Conditional Inference in Cointegrating Vector Autoregressive Models

A Vector Autoregressive model (VAR) with normally distributed innovations is a Curved Exponential Model (CEM). Cointegration imposes further curvature on the model and this means that in addition to the important reasons for conditioning in non-stationary time series as given by Johansen (1995, EJ), there are further reasons due to the curvature of the model. This paper investigates the effects of conditioning on the likelihood ratio test statistic for the cointegrating rank, which in this case is a natural approximate ancillary statistic. We investigate the effect of conditioning on this test statistic for inference on the long-run (beta) and also on the speed-of-adjustment (alpha) coefficients. We show that this conditioning gives virtually the same estimates of the estimator variance as using the observed information instead of the expected information. We examine the possibility of achieving asymptotic refinements for inference on alpha using a conditioning parametric bootstrap procedure