Testing for General Fractional Integration in the Time Domain

Abstract

In this paper we propose a family of least-squares based testing procedures that look to detect general forms of fractional integration at the long-run and/or the cyclical component of a time series, and which are asymptotically equivalent to Lagrange Multiplier tests. Our setting extends Robinson's (1994) results to allow for short memory in a regression framework and generalises the procedures in Agiakloglou and Newbold (1994), Tanaka (1999) and Breitung and Hassler (2002) by allowing for single or multiple fractional unit roots at any frequency in [0, π]. Our testing procedure can be easily implemented in practical settings and is flexible enough to account for a broad family of long-and shortmemory specifications, including ARMA and/or GARCH-type dynamics among others. Furthermore, these tests have power against different types of alternative hypotheses and enable inference to be conducted under critical values drawn from a standard Chi-squared distribution, irrespective of the long-memory parameters.

Detecting Multiple Breaks in Long Memory: The Case of U.S. inflation

“…We indicate generalizations without going into technical details and omit formal proof, as our test statistic is related to statistics handled in the papers referenced below. A valid set of conditions replacing Assumption 2 is now adopted from Hassler, Rodrigues, and Rubia (2009 Let us briefly comment on four generalizations going beyond the previous section (Assumption 2).…”

“…Although it lacks optimality properties it is widely used in applied econometrics. Demetrescu, Hassler, and Kuzin (2011) Hassler et al (2009). Hence, the zero starting value assumption can be discarded without loss.…”

Quasi-maximum likelihood estimation and bootstrap inference in fractional time series models with heteroskedasticity of unknown form

“…The strengthening of Assumption 1(a)(iii) to a summability condition on the eighth order cumulants of ε t in Assumption 5 would appear to be standard, whether stated directly or indirectly, in the fractional literature where asymptotic distribution theory is derived under (conditional) heteroskedasticity, the leading example being the literature on hypothesis testing on the long memory parameter, d; see, inter alia, Demetrescu, Kuzin and Hassler (2008), Hassler, Rodrigues and Rubia (2009) and Kew and Harris (2009). Also, the moment condition sup t E|z t | 8 < ∞, imposed for example by these authors, is necessary for Assumption 5 with q = 8 to hold and therefore is not stated explicitly.…”

“…We do so using a new framework which includes the general form of non-stationary volatility considered in Cavaliere (2005), Cavaliere andTaylor (2005, 2008), and Phillips and Xu (2006) as a special case and also includes a set of conditional heteroskedasticity conditions which are similar, but somewhat weaker, than those employed previously in the fractional integration literature by Robinson (1991), Demetrescu, Kuzin and Hassler (2008) and Hassler, Rodrigues and Rubia (2009), among others. Neither of these conditions involve specifying a parametric model for the volatility process.…”

“…The classical likelihood-based Wald, likelihood ratio (LR) and Lagrange multiplier (LM) tests for inference on the long memory parameter have been derived under the assumption of conditionally (and, hence, unconditionally) homoskedastic shocks; see, among others, Robinson (1994), Agiakloglou and Newbold (1994), Tanaka (1999), Nielsen (2004), Lobato and Velasco (2007), and Nielsen (2010, 2012). A small number of papers have considered the case where the shocks can display certain forms of conditional heteroskedasticity (but maintaining the assumption of unconditional homoskedasticity); see, for example, Robinson (1991), Baillie, Chung, Tieslau (1996), Ling and Li (1997), Ling (2003), Demetrescu, Kuzin and Hassler (2008) and Hassler, Rodrigues and Rubia (2009). Allowing for non-stationary volatility of a similar form to that considered in this paper, Kew and Harris (2009) extend the idea of Demetrescu, Kuzin and Hassler (2008) to use heteroskedasticity-robust White (1980)-type standard errors when computing regression-based tests for fractional integration.…”