The general takeaway is this: the Dumitrescu and Hurlin (2012) test achieves best size when regression lags $p$ are smallest (regardless of the underlying true AR structure), whereas it achieves best power when $p$ matches the true AR structure, where the penalty for underspecifying $p$ can be severe. In particular, if $k$ is the number of lags in the true DGP, and $p$ is the number of regression lags selected, the test is severely underpowered for all $p k$, the effect is not nearly as severe, and virtually unnoticeable. In contrast, the test can be grossly underpowered whenever the regression lag $p$ deviates from the lag structure characterizing the true DGP. In particular, in the case where $T,N \longrightarrow \infty$, observe that $W_N$ version of the test. In either case, the results follow from classical statistical concepts and central limit theorems (CLT). The latter ensures that the OLS regression in Step 1 above is valid, by preventing situations in which there are more parameters than observations. Where $0\leq N_1/N 5 + 3K$ as a necessary condition for the validity of results. Unlike traditional panel data in which each cross section $i = 1, \ldots, N$ is associated with $t=1, \ldots, T N_1$.} With data availability at its historical peak, time series panel econometrics is in the limelight.
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