Residual Based Test for Cointegration with GLS Detrented Data

We analyze di¤erent residual-based tests for the null of no cointegration using GLS detrended data. We find and simulate the limiting distributions of these statistics when GLS demeaned and GLS detrended data are used. The distributions depend of the number of right-hand side variables, the type of deterministic components used in the cointegration equation, and a nuisance parameter R2 which measures the long-run correlation between xt and yt. We present an extensive number of Figures which show the asymptotic power functions of the different statistics analyzed in this paper. The results show that GLS allows obtaining more asymptotic power in comparison with OLS detrending. The more simple residual-based tests (as the ADF) shows power gains for small values of R2 and for only one right-hand side variable. This evidence is valid for R2 less than 0.4. Figures show that when R2 is larger, the ECR statistics are better for any value of the right-hand side variables. In particular, evidence shows that the ECR statistic which assumes a known cointegration vector is the most powerful. A set of simulated asymptotic critical values are also presented. Unlike other references, in the present framework we use different c for different number of right-hand side variables (xt variables) and according to the set of deterministic components. In this selection, we use a R2 = 0:4, which appears to be a sensible choice.

Keywords
Cointegration, Residual-Based Unit Root Tests, ECR Tests, OLS and GLS Detrended Data, Hypothesis Testing.

JEL Classification

C2, C3, C5