Bootstrap Determination of the Co‐Integration Rank in VAR Models with Unrestricted Deterministic Components

<jats:p>In a recent paper, <jats:italic>Cavaliere et al.</jats:italic>, <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#jtsa12104-bib-0004" /> develop bootstrap implementations of the popular likelihood‐based co‐integration rank tests and associated sequential rank determination procedures of <jats:italic>Johansen</jats:italic> <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#jtsa12104-bib-0008" />. By using estimates of the parameters of the underlying co‐integrated VAR model obtained under the restriction of the null hypothesis, they show that consistent bootstrap inference can be obtained for processes whose deterministic component is either zero, a restricted constant or a restricted trend. In this article, we extend their bootstrap approach to allow the deterministic component to follow the practically relevant cases of either an unrestricted constant or an unrestricted trend from <jats:italic>Johansen</jats:italic> <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#jtsa12104-bib-0008" />. A full asymptotic theory is provided for these two cases, establishing the asymptotic validity of the resulting bootstrap tests. Our results, taken together with those in <jats:italic>Cavaliere et al.</jats:italic>, <jats:ext-link xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#jtsa12104-bib-0004" />, therefore show that the bootstrap approach based on imposing the reduced rank null hypothesis is valid for all five of these deterministic settings. Monte Carlo evidence demonstrates the improvements that the proposed bootstrap methods can deliver over the corresponding asymptotic procedures.</jats:p>