Asymptotic Theory for Maximum Likelihood Estimation in Stationary Fractional Gaussian Processes, Under Short-, Long- and Intermediate Memory

Consistency, asymptotic normality and e¢ ciency of the maximum likelihood estimator and the Whittle pseudo-maximum likelihood estimator for stationary Gaussian time series, were shown to hold in the short memory case by Hannan (1973) and in the long memory case by Dahlhaus (1989). In this paper, we extend these results to the entire stationarity region, including the case of intermediate memory and noninvertibility. To deal with the noninvertibility region, we adopt a generalization of the Whittle approximation to the inverse of the covariance matrix of a stationary process, recently proposed by Lieberman et. al. (2008). In the process of proving the main results, we provide a useful theorem on the limiting behavior of a product of Toeplitz matrices under strictly weaker conditions than those employed by Dahlhaus (1989). 1 Introduction Let Xt, t 2 Z; be a stationary Gaussian time series with mean and spectral density f (!) ; ! 2 [ ; ] and denote the true values of the parameters by 0 and 0. We are concerned with spectral densities f (!) that belong to the parametric family ff : 2 Rg, such that for all 2 f (!) j!j ( ) L (!) as ! ! 0; (1) where ( ) < 1 is the memory parameter of the series and L (!) is a positive function that varies slowly at ! = 0. Xt is said to have long memory (or long-range dependence) if 0 < ( ) < 1, short memory (or short-range dependence) if ( ) = 0 and intermediate memory (or anti-persistence) if ( ) < 0. The range ( ) 1 corresponds to noninvertibility and our results cover this case as well. Two examples of parametric models that are consistent with (1) are the fractional Gaussian noise (Mandelbrot and Van Ness, 1968) and the ARFIMA models (Granger and Joyeux 1980, Hosking 1981). The asymptotic properties of the Gaussian maximum likelihood estimator (MLE) for short memory dependent observations were derived by Hannan (1973). For the Gaussian ARFIMA(0; d; 0) model, Yajima (1985) proved consistency and asymptotic normality of the MLE when 0 < d < 1 2 and asymptotic normality of the least squares estimator when 0 < d < 1 4 . Dahlhaus (1989, 2006) established consistency, asymptotic normality and e¢ ciency for general Gaussian stationary processes with 1 long memory satisfying (1) and 0 < < 1. Similar results for the parametric Gaussian MLE under intermediate memory and noninvertibility do not appear to be documented in the literature. In the semiparametric framework, Robinson (1995a) established consistency and asymptotic normality of the log-periodogram estimator when 1 < < 1. Velasco (1999a) extended these results by showing that consistency still holds for the range 1 < < 2 and asymptotic normality for 1 < < 3=2. Moreover, with a suitable choice of data taper, a modi…ed version of this estimator was shown to be consistent and asymptotically normal for any real . For the Whittle MLE, Fox and Taqqu (1986) proved consistency and asymptotic normality under the condition 0 < < 1. Velasco and Robinson (2000) extended these results to the range 1 < < 2 and with adequate data tapers, to any degree of nonstationary. The local Whittle estimator was shown by Robinson (1995b) to be asymptotically normal for 1 < < 1, while Velasco (1999b) extended these results by proving consistency for 1 < < 2 and asymptotic normality for 1 < < 3=2. As with the ‘ordinary’Whittle MLE, with suitable tapering, the results are extended to any 1. For the exact local Whittle estimator, Shimotsu and Phillips’s (2004) proved asymptotic normality for any real , if the true mean of the series is known, and Shimotsu (2006) showed that similar results hold in the case where the process has an unknown mean and a linear time trend, for 2 ( 1; 4). 2 The purpose of the paper is to continue this line of literature and …ll the gap concerning the asymptotic properties of the Gaussianas well as Whittle MLE’s by extending it to the case < 1. While there are simulations studies that analyze the performance of these estimators in long-, shortand intermediate-memory, (see Sowell 1992, Cheung and Diebold 1994, Hauser 1999, and Nielsen and Frederiksen 2005), we emphasize that todate, consistency, asymptotic normality and e¢ ciency of the Gaussianas well as Whittle MLE’s in the intermediate memory and noninvertibility case have not been established. We prove these properties without making apriori assumptions on the memory-type of the series. By this it is meant that the researcher is free to …nd the MLE over the entire range < 1. In practice, todate, if the MLE for a given data set was found to be negative and the process was assumed to have positive memory, the value of the MLE was censored to zero. In various simulation experiments, this resulted in a pile-up of MLE values at zero, and essentially this amounts to restricted maximum likelihood estimation, rather than the unrestricted analogue. See, for instance, Lieberman and Phillips (2004a). By establishing a theory for the range < 1, the pile-up at zero is avoided. While our proofs make extensive use of the results of Dahlhaus (1989, 2005), who was restricted to the 0 < < 1 case, the extension to < 1 is not trivial. The main obstacle in the proofs is in the fact the the quadratic form appearing in the log-likelihood function do not have a uniform behavior over the entire range of . Speci…cally, when ( 0) ( ) < 1, the limiting behavior of the N -normalized 3 log-likelihood function is well de…ned and the results can be proven as in Dahlhaus’s (1989) work. However, when ( 0) ( ) 1, the N -normalized log-likelihood function converges to +1 and therefore the minimizer of this function will not be found in this region. For this case, a di¤erent approach is required. The reason for this break is that the limiting behavior of the trace of a product of Toeplitz matrices is very di¤erent when ( 0) ( ) < 1 and when ( 0) ( ) 1. These terms appear in the cumulants of the log-likelihood and its derivatives. In the process of proving the main results, we generalize Theorem 5.1 of Dahlhaus (1989) and prove it under strictly weaker conditions. This result is of use and interest by its own right. It continues a very long tradition on the limiting behavior of Toeplitz matrices. See, among others, Kac (1954), Grenander and Szego (1956), Taniguchi (1983), Fox and Taqqu (1987), Avram (1988), Bercu et. al. (1999) and Lieberman and Phillips (2004b). Our set of assumptions are not stronger than those of Dahlhaus (1989) and are satis…ed in the stationary ARFIMA (p; d; q) model, allowing for the possibility that d 1=2. The outline of the paper is as follows. Section 2 states the model and main results of the paper. Section 3 introduces some limit theorems for Toeplitz matrices. Section 4 contains the proof of consistency, asymptotic normality and e¢ ciency of the Gaussian MLE. In Section 5 analogous results are established for the Whittle MLE and Section 6 concludes. Some auxiliary technical lemmas and proofs are