Recent studies have suggested that stock markets' volatility has a type of long-range dependence that is not appropriately described by the usual Generalized Autoregressive Conditional Heteroskedastic (GARCH) and Exponential GARCH (EGARCH) models. In this paper, different models for describing this long-range dependence are examined and the properties of a Long-Memory Stochastic Volatility (LMSV) model, constructed by incorporating an Autoregressive Fractionally Integrated Moving Average (ARFIMA) process in a stochastic volatility scheme, are discussed. Strongly consistent estimators for the parameters of this LMSV model are obtained by maximizing the spectral likelihood. The distribution of the estimators is analyzed by means of a Monte Carlo study. The LMSV is applied to daily stock market returns providing an improved description of the volatility behavior. In order to assess the empirical relevance of this approach, tests for long-memory volatility are described and applied to an e...

This paper discusses the use of fractional exponential models (Robinson (1990), Beran (1994)) to model the spectral density f(x) of a covariance stationary process when f(x) may be decomposed as f(x) = x \Gamma2d f (x), where f (x) is bounded and bounded away from zero. A form of log-periodogram regression technique is presented both in the parametric context (i.e. f (x) is a finite order exponential model in the sense of Bloomfield (1973)) and the semi-parametric context (f (x) is regarded as a nuisance parameter). Assuming gaussianity and additional conditions on the regularity of f (x) which seem mild, asymptotic normality of the parameter estimates in the parametric and the semi-parametric context is established. As a by-product, some improvements over the results presented by Robinson (1994) have been obtained for the large sample distribution of log-periodogram ordinates for Gaussian processes.

We show that a class of microeconomic behavioral models with interacting agents, derived from Kirman (1991, 1993), can replicate the empirical long-memory properties of the two first conditional moments of financial time series. The essence of these models is that the forecasts and thus the desired trades of the individuals in the markets are influenced, directly, or indirectly by those of the other participants. These "field effects" generate "herding" behaviour which affects the structure of the asset price dynamics. The series of returns generated by these models display the same empirical properties as financial returns: returns are I(0), the series of absolute and squared returns display strong dependence, while the series of absolute returns do not display a trend. Furthermore, this class of models is able to replicate the common long-memory properties in the volatility and co-volatility of financial time series, revealed by Teyssière (1997, 1998a). These properties are investigated by using various model independent tests and estimators, i.e., semiparametric and nonparametric, introduced by Lo (1991), Kwiatkowski, Phillips, Schmidt and Shin (1992), Robinson (1995), Lobato and Robinson (1998), Giraitis, Kokoszka Leipus and Teyssière (2000, 2001). The relative performance of these tests and estimators for long-memory in a non-standard data generating process is then assessed.

This paper reviews several periodogram-based methods for estimating the long-memory parameter H in time series and suggests a way to robustify them. The high frequencies tend to bias the estimates. Using only low frequencies eliminates the bias but increases the variance. We hence suggest plotting the estimates of H as a function of a parameter which balances bias versus variance and, if the plot flattens in a central region, to use the flat part for estimating H. We apply this technique to the periodogram regression method, the Whittle approximation to maximum likelihood and to the local Whittle method. We investigate its effectiveness on several simulated fractional ARIMA series and also apply it to estimate the long-memory parameter H in computer network traffic. 1 Introduction Time series with long memory have been considered in many fields including hydrology, biology and computer networks. Unfortunately, estimating the long memory (long-range dependence) parameter H in a given d...

Consider the fractional ARIMA time series with innovations that have infinite variance. This is a finite parameter model which exhibits both long-range dependence (long memory) and high variability. We prove the consistency of an estimator of the unknown parameters which is based on the periodogram and derive its asymptotic distribution. This shows that the results of Mikosch, Gadrich, Kluppelberg and Adler (1995) for ARMA time series remain valid for fractional ARIMA with long-range dependence. We also extend the limit theorem for sample autocovariances of infinite variance moving averages developed in Davis and Resnick (1985) to moving averages whose coefficients are not absolutely summable. 1 Introduction and main results This paper is concerned with the estimation of the parameters of the fractional ARIMA time series fX n g defined by the equations \Phi(B)X n = \Theta(B )\Delta \Gammad Z n ; (1.1) where the innovations Z n have infinite variance and where d is a positive fracti...

The long range dependence paradigm appears to be a suitable description of the data generating process for many observed economic time series. This is mainly due to the fact that it naturally characterizes time series displaying a high degree of persistence, in the form of a long lasting effect of unanticipated shocks, yet exhibiting mean reversion. Whereas linear long range dependent time series models have been extensively used in macroeconomics, empirical evidence from financial time series prompted the development of nonlinear long range dependent time series models, in particular models of changing volatility.We discuss empirical evidence of long range dependence as well as the theoretical issues, both for economics and econometrics, such evidence has stimulated.

This paper proposes a model specification testing procedure for parametric specification of the conditional mean function in a nonlinear time series model with long–range dependence. An asymptotically normal test is established even when long–range dependence is involved. In order to implement the proposed test in practice using a simulated example, a bootstrap simulation procedure is established to find a simulated critical value to compute both the size and power values of the proposed test.

this paper we study the scale analysis of bivariate time series through use of the wavelet cross-covariance. The estimation of this quantity is carried out using the maximaloverlap discrete wavelet transform (MODWT). The MODWT (Greenhall, 1991; Percival and Guttorp, 1994) is also called the undecimated or translation invariant or shift invariant discrete wavelet transform (Shensa, 1992; Nason and Silverman, 1995; Coifman and Donoho, 1995). The MODWT carries out the same filtering steps as the ordinary discrete wavelet transform, but does not subsample. We need immediately to see what is meant by a scale analysis. Let

We analyze by simulation the properties of two time domain and two frequency domain estimators for low order autoregressive fractionally integrated moving average Gaussian models, ARFIMA (p; d; q). The estimators considered are the exact maximum likelihood for demeaned data, EML, the associated modified profile likelihood, MPL, and the Whittle estimator with, WLT, and without tapered data, WL. Length of the series is 100. The estimators are compared in terms of pile-up effect, mean square error, bias, and empirical confidence level. The tapered version of the Whittle likelihood turns out to be a reliable estimator for ARMA and ARFIMA models. Its small losses in performance in case of "well-behaved" models are compensated sufficiently in more "difficult" models. The modified profile likelihood is an alternative to the WLT but is computationally more demanding. It is either equivalent to the EML or more favorable than the EML. For fractionally integrated models, particularly, it dominate...

Moving from univariate to bivariate jointly dependent long memory time series introduces a phase parameter (), at the frequency of principal interest, zero; for short memory series = 0 automatically. The latter case has also been stressed under long memory, along with the "fractional di¤erencing " case = ( 2 1) =2; where 1; 2 are the memory parameters of the two series. We develop time domain conditions under which these are and are not relevant, and relate the consequent properties of cross-autocovariances to ones of the (possibly bilateral) moving average representation which, with martingale di¤erence innovations of arbitrary dimension, is used in asymptotic theory for local Whittle parameter estimates depending on a single smoothing number. Incorporating also a regression parameter ( ) which, when non-zero, indicates cointegration, the consistency proof of these implicitly-de…ned estimates is nonstandard due to the estimate converging faster than the others. We also establish joint asymptotic normality of the estimates, and indicate how this outcome can apply in statistical inference on several questions of interest. Issues of implemention are discussed, along with implications of knowing and of correct or incorrect speci…cation of; and possible extensions to higher-dimensional systems and nonstationary series.