Estimated by Two
Stage Least Squares," in Advances in Econometrics and Quantitative Economics, ed. G.S.
Maddala, P.C.B. Phillips, and TN. Srinivasan. Oxford: Blackwell, 66-87.

Wooldridge, J. M. (2002). Econometric Analysis of Cross Section and Panel Data:
The MIT Press: Cambridge, Massachusetts.

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