Statistical-Inference in Vector Autoregressions with Possibly Integrated Processes

This paper shows how we can estimate VAR's formulated in levels and test general restrictions on the parameter matrices even if the processes may be integrated or cointegrated of an arbitrary order. We can apply a usual lag selection procedure to a possibly integrated or cointegrated VAR since the standard asymptotic theory is valid (as far as the order of integration of the process does not exceed the true lag length of the model). Having determined a lag length k, we then estimate a (k + d(max))th-order VAR where d(max) is the maximal order of integration that we suspect might occur in the process. The coefficient matrices of the last d(max) lagged vectors in the model are ignored (since these are regarded as zeros), and we can test linear or nonlinear restrictions on the first k coefficient matrices using the standard asymptotic theory.