ESTIMATING THE RANK OF COINTEGRATION AFTER ESTIMATING THE ORDER OF A VECTOR AUTOREGRESSION

Abstract

Small sample properties are studied for the maximum likelihood test in determining the rank of cointegration. Firstly, some statistical methods are developed to determine the lag order of the vector auto-regressive (VAR) processes with unit roots. The asymptotic χ2 distribution of the likelihood ratio statistic is proved. Then the asymptotic standard normal distribution of the t-ratio is established for coefficients of differenced variables in the error correction representation. The t -ratio can be used to test the significance of individual coefficients in the highest order term of VAR processes without using any information on co-integration. The small sample properties of the likelihood ratio test, the t-test, AIC, and BIC are explored by simulations which are found indispensable in finding the order of VAR processes possibly with unit roots. Furthermore, and most importantly, our simulation shows that the trace test for finding the rank of co-integration does not depend much on the lag order selection criteria. Whichever procedure is used to find the lag order of a VAR process, the trace or the maximum eigen value test may on average give a similar rank of co-integration though this can be wrong.