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Standard methods, such as sequential procedures based on Johansen's(pseudo-)likelihood ratio (PLR) test, for determining the co-integration rankof a vector autoregressive (VAR) system of variables integrated of order onecan be significantly affected, even asymptotically, by unconditionalheteroskedasticity (non-stationary volatility) in the data. Known solutions tothis problem include wild bootstrap implementations of the PLR test or the useof an information criterion, such as the BIC, to select the co-integrationrank. Although asymptotically valid in the presence of heteroskedasticity,these methods can display very low finite sample power under some patterns ofnon-stationary volatility. In particular, they do not exploit potentialefficiency gains that could be realised in the presence of non-stationaryvolatility by using adaptive inference methods. Under the assumption of a knownautoregressive lag length, Boswijk and Zu (2022) develop adaptive PLR testbased methods using a non-parameteric estimate of the covariance matrixprocess. It is well-known, however, that selecting an incorrect lag length cansignificantly impact on the efficacy of both information criteria and bootstrapPLR tests to determine co-integration rank in finite samples. We show thatadaptive information criteria-based approaches can be used to estimate theautoregressive lag order to use in connection with bootstrap adaptive PLRtests, or to jointly determine the co-integration rank and the VAR lag lengthand that in both cases they are weakly consistent for these parameters in thepresence of non-stationary volatility provided standard conditions hold on thepenalty term. Monte Carlo simulations are used to demonstrate the potentialgains from using adaptive methods and an empirical application to the U.S. termstructure is provided.

Adaptive information-based methods for determining the co-integration rank in heteroskedastic VAR models