On the Use of GLS Demeaning in Panel Unit Root Testing

One of the most well-known facts about unit root testing in time series is that the Dickey–Fuller (DF) test based on ordinary least squares (OLS) demeaned data suffers from low power, and that the use of generalized least squares (GLS) demeaning can lead to substantial power gains. Of course, this development has not gone unnoticed in the panel unit root literature. However, while the potential of using GLS demeaning is widely recognized, oddly enough, there are still no theoretical results available to facilitate a formal analysis of such demeaning in the panel data context. The present article can be seen as a reaction to this. The purpose is to evaluate the effect of GLS demeaning when used in conjuncture with the pooled OLS <i>t</i>-test for a unit root, resulting in a panel analog of the time series DF–GLS test. A key finding is that the success of GLS depend critically on the order in which the dependent variable is demeaned and first-differenced. If the variable is demeaned prior to taking first-differences, power is maximized by using GLS demeaning, whereas if the differencing is done first, then OLS demeaning is preferred. Furthermore, even if the former demeaning approach is used, such that GLS is preferred, the asymptotic distribution of the resulting test is independent of the tuning parameters that characterize the local alternative under which the demeaning performed. Hence, the demeaning can just as well be performed under the unit root null hypothesis. In this sense, GLS demeaning under the local alternative is redundant.

时间序列单位根检验领域最为人熟知的结论之一便是：基于普通最小二乘（ordinary least squares, OLS）去均值数据的迪基-富勒（Dickey–Fuller, DF）检验功效偏低，而采用广义最小二乘（generalized least squares, GLS）去均值则可大幅提升检验功效。当然，这一进展在面板单位根检验的相关研究中并未被忽视。然而，尽管GLS去均值的应用潜力已得到广泛认可，但奇怪的是，目前仍未有理论成果可支撑面板数据场景下此类去均值操作的正式分析。本文正是针对这一研究空白展开的工作。本文旨在评估将GLS去均值与面板混合普通最小二乘t检验结合用于单位根检验时的效果，由此构建出时间序列DF-GLS检验的面板对应形式。核心研究发现表明，GLS去均值的效果严格取决于对被解释变量执行去均值与一阶差分的先后顺序：若先对变量进行去均值操作再执行一阶差分，则采用GLS去均值可实现检验功效最大化；而若先进行差分再去均值，则更适合选用OLS去均值。此外，即便采用前述先去均值后差分的操作流程（此时GLS去均值为最优选择），所得检验统计量的渐近分布仍与刻画该去均值操作所依托的局部备择假设的调谐参数无关。因此，去均值操作完全可以在单位根原假设下进行。从这一角度而言，基于局部备择假设的GLS去均值实属冗余。