A test of cross section dependence for a linear dynamic panel model

A test of cross section dependence for a linear dynamic panel model
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A test of cross section dependence for a linear
dynamic panel model with regressors
Sarafidis, Vasilis; Yamagata, Takashi; Robertson, Donald
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Sarafidis, Vasilis ; Yamagata, Takashi ; Robertson, Donald: A test of cross section dependence for a linear dynamic
panel model with regressors. In: Journal of Econometrics 148 (2009), 2, pp. 149-161. DOI: http://dx.doi.org/10.1016/
j.jeconom.2008.10.006
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Accepted Manuscript
A test of cross section dependence for a linear dynamic panel model
with regressors
Vasilis Sarafidis, Takashi Yamagata, Donald Robertson
PII:
DOI:
Reference:
S0304-4076(08)00182-6
10.1016/j.jeconom.2008.10.006
ECONOM 3119
To appear in:
Journal of Econometrics
Received date: 8 May 2006
Revised date: 23 September 2008
Accepted date: 10 October 2008
Please cite this article as: Sarafidis, V., Yamagata, T., Robertson, D., A test of cross section
dependence for a linear dynamic panel model with regressors. Journal of Econometrics
(2008), doi:10.1016/j.jeconom.2008.10.006
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Vasilis Sara…dis}
Takashi Yamagata4y
Donald Robertson|
Discipline of Econometrics and Business Statistics, University of Sydney
4
Department of Economics and Related Studies, University of York
|
Faculty of Economics, University of Cambridge
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}
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A Test of Cross Section Dependence for a Linear Dynamic
Panel Model with Regressors
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23 September 2008
Abstract
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This paper proposes a new testing procedure for detecting error cross section
dependence after estimating a linear dynamic panel data model with regressors
using the generalised method of moments (GMM). The test is valid when the crosssectional dimension of the panel is large relative to the time series dimension. Importantly, our approach allows one to examine whether any error cross section dependence remains after including time dummies (or after transforming the data in
terms of deviations from time-speci…c averages), which will be the case under heterogeneous error cross section dependence. Finite sample simulation-based results
suggest that our tests perform well, particularly the version based on the Blundell
and Bond (1998) system GMM estimator. In addition, it is shown that the system
GMM estimator, based only on partial instruments consisting of the regressors, can
be a reliable alternative to the standard GMM estimators under heterogeneous error
cross section dependence. The proposed tests are applied to employment equations
using UK …rm data and the results show little evidence of heterogeneous error cross
section dependence.
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Key Words: cross section dependence, generalised method of moments, dynamic panel
data, overidentifying restrictions test.
JEL Classi…cation: C12; C13; C15; C33.
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We would like to thank Tom Flavin, Hashem Pesaran and Neville Weber for helpful discussions. We
have also bene…ted from the constructive and most helpful comments of Cheng Hsiao and two anonymous
referees. Sara…dis gratefully acknowledges full …nancial support from the ESRC during his PhD studies
at Cambridge University (PTA-030-2002-00328). Yamagata gratefully acknowledges the …nancial support
from the ESRC (Grant No. RES-000-23-0135).
y
Corresponding author. E-mail: ty509@york.ac.uk
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Introduction
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During the past decade a substantial literature has been developed analysing the e¤ects
of cross section dependence as well as advancing ways of dealing with it in panel data
models. Cross section dependence may arise for several reasons often, due to spatial
correlations, economic distance and common unobserved shocks. In the case of spatial
attributes, where a natural immutable distance measure is available, the dependence
may be captured through spatial lags using techniques that are familiar from the time
series literature (Anselin, 1988, 2001). In economic applications, spatial techniques
are often adapted using alternative measures of economic distance (see e.g. Conley,
1999, Kapoor, Kelejian and Prucha, 2004, Lee, 2004, Lee, 2007, and others). There
are several contributions in the literature that allow for time-varying individual e¤ects
(Holtz-Eakin, Newey and Rosen, 1988, Ahn, Lee and Schmidt, 2001 and Han, Orea
and Schmidt, 2005). Recently, a number of researchers have modelled cross section
dependence by restricting the covariance matrix of the errors using a common factor
speci…cation with a …xed number of unobserved factors and individual-speci…c factor
loadings that give rise to heterogenous cross section dependence (see Forni and Reichlin,
1998, Robertson and Symons, 2000, Phillips and Sul, 2003, Stock and Watson, 2002,
Bai and Ng, 2004, Moon and Perron, 2004, Pesaran, 2006, among others). The factor
structure approach is widely used because it can approximate a wide variety of error
cross section dependence. For example, in a panel data set of …rms we may think of the
factors as capturing ‡uctuations in economic activity or changes in regulatory policy for
the industry as a whole, and so on. The impact of these factors will vary across …rms,
due to di¤erences in size, liquidity constraints, market share etc. In a macro panel
data model, the factors may represent a general demand shock or an oil price shock
with the factor loadings re‡ecting the relative openness of the economies, di¤erences in
technological constraints, and so on.1
In the literature of estimating linear dynamic panel data models with a large number
of cross-sectional units (N ) and a moderately small number of time series observations
(T ), generalised method of moments (GMM) estimators are widely used, such as those
proposed by Arellano and Bond (1991), Ahn and Schmidt (1995), Arellano and Bover
(1995) and Blundell and Bond (1998). These methods typically assume that the disturbances are cross-sectionally independent. On the other hand, in empirical applications
it is common practice to include time dummies, or, equivalently, to transform the observations in terms of deviations from time-speci…c averages (i.e. to cross-sectionally
demean the data) in order to eliminate any common time-varying shocks; see, for example, Arellano and Bond (1991) and Blundell and Bond (1998). This transformation
will marginal out these common e¤ects, unless their impact di¤ers across cross-sectional
units (heterogeneous cross section dependence). In this case, the standard GMM estimators used in the literature will not be consistent, as shown by Sara…dis and Robertson
(2007) and in the current paper.
Several tests for cross section dependence have been proposed in the econometric
1
Other examples are provided by Ahn, Lee and Schmidt (2001).
2
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literature. The most widely used test is perhaps the Lagrange Multiplier (LM) test proposed by Breusch and Pagan (1980), which is based on the squared pair-wise Pearson’s
correlation coe¢ cients of the residuals. This test requires T being much larger than
N . Frees (1995) proposed a cross section dependence test that is based on the squared
Spearman rank correlation coe¢ cients and allows N to be larger than T . Recently, Ng
(2006) has developed tools for guiding practitioners as to how much residual cross section correlation is in the data and which cross-sectional units are responsible for this in
particular, through tests that are based on probability integral transformations of the
ordered residual
correlations. However, the proposed procedures are valid only in panels
p
for which T -consistent estimates are available. Pesaran, Ullah and Yamagata (2006)
developed bias-adjusted normal approximation versions of the LM test of Breusch and
Pagan (1980), which are valid for large-N panel data models but with strictly exogenous
regressors only. Pesaran (2004) proposed another test for cross section dependence, called
the CD test, which is closely related to Friedman’s (1937) test statistic. Pesaran showed
that the CD test can also be applied to a wide variety of models, including heterogeneous dynamic models with multiple breaks and non-stationary dynamic models with
small/large N and T . However, as Frees (1995) implied and Pesaran (2004) pointed out,
the problem of the CD test is that in a stationary dynamic panel data model it will fail
to reject the null of error cross section independence when the factor loadings have zero
mean in the cross-sectional dimension. It follows that the CD test will have poor power
properties when it is applied to a regression with time dummies or on cross-sectionally
demeaned data.
This paper proposes a new testing procedure for error cross section dependence after
estimating a linear dynamic panel data model with covariates by the generalised method
of moments. This is valid when N is large relative to T . Importantly, unlike the CD test,
our approach allows one to examine whether any error cross section dependence remains
after including time dummies, or after transforming the data in terms of deviations from
time-speci…c averages, which will be the case under heterogeneous error cross section
dependence.
The small sample performance of our proposed tests is investigated by means of
Monte Carlo experiments and we show that they have correct size and satisfactory power
for a wide variety of simulation designs. Furthermore, the paper suggests a consistent
GMM estimator under heterogeneous error cross section dependence. Results on the
…nite sample properties of the estimator are reported and discussed.
Our proposed tests and estimators are applied to employment equations using UK
…rm data, and it is found that there is little evidence of heterogeneous cross section
dependence in this data set.
The remainder of the paper proceeds as follows. Section 2 reviews some relevant
existing tests for error cross section dependence. Section 3 proposes a new test for cross
section dependence and a consistent GMM estimator under these circumstances. Section
4 reports the results of our Monte Carlo experiments. Section 5 illustrates an empirical
application of our approach. Finally, Section 6 contains concluding remarks.
3
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Existing Tests for Cross Section Dependence
Consider a panel data model
i
+
0
xit + uit , i = 1; 2; : : : ; N , t = 1; 2; : : : ; T ,
CR
yit =
(1)
where the uit may exhibit cross section dependence. The hypothesis of interest is
H0 : E(uit ujt ) = 0 8 t for all i 6= j;
(2)
H1 : E(uit ujt ) 6= 0 for some t and some i 6= j;
(3)
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vs
where the number of possible pairings (uit ; ujt ) rises with N . In the literature several
tests for error cross section dependence have been proposed, and some relevant ones are
discussed in this section.
Breusch-Pagan (1980) Lagrange Multiplier Test
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2.1
Breusch and Pagan (1980) proposed a Lagrange multiplier (LM) statistic for testing the
null of zero cross-equation error correlations, which is de…ned as
LM = T
N
X1
N
X
^2ij ;
(4)
i=1 j=i+1
where ^ij is the sample estimate of the pair-wise Pearson correlation coe¢ cient of the
residuals
PT
t=1 eit ejt
^ij = ^ji =
;
(5)
PT 2 1=2 PT 2 1=2
e
e
t=1 it
t=1 jt
2.2
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where eit is the Ordinary Least Squares (OLS) estimate of uit in (1). LM is asymptotically distributed as chi-squared with N (N 1)=2 degrees of freedom under the null
hypothesis, as T ! 1 with N …xed.
Pesaran’s (2004) CD Test
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Recently Pesaran (2004) proposed another test for cross section dependence, called CD
test, which allows for a ‡exible model structure, including fairly general heterogeneous
dynamic models and nonstationary models. The test statistic is de…ned as
1
0
s
N
N
X1 X
2T
@
^ij A .
(6)
CD =
N (N 1)
i=1 j=i+1
4
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uit =
i ft
+ "it ,
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For su¢ ciently large N and T , the CD test statistic tends to a standard normal variate
under the null of cross section independence.2 The …nite sample evidence in Pesaran
(2004) shows that the estimated size of the test is very close to the nominal level for any
combinations of N and T considered. As Pesaran (2004) notes, there are two important
cases in which the CD test can be unreliable. Firstly, when the distribution of the
errors is not symmetric, the CD test becomes invalid and it may not have correct size.3
Secondly, the CD test may lack power towards some directions of alternatives. To see
this, consider the following single-factor structure for the error process
3
3.1
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where i is a factor loading that is …xed and bounded, ft is an unobserved common
factor such that ft
i:i:d:(0; 1), "it
i:i:d:(0; 2 ) and E(ft "it ) = 0 for all i and t.
The common factor ft generates error cross section dependence because of the fact that
cov(uit ; ujt ) = i j , and the power of the CD test hinges on this non-zero covariance.
Now suppose that i i:i:d:(0; 2 ) and i is uncorrelated with ft and "it . In this case,
cov(uit ; ujt ) = E( i )E( j ) = 0, even if there does exist (potentially large) error cross
section dependence.
In the next section, we elaborate on the stochastic properties of the factors and factor
loadings, and develop a new cross section dependence test.
Sargan’s Di¤erence Tests for Heterogeneous Error Cross
Section Dependence in a Linear Dynamic Model with
Regressors
Model Speci…cation
Consider the following model
i
+ yi;t
1
+
0
xit + uit , i = 1; 2; :::; N , t = 1; 2; :::; T ,
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yit =
(7)
where j j < 1, is a (K 1) parameter vector that is bounded and non-zero, xit is a
(K 1) vector of regressors with xit = (x1it ; x2it ; :::; xKit )0 , i is a random e¤ect with
…nite mean and …nite variance, and uit has a multi-factor structure such that
0
i ft
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uit =
+ "it ;
(8)
where i = ( 1i ; 2i ; :::; M i )0 is a (M 1) vector of factor loadings that is assumed to
be i:i:d:( ;
) with
being a positive semi-de…nite matrix, ft = (f1t ; f2t ; :::; fM t )0 is a
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2
As Frees (1995) pointed out, test statistics similar to the CD test of Pesaran were proposed by
Friedman (1937), based on the Spearman rank correlation coe¢ cient (which is expected to be robust
against non-normality). Although we do not consider the Friedman test in this paper, results that are
similar to the CD test would apply for this test.
3
However, the experimental results of Pesaran (2004) illustrate that the CD test is robust to skewed
errors.
5
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i
+
xi;t
1
+
xi ft
+ "i;t
1
+ vit , i = 1; 2; :::; N , t = 1; 2; :::; T ,
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xit =
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(M 1) vector of time-varying common factors that are assumed to be non-stochastic and
bounded, and "it is an independently distributed random variable over i with zero-mean
and …nite variance 2i .
The error (multi-) factor structure has been employed extensively in the economic
literature.4 For example, in a macro panel, a common factor could be an unobserved
technological shock, and the factor loadings can be thought of as capturing a crosssectionally heterogeneous response to such shock. Note that since our asymptotic is
N ! 1 with T …xed, ft is treated as non-stochastic here. In this paper we explicitly
employ a random coe¢ cient assumption for the factor loadings.
The process for xit is de…ned as
(9)
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where i = ( i1 ; i2 ; :::; iK )0 is a (K 1) vector of random e¤ects with …nite mean
and …nite variance,
signi…es the Hadamard product, = ( 1 ; 2 ; :::; K )0 such that
j k j < 1 for k = 1; 2; :::; K, xi = ( 1i ; 2i ; :::; Ki )0 with ki = ( ki1 ; ki2 ; :::; kiM )0 ,
such that xi i:i:d:( x ; x ), = ( 1 ; 2 ; :::; K )0 , and vit is a vector of independently
distributed random variables over i with mean vector zero and a …nite variance matrix
2
vi = diag( vki ), k = 1; 2; :::; K. The model de…ned by (7), (8), and (9) is general
enough to allow for a large variety of plausible speci…cations that are widely used in the
economic literature. Furthermore, this model accommodates more simple processes for
xit ; such as those where xit is strictly exogenous, or exogenous with respect to ft :
The null hypothesis of interest is then
H0 : var(
against the alternative
i)
H1 :
as opposed to (2) and (3).
=
6= 0,
=0
(10)
(11)
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Remark 1 Observe that error cross section dependence may occur under the null hypothesis when i = for all i, since E(uit ujt ) = 0 ft ft0 , which is not zero unless = 0.
However, such error cross section dependence can be eliminated simply by including time
dummies, or equivalently by cross-sectionally demeaning the data. This implies that the
null hypothesis in (10) can be interpreted as saying that the cross section dependence
is homogeneous across pairs of cross-sectional units, against the alternative hypothesis
(11) of heterogeneous error cross section dependence.
We make the following assumptions:
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4
See e.g. Robertson and Symons (2000), Phillips and Sul (2003), Bai and Ng (2004), Moon and
Perron (2004), and Pesaran (2006) among others. Notice that these methods are only justi…ed when T
is large. For related work that allows for time-varying individual e¤ects in the …xed T , large N case,
see Holtz-Eakin, Newey and Rosen (1988), Ahn, Lee, and Schmidt (2001) and Han, Orea and Schmidt
(2005).
6
Assumption 1: E( i "it ) = 0, E(
E (vit "is ) = 0 for all t and s.
i "it )
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= 0, and E( i vit ) = 0, for all i, t. Also
0 ) = 0 for all i and t 6= s.
Assumption 2: E("it "is ) = 0 and E(vit vis
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Assumption 3: E(yi0 "it ) = 0 and E(xi1 "it ) = 0 for all i and t = 1; 2; :::; T .
Assumption 4: Time-varying common factors, fm;t , m = 1; 2; :::; M ; t = 1; 2; :::; T are
non-stochastic and bounded.
Assumption 5:
(b) cov(
xi ;
i)
= 0 and cov(
Assumption 6:
0
i ; xi1 )
i)
= 0, cov( i ;
xi )
= 0,
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(a) E("it i ) = 0, E(vit 0i ) = 0, E("it xi ) = 0, cov( i ;
cov( i ; 0i ) = 0, cov( 0xi ; i ) = 0, for all i and t.
= 0.
6= 0.
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Assumptions 1-3 are standard in the GMM literature; see, for example, Ahn and
Schmidt (1995). Assumption 4 ensures that the initial observations of y and x are
bounded.
Assumption 5(a), a random coe¢ cient type assumption on the factor loadings, allows
cross-sectionally heterogeneous inter-dependence among both level and …rst-di¤erenced
variables (yit ; x1it ; x2it ; :::; xKit ), as well as cross section dependence of each variable
through the common factor ft . This contrasts to a simple time e¤ects assumption,
namely i = and xi = x for all i, which is stronger than ours. On the other hand,
Assumption 5(a) is stronger than assuming i and xi to be merely bounded, in that the
cross-sectionally demeaned variables of (yit ; x1it ; x2it ; :::; xKit ) become cross-sectionally
uncorrelated to each other, asymptotically.
Assumption 5(b) implies that the cross-sectionally demeaned xi;t 1 and (uit ui;t 1 )
or the cross-sectionally demeaned (xi;t 1 xi;t 2 ) and uit are uncorrelated under the alternative hypothesis (11). This will obviously hold true if the common factor components
have no impact on xit namely, when xi = 0 for all i in equation (9). This restriction
can be relaxed provided that the direct e¤ect of the factors on yit is uncorrelated with
the e¤ect of the factors on xit , which satis…es Assumption 5(b).5
Assumption 6, 6= 0, is also necessary in order to obtain a consistent estimator
under the alternative of heterogeneous error cross section dependence, although this is
probably not a restrictive assumption in many applications.
For further discussion, stacking (7) for each i yields
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yi =
i T
+ yi;
1
+ Xi + ui , i = 1; 2; :::; N ,
ui = F
5
i
+ "i ,
(12)
(13)
For su¢ ciently large N , the overidentifying restrictions test (de…ned by equation (28) below) is
expected to have enough power to detect the violation of Assumption 5(b).
7
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where yi = (yi1 ; yi2 ; :::; yiT )0 , g is a (g 1) vector of unity, yi; 1 = (yi0 ; yi1 ; :::; yiT 1 )0 ,
Xi = (xi1 ; xi2 ; :::; xiT )0 , ui = (ui1 ; ui2 ; :::; uiT )0 , F = (f 1 ; f2 ; :::; fT )0 , "i = ("i1 ; "i2 ; :::; "iT )0 ,
and the cross-sectionally demeaned and di¤erenced equation is de…ned by
yi;
+
Xi +
ui =
F
+
"i ,
yi =
Wi +
ui ,
1
i
ui , i = 1; 2; :::; N ,
(14)
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yi =
or
(15)
(16)
=( ;
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where the underline signi…es that the variables are cross-sectionally demeaned, and
“ " denotes the …rst-di¤erenced operator. For example, yi = ( y i2 ; y i3 ; :::; y iT )0 ,
P
y it = (y it y it 1 ), y it = (yit yt ) with yt = N 1 N
Wi = ( yi; 1 ; Xi ),
i=1 yit ,
0 0
) , with obvious notation.
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Remark 2 Pesaran’s CD test statistic as de…ned in (6) can fail to reject the null hypothesis of homogeneous error cross section dependence given in (11), when E( i ) = 0. This
arises when time dummies are introduced or the data are cross-sectionally demeaned,
since E( i ) = 0. On the other hand, the Breusch and Pagan (1980) LM test de…ned as
in (4) will have power, although its use is justi…ed only when T is much larger relative
to N .
Sargan’s Di¤erence Tests based on the First-di¤erenced GMM Estimator
De…ne the matrices of instruments
2
y
0
0
0
6 0i0 y
y
0
6
i0
i1
6
0
0
0 y i0
ZY i = 6
6 .
..
..
..
6 .
.
.
.
4 .
where hy = T (T
0
.
:::
y i0
0
:::
y iT
2
1)=2, and
2
6
6
6
=6
6
4
x0i1 0
0
0
0 x0i1 x0i2 0
0
0
0 x0i1
..
..
..
..
.
.
.
.
0
0
0
0
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ZXi
0
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0
..
3
0
0
0
..
.
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where hx = KT (T
vec(Xi )0 .6
T 1
..
.
:::
x0i1
:::
1)=2. If xit is strictly exogenous
3
0
0
0
..
.
x0iT
7
7
7
7 , (T
7
7
5
1
7
7
7
7 , (T
7
5
that is,
1
hy )
(17)
1
hx )
(18)
= 0 in (9), ZXi =
6
One can construct ZXi such that both strictly exogenous and predetermined regressors are present.
See, for example, Arellano (2003).
8
Proposition 3 Consider two sets of moment conditions
0
Moment Conditions I: E[ZY i ui ] = 0
and
0
(19)
(20)
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Moment Conditions II: E[ZXi ui ] = 0,
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Proof. See Appendix A.
Now de…ne the full set of moment conditions
0
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where ZY i , ZXi , and ui are de…ned as in (17), (18), and (15), respectively. Under
Assumptions 1-4, 5(a)-(b), 6 and model (7), both sets of moment conditions given in (19)
and (20) hold under the null hypothesis (10). However, under the alternative hypothesis
given in (11), (20) holds but (19) does not.
E[Zi ui ] = 0,
where
Zi =
ZY i ZXi
(T
1
(21)
h);
(22)
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and h = hy + hx .
Based on Proposition 3, the hypothesis testing for (10) reduces to a test for the
validity of the subset of the moment conditions in (21), as given in (19). There are at
least three possible approaches. The …rst one is to take a Hausman-type (1978) approach.
Newey (1985) showed that the Hausman test may be inconsistent against some local
alternatives. In our application, this may happen when hy = T (T
1)=2 > K + 1,
which is the most likely empirical situation.7 In addition, Arellano and Bond (1991)
examined a Hausman-type test for serial correlation and found that it has poor …nite
sample performance, in that it rejects the null hypothesis far too often. The second
approach is to adopt Newey’s (1985) optimal test. The third approach is to use Sargan’s
(1988) di¤erence test, which we are adopting, since it is relatively easier to compute
than the Newey’s test statistic, and furthermore, they are asymptotically equivalent.
We consider two versions of Sargan’s di¤erence test. One is based on the Arellano and
Bond (1991) two-step …rst-di¤erenced GMM estimator hereafter DIF and the other
is based on the Blundell and Bond (1998) two-step system GMM estimator hereafter,
SYS.
Initially, the test based on DIF is considered. This is de…ned as
! 1 N
N
N
N
X
X
1X
1X
e
e
0
0
_
•
W Zi
Z Wi
W0 Z 1 e_
Z0 y ;
(23)
DIF 2 =
EP
i
i=1
i
i
i=1
i=1
i
i
i
i=1
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e
0 e
e_ 0 Z with u
e_ = y
where e_ = N 1 N
_i u
Wi _ DIF 1 , which is the residual
i
i i
i=1 Zi u
i
vector based on the one-step DIF estimator (denoted by DIF 1),
! 1 N
N
N
N
X
X
X
X
e_
1e 1
0
0
Wi Zi
Zi Wi
W0i Zi e 1
Z0i yi ;
(24)
DIF 1 =
7
i=1
i=1
i=1
i=1
See Proposition 5 and its proof in Newey (1985), under the assymption of i.i.d. errors.
9
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i=1
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P
0
where e = N 1 N
1, with 2’s on the
i=1 Zi HZi , and H is the square matrix of order T
main diagonal, -1’s on the …rst o¤-diagonals, and zeros elsewhere.
Now, the two-step DIF estimator based only on the restricted set of moment conditions in (20), which is consistent both under the null and the alternative, is de…ned
as
! 1 N
N
N
N
X
X
1X
1X
b
b
0
0
_
•
W0i ZXi b_ X
Wi ZXi X
ZXi Wi
Z0Xi y ; (25)
DIF X2 =
i=1
i=1
i
i=1
b_
DIF X1 =
N
X
i=1
W0i ZXi b X1
N
X
Z0Xi
Wi
i=1
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b
0
b_ u
b0
b_ = y
b_ is
Wi _ DIF X1 , where u
where b_ X = N 1 N
u
i _ i ZXi with
i
i
i=1 ZXi u
i
the residual vector based on the one-step DIF estimator that exploits (20), denoted by
DIF X1,
!
1 N
X
i=1
W0i ZXi b X1
N
X
Z0Xi yi ;
(26)
i=1
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0
where b X = N 1 N
i=1 ZXi HZXi .
Sargan’s (1958), or Hansen’s (1982) test statistic of overidentifying restrictions for
the full set of moment conditions in (21) is based on DIF 2 and is given by
!
!
N
N
X
1 X
0
e
1
0 e
e
_
SDIF 2 = N
u
• Zi
•i ,
(27)
Z u
i
i=1
e =
where u
•
i
given by
i
i=1
e
Wi •DIF 2 . Sargan’s statistic for (20) that is based on DIF X2 is
yi
SDIF X2 = N
1
N
X
i=1
b0 Z
u
•
i Xi
!
b_ 1
X
N
X
Z0Xi
i=1
b
u
•
i
!
;
(28)
TE
b
b = y
Wi •DIF X2 .
where u
•
i
i
We are now ready to state a proposition on Sargan’s di¤erence test of heterogeneous
cross section dependence:
d
Proposition 4 Under Assumptions 1-4, 5(a)-(b), 6 and model (7), SDIF 2 !
d
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under the null hypothesis, SDIF X2 !
native hypothesis, and
2
hx (K+1)
DDIF 2 = (SDIF 2
2
h (K+1)
under the null hypothesis and the alterd
SDIF X2 ) !
2
hy
(29)
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under the null hypothesis, where SDIF 2 , SDIF X2 are as de…ned in (27) and (28) respectively, while h = hy + hx with hy and hx de…ned as in (17) and (18) respectively.
Proof. See Appendix B.
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e ; which
Remark 5 The cross section correlation of the sample moment conditions Z0i u
•
i
is induced as a result of the cross-sectional demeaning of the series, is asymptotically
negligible,P
and standard central limit theorems can be applied to the asymptotic expansion
0 e
of N 1=2 N
• i ; See proof in Appendix B.
i=1 Zi u
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Remark 6 Sargan’s di¤ erence test has non-trivial asymptotic local power; See proof in
Appendix B.
3.3
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Remark 7 Proposition 4 holds with unbalanced panel data, so long as min Nt ! 1,
where Nt is the number of cross-sectional units for a given time t. In this case, P
the crosst
8
sectional demeaning for y t , say, is now de…ned as y t = yt yt , where yt = Nt 1 N
i=1 yit .
Sargan’s Di¤erence Tests based on the System GMM Estimator
DM
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It has been well documented (see, for example, Blundell and Bond, 1998) that the DIF
estimator can su¤er from a weak instruments problem when is close to unity and/or
the variance of the individual e¤ects is large relative to that of the idiosyncratic errors.
Thus, we also consider another version of Sargan’s di¤erence test based on the Blundell
and Bond (1998) SYS estimator, which is known to be more robust to the problem of
weak instruments under certain conditions.
Arellano and Bover (1995) proposed the use of lagged di¤erences as possible instruments for the equations in levels, E (uit wit ) = 0 with wit = (yt 1 ; x0t )0 for t = 3; 4; :::; T ,
which is valid under the null hypothesis and Assumptions 1-3. In addition, Blundell
and Bond (1998, 2000) proposed using an additional condition E (ui2 wi2 ) = 0 under
mild conditions upon the initial observations, which would follow from joint stationarity
of the y and x processes. Accordingly, we add Assumption 5(c):
Assumption 5(c): cov( i ; yi1 ) = 0 and cov( i ; xi2 ) = 0.
De…ne
ZY i 0
0 ZL
Yi
TE
Z+
Yi =
where ZL
Y i = diag( y i;t
1
(2(T
1)
hys );
) for t = 2; 3; :::; T , hys = hy + (T
ZXi
0
0 ZL
Xi
EP
Z+
Xi =
(2(T
1)
(30)
1); and
hxs );
where ZL
1 K(T
1)) matrix whose sth diagonal raw vector is
Xi is a (T
s = 2; 3; :::; T , otherwise zeros, and hxs = hx + K(T 1).
(31)
x0is ,
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Proposition 8 Consider two sets of moment conditions
Moment Conditions I : E[Z+0
Yi
8
i]
=0
(32)
For more details on the computations of the Arellano and Bond (1991) and Blundell and Bond (1998)
estimators with unbalanced panels, see Arellano and Bond (1999).
11
and
Moment Conditions II : E[Z+0
Xi
where
Z+
Yi
and
Z+
Xi
i]
= 0,
are de…ned as in (30) and (31) respectively,
+
0
0
= ( u0i ; u+0
i ) ; and ui = (ui2 ; ui3 ; :::; uiT ; ) .
(33)
(34)
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Under Assumptions 1-4, 5(a)-(c), 6 and model (7), both (32) and (33) hold under the
null hypothesis in (10). However, under the alternative hypothesis given in (11), (33)
holds but (32) does not.
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Proof. See Appendix C.
In the same manner as DDIF 2 , we de…ne the Sargan’s di¤erence test statistic based
on SYS estimator
DSY S2 = (SSY S2 SSY SX2 )
(35)
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where SSY S2 is Sargan’s test statistic of overidentifying restrictions based on the twostep SYS estimator that makes use of both sets of moment conditions I and II in
Proposition 8 (denoted by SY S2), namely
E[Z+0
i
where
Z+
i =
i]
+
Z+
Y i ZXi
= 0,
(2(T
1)
(36)
hs );
(37)
hs = hsy +hsx , and SSY SX2 is Sargan’s test statistic based on the two-step SYS estimator
that exploits only Moment Conditions II , as de…ned in (33), denoted by SY SX2.9
e
One-step and two-step SYS estimators based on the moment conditions (36), _ SY S1 and
e
b
b
•SY S2 , and those based on (33), _ SY SX1 and •SY SX2 , are de…ned accordingly.10 It is
d
3.4
Discussion
2
hys
as N ! 1, under the null hypothesis.
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straightforward to see that DSY S2 !
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The overidentifying restrictions test can be regarded as a misspeci…cation test, in a sense
that it is designed to detect violations of moment conditions, which are the heart of GMM
methods. Thus, it will have power under the alternative hypothesis of heterogenous error
cross section dependence. Nonetheless, the proposed Sargan’s di¤erence test is expected
to have higher power than the overidentifying restrictions test, so long as Assumption
6,
6= 0, holds, since the former exploits extra information about the validity of the
moment conditions under the alternative hypothesis, which the latter does not use. This
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9
With small samples DDIF 2 may not be positive, but it can be patched easily. See, for example,
Hayashi (2000, p.220). However, we did not adopt this modi…cation here since one of our aims is to
show the properties of a consistent estimator based only on orthogonality conditions E[Z0Xi ui ] = 0 or
E[Z+0
Xi i ] = 0.
10
The initial weighting matrix for one-step GMM-SYS estimator is de…ned as a block diagnoal matrix
of order 2(T 1), whose diagnoal blocks are H and IT 1 .
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also implies that, when = 0, the overidentifying restrictions test should replace our
approach.11
Now consider a violation of Assumption 2, E("it "is ) = 0 for t 6= s, no error serial
correlation. Under the alternative of error cross section dependence, the composite error
uit = 0i ft + "it will be serially correlated, since E(uit uis ) = ft0 E( i 0i )fs 6= 0 for all
i. This means that the second-order serial correlation test based on DIF2 or SYS2,
the m2 test, proposed by Arellano and Bond (1991), is likely to reject the hypothesis
of no error serial correlation, under the alternative of heterogeneous error cross section
dependence.12 Then, a question that may arise is how to distinguish between error cross
section dependence and serial correlation in the idiosyncratic errors. To answer this
question, consider two scenarios. First, suppose that there is …rst-order autoregressive
serial correlation but no heterogeneous error cross section dependence, such that "it =
i:i:d:(0; 2 ). In this case, the problem can be solved
" "it 1 + it with j " j < 1 and it
in a straightforward manner by adding a further lag of the dependent variable on the
right hand side of (7) and using (up to) yit 3 as instruments for yit 1 and yit 2 .
Second, suppose there is both …rst-order autoregressive error serial correlation as above
and heterogeneous error cross section dependence. Clearly, the m2 test based on DIFX2
or SYSX2 is likely to reject the null even when E("it "is ) = 0 for t 6= s. Meanwhile, the
probability of rejecting the null by the overidentifying restrictions test for the restrictions
based only on the subset of Xi (de…ned by (20) or (33)), tends to its signi…cance level
when E("it "is ) = 0 for t 6= s, but such a probability goes to one when E("it "is ) 6= 0 for
t 6= s. Therefore, the solution given in the …rst case applies, but the test statistic to
employ is the overidentifying restrictions test based only on the subset of Xi , not the
m2 test based on DIFX2 or SYSX2.
Finally, we have shown that the moment conditions (20) and (33) hold under the alternative of error cross section dependence, therefore, the DIFX2 and SYSX2 estimators
are consistent. However, in …nite samples there could be a trade-o¤ between e¢ ciency
and bias. If the degree of heterogeneity of the error cross section dependence is relatively
small, then the bias of the standard GMM estimators exploiting moment conditions including (19) or (20) which are invalid, may be small enough so that these estimators are
preferred (in root mean square errors terms) to a consistent estimator based only on the
valid moment conditions (20) or (33). We will investigate the …nite sample performance
of these estimators in the next section.13
11
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In Appendix B, it is formally shown that when hx > (K + 1), the DDIF 2 test is asymptotically more
powerful than the SDIF 2 test under the local alternatives.
12
Heterogeneity of would also render the error term serially correlated, as discussed in Pesaran and
Smith (1995).
13
Other solutions have been proposed in the literature, such as a panel feasible generalized median
unbiased estimator, proposed by Phillips and Sul (2003), or the common correlated e¤ects (CCE) estimator proposed by Pesaran (2006). However, both estimators require a larger value for T than that
considered in this paper.
13
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Small Sample Properties of Cross Section Dependence
Tests
4.1
Design
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This section investigates by means of Monte Carlo experiments the …nite sample performance of our tests, the Breusch and Pagan (1980) LM test and Pesaran’s (2004) CD
test, all based on cross-sectionally demeaned variables. Our main focus is on the e¤ects
of (i) the degree and heterogeneity of error cross section dependence, (ii) the relative
importance of the variance of the factor loadings and the idiosyncratic errors, and (iii)
di¤erent values of and . In order to make the results comparable across experiments,
we control the population signal-to-noise ratio and the impact of the ratio between the
variance of the individual-speci…c time-invariant e¤ects and the variance of the idiosyncratic errors and the common factor on yit . To this end, we extend the Monte Carlo
design of Kiviet (1995) and Bun and Kiviet (2006) to accommodate a factor structure
in the error process.
Recently Bowsher (2002) reports …nite sample evidence that Sargan’s overidentifying
restrictions test exploiting all moment conditions available can reject the null hypothesis
too infrequently in linear dynamic panel models. Thus, we only make use of y it 2 and
y it 3 as instruments for y it 1 and we use xit 2 and xit 3 as instruments for xit 1 .
The data generating process (DGP) we consider is given by
yit =
i
+ yi;t
uit =
i ft
1
+ xit + uit ;
+ "it , i = 1; 2; :::; N ; t =
(38)
48; 47; :::; T ,
TE
where i i:i:d:N (1; 2 ) and ft i:i:d:N (0; 2f ).14 "it is drawn from (i) i:i:d:N (0; 2" )
p
and (ii) i:i:d:( 21 1)= 2, in order to investigate the e¤ect of non-normal errors. yi; 49 =
0 and the …rst 49 observations are discarded.15 To control the degree and heterogeneity
of cross section dependence three speci…cations for the distribution of i are considered:
8
i:i:d:U [ 0:3; 0:7]
< Low cross section dependence:
i
Medium cross section dependence:
i:i:d:U
[ 1; 2]
i
:
High cross section dependence:
i:i:d:U
[
1; 4]:
i
EP
Also, as we change the value of = 0:2; 0:5; 0:8,
the long run e¤ect of x on y constant.
The DGP of xit considered here is given by
xit = xi;t
1
+ "i;t
1
+
i ft
is equated to 1
+ vit , i = 1; 2; :::; N ; t =
in order to keep
48; 47; :::; T ,
(39)
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14
Note that the unobserved common factor, ft , is randomly drawn to control the signal-to-noise ratio
without loss of generality.
15
We do not report the results based on non-normal errors in this paper, since the results were very
similar. They are available from the authors upon request.
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where = 0:5, i i:i:d:U [ 1; 2], vit i:i:d:N (0; 2v ). is set to 0:5.16 xi; 49 = 0 and
the …rst 50 observations are discarded.
Since our focus is on the performance of the tests and estimators, we pay careful
attention to the main factors that a¤ect it
namely, (i) the signal-to-noise ratio, (ii)
the relative importance of the variance of the factor loadings and the idiosyncratic errors,
and (iii) the impact of the ratio between the variance of the individual-speci…c e¤ects
and the variance of the idiosyncractic error and factor loadings on yit . To illustrate, we
de…ne the signal as 2s = var (yit uit ), where yit = yit
). Then, denoting the
i =(1
(2) 2
(2)
2
2
variance of the composite error by u = var(uit ) =
= E( 2i ), we
f + " with
2
2
de…ne the signal-to-noise ratio as = s = u . We set = 3. The relative importance
(2) 2
2
in terms of the magnitude of the variance of i ft and "it , as measured by
f = ",
is thought in the literature to be an important factor to control for and we achieve
(2)
this by changing
and applying the normalisation 2f = 2" = 1. As it has been
discussed by Kiviet (1995), Blundell and Bond (1998), and Bun and Kiviet (2006), in
order to compare the performance of estimators across di¤erent experimental designs it
is important to control the relative importance of i and ("it , ft ). We choose 2 such
that the ratio of the impact on var(yit ) of the two variance components i and ("it , ft )
is constant across designs.17
We consider all combinations of N = 50; 100; 200; 400, and T = 5; 9. All experiments
are based on 2,000 replications.
Results
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Tables 1 reports the size and power of the tests for T = 5.18 LM denotes Breusch and
Pagan’s LM test, as de…ned in (4), and CD denotes Pesaran’s CD test, de…ned in (6),
both of which are based on the …xed e¤ects estimator. DDIF 2 is Sargan’s di¤erence test
based on the two-step DIF estimator de…ned in (29), and DSY S2 is Sargan’s di¤erence
test based on the two-step SYS estimator de…ned in (35). The size of the LM test is
always indistinguishable from 100% and therefore it is not recommended. The CD test
does not reject the null in all experiments, and has no power across experiments. On
the other hand, although the size of DDIF 2 and DSY S2 is below the nominal level for
N = 50 (especially for the latter), as N becomes larger the size quickly approaches its
nominal size. In addition, our proposed tests have satisfactory power. DSY S2 has more
power than DDIF 2 in general, unless DSY S2 rejects the null too infrequently. Di¤erent
values of seem to have very little e¤ect on the performance of DDIF 2 and DSY S2 .
We now turn our attention to the performance of the estimators. Table 2 reports
the bias of the estimators for .19 DIF 1 and DIF 2 are the one-step and two-step DIF
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16
xit has zero mean without loss of generality, since we cross section-demean all data before computing
the statistics.
17
See Appendix D for the details of the way of controlling these parameters.
18
We do not report the results for T = 9 in this paper, since these were similar to those for T = 5.
They are available from the authors upon request.
19
We do not report the performance of the estimators for , since it has a similar pattern to that for
; although it is not as much a¤ected by error cross section dependence.
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estimators respectively, de…ned by (24) and (23), and they are based on the full set of
moment conditions I and II in Proposition 3. DIF X1 and DIF X2 denote the one-step
and two-step DIF estimators de…ned by (26) and (25), and they are based only on the
subset of the moment conditions II. SY S1 and SY S2 are the one-step and two-step SYS
estimators respectively, and they are based on the full set of moment conditions I and
II in Proposition 8, and SY SX1 and SY SX2 denote the one-step and two-step SYS
estimators based only on the subset of moment conditions II . The bias of all GMM
estimators under low cross section dependence is not noticeably di¤erent from that under
zero cross section dependence. As the degree of error cross section dependence rises, the
bias of the GMM estimators based on the full set of moment conditions increases, which
is expected as only those estimators based on Moment Conditions II or II are consistent.
As a result, the relative bias between those estimators that use the full set of moment
conditions and those that use only Moment Conditions II or II increases. Table 3 reports
root mean square errors of the estimators for . Under no error cross section dependence
and low cross section dependence, DIF 2 and SY S2 outperform DIF X2 and SY SX2
respectively in terms of root mean square error. However, under moderate and high
cross section dependence, DIF X2 and SY SX2 have a smaller root mean square error
compared to DIF 2 and SY S2 respectively, in most cases.
5
An Empirical Example: Employment Equations of U.K.
Firms
yit =
TE
In this section we examine the homogeneity of error cross section dependence of the
employment equations using (unbalanced) panel data for a sample of UK companies,
which is an updated version of that used by Arellano and Bond (1991), and it is contained
in the DPD-Ox package.20 Brie‡y, these authors select a sample of 140 companies that
operate mainly in the UK with at least 7 continuous observations during the period
1976-1984.
We apply our test to the model speci…cations of Blundell and Bond (1998). The
model we estimated is given by
i
+
1 yit 1
+
0 wit
+
1 wit 1
+ '0
it
+ '1
it 1
+ uit ,
20
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where yit is log of the number of employees of company i, wit is log of real product wage,
it is the log of gross capital stock.
Table 4 presents estimation and test results.21 Observe that year dummies are included to remove possible time e¤ects, therefore no cross-sectional demeaning of the
series is implemented. Our estimation results based on the full sets of instruments, Zi
and Z+
i , as de…ned in (22) and (37) but without cross-sectional demeaning, resemble
The data set used is available at http://www.doornik.com/download/dpdox121.zip
The GMM estimates of the parameters have been obtained using the xtabond2 command in Stata; see
Roodman, D., (2005). xtabond2: Stata module to extend xtabond dynamic panel data estimator. Center
for Global Development, Washington. http://econpapers.repec.org/software/bocbocode/s435901.htm
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those reported in the last two columns of Table 4 in Blundell and Bond (1998), although
the values do not match exactly due to di¤erences in computations and the data set
used. First, all m2 tests suggest that there is no evidence of error serial correlation and
this implies possibly no heterogeneous error cross section dependence. This is con…rmed
by the fact that both Sargan’s di¤erence tests based on DIF and SYS for heterogeneous
error cross section dependence safely fail to reject the null hypothesis of homogeneous
error cross section dependence. The estimation results based only on partial instruments
consisting of the covariates, ZXi and Z+
Xi , as de…ned by (18) and (31) but without crosssectional demeaning, are largely downward biased for the DIF estimator and less so for
the SYS estimator. This indicates that the e¢ ciency loss of SYS that does not contain
Z+
Y i in the instrument set is much smaller compared to the e¢ ciency loss of DIF. This
feature seems to have some e¤ect on the testing results. For example, the p-value of the
…rst-order serial correlation test, m1 , for DIF with the full set of instruments, Zi , is zero
up to three decimal points, but it goes up to 0.028 with the subset of instruments ZXi .
On the other hand, the p-value of m1 for SYS is zero up to three decimal points in both
cases.
Concluding Remarks
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This paper has proposed a new testing procedure for error cross section dependence
after estimating a linear dynamic panel data model with regressors by the generalised
method of moments (GMM). The procedure is valid when the cross-sectional dimension
is large and the time series dimension of the panel is small. Importantly, our approach
allows one to examine whether any error cross section dependence remains after including
time dummies, or after transforming the data in terms of deviations from time-speci…c
averages, which will be the case under heterogeneous error cross section dependence. The
…nite sample simulation-based results suggest that our tests perform well, particularly
the version based on the Blundell and Bond (1988) system GMM estimator. On the
other hand, the LM test of Breusch and Pagan (1980) overrejects the null hypothesis
substantially and Pesaran’s (2004) CD test lacks power. Also it is shown that the system
GMM estimator, based only on partial instruments consisting of the regressors, can be
a reliable alternative to the standard GMM estimators under heterogeneous error cross
section dependence. The proposed tests are applied to employment equations using
UK …rm data, and the results show little evidence of heterogeneous error cross section
dependence.
17
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Appendices
A
Proof of Proposition 3
t j
j
=
i
X2
t j
`
+
t j
1
xi1 +
`=0
and for t
0
i
X2
t j
`
ft
j
`
X2
+
`=0
"i;t
`=0
i
t 1
X
j
t
+
y i0 +
t 1
X
j
xi;t
j
0
i
+
j=0
j=0
t 1
X
+
1 `
`
v i;t
j
`;
(40)
`=0
1
y it =
j
X2
2 + j,
t j
`
j
ft
+
t 1
X
j
US
xit
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To simplify the analysis without loss of generality we consider the case where K = 1. For t
given j such that 0 j t 1, it can be shown that
j
j=0
"i;t
j:
(41)
j=0
Firstly we consider E(y i;t s uit ) for s t T , given 2 s T , under the alternative hypothesis of
6= . But this is equivalent to considering E(y it ui;t+s ) for 0 t T s, given 2 s T . Initially
we focus on the case of s = 2. When t = 0, E(y i0 ui2 ) = E(y i0 0i ) f2 by Assumption 3, which is not
necessarily zero under the alternative. When t 1, using (40) and (41), together with uit = 0i ft + "it ,
E(y it ui;t+2 ) =
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i
t
E y i0
0
i
ft+2 +
0
ft+2
E
i
0
i
t 1
X
j
j=0
ft
j
(42)
6= 0;
under Assumptions 1-4, 5(a)-(b), 6 and model (7). A similar approach for the case of s > 2 leads to the
conclusion that E(y it ui;t+s ) 6= 0 for t 0, s 2 under the alternative, as required. Under the null of
, it follows immediately that E(y it ui;t+s ) = 0 for t 0, s 2.
i =
Now we consider E(xi;t s uit ) for s < t
T , given 1
s
T 1, under the alternative, which
is equivalent to considering E(xit uit+s ) for 1 t < T s, given 1 s T 1. Initially we focus on
2 and using
the case of s = 1. When t = 1, E(xi1 ui2 ) = 0 due to Assumption 3 and 5(b). For t
(40), together with uit = 0i ft + "it , we have E(xit uit+1 ) = 0, under Assumptions 1-4, 5(a)-(b), 6 and
model (7), under the alternative. A similar approach for the case where s > 1 leads to the conclusion
that E(xit ui;t+s ) 6= 0 for t 0, s 1 under the alternative, as required. Under the null of i = , it
follows immediately that E(y it ui;t+s ) = 0 for t 0, s 1, which completes the proof. In addition, it
is straightforward to show that E(xi;t 1 y i;t 1 ) 6= 0.
Proof of Proposition 4
TE
B
d
Firstly we establish that SDIF X2 ! 2hx (K+1) under the alternative of H1 :
P
0
b gives
de…ned in (28). Rewriting p1N N
ui
u
•
i
i=1 ZXi
EP
N
N
X
1 X 0
b = p1
p
ZXi u
•
Z0Xi ui
i
N i=1
N i=1
N
1
N
X
i=1
b 1=2 PN
0
b gives
Next, orthogonally decomposing _ X
•
i
i=1 ZXi u
1=2 X
1
b
p b_ X
Z0Xi u
•
i
N
i=1
AC
C
N
p
e
Z0Xi Wi N •DIF 2
.
(43)
1=2 X
1 ^ b_ 1=2 X 0
b + p1 M
b
^ B b_ X
p P
ZXi u
•
Z0Xi u
•
B X
i
i
N
N
i=1
i=1
N
=
6= 0, where SDIF X2 is
1 ^ b_ 1=2 X 0
b,
p M
ZXi u
•
B X
i
N
i=1
N
N
=
18
(44)
^ 0, M
^B =I
B
b_ 1=2
N
X
^B, B
^ =
P
1
PN
i=1
Z0Xi Wi , and the last line follows
b = 0 by the de…nition of the GMM estimator. Substituting (43) into (44)
u
•
i
1=2 X
1=2 X
1
b = p1 M
^ B b_ X
p b_ X
•
Z0Xi u
Z0Xi ui ,
i
N
N
i=1
i=1
N
N
(45)
CR
1
^B = B
^ B
^ 0B
^
where P
1=2 P
N
0
^ 0 b_ X
from B
i=1 ZXi
yields
IPT
ACCEPTED MANUSCRIPT
^ BB
^ = 0.
since M
We can express the instruments as deviations from their cross-sectional averages:
ZXi = ZXi
ZX ;
x0it s
(46)
ZoXi = ZXi
US
where ZXi is de…ned similarly to ZXi but all
are replaced with xit , and ZX is de…ned similarly but
all non-zero elements are replaced with their cross-sectional averages, xt . Also de…ne the instruments in
terms of deviations from their mean as
and
uoi =
where mZX = E(ZXi ),
F(
i
"i
)+
i ). Using (46)-(48)
DM
AN
= E(
N
1 X 0
p
ZXi ui
N i=1
N
1 X o0
p
ZXi ( ui
N i=1
=
ZX
(47)
mZX ;
mZX
0
(48)
(49)
u)
N
1 X
p
( ui
N i=1
u) .
It is easily seen that the second term of (49) is asymptotically negligible. Consider the …rst term of (49).
Reminding ourselves that ui = F i + "i , we have
N
1 X o0
p
ZXi ( ui
N i=1
=
N
1 X o0
p
ZXi F (
N i=1
i
N
1 X o0
p
ZXi ( "i
N i=1
)
i:i:d:( ;
), "it
i:i:d:(0; 2i ) above,
PN
o
=
O
(1).
Then it follows that
Z
p
i=1 Xi
") .
= Op (N
TE
By the assumptions i
Op (N 1=2 ) as well as N
u)
1=2
) and
" =
1=2
N
N
1 X 0
1 X o0
p
ZXi ui = p
ZXi uoi + op (1).
N i=1
N i=1
(50)
EP
P
1=2
b
o0
o
^ B = op (1) where B =
Since _ X
= op (1), B
N 1 N
and
X
X
i=1 ZXi Wi
P
N
1
o0
o
o0 o
=
p
lim
N
E(Z
u
u
Z
)
with
obvious
notations,
together
with
(50),
(45)
can
X
N !1
Xi
i
i
Xi
i=1
be written as
N
N
1=2 X
X
1
1=2 1
o
b = p1 MB
p b_ X
p
Z0Xi u
•
Zo0
(51)
i
Xi ui + op (1).
X
N
N
N
i=1
i=1
AC
C
o
As Zo0
Xi ui are independent across i, a suitable Central Limit Theorem ensures that
1=2 1
p
X
N
N
X
i=1
d
o
Zo0
Xi ui ! N (0; Ihx ).
19
SDIF X2
=
(K + 1) we have
!
N
1 X
o
Z
uo0
i
Xi
N i=1
d
2
hx
!
1=2
MB
X
N
X
1=2
X
o
Zo0
Xi ui
i=1
!
(K+1) ,
under the alternative hypothesis of H1 :
+ op (1)
CR
Noting that rank(MB ) = hx
IPT
ACCEPTED MANUSCRIPT
6= 0, as required. Under the null hypothesis of H0 :
(52)
= 0,
d
(52) follows immediately. Also it is straightforward to establish that SDIF 2 ! 2h (K+1) , where SDIF 2
is de…ned as in (27), in line with the proof provided for (52).
Now we provide the asymptotic distribution of SDIF 2 SDIF X2 . Consider the local alternative
i
=
i
+
N 1=4
,
US
HN :
where 0 < jj i jj < 1 for all i, which are assumed to be non-stochastic for expositional convenience. Here
the analysis is based on the instruments in terms of deviations from their true mean, rather than from the
cross-sectional average, since we have already shown that the e¤ect of such replacement is asymptotically
negligible. Without loss of generality, consider ZoY i = diag(y oi;t 2 ), t = 2; 3; :::; T , hy = T 1. Also de…ne
Zoi =
ZoXi
,
1
= p lim N
N !1
N
X
uoi uo0 Zoi ,
E Zo0
i
where
=
(53)
i=1
DM
AN
ZoY i
Y
YX
XY
X
,
(54)
with block elements that are conformable with ZoY i and ZoXi . By using (42) we have
!
N
X
1=2
o0
o
ZY i u i = N ,
E N
i=1
P
0
= O(1) is a (T 1 1) vector whose …rst element is N 1 N
1)th
i=1 E(y i0 i ) f2 and the (t
P
P
t 1 j
N
0
0
0
1
t
ft+2 + ft+2 i i
ft j , for t = 2; 3; :::; T 1. De…ne
elements are N
E y i0 i
j=0
i=1
where
N
Ihy
0
Zoi = Zoi L0 , with L =
TE
where L is non-singular, so that
N
1 X o
p
Zi
N i=1
where
0
N
uoi
0
with
Y
=
EP
= L L0 =
Y
YX
SDIF 2 =
XY
. It follows that
!
N
X
1
N
B (B 0 B )
AC
C
with MB = I
1
X
o
uo0
i Zi
1=2
YX
d
),
! N (0;
0
Y
0
;
X
1=2
MB
N
X
Zoi
0
i=1
i=1
1
1
X
Ihx
B 0,
B =
BY
B
=
1=2
N
1
N
X
i=1
20
Zoi
0
Woi ,
uoi
!
+ op (1)
so that
SDIF X2
=
i=1
N
X
1
N
o
uo0
i ZXi
i=1
N
X
1
N
=
!
o
uo0
i Zi
!
o
uo0
i Zi
i=1
1=2
MB
0
0
Zoi 0
i=1
!
1=2
MB
X
N
X
1=2
X
1=2
1=2
M
N
X
Zoi
0
0
MB
M11
M21
=
M12
M22
2
(hy ; )
!
(55)
!
+ op (1)
!
+ op (1)
,
US
d
uoi
i=1
which is a symmetric and idempotent matrix of rank hy . Finally, we have
!
N
N
X
1 X
o 0
1=2
1=2
Z
M
SDIF 2 SDIF X2 =
uo0
Zoi
i
i
N i=1
i=1
!
uoi
o
Zo0
Xi ui
i=1
where
M = MB
N
X
1=2
CR
SDIF 2
N
X
1
N
IPT
ACCEPTED MANUSCRIPT
0
uoi
!
(56)
+ op (1)
DM
AN
which is a non-central chi-squared distribution with hy degrees of freedom and non-centrality parameter
d
1=2
1=2
M11 Y
= p limN !1 0N Y
N > 0, so long as N 6= 0. Therefore, the result SDIF 2 SDIF X2 !
2
hy readily follows under the null hypothesis, as required.
d
Furthermore, using (55) and (56), it is easily seen that SDIF 2 ! 2 (h (K + 1); ) under the local
alternatives, where is the same non-centrality parameter of the asymptotic distribution of (SDIF 2
SDIF X2 ). Therefore, when h (K + 1) > hy , or subtracting hy from both sides hx > (K + 1), the
Sargan’s di¤erence test is locally more powerful than the overidentifying restrictions test.
C
Proof of Proposition 8
In line with the proof of Proposition 3 in Appendix A, consider E( xit ( i + uit )). For t 3 and using
(40) together with uit = 0i ft + "it , we have under the alternative hypothesis of i 6= , E( xit ( i +
uit )) = 0, under Assumptions 1-4, 5(a)-(c), 6 and model (7). A similar line of argument proves that
E( xi2 ( i + ui2 )) = 0. However, for t 2 and using (40) and (41) we have under the alternative
i
+ uit+1 )) =
t
E
TE
E( y it (
y i1
0
i
0
ft+1 + ft+1
E
i
0
i
t 1
X
j=0
j
ft
j
6= 0;
D
EP
under Assumptions 1-4, 5(a)-(c), 6 and model (7). A similar line of argument will prove that E( y i1 ( i +
ui2 )) 6= 0 under the alternative. Finally, under the null hypothesis, it is also easily seen that E( y it ( i +
uit+1 )) = 0 for t
1 and E( xit ( i + uit )) = 0 for t
2, which completes the proof. Furthermore, it is straightforward to show that E( xi;t 1 xit ) 6= 0, E( xi;t 1 y i;t 1 ) 6= 0, E( y i;t 1 xit ) 6= 0,
E( y i;t 1 y i;t 1 ) 6= 0.
Derivations of Parameters in Monte Carlo Experiments
AC
C
Using the lag operator, L, we can write yit and xit as
yit =
i
+
1
xit =
1
L
L
1
L
i
xit +
"it +
1
i
1
21
L
L
ft +
ft +
1
1
1
1
L
L
vit
"it
(57)
(58)
and thereby substituting (58) into (57) yields
i
+
1
De…ne yit = yit
vit
L) (1
(1
L)
+
i (1
L) (1
L)
1+(
ft +
L)
(1
L) (1
), such that (57) can be rewritten as
i =(1
yit = yit
= var (yit
+ xit + uit
1
2
2
s = u;
and let the signal-to-noise ratio be denoted by
2
s
+
i
(1
where
2
s
)L
"it .
L)
(59)
(60)
CR
yit =
IPT
ACCEPTED MANUSCRIPT
is the variance of the signal,
2cov (yit ; uit ) .
uit ) = var (yit ) + var (uit )
2
s
(61)
2
s
where a1 =
2(
(1+
(1
2 )(1
)
2 )(1
2
=
)
2
v
2
f b1
+
, b1 =
+
2
" b2
2
+
a1
2
=
)(
2
"
= 1, substituting
2
u
(1 + ) + 2
2
s
=
2
u
+ )
=a1
)2 +
and b2 = 1 + (
, and solving for
DM
AN
2
v
=
2(
2
u,
1+
1+
2
f
2
f
2
2
+
)( + )
.
Applying the normalisation
US
varies across designs, with the aim being to keep the signal-to-noise ratio constant over changes in
and the distribution of i , so that the explanatory power of the model does not change. In particular,
we set 2s = 2u =
= 3, where 2u = var (uit ). We normalise 2f = 2" = 1 and we keep the total
signal-to-noise ratio …xed by modifying 2s accordingly through changes in 2v . It can be shown that
2
v
yields
(b1 + b2 ) .
In line with the simulation design of Bun and Kiviet (2006), we choose 2 such that the ratio of the
impacts on var(yit ) of the two variance components i and ("it , ft ) is 2 . By (59)
var(yit )
=
var
1
+var
i
+
(1
i (1
L) (1
2
=
Now de…ne
=
2
(1
such that
2
(1
)2
=
2
2
(
+(
2
f b1
) (b1 + b2 )a1 : We choose
2
+
2
v
(1
2
f b1
+
2
" b2 )a1 .
= 1.
EP
AC
C
L) (1
L)
vit
L)
1+(
)L
ft + var
"it
L)
(1
L) (1
L)
TE
set
2
2
)2
(1
+ var
i
1
22
+
2
" b2 )a1 .
By applying the normalisation
2
"
=
2
f
= 1, we
IPT
ACCEPTED MANUSCRIPT
References
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CR
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US
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DM
AN
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TE
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C
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CR
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C
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TE
[42] Stock, J. and M.W. Watson, 2002, Macroeconomic Forecasting Using Di¤usion
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25
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Table 1: Size and Power of the Cross Section Dependence Tests in the Case with
Predetermined Regressors with T = 5
= 0:2
100
200
400
LM
100.00 100.00 100.00 100.00
CD
0.00
0.00
0.00
0.00
DDIF 2
3.55
4.20
5.30
5.35
DSY S2
3.65
5.20
5.30
5.15
LM
100.00 100.00 100.00 100.00
CD
0.00
0.00
0.00
0.00
DDIF 2
4.80
9.10 18.15 29.25
DSY S2
4.00 10.90 21.65 36.40
DM
AN
LM
100.00 100.00 100.00 100.00
CD
0.00
0.00
0.00
0.00
DDIF 2 38.65 70.10 86.40 94.00
DSY S2 29.80 77.25 92.35 96.95
= 0:5
100
200
400
Size: i = = 0
100.00 100.00 100.00 100.00
0.00
0.00
0.00
0.00
4.40
4.45
4.60
5.40
4.15
5.35
5.55
5.80
Power: i i:i:d:U [ 0:3; 0:7]
100.00 100.00 100.00 100.00
0.00
0.00
0.00
0.00
6.00
9.90 18.70 32.10
5.35 10.75 22.95 41.60
Power: i i:i:d:U [ 1; 2]
100.00 100.00 100.00 100.00
0.00
0.00
0.00
0.00
43.25 74.45 88.45 95.25
34.50 79.75 93.30 97.80
Power: i i:i:d:U [ 1; 4]
100.00 100.00 100.00 100.00
0.00
0.00
0.00
0.00
67.70 90.30 96.40 98.55
56.35 94.10 98.30 99.65
50
LM
100.00 100.00 100.00 100.00
CD
0.00
0.00
0.00
0.00
DDIF 2 63.30 89.70 95.15 98.20
DSY S2 55.25 93.80 98.60 99.30
50
= 0:8
100
200
400
100.00 100.00 100.00 100.00
0.00
0.00
0.00
0.00
6.45
7.05
6.30
5.40
5.10
6.25
6.35
6.30
CR
50
100.00 100.00 100.00 100.00
0.00
0.00
0.00
0.00
7.65 13.95 21.75 35.30
5.50 13.55 28.05 46.55
US
Test,N
100.00 100.00 100.00 100.00
0.00
0.00
0.00
0.00
46.25 77.85 91.30 95.85
36.80 83.30 95.90 98.80
100.00 100.00 100.00 100.00
0.00
0.00
0.00
0.00
69.00 92.65 98.00 99.25
62.20 96.35 98.95 99.80
AC
C
EP
TE
Notes: LM and CD denote the Breusch-Pagan LM test and Pesaran’s (2004) CD test, respectively. Both
are based on the residuals of the Fixed E¤ects estimator. DDIF 2 and DSY S2 denote Sargan’s di¤erence
tests based on the two-step Arellano Bond (1991) DIF estimator, and on the two-step Blundell and Bond
(1998) SYS estimator respectively. The data generating process (DGP) is yit = i + yi;t 1 + xit +
48; 47; :::; T with yi; 49 = 0. The initial 49 observations are discarded.
i ft + "it , i = 1; 2; :::; N , t =
i:i:d:N (1; 2 ), and i are as speci…ed in the Table, = 1
, ft i:i:d:N (0; 1), "it i:i:d:N (0; 1);
i
xit = xi;t 1 + "i;t 1 + i ft + vit , i = 1; 2; :::; N , t = 48; 47; :::; T with xi; 49 = 0 and the initial
50 observations being discarded.
= = 0:5, = 0:5; i
i:i:d:U [ 1; 2], vit
i:i:d:N (0; 2v ), 2v is
chosen such that the signal-to-noise ratio equals 3. 2 is chosen such that the impact of the two variance
components i and (ft , "it ) on var(yit ) is constant. All variables are cross-sectionally demeaned before
computing statistics. All experiments are based on 2,000 replications.
26
Table 2: Bias ( 1000) of Fixed E¤ects and GMM estimators for
Predetermined Regressors with T = 5
FE
-115.62 -114.70
DIF1
-23.68 -10.76
DIF2
-22.83 -10.11
DIFX1 -20.58 -10.07
DIFX2 -19.93 -10.18
SYS1
13.43
7.31
SYS2
11.58
6.06
SYSX1
9.57
4.71
SYSX2
7.94
3.37
FE
-107.82 -106.72
DIF1
-22.29 -10.66
DIF2
-20.85
-9.55
DIFX1 -18.53
-8.89
DIFX2 -17.71
-8.66
SYS1
12.24
6.24
SYS2
10.12
4.93
SYSX1
9.36
4.72
SYSX2
7.18
3.42
-87.17
-30.36
-27.14
-14.25
-12.92
-0.15
-1.04
6.00
3.96
-85.41
-21.66
-16.75
-6.81
-5.71
-2.22
-1.85
3.00
2.14
FE
-63.49
-34.91
-30.25
-9.91
-8.17
-0.25
-1.63
9.23
4.93
-61.52
-29.04
-22.18
-4.75
-3.38
-3.56
-3.71
4.76
2.10
EP
DIF1
DIF2
DIFX1
DIFX2
SYS1
SYS2
SYSX1
SYSX2
= 0:5
= 0:8
400
50
100
200
400
50
100
200
No Cross Section Dependence: i = = 0
-116.66 -115.46 -187.03 -183.89 -185.06 -184.63 -392.65 -389.05 -388.91
-7.76
-2.46 -46.94 -24.22 -10.82
-5.29 -205.25 -105.79 -59.39
-7.65
-1.98 -48.67 -24.47 -10.48
-5.25 -223.06 -115.40 -61.99
-7.70
-2.30 -65.99 -32.32 -15.45
-7.80 -438.04 -326.91 -209.79
-7.62
-2.11 -66.16 -33.28 -14.50
-7.79 -457.19 -339.03 -214.87
1.92
2.04
6.89
2.96
1.59
1.42 -12.35
-3.91
-5.17
0.34
1.51
7.24
4.18
2.20
1.42
-4.91
2.04
-0.20
0.54
1.47
5.67
5.55
3.01
2.37 -50.06 -42.01 -30.40
-0.59
1.45 10.07
8.00
6.87
4.32 -37.91 -28.31 -15.51
Low Cross Section Dependence: i i:i:d:U [ 0:3; 0:7]
-108.15 -107.89 -171.34 -168.69 -169.18 -168.50 -325.46 -321.39 -321.04
-7.80
-3.68 -42.94 -23.58 -11.97
-6.50 -140.41 -75.11 -45.60
-7.44
-2.92 -43.64 -23.37 -11.12
-6.08 -147.90 -77.45 -46.13
-7.33
-2.29 -60.43 -29.10 -13.46
-7.00 -353.16 -251.67 -159.44
-7.26
-2.05 -60.16 -29.79 -12.58
-6.89 -365.60 -254.65 -158.36
1.06
0.53
2.80
-0.67
-1.42
-0.59 -18.63 -11.68 -10.26
-0.41
0.12
3.05
1.22
-0.17
-0.37 -12.05
-5.93
-5.40
0.38
1.27
2.25
3.82
2.44
1.75 -53.49 -43.00 -30.35
-0.89
1.29
6.42
6.01
6.09
3.68 -43.81 -31.09 -18.55
Moderate Cross Section Dependence: i i:i:d:U [ 1; 2]
-85.48 -87.94 -143.05 -141.46 -140.35 -138.30 -281.73 -278.38 -277.75
-18.81 -19.59 -60.73 -50.04 -42.90 -35.16 -202.24 -165.42 -151.49
-15.14 -14.83 -57.18 -44.98 -36.37 -29.22 -204.32 -163.72 -150.64
-5.99
-1.91 -51.31 -24.03 -11.00
-5.72 -289.33 -199.89 -124.44
-5.65
-1.29 -46.85 -21.95
-8.43
-4.37 -281.22 -184.10 -106.67
-6.09
-9.19 -15.90 -19.55 -20.10 -15.63 -50.84 -48.19 -46.00
-6.40
-8.35 -13.94 -16.42 -17.16 -13.89 -45.48 -43.31 -41.70
-0.28
0.67
-3.87
1.04
1.15
0.76 -56.04 -40.74 -28.56
-1.32
1.02
0.77
3.29
4.38
2.87 -43.59 -25.70 -16.02
High Cross Section Dependence: i i:i:d:U [ 1; 4]
-60.98 -64.34 -119.29 -118.17 -116.32 -113.69 -374.01 -371.82 -372.45
-26.18 -29.00 -83.57 -77.34 -72.34 -64.51 -459.56 -434.21 -438.01
-20.73 -22.36 -76.68 -68.73 -62.60 -54.55 -466.73 -441.78 -447.79
-4.25
-1.22 -41.29 -19.10
-8.88
-4.72 -453.74 -344.77 -244.99
-3.57
-0.52 -33.56 -14.40
-4.84
-2.42 -436.91 -313.41 -198.66
-8.64 -13.43 -17.11 -26.10 -29.33 -25.59 -79.86 -95.60 -100.75
-8.97 -12.56 -15.61 -24.64 -27.43 -24.05 -77.14 -94.71 -100.97
1.29
1.22 11.12
9.72
5.95
3.11
-5.01
2.03
0.03
-0.78
0.87 12.32
8.56
6.71
3.56
4.83 18.18 17.69
TE
FE
DIF1
DIF2
DIFX1
DIFX2
SYS1
SYS2
SYSX1
SYSX2
; in the Case with
CR
= 0:2
100
200
US
50
DM
AN
Te st,N
IPT
ACCEPTED MANUSCRIPT
400
-388.59
-26.46
-27.85
-116.42
-118.03
-1.46
0.54
-13.39
-2.08
-319.85
-24.33
-24.42
-87.67
-85.55
-6.72
-3.88
-15.23
-6.48
-274.77
-136.35
-135.37
-68.86
-55.71
-48.00
-42.90
-16.86
-5.77
-366.96
-414.08
-427.26
-148.10
-106.04
-114.51
-113.50
1.38
22.97
AC
C
Notes: See notes to Table 1. FE is the …xed e¤ects estimator, DIF1 and DIF2 are the Arellano and
Bond (1991) one-step and two-step …rst di¤erenced GMM (DIF) estimators, respectively. DIFX1 and
DIFX2 are the one-step and two-step DIF estimators, respectively, which are based on the instruments
consisting of subsets of Xi only. SYS1 and SYS2 are the Blundell and Bond (1998) one-step and two-step
system GMM (SYS) estimators, respectively. SYSX1 and SYSX2 are the one-step and two-step SYS
estimators, respectively, which are based on the instruments consisting of subsets of Xi only.
27
IPT
ACCEPTED MANUSCRIPT
Table 3: Root Mean Square Errors ( 1000) of Fixed E¤ects and GMM estimators for
; in the Case with Predetermined Regressors with T = 5
FE
13.70
DIF1
4.26
DIF2
4.72
DIFX1 6.38
DIFX2 7.27
SYS1
3.78
SYS2
3.95
SYSX1 6.57
SYSX2 7.05
FE
11.92
DIF1
7.20
DIF2
6.34
DIFX1 5.14
DIFX2 5.11
SYS1
7.18
SYS2
6.26
SYSX1 5.23
SYSX2 4.87
9.38
7.95
6.24
3.80
2.86
9.28
7.63
4.18
2.93
EP
FE
DIF1
DIF2
DIFX1
DIFX2
SYS1
SYS2
SYSX1
SYSX2
See Notes to Table 2.
AC
C
50
= 0:8
100
200
= =0
159.49 153.85 152.75
78.33 29.52 12.20
98.83 37.00 14.03
315.62 204.34 106.17
362.43 230.90 115.28
8.64
5.38
3.16
9.23
5.41
2.68
41.18 32.94 22.05
47.67 37.96 23.18
i:i:d:U [ 0:3; 0:7]
110.86 105.91 104.80
44.25 19.17
9.75
53.34 21.98 10.06
219.80 133.09 71.42
248.84 143.90 74.07
6.91
5.07
3.37
7.32
4.65
2.64
33.21 29.45 20.42
36.72 29.89 20.00
i:i:d:U [ 1; 2]
i
93.31 89.68 88.77
98.86 83.78 81.96
102.60 81.07 77.90
176.18 98.94 52.60
186.10 97.06 47.46
22.59 22.96 24.48
20.06 19.19 19.07
32.33 27.07 18.39
32.23 21.75 14.97
i:i:d:U [ 1; 4]
185.46 183.37 184.96
364.79 357.52 380.38
378.32 369.79 395.92
362.81 233.76 141.53
370.67 223.07 119.02
59.51 67.97 79.84
57.14 64.93 74.88
43.40 34.61 28.51
46.88 34.68 26.44
i
28
400
151.86
5.26
5.66
52.48
54.88
1.66
1.26
13.32
12.41
CR
= 0:5
400
50 100 200 400
No Cross Section Dependence:
14.24 14.27 13.75 37.79 35.41 35.12 34.69
1.98 1.09 0.47 9.06 4.18 1.81 0.92
2.34 1.18 0.50 11.00 4.82 1.97 0.97
3.48 1.91 0.89 22.98 11.60 5.92 3.16
3.91 1.99 0.91 27.12 12.61 6.30 3.23
1.84 0.94 0.48 4.82 2.48 1.23 0.65
1.84 0.88 0.39 5.14 2.50 1.18 0.58
3.44 1.83 0.95 15.09 8.64 4.80 2.58
3.66 1.80 0.84 16.82 9.33 4.83 2.49
Low Cross Section Dependence: i
12.54 12.43 12.12 32.16 30.03 29.56 29.09
1.99 1.13 0.57 8.33 4.04 1.94 1.16
2.29 1.17 0.56 9.97 4.47 2.02 1.11
3.17 1.70 0.81 20.57 9.96 5.19 2.73
3.53 1.74 0.83 24.14 10.70 5.50 2.74
1.95 1.04 0.63 4.58 2.55 1.44 0.95
1.88 0.95 0.50 4.88 2.41 1.25 0.71
3.19 1.65 0.86 13.60 7.63 4.27 2.29
3.31 1.59 0.77 14.81 8.02 4.26 2.19
Moderate Cross Section Dependence:
10.98 10.77 10.63 27.41 25.10 24.52 23.50
5.57 4.98 4.40 18.59 14.22 12.47 10.62
4.24 3.44 2.91 16.57 11.09 8.86 7.00
2.54 1.32 0.65 17.80 8.07 4.06 2.12
2.40 1.15 0.56 17.92 7.35 3.74 1.78
6.56 5.99 5.53 12.54 11.26 10.69 10.06
4.70 3.90 3.38 10.61 7.93 6.74 5.93
2.60 1.31 0.67 11.61 6.48 3.54 1.91
2.25 1.06 0.53 10.82 5.52 2.95 1.53
High Cross Section Dependence: i
8.73 8.62 8.33 24.88 22.38 21.96 20.60
6.81 6.48 6.03 28.58 24.28 23.09 19.77
4.69 4.26 3.92 24.40 19.06 17.33 14.08
1.81 0.95 0.47 15.21 6.73 3.29 1.74
1.30 0.62 0.30 11.89 4.63 2.31 1.05
9.28 9.10 8.47 19.20 18.75 18.68 17.70
6.62 6.10 5.55 16.35 14.20 13.37 12.27
2.02 1.02 0.52 10.96 6.02 3.22 1.68
1.27 0.60 0.29 8.95 4.09 2.08 1.00
US
FE
15.45
DIF1
4.42
DIF2
5.03
DIFX1 7.01
DIFX2 8.05
SYS1
3.91
SYS2
4.11
SYSX1 7.09
SYSX2 7.79
= 0:2
100 200
DM
AN
50
TE
Test,N
103.60
5.98
5.60
33.30
33.74
2.46
1.71
12.20
10.27
87.15
72.86
66.73
25.59
21.20
24.15
18.60
12.57
8.52
180.72
358.02
376.19
69.36
48.29
81.53
76.46
22.76
17.26
IPT
ACCEPTED MANUSCRIPT
Table 4: Homogeneity Error Cross Section Dependence Tests and Estimates of Employment Equation, 140 Firms with 9-Year Observations
(79)
m1
m2
DDIF 2
-4.46
-0.17
26:84
(28)
US
CR
A: Two-Step DIF Estimator, 1976-84
Estimation Results
Based on Zi
Based on ZXi
Coef. Std.Err. Coef. Std.Err.
yi;t 1
0.679
(0.084)
0.401
(0.124)
wit
-0.720
(0.117)
-0.551
(0.130)
wit 1
0.463
(0.111)
0.347
(0.112)
0.454
(0.101)
0.447
(0.110)
it
-0.191
(0.086)
-0.079
(0.105)
it 1
cons
0.005
(0.017)
0.003
(0.014)
Test Results
statistics p-values statistics p-values
Sargan
88:8
[0.211]
62:0
[0.140]
(51)
[0.000]
[0.866]
[0.527]
-2.19
-0.47
-
[0.028]
[0.641]
-
DM
AN
B: Two-Step SYS Estimator, 1976-84
Estimation Results
Based on Z+
Based on Z+
i
Xi
Coef. Std.Err. Coef. Std.Err.
yi;t 1
0.873
(0.044)
0.825
(0.071)
wit
-0.780
(0.116)
-0.717
(0.105)
wit 1
0.527
(0.168)
0.560
(0.149)
0.470
(0.071)
0.395
(0.088)
it
-0.358
(0.072)
-0.253
(0.092)
it 1
cons
0.948
(0.390)
0.720
(0.402)
Test Results
statistics p-values statistics p-values
Sargan
111:6
[0.201]
77:3
[0.142]
(100)
-5.81
-0.15
34:30
(35)
(65)
[0.000]
[0.883]
[0.502]
TE
m1
m2
DSY S2
-5.19
-0.12
-
[0.000]
[0.906]
-
AC
C
EP
Notes: The estimated model is yit = i + t + 1 yit 1 + 0 wit + 1 wit 1 + '0 it + '1 it 1 + uit ,
where yit is the log of the number of employees of company i, wit is the log of real product wage and
it is the log of gross capital stock. Year dummies are included in all speci…cations. The standard
errors reported are those of the robust one-step GMM estimator. The …rst row of the test results
reports Sargan’s statistic for overidentifying restrictions. m1 and m2 are the …rst-order and secondorder serial correlation tests in the …rst-di¤erenced residuals. DDIF 2 denotes Sargan’s Di¤erence test
for heterogeneous error cross section dependence based on the two-step Arellano and Bond (1991) DIF
GMM estimator. DSY S2 denotes Sargan’s Di¤erence test based on the two-step Blundell and Bond
(1998) SYS GMM estimator. Sargan test and Sargan’s di¤erence test are distributed as 2 under the
null with degrees of freedom reported in parentheses. Instruments used in each equation are for DIF:
yi;t 2 ; yi;t 3 ; :::; yi;0 ;wi;t 2 ; wi;t 3 ; :::; wi;0 ; i;t 2 ; i;t 3 ; :::; i;0 , and for SYS: yi;t 1 , wi;t 1 ,
i;t 1 .
+
Zi , ZXi , Z+
i , ZXi are de…ned by (22), (18), (37), (31) but without cross-sectional demeaning, respectively.
29
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