Time-specific disturbances and cross-sectional dependency in a small-sample heterogeneous panel data unit root test

Abstract

In their seminal work, Im et al. (1997, 2003) suggested that time series for several cross-sectional units could be used to increase the power of the Dickey--Fuller unit root test. They argued that when cross-sectional correlation is a problem that can be modelled by a time-specific factor, demeaning across the cross-sectional units can solve the problem. In this study, this proposition is proven valid, but it is also shown that previously supplied standardizing moments are inappropriate when the number of cross-sections are small, causing size to differ from the significance level. To correct this size distortion, the current paper supplies response surface parameters that can be used to obtain moments that are valid when a time-specific factor suffices for modelling cross-sectional correlation in the heterogeneous panel data unit root framework. The correct size of the panel data unit root test comes at the cost of a somewhat lower power against a stationary alternative.

Acknowledgements

The author would like to thank Tommy Andersson, David Edgerton and Klas Fregert for useful discussions on the topics covered in this study. Valuable comments have been received from an anonymous referee and the Editor. All simulations in the current study are performed on the LUNARC cluster. The author would like to thank Thomas Elger, Ulf Erlandsson and Per-Anders Wernberg for technical advice. Financial support from The Crafoord Foundation and the Jan Wallander and Tom Hedelius Foundation, research grant number P2005-0117 : 1, is gratefully acknowledged. The responsibility for the remaining errors rests solely with the author.

Notes

¹ T denotes the number of observations available when having constructed the lagged level and the lagged first differences needed in the test. This is in accordance with the convention adopted by Im et al. (2003).

² Homogeneity on behalf of the error variances is not needed in the panel data test, but assumed here to illustrate how the time-specific disturbance causes cross-sectional correlation.

³Throughout the simulations the test is applied where no lagged first differences are included in the test regression. Results for other cases are available upon request.

⁴ The approximate confidence 95% confidence interval for a p-value of 0.10 is given by (0.092; 0.108) and is calculated as , where π is the p-value and K is the number of replications in the Monte Carlo simulation.

⁵ It can be noted that it would be possible to include the different values of k in one response surface regression. However, it would be an impractical expression to handle, sacrificing much of the simplicity inherent in the current specification when it comes to calculating moments.

⁶ It can be noted that one should interpret the intercepts in the response surface regressions cautiously. Since they are obtained from a sample where the largest cross-sectional dimension is 15 and the largest time-series dimension is 250, one should not interpret the intercept as asymptotic moments. The choice to let the largest cross-sectional dimension considered be 15 is motivated by the fact that the results in Table 1 indicate that previously supplied moments perform reasonably well in cases where N ≥ 10.

⁷The results presented are restricted to the case of k=0 to save space. Results for other lag specifications can be obtained upon request.