Unit root tests in panel data: asymptotic and finite-sample properties

https://doi.org/10.1016/S0304-4076(01)00098-7Get rights and content

Abstract

We consider pooling cross-section time series data for testing the unit root hypothesis. The degree of persistence in individual regression error, the intercept and trend coefficient are allowed to vary freely across individuals. As both the cross-section and time series dimensions of the panel grow large, the pooled t-statistic has a limiting normal distribution that depends on the regression specification but is free from nuisance parameters. Monte Carlo simulations indicate that the asymptotic results provide a good approximation to the test statistics in panels of moderate size, and that the power of the panel-based unit root test is dramatically higher, compared to performing a separate unit root test for each individual time series.

Introduction

A large body of literature during the past two decades has considered the impact of integrated time series in econometric research (cf. surveys by Diebold and Nerlove, 1990; Campbell and Perron, 1991). In univariate analysis, the Box–Jenkins (Box and Jenkins, 1970) approach of studying difference-stationary ARMA models requires a consistent and powerful test for the presence of unit roots. In general, such tests have non-standard limiting distributions; for example, the original Dickey–Fuller test (Dickey and Fuller, 1979) and the subsequent augmented Dickey–Fuller (ADF) test statistic (Dickey and Fuller, 1981) converge to a function of Brownian motion under quite general conditions (Said and Dickey, 1984). The critical values of the empirical distribution were first tabulated by Dickey (cf. Fuller, 1976). Semi-parametric test procedures have also been proposed (i.e. Phillips, 1987; Phillips and Perron, 1988), with improved empirical size and power properties under certain conditions (cf. Diebold and Nerlove, 1990).

In finite samples, these unit root test procedures are known to have limited power against alternative hypotheses with highly persistent deviations from equilibrium. Simulation exercises also indicate that this problem is particularly severe for small samples (see Campbell and Perron, 1991). This paper considers pooling cross-section time series data as a means of generating more powerful unit root tests. The test procedures are designed to evaluate the null hypothesis that each individual in the panel has integrated time series versus the alternative hypothesis that all individuals time series are stationary. The pooling approach yields higher test power than performing a separate unit root test for each individual.

Some earlier work has analyzed the properties of panel-based unit root tests under the assumption that the data is identically distributed across individuals. Quah 1990, Quah 1994 used random field methods to analyze a panel with i.i.d. disturbances, and demonstrated that the Dickey–Fuller test statistic has a standard normal limiting distribution as both the cross-section and time series dimensions of the panel grow arbitrarily large. Unfortunately, the random field methodology does not allow either individual-specific effects or aggregate common factors (Quah, 1990, p. 17). Breitung and Meyer (1991) have derived the asymptotic normality of the Dickey–Fuller test statistic for panel data with an arbitrarily large cross-section dimension and a small fixed time series dimension (corresponding to the typical microeconomic panel data set). Their approach allows for time-specific effects and higher-order serial correlation, as long as the pattern of serial correlation is identical across individuals, but cannot be extended to panel with heterogeneous errors. More recent advances in nonstationary panel analysis include Im et al. (1995), Harris and Tzavalis (1996) and Phillips and Moon (1999), among others.

What type of the asymptotics considered in the panel unit root test is a delicate issue. Earlier work by Anderson and Hsiao (1982) consider a stationary panel with fixed time series observations while letting the cross sectional units grow arbitrarily large. Similar asymptotic method is used in nonstationary panel by Breitung and Meyer (1991) and Harris and Tzavalis (1996). Im et al. (1995) and Quah 1990, Quah 1994 explore the case of joint limit in which both time series and cross sectional dimension approach infinity with certain restrictions. The precise meaning regarding in what way the cross sectional and time series dimension approach infinity has been clearly defined in Phillips and Moon (1999). This article is one of the earlier research that consider the joint limit asymptotics in which both N and T approach infinity subjecting to certain conditions such as N/T→0 in some model, and N/T→0 in others.

The panel-based unit root test proposed in this article allows for individual-specific intercepts and time trends. Moreover, the error variance and the pattern of higher-order serial correlation are also permitted to vary freely across individuals. Our asymptotic analysis in Section 3 indicates that the proposed test statistics have an interesting mixture of the asymptotic properties of stationary panel data and the asymptotic properties of integrated time series data. In contrast to the non-standard distributions of unit root test statistics for a single time series, the panel test statistics have limiting normal distributions, as in the case of stationary panel data (cf. Hsiao, 1986). However, in contrast to the results for stationary panel data, the convergence rate of the test statistics is higher with respect to the number of time periods (referred to as “super-consistency” in the time series literature) than with respect to the number of individuals in the sample. Furthermore, whereas regression t-statistics for stationary panel data converge to the standard normal distribution, we find that the asymptotic mean and variance of the unit root test statistics vary under different specification of the regression equation (i.e. the inclusion of individual-specific intercepts and time trends).

For practical purposes, the panel based unit root tests suggested in this paper are more relevant for panels of moderate size. If the time series dimension of the panel is very large then existing unit root test procedures will generally be sufficiently powerful to be applied separately to each individual in the panel, though pooling a small group of individual time series can be advantageous in handling more general patterns of correlation across individuals (cf. Park, 1990; Johansen, 1991). On the other hand, if the time series dimension of the panel is very small, and the cross-section dimension is very large, then existing panel data procedures will be appropriate (cf. MaCurdy, 1982; Hsiao, 1986; Holtz-Eakin et al., 1988; Breitung and Meyer, 1991). However, panels of moderate size (say, between 10 and 250 individuals, with 25–250 time series observations per individual) are frequently encountered in industry-level or cross-country econometric studies. For panels of this size, standard multivariate time series and panel data procedures may not be computationally feasible or sufficiently powerful, so that the unit root test procedures outlined in this paper will be particularly useful.

The remainder of this paper is organized as follows: Section 2 specifies the assumptions and outlines the panel unit root test procedure. Readers who are interested mainly in empirical application can skip the rest of the paper. Section 3 analyzes the limiting distributions of the panel test statistics. Section 4 briefly discusses the Monte Carlo simulations. Concluding remarks regarding the limitations of the proposed panel unit root test are offered in Section 5. All proofs are deferred to the appendix.

Section snippets

Model specifications

We observe the stochastic process {yit} for a panel of individuals i=1,…,N, and each individual contains t=1,…,T time series observations. We wish to determine whether {yit} is integrated for each individual in the panel. As in the case of a single time series, the individual regression may include an intercept and time trend. We assume that all individuals in the panel have identical first-order partial autocorrelation, but all other parameters in the error process are permitted to vary freely

Asymptotic properties

Define the following sample statistics for each individual:ξ1iT=1σεi2(T−pi−1)t=pi+2Tv̂i,t−1êit,ξ2iT=1σεi2(T−pi−1)2t=pi+2Tv̂i,t−12,ξ3iT=1σεi2(T−pi−1)t=pi+2Têit2.Next, define the following two ratios for each individual:γ1iT=(T−pi−1)σεi2T̃σ̂εi2,γ2iT=(T−pi−1)2σεi2T̃2σ̂εi2.Given the above definitions, the statistics of interest in , , can be rewritten, respectively, asδ̂=N−1i=1Nγ1iTξ1iTT̃N−1i=1Nγ2iTξ2iT,σ̂ε̃2=1Ni=1Nγ1iTξ3iT−2δ̂i=1Nγ1iTξ1iT+T̃δ̂2i=1Nγ2iTξ2iT,tδ=N−1/2i=1Nγ1iTξ1iTσ̂ε̃[N−1

Monte Carlo simulations

In this section, we briefly discuss the results of three Monte Carlo experiments. The first set of simulations were used to determine appropriate values of the mean and standard deviation adjustments, μmT and σmT, used in the adjusted t-statistic given in Eq. (12) for a particular deterministic specification (m=1,2,3) and time series dimension . The lag truncation parameter was selected according to the formula K̄=3.21T1/3. Due to computational limitations, the ADF lag length pi was set

Conclusion

In this paper, we have developed a procedure utilizing pooled cross-section time series data to test the null hypothesis that each individual time series contains a unit root against the alternative hypothesis that each time series is stationary. As both the cross-section and time series dimensions of the panel grow large, the panel unit root test statistic has a limiting normal distribution. The Monte Carlo simulations indicate that the normal distribution provides a good approximation to the

Acknowledgements

This is a revised version of the Levin and Lin's (1992) earlier work. We would like to thank Clive Granger, Rob Engle, Neil Ericsson, Soren Johansen, Cheng Hsiao, Thomas MaCurdy, James McKinnon, Max Stinchcombe, C.-R. Wei, Hal White, and an anonymous referee for helpful comments and suggestions. We appreciate the assistance of Paul Hoffman and the staff of the UCLA supercomputer center.

References (29)

  • F.X. Diebold et al.

    Unit roots in economic time series: a selective survey

    Advances in Econometrics

    (1990)
  • Fuller, W., 1976. Introduction to Statistical Time Series. John...
  • Hall, A., 1990. Testing for a Unit Root in Time Series with Pretest Data-based Model Selection. North Carolina State...
  • Harris, R., Tzavalis, E., 1996. Inference for unit root in dynamic panels. Unpublished...
  • Cited by (7462)

    View all citing articles on Scopus
    View full text