The Wold Representation, Cointegration and the Johansen Trace Test +,*

This paper considers the importance of moving average errors for the Johansen trace test of cointegrating rank based upon approximating vector autoregressions (VARs). Two cases are emphasized and explored, both taking as their starting point the Wold decomposition. Though each case defined satisfies the same cointegrating vector, one is a case of multicointegration while the other, based on results associated with the Sargan-Bézout factorization, negating the possibility of multicointegration, inverts to an exact VAR(1). Both cases are controlled by a single parameter which exploits the key distinction between dependent and independent trends. The results show both size and power to be highly sensitive to the order of vector autoregressive model fitted and depend on sample size, closeness to cointegration, and closeness of the VAR approximation to the process generating the data. In most cases, the null distributions from which critical values are conventionally drawn are poor approximations to the empirical null distributions.