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 = y t - β ^ x t (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 u t . 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 x t 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 χ 2 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.