PUB - Publikationen an der Universität Bielefeld

In this paper the theory on the estimation of vector autoregressive (VAR) models for I(2)processes is extended to the case of long VAR approximation of more general processes. Hereby theorder of the autoregression is allowed to tend to infinity at a certain rate depending on the samplesize. We deal with unrestricted OLS estimators (in the model formulated in levels as well as in vectorerror correction form) as well as with two stage estimation (2SI2) in the vector error correction model(VECM) formulation. Our main results are analogous to the I(1) case: We show that the long VARapproximation leads to consistent estimates of the long and short run dynamics. Furthermore, testson the autoregressive coefficients follow standard asymptotics. The pseudo likelihood ratio testson the cointegrating ranks (using the Gaussian likelihood) used in the 2SI2 algorithm show underthe null hypothesis the same distributions as in the case of data generating processes followingfinite order VARs. The same holds true for the asymptotic distribution of the long run dynamicsboth in the unrestricted VECM estimation and the reduced rank regression in the 2SI2 algorithm.Building on these results we show that if the data is generated by an invertible VARMA process,the VAR approximation can be used in order to derive a consistent initial estimator for subsequentpseudo likelihood optimization in the VARMA model.