The Estimation and Application of Long Memory Time Series Models

Abstract

The definitions of fractional Gaussian noise and integrated (or fractionally differenced) series are generalized, and it is shown that the two concepts are equivalent. A new estimator of the long memory parameter in these models is proposed, based on the simple linear regression of the log periodogram on a deterministic regressor. The estimator is the ordinary least squares estimator of the slope parameter in this regression, formed using only the lowest frequency ordinates of the log periodogrqm. Its asymptotic distribution is derived, from which it is evident that the conventional interpretation of these least squares statistics is justified in large samples. Using synthetic data the asymptotic theory proves to be reliable in samples of 50 observations or more. For three postwar monthly economic time series, the estimated integrated series model provides more reliable out-of-sample forecasts than do more conventional procedures.

Non-stationary log-periodogram regression

“…(7) is an approximate linear regression model with response variable log I( H ), regressor !2log H , and slope d, and standard methods could be used to estimate d if the above properties of the normalized periodogram extend to long memory (d'0) or antipersistent (d(0) processes. The log-periodogram regression estimate is just the ordinary least squares (OLS) solution as proposed by Geweke and Porter-Hudak (1983) with H substituted by the asymptotically equivalent quantity 2 sin H /2. Among other issues, Robinson (1995) modi"ed the OLS estimate considering the logs of a pooled periodogram of J"1, 2, 2 (a "xed number of) periodogram ordinates,…”

“…One of most popular semiparametric estimates in the frequency domain is the log-periodogram regression, proposed initially by Geweke and Porter-Hudak (1983). Robinson (1995) showed the consistency and asymptotic normality of a version of that estimate for stationary and invertible Gaussian vector time series.…”

Long memory in return structures from developed markets

“…In this paper we tested the stationary properties of all the data series using Augmented Dickey-Fuller (ADF) test, Phillips-Perron (PP) test. We have tried to capture the long memory property of financial data using classical rescaled-range (R/S) analysis (Hurst, 1951;Mandelbrot, 1972), modified rescaled-range (R/S) analysis introduced by Lo (1991) and the spectral regression method suggested by Geweke and Porter-Hudak (1983). The above tests were applied on logarithmic return series, absolute return series and squared return series.…”

“…In order to estimate the fractional differencing estimator d, Geweke and Porter-Hudak (1983) proposed a semi-parametric method of the long memory parameter d which can capture the slope of the sample spectral density through a simple OLS regression based on the periodogram, as follows: logI(λ) = β 0 -d log {4sin 2 (λ j / 2)}+υ j , j=1,…M; where I(λ) is the j th periodogram point; λ j = 2πj / T; T is the number of observations; β 0 is a constant; and υ j is an error term, asymptotically i.i.d, across harmonic frequencies with zero mean and variance known to be equal to π 2 / 6. M = g (T) = T µ with 0 < µ < 1 is the number of Fourier frequencies included in the spectral regression and is an increasing function of T. As argued by GPH the inclusion of improper periodogram ordinates M, causes bias in the regression which result in an imprecise value of d. To achieve the optimal choice of T, several choices are established in terms of the bandwidth parameter M = T 0.45 ; T 0.50 ; …, T 0.7 .…”

Workload Modeling for Computer Systems Performance Evaluation

“…So if we calculate the periodogram and graph it as a function of frequency on log-log axes, fitting a straight line to the low frequencies will enable us to estimate H: if the slope of the line is α, then H = 1−α 2 [282]. And now for the details.…”