Detecting true relationships in time series data with different orders of integration

It is fairly well-known that proper time series analysis requires that estimated equations be balanced. Numerous scholars mistake this to mean that one cannot mix orders of integration. Previous studies have clarified the distinction between equation balance and having different orders of integration, and shown that mixing orders of integration does not increase the risk of type I error when using the general error correction/autoregressive distributed lag (GECM/ADL) models, that is, so long as equations are balanced (and other modeling assumptions are met). This paper builds on that research to assess the consequences for type II error when employing those models. Specifically, we consider cases where a true relationship exists, the left- and right-hand sides of the equation mix orders of integration, and the equation still is balanced. Using the asymptotic case, we find that the different orders of integration do not preclude identification of the true relationship using the GECM/ADL. We then highlight that estimation is trickier in practice, over finite time, as data sometimes do not reveal the underlying process. But, simulations show that even in these cases, researchers will typically draw accurate inferences as long as they select their models based on the observed characteristics of the data and test to be sure that standard model assumptions are met. We conclude by considering the implications for researchers analyzing or conducting simulations with time series data.