Testing cointegration in infinite order vector autoregressive processes
Tests are investigated to find the cointegration rank in infinite order \(n\)-dimensional autoregressive processes: \[ y_{1t}= Ay_{2t}+ u_{1t}, \quad\Delta y_{2t}= u_{2t}, \] where \(\dim y_{1t} =n_1\), \(\dim y_{2t}= n_2\) \((n_1+n_2+n)\), and residuals have an infinite order autoregressive representation. On tests the null hypothesis \(H(n_1)\): ``cointegrating rank equal to \(n_1\)'' against the alternative hypothesis \(\overline H(n_1)\): ``cointegrating rank larger than \(n_1\)''. It is shown that the original versions of the likelihood ratio tests for finite order Gaussian VAR processes [see, e.g., \textit{S. Johanson}, Econometrica 59, No. 6, 1551-1580 (1991; Zbl 0755.62087)] can be extended to infinite order non-Gaussian processes. Moreover, new tests are suggested for the cases where intercept terms are included in the cointegration relations \(z_t=m+y_r\). The intercept terms are estimated using generalised least squares. The resulting test statistics have under the null hypothesis the same limiting distributions as in the case where the model contains no intercept term or when the values of the intercept term are a priori known. The new tests have good power properties compared with their previous counterparts. These properties are confirmed by a simulation study.
scientific article; zbMATH DE number 1083050