Combining p-Values in Non-Stationary Panels

Abstract

This article summarizes and discusses the existing p-value pooling approaches and compares their performances in the context of panel unit root tests. When the data are free of contemporaneous correlation, most tests achieve very high power. However, in the presence of contemporaneous correlation, most tests suffer from moderate to severe size distortions. When the panel contains both stationary and nonstationary series, the power of tests increases as the cross-sectional units grows. Among all the tests under study, the mean-of-Z test yields the highest power for the benchmark model, while the Fisher test is most robust for complicated model structures.

Keywords:

• Combining p-values
• Meta analysis
• Non-stationary panel
• Unit root tests

Mathematical Subject Classification::

• 37M10
• 62M10

Acknowledgments

We thank Isaac Ehrlich for detailed comments and suggestions on this paper.

Notes

For a recent survey paper, please see Baltagi and Kao (2000).

Note that Moon and Phillips (1999) also discuss different approaches in dynamic panels, but with a different objective. They focus on different ways to establish the asymptotic results due to the two-dimensional setup in dynamic panel models.

It is not necessary that the same continuous test be applied throughout the analyses. The only requirement here is that the tests used are continuous tests. This can be regarded as a strong feature of p-value pooling, as argued in Maddala and Wu (1999).

This turns out to be a key assumption in panel data unit root tests. Almost all early proposed tests suffer from moderate to serious size distortions when this assumption does not hold. We will later discuss this assumption in our Monte Carlo simulation designs in more details. For now, we just assume there is no contemporaneous correlation.

Both balanced and unbalanced panel are considered here by allowing for different T_i's across the cross-sections.

It should be noted that in general one will only apply a given univariate unit root test to all units, for example, the ADF test. But in principle, one can apply different univariate unit root tests to different cross-sectional units. Of course, one would need to justify this strategy in practice. One scenario might be the existence of two types of serial correlations for the cross-sections in the panel which call for applications of two different types of unit root tests.

However, later in their paper Maddala and Wu (1999) did report the results for the Dufour and Torres tests based on the Bonferroni inequality. The MP test can be viewed as the simplest form of the Dufour and Torres tests.

This is the reason that Maddala and Wu (1999) used it in their paper.

For an introduction to this issue, see Maddala and Wu (1999) and Baltagi and Kao (2000). It should be noted that some of the methods of pooling p-values are based on the asymptotic distribution as well. So the argument of exact tests vs. asymptotic tests used in Maddala and Wu (1999) is not completely applicable here.

Technically, we can generate p-values based on the uniform distribution. But then in this case we would be able to investigate the performances of the different pooling method in a general regression model instead of in the context of panel unit root tests. Doing so would be inconsistent with our objectives in this paper.

This is what has been used in the literature. See, for example, Im et al. (2003) and Maddala and Wu (1999).

This is the same design used in Maddala and Wu (1999).

The powers are adjusted by using the 5% critical values derived from the simulations in Design 1.

The Dicky-Fuller test has power of 0.8680 and 0.9970, respectively, in Designs 5 and 6 when T = 100.

The reason we experiment with a Φ of 0.9 instead of 0.8 as we did in Design 3 is that we found the powers go to 1 too quickly with Φ of 0.8.