Estimation for Partially Nonstationary Multivariate Autoregressive Models

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 detΦ (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 Gaussian partial reduced rank estimation procedure that explicitly incorporates the unit root structure in the model is then presented, and an asymptotically equivalent two-step reduced rank estimation procedure is also considered. A numerical example is presented to illustrate the methods and concepts. The finite sample properties of the estimators are also briefly examined through a small Monte Carlo sampling experiment.