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 − Φ 1 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 = [Z t 1 t , Z 2 2 t ] t is such that the d × 1 process Z 1 t is nonstationary with Z 1 t − Z 1 t -1 stationary while Z 2 t 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.