Broadband Semiparametric Estimation of the Memory Parameter of a Long‐Memory Time Series Using Fractional Exponential Models

Abstract

Abstract. We consider a fractional exponential, or FEXP estimator of the memory parameter of a stationary Gaussian long-memory time series. The estimator is constructed by ®tting a FEXP model of slowly increasing dimension to the log periodogram at all Fourier frequencies by ordinary least squares, and retaining the corresponding estimated memory parameter. We do not assume that the data were necessarily generated by a FEXP model, or by any other ®nite-parameter model. We do, however, impose a global differentiability assumption on the spectral density except at the origin. Because of this, and its use of all Fourier frequencies, we refer to the FEXP estimator as a broadband semiparametric estimator. We demonstrate the consistency of the FEXP estimator, and obtain expressions for its asymptotic bias and variance. If the true spectral density is suf®ciently smooth, the FEXP estimator can strongly outperform existing semiparametric estimators, such as the Geweke±Porter-Hudak (GPH) and Gaussian semiparametric estimators (GSE), attaining an asymptotic mean squared error proportional to (log n)an, where n is the sample size. In a simulation study, we demonstrate the merits of using a ®nite-sample correction to the asymptotic variance, and we also explore the possibility of automatically selecting the dimension of the exponential model using Mallows' C L criterion.

Estimating fractional cointegration in the presence of polynomial trends

“…We fail to reject the unit root null at both levels when the longer maturity is at least ÿve years, reject at 5% level for the 3 month-3 year series and reject at both levels for the remaining pairs. To investigate the strength of the fractional cointegration among interest rates thoroughly, we apply both OLS and the tapered NBLS for ÿ and three estimators for d U the GSE, the GPH (Geweke and Porter-Hudak, 1983) and the FEXP (Moulines and Soulier, 1999;Hurvich and Brodsky, 2001). The GPH estimator is the least squares estimator of d in the regression model, log I ; j = C 1 − 2d log sin( j =2) + j ; j = 1; 2; : : : ; m;…”

Semiparametric inference in multivariate fractionally cointegrated systems

“…Various devices are available for achieving this. With respect to f , Parzen (1957) employed higherorder kernels in the frequency domain, while with respect to the δ's Andrews and Sun (2004), Hurvich and Brodsky (2001), Moulines and Soulier (1999) and Robinson and Henry (2003) employed various methods (albeit justified only in case of integration orders falling in the stationarity region). All these methods entail an increase in computation, they rely on greater smoothness in f , and in practice they can sometimes exhibit disappointing small-sample properties.…”

Estimators for the long-memory parameter in LARCH models, and fractional Brownian motion

“…As an example, consider condition (22). Recall that the conditional standard deviation of the LARCH(1) process is de…ned as t = a+ P 1 j=1 b j r t j where a 6 = 0: Giraitis, Robinson and Surgailis (2000) established the existence of the weakly stationary LARCH(1) process under the condition…”

“…This means that the Ejr 2 l t ( 0 )j need not be …nite and, therefore, the sum (22) does not satisfy conditions of the ergodic theorem. Because of this, it is hard to see how the condition (22) can be enforced in the LARCH (1) case. This opinion had also been conveyed to us by Prof. L. Giraitis in personal communication; he also suggested that there may exist a modi…cation of the LARCH(1) process for which these conditions are true but that they are almost certainly cannot be validated in the "classical" version of the LARCH (1) process considered here.…”

“…They were explored in great detail by Robinson (1995a) [36], and Robinson (1995b) [37]. Closely related are the broad-band estimators of Moulines and Soulier (1999) [33] and of Hurvich and Brodsky (2001) [22], as well as the exact local Whittle estimation method of Shimotsu and Phillips (2005) [38]. There has been relatively little work done on the semiparametric estimation of the long-memory parameter for nonlinear time series.…”