Vector Autoregressive Models With Unit Roots and Reduced Rank Structure:estimation. Likelihood Ratio Test, and Forecasting

Abstract. The nonstationary multivariate autoregressive (AR) model Φ (L) Y t =ε t is considered for an m ‐dimensional process Y t, where it is assumed that det Φ (L) = 0 has d < m unit roots and all other roots are outside the unit circle, and also that rank Φ (1) = r (r = m – d). Limiting distribution results obtained by Ahn and Reinsel for the least‐squares and the Gaussian reduced rank (unit roots imposed) estimators for this AR model are extended to a model where the AR parameters possess additional structure such as nested reduced rank, and based on these results the asymptotic distribution of the likelihood ratio test statistic for testing the number d of unit roots is obtained. An analysis of three US monthly interest rate series is presented to illustrate the testing and estimation procedures. A small simulation study is also performed to examine the finite‐sample properties of the likelihood ratio test and the prediction performance of models which impose different numbers of unit roots.