Upload
nguyenthien
View
212
Download
0
Embed Size (px)
Citation preview
The University of Adelaide School of Economics
Research Paper No. 2009-16 October 2009
A New Diagnostic Test for Cross–Section Independence in Nonparametric Panel Data
Model
Jia Chen, Jiti Gao and Degui Li
1
The University of Adelaide, School of EconomicsWorking Paper Series No: 0075 (2009-16)
A New Diagnostic Test for Cross–Section
Independence in Nonparametric Panel Data Models
Jia Chen, Jiti Gao1 and Degui Li
The University of Adelaide, SA 5005, Australia
Abstract
In this paper, we propose a new diagnostic test for residual cross–section in-
dependence in a nonparametric panel data model. The proposed nonparametric
cross–section dependence (CD) test is a nonparametric counterpart of an existing
parametric CD test proposed in Pesaren (2004) for the parametric case. We establish
an asymptotic distribution of the proposed test statistic under the null hypothesis.
As in the parametric case, the proposed test has an asymptotically normal distribu-
tion. We then analyze the power function of the proposed test under an alternative
hypothesis that involves a nonlinear multi–factor model. We also provide several
numerical examples. The small sample studies show that the nonparametric CD
test associated with an asymptotic critical value works well numerically in each in-
dividual case. An empirical analysis of a set of CPI data in Australian capital cities
is given to examine the applicability of the proposed nonparametric CD test.
Keywords: Cross–section independence; local linear smoother; nonlinear panel data
model; nonparametric diagnostic test, size and power function
1Jiti Gao is from the School of Economics, The University of Adelaide. Adelaide SA 5005, Australia. Email:
2 J. Chen, J. Gao and D. Li
1. Introduction
Panel data analysis has become increasingly popular in many fields, such as economics,
finance and biology, since it provides the researcher with a wide variety of double–index models
rather than just purely cross–section or time series data models. There exists a rich literature
on parametric linear and nonlinear panel data models. For an overview of statistical inference
and econometric analysis of the parametric panel data models, we refer to the books by Baltagi
(1995), Arellano (2003) and Hsiao (2003). As in both the cross–sectional and time series
cases, parametric models may be too restrictive in some cases. As a consequence, existing
parametric tests may not be applicable in such cases. To address such issues, nonparametric
and semiparametric methods have been used in both model estimation and specification testing.
Recent studies include Li and Hsiao (1998), Ullah and Roy (1998), Hjellvik, Chen and Tjøstheim
(2004), Li and Racine (2007), Cai and Li (2008), and Henderson, Carroll and Li (2008).
Existing studies in nonparametric and semiparametric estimation and model specification
testing mainly assume cross–section independence. Such an assumption is far from realistic,
since cross–section dependence may arise in practice due to the presence of common shocks,
unobserved components that become part of the error term ultimately, economic distance and
spatial correlations. If observations are cross–section dependent, parametric and nonparametric
estimators based on the assumption of cross–section independence may be inconsistent. As
pointed out by Hsiao (2003), meanwhile, there is no natural ordering for cross–section indices,
and appropriate modelling and estimation of cross–section dependence is difficult particularly
when the dimension of cross–section observations N is large. Hence, it is appealing to test for
cross–section independence before one attempts to make some statistical inference for a panel
data model.
There is a substantial literature on diagnostic tests for cross–section independence in para-
metric panel data models. Breusch and Pagan (1980) proposed an Lagrange multiplier (LM)
test statistic, which is based on the average of the squared pair–wise correlation coefficients of
the residuals. The LM test requires that T is much larger than N , where T and N are the time
dimension and the cross–section dimension, respectively. Note that the mean of the squared
correlation coefficients is, however, not correctly centered when T is small. Frees (1995) thus
proposed a test statistic that is based on the squared Spearman rank correlation coefficients
and allows N to be larger than T . Recently, Pesaran (2004) introduced the so–called cross–
section dependence (CD) test. The main idea of proposing the CD test is to use the simple
average of all pair–wise correlation coefficients of the residuals from the individual parametric
linear regressions in the panel. The advantage of the CD test is that it is correctly centered
when both N and T are fixed. Ng (2006) employed spacing variance ratio statistics to test the
severity of cross–section correlation in panels by partitioning the pair–wise cross–correlations
into groups from high to low. Ng (2006)’s test statistics are proposed as agnostic tools for
Nonparametric Test for Cross–Section Independence 3
identifying and characterizing correlations across groups. More recently, Hsiao, Pesaran and
Pick (2007) extended the LM and CD tests from parametric linear panel data models to para-
metric nonlinear models. For other recent contributions to diagnostic tests of cross–section
independence, we refer to Huang, Kab and Urga (2008), Pesaran, Ullah and Yamagata (2008),
and Sarafidis, Yamagata and Robertson (2009).
By contrast, there is little study on diagnostic testing of the null hypothesis that the
residuals are cross–section independence in a nonparametric nonlinear panel data model. We
therefore propose a new diagnostic test for cross–section independence in a nonparametric
nonlinear panel data model. The main contributions of this paper can be summarized as
follows.
(i) We construct a local linear estimator of an individual regression function in the case where
T → ∞ and then propose a nonparametric CD test statistic in a similar fashion to that
proposed in Pesaran (2004) for the parametric case. As a sequence of using the local linear
estimation method, the first order biases involved are all eliminated in the construction
of the proposed test. As shown in Sections 3 and 5, respectively, the proposed test has
both sound large and small sample properties.
(ii) We then establish an asymptotically normal distribution under the null hypothesis, and
also an asymptotically normal distribution under a sequence of local alternatives in Section
3 below. In the small sample studies in Section 5 below, we examine the performance of
both the size and the power functions under various cases where the conditional mean
function and the residual may take the form of either linear, nonlinear or a mixture of
both.
(iii) We conclude from the small sample studies in Section 5 that the proposed nonparametric
CD test performs well when the data satisfy a nonparametric panel data model. By
comparison, existing tests for the parametric case are not applicable. In addition, the
proposed nonparametric CD test also performs well in both the size and power even when
the conditional mean function is of a parametric form. In this case, the nonparametric
CD test is just slightly less powerful that the parametric CD test.
(iv) In summary, the construction in Section 2 and the small sample analysis in Section 5 both
show that the proposed nonparametric CD test is easily computable and implementable.
The simulation study in Section 5 shows that the proposed nonparametric CD test is
generally more applicable than the corresponding parametric CD test. As an empirical
application, we apply the proposed test for testing the cross–section independence of a
set of CPI data in Australian capital cities.
4 J. Chen, J. Gao and D. Li
The rest of this paper is organized as follows. A nonparametric test for cross–section inde-
pendence in a nonlinear panel data model is proposed in Section 2. An asymptotic distribution
of the proposed nonparametric CD test statistic is established in Section 3. Section 3 also es-
tablishes an asymptotic normality under an alternative hypothesis. Section 4 discusses possible
extensions. Several simulated examples are given in Section 5. An empirical analysis of a set
of CPI data in Australian capital cities is given in Section 6. All the mathematical proofs of
the asymptotic results are given in Appendix A.
2. Nonparametric panel data model and CD test statistic
Consider a nonparametric nonlinear panel data model of the form
Yit = gi(Xit) + uit, i = 1, · · · , N ; t = 1, · · · , T, (2.1)
where gi(·) is the individual regression function, Xit is random and satisfies some mild con-
ditions (see A2 below), and uit is independent of Xit with E[uit] = 0.
The aim of this paper is to test the null hypothesis
H0 : uit is independent of ujt for all i 6= j. (2.2)
The above testing problem has been studied by many authors in the context of parametric
panel data models. In the parametric case, the so–called CD test statistic was introduced
by Pesaran (2004) in the parametric linear panel data case. The main idea is to use the
simple average of all pair–wise correlation coefficients of the residuals from the individual
nonparametric nonlinear regression in the panel.
Before proposing a nonparametric CD test statistic, we need to decide which kernel method
should be used in the construction of our nonparametric CD test. Existing studies (see, for
example, Chapter 3 of Gao 2007) already show that the use of the Nadaraya–Watson kernel
estimation method in the construction of a nonparametric kernel test may have severe size
distortion due to the first order bias issue inherited from the Nadaraya–Watson kernel esti-
mation method. In this paper, we thus choose to use a local linear estimation method in the
construction of our nonparametric CD test. As shown in Section 3, the proposed nonparamet-
ric CD test has sound large sample theory under some mild conditions. Section 5 shows that
the proposed nonparametric CD test also has good small sample properties without using a
bootstrap method.
We now introduce the local linear estimator of the individual regression function gi(·).Assume that gi(·) has derivatives up to the second order at the point x0. By Taylor’s expansion,
for x in a neighborhood of x0, we have
gi(x) = gi(x0) + g′i(x0)(x− x0) +O((x− x0)2
). (2.3)
Nonparametric Test for Cross–Section Independence 5
Then, we find (α0, α1) to minimize
T∑t=1
(Yit − α0 − α1(Xit − x0))2K
(Xit − x0
h
), (2.4)
where K(·) is some kernel function and h := hT is the bandwidth. The local linear estimator for
gi(x0) is defined as gi(x0) = α0i, where (α0i, α1i) is the unique pair that minimizes (2.4). For
more details about the local linear estimators, we refer to Fan and Gijbels (1996). In general,
one probably should use a kernel function and a bandwidth indexed by i for each cross section.
For notational simplicity, this paper uses the same kernel and bandwidth for both the large
and small sample discussion. In practice, the bandwidth can be chosen using the conventional
leave–one–out cross–validation method.
By an elementary calculation, the local linear estimator of gi(x0) can be expressed as
gi(x0) =T∑t=1
wit(x0)Yit, 1 ≤ i ≤ N, (2.5)
where wit(x0) = Kx0,h(Xit)T∑
t=1
Kx0,h(Xit)
, in which
Kx0,h(Xit) =1hK
(Xit − x0
h
)(Si2(x0)−
(Xit − x0
h
)Si1(x0)
)
with Sij(x0) = 1Th
∑Tt=1
(Xit−x0
h
)jK(Xit−x0
h
)for j = 0, 1, 2.
With the help of the local linear smoother defined above, we estimate uit by uit = Yit −gi(Xit).
We are now ready to propose a nonparametric CD test statistic of the form
NCD =
√T
N(N − 1)
N∑i=1
N∑j 6=i
ρij , (2.6)
where
ρij = ρji =
T∑t=1
uitujt√T∑t=1
u2it
√T∑t=1
u2jt
=
1T
T∑t=1
uitujt√1T
T∑t=1
u2it
√1T
T∑t=1
u2jt
,
in which uit = uitfi(Xit) and fi(x0) = 1T
T∑t=1
Kx0,h(Xit). Note that the test statistic NCD is
invariant to σ2ui = E[u2
i1].
The aim of using uit instead of uit in the test statistic (2.6) is to eliminate the random
denominator problem involved in the nonparametric estimator gi. The construction of the
nonparametric CD test in (2.6) is motivated by a similar form proposed in Pesaran (2004) for
6 J. Chen, J. Gao and D. Li
the parametric case. The main step is the involvement of a nonparametric estimate uit, which
is equivalent to the OLS estimate in the parametric case. As shown in Sections 3 and 5 below,
the nonparametric CD test has both good large and small sample properties.
In Section 3 below, we show that the nonparametric CD test statistic (2.6) is asymptotically
centered when T → ∞ first and then N → ∞. Furthermore, asymptotic distributions of the
test statistic are established under either the null hypothesis or a sequence of local alternatives.
3. Large sample theory
3.1 Asymptotic theory under the null hypothesis
To study the asymptotic theory of the test statistic, we need the following conditions.
A1 (i). The probability kernel function K(·) is a symmetric and continuous function with
some compact support.
(ii). The individual regression function gi(·), 1 ≤ i ≤ N , has derivatives up to the second
order and the derivatives are continuous. Furthermore, maxi≥1E[|g′′i (Xi1)|2
]<∞, where
g′′i (·) is the second order derivative of gi(·).
A2 (i). For each individual series (for each fixed 1 ≤ i ≤ N), Xit is a sequence of station-
ary α–mixing random regressors with maxi≥1E[|Xi1|2
]< ∞ and the mixing coefficient
αxi(·) satisfying αxi(k) ≤ C0k−β uniformly in i ≥ 1 and for some 0 < C0 < ∞ and
β > 3.
(ii). Let fi(·) be the density function of Xit. Suppose that fi(x) is continuous and
bounded in x ∈ R. There exists a joint density function fis1,is2,···,isl,jt1,jt2,···,jtk(·, · · · , ·) of
(Xis1 , Xis2 , · · · , Xisl, Xjt1 , Xjt2 , · · · , Xjtk), 1 ≤ i, j ≤ N, 1 ≤ l, k ≤ 4,
such that fis1,is2,···,isl,jt1,jt2,···,jtk(·, · · · , ·) is continuous and bounded.
(iii). Let uis, 1 ≤ i ≤ N, 1 ≤ s ≤ T and Xjt, 1 ≤ j ≤ N, 1 ≤ t ≤ T be inde-
pendent for all (i, j) and (s, t). For each individual series (for each fixed 1 ≤ i ≤ N),
uit is a sequence of stationary α–mixing random errors with the mixing coefficient
αui(·) satisfying∞∑k=1
αδ0
2+δ0ui (k) <∞ for some δ0 > 0. In addition, E
[u2i1
]= σ2
ui > 0 and
maxi≥1E[|ui1|2+δ0
]<∞.
(iv). Let τ2i,j = µ4
2µ40
(κi,jσ
2uiσ
2uj + 2
∞∑t=2
E[ui1uit]E[uj1ujt]κi,j(t))
, where µk =∫ukK(u)du,
κi,j =∫ ∫
f4i (x)f
4j (y)fi1,j1(x, y)dxdy and
κi,j(t) =∫ ∫ ∫ ∫
f2i (x1)f2
i (x2)f2j (y1)f2
j (y2)fi1,it,j1,jt(x1, x2, y1, y2)dx1dx2dy1dy2,
Nonparametric Test for Cross–Section Independence 7
and σ2i = µ2
2µ20σ
2ui
∫f5i (x)dx. Let 0 < τ2
i,j < ∞ and σ2i > 0 for all 1 ≤ i, j ≤ N . Suppose
that there exists some τ0 > 0 such that as N →∞,
1N(N − 1)
N∑i=1
N∑j 6=i
τ2i,j
σ2i σ
2j
→ τ0. (3.1)
A3. The bandwidth h satisfies
T θh
log T→∞ as T →∞ and N2Th8 → 0 as T →∞ and N →∞, (3.2)
where θ = β−3β+2 .
Remark 3.1. The above assumptions are mild and can be satisfied in many cases. For
example, A1(i) is a mild condition on the kernel function and is assumed by many authors in
nonparametric inference of both stationary time series and panel data (see, for example, Fan
and Yao 2003; Gao 2007; Cai and Li 2008). A1(ii) and A2(ii) are some mild conditions on
the individual regression functions and density functions. The α–mixing condition assumed
in A2(i) and A2(iii) is a commonly used condition in the time series case (see, for example,
Auestad and Tjøstheim 1990, Chen and Tsay 1993, Fan and Yao 2003; Gao 2007; Li and Racine
2007). It is introduced in this paper for the nonparametric panel data case. Note that when
uis and uit are mutually independent for all s 6= t and each fixed i, and Xit and Xjtare independent for all i 6= j and each given t, we have κi,j =
∫ ∫f5i (x)f
5j (y)dxdy, κi,j(t) ≡ 0
and τ2ij = µ4
2µ40κi,jσ
2uiσ
2uj = σ2
i σ2j . Thus, τ0 ≡ 1.
Condition A3 is a set of conditions on the bandwidth as well as on the restriction on T and
N . The first bandwidth condition in A3 is proposed in order to apply the uniform consistency
of the nonparametric kernel estimator in the proofs of Theorems 3.1 and 3.2 below. The second
bandwidth condition in A3 is also needed in the proofs of Theorems 3.1 and 3.2. In addition, the
second part of A3 allows for the case where rate of T →∞ is slower than that of N →∞. This
basically implies that condition A3 allows for both medium and small integers for T in practice
while the asymptotic theory requires both N → ∞ and T → ∞ in theory. The simulation
studies in Section 5 support that the nonparametric CD test works well even when T as small
as T = 10, although it cannot be shown at this stage that the conclusions of Theorems 3.1 and
3.2 remain true when T is fixed.
In the following theorem, we show that the nonparametric CD test statistic, defined by
(2.6), has an asymptotically normal distribution as that obtained by Pesaran (2004) and Hsiao,
Pesaran and Pick (2007), who considered similar testing problems in the context of parametric
linear and nonlinear panel data models.
8 J. Chen, J. Gao and D. Li
Theorem 3.1. Assume that (2.1) and the conditions A1–A3 are satisfied. Then under H0,
NCD =
√T
N(N − 1)
N∑i=1
N∑j 6=i
ρij
d−→ N(0, τ0) (3.3)
as T →∞ first and then N →∞.
The proof of Theorem 3.1 is given in Appendix A below.
Remark 3.2. (i) Note that τ0 = 1 when uis and uit are mutually independent for all s 6= t
and each fixed i, and Xit and Xjt are independent for all i 6= j and each given t.
(ii) In general, τ0 is an unknown parameter to be estimated. Define
ρij =ρij σiσjτi,j
,
where τi,j and σi are consistent estimators of τi,j and σi, respectively. In this case, it can be
shown that under H0
NCD =:
√T
N(N − 1)
N∑i=1
N∑j 6=i
ρij
d−→ N(0, 1)
as T →∞ first and then N →∞.
Remark 3.3. The asymptotic distribution in Theorem 3.1 is obtained by letting T →∞ first
and then N → ∞. A natural question is what will happen if either N → ∞ first and then
T →∞ or T →∞ and N →∞ simultaneously.
(i) To see this, we define Zit = uitf2i (Xit)µ0µ2
σi. Following the proof of Theorem 3.1 in
Appendix A, we can find that the leading term of NCD is
1√N(N − 1)T
N∑i=1
N∑j 6=i
T∑t=1
ZitZjt =1√T
T∑t=1
1√N(N − 1)
N∑i=1
N∑j 6=i
ZitZjt
=:1√T
T∑t=1
ωN (t),
where ωN (t) = 1√N(N−1)
N∑i=1
N∑j 6=i
ZitZjt = 1√N(N−1)
(∑Ni=1 Zit
)2−∑Ni=1 Z
2it
.
It is obvious that, for each fixed 1 ≤ t ≤ T , ωN (t) is a sequence of U–statistics. By
Theorem 5.5.2 in Serfling (1980), under H0,
ωN (t) d−→ χ2t,1 − 1 as N →∞,
where χ2t,1 is the chi–square distribution with one degree of freedom. If both uit and Xit
are i.i.d. for all (i, t), then it can be seen that χ2t,1, 1 ≤ t ≤ T is a sequence of i.i.d. chi–
square random variables. By the conventional central limit theorem for the i.i.d. case (see, for
example, Chow and Teicher 1988), the conclusion of Corollary 3.1 remains true when N →∞first and then T →∞.
Nonparametric Test for Cross–Section Independence 9
(ii) If both uit and Xit are i.i.d. for all (i, t), moreover, ωN (t is also a sequence of
i.i.d. errors. Thus, it follows from the conventional central limit theorem for the i.i.d. case
that as both N →∞ and T →∞ simultaneously
1√T
T∑t=1
ωN (t) d−→ N(0, 1),
which implies that the conclusion of Corollary 3.1 remains true.
In summary, Theorem 3.1 and the discussion given in Remarks 3.2(i) and 3.3 imply the
following corollary; its implementation is given through the simulation studies in Section 5 and
the empirical application in Section 6.
Corollary 3.1. Assume that (2.1) and the conditions A1 and A3 are satisfied. If, in addition,
uis and uit are mutually independent for all s 6= t and each fixed i, and Xit and Xjtare independent for all i 6= j and each given t, then under H0,
NCD =
√T
N(N − 1)
N∑i=1
N∑j 6=i
ρij
d−→ N(0, 1) (3.4)
as either T → ∞ first and then N → ∞, or N → ∞ first and then T → ∞, or both N → ∞and T →∞ simultaneously.
In summary, the limiting distribution of the suitably normalized test statistic depends
on the independence or dependence assumption on Xit and uit as well as the treatment
of the two indices N and T . Phillips and Moon (1999) introduced three limit approaches:
sequential limit theory, diagonal path limit theory and joint limit theory. They also discuss some
relations between sequential and joint limits. The asymptotic distribution given in Theorem
3.1 is obtained by a sequential limit approach (T →∞ first and then N →∞). It is not clear
whether the conclusion of Theorem 3.1 remains true when either N →∞ first and then T →∞or both N →∞ and T →∞ simultaneously. Such issues are thus left to future research.
3.2. Asymptotic theory under an alternative hypothesis
In this section, we analyze the power of the proposed test under a sequence of local alterna-
tives. Naturally, the power of the proposed test for the cross–section dependence relies on the
form of an alternative hypothesis. We now consider a sequence of cross–sectional dependence
alternatives via a nonlinear multi–factor model of the form
H1 : uit = FNT (zt, βi) + εit with FNT (zt, βi) =1(
N1/2T 1/4)kG(zt, βi) (3.5)
for k = 0, 1, where G(zt, βi) is a sequence of known parametric linear or nonlinear functions
indexed by βi, zt, 1 ≤ t ≤ T is a sequence of stationary α–mixing random variables,
10 J. Chen, J. Gao and D. Li
βi, 1 ≤ i ≤ N is a sequence of common factors, εit is a sequence of stationary α–mixing
random variables for fixed i and is independent of zt, and εit is independent of εjt for
all t and i 6= j. Note that form (3.5) defines a global alternative when k = 0, while it gives a
sequence of local alternatives when k = 1.
Before establishing an asymptotic distribution of the nonparametric CD test statistic under
the alternative hypothesis H1, we need the following set of conditions.
A4 (i) zt is a sequence of stationary α–mixing random variables with the mixing coefficient
αz(·) satisfying∞∑t=1
αδ1/(2+δ1)z (t) <∞ for some δ1 > 0.
(ii) The nonlinear function G(·, ·) satisfies the following conditions,
E[G(zt, βi)] = 0, (3.6)
E[G(zt, βi)]2+δ1 <∞. (3.7)
In addition, there exists an array of constants ψij ; 1 ≤ i ≤ N, 1 ≤ j ≤ N with ψij = ψji
such that
E[G(zt, βi)G(zt, βj)] = ψij (3.8)
1N(N − 1)
N∑i=1
N∑j 6=i
ψij → ψ as N →∞, (3.9)
where ψ is a constant.
(iii) A2(iii) and A.2(iv) are both satisfied when uit is replaced by εit. Moreover,
εit is independent of zt. Let τ1 be defined in the same way as for τ0 with uit being
replaced by εit.
Condition A4 allows for a general class of forms for G(zt, βi). It obviously covers the linear
multi–factor case: G(zt, βi) = ztβi, which was studied by Pesaran (2004).
When the alternative hypothesis H1 holds, we have the following asymptotic distribution
for the nonparametric CD test statistic NCD. The proof of Theorem 3.2 below is given in
Appendix A below.
Theorem 3.2. Assume that (2.1) and the conditions A1, A2 (i), A3 and A4 are satisfied.
(i) Under H1 with k = 0, we have as T →∞ first and then N →∞
NCDP−→∞. (3.10)
Nonparametric Test for Cross–Section Independence 11
(ii) Under H1 with k = 1, we have as T →∞ first and then N →∞
NCDd−→ N(ψ, τ1). (3.11)
The divergence result in (3.10) is quite common in the case where we assume this kind of
global alternative. The asymptotic distribution in (3.11) is similar to the result obtained by
Pesaran (2004). FNT (zt, βi) can be viewed as the measure of the dependence between individual
time series. By an elementary calculation and (3.8), we can show that E[uitujt] = ψij
N√T
.
Furthermore, (3.9) implies that the nonparametric CD test statistic allows the detec-
tion of the alternatives when the nonlinear multi–factor function has a decreasing rate of
O(T−1/4N−1/2
), which is the same as that in Pesaran (2004).
The simulated examples in Section 5 show that the power of the proposed test is satisfactory
when ψ > 0 (or ψ < 0). However, when ψ = 0, the asymptotic distribution in (3.11) is the
same as that in Theorem 3.1, which implies that the test will not have a satisfactory power. In
the context of parametric panel data models, Pesaran, Ullah and Yamagata (2008) proposed a
bias adjusted LM test to avoid the problem of poor power for the case of ψ = 0. It is interesting
to consider a nonparametric type of bias adjusted LM test statistic. Such an issue is left for
our future study.
4. Some extensions
In both theory and practice, there are cases where we need to consider a nonlinear autore-
gressive panel data model of the form
Yit = gi(Yi,t−1) + uit, i = 1, · · · , N ; t = 1, · · · , T. (4.1)
Pesaran (2004) and Sarafidis, Yamagata and Robertson (2008) studied the test of error
cross–section dependence when all gi(·), 1 ≤ i ≤ N , are of some linear form. As far as we
are aware, however, there is little study on diagnostic testing of cross–section independence for
model (4.1). It seems that we may apply the nonparametric CD test statistic NCD to test
whether H0 holds. In theory, establishing an asymptotic distribution for the nonparametric
CD test NCD in this case is not straightforward. Further discussion is left for our future study.
Meanwhile, nonparametric approaches are useful for exploring hidden structures. When
there are multiple regressor variables, however, the nonparametric approaches face a serious
problem of the so–called “curse of dimensionality”. To address this issue, some dimensional
reduction methods have been discussed in both the cross–section data and the time series data
cases (see, for example, Hardle, Liang and Gao 2000; Gao 2007; Li and Racine 2007).
In the panel data case, we may consider a partially linear model of the form
Yit = Xτitα+m(Zit) + εit, i = 1, · · · , N ; t = 1, · · · , T, (4.2)
12 J. Chen, J. Gao and D. Li
where Xit is a vector of regressors, α is a vector of unknown parameters and the coefficient
functions m(·) are all unknown. Recently, there have been some attempts on both theoretical
studies and empirical applications of this type of partially linear models in the panel data case
(see, for example, Li and Hsiao 1998; Henderson, Carroll and Li 2008).
To the best of our knowledge, there is little study on the testing of cross–section indepen-
dence for a partially linear panel data model of the form (4.2). We will extend the proposed
test statistic to the partially linear model case and establish an asymptotic distribution of
the proposed test statistic. Since different methods and more technicalities are likely to get
involved, such an issue is therefore left for future research.
5. Small sample simulation studies
In this section, we give some simulated examples to show the finite sample performance
of the nonparametric CD test. In addition, we also compare its performance with that of a
parametric CD test. Since both the sizes and power values of the proposed nonparametric CD
test associated with an asymptotic critical value in each case are already comparable with those
of the parametric CD test based on an asymptotic critical value, our experience suggests that
there is no need to introduce a bootstrap simulation procedure to improve the finite sample
performance of the proposed nonparametric CD test.
In the following experiments, the uniform kernel K(u) = 12I|u| ≤ 1 is used in the
implementation of the proposed nonparametric CD test. The bandwidth is chosen using the
conventional leave–one–out cross–validation method.
We first examine the finite sample performance of the proposed nonparametric test when
the data set is simulated from a parametric linear panel data model of the form
Yit = ai + biXit + uit, i = 1, 2, · · · , N ; t = 1, 2, · · · , T, (5.1)
where aii.i.d.∼ U(0, 1), bi
i.i.d.∼ N(1, 0.04), Xiti.i.d.∼ N(0, 1), uit = f(ri, βt) + eit, βt is the time–
specific common effect and βti.i.d.∼ N(0, 1), eit
i.i.d.∼ N(0, 1), and ri is a sequence of non–random
numbers indicating the degree of cross–section error correlations. Note that uit and Xitare generated independently.
Under the null hypothesis of cross–section independence, we have ri = 0, and under the
alternative hypothesis, we experiment with rii.i.d.∼ U(0.1, 0.3). The parameters ai, bi, and ri
are drawn once for each i = 1, 2, · · · , N , and then fixed throughout the replications. Xit, βt,
and eit are newly drawn for each replication, independently of each other.
We experiment with both linear and nonlinear forms for the function f(·, ·). For the linear
case, we set f(ri, βt) = riβt, and for the nonlinear case, we set f(ri, βt) = riβt
1+r2i β2t. It can be
Nonparametric Test for Cross–Section Independence 13
easily seen that the pair–wise correlation coefficients are given by
corr(ui,t, ujt) =rirj√
(1 + r2i )(1 + r2j ),
for the parametric linear case, and
corr(uit, ujt) =E(rirjβ
2t /((1 + r2i β
2t )(1 + r2jβ
2t )))
√(1 + E
(r2i β
2t /(1 + r2i β
2t )2)) (
1 + E(r2jβ
2t /(1 + r2jβ
2t )2))
for the parametric nonlinear case.
Using an asymptotic critical value, we computed the two–sided simulated sizes and power
values of the proposed nonparametric CD test and the parametric counterpart in each case.
The experiments are carried out for N , T = 10, 20, 30, 50, 100. The number of replications
is 1000, and the significance level is p = 1%, 5%, and 10%, respectively. The simulated sizes
of the parametric and the nonparametric CD tests for the linear model (5.1) are reported in
Table 5.1 below.
Table 5.1(a) Size of the tests for linear model (5.1) at the 1% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
10 0.023 0.019 0.025 0.024 0.024 0.025 0.032 0.029 0.022 0.027
20 0.021 0.011 0.016 0.013 0.010 0.020 0.009 0.016 0.024 0.012
30 0.011 0.018 0.012 0.015 0.014 0.014 0.011 0.012 0.014 0.013
50 0.013 0.011 0.013 0.013 0.019 0.006 0.014 0.010 0.010 0.005
100 0.010 0.011 0.011 0.010 0.015 0.014 0.011 0.015 0.009 0.010
Table 5.1(b) Size of the tests for linear model (5.1) at the 5% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
10 0.053 0.052 0.069 0.048 0.057 0.044 0.062 0.058 0.053 0.051
20 0.064 0.049 0.054 0.052 0.041 0.058 0.042 0.055 0.050 0.044
30 0.048 0.051 0.054 0.060 0.048 0.047 0.041 0.053 0.057 0.042
50 0.063 0.054 0.043 0.049 0.052 0.046 0.058 0.044 0.048 0.046
100 0.047 0.049 0.049 0.042 0.055 0.047 0.044 0.048 0.041 0.046
Table 5.1(c) Size of the tests for linear model (5.1) at the 10% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
10 0.100 0.104 0.100 0.082 0.089 0.094 0.102 0.103 0.090 0.095
20 0.111 0.096 0.094 0.102 0.088 0.110 0.105 0.100 0.093 0.098
30 0.104 0.107 0.106 0.111 0.092 0.094 0.103 0.101 0.111 0.084
50 0.103 0.108 0.084 0.096 0.106 0.096 0.107 0.088 0.096 0.101
100 0.102 0.099 0.098 0.101 0.114 0.103 0.097 0.094 0.081 0.098
14 J. Chen, J. Gao and D. Li
Tables 5.1(a)–5.1(c) show that the simulated sizes look quite reasonable in each case re-
gardless of whether using the nonparametric CD test or using the parametric CD test. This
implies that the nonparametric CD test is still applicable even when the data follow a paramet-
ric linear model. In addition, the results in Tables 5.1(a)–5.1(c) show that the nonparametric
CD test associated with an asymptotic critical value works well numerically even when T and
N are as small as T = N = 20. In addition, the tables also show that the sizes of the parametric
CD test are slightly more stable than those of the nonparametric CD test, mainly because the
true model is just parametric and the parametric CD test is supposed to perform better.
The power values of the tests for model (5.1) with linear (f(ri, βt) = riβt) and nonlinear
(f(ri, βt) = riβt/(1 + r2i β2t )) forms of f(·, ·) are given in Table 5.2 below.
Table 5.2(a) Power of the tests for linear model (5.1) at the 1% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
f(ri, βt) = riβt
10 0.098 0.234 0.336 0.649 0.896 0.076 0.163 0.256 0.484 0.815
20 0.116 0.509 0.657 0.911 0.995 0.080 0.366 0.503 0.797 0.967
30 0.122 0.685 0.688 0.948 0.997 0.079 0.529 0.567 0.876 0.992
50 0.343 0.676 0.978 0.997 1.000 0.247 0.528 0.941 0.987 1.000
100 0.432 0.902 1.000 1.000 1.000 0.320 0.804 0.995 1.000 1.000
f(ri, βt) = riβt/(1 + r2i β2
t )
10 0.075 0.184 0.253 0.448 0.829 0.058 0.143 0.183 0.334 0.705
20 0.091 0.182 0.433 0.822 0.990 0.059 0.126 0.304 0.678 0.965
30 0.092 0.324 0.575 0.913 0.998 0.061 0.223 0.436 0.804 0.991
50 0.266 0.564 0.839 0.963 1.000 0.189 0.411 0.725 0.892 1.000
100 0.304 0.797 0.997 1.000 1.000 0.205 0.667 0.982 1.000 1.000
Table 5.2(b) Power of the tests for linear model (5.1) at the 5% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
f(ri, βt) = riβt
10 0.170 0.348 0.450 0.742 0.942 0.147 0.255 0.364 0.616 0.884
20 0.234 0.663 0.773 0.958 0.997 0.174 0.525 0.646 0.879 0.988
30 0.260 0.795 0.807 0.976 0.999 0.190 0.665 0.691 0.936 0.996
50 0.527 0.812 0.996 0.999 1.000 0.406 0.692 0.977 0.995 1.000
100 0.624 0.955 1.000 1.000 1.000 0.510 0.894 0.997 1.000 1.000
f(ri, βt) = riβt/(1 + r2i β2
t )
10 0.146 0.282 0.394 0.575 0.883 0.139 0.239 0.297 0.453 0.780
20 0.174 0.286 0.589 0.898 0.995 0.129 0.232 0.454 0.787 0.982
30 0.207 0.477 0.708 0.953 1.000 0.148 0.372 0.595 0.898 0.994
50 0.435 0.704 0.921 0.984 1.000 0.346 0.591 0.847 0.947 1.000
100 0.488 0.899 0.999 1.000 1.000 0.385 0.811 0.995 1.000 1.000
Nonparametric Test for Cross–Section Independence 15
Table 5.2(c) Power of the tests for linear model (5.1) at the 10% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
f(ri, βt) = riβt
10 0.225 0.393 0.520 0.795 0.957 0.211 0.319 0.429 0.677 0.904
20 0.322 0.723 0.838 0.970 0.998 0.246 0.603 0.717 0.923 0.995
30 0.338 0.839 0.851 0.987 0.999 0.274 0.751 0.755 0.956 0.999
50 0.622 0.859 0.996 0.999 1.000 0.506 0.772 0.984 0.995 1.000
100 0.732 0.968 1.000 1.000 1.000 0.601 0.934 0.998 1.000 1.000
f(ri, βt) = riβt/(1 + r2i β2
t )
10 0.198 0.365 0.470 0.625 0.908 0.190 0.296 0.369 0.529 0.827
20 0.242 0.369 0.669 0.927 0.998 0.205 0.303 0.546 0.852 0.992
30 0.277 0.555 0.771 0.974 1.000 0.220 0.451 0.675 0.934 0.997
50 0.525 0.783 0.950 0.991 1.000 0.437 0.679 0.888 0.966 1.000
100 0.581 0.934 0.999 1.000 1.000 0.484 0.867 0.996 1.000 1.000
Tables 5.2(a)–5.2(c) show that the simulated power values are quite satisfactory in each
of the cases concerned. Meanwhile, the simulated power values of the nonparametric CD test
associated with an asymptotic critical value are quite comparable with those of the parametric
CD test based on the use of an asymptotic critical value. This may be due to the fact that the
asymptotic normality can be used as a good approximation to the sample distribution of the
proposed nonparametric test in each of the cases considered.
In addition, Tables 5.2(a)–5.2(c) show that the parametric CD test is more powerful than
the nonparametric CD test. This is not surprising, since the true model is just parametric and
the parametric CD test is supposed to be more powerful.
In the following simulation studies, we examine the finite sample performance of the pro-
posed nonparametric test when the data set is simulated from a parametric nonlinear panel
data model of the form
Yit =1
1 + θ2iX
2it
+ uit, i = 1, 2, · · · , N ; t = 1, 2, · · · , T, (5.2)
where θii.i.d.∼ N(1, 0.04), Xit
i.i.d.∼ U(0.1, 0.7), and uit is the same as in model (5.1). When
ri = 0, the simulated sizes of the parametric and nonparametric CD test for this model are
reported in Table 5.3 below at different significance levels, and when rii.i.d.∼ U(0.1, 0.3), the
power values of the test are reported in Table 5.4 below.
16 J. Chen, J. Gao and D. Li
Table 5.3(a) Size of the tests for nonlinear model (5.2) at the 1% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
10 0.027 0.022 0.031 0.053 0.029 0.021 0.016 0.018 0.026 0.024
20 0.019 0.018 0.015 0.023 0.024 0.020 0.013 0.014 0.016 0.009
30 0.008 0.012 0.009 0.019 0.013 0.007 0.013 0.011 0.021 0.011
50 0.010 0.019 0.009 0.008 0.012 0.011 0.018 0.011 0.009 0.012
100 0.005 0.007 0.016 0.011 0.011 0.010 0.004 0.016 0.013 0.013
Table 5.3(b) Size of the tests for nonlinear model (5.2) at the 5% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
10 0.061 0.056 0.075 0.110 0.240 0.051 0.045 0.047 0.051 0.052
20 0.055 0.060 0.062 0.068 0.059 0.060 0.059 0.055 0.057 0.039
30 0.047 0.052 0.052 0.060 0.056 0.046 0.053 0.049 0.066 0.048
50 0.046 0.048 0.052 0.052 0.051 0.049 0.054 0.057 0.056 0.045
100 0.050 0.053 0.058 0.047 0.051 0.047 0.046 0.058 0.050 0.058
Table 5.3(c) Size of the tests for nonlinear model (5.2) at the 10% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
10 0.101 0.089 0.120 0.165 0.316 0.106 0.095 0.085 0.095 0.092
20 0.096 0.115 0.104 0.120 0.099 0.101 0.117 0.101 0.109 0.079
30 0.093 0.101 0.104 0.113 0.095 0.095 0.102 0.099 0.118 0.095
50 0.100 0.091 0.106 0.108 0.102 0.095 0.094 0.112 0.113 0.094
100 0.105 0.101 0.100 0.091 0.103 0.101 0.096 0.090 0.101 0.098
Tables 5.3(a)–5.3(c) show that both the parametric CD test and the nonparametric CD
test already have reasonable simulated sizes when using an asymptotic critical value in each
case. As in Tables 5.1(a)–5.1(c), the simulated sizes of the nonparametric CD test are very
comparable with those of the parametric CD test.
Table 5.4 gives the corresponding power values for both the parametric and nonparametric
CD tests.
Nonparametric Test for Cross–Section Independence 17
Table 5.4(a) Power of the tests for nonlinear model (5.2) at the 1% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
f(ri, βt) = riβt
10 0.086 0.251 0.417 0.799 0.954 0.084 0.229 0.358 0.735 0.915
20 0.128 0.322 0.583 0.884 0.996 0.130 0.304 0.567 0.852 0.995
30 0.152 0.571 0.736 0.973 1.000 0.144 0.573 0.722 0.968 0.999
50 0.265 0.883 0.958 0.993 1.000 0.261 0.879 0.952 0.992 1.000
100 0.322 0.988 0.998 1.000 1.000 0.299 0.985 0.993 1.000 1.000
f(ri, βt) = riβt/(1 + r2i β2t )
10 0.090 0.221 0.340 0.665 0.940 0.086 0.204 0.297 0.560 0.869
20 0.084 0.376 0.510 0.807 0.984 0.090 0.369 0.504 0.789 0.978
30 0.152 0.408 0.581 0.934 0.999 0.152 0.414 0.574 0.925 0.999
50 0.167 0.621 0.911 0.981 1.000 0.164 0.603 0.908 0.978 1.000
100 0.397 0.811 0.998 1.000 1.000 0.391 0.804 0.998 1.000 1.000
Table 5.4(b) Power of the tests for nonlinear model (5.2) at the 5% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
f(ri, βt) = riβt
10 0.154 0.385 0.544 0.863 0.969 0.143 0.343 0.486 0.816 0.940
20 0.244 0.482 0.729 0.937 0.999 0.234 0.444 0.705 0.919 0.998
30 0.276 0.727 0.848 0.987 1.000 0.271 0.727 0.831 0.983 1.000
50 0.437 0.954 0.979 0.996 1.000 0.427 0.950 0.977 0.995 1.000
100 0.522 0.998 1.000 1.000 1.000 0.503 0.995 1.000 1.000 1.000
f(ri, βt) = riβt/(1 + r2i β2t )
10 0.159 0.360 0.477 0.762 0.972 0.152 0.312 0.413 0.688 0.916
20 0.167 0.533 0.650 0.892 0.990 0.170 0.517 0.626 0.869 0.986
30 0.287 0.563 0.716 0.971 1.000 0.278 0.560 0.708 0.967 1.000
50 0.304 0.750 0.967 0.991 1.000 0.297 0.743 0.961 0.990 1.000
100 0.604 0.914 0.999 1.000 1.000 0.569 0.915 0.999 1.000 1.000
18 J. Chen, J. Gao and D. Li
Table 5.4(c) Power of the tests for nonlinear model (5.2) at the 10% level
parametric test nonparametric test
T\N 10 20 30 50 100 10 20 30 50 100
f(ri, βt) = riβt
10 0.205 0.461 0.620 0.893 0.977 0.200 0.419 0.547 0.853 0.956
20 0.327 0.558 0.797 0.958 1.000 0.310 0.544 0.776 0.950 0.998
30 0.357 0.802 0.883 0.990 1.000 0.350 0.799 0.876 0.990 1.000
50 0.529 0.966 0.987 0.997 1.000 0.517 0.970 0.989 0.997 1.000
100 0.608 0.999 1.000 1.000 1.000 0.607 1.000 1.000 1.000 1.000
f(ri, βt) = riβt/(1 + r2i β2t )
10 0.216 0.429 0.550 0.820 0.984 0.215 0.381 0.479 0.742 0.932
20 0.247 0.606 0.729 0.917 0.995 0.255 0.589 0.692 0.900 0.990
30 0.364 0.664 0.768 0.983 1.000 0.362 0.648 0.764 0.979 1.000
50 0.396 0.812 0.984 0.994 1.000 0.394 0.807 0.978 0.994 1.000
100 0.691 0.943 0.999 1.000 1.000 0.668 0.941 0.999 1.000 1.000
Tables 5.4(a)–5.4(c) show that the simulated power values are quite satisfactory in each of
the cases concerned. Meanwhile, the simulated power values of the nonparametric test show
that the nonparametric CD test is only slightly less powerful than the parametric CD test.
In summary, we can conclude that in both the parametric linear and nonlinear models,
the nonparametric CD test has the correct size even for small N and T . While the power of
the proposed nonparametric CD test increases as N or T increases. it increases faster with N
than with T . Similar findings have been drawn from Hsiao, Pesaran and Pick (2007) for the
parametric CD test.
This shows that the proposed nonparametric CD test is a generally applicable test in this
kind of testing for cross–section independence, as the applicability does not require a model to
be parametrically specified. In other words, it still works well without necessarily pre–specifying
the conditional mean function.
In the following example, we show that the proposed nonparametric CD test is needed
when the data follow a nonparametric panel data model, since existing tests for the parametric
case are not applicable.
Consider a nonparametric panel data model of the form
Yit =Xit
1 +X2it
+ uit, i = 1, 2, · · · , N ; t = 1, 2, · · · , T, (5.3)
where Xiti.i.d.∼ N(0, 1), and uit is the same as used in model (5.1). For ri = 0, the sizes of
the proposed nonparametric CD test are reported in Table 5.5, and for ri ∼ U(0.1, 0.3), the
power values are given in Table 5.6.
Nonparametric Test for Cross–Section Independence 19
Table 5.5(a) Size of the nonparametric test for model (5.3) at the 1% level
T\N 10 20 30 50 100
10 0.022 0.022 0.015 0.028 0.020
20 0.013 0.014 0.014 0.020 0.011
30 0.012 0.017 0.011 0.008 0.014
50 0.012 0.012 0.009 0.010 0.015
100 0.013 0.011 0.011 0.009 0.007
Table 5.5(b) Size of the nonparametric test for model (5.3) at the 5% level
T\N 10 20 30 50 100
10 0.053 0.049 0.044 0.062 0.046
20 0.049 0.049 0.046 0.057 0.046
30 0.040 0.053 0.046 0.041 0.049
50 0.044 0.044 0.041 0.052 0.045
100 0.052 0.046 0.047 0.052 0.047
Table 5.5(c) Size of the nonparametric test for model (5.3) at the 10% level
T\N 10 20 30 50 100
10 0.088 0.098 0.079 0.096 0.084
20 0.100 0.091 0.090 0.092 0.087
30 0.105 0.098 0.088 0.098 0.103
50 0.091 0.095 0.091 0.088 0.105
100 0.106 0.099 0.091 0.096 0.095
Table 5.6(a) Power of the nonparametric test for model (5.3) at the 1% level
f(ri, βt) = riβt f(ri, βt) = riβt/(1 + r2i β2t )
T\N 10 20 30 50 100 10 20 30 50 100
10 0.083 0.232 0.250 0.473 0.771 0.051 0.110 0.184 0.325 0.694
20 0.061 0.255 0.500 0.795 0.966 0.080 0.185 0.348 0.648 0.942
30 0.105 0.422 0.603 0.928 0.999 0.073 0.264 0.555 0.836 0.994
50 0.189 0.381 0.865 0.984 1.000 0.104 0.427 0.678 0.917 0.998
100 0.440 0.781 0.944 1.000 1.000 0.136 0.698 0.952 0.999 1.000
20 J. Chen, J. Gao and D. Li
Table 5.6(b) Power of the nonparametric test for model (5.3) at the 5% level
f(ri, βt) = riβt f(ri, βt) = riβt/(1 + r2i β2t )
T\N 10 20 30 50 100 10 20 30 50 100
10 0.149 0.326 0.374 0.581 0.848 0.117 0.192 0.296 0.450 0.787
20 0.136 0.376 0.652 0.879 0.986 0.161 0.308 0.486 0.757 0.969
30 0.201 0.584 0.723 0.960 0.999 0.169 0.418 0.711 0.920 0.999
50 0.338 0.556 0.927 0.997 1.000 0.210 0.602 0.823 0.964 1.000
100 0.653 0.898 0.979 1.000 1.000 0.261 0.846 0.982 1.000 1.000
Table 5.6(c) Power of the nonparametric test for model (5.3) at the 10% level
f(ri, βt) = riβt f(ri, βt) = riβt/(1 + r2i β2t )
T\N 10 20 30 50 100 10 20 30 50 100
10 0.215 0.391 0.438 0.646 0.876 0.180 0.266 0.355 0.516 0.826
20 0.217 0.466 0.730 0.908 0.987 0.212 0.405 0.562 0.811 0.981
30 0.274 0.655 0.786 0.970 1.000 0.231 0.503 0.776 0.944 1.000
50 0.434 0.666 0.959 0.998 1.000 0.289 0.694 0.882 0.980 1.000
100 0.743 0.931 0.988 1.000 1.000 0.351 0.893 0.991 1.000 1.000
Tables 5.5(a)–5.5(c) show that the nonparametric CD test has the correct sizes for the
simulated nonparametric panel data model (5.3). Meanwhile, Tables 5.6(a)–5.6(c) show that
the simulated power values of the nonparametric CD test are also satisfactory.
6. Empirical application: An analysis of CPI in Australian capital cities
As an application of our testing method, we test for the cross–sectional independence of
CPI (consumer price index) between eight Australian capital cities during the period 1989–
2008. The data set, which is obtained from the website of the Australian Bureau of Statistics,
is recorded quarterly each year. Hence, it consists of the CPI numbers for eight cities (N = 8)
at 80 different times (T = 80). We chose Yit as the log of the food CPI for city i at time
t and Xit as the log of all group CPI for city i at time t. For each city i, we computed
the nonparametric regression function of Yit on Xit (t = 1, 2, · · · , T ) using the nonparametric
local linear estimation method. Then, we used the estimation residuals uit to compute the
nonparametric CD test statistic. In a similar way, we also computed the regression of log of
the transportation CPI on log of all group CPI for each city. The results are summarized in
Table 6.1.
Nonparametric Test for Cross–Section Independence 21
Table 6.1 Cross section dependence of CPI in Australian capital cities
food transportation
nonparametric CD test 47.2378 47.0227
bootstrap 1% critical values [−2.3130, 2.6100] [−2.4895, 2.7300]
bootstrap 5% critical values [−1.8796, 1.8517] [−1.8786, 1.8899]
bootstrap 10% critical values [−1.6086, 1.5584] [−1.6532, 1.6203]
Note that the two-sided bootstrap critical values were calculated using 1000 iterations.
It follows from Table 6.1 that there is some evidence to suggest rejecting the null hypothesis
that the cross–section independence is true for both the food and transportation indexes.
Meanwhile, based on the bootstrap simulated critical value in each case, the cross–section
independence should be rejected at all the levels of 1%, 5% and 10%.
This suggests that the assumption of cross–section independence in such empirical studies
may not be appropriate. Further studies are needed to find ways of defining a suitable cross–
section dependence structure in order to deal with panel data analysis when there is some
cross–section dependence.
7. Conclusions and discussion
We have proposed a new diagnostic test for residual cross–section independence in a non-
parametric panel data model. The proposed test is a nonparametric counterpart of an existing
test proposed in Pesaren (2004) for the parametric case. The asymptotic distribution under
either the null or a sequence of local alternatives has been established. The small sample per-
formance of the proposed test has been examined in Section 5. Section 6 has given an example
of empirical application.
Future research in this field includes discussion about how to choose a data–driven band-
width such that both the resulting size and power functions are appropriately assessed. As
pointed out in Section 4, certain extensions of the model may also be considered. Since study
of such topics is not trivial, they are left for future research.
8. Acknowledgments
This work was motivated by a keynote presentation by Professor Cheng Hsiao at an In-
ternational Conference on Time Series Econometrics at WISE in Xiamen, China in May 2008
when the second author was a participant at the conference. The second author would like to
thank Professor Yongmiao Hong for his invitation to participate in the conference. The authors
would all like to acknowledge the Australian Research Council Discovery Grants Program for
its financial support under Grant Numbers: DP0558602 and DP0879088.
Appendix A: Proofs of the main results
22 J. Chen, J. Gao and D. Li
Before proving the main results, we need the following lemma on the uniform consistency
of nonparametric estimators. Since the proposed test statistic is invariant to σ2ui = E[u2
i1], we
assume without loss of generality that σui ≡ 1 throughout this appendix. In addition, we use a
double sum of the form∑Tt=1
∑s6=t to replace either
∑Tt=1
∑Ts=1,6=t or
∑Tt=1
∑Ts6=t for notational
simplicity throughout this appendix.
Lemma A.1. Assume that A1(i) and A2(i) are satisfied. If, in addition, T θhlog T → ∞ as
T →∞, where θ = β−3β+2 , then we have for k = 0, 1, 2,
supx∈R
|Sik(x)− fi(x)µk| = oP (1),
uniformly in i ≥ 1, where µk =∫ukK(u)du.
Proof. Observe that
supx∈R
|Sik(x)− fi(x)µk| ≤ sup|x|≤cT
|Sik(x)− fi(x)µk|+ sup|x|≥cT
|Sik(x)− fi(x)µk|
uniformly in i ≥ 1, where cT = T 1/2 log T .
For any ε > 0, by Theorem 6 in Hansen (2008), we know that there exists an integer T0
such that when T > T0, we have
P
(maxi≥1
sup|x|≤cT
|Sik(x)− fi(x)µk| ≤ ε/2
)→ 1.
Since fi(·) is continuous and integrable, maxi≥1 sup|x|≥cT
|fi(x)| → 0 as T →∞. Hence, when
µk 6= 0,
P
(maxi≥1 sup
|x|≥cT|Sik(x)− fi(x)µk| > ε/2
)
≤ P
(maxi≥1 sup
|x|≥cT|Sik(x)| > ε/4
)+ P
(maxi≥1 sup
|x|≥cT|fi(x)| > ε/(4|µk|)
)
= P
(maxi≥1 sup
|x|≥cT
1Th
∣∣∣∣∣ T∑t=1
(Xit−xh
)kK(Xit−xh
)∣∣∣∣∣ > ε/4
)
+ P
(maxi≥1 sup
|x|≥cT|fi(x)| > ε/(4|µk|)
)
≤T∑t=1
P (maxi≥1 |Xit| ≥ cT − Ch) + P
(maxi≥1 sup
|x|≥cT|fi(x)| > ε/(4|µk|)
)
≤ CT (cT )−2 maxi≥1E|Xi1|2 + P
(maxi≥1 sup
|x|≥cT|fi(x)| > ε/(4|µk|)
)→ 0,
where C is some positive constant. When µk = 0, from the above argument we can see that
Nonparametric Test for Cross–Section Independence 23
P
(maxi≥1 sup
|x|≥cT|Sik(x)| > ε/2
)→ 0. Therefore, we have
P
(maxi≥1
supx∈R
|Sik(x)− fi(x)µk| > ε
)→ 0,
which completes the proof of Lemma A.1.
We then give the well–known Davydovs inequality for α–mixing sequence, which follows
from Corollary A2 of Hall and Heyde (1980). An updated version is given in Lemma A.1 of
Gao (2007).
Lemma A.2. Suppose that E|X|p <∞ and E|Y |q <∞, where p, q > 1, p−1 + q−1 < 1. Then
|E(XY )− (EX)(EY )| ≤ 8(E|X|p)1/p(E|Y |q)1/qα1−p−1−q−1,
where α = supA∈σ(X),B∈σ(Y )
|P (AB)− P (A)P (B)|.
Proof of Theorem 3.1. Note that
uit = (Yit − gi(Xit))fi(Xit) = uitfi(Xit) + (gi(Xit)− gi(Xit))fi(Xit).
Hence, by a standard decomposition, we have
N∑i=1
∑j 6=i
T∑t=1
uitujt =N∑i=1
∑j 6=i
T∑t=1
uitfi(Xit)ujtfj(Xjt)−N∑i=1
∑j 6=i
T∑t=1
uitfi(Xit)
(1T
T∑s=1
ujsKjst
)
+N∑i=1
∑j 6=i
T∑t=1
uitfi(Xit)
(1T
T∑s=1
(gj(Xjt)− gj(Xjs)) Kjst
)
−N∑i=1
∑j 6=i
T∑t=1
ujtfj(Xjt)
(1T
T∑s=1
uisKist
)
+N∑i=1
∑j 6=i
T∑t=1
ujtfj(Xjt)
(1T
T∑s=1
(gi(Xit)− gi(Xis)) Kist
)+
N∑i=1
∑j 6=i
T∑t=1
(gi(Xit)− gi(Xit)) (gj(Xjt)− gj(Xjt)) fi(Xit)fj(Xjt)
=:N∑i=1
∑j 6=i
6∑k=1
ρT (i, j, k),
(A.1)
where Kist = KXit,h(Xis).
In the following, we complete the proof of Theorem 3.1 through using Lemmas A.3–A.7
below.
Lemma A.3. Assume that the conditions of Theorem 3.1 are satisfied. Then under H0, we
haveN∑i=1
∑j 6=i
ρT (i, j, 2) = oP (N√T ),
N∑i=1
∑j 6=i
ρT (i, j, 4) = oP (N√T ). (A.2)
24 J. Chen, J. Gao and D. Li
Proof. We only give the detailed proof for the case ofN∑i=1
∑j 6=i
ρT (i, j, 2) since the proof for
N∑i=1
∑j 6=i
ρT (i, j, 4) is similar. Observe that under H0,
E
N∑i=1
∑j 6=i
ρT (i, j, 2)
2
=1T 4E
N∑i=1
∑j 6=i
T∑t=1
uit
(T∑s=1
Kist
)(T∑l=1
ujlKjlt
)2
=1T 4E
N∑i=1
∑j 6=i
T∑t1,t2=1
T∑l1,l2=1
T∑s1,s2=1
uit1uit2ujl1ujl2Kis1t1K
is2t2K
jl1t1
Kjl2t2
≤ C
T 4E
N∑i=1
∑j 6=i
T∑t=1
T∑l=1
T∑s=1
u2itu
2jl(K
ist)
2(Kjlt)
2
+
C
T 4E
N∑i=1
∑j 6=i
T∑t=1
T∑l=1
T∑s1 6=s2
u2itu
2jlK
is1tK
is2t(K
jlt)
2
+
C
T 4E
N∑i=1
∑j 6=i
T∑t1 6=t2
T∑l=1
T∑s=1
uit1uit2u2jlK
ist1K
ist2(K
jlt)
2
+
C
T 4E
N∑i=1
∑j 6=i
T∑t=1
T∑l1 6=l2
T∑s=1
u2itujl1ujl2(K
ist)
2Kjl1tKjl2t
+
C
T 4E
N∑i=1
∑j 6=i
T∑t1 6=t2
T∑l=1
T∑s1 6=s2
uit1uit2u2jlK
is1t1K
is2t2(K
jlt)
2
+
C
T 4E
N∑i=1
∑j 6=i
T∑t=1
T∑l1 6=l2
T∑s1 6=s2
u2itujl1ujl2K
is1tK
is2tK
jl1tKjl2t
+
C
T 4E
N∑i=1
∑j 6=i
T∑t1 6=t2
T∑l1 6=l2
T∑s=1
uit1uit2ujl1ujl2Kist1K
ist2K
jl1t1
Kjl2t2
+
C
T 4E
N∑i=1
∑j 6=i
T∑t1 6=t2
T∑l1 6=l2
T∑s1 6=s2
uit1uit2ujl1ujl2Kis1t1K
is2t2K
jl1t1
Kjl2t2
=:
8∑k=1
Πk.
By Lemma A.1 and µ1 = 0, we have
supx∈R
∣∣∣∣hKx,h(Xit)−K
(Xit − x
h
)(fi(x)µ2 − oP (1) ·
(Xit − x
h
))∣∣∣∣ = oP (1),
Nonparametric Test for Cross–Section Independence 25
which, by A1 (iii), implies that
Kx,h(Xit) ≤ C1h−1K
(Xit − x
h
), (A.3)
where C1 is independent of x and Xit. Let Kist = K
(Xit−Xis
h
).
For Π1, by (A.3), we have
Π1 ≤ C
T 4h4E
N∑i=1
∑j 6=i
T∑t=1
u2itu
2jt
+C
T 4h4E
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
u2itu
2jt
((Ki
st)2 + (Kj
st)2)
+1
T 4h4E
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
∑l 6=t
u2itu
2jl(K
ist)
2(Kjlt)
2
= O
(N2T−3h−4
)+O
1T 4h4
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
E[(Ki
st)2 + (Kj
st)2]
+ O
1T 4h4
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
∑l 6=t
E[(Ki
st)2(Kj
lt)2]
= O(N2T−3h−4
)+O
1T 4h4
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
∫ ∫K2
(w − v
h
)fis,it(v, w)dvdw
+ O
1T 4h4
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
∫ ∫K2
(w − v
h
)fjs,jt(v, w)dvdw
+ O
1T 4h4
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
∑l 6=t
∫ ∫ ∫ ∫K2
(v1 − u1
h
)
× K2(u2 − v2h
)fis,it,jl,jt(u1, v1, u2, v2)du1du2dv1dv2
)= O
(N2
T 3h4+
N2
T 2h3+
N2
Th2
).
Therefore,
Π1 = O
(N2
Th2
). (A.4)
On the other hand, for Π2, we have
Π2 ≤ C
T 4h4E
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
T∑l=1
u2itu
2jlK
ist(K
jlt)
2
+
C
T 4h4E
N∑i=1
∑j 6=i
T∑t=1
∑s1 6=t
∑s2 6=t,s1
T∑l=1
u2itu
2jlK
is1tK
is2t(K
jlt)
2
= O
1T 4h4
E
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
Kist
+O
1T 4h4
E
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
∑l 6=t
Kist(K
jlt)
2
26 J. Chen, J. Gao and D. Li
+ O
1T 4h4
E
N∑i=1
∑j 6=i
T∑t=1
∑s1 6=t
∑s2 6=t,s1
Kis1tK
is2t
+ O
1T 4h4
E
N∑i=1
∑j 6=i
T∑t=1
∑s1 6=t
∑s2 6=t,s1
∑l 6=t
Kis1tK
is2t(K
jlt)
2
≤ O
N
T 4h4
N∑i=1
T∑t=1
∑s6=t
∫ ∫K
(w − v
h
)fis,it(v, w)dvdw
+ O
1T 4h4
N∑i=1
∑j 6=i
T∑t=1
∑s6=t
∑l 6=t
∫ ∫ ∫ ∫K
(v1 − u1
h
)
× K2(v2 − u2
h
)fis,it,jl,jt(u1, v1, u2, v2)du1du2dv1dv2
)
+ O
1T 4h4
N∑i=1
∑j 6=i
T∑t=1
∑s1 6=t
∑s2 6=t,s1
∫ ∫ ∫K
(v − u
h
)
× K
(w − u
h
)fis1,is2,it(u, v, w)dudvdw
)
+ O
1T 4h4
N∑i=1
∑j 6=i
T∑t=1
∑s1 6=t
∑s2 6=t,s1
∑l 6=t
∫ ∫ ∫ ∫ ∫K
(v1 − u1
h
)K
(w1 − u1
h
)
× K2(v2 − u2
h
)fis1,is2,it,jl,jt(u1, v1, w1, u2, v2)du1du2dv1dv2dw1
)= O
(N2
T 2h3+
N2
Th2+N2
h
).
Therefore, we have
Π2 = O
(N2
h
). (A.5)
By A2 (ii) and Lemma A.2, we have
|E [uit1uit2 ]− E [uit1 ]E [uit2 ]| ≤ C0αδ0
2+δ0u (|t1 − t2|), (A.6)
where C0 is some positive constant. Hence, by the α–mixing coefficient condition in A2 (ii),
(A.6) and following the calculation of Π2, we have
Πk = O
(N2
h
), k = 3, · · · , 8. (A.7)
In view of (A.4), (A.5) and (A.7), we have
N∑i=1
∑j 6=i
ρT (i, j, 2) = OP (Nh−1/2) = oP (N√T ) (A.8)
since Th→∞ by A3.
Nonparametric Test for Cross–Section Independence 27
Lemma A.4. Assume that the conditions of Theorem 3.1 are satisfied. Then under H0, we
haveN∑i=1
∑j 6=i
ρT (i, j, k) = oP (N√T ), k = 3, 5, 6. (A.9)
Proof. For any x, by A1 (ii) and the definition of the local linear estimator, we have
(gi(x)− gi(x))ft(x) =1T
T∑t=1
Kx,h(Xit)uit +g′′i (x)2T
T∑t=1
(Xit − x)2Kx,h(Xit). (A.10)
ForN∑i=1
∑j 6=i
ρT (i, j, 6), note that, by (A.10),
N∑i=1
∑j 6=i
ρT (i, j, 6)
=N∑i=1
∑j 6=i
T∑t=1
(gi(Xit)− gi(Xit))(gj(Xjt)− gj(Xjt))fi(Xit)fj(Xjt)
=1T 2
N∑i=1
∑j 6=i
T∑t=1
T∑s1=1
T∑s2=1
Kis1tK
js2tuis1ujs2
+1
2T 2
N∑i=1
∑j 6=i
T∑t=1
g′′j (Xjt)T∑
s1=1
T∑s2=1
(Xjs2 −Xjt)2Kis1tK
js2tuis1
+1
2T 2
N∑i=1
∑j 6=i
T∑t=1
g′′i (Xit)T∑
s1=1
T∑s2=1
(Xis1 −Xit)2Kis1tK
js2tujs2
+1
4T 2
N∑i=1
∑j 6=i
T∑t=1
g′′i (Xit)g′′j (Xjt)T∑
s1=1
T∑s2=1
(Xis1 −Xit)2Kis1t(Xjs2 −Xjt)2K
js2t
=:N∑i=1
∑j 6=i
4∑k=1
ρT (i, j, 6, k).
Similarly to the calculation ofN∑i=1
∑j 6=i
ρT (i, j, 2), we have
N∑i=1
∑j 6=i
ρT (i, j, 6, 1) = oP (N√T ). (A.11)
By A1(i)–(iii), we have
E
∣∣∣∣∣∣N∑i=1
∑j 6=i
ρT (i, j, 6, 4)
∣∣∣∣∣∣≤ Ch4
T 2h2E
∣∣∣∣∣∣N∑i=1
∑j 6=i
T∑t=1
g′′i (Xit)g′′j (Xjt)∑s1 6=t
(Xis1 −Xit
h
)2
K
(Xis1 −Xit
h
)
28 J. Chen, J. Gao and D. Li
×∑s2 6=t
(Xjs2 −Xjt
h
)2
K
(Xjs2 −Xjt
h
)∣∣∣∣∣∣ (1 + o(1))
= O(N2Th4).
Since N2Th8 → 0 by A3, we have
N∑i=1
∑j 6=i
ρT (i, j, 6, 4) = OP (N2Th4) = oP (N√T ). (A.12)
By (A.11), (A.12) and the Cauchy–Schwarz inequality, we have
N∑i=1
∑j 6=i
ρT (i, j, 6, 2) = oP (N√T ) (A.13)
andN∑i=1
∑j 6=i
ρT (i, j, 6, 3) = oP (N√T ). (A.14)
It then follows from (A.11)–(A.14) that
N∑i=1
∑j 6=i
ρT (i, j, 6) = oP(N√T). (A.15)
By the arguments and derivations as inN∑i=1
∑j 6=i
ρT (i, j, 6), we have
N∑i=1
∑j 6=i
ρT (i, j, 3) = oP(N√T)
andN∑i=1
∑j 6=i
ρT (i, j, 5) = oP(N√T). (A.16)
Lemma A.5. Assume that the conditions of Theorem 3.1 are satisfied. Then under H0, we
haveN∑i=1
∑j 6=i
ρT (i, j, 1) =N∑i=1
∑j 6=i
T∑t=1
uitujtf2i (Xit)f2
j (Xjt)µ22µ
20 + oP (N
√T ). (A.17)
Proof. The proof can be done in a similar way to that of Lemma A.4 above. In the following,
we adopt a simplified way to complete the proof. Note that
ρT (i, j, 1) =T∑t=1
uitfi(Xit)ujtfj(Xjt) =T∑t=1
uitujt
×[f2i (Xit)µ2µ0 + fi(Xit)− f2
i (Xit)µ2µ0
] [f2j (Xjt)µ2µ0 + fj(Xjt)− f2
j (Xjt)µ2µ0
]=
T∑t=1
uitujtf2i (Xit)f2
j (Xjt)µ22µ
20
Nonparametric Test for Cross–Section Independence 29
+T∑t=1
uitujt(fi(Xit)− f2
i (Xit)µ2µ0
)f2j (Xjt)µ2µ0
+T∑t=1
uitujt(fj(Xjt)− f2
j (Xjt)µ2µ0
)f2i (Xit)µ2µ0
−T∑t=1
uitujt(fi(Xit)− f2
i (Xit)µ2µ0
) (fj(Xjt)− f2
j (Xjt)µ2µ0
)
=:4∑
k=1
ρT (i, j, 1, k).
For any ε > 0,
P
N∑i=1
N∑j=1,6=i
ρT (i, j, 1, 4) > εN√T
= P
N∑i=1
N∑j=1,6=i
ρT (i, j, 1, 4) > εN√T , Ω(η)
+ P
N∑i=1
N∑j=1,6=i
ρT (i, j, 1, 4) > εN√T , Ωc(η)
, (A.18)
where Ω(η) :=
maxi≥1 sup
x∈R
∣∣∣fi(x)− f2i (x)µ2µ0
∣∣∣ < η, maxj≥1 supx∈R
∣∣∣fj(x)− f2j (x)µ2µ0
∣∣∣ < η
.
By Lemma A.1,
P
N∑i=1
N∑j=1,6=i
ρT (i, j, 1, 4) > εN√T , Ωc(η)
≤ P (Ωc(η)) → 0. (A.19)
Let δi(Xit) = fi(Xit)− f2i (Xit)µ2µ0. Meanwhile, for each fixed (i, j)
E [ρT (i, j, 1, 4)I(Ω(η))]2 =T∑t=1
E[u2itu
2jtδ
2i (Xit)δ2j (Xjt)I(Ω(η))
]
+T∑t=1
T∑s=1,6=t
E [uituisujtujs]E [δi(Xit)δi(Xis)δj(Xjt)δj(Xjs)I(Ω(η))]
≤T∑t=1
E[u2itu
2jtδ
2i (Xit)δ2j (Xjt)I(Ω(η))
]
+12
T∑t=1
T∑s=1,6=t
|E [uituisujtujs]|E[(δ2i (Xit)δ2j (Xjt) + δ2i (Xis)δ2j (Xjs)
)I(Ω(η))
]
≤T∑t=1
E[u2itu
2jtδ
2i (Xit)δ2j (Xjt)I(Ω(η))
]
+
(T∑t=1
|E [uitui1ujtuj1]|)(
T∑s=1
E[δ2i (Xis)δ2j (Xjs)I(Ω(η))
])
≤ Cη4T∑t=1
E[u2itu
2jt
]= O(η2T ), (A.20)
30 J. Chen, J. Gao and D. Li
where I(A) is the indicator function of a set A, and we have used that uit and Xjt are
mutually independent as well as the fact that∑Tt=1 |E [uitui1ujtuj1]| < ∞ for all (i, j), which
all follow from condition A2 and Lemma A.2.
Let vit = uitδi(Xit)I(Ω(η)). In a similar way, we can derive that for T and N large enough
E
N∑i=2
i−1∑j=1
T∑t=1
vitvjt
2
=N∑i=2
i−1∑j=1
T∑t=1
E[u2itu
2jtδ
2i (Xit)δ2j (Xjt)I(Ω(η))
]
+ 2N∑i=2
i−1∑j=1
T∑t=2
t−1∑s=1
E [uisuitujsujt]E [δi(Xit)δi(Xis)δj(Xjt)δj(Xjt)I(Ω(η))]
+ 2N∑i=3
i−1∑j1=1
j1−1∑j2=1
T∑t=1
E[u2ituj1tuj2t
]E[δ2i (Xit)δj1(Xj1t)δj2(Xj2t)I(Ω(η))
]
+ 4N∑i=3
i−1∑j1=1
j1−1∑j2=1
T∑t=2
t−1∑s=1
E [uisuituj1suj2t]E [δi(Xis)δi(Xit)δj1(Xj1s)δj2(Xj2t)I(Ω(η))]
+ 4N∑i1=4
i1−1∑i2=3
i1−1∑j1=1
i2−1∑j2=1
T∑t=1
E [ui1tui2tuj1tuj2t]E [δi1(Xi1t)δi2(Xi2t)δj1(Xj1t)δj2(Xj2t)I(Ω(η))]
+ 8N∑i1=4
i1−1∑i2=3
i1−1∑j1=1
i2−1∑j2=1
T∑t=2
t−1∑s=1
E [ui1sui2tuj1suj2t]
× E [δi1(Xi1s)δi2(Xi2t)δj1(Xj1s)δj2(Xj2t)I(Ω(η))]
≤ Cη4N∑i=2
i−1∑j=1
T∑t=1
E[u2itu
2jt
]= O(η4N2T ), (A.21)
where condition A.2 and Lemma A.2 have been repeatedly used under H0, under which
E[ukituljt] = E[ukit]E[uljt] hold for all i 6= j, all t ≥ 1 and k, l = 1, 2.
Letting η → 0, it follows from (A.18)–(A.21) that
N∑i=1
∑j 6=i
ρT (i, j, 1, 4) = oP (N√T ). (A.22)
Note thatN∑i=1
∑j 6=i
ρT (i, j, 1, 1) = OP (N√T ).
By the above equation, (A.22) and the Cauchy–Schwarz inequality, we have
N∑i=1
∑j 6=i
ρT (i, j, 1, 2) = oP (N√T ) and
N∑i=1
∑j 6=i
ρT (i, j, 1, 3) = oP (N√T ). (A.23)
Then (A.17) follows from (A.22) and (A.23).
Lemma A.6. Assume that the conditions of Theorem 3.1 are satisfied. Then under H0, we
Nonparametric Test for Cross–Section Independence 31
have for each fixed i and as T →∞,
1T
T∑t=1
u2it = σ2
i + oP (1), (A.24)
where σ2i = µ2
2µ20
∫f5i (x)dx for each fixed i.
Proof. Observe that
1T
T∑t=1
u2it =
1T
T∑t=1
(uitf
2i (Xit)µ2µ0 + uit − ui,tf
2i (Xit)µ2µ0
)2
=1T
T∑t=1
(uitf
2i (Xit)µ2µ0 + ui,t(fi(Xit)− f2
i (Xit)µ2µ0)
− 1T
T∑s=1
KXit,h(Xis)uis −1T
T∑s=1
KXit,h(Xis)(gi(Xis)− gi(Xit)))2
By A1 (iii) and the law of large numbers for α–mixing sequence (cf. Lin and Lu 1996), we
have1T
T∑t=1
u2itf
4i (Xit)µ2
2µ20 = σ2
i + oP (1). (A.25)
Similarly to the calculation ofN∑i=1
∑j 6=i
ρT (i, j, 1, 4), we have
E
[T∑t=1
u2it
(fi(Xit)− f2
i (Xit)µ2µ0
)2]
= o(T ). (A.26)
Furthermore, by (A.3) and standard calculation, we have
E
T∑t=1
(1T
T∑s=1
KXit,h1(Xis)uis
)2 ≤ C
T 2h2
T∑t=1
T∑s=1
E(K2
(Xis−Xit
h
)u2is
)= O(h−1) = o(T ).
(A.27)
By Taylor expansion and a calculation similar to ρT (i, j, 6, 4),
E
T∑t=1
(1T
T∑s=1
KXit,h(Xis)(gi(Xis)− gi(Xit))
)2 = O(Th2) = o(T ). (A.28)
By (A.25)–(A.28), we have
T∑t=1
u2it
(fi(Xit)− f2
i (Xit)µ2µ0
)2= oP (T ),
T∑t=1
(1T
T∑s=1
KXit,h(Xis)uis
)2
= oP (T ),
T∑t=1
(1T
T∑s=1
KXit,h(Xis)(gi(Xis)− gi(Xit))
)2
= oP (T ),
32 J. Chen, J. Gao and D. Li
which imply that (A.24) holds.
Lemma A.7. Define Zit = uitf2i (Xit)µ0µ2
σi. Let the conditions of Theorem 3.1 hold. Then under
H0, we have1√
N(N − 1)T
N∑i=1
N∑j 6=i
T∑t=1
ZitZjtd−→ N(0, τ0) (A.29)
as T →∞ first and then N →∞, where τ0 is as defined in Theorem 3.1.
Proof. Let Zit = σi√τi,jZit. By the central limit theorem for stationary α–mixing sequence (cf.
Lin and Lu 1996), we have for each fixed (i, j) and as T →∞
1√T
T∑t=1
ZitZjtd−→ N(0, 1), (A.30)
in view of the fact that under H0 we have E[ZitZjt
]= 0 and
E
[1√T
T∑t=1
ZitZjt
]2
=1T
T∑t=1
E[ZitZjt
]2+
1T
T∑t=1
T∑s=1,6=s
E[ZitZjtZitZjt
]
=µ4
0µ42
τ2ij
1T
T∑t=1
E[u2it]E[u2
jt]E[f4i (Xit)f4
j (Xjt)]
+µ4
0µ42
τ2ij
1T
T∑t=1
T∑s=1,6=t
E[uituis]E[ujtujs]E[fi(Xitfi(Xis)fj(Xjtfj(Xjs)]
= 1 + o(1)
for large enough T and all fixed (i, j).
Let Wij = 1√T
τijσiσj
T∑t=1
ZitZjt. Note that under H0, Wij and Wkl are uncorrelated for
all (i, j) 6= (k, l). Thus, by equation (A.30) and the continuous mapping theorem (see, for
example, Corollary 2 of Billingsley 1968, p. 31), we have as N →∞
1√N(N − 1)
N∑i=1
N∑j=1,6=i
Wijd−→ N(0, τ0), (A.31)
which implies that (A.29) holds.
Proof of Theorem 3.2. We start the proof of Theorem 3.2(ii). Following the proof of
Theorem 3.1 as above, we need only to show that
1N(N − 1)T
N∑i=1
∑j 6=i
T∑t=1
G(zt, βi)G(zt, βj)P−→ ψ, (A.32)
N∑i=1
∑j 6=i
T∑t=1
F (zt, βi)
(1T
T∑s=1
εjsKjst
)= oP (N
√T ), (A.33)
Nonparametric Test for Cross–Section Independence 33
N∑i=1
∑j 6=i
T∑t=1
F (zt, βi)
(1T
T∑s=1
F (zs, βj)Kjst
)= oP (N
√T ), (A.34)
N∑i=1
∑j 6=i
T∑t=1
1T
T∑s1=1
F (zs1 , βi)Kis1t
1T
T∑s2=1
F (zs2 , βj)Kjs2t
= oP (N√T ), (A.35)
N∑i=1
∑j 6=i
T∑t=1
1T
T∑s1=1
F (zs1 , βi)Kis1t
1T
T∑s2=1
εjs2Kjs2t
= oP (N√T ), (A.36)
N∑i=1
∑j 6=i
T∑t=1
εit
(1T
T∑s=1
F (zs, βj)Kjst
)= oP (N
√T ), (A.37)
1T
T∑t=1
F 2(zt, βi) → 0, (A.38)
where F (zt, βi) := FNT (zt, βi).
By the law of large numbers for stationary α–mixing sequences, under A4(i) and (ii) we
have1T
T∑t=1
G(zt, βi)G(zt, βj) → ψij , (A.39)
which, together with (3.8) and (3.9), implies that (A.32) holds.
By (3.8) and the law of large numbers for stationary α–mixing sequences, we can show
that (A.38) holds analogously.
We now start to prove (A.33). By A3, A4 and Lemma A.2, we have
E
N∑i=1
∑j 6=i
T∑t=1
F (zt, βi)
(1T
T∑s=1
εjsKjst
)2
≤ C
T 2h2
N∑i1=1
N∑i2=1
∑j 6=i1,i2
T∑t1=1
T∑t2=1
T∑s=1
E[F (zt1 , βi1)F (zt2 , βi2)]E[ε2js
]E[Kjst1K
jst2
]
≤ C
T 2h2
N∑i1=1
N∑i2=1
∑j 6=i1,i2
T∑t1=1
T∑t2=1
T∑s=1
h2E [F (zt1 , βi1)F (zt2 , βi2)]E[ε2js
]
≤ C
T 2h2
N∑i1=1
N∑i2=1
∑j 6=i1,i2
T 2h2
NT 1/2
= O(N2T−1/2
),
which, by Markov inequality, implies that (A.33) holds. The proofs of (A.34) and (A.37) are
similar to that of (A.33).
We then prove (A.35). By A4 and Lemma A.2, we have
E
1T
T∑s1=1
F (zs1 , βi)Kis1,t
1T
T∑s2=1
F (zs2 , βj)Kjs2,t
2
34 J. Chen, J. Gao and D. Li
≤ 1T 4h4
T∑s1=1
T∑t1=1
T∑s2=1
T∑t2=1
E [F (zs1 , βi)F (zt1 , βi)F (zs2 , βj)F (zt2 , βj)]
× E[Kis1tK
it1tK
js2tK
jt2t
]= O
(1
N2T 3
),
which, by Markov inequality, implies that (A.35) holds. By the same argument, we can show
that (A.36) holds. The proof of Theorem 3.2(ii) is therefore completed.
Let G(zt, βi) =(N1/2T 1/4G(zt, βi)
). In view of
FNT (zt, βi) = G(zt, βi) =1
N1/2T 1/4
(N1/2T 1/4G(zt, βi)
)=
1N1/2T 1/4
G(zt, βi),
the proof of Theorem 3.2(i) follows trivially from (A.32) with G(zt, βi) being replaced by
G(zt, βi).
References
Arellano, M. (2003). Panel Data Econometrics. Oxford University Press: Oxford.
Auestad, B. and Tjøstheim, D. (1990). Identification of nonlinear time series: first order characteriza-tion and order determination. Biometrika 77, 669-687.
Baltagi, B. H. (1995). Econometrics Analysis of Panel Data. John Wiley: New York.
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley: New York.
Breusch, T. S. and Pagan, A. R. (1980). The Lagrange multiplier test and its application to modelspecifications in econometrics. Review of Economic Studies 47, 239-253.
Cai, Z. and Li, Q. (2008). Nonparametric estimation of varying coefficient dynamic panel data models.Econometric Theory 24, 1321-1342.
Chen, R. and Tsay, R. S. (1993). Functional–coefficient autoregressive models. Journal of the AmericanStatistical Association 88, 298-308.
Chow, Y. S. and Teicher, H. (1988). Probability Theory. Springer–Verlag: New York.
Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman & Hall:London.
Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer:New York.
Frees, E. W. (1995). Assessing cross sectional correlation in panel data. Journal of Econometrics 69,393-414.
Gao, J. (2007). Nonlinear Time Series: Semiparametric and Nonparametric Methods. Chapman &Hall CRC: London.
Nonparametric Test for Cross–Section Independence 35
Hall, P. and Heyde, C. (1980). Martingale Limit Theory and Its Applications. Academic Press: NewYork.
Hansen, B. E. (2008). Uniform convergence rates for kernel estimation with dependent data. Econo-metric Theory 24, 726-748.
Hardle, W., Liang, H. and Gao, J. (2000). Partially Linear Models. Springer Series: Contributions toStatistics. Physica-Verlag: New York.
Henderson, D., Carroll, R. and Li, Q. (2008). Nonparametric estimation and testing of fixed effectspanel data models. Journal of Econometrics 144, 257-275.
Hjellvik, V., Chen, R. and Tjøstheim, D. (2004). Nonparametric estimation and testing in panels ofintercorrelated time series. Journal of Time Series Analysis 25, 831-872.
Hsiao, C. (2003). Analysis of Panel Data. Cambridge University Press: Cambridge.
Hsiao, C., Pesaran, M. H. and Pick, A. (2007). Diagnostic tests of cross section independence fornonlinear panel data models. IZA discussion paper No. 2756.
Huang, H., Kab, C. and Urga, G. (2008). Copula-based tests for cross-sectional independence in panelmodels. Economics Letter 100, 224-228.
Li, Q. and Hsiao, C. (1998). Testing serial correlation in semiparametric panel data models. Journalof Econometrics 87, 207-237.
Li, Q. and Racine, J. (2007). Nonparametric Econometrics: Theory and Practice. Princeton UniversityPress: Princeton.
Lin, Z. and Lu, C. (1996). Limit Theorems of Mixing Dependent Random Variables. Science Press,Kluwer, Academic Pub: New York, Dordrecht.
Ng, S. (2006). Testing cross section correlation in panel data using spacing. Journal of Business andEconomic Statistics 24, 12-23.
Pesaran, M. H. (2004). General diagnostic tests for cross section dependence in panels. CambridgeWorking Paper in Economics No. 0435.
Pesaran, M. H., Ullah, A. and Yamagata, T. (2008). A bias adjusted LM test of error cross sectionindependence. Econometrics Journal 11, 105-127.
Phillips, P. C. B. and Moon, H. (1999). Linear regression limit theory for nonstationary panel data.Econometrica 67, 1057-1111.
Sarafidis, V., Yamagata, T. and Robertson, D. (2009). A test of cross section dependence for a lineardynamic panel model with regressors. Journal of Econometrics 148, 149–161.
Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley: New York.
Ullah, A. and Roy, N. (1998). Nonparametric and semiparametric econometrics of panel data. Hand-book of Applied Economics Statistics, Ullah, A., Giles, D.E.A. (Eds.). Marcel Dekker, New York,pp. 579C604.