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On the limit theory of mixed to unity VARs: Panel setting with weakly dependent errors

In this article, we re‐visit a recent idea of Phillips and Lee (2015. Econometric Reviews 34: 1035 ‐ 1056). They examine an empirically relevant situation when two time series exhibit different degrees of non‐stationarity and one need to learn whether their persistence properties are the same. By bridging the asymptotic theory of local to unity and mildly explosive processes, they construct a Wald test for the commonality of the long‐run behavior of the series. However, inference is complicated by the fact that their statistic does not converge in distribution under the null and diverges under the alternative. This is true if the parameters of the data generating process are known and a re‐normalizing function can be constructed. If the parameters are unknown, which will be the case in practice, the test statistic may be divergent even under the null. We solve this problem by converting the original setting of vector time series into a panel setting with N individual vector series. We show that the proposed panel Wald test statistics converge to chi‐squared distribution which is free of nuisance parameters under the null hypothesis of common local to unity behavior. The result is an extreme example of simplified asymptotics brought about by panel data.