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...

We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.

We develop an ordinary least squares estimator of the long memory parameter from a fractionally integrated process that is an alternative to the Geweke Porter-Hudak estimator. Using the wavelet transform from a fractionally integrated process, we establish a log-linear relationship between the wavelet coe cients ' variance and the scaling parameter equal to the long memory parameter. This log-linear 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 Porter-Hudak 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 Porter-Hudak 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

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 chapter is concerned with methodological issues related to estimation, testing and computation in the context of structural changes in the linear models. A central theme of the review is the interplay between structural change and unit root and on methods to distinguish between the two. The topics covered are: methods related to estimation and inference about break dates for single equations with or without restrictions, with extensions to multi-equations systems where allowance is also made for changes in the variability of the shocks; tests for structural changes including tests for a single or multiple changes and tests valid with unit root or trending regressors, and tests for changes in the trend function of a series that can be integrated or trendstationary; testing for a unit root versus trend-stationarity in the presence of structural changes in the trend function; testing for cointegration in the presence of structural changes; and issues related to long memory and level shifts. Our focus is on the conceptual issues about the frameworks adopted and the assumptions imposed as they relate to potential applicability. We also highlight the potential problems that can occur with methods that are commonly used and recent work that has been done to overcome them.

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This thesis describes describes a small number of problems arising from the applied study of networks in various contexts. The work can be split into two main areas: telecommunications networks (particularly the Internet) and road networks.

. By design a wavelet's strength rests in its ability to localize a process simultaneously in time-scale space. The wavelet's ability to localize a time series in time-scale space directly leads to the computational efficiency of the wavelet representation of a N \Theta N matrix operator by allowing the N largest elements of the wavelet represented operator to represent the matrix operator [Devore, et al. (1992a) and (1992b)]. This property allows many dense matrices to have sparse representation when transformed by wavelets. In this paper we generalize the long-memory parameter estimator of McCoy and Walden (1996) to estimate simultaneously the short and long-memory parameters. Using the sparse wavelet representation of a matrix operator, we are able to approximate an ARFIMA models likelihood function with the series's wavelet coefficients and their variances. Maximization of this approximate likelihood function over the short and long-memory parameter space results in the approximat...

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.

Abstract: We study so–called augmented GARCH sequences, which include many sub-models of considerable interest, such as polynomial and exponential GARCH. To model the returns of speculative assets, it is particularly important to understand the behaviour of the squares of the observations. The main aim of this paper is to present a strong approximation for the sum of the squares. This will be achieved by an approximation of the volatility sequence with a sequence of blockwise independent random variables. Furthermore, we derive a necessary and sufficient condition for the existence of a unique (strictly) stationary solution of the general augmented GARCH equations. Also, necessary and sufficient conditions for the finiteness of moments are provided.