Bootstrap unit root tests in panels with cross-sectional dependency
Introduction
Recently, nonstationary panels have drawn much attention in both theoretical and empirical research, as a number of panel data sets covering relatively long time periods have become available. Various statistics for testing unit roots and cointegration for panel models have been proposed, and frequently used for testing growth convergence theories, purchasing power parity hypothesis and for estimating long-run relationships among many macroeconomic and international financial series including exchange rates and spot and future interest rates. Such panel data based tests appeared attractive to many empirical researchers, since they offer alternatives to the tests based only on individual time series observations that are known to have low discriminatory power. A number of unit roots and cointegration tests have been developed for panel models by many authors. See Levin and Lin 1992, Levin and Lin 1993, Quah (1994), Im et al. (1997) and Maddala and Wu (1996) for some of the panel unit root tests, and Pedroni 1996, Pedroni 1997 and McCoskey and Kao (1998) for the panel cointegration tests available in the current literature. Banerjee (1999) gives a good survey on the recent developments in the econometric analysis of panel data whose time series component is nonstationary.1
Since the work by Levin and Lin (1992), a number of unit root tests for panel data have been proposed. Levin and Lin 1992, Levin and Lin 1993 consider unit root tests for homogeneous panels, which are simply the usual t-statistics constructed from the pooled estimator with some appropriate modifications. Such unit root tests for homogeneous panels can therefore be represented as a simple sum taken over i=1,…,N and t=1,…,T. They show under cross-sectional independency that the sequential limit of the standard t-statistics for testing the unit root is the standard normal distribution. For heterogeneous panels, the unit root test can no longer be represented as a simple sum since the pooled estimator is inconsistent for such heterogeneous panels as shown in Pesaran and Smith (1995). Consequently, the second stage N-asymptotics in the above sequential asymptotics does not work here. Im et al. (1997) look at the heterogeneous panels and propose unit root tests which are based on the average of the independent individual unit root tests, t-statistics and LM statistics, computed from each individual unit. They show that their tests also converge to the standard normal distribution upon taking sequential limits. Though they allow for the heterogeneity, their limit theory is still based on the cross-sectional independency, which can be quite a restrictive assumption for most of the economic panel data we encounter.
The tests suggested by Levin and Lin (1993) and Im et al. (1997) are not valid in the presence of cross-correlations among the cross-sectional units. The limit distributions of their tests are no longer valid and unknown when the independency assumption is violated. Indeed, Maddala and Wu (1996) show through simulations that their tests have substantial size distortions when used for cross-sectionally dependent panels. As a way to deal with such inferential difficulty in panels with cross-correlations, they suggest to bootstrap the panel unit root tests, such as those proposed by Levin and Lin (1993), Im et al. (1997) and Fisher (1933), for cross-sectionally dependent panels. They show through simulations that the bootstrap versions of such tests perform much better, but do not provide the validity of using bootstrap methodology.
In this paper, we apply bootstrap methodology to unit root tests for cross-sectionally dependent panels. More specifically, we let each panel be driven by a general linear process which may differ across cross-sectional units, and approximate it by a finite order autoregressive integrated process of order increasing with T. As we allow the dependency among the innovations generating the individual series, we construct our unit root tests from the estimation of the system consisting of the entire N cross-sectional units. The limit distributions of the tests are derived by passing T to infinity, with N fixed. We then apply the bootstrap method to the approximated autoregressions to obtain the critical values for the panel unit root tests based on the original sample, and establish the asymptotic validity of such bootstrap panel unit root tests under general conditions.
The rest of the paper is organized as follows. Section 2 introduces the unit root tests for cross-sectionally dependent panels based on the original sample, and constructs the bootstrap tests by applying the sieve bootstrap methodology to the sample tests. Also discussed in Section 2 are the practical issues arising from the implementation of the sieve bootstrap methodology and the extension of our method to models with deterministic trends. Section 3 derives the limit theories for the asymptotic tests and establishes asymptotic validity of the sieve bootstrap unit root tests. In Section 4, we conduct simulations to investigate finite sample performance of the bootstrap unit root tests. Section 5 concludes, and mathematical proofs are provided in the appendix.
Section snippets
Unit root tests for dependent panels
We consider a panel model generated as the following first-order autoregressive regression:As usual, the index i denotes individual cross-sectional units, such as individuals, households, industries or countries, and the index t denotes time periods. We are interested in testing the unit root null hypothesis, αi=0 for all yit given as in (1), against the alternative, αi<0 for some yit, i=1,…,N. Thus, the null implies that all yit's have unit roots, and is
Limit theories for panel unit root tests
It is well known that an invariance principle holds for a partial sum process of (εt) defined in (3) under Assumption 1. That is,as T→∞, where B=(B1,…,BN)′ is an N-dimensional Brownian motion with covariance matrix Σ, and [x] denotes the maximum integer which does not exceed x.
Let σij and σij denote, respectively, the (i,j)-elements of the covariance matrix Σ and its inverse Σ−1. The limit theories for the F-type tests, FGT and FOT defined in , , are given in Theorem A.1 Under the null
Simulations
We conduct a set of simulations to investigate the finite sample performance of the bootstrap panel unit root tests, and , proposed in the paper. For the simulations, we consider two classes of models: (M) the models with heterogeneous fixed effects only and (T) the models with individual time trends as well as fixed effects. More specifically, we consider the models given in , with the series (yit) defined by (1). For each class of models, errors (uit) in (1) are
Conclusion
There has been much recent empirical and theoretical econometric work on models with nonstationary panel data. In particular, much attention has been paid to the development and implementation of the panel unit root tests which have been used frequently to test for various covergence theories, such as growth covergence theories and purchasing power parity hypothesis. A variety of tests have been proposed, including the tests proposed by Levin and Lin (1993) and Im et al. (1997) that appear to
Acknowledgements
I am very grateful to an Associate Editor for encouragement and helpful suggestions, and to two anonymous referees for their constructive comments. This paper was written while I was visiting the Cowles Foundation for Research in Economics at Yale University during the fall of 1999. I would like to thank Don Andrews, Bill Brown, Joon Park, Peter Phillips and Donggyu Sul for helpful discussions and comments. My thanks also go to the seminar participants at Yale, the Texas Econometrics Camp 2000,
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