Estimation for Partially Nonstationary Multivariate Autoregressive Models

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Abstract
For an m-variate (“partially”) nonstationary vector autoregressive process {Y t }, we consider the autoregressive model Φ(L)Y t = ε t , where Φ(L) = I − Φ1 L − … − Φ p L p and det{Φ(z)} = 0 has d < m roots equal to unity and all other roots are outside the unit circle. It is also assumed that rank {Φ(1)} = r, r = m − d, so that each component of the first differences W t = Y t − Y t − 1 is stationary. The relation of the model to error correction models and co-integration (Engle and Granger 1987) is discussed. The process {Y t } has the error correction representation Φ*(L)(1 − L)Y t = −P 2(I r − Λ r )Q 2 1 Y t-1 + ε t , where Q(I m − Φ(1))P = J = diag(I d , Λ r ) is in Jordan canonical form and Q' = [Q 1, Q 2]. It follows that the transformation Z t = QY t = [Zt 1t , Z2 2t ] t is such that the d × 1 process Z 1t is nonstationary with Z 1t − Z 1t-1 stationary while Z 2t is stationary. Asymptotic distribution theory for least squares parameter estimators of the model is first considered. A Gaussia...

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