A Bias-Adjusted LM Test of Error Cross Section Independence∗
M. Hashem Pesaran†
Cambridge University & USC
Aman Ullah‡
University of California, Riverside
Takashi Yamagata§
Cambridge University
First Version May 2006, This Version July 2007
Abstract
This paper proposes a bias-adjusted version of Breusch and Pagan (1980) Lagrange
multiplier (LM) test statistic of error cross section independence, in the case of panel models
with strictly exogenous regressors and normal errors. The exact mean and variance of the
test indicator of the Lagrange multiplier (LM) test statistic are provided for the purpose of
the bias-adjustments. It is shown that the centering of the LM statistic is correct for fixed
T and N . Importantly, the proposed bias-adjusted LM test is consistent even when the
Pesaran’s (2004) CD test is inconsistent. Also an alternative bias-adjusted LM test, which
is consistent under local error cross section dependence of any fixed order p, is proposed.
The finite sample behavior of the proposed tests are investigated and compared to that of
the LM and CD tests. It is shown that the bias-adjusted LM tests successfully control the
size, maintaining satisfactory power in panel with exogenous regressors and normal errors.
However, it is also shown that the bias-adjusted LM test is not as robust as the CD test to
non-normal errors and/or in the presence of weakly exogenous regressors.
JEL-Classification: C12, C13, C33
Keywords: Cross Section Dependence, Spatial Dependence, LM test, Panel Model, Bias-adjusted Test
∗
We would like to thank the participants of seminar at USC for their helpful comments. We are particularly
grateful to two anonymous referees for their suggestions and useful comments. Pesaran and Yamagata gratefully
acknowledge the financial supports from the ESRC (Grant No. RES-000-23-0135). Ullah gratefully acknowledges
the financial supports from the Academic Senate, UCR.
†
Faculty of Economics, University of Cambridge, Cambridge, CB3 9DD, UK. mhp1@cam.ac.uk.
‡
Department of Economics, University of California, Riverside, CA, 92521, USA. aman.ullah@ucr.edu
§
CFAP, Judge Business School, University of Cambridge, Cambridge, CB2 1AG, UK. ty228@cam.ac.uk
1
1
Introduction
A number of different approaches already exist for testing cross section independence in panel
data models. An early contribution is due to Moran (1948) who provides a test of spatial
independence in the context of a pure cross section model. Further developments of Moran’s
test are reviewed in Anselin (1988, 2001). This approach depends on the choice of the spatial
matrix, and may not be appropriate for many panels in economics and finance where space is
not a natural metric for modelling of cross section dependence. An alternative procedure would
be to use the Lagrange multiplier (LM) correlation test of Breusch and Pagan (1980), which
does not require a priori specification of a spatial matrix. The LM test is based on the average
of the squared pair-wise correlation coefficients of the residuals and is applicable in the case of
panel data models where the cross section dimension (N) is small relative to the time dimension
(T ), and where Zellner’s (1962) seemingly unrelated regression equation (SURE) method can
be used. But the LM test exhibit substantial size distortion when N is large relative to T , a
situation frequently encountered in empirical applications. (Pesaran, 2004).
2
Frees (1995) has proposed a version of the Breusch and Pagan LM test, RAV
E , based on
squared pair-wise Spearman rank correlation coefficients which is applicable to panel data models where N is large relative to T . However, Frees only provides the distribution of the test in
the case of models with only one regressor (intercept), and its generalization for models with
2
additional explanatory variables is not known. For example, the mean of RAV
E appearing in
Corollary 1 and Theorem 2 of Frees (1995) may not be valid for the models with explanatory
variables.1 Frees (1995) also proposes tests based on average pair-wise sample correlations of
the series across the different cross section units. His RAV E test statistic is based on Spearman
rank correlations, and his CAV E test statistic is based on Pearson rank correlations. The latter
is closely related the CD test also considered in Pesaran (2004).
Pesaran (2004) shows that unlike the LM test statistic, the CD statistic has exactly mean
zero for fixed values of T and N, under a wide class of panel data models, including heterogeneous dynamic models subject to multiple breaks in their slope coefficients and error variances,
so long as the unconditional means of yit and xit are time-invariant and their innovations are
symmetrically distributed. However, the CD test has an important drawback; namely it will lack
power in certain situations where the population average pair-wise correlations is zero, although
the underlying individual population pair-wise correlations are non-zero. This could arise, for
example, where under the alternative hypothesis cross dependence can be characterized as a
factor model with mean zero factor loadings (Pesaran, 2004, p.14).
In this paper we propose bias-adjusted versions of the LM tests, which use the exact mean
and variance of the LM statistic in the case of panel data models with strictly exogenous
regressors and normal errors. The adjustments are obtained using the results in Ullah (2004),
so that the centering of the LM statistic is correct for fixed T and N. Importantly, it will be
shown that the bias-adjusted test is consistent even when the cross section mean of the factor
loadings is near zero, under which Pesaran’s CD test is not consistent.
In cases where the cross section units can be ordered a priori, as with spatial observations,
the proposed LM tests might not be sufficiently powerful as they do not exploit the spatial information. To deal with this problem, following to Pesaran (2004), we also propose a generalization
of the bias-adjusted LM tests, which capture the spatial patterns.
The finite sample behavior of the bias-adjusted LM test is investigated by means of Monte
Carlo experiments, and compared to that of the (non-bias-adjusted) LM test, as well as to
the CD test. It will be shown that the bias-adjusted LM test successfully controls the size,
maintaining reasonable power in panels with exogenous regressors and normal errors, even when
1
2
In fact using Monte Carlo experiments we found that the uncorrected version of the RAV
E test tends to
behave similarly to the uncorrected version of the Breusch and Pagan LM test when N is large for models with
explanatory variables. These results are available from the authors upon request.
2
cross section mean of the factor loadings is close to zero, where the CD test has little power. Also
their spatial versions perform similarly in the case of spatial cross section dependence. However,
it is shown that the bias-adjusted LM test is not as robust as the CD test to non-normal errors
and/or in the presence of weakly exogenous regressors.
The plan of the paper is as follows. Section 2 presents the panel data model and the Breusch
and Pagan LM test and formulate the bias-adjusted LM tests. Section 3 discuses Pesaran’s
(2004) CD test and its potential inconsistency. Section 4 reports the results of the Monte Carlo
experiments. Section 5 provides some concluding remarks.
2
Model and Tests
Consider the following panel data model
yit = βi xit + uit , for i = 1, 2, ..., N; t = 1, 2, ..., T,
(1)
where i indexes the cross section dimension and t the time series dimension, xit is a k × 1 vector
of strictly exogenous regressors with unity on its first row. The coefficients, βi , are defined on
a compact set and allowed to vary across i. For each i, uit ∼ IIDN(0, σ2ui ), for all t, although
they could be cross-sectionally correlated. We first review the Breusch and Pagan (1980) test.
2.1
Breusch and Pagan’s Test of Cross Section Independence
In the SURE context with N fixed and as T → ∞, Breusch and Pagan (1980) proposed a
Lagrange multiplier (LM) statistic for testing the null of zero cross equation error correlations
which is particularly simple to compute and does not require the system estimation of the SURE
model. The test is based on the following LM statistic
LM = T
N
−1
N
ρ̂2ij ,
(2)
i=1 j=i+1
where ρ̂ij is the sample estimate of the pair-wise correlation of the residuals. Specifically,
ρ̂ij = ρ̂ji = T
T
2
t=1 eit
t=1 eit ejt
1/2 T
2
t=1 ejt
1/2 ,
(3)
and eit is the Ordinary Least Squares (OLS) estimate of uit defined by
eit = yit − β̂i xit ,
(4)
with β̂i being the estimates of βi computed using the OLS regression of yit on xit for each i,
separately. This LM test is generally applicable and does not require a particular ordering of
the cross section units. In this setting where uit ∼ IIDN(0, σ2ui ) it can be shown that under
the null hypothesis
H0 : Cov (uit , ujt ) = 0, for all t and i = j,
(5)
the LM statistic is asymptotically distributed as chi-squared with N(N −1)/2 degrees of freedom.
However, the LM test is likely to exhibit substantial size distortions for N large and T small, a
situation that can frequently arise in empirical applications.
Recently Ullah (2004) provides unified techniques to obtain the exact and approximate
moments of econometric estimators and test statistics. We make use of this approach to correct
for the small sample bias of the LM statistic.
3
2.2
Finite Sample Adjustments
To obtain the bias-adjusted LM test we make the following assumptions:
Assumption 1: For each i, the disturbances, uit , are serially independent with the mean 0
and the variance, 0 < σ 2i < ∞.
Assumption 2: Under the null hypothesis defined by H0 : uit = σi εit , εit ∼ IIDN(0, 1)
for all i and t.
Assumption 3: The regressors, xit , are strictly exogenous such that E(uit |Xi ) = 0, for all
i and t, where Xi = (xi1 , . . . , xiT ) , and Xi Xi is a positive definite matrix.
Now we introduce the following idempotent matrix of rank T − k,
Mi = IT − Hi ;
Hi = Xi (Xi Xi )−1 Xi ,
(6)
such that T r(Mi ) = T −k, where IT is an identity matrix of order T . Similarly Mj = IT −Hj is
the same as Mi with Xi replaced by Xj , and T r(Mj ) = T − k. Then we can state the following
theorem.
Theorem 1 : Consider the panel data model (1), and suppose that Assumptions 1-3 hold.
Then the exact mean and variance of (T − k)ρ̂2ij are, respectively, given by
µT ij = E (T − k) ρ̂2ij =
and
υ 2T ij
where
1
T r [E (Mi Mj )]
T −k
2
2
= V ar (T − k) ρ̂ij = {T r [E (Mi Mj )]} a1T + 2T r E (Mi Mj )2
a1T = a2T −
1
(T − k − 8) (T − k + 2) + 24
2 , a2T = 3 (T − k + 2) (T − k − 2) (T − k − 4)
(T − k)
(7)
a2T ,
(8)
.
(9)
2
Proof is given in Appendix A.2.
Using (7) and (8), the bias-adjusted LM test statistic is defined as
LMadj =
N−1
N
(T − k)ρ̂2ij − µT ij
2
.
N(N − 1)
υT ij
(10)
i=1 j=i+1
Under Assumptions 1-3, and assuming that H0 defined by (5) holds, first T → ∞ then N → ∞
we have
LMadj →d N(0, 1).
(11)
Remark 2 For a fixed T (> k + 8), it is easily seen that LMadj has exactly mean zero,
√ unlike
the LM test. However, even under the normality assumption, the covariance between T − kρ̂2ij
√
and T − kρ̂2ik for j = k does not disappear unless T → ∞. Therefore, for (11), the sequential
asymptotic, where T → ∞ first followed by N → ∞, is required.
2.3
Tests for Local Cross Section Independence
The power of the LM test is adversely affected when the dependence under the alternative
hypothesis is spatial.2 The spatial dependence of the errors can be modelled using the spatial
weight matrix, W = (wij ), which is applied to a particular ordering of the cross section units.
It is often convenient to order the cross section units by their topological position, so that the
pth order neighbors of the ith cross section unit can be defined as the i + p and the i − p cross
section units.
2
See Pesaran (2004).
4
For such spatial alternatives the following pth order bias-adjusted LM test can be used
p
LM(p)adj =
N−s
(T − k)ρ̂2i,i+s − µT i,i+s
2
,
p (2N − p − 1)
υ T i,i+s
(12)
s=1 i=1
where
µT i,i+s =
3
T r [E (Mi Mi+s )]
, υT i,i+s = {T r [E (Mi Mi+s )]}2 a1T + 2T r E (Mi Mi+s )2
T −k
a2T .
The CD Test and its Potential Inconsistency
To deal with the large N bias of the LM test, Pesaran (2004) suggests using the CD statistic
defined by


N−1
N
2T

(13)
CD =
ρ̂ij  .
N(N − 1)
i=1 j=i+1
He shows that under H0 and for T sufficiently large, CD →d N(0, 1) as N → ∞.3 Unlike the
LM test statistic, the above statistic has exactly mean zero for fixed values of T and N, under
a wide class of panel data models, including heterogeneous dynamic models subject to multiple
breaks in their slope coefficients and error variances, so long as the unconditional means of yit
and xit are time-invariant and their innovations are symmetrically distributed.
However, as pointed out in Pesaran (2004), the CD test need not be consistent for alternatives of interest. To see this, consider the following alternatives
H1 : uit = γ i ft + εit ,
(14)
where γ i ∼ IID 0, σ2γ , 0 < σ2γ < ∞, and the factors ft ∼ IID (0, 1) are unobserved common
effects. Also assume that E ft4 = µf 4 with 0 < µf 4 < ∞, εit ∼ IIDN(0, 1), and E (εit fs ) = 0
for all i, t, and s. Under H1
Cov (uit , ujt ) = E (γ i ) E γ j ,
and the CD test statistic is centred at 0 if E (γ i ) = 0, even when γ i = 0 for some i and ft = 0.
Therefore, under the alternatives with E(γ i ) = 0 the power of the CD test would not increase
with N. But, the power of the LM type tests involves the terms
Cov u2it , u2jt = E γ 2i E γ 2j µf 4
(15)
which continue to differ from zero even when E(γ i ) = 0. Hence, the power of LM type tests will
increase with N even under alternatives with E(γ i ) = 0. These results will continue to hold
under multi-factor alternatives.
Under the alternative hypothesis of a pth order local dependence, Pesaran (2004) proposes
a pth order generalization of the CD test defined by
p N−s
2T
CD (p) =
ρ̂
.
(16)
p (2N − p − 1) s=1 i=1 i,i+s
3
The CD test is based on a less restrictive version of Assumption 2, and only requires the errors, εit , to be
symmetrically i.i.d. distributed around zero with a unit variance.
5
4
Finite Sample Behavior of the Tests of Cross Section Independence
In this section we investigate the finite sample behavior of alternative tests of cross section
independence by means of Monte Carlo experiments. We shall focus on our proposed biasadjusted LM test, LMadj , which is defined by (10), and compare its performance to that of
the LM test and the CD test, which are defined by (2) and (13), respectively. Initially we
consider the experiments in panels with exogenous regressors. We shall also investigate the
small sample properties of the proposed test in the case of dynamic panels with and without
parameter heterogeneity and structural breaks. Recall that the bias-adjusted LM test need not
be valid for dynamic panels, and it would be interesting to see how it performs as compared to
the CD test. Finally, we provide small sample evidence on the performance of the tests against
spatial alternatives.
4.1
Experimental Designs
Initially, we consider the data generating process (DGP) specified as
yit = αi +
k
xit β i + uit , i = 1, 2, ..., N; t = 1, 2, ..., T,
(1)
=2
where αi ∼ IIDN (1, 1), β i ∼ IIDN (1, 0.04). The covariates are generated as
xit = 0.6xit−1 + vit , i = 1, 2, ..., N; t = −50, ..., 0, ..., T ; ) = 2, 3, ..., k
(2)
with xi,−51 = 0 where vit ∼ IIDN(0, τ 2i / 1 − 0.62 ), τ 2i ∼ IIDχ2 (6) /6. The disturbances
are generated as
uit = c(γ,k) (γ i ft + σ i εit ) , i = 1, 2, ..., N; t = 1, 2, ..., T,
where ft ∼ IIDN (0, 1), and σ2i ∼ IIDχ2 (2) /2. The idiosyncratic errors, εit , are generated under twodifferent
schemes, (i) normal errors, IIDN (0, 1), and (ii) chi-squared errors,
√
IID χ2 (1) − 1 / 2. The latter is to check the robustness of the tests to non-normal errors.
The values of αi , xit , σ2i are drawn for each i = 1, 2, ..., N, and then fixed across replications.
Under the null hypothesis we have γ i = 0 for all i, and under the alternatives we consider
(i) γ i ∼ IIDU[0.1, 0.3],
(ii) γ i ∼ IIDN (0, 0.1) ,
where under (ii), the CD test is inconsistent, as shown above. In order to examine the effects
of changing the number of regressors, k = 2, 4, 6 are considered. Meanwhile the same average
population explanatory power of each cross section regression and the same degree of error cross
section correlation are to be maintained for all k. To this end, c2(γ,k) is set
c2(γ,k)

 1.04(k − 1) for γ i = 0
12.48
=
(k − 1), for γ i ∼ IIDU[0.1, 0.3],
 12.13
10.4
(k
− 1), for γ i ∼ IIDN (0, 0.1) ,
11.0
so that R̄2 = 0.5 across experiments, where R̄2 = E(σ 2ui )/V ar(yit ) with σ 2ui = V ar(uit ) and
V ar(yit ) = (k − 1)E(β 2i ) + E(σ2ui ).
For examining the power of the first order cross section independence tests, the DGP defined
by (1) for k = 2 but with spatially correlated errors are considered:
uit = λ (0.5ui−1,t + 0.5ui+1,t ) + σi εit ,
6
(3)
with end points set at u1t = u2t + ε1t and uN t = uN−1t + εNt , where σ2i ∼ IIDχ2 (2) /2,
εit ∼ IIDN(0, 1). For this DGP, the finite sample performance of the spatial version of the
tests, defined by (16) and (12), are examined in the case of p = 1, and for the values of λ = 0,
−0.1 and 0.1.
In the case of dynamic models, following Pesaran (2004), five specifications are considered.
The first is the heterogeneous first order autoregressive (AR(1)) panel data model:
yit = µi (1 − β i ) + β i yi,t−1 + uit ,
(4)
uit = γ i ft + σit εit , i = 1, 2, ..., N; t = −50, −49, ..., T,
with yi,−51 = 0. The idiosyncratic errors, εit , are generated under two
as
√
2different schemes
above, (i) normal errors, IIDN(0, 1), and (ii) chi-squared errors, IID χ (1) − 1 / 2. Here
we focus on the heterogeneous slope experiments where β i ∼ IIDU[0, 1). The fixed effects, µi ,
are drawn as εi0 + ηi , with ηi ∼ IIDN(1, 2), thus allowing for the possibility of correlations
between fixed effects and the initial values, yi0 . γ i , σit = σ i , and ft are generated in the same
manner as specified for the DGP with exogenous regressors. The parameters η i , β i and σit are
fixed across replications.
For examining the empirical size of the tests in the case of structural break(s), two specifications are considered. The first dynamic DGP is subject to single break, specified as (4)
except µi ∼ √
IIDN (1, 1), β it = β t = 0.6 for t = −50, ..., T /2, β t = 0.8 for t = T /2 + 1, ..., T ;
σit = σt = 1.5 for t = −50, ..., T /2, σ t = 1 for t = T /2 + 1, ..., T , and εit ∼ IIDN (0, 1).
The second dynamic DGP is subject to multiple breaks, specified as (4) except β it = 0.5 for
t = −50, ..., 0 and all i, β it ∼ IIDU[0, 1) for t = 1, ..., T , i = 1, ..., N; σ2it ∼ IIDχ2 (2)/2 for
t = −50, ..., T , i = 1, ..., N. For both designs, the first 50 observations are discarded.
Finally, the DGP subject to unit root, which is specified as (4) except β it = β = 1 for all i
and t, σ2it ∼ IIDχ2 (2)/2, is considered.
The test statistics are computed using the OLS residuals from the individual regressions.
For all experiments the combinations of sample sizes N = 10, 20, 30, 50, 100, 200 and T =
20, 30, 50, 100 are considered. The nominal size of the tests is set at the 5% significance level.
All experiments are based on 2000 replications.
4.2
Monte Carlo Outcomes
Table 1 reports the size of the tests for the DGP with different number of exogenous regressors
(k = 2, 4, 6) and normal errors. As shown in Pesaran (2004), the CD test has the correct size,
and the LM test severely over-rejects the null particularly for N ≥ T . In contrast the biasadjusted LM test successfully controls the size for all combinations of N and T , except when
both k and N are large and T small. In the case of γ i ∼ IIDU[0.1, 0.3], whose results are
reported in Table 2, the bias-adjusted LM test and the CD test seem to have reasonable power.
In the case of γ i ∼ IIDN (0, 0.1), whose results are reported in Table 3, as theory predicts the
CD test has little power. The power of the CD test increases with T very slowly, but it does
not increase with N for given T . This is because the sample average of factor loadings for finite
N cannot be different from zero far enough for the test to reject the null, for some replications,
and the precision of this happening increases as T rises. On the other hand, the bias-adjusted
LM test maintains reasonable power under the same design. Overall, the powers of both the
CD and the bias-adjusted LM test increase faster with N than T , and it seems that the number
of regressors does not affect the power of these tests
much4 .
√
The results for the case with IID χ2 (1) − 1 / 2 errors are given in Table 4, and show
that the bias-adjusted LM test is generally not as robust to non-normal errors as the CD test.
It tends to over-reject (moderately) for all combinations of N and T .
4
It is worth emphasizing that we keep the population goodness of fit of the model fixed for different number
of exogenous regressors in our Monte Carlo experiments.
7
Table 5 summarizes the results of the spatial first order tests. The bias-adjusted LM test,
LM(1)adj , which defined by (12), successfully controls the size. The CD(1) test has the correct
size, which is consistent to the results in Pesaran (2004). Both the bias-adjusted LM test and
the CD(1) test seem to have reasonable power under the alternatives, where λ = ±0.1.
Tables 6 to 10 provide the results for the various dynamic DGPs. For all experiments, the
CD test has the correct size and the LM test severely over-rejects the null when N ≥ T . This
is to be expected, as discussed above, since for small T relative to N, the mean approximation
of ρ̂2ij ≈ 1 will not be correct, and with N large the incorrect centering of the test indicator is
likely to be accentuated, resulting in size distortions that tend to get worse with N.
Unlike in the case of DGP with exogenous regressors and normal errors, in the case of
heterogenous dynamic AR(1) specifications with IIDN (0, 1) errors (Table 6), the bias-adjusted
LM test tends to over-reject when N is much larger than T . With respect to the power, as
was in the case of DGP with exogenous regressors,
the
√CD test has little power in the case of
γ j ∼ IIDN (0, 0.1). The results for IID χ2 (1) − 1 / 2 errors (Table 7) are similar to those
in Table 4. For the DGP with a single structural break (Table 8), the bias-adjusted LM test
rejects the null too often. For example, when N = 200, the estimated size of the LMadj test is
100% for all T . In the case of multiple structural breaks (Table 9), the bias-adjusted LM test
tends to over-reject, especially for N ≥ T . Finally, in the case of models with unit roots (Table
10) the bias-adjusted LM tests tends to over-reject, with the extent of over-rejection increasing
with N.
5
Concluding Remarks
This paper has proposed a bias-adjusted LM test of cross section independence. For the biasadjustment, we derived the exact mean and variance of the test indicator of the LM statistic in
the case of the model with strictly exogenous regressors and normal errors, based on the work
in Ullah (2004), so that the centering of the LM statistic is correct for fixed T and large N.
Importantly, the proposed bias-adjusted LM test is consistent even when the Pesaran’s (2004)
CD test is inconsistent. Small sample evidence based on Monte Carlo experiments suggests
that the bias-adjusted LM test successfully controls the size, maintaining reasonable power in
panels with exogenous regressors and normal errors, even when cross section mean of the factor
loadings is close to zero, where the CD test has little power. Also their spatial versions perform
similarly in the case of spatial cross section dependence. However, it is shown that the biasadjusted LM tests are not as robust as the CD test to non-normal errors and/or in the presence
of weakly exogenous regressors.
Clearly, it would be worth deriving the mean and variance of the LM test statistic in the case
of dynamic models, and in the case where the errors are non-normal. This will be the subject
of our future research.
8
A
A.1
Appendix
Evaluation of the First Two Derivatives of E (W −r )
Let us consider a quadratic form W = u Mu, where the T × 1 vector u ∼ N (µ, IT ) and M is an idempotent
matrix of rank m ≤ T . Then W is distributed as a non-central chi-squared distribution with the non-centrality
parameter θ = µ Mµ/2. When
µ = 0, hence θ = 0,
W is distributed as a central chi-square distribution. In
what follows we evaluate dE W −r and dd E W −r , where d = µ + ∂/∂µ and r = 1, 2, ....
Now
−r −r −r ∂
∂ −r dE W
= µ +
E W
=µE W
+
E W
µ M,
(A.1)
∂µ
∂θ
dd E W −r
−r ∂
∂
µ E W −r +
E
W
∂µ
∂µ
−r −r ∂ −r E W
+ µµ E W
+
E W
µµ M + Mµµ + M
∂θ
2
−r ∂
+
E
W
Mµµ M,
∂θ2
=
=
where we use
∂
E W −r
∂µ
∂2
E W −r
∂µ∂µ
=
=
µ+
∂ −r E W
Mµ,
∂θ
2
−r ∂ −r ∂
∂ −r ∂
E
W
Mµµ
M
+
E W
µM =
E
W
M.
∂µ ∂θ
∂θ
∂θ2
First we note from Ullah (1974, p.147) that for m > 2r
i
∞
−r Γ m
1 −θ
2 −r+i θ
E W
=
e
, when θ = 0
2r
i!
Γ m
2 +i
i=0
m
1
1 Γ 2 −r
=
, when θ = 0.
=
2r Γ m
(m
−
2)
(m
−
4) ... (m − 2r)
2
(A.2)
(A.3)
Further, for s = 1, 2, ...,
∂s
EW −r
∂θs
=
=
∞
Γ m
− r + i θi
1 Γ(r + s) −θ
2
, when θ = 0
(−1) r
e
2
Γ (r)
Γ m
+ 1s + i i!
2
i=0
m
s 1 Γ(r + s) Γ 2 − r
, when θ = 0.
(−1) r
2
Γ (r) Γ m
2 +s
s
(A.4)
−r Substituting
and
−r r = 1, 2 and s = 1, 2 in (A.3) and (A.4), and using these results in (A.1) we get d E W
dd E W
for θ = 0. When µ = 0, hence θ = 0, we obtain
−r −r −r ∂ −r dE W
= 0 , dd E W
=E W
+
E W
M,
(A.5)
∂θ
where ∂E W −r /∂θ is given by (A.4) for s = 1 and θ = 0.
A.2
Proof of Theorem 1
From (3) and (4)
2
ρ̂2ij =
=
=
(ui Mi Mj uj )
(ui Mi ui ) uj Mj uj
(A.6)
ui Mi Mj uj uj Mj Mi ui
(ui Mi ui ) uj Mj uj
Mi Mj uj uj Mj Mi
ui Aij ui
with Aij =
,
ui Mi ui
uj Mj uj
where Mi and Mj are as defined in (6), ui ∼ N(0, IT ), uj ∼ N(0, IT ). First we consider the case where Xi and
Xj are fixed so that the Mi and Mj are fixed matrices. Taking the iterative expectations on both sides of (A.6)
we can write
E ρ̂2ij = Euj E ρ̂2ij |uj ,
[A.1]
in which
E ρ̂2ij |uj
=
=
=
=
ui Aij ui
uj
ui Mi ui
E ui Aij ui Wi−1 |uj
di Aij di E Wi−1 = T r Aij di di E Wi−1
∂
T r (Aij ) E Wi−1 + µi Aij µi E Wi−1 +
E Wi−1 ×
∂θi
2
−1 ∂
µi Mi Aij µi + µi Aij Mi µi + T r (Aij Mi ) +
E
W
µi Mi Aij Mi µi ,
∂θ2i
E
(A.7)
where we use (A.1), Wi = ui Mi ui , di = µi + ∂/∂µi with µi = Xi βi , the third equality follows by using the
results in Ullah (2004, (2.28)), and θi = µi Mi µi /2. We note that all the expectations on the right hand equalities
in (A.7) are with respect to ui for conditional uj . Now
uj Mj Mi Mj uj
E ρ̂2ij
= E Wi−1 E
+ µi E (Aij ) µi E Wi−1
(A.8)
uj Mj uj
uj Mj Mi Mj uj
∂
+
E Wi−1
2µi E (Aij ) µi + E
∂θi
uj Mj uj
2
∂
+
E Wi−1 µi E (Aij ) µi ,
∂θ2i
where all the right hand expectations, except E(Wi−1 ),are with respect to uj . But for µi = 0
−1 uj Mj Mi Mj uj
uj Mj Mi Mj uj
∂
+
E
W
E
E ρ̂2ij
= E Wi−1 E
i
uj Mj uj
∂θi
uj Mj uj
uj Mj Mi Mj uj
∂
=
E Wi−1 +
E Wi−1 E
.
∂θi
uj Mj uj
Now, writing Bij = Mj Mi Mj ,Wj = uj Mj uj , θj = µj Mj µj /2 with µj = Xj βj and using (A.7),
uj Bij uj
= T r (Bij ) E Wj−1 + µj Bij µj E Wj−1
E
uj Mj uj
∂
+
E Wj−1
2µj Mj Bij µj + T r (Bij Mj )
∂θj
2
−1 ∂
E
W
µj Mj Bij Mj µj ,
+
∂θ2j
which can be written for µj = 0 as
−1 −1 uj Bij uj
∂
E
= E Wj
+
E Wj
T r (Bij ) , T r (Bij ) = T r (Mi Mj ) .
uj Mj uj
∂θj
Substituting (A.11) in (A.9) we get
∂
∂
E ρ̂2ij = E Wi−1 +
E Wi−1
E Wj−1 +
E Wj−1 T r (Mi Mj )
∂θ i
∂θj
=
=
1
2
−
m−2
m (m − 2)
1
T r (Mi Mj ) ,
m2
2
(A.9)
(A.10)
(A.11)
(A.12)
T r (Mi Mj )
and
1
E mρ̂2ij = T r (Mi Mj ) ,
m
where m = T − k and we use the results (A.3) and (A.4) for r = 1 and s = 1.
Next we consider
2 2
ρ̂4ij = ui Aij ui / ui Mi ui ,
and taking the expectations on both sides of (A.14) we get
E ρ̂4ij = Euj E ρ̂4ij |uj ,
[A.2]
(A.13)
(A.14)
in which using Ullah (2004, (2.28))
2
E ρ̂4ij |uj = E ui Aij ui Wi−2 |uj
=
=
=
(A.15)
2 −2 E Wi
di Aij di
di Aij di E Wi−2
di Aij di T r Aij di di E Wi−2
T r Aij di di c(µi ) ,
di Aij di
=
−2 where c(µi ) = T r Aij di di E Wi
, and using (A.7) (see also Ullah (2004, Chapter 2)),
∂
c(µi ) = T r (Aij ) E Wi−2 + µi Aij µi E Wi−2 +
E Wi−2
2µi Aij µi + T r (Aij )
∂θi
2
−2 ∂
+
E
W
µi Aij µi .
i
∂θ 2i
In order to evaluate the term in the last equalities of (A.15) we note that
∂
c (µi ) ,
di c (µi ) = µi +
∂µi
∂
∂
di di c (µi ) = µi +
µi c(µi ) +
c
(µ
)
i
∂µi
∂µi
∂
∂
∂2
= µi µi c (µi ) + µi
c
(µ
)
+
c
(µ
)
+
c
(µ
)
µi +
c (µi ) .
i
i
i
∂µi
∂µi
∂µi ∂µi
Using (A.18) in (A.15) we then get
T r Aij di di c(µi ) = µi Aij µi c (µi ) + T r (Aij ) c (µi )
(A.16)
(A.17)
(A.18)
(A.19)
∂
∂
+ T r Aij µi
c
(µ
)
+
T
r
A
c
(µ
)
µ
ij
i
i
i
∂µi
∂µi
2
∂
+T r Aij
c (µi ) ,
∂µi ∂µi
in which, using (A.16), we can verify that
∂c(µi )
∂µi
=
=
where
∂
∂
T r (Aij )
E Wi−2 Mi µi + 2Aij µi EWi−2 +
E Wi−2 Mi µi µi Aij µi (A.20)
∂θi
∂θi
2
−2 ∂
E Wi
Mi µi 2µi Aij µi + T r (Aij )
+
∂θ 2i
3
−2 ∂
∂
+
E Wi−2 (4Aij µi ) +
E
W
Mi µi µi Aij µi
i
3
∂θ i
∂θi
2
−2 ∂
+ (2Aij µi )
E Wi
∂θ 2i
a1 (θ) (2Aij µi ) + a2 (θ) µi Aij µi Mi µi + a3 (θ)T r (Aij ) Mi µi ,
a1 (θ)
=
a2 (θ)
=
a3 (θ)
=
and
∂2
c (µi )
∂µi ∂µi
∂
∂2 E Wi−2 + 2
E Wi−2 + 2 E Wi−2 ,
∂θi
∂θi
−2 ∂
∂ 2 −2 ∂3 E Wi
+ 2 2 E Wi
+ 3 E Wi−2 ,
∂θi
∂θi
∂θi
∂
∂2 E Wi−2 + 2 E Wi−2 ,
∂θi
∂θi
=
2Aij a1 (θ) + 2Mi µi µi Aij a2 (θ)
+ 2Aij µi µi Mi + µi Aij µi Mi a2 (θ)
+ µi Aij µi Mi µi µi Mi a4 (θ)
+T r (Aij ) Mi a3 (θ) + Mi µµi Mi T r (Aij ) a5 (θ),
[A.3]
(A.21)
where
a4 (θ)
=
a5 (θ)
=
∂ 2 −2 ∂ 3 −2 ∂ 4 −2 E
W
+
2
E
W
+
E Wi
,
i
i
∂θ2i
∂θ3i
∂θ4i
∂3 ∂ 2 −2 + 3 E Wi−2 .
2 E Wi
∂θi
∂θi
In order to evaluate (A.19) for µi = 0 we first note that
∂
c (0) = T r (Aij ) E Wi−2 +
E Wi−2 ,
∂θi
∂
c (µi ) = 0,
∂µi
−2 −2 ∂2
∂
∂ 2 −2 c
(µ
)
=
2A
E
W
+
2
E
W
+
E
W
ij
i
i
i
i
∂µi ∂µi
∂θi
∂θ2i
2
∂
∂
+T r (Aij ) Mi
E Wi−2 + 2 E Wi−2 .
∂θi
∂θi
(A.22)
Therefore, for µi = 0, we can simplify
T r Aij di di c(µi )
=
=
−2 2 −2 ∂
∂ 2 −2 2
(A.23)
[T r (Aij )] + 2T r Aij
E Wi
+2
E Wi
+ 2 E Wi
∂θi
∂θi
∂
∂2 3 [T r (Aij )]2 E Wi−2 + 2
E Wi−2 + 2 E Wi−2 ,
∂θi
∂θi
because [T r (Aij )]2 = T r A2ij . Further, using (A.23), (A.3) and (A.4) in (A.15) we get, for µi = 0,
E ρ̂4ij
= 3E [T r (Aij )]2 a1m
= 3 [T r (Mi Mj )]2 + 2T r (Mi Mj )2 a21m ,
(A.24)
where the right hand expectation is with respect to uj and it is
2 uj Mj Mi Mj uj
2
E [T r (Aij )]
= E
uj Mj uj
=
[T r (Mi Mj )]2 + 2T r (Mi Mj )2 a1m ,
and
a1m =
(m − 8)(m + 2) + 24
.
(m + 2) m (m − 2) (m − 4)
Next, using (A.12) and (A.24), we get
2
V ρ̂2ij
= E ρ̂4ij − E ρ̂2ij
1
= [T r (Mi Mj )]2 3a21m − 4 + 6T r (Mi Mj )2 a21m ,
m
which gives
where
1
V mρ̂2ij = [T r (Mi Mj )]2 bm − 2 + 2bm T r (Mi Mj )2 ,
m
bm = 3
(m − 8)(m + 2) + 24
(m + 2) (m − 2) (m − 4)
2
(A.25)
(A.26)
= 3m2 a21m .
When Xi and Xj are random then the results in (A.12) and (A.25) are the same by replacing T r(Mi Mj ),
T r[(Mi Mj )2 ], [T r(Mi Mj )]2 with their expected values.
[A.4]
References
Anselin, L., (1988). Spatial Econometrics: Methods and Models. Dorddrecht: Kluwer Academic
Publishers.
Anselin, L., (2001). Spatial econometrics, In: B. Baltagi (ed.)
Econometrics, Oxford: Blackwell.
A companion to Theoretical
Breusch, T.S., Pagan, A.R., (1980). The Lagrange multiplier test and its applications to model
specification tests in econometrics. Review of Economic Studies 47, 239-53.
Frees, E.W., (1995). Assessing cross-sectional correlation in panel data. Journal of Econometrics 69, 393-414.
Moran, P.A.P., (1948). The interpretation of statistical maps. Biometrika 35, 255-60.
Pesaran, M.H., (2004). General diagnostic tests for cross section dependence in panels. CESifo
Working Papers No.1233.
Ullah, A., (1974). On the sampling distribution of improved estimators for coefficients in linear
regression. Journal of Econometrics 2, 143-150.
Ullah, A., (2004). Finite Sample Econometrics. New York: Oxford University Press.
Zellner, A. (1962). An efficiency method of estimating seemingly unrelated regression equations
and tests for aggregation bias. Journal of the American Statistical Association 57, 348-368.
[R.1]
Table 1
Size of Cross Section Independence Tests with Exogenous Regressors, γ i = γ = 0
Normal Errors with Different Number of Regressors (k)
(T,N)
LM
20
30
50
100
LM adj
20
30
50
100
CD
20
30
50
100
10
20
k=2
30
50
100
200
10
20
k=4
30
50
100
200
10
20
k=6
30
50
100
200
7.50 11.65 17.85 35.90 85.70 100.00
5.40
8.05 11.90 19.70 54.25
97.70
5.40
6.35
8.95 11.65 26.25
66.15
4.80
5.90
6.10
8.80 11.20
26.95
8.65 15.65 25.80 50.20 96.95 100.00
7.60 11.10 14.85 25.35 67.25
99.45
6.35
8.30
9.60 14.00 32.40
77.35
5.50
5.05
6.15
8.65 11.60
27.55
6.30
5.10
5.65
5.75
5.85
5.40
5.25
5.55
6.25
6.05
5.55
5.45
5.10
5.50
4.95
5.90
5.75
4.70
5.00
3.60
5.25
4.95
5.00
5.30
5.80
6.90
6.40
6.15
6.30
6.65
6.15
4.65
5.30
5.95
5.85
4.75
5.50
5.45
5.25
5.45
6.45
5.95
5.85
4.50
6.80
7.05
5.00
4.55
6.00
5.60
4.95
5.40
6.65
5.00
5.90
5.40
6.25
5.95
5.75
5.35
7.30
5.90
5.20
5.75
8.30
6.55
5.45
5.20
8.85
6.85
6.15
5.75
4.70
4.75
4.45
5.05
5.05
5.75
5.00
5.25
5.35
4.55
5.15
5.30
4.10
5.10
5.45
4.85
4.80
5.35
4.85
4.70
4.90
5.10
5.15
6.10
4.50
5.20
5.50
5.10
5.75
5.00
5.30
4.90
4.60
4.30
5.25
5.15
4.05
5.40
5.00
4.90
6.40
5.50
4.95
4.80
5.50
4.20
5.75
5.25
4.95
5.05
5.20
4.95
5.60
5.85
5.20
4.15
4.60
5.55
4.75
4.65
5.50
5.35
5.85
5.20
5.30
5.50
4.90
4.30
5.65
5.90
4.65
4.75
Notes: Data are generated as yit = αi +
10.85 22.85 42.15 71.35 99.95 100.00
7.80 12.45 21.35 38.60 87.45 100.00
5.05
8.00 11.75 17.85 43.40
90.20
4.75
5.55
6.95
9.70 15.40
34.15
k
xit β i + uit , uit = c(γ,k) (γ i ft + σi εit ), i = 1, 2, ..., N , t = 1, 2, ..., T , where αi ∼ IIDN (1, 1), with xit = 0.6xit−1 + vit ,
= 2, 3, ..., k, i = 1, 2, ..., N , t = −50, −49, ..., T , xi,−51 = 0, where vit ∼ IIDN(0, τ 2i / 1 − 0.62 ), τ 2i ∼ IIDχ2 (6) /6. β i ∼ IIDN (1, 0.04), ft ∼ IIDN (0, 1),
=2
σ 2i ∼ IIDχ2 (2) /2, and εit ∼ IIDN (0, 1). αi , xit , σ2i are fixed across replications. c2(γ,k) is chosen so that R̄2 = E(σ2ui )/V ar(yit ) = 0.5 with σ 2ui = V ar(uit ) and
V ar(yit ) = (k − 1)E(β 2i ) + E(σ 2ui ). LM is the Breusch-Pagan (1980) LM test, LMadj is the bias-adjusted LM test and CD is the Pesaran’s (2004) CD test. LM test is based
on χ2N (N−1)/2 distribution. LMadj is based on positive-one-sided N (0, 1) test. CD test is based on two-sided N(0, 1) test. All tests are conducted at 5% nominal level. All
experiments are based on 2,000 replications.
[T.1]
Table 2
Power of Cross Section Independence Tests with Exogenous Regressors, γ i ∼ IIDU[0.1, 0.3]
Normal Errors with Different Number of Regressors (k)
(T,N)
LM
20
30
50
100
LMadj
20
30
50
100
CD
20
30
50
100
10
20
k=2
30
50
100
200
10
20
k=4
30
50
100
200
10
20
k=6
30
50
100
200
27.75
40.35
53.10
76.75
57.30
72.15
86.60
96.70
79.15 95.70
88.40 98.45
96.55 99.65
99.90 100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
26.75
35.30
54.85
74.40
55.80
69.60
83.95
96.45
75.60 95.70
86.30 98.40
95.90 99.65
99.60 100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
27.35
34.35
50.95
72.90
54.95
65.95
83.25
97.10
78.95 97.80
83.55 98.10
95.95 99.65
99.80 100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
25.75
39.85
53.55
77.55
46.75
67.15
85.45
96.45
64.35 85.35 97.25 99.80
82.65 95.65 99.90 100.00
95.80 99.40 100.00 100.00
99.90 100.00 100.00 100.00
21.65
36.80
55.20
76.15
39.70
60.35
83.55
96.75
55.90 76.30 95.40 99.45
78.30 92.55 99.70 100.00
93.75 99.40 100.00 100.00
99.75 100.00 100.00 100.00
18.75
32.55
52.40
75.95
33.75
56.15
80.10
96.90
46.70 65.05 89.40 98.00
73.35 90.15 99.30 100.00
92.20 99.30 100.00 100.00
99.20 100.00 100.00 100.00
50.15
65.05
78.55
92.55
85.40 95.95 99.80
93.50 99.20 100.00
98.35 100.00 99.95
99.95 100.00 100.00
47.65
62.00
77.05
91.55
80.85 94.00 99.30 100.00 100.00
92.20 98.60 100.00 100.00 100.00
98.15 99.95 100.00 100.00 100.00
99.95 100.00 100.00 100.00 100.00
43.15
58.65
72.60
89.85
77.20 90.55 98.70
89.95 98.10 100.00
97.90 100.00 100.00
99.90 100.00 100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
Notes: The design is the same as that of Table 1 except γ i ∼ IIDU [0.1, 0.3].
[T.2]
100.00
100.00
100.00
100.00
100.00
100.00
100.00
100.00
Table 3
Power of Cross Section Independence Tests with Exogenous Regressors, γ i ∼ IIDN(0, 0.1)
Normal Errors with Different Number of Regressors (k)
(T,N)
LM
20
30
50
100
LMadj
20
30
50
100
CD
20
30
50
100
10
20
45.10
56.05
72.70
88.60
42.50
55.55
73.05
88.90
k=2
30
50
100
200
10
77.80
86.15
95.85
99.35
92.25 99.30 100.00
96.95 99.85 100.00
99.55 100.00 100.00
99.95 100.00 100.00
100.00
100.00
100.00
100.00
70.15
83.50
95.35
99.30
85.00 95.90 99.80
95.00 99.50 100.00
99.45 100.00 100.00
99.95 100.00 100.00
100.00
100.00
100.00
100.00
7.85 8.00 7.40
10.75 9.10 9.45
11.55 12.30 13.10
20.00 19.80 17.80
7.60
9.00
12.45
17.00
7.25
9.10
11.35
17.15
8.05
9.20
11.55
18.50
20
k=4
30
50
k=6
100
200
10
20
37.95
54.40
71.85
87.55
73.20 90.45 99.40 100.00
85.40 95.85 99.85 100.00
94.90 99.40 100.00 100.00
99.10 100.00 100.00 100.00
100.00
100.00
100.00
100.00
36.60
51.05
69.40
87.50
36.90
53.50
70.45
88.20
61.00 79.30 93.60 99.70 99.90
80.05 94.05 99.10 100.00 100.00
93.50 98.95 100.00 100.00 100.00
99.35 100.00 100.00 100.00 100.00
29.35
47.50
68.65
87.95
6.95 7.70
9.60 9.40
11.65 10.60
18.65 19.05
6.00
8.55
11.10
18.50
Notes: The design is the same as that of Table 1 except γ i ∼ IIDN(0, 0.1).
[T.3]
6.80
8.85
12.10
17.05
6.65
7.80
11.45
18.05
6.40
8.60
11.50
17.45
30
100
200
71.95
83.80
94.90
99.00
90.35 99.25 100.00
95.45 99.80 100.00
99.35 100.00 100.00
99.90 100.00 100.00
100.00
100.00
100.00
100.00
52.55
76.10
93.25
99.00
69.85 88.25 98.15 99.90
90.15 98.50 99.95 100.00
98.90 100.00 100.00 100.00
99.90 100.00 100.00 100.00
7.05 6.45 7.00
7.90 8.05 9.00
11.95 11.45 12.60
17.65 18.00 17.85
50
6.95
8.90
11.85
16.95
7.60
8.90
11.25
20.35
6.50
8.20
11.55
18.50
Table 4
Size of Cross Section Independence Tests with Exogenous Regressors, γ i = γ = 0
Non-normal Errors with Different Number of Regressors (k)
(T,N)
LM
20
30
50
100
LMadj
20
30
50
100
CD
20
30
50
100
10
20
k=2
30
50
100
200
10
20
k=4
30
50
100
200
10
20
k=6
30
50
100
200
8.85 12.10 22.40 37.65 82.05 100.00
9.75 11.00 15.30 22.10 51.50 95.45
7.95 9.75 12.50 15.00 28.30 63.10
8.15 8.95 8.25 10.85 16.50 30.10
9.70 16.80 27.95 48.05 95.95 100.00
8.60 12.40 17.45 27.90 65.40 99.45
9.60 9.70 13.75 15.85 35.05 73.95
8.35 7.90 8.95 10.10 16.40 30.10
9.65 24.45 41.10 70.05 99.80 100.00
9.35 14.20 23.10 39.15 85.05 100.00
7.90 10.65 13.45 19.70 43.25 87.60
8.25 9.00 8.85 10.90 17.85 37.55
8.10
9.40
8.65
8.80
6.35
7.65
8.15
8.45
8.65
8.25
9.00
7.30
8.00
7.40
8.05
8.00
8.10
8.15
7.60
7.95
7.80
7.80
9.50
8.90
6.40
7.55
7.50
9.10
5.15
4.70
5.60
4.70
4.85
4.30
4.85
5.30
4.45
4.85
4.70
4.35
4.70
4.60
5.05
5.00
5.00
5.20
4.75
4.95
5.85
4.75
5.00
5.30
6.15
4.60
5.75
5.50
6.50
7.30
7.80
8.20
5.00
5.65
5.95
5.55
√
Notes: The design is the same as that of Table 1 except εit ∼ IID χ2 (1) − 1 / 2.
6.20
8.10
7.75
7.60
6.75
7.30
8.05
7.60
9.60
7.60
7.45
7.30
10.65
8.00
7.10
7.45
7.05
7.80
9.70
9.20
6.90
7.80
7.55
7.55
8.50
8.40
8.60
7.80
7.90
8.10
7.45
7.25
8.65
8.00
7.70
8.55
9.80
9.05
8.15
7.30
5.30
4.60
5.45
5.05
5.05
5.60
5.85
4.95
4.05
5.20
4.90
5.80
5.40
4.85
5.20
4.60
5.50
5.35
5.25
4.70
5.20
4.65
4.90
4.55
6.65
5.70
5.50
4.30
5.85
4.30
5.00
4.85
5.35
6.05
4.80
4.65
4.90
5.10
5.65
4.25
[T.4]
Table 5
Size and Power of First Order Cross Section Independence Tests with Exogenous Regressors, Spatially Correlated Errors
(T,N)
LM(1)adj
20
30
50
100
λ=0
30
50 100 200
10
20
6.95
6.40
6.80
5.95
7.15
5.95
6.35
6.10
5.35
6.60
5.50
7.05
5.95
5.85
6.30
5.40
6.45
5.65
5.60
6.05
5.20
5.30
4.95
4.55
5.55
5.05
4.35
4.95
5.35
5.75
4.30
5.20
6.50
5.30
4.80
5.25
4.95
4.65
5.05
5.20
λ = 0.1
30
50
10
20
6.45
5.30
5.15
5.70
13.95
20.15
35.55
67.30
16.20
24.80
46.40
83.90
18.20
30.65
55.35
91.95
5.75
5.10
5.65
4.85
28.80
42.05
64.05
90.80
46.70
66.15
88.70
99.40
λ = −0.1
30
50
200
10
20
33.80 48.30
55.80 78.45
89.30 99.05
99.90 100.00
13.60
19.30
34.90
66.35
18.05
26.50
44.85
83.70
63.75 83.25 98.55 100.00
82.55 95.50 99.90 100.00
97.05 99.80 100.00 100.00
99.90 100.00 100.00 100.00
28.60
40.90
62.20
90.30
49.30 64.35 83.90 98.35 100.00
67.30 81.65 96.15 99.95 100.00
88.50 96.60 99.80 100.00 100.00
99.55 100.00 100.00 100.00 100.00
22.55
38.80
70.45
98.15
100
18.80
29.00
53.85
91.55
23.90
39.30
67.95
97.75
100
200
33.75 48.05
55.45 79.05
90.15 99.00
99.90 100.00
CD(1)
20
30
50
100
Notes: The design is the same as that of Table 1 for k = 2, except errors are spatially correlated such that uit = λ (0.5ui−1,t + 0.5ui+1,t ) + σ i εit , with end points set at
u1t = u2t + ε1t and uN t = uN −1t + εNt .
[T.5]
Table 6
Size and Power of Cross Section Independence Tests, Heterogeneous AR(1) Specification with Normal Errors
10
20
γi = γ = 0
30
50
100
γ i ∼ IIDU [0.1, 0.3]
30
50
100
200
10
20
γ i ∼ IIDN (0, 0.1)
30
50
100
200
69.00
60.35
69.30
92.40
98.95 100.00
96.15 100.00
96.35 99.95
99.90 100.00
17.50
21.75
33.95
57.20
41.10
49.60
66.45
90.00
62.90 90.20 99.90
71.95 92.30 99.85
86.05 98.10 99.95
97.65 100.00 100.00
100.00
100.00
100.00
100.00
24.25
33.40
56.30
90.00
46.85 80.95
62.15 90.90
87.25 99.20
99.70 100.00
14.95
20.85
34.45
58.45
28.85
42.85
63.80
89.65
42.85 65.10 89.60 98.75
61.10 81.90 97.90 99.70
82.60 96.65 99.85 100.00
97.45 100.00 100.00 100.00
49.25 71.90 90.90 99.55 100.00
63.20 84.60 97.40 99.95 100.00
83.15 96.65 99.90 100.00 100.00
97.25 100.00 100.00 100.00 100.00
5.75
6.25
7.10
9.50
5.85
6.35
8.55
9.15
(T,N)
LM
20
30
50
100
200
10
20
9.05 14.45 22.70 45.90 95.55 100.00
6.95 9.90 14.55 24.90 65.20 99.65
5.65 7.85 9.15 14.30 30.50 75.95
5.90 6.40 6.70 8.55 14.10 26.35
11.60
9.75
12.30
21.60
19.45
20.20
25.10
46.25
36.30
33.75
39.75
67.75
LMadj
20
30
50
100
7.05
6.60
6.00
6.70
6.55
6.50
6.85
5.95
7.10
7.55
5.95
5.55
7.75 13.05
6.45 8.45
6.60 6.00
5.95 6.05
22.00
13.05
7.70
5.65
9.40
8.85
12.70
23.15
11.55
14.05
21.70
45.45
15.15
20.85
31.45
65.75
5.80
5.40
5.45
4.90
4.80
5.45
5.15
5.15
5.25
4.45
5.25
4.95
4.80
5.50
5.15
4.45
5.10
5.00
4.90
4.45
22.85
28.95
42.10
63.05
CD
20
30
50
100
5.50
5.20
6.05
4.20
5.00
6.90
7.25
8.70
6.45
5.80
7.90
9.65
5.60
7.35
7.00
10.05
4.95
5.95
7.10
9.70
Notes: See notes to Table 1. The DGP is specified as yit = µi (1 − β i ) + β i yi,t−1 + uit , uit = γ i ft + σ it εit , i = 1, 2, ..., N; t = −50, ..., T , where β i ∼ IIDU [0, 1), µi ∼ εi0 + η i ,
η i ∼ IIDN (1, 2), fi ∼ IIDN (0, 1), σ2it = σ2i ∼ IIDχ2 (2) /2, and εit ∼ IIDN (0, 1). η i , β i and σ it are fixed across replications. yi,−51 = 0 and the first 50 observations are
discarded.
[T.6]
Table 7
Size of Cross Section Independence Tests
Heterogeneous AR(1) with Non-normal Errors
γi = γ = 0
(T,N)
10
20
30
50 100
200
LM
20 11.05 17.15 26.30 45.80 93.30
100.00
30 9.15 12.55 17.20 27.30 62.00
99.00
50 8.35 10.50 13.00 17.35 34.35
70.30
100 7.60 10.20 9.30 10.45 14.85
30.85
Table 8
Size of Cross Section Independence Tests
DGP Subject to a Single Break
γi = γ = 0
(T,N)
10
20
30
50
100
200
LM
20 12.10 29.15 51.55 89.90 100.00 100.00
30 10.90 23.25 40.60 74.30 99.85 100.00
50 10.45 20.90 34.15 64.55 99.50 100.00
100 9.65 19.65 30.80 59.80 97.65 100.00
LMadj
20
30
50
100
LMadj
20 9.90 15.80 24.95
30 10.35 15.60 24.50
50 10.55 17.45 26.15
100 10.70 18.90 27.35
9.50
8.65
8.35
8.05
9.45 10.10
8.50 9.20
8.95 9.35
9.75 7.95
9.75 13.90
8.45 11.65
8.60 9.95
7.65 7.40
19.95
13.25
11.10
9.45
47.00
43.80
46.20
51.70
88.80
88.85
91.35
94.15
100.00
100.00
100.00
100.00
CD
CD
20 5.75 5.15 4.85 4.35 3.90
5.15
30 4.70 5.35 4.45 4.35 5.20
5.30
50 5.00 5.40 5.55 4.25 4.85
5.65
100 4.70 5.45 5.25 4.70 4.85
5.15
Notes: See the notes to Table 6. The design isthe
√ same
as that of Table 6 except εit ∼ IID χ2 (1) − 1 / 2.
20 6.15 4.80 5.75 5.80
5.80
4.70
30 5.10 5.30 6.10 4.70
6.10
5.75
50 5.50 6.10 5.95 5.90
5.40
5.70
100 6.50 5.15 6.75 5.25
6.05
4.25
Notes: The DGP is specified as yit − µi = β it (yit−1 − µi ) + uit ,
uit = γ i ft +√σ it εit , where µi ∼ IIDN (1, 1); β it = β t = 0.6 and
σ it = σ t = 1.5 for t = −50, ..., T /2; β t = 0.8 and σt = 1
for t = T /2 + 1, ..., T ; εit ∼ IIDN (0, 1). yi,−51 = 0 and
the first 50 observations are discarded. See also the notes to Table 6.
[T.7]
Table 9
Size of Cross Section Independence Tests
DGP Subject to Multiple Structural Break
γi = γ = 0
(T,N)
10
20
30
50 100
200
LM
20 9.25 15.40 34.65 46.60 95.30 100.00
30 8.60 15.95 14.20 26.95 66.70 99.80
50 5.80 7.85 9.55 18.60 30.30 74.60
100 5.10 6.50 8.35 7.35 13.25 32.60
Table 10
Size of Cross Section Independence Tests
DGP Subject to Unit Root
γi = γ = 0
(T,N)
10
20
30
50 100
200
LM
20 9.70 15.80 37.75 55.80 96.45 100.00
30 9.45 16.85 18.40 33.90 78.55 100.00
50 6.75 8.65 13.15 22.10 43.70 89.65
100 5.40 7.50 9.60 9.20 23.20 50.15
LMadj
20
30
50
100
LMadj
20
30
50
100
7.40 8.15 13.65
7.90 11.10 6.35
6.00 6.60 6.20
6.15 6.10 6.75
9.35 13.45
7.55 9.10
9.45 6.75
4.85 5.85
30.30
13.80
8.25
7.60
7.10 6.50 13.10
8.45 10.30 7.35
6.75 6.65 7.90
5.90 6.90 8.10
9.25 10.35
9.55 11.35
9.60 8.65
6.40 8.80
22.45
22.50
11.90
9.60
CD
CD
20 5.05 5.45 5.65 4.65 5.30
4.95
30 5.10 5.15 4.55 5.05 5.50
4.75
50 5.45 5.85 4.60 5.10 4.30
5.25
100 4.45 5.25 5.40 4.70 5.00
4.55
Notes: The design is the same as that of Table 8,
except β it = 0.5 for t = −50, ..., 0 and β it ∼ IIDU [0, 1)
for t = 1, ..., T , i = 1, ..., N ; σ2it ∼ IIDχ2 (2)/2 for
t = −50, ..., T , i = 1, ..., N. See also the notes to Table 6.
20 4.95 4.55 5.15 6.00 5.45
5.15
30 5.05 5.00 4.55 4.40 5.25
4.80
50 5.90 5.40 5.10 5.40 5.70
4.35
100 5.60 5.60 5.00 5.65 4.80
5.05
Notes: The DGP is the same as that of Table 10
except β it = β = 1 for all i and t. See also the notes
to Table 6.
[T.8]