Rank‐based Tests for Cross‐sectional Dependence in Large (N, T) Fixed Effects Panel Data Models
| Author | Long Feng,Binghui Liu,Yanling Ding |
| Published date | 01 October 2020 |
| Date | 01 October 2020 |
| DOI | http://doi.org/10.1111/obes.12378 |
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©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 82, 5 (2020) 0305–9049
doi: 10.1111/obes.12378
Rank-based Tests for Cross-sectional Dependence in
Large (N,T) Fixed Effects Panel Data Models*
Long Feng†, Yanling Ding‡ and Binghui Liu§
†KLAS and School of Mathematics and Statistics, Northeast Normal University, Changchun,
China (e-mail: fengl100@nenu.edu.cn)
‡School of Science, Changchun Institute of Technology, Changchun, China (e-mail:
dingyanling0@163.com)
§KLAS and School of Mathematics and Statistics, Northeast Normal University, Changchun,
China (e-mail:liubh100@nenu.edu.cn)
Abstract
Most existing methods for testing cross-sectional dependence in fixed effects panel data
models are actually conducting tests for cross-sectional uncorrelation, which are not robust
to departures of normality of the error distributions as well as nonlinear cross-sectional
dependence. To this end, we construct two rank-based tests for (static and dynamic) fixed
effects panel data models, based on two very popular rank correlations, that is, Kendall’s
tau and Bergsma–Dassios’*, respectively, and derive their asymptotic distributions under
the null hypothesis. Monte Carlo simulations demonstrate applicability of these rank-based
tests in large (N,T) case, and also the robustness to departures of normality of the error
distributions and nonlinear cross-sectional dependence.
I. Introduction
In economics, panel data are widely used to analyse complex phenomena, which contain
observations of multiple phenomena obtained over multiple time periods for the same
units. Traditionally, panel data analysis focussed on data sets with large cross-sectional
dimension Nand smaller time series dimension T. With the emergence of rich data sets
both in Nand T, the theoretical analysis of large (N,T) panel data is becoming more and
more important. Furthermore, economic panel data are generally not normally distributed
and sometimes have nonlinear cross-sectional dependence between units. However, most
existing methods for testing cross-sectional dependence in panel data models are actually
conducting tests for cross-sectional uncorrelation, which are not robust to departures of
normality of the error distributions as well as nonlinear cross-sectional dependence. Hence
JEL Classification numbers: C12 (Hypothesis Testing: General); C14 (Semiparametric and Nonparametric Meth-
ods: General); C01 (Econometrics)
*This work has been funded under NSFC grants 11501092, 11571068, 11671073, the Fundamental Research
Funds for the Central Universities grant 2412017BJ002, the Key Laboratory of Applied Statistics of MOE (KLAS)
grants 130026507 and 130028612, Jilin Education Science Planning Project grands GH180346, JKBLX2018-077.
Rank-based tests for cross-sectional dependence 1199
it is very necessary to develop robust tests for cross-sectional dependence in large (N,T)
panel data models to departures of normality and nonlinear cross-sectional dependence.
On this ground, in this paper, we study the testing problem of cross-sectional dependence in
panel data models with fixed effects, and developsome rank-based testing methods, which
are applicable in large (N,T) case and robust to departures of normality and nonlinear
dependence.
Cross-sectional dependence is described as the interaction between cross-sectional
units, which could arise from the behavioural interaction between units, probably due
to unobservable common factors or common shocks popular in macroeconomics. Cross-
sectional dependence leads to efficiency loss for least squares and invalidatesconventional
tests using standard variance–covarianceestimators. To test the existenceof cross-sectional
dependence, there has been a large number of researches in the spatial econometrics lit-
erature. Originally, Breusch and Pagan (1980) derived the traditional Lagrange Multiplier
(LM) test, written as the LMBP test, for a number of model specifications. In the fixed
Ncase and as T→∞, the LMBP test statistic is asymptotically Chi-square distributed
with N(N−1)/2 degrees of freedom under the null hypothesis. However, this test is not
applicable when N→∞. Therefore, Pesaran (2004) proposed a scaled version based on
the pairwise average of the off-diagonal sample correlation coefficients in a seemingly un-
related regression (SUR) model, written as the CDLM test, which has a N(0, 1) distribution
as T→∞first, followed by N→∞. As pointed out by Pesaran (2004), the CDLM test
is not correctly centred at zero for finite T, and is likely to exhibit large size distortions
as Nincreases. To solve this problem, Pesaran (2004) also proposed a test based on the
average of the sample correlations, written as the CD test, which is valid in the large N
case. Additionally, Pesaran, Ullah andYamagata(2008) developed a bias-adjusted LM test,
written as the LMPUY test, using finite sample approximations in panel models, and derived
its asymptotic property as first T→∞then N→∞, where the analytical bias corrections
can be obtained only under the assumption that the regressors are strictly exogenous and
the errors are normally distributed.
These methods are designed for heterogeneous panel data models, which can be directly
applied to fixed effects panel data models but with serious loss of information on fixed
effects. To this end, Baltagi, Feng and Kao (2012) proposed a special LM test for cross-
sectional dependence in fixed effects panel data models, written as the LMBFK test, and
derived that the asymptotic bias of this scaled version is a constant related to Nand T. One
drawback of the LMBFK test, like the methods mentioned above for heterogeneous panel
data models, is that the asymptotic distribution of the test statistic depends on the error
population, even under the hypothesis of no cross-sectional correlations. On the contrary,
Frees (1995) advanced a non-parametric and distribution-free version of the LM statistic,
named R2
AVE , which is robust to departures of normality of error distribution and applicable
to panel data models in case that Nis large relative to T.
In particular, R2
AVE is a rank-based test, which is constructed based on the sum of squared
pair-wise Spearman’s rho correlation coefficients (Spearman, 1904; Wang, Zou andWang,
2013). Spearman’s rho correlation coefficient as well as Kendall’s tau correlation coefficient
(Kendall, 1938) are the most commonly used measures of monotone association for non-
normally distributed random variables. Unfortunately, both of them have the undesirable
property that they may be equal to zero even when the two random variables are not
©2020 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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