Residual log-periodogram inference for long-run relationships

We assume that some consistent estimator β^ of an equilibrium relation between non-stationary series integrated of order d∈(0.5,1.5) is used to compute residuals u^t=yt-β^xt (or differences thereof). We propose to apply the semiparametric log-periodogram regression to the (differenced) residuals in order to estimate or test the degree of persistence δ of the equilibrium deviation ut. Provided β^ converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of δ. At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on δ. This requires that d-δ>0.5 for superconsistent β^, so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0⩽δ<0.5, as well as for non-stationary but transitory equilibrium errors, 0.5<δ<1. In particular, if xt contains several series we consider the joint estimation of d and δ. Wald statistics to test for parameter restrictions of the system have a limiting χ2 distribution. We also analyse the benefits of a pooled version of the estimate. The empirical applicability of our general cointegration test is investigated by means of Monte Carlo experiments and illustrated with a study of exchange rate dynamics.