Testing for unit roots in heterogeneous panels

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Abstract

This paper proposes unit root tests for dynamic heterogeneous panels based on the mean of individual unit root statistics. In particular it proposes a standardized t-bar test statistic based on the (augmented) Dickey–Fuller statistics averaged across the groups. Under a general setting this statistic is shown to converge in probability to a standard normal variate sequentially with T (the time series dimension) →∞, followed by N (the cross sectional dimension) →∞. A diagonal convergence result with T and N→∞ while N/T→k,k being a finite non-negative constant, is also conjectured. In the special case where errors in individual Dickey–Fuller (DF) regressions are serially uncorrelated a modified version of the standardized t-bar statistic is shown to be distributed as standard normal as N→∞ for a fixed T, so long as T>5 in the case of DF regressions with intercepts and T>6 in the case of DF regressions with intercepts and linear time trends. An exact fixed N and T test is also developed using the simple average of the DF statistics. Monte Carlo results show that if a large enough lag order is selected for the underlying ADF regressions, then the small sample performances of the t-bar test is reasonably satisfactory and generally better than the test proposed by Levin and Lin (Unpublished manuscript, University of California, San Diego, 1993).

Introduction

Since the seminal work of Balestra and Nerlove (1966), dynamic models have played an increasingly important role in empirical analysis of panel data in economics. Given the small time dimension of most panels, the emphasis has been put on models with homogeneous dynamics, and until recently little attention has been paid to the analysis of dynamic heterogeneous panels. However, over the past decade a number of important panel data set covering different industries, regions, or countries over relatively long time spans have become available, the most prominent example of which is the Summers and Heston (1991) data. The availability of this type of “pseudo” panels raises the issue of the plausibility of the dynamic homogeneity assumption that underlies the traditional analysis of panel data models, and poses the problem of how best to analyze them. The inconsistency of pooled estimators in dynamic heterogeneous panel models has been demonstrated by Pesaran and Smith (1995), and Pesaran et al. (1996). The present paper builds on these works and considers the problem of testing for unit roots in such pseudo panels.

Panel based unit root tests have been advanced by Quah 1992, Quah 1994 and Levin and Lin (1993, LL hereafter). The tests proposed by Quah do not accommodate heterogeneity across groups such as individual specific effects and different patterns of residual serial correlations. LL's test is more generally applicable, allows for individual specific effects as well as dynamic heterogeneity across groups, and requires N/T→0 as both N (the cross section dimension) and T (the time series dimension) tend to infinity.1

Using the likelihood framework, this paper proposes an alternative testing procedure based on averaging individual unit root test statistics for panels. In particular, we propose a test based on the average of (augmented) Dickey–Fuller (Dickey and Fuller, 1979) statistics computed for each group in the panel, which we refer to as the t-bar test. Like the LL procedure, our proposed test allows for residual serial correlation and heterogeneity of the dynamics and error variances across groups. Under very general settings this statistic is shown to converge in probability to a standard normal variate sequentially with T→∞, followed by N→∞. A diagonal convergence result with T and N→∞ while N/T→k,k being a finite non-negative constant, is also conjectured.

In the special case where errors in individual Dickey–Fuller (DF) regressions are serially uncorrelated a modified version of the (standardized) t-bar statistic, denoted by Zt̃bar, is shown to be distributed as standard normal as N→∞ for a fixed T, so long as T>5 in the case of DF regressions with intercepts, and T>6 in the case of DF regressions with intercepts and linear time trends. An exact fixed N and T test is also developed using the simple average of the DF statistics. Based on stochastic simulations it is shown that the standardized t-bar statistic provides an excellent approximation to the exact test even for relatively small values of N.

Mean and variance of the individual t statistics needed for the implementation of the test for N and T large are also provided. The finite sample performances of the proposed t-bar test and the LL test are examined using Monte Carlo methods. The simulation results clearly show that if a large enough lag order is selected for the underlying ADF regressions, then the finite sample performance of the t-bar test is reasonably satisfactory and generally better than that of the LL test.

The plan of the paper is as follows. Section 2 sets out the model and provides a brief review of the previous studies. Section 3 sets out the likelihood framework for heterogeneous panels, and derives test statistics in the case where errors in individual Dickey–Fuller regressions are serially uncorrelated. Section 4 considers a more general case with serially correlated errors. Section 5 presents the Monte Carlo evidence. Section 6 provides some concluding remarks.

Section snippets

The basic framework

Consider a sample of N cross sections (industries, regions or countries) observed over T time periods. We suppose that the stochastic process, yit, is generated by the first-order autoregressive process:yit=(1−φiiiyi,t−1it,i=1,…,N,t=1,…,T,where initial values, yi0, are given. We are interested in testing the null hypothesis of unit roots φi=1 for all i. (2.1) can be expressed asΔyitiiyi,t−1it,where αi=(1−φii,βi=−(1−φi) and Δyit=yityi,t−1. The null hypothesis of unit roots then

Fixed T unit root tests for heterogeneous panels with serially uncorrelated errors

In this section we develop our proposed test in the context of the panel data model, (2.1), where the errors are serially uncorrelated but T is fixed. For this purpose we make the following assumption:

Assumption 3.1

εit,i=1,…,N,t=1,…,T, in (2.1) are independently and normally distributed random variables for all i and t with zero means and finite heterogeneous variances, σi2.

In this case the relevant Dickey–Fuller (1979) regressions are given by (2.2) with the following pooled log-likelihood function:NT(β,ϕ)=

Unit root tests for heterogeneous panels with serially correlated errors

In this section we consider the more general case where the errors in (2.1) may be serially correlated, possibly with different serial correlation patterns across groups, but with T and N sufficiently large.

Suppose that yit are generated according to the following finite-order AR(pi+1) processes:yitiφi(1)+j=1pi+1φijyi,t−jit,i=1,…,N,t=1,…,T,which can be written equivalently as the ADF(pi) regressions:Δyitiiyi,t−1+j=1piρijΔyi,t−jit,i=1,…,N,t=1,…,T,where φi(1)=1−∑j=1pi+1φij,αiiφi(1),β

Monte Carlo simulation results

In this section we use Monte Carlo experiments to examine finite sample properties of the alternative panel-based unit root tests. We consider four sets of Monte Carlo experiments. The first set focuses on the benchmark model,yit=(1−φiiiyi,t−1it,t=1,…,T,i=1,…,N,where εit∼N(0,σi2). The second set of experiments allows for the presence of positive (heterogeneous) AR(1) serial correlations in εit,εitiεi,t−1+eit,t=1,…,T,i=1,…,N,where eitN(0,σi2),ρiU[0.2,0.4],U stands for a uniform

Concluding remarks

In this paper we have developed a computationally simple procedure for testing the unit root hypothesis in heterogeneous panels. The small sample properties of the proposed tests are investigated via Monte Carlo methods. It is found that when there are no serial correlations, the t-bar test performs very well even when T=10. In this case it is possible to substantially augment the power of the unit root tests applied to single time series.

The situation is more complicated when the disturbances

Acknowledgements

We are grateful to Ron Smith, Stephen Satchell, Michael Binder, James Chu, Cheng Hsiao, Karim Abadir, Rolf Larsson, Clive Granger, Kevin Lee, Bent Nielson, Jan Magnus, Jeffrey Woodridge, Robert de Jong, Peter Robinson and a number of anonymous referees for helpful comments. Partial financial support from the ESRC (Grant No. H519255003) and the Isaac Newton Trust of Trinity College, Cambridge is gratefully acknowledged.

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    This is a substantially revised version of the DAE Working Papers Amalgamated Series No. 9526, University of Cambridge.

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