Residual log-periodogram inference for long-run relationships

• Published in: Journal of econometrics Vol. 130; no. 1; pp. 165 - 207
• Main Authors: Hassler, U., Marmol, F., Velasco, C.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 2006; Elsevier; Elsevier Sequoia S.A
• Series: Journal of Econometrics
• Subjects: Applications; Correlation analysis; Econometrics; Exact sciences and technology; Exchange rates; Foreign exchange rates; Fractional cointegration; Inference from stochastic processes; time series analysis; Insurance, economics, finance; Limiting normality; Long memory; Mathematics; Monte Carlo simulation; Non-stationarity; Nonparametric inference; Parametric inference; Probability and statistics; Sciences and techniques of general use; Semiparametric inference; Statistics; Studies; Exchange rates; C22; C14; Non-stationarity; Semiparametric inference; Fractional cointegration; Limiting normality; Long memory; Monte Carlo method; Data analysis; Wald test; Error estimation; C22 Fractional cointegration: Semiparametric inference; Long memory process; Statistical estimation; Non stationary process; Regression residual; Semiparametric method; Economic data; Transitory; Periodogram; Confidence interval; Hypothesis test; Non- stationarity; Consistent estimator; Econometrics; Exchange rate
• ISSN: 0304-4076; 1872-6895
• DOI: 10.1016/j.jeconom.2005.03.001

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.