Residual log-periodogram inference for long-run relationships

Abstract

We assume that some consistent estimator b b of an equilibrium relation between non-stationary series integrated of order d 2 ð0:5; 1:5Þ is used to compute residualsû t ¼ y t b bx 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 d of the equilibrium deviation u t : Provided b b converges fast enough, we describe simple semiparametric conditions around zero frequency that guarantee consistent estimation of d: At the same time limiting normality is derived, which allows to construct approximate confidence intervals to test hypotheses on d: This requires that d d40:5 for superconsistent b b; so the residuals can be good proxies of true cointegrating errors. Our assumptions allow for stationary deviations with long memory, 0pdo0:5; as well as for non-stationary but transitory equilibrium errors, 0:5odo1: In particular, if x t contains several series we consider the joint estimation of d and d: Wald statistics to test for parameter restrictions of the system have a limiting w 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. r 2005 Elsevier B.V. All rights reserved.JEL classification: C14; C22

Determining the cointegrating rank in nonstationary fractional systems by the exact local Whittle approach

“…Breitung and Hassler (2002), Hurvich (2003a,b, 2004), Robinson and Hualde (2003), Robinson and Marinucci (2003), Velasco (2003a,b), Dolado and Marmol (2004), Nielsen (2004b), Marmol and Velasco (2004), Hassler and Breitung (2005), Christensen and Nielsen (2006), and Hassler et al (2006). Recent applications of fractional cointegration can be found, for example, in Dueker and Startz (1998), Brunetti and Gilbert (2000), Kim and Phillips (2001), Marinucci and Robinson (2001), and Henry and Zaffaroni (2003).…”

Foreign exchange, fractional cointegration and the implied–realized volatility relation

“…To examine these issues, we construct two RV series from intra-day 5 and daily data. Thirdly, the possibility of fractional cointegration is examined formally using a new adaptation of the recently developed semiparametric technique of Hassler et al (2006) [hereafter HMV]. Under certain assumptions HMV prove that a residual-based log periodogram estimator, where the first few harmonic frequencies have been trimmed, has a limiting normality property.…”

Cointegration in fractional systems with deterministic trends

“…Though the fractional cointegration literature is dwarfed by the I(1)/I(0) literature, there are now a number of references dealing with theoretical issues (see e.g. Dolado and Marmol (1997), Chan and Terrin (1995), Jeganathan (1999), Kim and Phillips (2000), Velasco (2000), Hassler, Marmol and Velasco (2002)) and empirical applications (see e.g. Cheung and Lai (1993)).…”