Testing for serial independence of panel errors

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Abstract

A test for the serial independence of errors in panel data models is proposed. The test is based on the difference between the joint empirical characteristic function of residuals at different lags and the product of their marginal empirical characteristic functions. The test is nuisance-parameter-free and powerful against any type of pairwise dependence at all lags. A simple random permutation procedure is used to approximate the limit distribution of the test. A Monte Carlo experiment illustrates the finite sample performance of the test, and supports that the test statistic based on the estimated residuals has the same asymptotic distribution as the corresponding statistic based on the unobservable true errors.

Introduction

The serial independence of unobservable errors is a key assumption for the validity of many statistical inferences and asymptotic results. It is also a useful identification assumption in many econometric models for consumer choice, treatment effects and binary choice, to give a few examples. Brown and Wegkamp (2002) propose a minimum distance from independence estimator, where the independence between the unobserved errors and the exogenous variables is the crucial condition for identification. When one applies Efron’s (1979) bootstrap to residuals, one often assumes the independence of errors for the bootstrap to be valid, see e.g.  Singh (1981). Peretti (2005) shows that testing the significance of the departures from utility maximization boils down to testing the independent and identically distributed (iid) assumption of some unobserved errors. For some other cases where the independence of errors is crucial, see e.g.  Diebold et al. (1998), Clement and Smith (2000) and Brown et al. (2007).

Despite its importance, there are only a few studies on testing independence of errors. Box and Pierce (1970) and Ljung and Box (1978), BLP hereafter, propose a test for the autocorrelation of errors in ARMA models. Brock et al., 1991, Brock et al., 1996 construct a test, BDS henceforth, for the independence of errors using chaos theory. Their test is nuisance-parameter-free only under conditional mean models, and it has no power against certain types of dependence. Hong and Lee (2003) overcome the aforementioned problems of BDS test, but they use kernel smoothing techniques and their test is affected by the choice of bandwidth. Du and Escanciano (in press) propose a distribution-free test based on the Hoeffding–Blum–Kiefer–Rosenblatt-type empirical process (HBKR hereafter, see Hoeffding, 1948, Blum et al., 1961, Delgado, 1999) applied to residuals at different lags.

There are no existing tests for serial independence of panel errors, to the best of our knowledge. Instead, there are only some tests for serial correlation of panel errors. Many of them are just for first-order serial correlation, see Breusch and Pagan (1980) and Wooldridge (2002, p. 275) for example. Arellano and Bond (1991) propose a test for lack of second-order serial correlation in the first-differenced errors. There are also some portmanteau tests for panel errors, such as Hong and Kao’s (2004) wavelet-based test, Inoue and Solon’s (2006) Lagrange multiplier test and Okui (2009), which is an extension of BLP test to panels. All these tests assume the cross-sectional independence of the errors, which is problematic sometimes, see e.g.  Ng (2006), Baltagi et al., 2009, Baltagi et al., 2012 and Shin et al. (2009).

In this paper we propose a test for serial independence of panel errors. Our test is nuisance-parameter-free and powerful against any type of pairwise dependence at all lags. It does not involve bandwidth selection. Moreover, unlike the previous tests, it is well defined even under the presence of cross-sectional dependence in the errors.

We consider a balanced linear panel data model with individual specific effects yit=Xitθ0+αi+uit,i=1,,N;t=1,,T, where yit is the dependent variable of interest, Xit is a vector of covariates, αi is the unobserved individual effect that could be correlated with Xit, uit is the error term, and θ0 is some unknown parameter in a compact set ΘRp.

The null hypothesis that we test in this paper is H0:{ut(θ0)}tZ  is iid for some  θ0ΘRp, where henceforth ut(θ)=(u1t(θ),u2t(θ),,uNt(θ))RN,θΘ, with uit(θ)=yity¯i(XitX¯i)θ,y¯i=1Tt=1Tyit;X¯i=1Tt=1TXit. The alternative hypothesis is the negation of the null (2).

Our test statistic is based on the difference between the joint characteristic function of (ut(θ0),utj(θ0)) and the product of their marginal characteristic functions. As θ0 is unknown, we replace it with a suitable estimate θ̂ and construct residuals ût=ut(θ̂). In contrast to the general theory on empirical processes with estimated parameters, see e.g.  Durbin (1973), our test enjoys the “nuisance-parameter-free” property. We actually show that our test statistic based on the residuals has the same asymptotic distribution as the corresponding statistic based on the true errors. By choosing a proper weighting function in our test statistic, we avoid the multivariate numerical integration in 2N dimensions. The limit distribution of our test statistic is approximated directly by a simple random permutation procedure, see e.g.Delgado (1996).

Our test here extends (Du and Escanciano, in press) to panel data models. Their test based on the HBKR process suffers from the curse of dimensionality as the indicators {I(utx)}t=1T are essentially all 0 when N gets large in panels, where hereinafter the indicator function I()=1, if the statement in the parentheses is true, and I()=0 otherwise. Our use of the characteristic function alleviates this problem.

To assess the finite sample performance of our proposed test, we do some Monte Carlo studies. We compare our test with Wooldridge (2002, p. 275) and Okui (2009). Generally, our test has good size and power performance. We also find that our test based on residuals has sizes and powers very close to its counterpart based on the true errors.

We illustrate our method under large T small N setup. We then discuss how one can modify our test to large N large T case if one is ready to assume iid across individuals.

The rest of the paper is organized as follows: in Section  2 we introduce our test statistic and derive its limit distribution. In Section  3 we do some Monte Carlo simulations to study the finite sample performance of the proposed test. In Section  4 we extend our test to large N large T case. In Section  5 we conclude and suggest some directions for future research. The proofs are gathered in the Appendix.

Section snippets

Test statistic

In the sequel, we simplify the notations as follows: Yt=(y1t,y2t,,yNt),Xt=(X1t,X2t,,XNt),ut=ut(θ0)=(u1t,u2t,,uNt), and ût=ut(θ̂T), where θ̂T is a T-consistent estimator for θ0. Our asymptotic results in this section are obtained as T goes to infinity. In Section  4 we discuss the case where both N and T go to infinity. Furthermore, let φj(x,y)=E[exp(ixut+iyutj)] and φ(x)=E[exp(ixut)] denote the joint and marginal characteristic functions of (ut,utj), respectively. Let be the

Monte Carlo simulations

To assess the finite sample performance of our proposed test in the previous section, especially the effect of replacing errors by residuals, we do some Monte Carlo studies. The software that we use is R version 2.13.0 for Windows.

We consider the following data-generating process in this section

yit=αi+Xitβ+uit,i=1,,N;t=1,,T, where the individual effect αi iid χ2(2), a chi-square distribution with 2 degrees of freedom, β1=β2=β3=1,X1,it=e1,it,X2,it=b1if1t+b2if2t+e2,it,X3,it=b3if3t+e3,it with bj

Extension to large N large T case

In this section we extend our test to large N large T case. In this setup, as Inoue and Solon (2006) and Okui (2009) did, we assume that uk=(uk1,uk2,,ukT) is iid across k, k=1,2,,N, and our asymptotic results are derived as both N and T go to infinity.

Our dependence measure in this case is ρj(x,y)=Cov[exp(ixukt),exp(iyuktj)],j1, whose sample counterpart is ρNj(x,y)=1N(Tj)t=1+jTk=1Nexp(ixukt+iyuktj)ψj,T(x)ψ1,Tj(y), where ψj,T(x)=1/(N(Tj))t=1+jTk=1Nexp(ixukt)and ψ1,Tj(y)=1/(N(T

Conclusions

In this paper, we propose a test for the serial independence of errors in panel data models. Our test is nuisance-parameter-free, powerful against any type of pairwise dependence at all lags, and does not involve bandwidth selection. We show that, provided some conditions are satisfied, our test statistic based on the estimated residuals has the same asymptotic distribution as the corresponding statistic based on the unobservable true errors. Some Monte Carlo simulations illustrate the good

Acknowledgments

I would like to thank Juan Carlos Escanciano, the Co-Editor, Erricos Kontoghiorghes, the associate editor and two anonymous referees for their helpful comments and suggestions. I acknowledge the computational resources provided by Bing Li, Yan Dong, Chunlin Chen, Pei Pei, Qing Lin and the Libra High Performance Cluster at Indiana University. The financial support from the Western and Border Areas Program, Humanities and Social Sciences, Ministry of Education of China, 11XJC790002 is greatly

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