Likelihood ratio tests for a unit root in panels with random effects

ABSTRACT

Because of the fixed heterogeneity of their models, most panel unit root tests impose restrictions on the rate at which the number of time periods, T, and the number of cross-section units, N, go to infinity. A common example of such a restriction is $N/T\to 0$, which in practice means that $T\gg N$, a condition that is not always met. In the current paper the heterogeneity is given a parsimonious random effects specification, which is used as a basis for developing a new likelihood ratio test for a unit root. The asymptotic analysis shows that the new test is valid for all $(N,T)$ expansion paths satisfying $N/T^5\to 0$, which represents a substantial improvement when compared to the existing fixed effects literature.

KEYWORDS:

• Unit root
• panel data
• random effects

JEL CLASSIFICATION:

• C12
• C13
• C33

Acknowledgments

The authors would like to thank Roland Fried (Editor in Chief), one Associate Editor and one anonymous referee for many useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. One exception is Pesaran [22], who assumes that $T/N\to c>0$ as $N,T\to \mathrm{\infty }$ (see also [23]). The reason for this rather unusual assumption is that the panel unit root test statistic considered has dependent cross-sections, which invalidates the use of the conventional central limit law argument to asymptotic normality as $N\to \mathrm{\infty }$. The normalization with respect to N, and hence the required expansion rate of N and T, are therefore not the usual ones.

2. Hahn and Kuersteiner [19] only consider the case when ${\sigma }_{\lambda }^2=0$. However, as Hahn and Moon [18] show, their bias correction formula is valid even when ${\sigma }_{\lambda }^2>0$.

3. The LR test statistic based on the exact (numerical) MLE was also simulated. In this case, we used OPTMUM, which was implemented in its default setting (using the BFGS algorithm). The results were, however, almost identical to those obtained for ${\hat{\alpha }}_{\mathrm{BC}}$. We therefore omit them.

4. Unreported results suggest that N>1500 is required for ${\mathrm{LR}}_3$ to be correctly sized.

5. Unreported results suggest that the size accuracy of all three tests is very good when $T=N^{3/2}$. In order to keep our figure uncluttered, however, we do not plot these results.

Additional information

Funding

Westerlund thanks the Knut and Alice Wallenberg Foundation for financial support through a Wallenberg Academy Fellowship, and the Jan Wallander and Tom Hedelius Foundation for financial support under research grant number [P2014–0112:1].