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112
On the Detection and Estimation of Long Memory in Stochastic Volatility
, 1995
"... Recent studies have suggested that stock markets' volatility has a type of longrange 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 ..."
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Cited by 132 (6 self)
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Recent studies have suggested that stock markets' volatility has a type of longrange 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 longrange dependence are examined and the properties of a LongMemory 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 longmemory volatility are described and applied to an e...
Estimators for LongRange Dependence: An Empirical Study
, 1995
"... Various methods for estimating the selfsimilarity parameter and/or the intensity of longrange dependence in a time series are available. Some are more reliable than others. To discover the ones that work best, we apply the different methods to simulated sequences of fractional Gaussian noise and f ..."
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Cited by 102 (5 self)
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Various methods for estimating the selfsimilarity parameter and/or the intensity of longrange dependence in a time series are available. Some are more reliable than others. To discover the ones that work best, we apply the different methods to simulated sequences of fractional Gaussian noise and fractional ARIMA(0, d, 0). We also provide here a theoretical justification for the method of residuals of regression.
Central Limit Theorems for Quadratic Forms with Time Domain Conditions
, 1996
"... We establish the central limit theorem for quadratic forms P N t;s=1 b(t\Gammas)P m;n (X t ; X s ) of the bivariate Appell polynomials P m;n (X t ; X s ) under time domain conditions. These conditions relate the weights b(t) and the covariances of the sequences (P m;n (X t ; X s )) and (X t ). The ..."
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Cited by 63 (5 self)
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We establish the central limit theorem for quadratic forms P N t;s=1 b(t\Gammas)P m;n (X t ; X s ) of the bivariate Appell polynomials P m;n (X t ; X s ) under time domain conditions. These conditions relate the weights b(t) and the covariances of the sequences (P m;n (X t ; X s )) and (X t ). The time domain approach, together with the spectral domain approach developed earlier, yield a general set of conditions for central limit theorems. 1 Introduction We study the central limit theorem (CLT) for quadratic forms QN := N X t;s=1 b(t \Gamma s)P m;n (X t ; X s ) (1.1) where P m;n (X t ; X s ) =: X t ; : : : ; X t  z m ; X s ; : : : ; X s  z n : is a bivariate Appell polynomial (Wick power) of the linear variables X t and X s , m;n 0; m+ n 1. Here X t = X s2Z a(t \Gamma u)¸ u ; t 2 Z (1.2) is a linear process, that is, the random variables ¸ t ; t 2 Z are independent and identically distributed, E¸ 0 = 0 E¸ 2 0 = 1, the sequence a(t); t 2 Z of realvalu...
On Estimating the Intensity of LongRange Dependence in Finite and Infinite Variance Time Series
, 1996
"... The goal of this paper is to provide benchmarks to the practitioner for measuring the intensity of longrange dependence in time series. It provides a detailed comparison of eight estimators for longrange dependence, using simulated FARIMA(p; d; q) time series with different finite and infinite var ..."
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Cited by 42 (3 self)
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The goal of this paper is to provide benchmarks to the practitioner for measuring the intensity of longrange dependence in time series. It provides a detailed comparison of eight estimators for longrange dependence, using simulated FARIMA(p; d; q) time series with different finite and infinite variance innovations. FARIMA time series model both longrange dependence (through the parameter d) and shortrange dependence (through the parameters p and q). We evaluate the biases and standard deviations of several estimators of d and compare them for each type of series used. We consider Gaussian, exponential, lognormal, Pareto, symmetric and skewed stable innovations. Detailed tables and graphs have been included. We find that the estimators tend to perform less well when p and q are not zero, that is, when there is additional shortrange dependence structure. For most of the estimators, however, the use of infinite variance instead of finite variance innovations does not cause a great dec...
NarrowBand Analysis Of Nonstationary Processes
, 1999
"... The behaviour of averaged periodograms and crossperiodograms of a broad class of nonstationary processes is studied. The processes include nonstationary ones that are fractional of any order, as well as asymptotically stationary fractional ones, and the crossperiodogram can involve two nonstationa ..."
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Cited by 37 (13 self)
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The behaviour of averaged periodograms and crossperiodograms of a broad class of nonstationary processes is studied. The processes include nonstationary ones that are fractional of any order, as well as asymptotically stationary fractional ones, and the crossperiodogram can involve two nonstationary processes of possibly di#erent orders, or a nonstationary and an asymptotically stationary one. The averaging takes place either over the whole frequency band, or on one that degenerates slowly to zero frequency as sample size increases. In some cases it is found to make no asymptotic di#erence, and in particular we indicate how the behaviour of the mean and variance changes across the twodimensional space of integration orders. The results employ only localtozero assumptions on the spectra of the underlying weakly stationary sequences. It is shown how the results can be readily applied in case of fractional cointegration with unknown integration orders. 1 1. INTRODUCTION In the analy...
Using Wavelets to Obtain a Consistent Ordinary Least Squares Estimator of the Longmemory Parameter
 Journal of Forecasting
, 1999
"... We develop an ordinary least squares estimator of the long memory parameter from a fractionally integrated process that is an alternative to the Geweke PorterHudak estimator. Using the wavelet transform from a fractionally integrated process, we establish a loglinear relationship between the wavel ..."
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Cited by 31 (6 self)
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We develop an ordinary least squares estimator of the long memory parameter from a fractionally integrated process that is an alternative to the Geweke PorterHudak estimator. Using the wavelet transform from a fractionally integrated process, we establish a loglinear relationship between the wavelet coe cients ' variance and the scaling parameter equal to the long memory parameter. This loglinear relationship yields a consistent ordinary least squares estimator of the long memory parameter when the wavelet coe cients ' population variance is replaced by their sample variance. We derive the small sample bias and variance of the ordinary least squares estimator and test it against the Geweke PorterHudak estimator and the McCoy Walden maximum likelihood wavelet estimator by conducting a numberofMonte Carlo experiments. Based upon the criterion of choosing the estimator which minimizes the mean squared error, the wavelet OLS approach was superior to the Geweke PorterHudak estimator, but inferior to the McCoy Walden wavelet estimator for the processes simulated. However, given the simplicity of programming and running the wavelet OLS estimator and its statistical inference of the long memory parameter we feel the general practitioner will be attracted to wavelet OLS estimator. Keywords
Microeconomic Models for LongMemory in the Volatility of Financial Time Series
"... We show that a class of microeconomic behavioral models with interacting agents, derived from Kirman (1991, 1993), can replicate the empirical longmemory properties of the two first conditional moments of financial time series. The essence of these models is that the forecasts and thus the desired ..."
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Cited by 31 (2 self)
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We show that a class of microeconomic behavioral models with interacting agents, derived from Kirman (1991, 1993), can replicate the empirical longmemory 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 longmemory properties in the volatility and covolatility 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 longmemory in a nonstandard data generating process is then assessed.
Whittle pseudomaximum likelihood estimation for nonstationary time series
 J. Am. Statist. Assoc
, 2000
"... Exact and approximate maximum likelihood estimation for the parameters of stationary time series has been justi�ed under various sets of conditions, including spectral densities with a peak at the origin due to a persistent behaviour �e.g. Fox and Taqqu �1986��. Thus it is often assumed that the spe ..."
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Cited by 29 (9 self)
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Exact and approximate maximum likelihood estimation for the parameters of stationary time series has been justi�ed under various sets of conditions, including spectral densities with a peak at the origin due to a persistent behaviour �e.g. Fox and Taqqu �1986��. Thus it is often assumed that the spectral density f�� � of an observed covariance stationary sequence satis�es, for 0 �G�1, �1 � f�� � � Gj�j,2d as � ! 0; where d 2 �, 1 1; � is the parameter that governs the degree of memory of the series. This is 2 2 the interval of values of d for which the process is stationary and invertible. If d 2 �0; 1 � then 2 we say that the series exhibits long memory or long range dependence. When the observations are nonstationary, they are usually di�erenced an integer number of times to achieve stationarity. If an observed processes fXtg has covariance stationary increments �X t, with spectral density satisfying f �X�� � � G j�j,2�d,1 � as � ! 0, d � 1
LongRange Dependence and Data Network Traffic
, 2001
"... This is an overview of a relatively recent application of longrange dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in highspeed data networks such as the Internet. We demonstrate that this new application area off ..."
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Cited by 24 (1 self)
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This is an overview of a relatively recent application of longrange dependence (LRD) to the area of communication networks, in particular to problems concerned with the dynamic nature of packet flows in highspeed data networks such as the Internet. We demonstrate that this new application area offers unique opportunities for significantly advancing our understanding of LRD and related phenomena. These advances are made possible by moving beyond the conventional approaches associated with the widespread "blackbox" perspective of traditional time series analysis and exploiting instead the physical mechanisms that exist in the networking context and that are intimately tied to the observed characteristics of measured network traffic. In order to describe this complexity we provide a basic understanding of the design, architecture and operations of data networks, including a description of the TCP/IP protocols used in today's Internet. LRD is observed in the large scale behavior of the data traffic and we provide a physical explanation for its presence. LRD tends to be caused by user and application characteristics and has little to do with the network itself. The network affects mostly small time scales, and this is why a rudimentary understanding of the main protocols is important. We illustrate why multifractals may be relevant for describing some aspects of the highly irregular traffic behavior over small time scales. We distinguish between a timedomain and waveletdomain approach to analyzing the small time scale dynamics and discuss why the waveletdomain approach appears to be better suited than the timedomain approach for identifying features in measured traffic (e.g., relatively regular traffic patterns over certain time scales) that have a direct networking interpretation (e....