Long Memory Time Series Research Articles

Computation of the forecast coefficients for multistep prediction of long-range dependent time series

Three different linear methods, called the truncation, type-II plug-in and type-II direct, for constructing multistep forecasts of a long-range dependent time series are discussed, all three methods being based on a stochastic model fitted to the time series for characterizing its long-memory as well as short-memory components. However, while the forecast coefficients for the truncation method may be obtained from the model equation itself, those for the type-II plug-in and type-II direct methods involve the autocorrelation function of the fitted stochastic model, analytic exact expressions for which may be cumbersome to evaluate. A numerical quadrature procedure, based on the fast Fourier transform algorithm, for computing the autocorrelation function of a long-memory process, as well as the forecast coefficients for the truncation method, is suggested and the computational accuracy of the approximations is investigated for several ARFIMA models. The three methods of constructing the forecasts of a long-memory time series apply when the innovations, ϵt, of the fitted model are postulated to follow a Gaussian distribution, and, also, when they follow an infinite variance stable distribution with characteristic exponent τ, 1<τ<2. A comparison of the multistep forecasts produced by these three methods is carried out by simulating several ARFIMA models with both Gaussian and stable innovations, and two FEXP models with Gaussian innovations. In addition, their relative behaviour with an actual time series, namely, the mean temperature in England, 1659–1976, is examined.

A class of nearly long-memory time series models

We consider an autoregressive regime-switching model for the dynamic mean structure of a univariate time series. The model allows for a variety of stationary and nonstationary alternatives, and includes the possibility of approximate long-memory behavior. The proposed model includes as special cases white noise, first-order autoregression, and random walk models as well as regime-switching models and the random level-shift model proposed by Chen and Tiao, Journal of Business and Economic Statistics, 8 (1990) p. 83. We describe properties of the model, focusing on its resemblance to long-memory under a certain asymptotic parameterization. We develop a reversible-jump Markov chain Monte Carlo method for Bayesian inference on unknown model parameters and apply the methodology to the Nile River data.

The FEXP estimator for potentially non-stationary linear time series

We consider semiparametric fractional exponential (FEXP) estimators of the memory parameter d for a potentially non-stationary linear long-memory time series with additive polynomial trend. We use differencing to annihilate the polynomial trend, followed by tapering to handle the potential non-invertibility of the differenced series. We propose a method of pooling the tapered periodogram which leads to more efficient estimators of d than existing pooled, tapered estimators. We establish asymptotic normality of the tapered FEXP estimator in the Gaussian case with or without pooling. We establish asymptotic normality of the estimator in the linear case if pooling is used. Finally, we consider minimax rate-optimality and feasible nearly rate-optimal estimators in the Gaussian case.

The approximation of long‐memory processes by an ARMA model

AbstractA mean square error criterion is proposed in this paper to provide a systematic approach to approximate a long‐memory time series by a short‐memory ARMA(1, 1) process. Analytic expressions are derived to assess the effect of such an approximation. These results are established not only for the pure fractional noise case, but also for a general autoregressive fractional moving average long‐memory time series. Performances of the ARMA(1,1) approximation as compared to using an ARFIMA model are illustrated by both computations and an application to the Nile river series. Results derived in this paper shed light on the forecasting issue of a long‐memory process. Copyright © 2001 John Wiley &amp; Sons, Ltd.

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

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 fitting 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 finite‐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 sufficiently 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)/n, where n is the sample size. In a simulation study, we demonstrate the merits of using a finite‐sample correction to the asymptotic variance, and we also explore the possibility of automatically selecting the dimension of the exponential model using Mallows’CL criterion.

An Efficient Taper for Potentially Overdifferenced Long‐memory Time Series

We propose a new complex‐valued taper and derive the properties of a tapered Gaussian semiparametric estimator of the long‐memory parameter dε (−0.5, 1.5). The estimator and its accompanying theory can be applied to generalized unit root testing. In the proposed method, the data are differenced once before the taper is applied. This guarantees that the tapered estimator is invariant with respect to deterministic linear trends in the original series. Any detrimental leakage effects due to the potential noninvertibility of the differenced series are strongly mitigated by the taper. The proposed estimator is shown to be more efficient than existing invariant tapered estimators. Invariance to kth order polynomial trends can be attained by differencing the data k times and then applying a stronger taper, which is given by the kth power of the proposed taper. We show that this new family of tapers enjoys strong efficiency gains over comparable existing tapers. Analysis of both simulated and actual data highlights potential advantages of the tapered estimator of d compared with the nontapered estimator.

Broadband log-periodogram regression of time series with long-range dependence

This paper discusses the properties of an estimator of the memory parameter of a stationary long-memory time-series originally proposed by Robinson. As opposed to ‘‘narrow-band’’ estimators of the memory parameter (such as the Geweke and Porter-Hudak or the Gaussian semiparametric estimators) which use only the periodogram ordinates belonging to an interval which degenerates to zero as the sample size $n$ increases, this estimator builds a model of the spectral density of the process over all the frequency range, hence the name, “broadband.” This is achieved by estimating the ‘‘short-memory’’ component of the spectral density, $f*(x) = |1 - e^{ix}|^{2d}f(x)$, where $d \epsilon (-1/2, 1/2)$ is the memory parameter and $f(x)$ is the spectral density, by means of a truncated Fourier series estimator of log $f*$. Assuming Gaussianity and additional conditions on the regularity of $f*$ which seem mild, we obtain expressions for the asymptotic bias and variance of the long-memory parameter estimator as a function of the truncation order. Under additional assumptions, we show that this estimator is consistent and asymptotically normal. If the true spectral density is sufficiently smooth outside the origin, this broadband estimator outperforms existing semiparametric estimators, attaining an asymptotic mean-square error $O(\log(n)/n)$ .

Some simulations and applications of forecasting long-memory time-series models

In this paper, we show some results of forecasting based on the ARFIMA(p,d,q) and ARIMA(p,d,q) models. We show, by simulation, that the technique of forecasting of the ARIMA(p,d,q) model can also be used when d is fractional, i.e., for the ARFIMA(p,d,q) model. We also conduct a simulation study to compare the two estimators of d obtained through regression methods. They are used in the hypothesis test to decide whether or not the series has long memory property and are compared on the basis of their k-step ahead forecast errors. The properties of long-memory models are also investigated using an actual set of data.

Gaussian and non-Gaussian limit distributions of estimates of the regression coefficients of a long-memory time series

We obtain Gaussian and non-Gaussian distributions of estimates of regression coefficients of a long-memory time series.

Plug‐in Selection of the Number of Frequencies in Regression Estimates of the Memory Parameter of a Long‐memory Time Series

We consider the problem of selecting the number of frequencies, m, in a log‐periodogram regression estimator of the memory parameter d of a Gaussian long‐memory time series. It is known that under certain conditions the optimal m, minimizing the mean squared error of the corresponding estimator of d, is given by m(opt)=Cn4/5, where n is the sample size and C is a constant. In practice, C would be unknown since it depends on the properties of the spectral density near zero frequency. In this paper, we propose an estimator of C based again on a log‐periodogram regression and derive its consistency. We also derive an asymptotically valid confidence interval for d when the number of frequencies used in the regression is deterministic and proportional to n4/5. In this case, squared bias cannot be neglected since it is of the same order as the variance. In a Monte Carlo study, we examine the performance of the plug‐in estimator of d, in which m is obtained by using the estimator of C in the formula for m(opt) above. We also study the performance of a bias‐corrected version of the plug‐in estimator of d. Comparisons with the choice m=n1/2 frequencies, as originally suggested by Geweke and Porter‐Hudak (The estimation and application of long memory time series models. Journal of Time Ser. Anal. 4 (1983), 221–37), are provided.

Estimation and forecasting of long-memory processes with missing values

This paper addresses the issues of maximum likelihood estimation and forecasting of a long-memory time series with missing values. A state-space representation of the underlying long-memory process is proposed. By incorporating this representation with the Kalman filter, the proposed method allows not only for an efficient estimation of an ARFIMA model but also for the estimation of future values under the presence of missing data. This procedure is illustrated through an analysis of a foreign exchange data set. An investment scheme is developed which demonstrates the usefulness of the proposed approach. © 1997 John Wiley & Sons, Ltd.

The change-point problem for dependent observations

We consider the change-point problem for the marginal distribution function of a strictly stationary time series. Asymptotic behavior of Kolmogorov-Smirnov type tests and estimators of the change point is studied under the null hypothesis and converging alternatives. The discussion is based on a general empirical process' approach which enables a unified treatment of both short-memory (weakly dependent) and long-memory time series. In particular, the case of long-memory moving-average process X j = Σ s ⩽ j b j− s ξ s is studied, using the recent results of Giraitis and Surgailis (1994).

The quasi-likelihood approach to statistical inference on multiple time-series with long-range dependence

The paper gives a unified approach to dealing with multiple long-memory time-series possessing a variety of singularities in their spectrum, based on the quasi-likelihood function. It proposes quasi-maximum-likelihood estimation and the quasi-likelihood ratio test for statistical inference purposes. A large-sample theory is given by means of a bracketing function approach under very general conditions, without the usual assumptions of Gaussianity or exact martingale differences for innovation processes. The paper also discusses the particular characteristic of modelling long-memory time-series which influences the type of the quasi-likelihood function and produces distinct differences in the asymptotic properties of related statistics.

Unemployment hysteresis in Canada: an approach based on long-memory time series models

Long-memory univariate time series models, known as ARFIMA, are applied to measure shock persistence for Canada and the USA. Both output and unemployment exhibit higher persistence in Canada than in the USA. The finding that the rate of unemployment in Canada exhibits high shock persistence supports the notion of hysteresis in unemployment. The analysis of disaggregated Canadian unemployment rates indicates that hysteresis is most pronounced for adult male workers. The same class of models is used to obtain a generalization of the Beveridge-Nelson decomposition of time series into a permanent and a cyclical component. The permanent component of the Canadian unemployment rate exhibits a kink in the early 1970s.

Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series

We derive the asymptotic distributions of the sample mean, autocovariances, and autocorrelations for a time series whose autocovariance function “γ κ” has the powerlaw decay γ κ ∼ λκ − Σ , λ > 0, o < α < 1, as κ → ∞. The results differ in important respects from the corresponding results for short-memory processes, whose autocovariance functions are absolutely summable. For long-memory processes the variances of the sample mean, and of the sample autocovariances and autocorrelations for 0 < α ⩽ 1 2 , are not of asymptotic order n −1. When 0 < α < 1 2 the asymptotic distributions of the sample autocovariances and autocorrelations are not Normal.

The Estimation of Continuous Parameter Long-Memory Time Series Models

A class of univariate fractional ARIMA models with a continuous time parameter is developed for the purpose of modeling long-memory time series. The spectral density of discretely observed data is derived for both point observations (stock variables) and integral observations (flow variables). A frequency domain maximum likelihood method is proposed for estimating the longmemory parameter and is shown to be consistent and asymptotically normally distributed, and some issues associated with the computation of the spectral density are explored.

Semiparametric exploration of long memory in stock prices

New or modified methods for semiparametric analysis of fractional long memory in time series are described and applied to twenty-six stock prices and two stock indices. Evidence is found that some, but not all, of the stocks have long memory, while one of the indices exhibits mean reversion.

A generalized fractionally differencing approach in long-memory modeling

We extend the class of fractional ARIMA models to the class of fractional ARUMA models, which describe long-memory time series with long-range periodical behavior at a finite number of spectrum frequencies. The exact asymptotics of the covariance function and the spectrum at the points of peaks and zeros are given. To obtain asymptotic expansions, Gegenbauer polynomials are used. Consistent parameter estimation is discussed using Whittle's estimate.

Semiparametric estimation from time series with long-range dependence

Abstract This paper studies the behaviour, in the presence of long-memory time-series dependence, of semaparametric averaged derivative statistics, which are useful in statistical inference on index models. They were shown to be asomptotically normal under weak dependence conditions by Robinson (1989) and under serial independence by Powell et al. (1989). We find that an element of long-range dependence can lead either to a nonnormal limiting distribution, or else to a normal one with a limiting variance which differs from that which obtains in case of weak dependence, implying that inferences incorrectly based on weak-dependence assumptions will be invalid.

Semiparametric Analysis of Long-Memory Time Series

We study problems of semiparametric statistical inference connected with long-memory covariance stationary time series, having spectrum which varies regularly at the origin: There is an unknown self-similarity parameter, but elsewhere the spectrum satisfies no parametric or smoothness conditions, it need not be in $L_p$, for any $p > 1$, and in some circumstances the slowly varying factor can be of unknown form. The basic statistic of interest is the discretely averaged periodogram, based on a degenerating band of frequencies around the origin. We establish some consistency properties under mild conditions. These are applied to show consistency of new estimates of the self-similarity parameter and scale factor. We also indicate applications of our results to standard errors of least squares estimates of polynomial regression with long-memory errors, to generalized least squares estimates of this model and to estimates of a cointegrating relationship between long-memory time series.