To motivate this examination, consider this critique of my modeling Palmer Drought Severity Index (PDSI) as an I(1) series cointegrated with Kansas GDP by reader Rick Stryker:
Cointegration methods are bedeviled with specification error problems. In Johansen, how you set up the form of the VECM matters. How you specify the constant terms matters. Often the multivariate trace and maximum eigenvalue tests disagree. The procedure is a multi-step process in which an error in one stage can pollute the results in the next. Small sample bias is a huge problem to deal with. It’s very easy to misspecify one of these regressions. Even if you do all this specification analysis and are satisfied, there is the rather huge leap in treating government spending as exogenous and causing Kansas GDP. This is a much bigger issue.
Because of the potential specification error problems, cointegration is not the first choice of many time series experts. It’s not necessarily the “correct” way. That’s not to say that no one ever should use it. If you do decide to use it, you just have to be pretty methodical and cautious, recognizing all the pitfalls. …
Mebbe. Tim Duy and Mark Thoma, in JEBS (1998) write:
Although the issue of identifying cointegrating relationships between time-series variables has become increasingly important in recent years, economists have yet to reach an agreement on the appropriate manner of modeling such relationships. In this paper, we attempt to distinguish between modeling techniques through a comparison of forecast statistics, while focusing on the issue of whether or not imposing cointegrating restrictions via an error-correction model improves long-run forecasts. We find that imposing cointegrating restrictions often improves forecasting power, and that these improvements are most likely to occur in models which exhibit strong evidence of cointegration between variables.
The findings of Chinn and Meese (1995) regarding exchange rate prediction conform to Duy-Thoma; to a lesser extent, the sequel (Cheung, Chinn, Garcia Pascual, Zhang, 2019) also shows that a PPP error correction term “works” well. So, I think issues of integration/cointegration are usefully considered in many time series contexts.