Reducing the size distortions of the panel LM Test for cointegration☆
Introduction
The problem of testing for cointegration, or the absence thereof, using residual-based procedures has received considerable attention in the recent panel data literature. Unit root tests for the null hypothesis of no cointegration have been used extensively for this purpose. However, these tests have been criticized because it is usually the opposite null that is of primary interest in empirical research. Thus, it would seem more natural to consider residuals-based procedures that seek to test the null hypothesis of cointegration rather than the opposite. The by far most popular test that can be applied for this purpose is that of McCoskey and Kao (1998), which generalizes the univariate LM test for cointegration proposed by Harris and Inder (1994) and Shin (1994) to panel data. As such, the test is based on a components model in which the unobserved regression errors can be written as a sum of a white noise and a unit root component. In this model, the null hypothesis of cointegration is equivalent to restricting the unit root component to a constant.
While this components representation of the regression errors certainly is attractive in the sense that the null hypothesis arises naturally as a restricted version of the model, it does not lend itself to the analysis of most panel time series data. This is so because neither the null nor the local alternative model is able to generate errors with even nearly the same amount of persistence as in the series usually observed in empirical work. To accommodate for more general data generating processes, McCoskey and Kao (1998) employ a semiparametric correction whereby the persistency of the errors is eliminated asymptotically using kernel estimation of the associated nuisance parameters. As shown by Westerlund (2005), however, this can deliver very poor small-sample performance with massive size distortions as a result. In spite of this, the test continues to be routinely applied to regressions with highly persistent errors.
The purpose of this paper is to develop a new test procedure that can be employed to alleviate these size distortions. The procedure is based on the Bonferroni principle recently employed by Choi (2004) to test for stationarity in time series. The sample is first split into even and odd numbered observations, and the panel LM test is applied to each subsample. The two tests are then combined using the Bonferroni principle. Our Monte Carlo results suggest that this procedure can be successfully applied to significantly reduce the size distortions of the panel LM test. It should be noted that, although this paper focuses on the test of McCoskey and Kao (1998), the new procedure can be applied to any cointegration test.
The remainder of this paper is organized as follows. Section 2 presents the new test, Section 3 contains the Monte Carlo study and Section 4 concludes.
Section snippets
Test procedure
For the purpose of this paper, we will assume that the multidimensional time series variable yit is generated by the following system of equationswhere xit is a K dimensional vector of regressors with βi being the corresponding vector of slope parameters. The error process is stationary when yit and xit are cointegrated and it has a unit root when they are not. To this end, consider the following autoregression
In this model, the problem
Monte Carlo simulations
To examine the small-sample properties of the test, we engage in a simple Monte Carlo simulation exercise. The data were generated by (1) through (3) with wit∼N(0, I2). For initiation of xit and eit, we use the value zero and we assume that there is a single regressor with αi = βi = 1. We then generate 1000 panels with N cross-sectional and T + 50 time series observations, where we disregard the first 50 observations for each cross-section to reduce the effect of the initial condition.
Simulations
Conclusions
In this paper, we propose a new procedure that can be employed to reduce the size distortions of the panel LM cointegration test proposed by McCoskey and Kao (1998). The new test is easy to implement and it is shown through Monte Carlo simulations to be able to significantly reduce the distortions of the panel LM test.
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The author would like to thank Eric Maskin and one anonymous referee for valuable comments and suggestions. Financial support from the Jan Wallander and Tom Hedelius Foundation, research grant number P2005-0117:1, is gratefully acknowledged.