The effects of cross-section dimension n in panel co-integration test

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Abstract

Most studies of panel unit test (PUT) believe that their tests could be extended to panel co-integration test (PCT) directly since PUT and PCT converge sequentially to standard normal distribution in the absence of trends and autocorrelation. But the difference between PUT and PCT cannot be ignored when the cross-section dimensions n is small, especially when n equals 1. In this paper, we examine the effects of cross-sectional dimension n in the translation from PUT to PCT. Based on the numerical simulations, it turns out that although the difference between PUT and PCT decreases with increased value of n, the difference could be ignored in general application only when n is larger than 90.

Introduction

In recent years, the analysis of panel unit test (PUT) and panel co-integration test (PCT) in panel data has developed rapidly as a fruitful area of theoretical and empirical study. Now, PUT has developed from the cases which do not consider cross-sectional dependence [2], [6], [9], [11], to the cases in which cross-sections are dependent [1], [4], [5], [13], [16], [17]. Meanwhile, PCT has developed from the cases which consider cross-sectional independence [10], [14] to the cases in which cross-sections are dependent [3], [19].

PUT and PCT could be regarded as the developments of unit root test and co-integration test in time series, whose limit distributions depend on two dimensions, i.e., time dimension T and cross-section dimension n. Most PUT and PCT studies are constructed on sequential limit theorem, i.e., T  ∞ followed by n  ∞ [18]. The introduction of cross-section dimensions n makes the relationship between PUT and PCT in panel data different from that between unit root test and co-integration test in time series.

When n = 1, PUT and PCT degenerate to simple unit root test and co-integration test in time series. The general method of unit root test in time series is DF test [7]. Moreover, the first co-integration test, EG two steps method [8], in time series was constructed on DF test. However, there are significant differences between the small sample properties of DF test and EG test because the residuals are generated from a regression function by which the errors cannot be determined but only be estimated. Consequently, the corrected critical value is used instead of the Dickey–Fuller tables when EG test is conducted [12]. The differences between PUT and PCT are studied based on this consideration.

While taking the inner relationship between unit root test and co-integration test in time series into account, most studies on PUT claimed that their test statistics could be applied to PCT directly [6]. There is also a lot of literature treats PUT as PCT directly in empirical research.

The effect of central limit theorem becomes more and more significant along with the increase in cross-section dimension. A theoretical foundation for the translation from PUT to PCT is provided while the two methods have large sample properties. Although in the panel, the small sample differences between the two methods are not as significant as those in time series, this relatively small differences should not be ignored either. In this paper, we examine the trend of the differences between PUT and PCT in small sample according to the variance of cross-section dimensions n and present the critical conditions under which the differences could be ignored.

The remainder of this paper is organized as follows. Section 2 presents the underlying theory and asymptotic results for each of the test statistics under simple assumptions. Section 3 studies the small sample properties of these tests under a variety of scenarios, especially under the variances of cross-section dimensions n. Section 4 ends with conclusions and suggestions for further research.

Section snippets

Models and limit distributions of PUT and PCT – a simple case

On the basis of the idea of EG two steps method, Kao studied residual-based co-integration test in the panel data for the null of no co-integration [10], which could be considered as a direct development of EG two steps method in panel data. In his study, Kao considered a general case which included the deterministic trend and the autocorrelation of error. However, the goal of the present study is to point out the importance of cross-section dimension n in PCT. For mathematical tractability,

Numerical simulation

In this section, the differences between PUT and PCT in small sample are examined through numerical simulation. Based on the above theory, consider the following data generating process (DGP):Δyit=uit,Δxit=vit,where uitvitIIN00,1θσθσσ2, i = 1,2,  , n; t = 1,2,  , T.

The normal procedure in SAS9.0 was used to generate the random numbers, where the initial seed was set to be 1234567. The data were generated by creating T + 1000 observations and discarding the first 1000 observations to remove the influence

Concluding remarks

In time series, DF test and EG test have different distributions. EG test uses the critical values corrected by MacKinnon instead of the critical values of DF. PUT and PCT inherited the intrinsic link between unit root and co-integration test in time series. Moreover, they have the same limit distribution because of the existence of cross-section dimension n, in the absence of determinant trends and autocorrelation. Thus compared with that in time series, the difference between PUT and PCT are

Acknowledgements

The authors thank the anonymous referees for their constructive comments. The research of Zhanming Chen has been supported by the State Key Program for Basic Research (973 Program, Grant No. 2005CB724204).

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