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An approximate wavelet MLE of short and long memory parameters
 Studies in Nonlinear Dynamics and Econometrics
, 1999
"... Abstract. By design a wavelet's strength rests in its ability to localize a process simultaneously in timescale space. The wavelet's ability to localize a time series in timescale space directly leads to the computational e ciency of the wavelet representation of a N N matrix operator by allowing ..."
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Cited by 11 (3 self)
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Abstract. By design a wavelet's strength rests in its ability to localize a process simultaneously in timescale space. The wavelet's ability to localize a time series in timescale space directly leads to the computational e ciency of the wavelet representation of a N 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 longmemory parameter estimator of McCoy and Walden (1996) to estimate simultaneously the short and longmemory 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 coe cients and their variances. Maximization of this approximate likelihood function over the short and longmemory parameter space results in the approximate wavelet maximum likelihood estimates of the ARFIMA model. By simultaneously maximizing the likelihood function over both the short and longmemory parameters and using only the wavelet coe cient's variances, the approximate wavelet MLE provides a fast alternative to the frequencydomain MLE. Furthermore, the simulation studies found herein reveal the approximate wavelet MLE to be robust over the invertible parameter region of the ARFIMA model's moving average parameter, whereas the frequencydomain MLE dramatically deteriorates as the moving average parameter approaches the boundaries of invertibility.
Estimating the Fractional Order of Integration of Interest Rates Using a Wavelet OLS Estimator
 PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NEURAL INFORMATION PROCESSING, HONG KONG
, 2001
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Wavelet Estimation of a Local Long Memory Parameter
, 2000
"... There are a number of estimators of a longmemory process' longmemory parameter when the parameter is assumed to hold constant over the entire data set, but currently no estimator exists for a timevarying longmemory parameter. In this paper we construct an estimator of the timevarying longmemor ..."
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Cited by 7 (1 self)
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There are a number of estimators of a longmemory process' longmemory parameter when the parameter is assumed to hold constant over the entire data set, but currently no estimator exists for a timevarying longmemory parameter. In this paper we construct an estimator of the timevarying longmemory parameter that is based on the timescale properties of the wavelet transform. Because wavelets are localised in time they are able to capture the timevarying statistical properties of a locally stationary longmemory process, and since wavelets are also localised in scale they identify the selfsimilarity scaling behaviour found in the statistical properties of the process. Together the time and scale properties of the wavelet produce an approximate loglinear relationship between the timevarying variance of the wavelet coefficients and the wavelet scale proportional to the local longmemory parameter. To obtain a leastsquares estimate of the local longmemory parameter, we replace the ...
Bayesian Inference of LongMemory Stochastic Volatility via Wavelets
"... In this paper we are concerned with estimating the fractional order of integration associated with a longmemory stochastic volatilitymodel. Wedevelop a new Bayesian estimator based on the Markov chain Monte Carlo sampler and the wavelet representation of the logsquared returns to drawvalues of the ..."
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Cited by 6 (2 self)
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In this paper we are concerned with estimating the fractional order of integration associated with a longmemory stochastic volatilitymodel. Wedevelop a new Bayesian estimator based on the Markov chain Monte Carlo sampler and the wavelet representation of the logsquared returns to drawvalues of the fractional order of integration and latent volatilities from their joint posterior distribution. Unlike short memory stochastic volatility models, longmemory stochastic volatility models do not have a statespace representation, and thus their sampler cannot employ the Kalman filters simulation smoother to update the chain of latentvolatilities. Instead, we design a simulator where the latent longmemory volatilities are drawn quickly and efficiently from the near independen tmultivariate distribution of the longmemory volatility's wavelet coefficients. We find that sampling volatility in the wavelet domain, rather than in the time domain, leads to a fast and simulationefficient sampler of the posterior distribution for the volatility's longmemory parameter and serves as a promising alternative estimator to the existing frequentist based estimators of longmemory volatility.
WaveletBased Estimation for Seasonal LongMemory Processes
, 2001
"... We introduce the multiscale analysis of seasonal persistent processes; i.e., time series models with a singularity in their spectral density function at one or more frequencies in [0, 1/2]. The discrete wavelet packet transform (DWPT) and a nondecimated version of it known as the maximal overlap ..."
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Cited by 4 (1 self)
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We introduce the multiscale analysis of seasonal persistent processes; i.e., time series models with a singularity in their spectral density function at one or more frequencies in [0, 1/2]. The discrete wavelet packet transform (DWPT) and a nondecimated version of it known as the maximal overlap DWPT (MODWPT) are introduced as alternative methods to spectral techniques for analyzing time series that exhibit seasonal longmemory. The approximate loglinear relationship between the wavelet packet variance and frequency is utilized to produce a leastsquares estimator of the fractional di#erence parameter. Approximate maximum likelihood estimation is performed by replacing the variance/covariance matrix with a diagonalized matrix based on the DWPT. Simulations are performed to compare the waveletbased techniques with those based on the spectral estimates, for both leastsquares and maximum likelihood procedures. We find that leastsquares estimation is quite poor when assuming a loglinear relationship between the spectral density function (wavelet packet variance) and frequency (scale) around the Gegenbauer frequency for spectral (waveletbased) procedures. Maximum likelihood is more robust when using either the true spectral density function or the loglinear approximation. An application of this methodology to atmospheric CO 2 measurements is also presented.
TimeVarying LongMemory in Volatility: Detection and Estimation with Wavelets
, 2000
"... Previous analysis of high frequency financial time series data has found volatility to follow a longmemory process and to display an intradaily Ushape pattern. These findings implicitly assume that a stable environment exists in the financial world. To better capture the nonstationary behavior ..."
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Cited by 3 (0 self)
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Previous analysis of high frequency financial time series data has found volatility to follow a longmemory process and to display an intradaily Ushape pattern. These findings implicitly assume that a stable environment exists in the financial world. To better capture the nonstationary behavior associated with market collapses, political upheavals and news annoucements, we propose a nonstationary class of stochastic volatility models that features timevarying parameters. The generality of our nonstationary stochastic volatility model better accommodates several empirical features of volatility and nests stationary stochastic volatility models within it. To estimate the timevarying longmemory parameter, we use the log linear relationship between the local variance of the maximum overlap discrete wavelet transform's coefficients and their scaling parameter to produce a semiparameteric, OLS estimator. Because wavelets are a set of well localized basis functions in time and s...
The Empirical Properties of Some Popular Estimators of Long Memory Processes
, 2008
"... We present the results of a simulation study into the properties of 12 different estimators of the Hurst parameter, H, or the fractional integration parameter, d, in long memory time series. We compare and contrast their performance on simulated Fractional Gaussian Noises and fractionally integrated ..."
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Cited by 1 (0 self)
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We present the results of a simulation study into the properties of 12 different estimators of the Hurst parameter, H, or the fractional integration parameter, d, in long memory time series. We compare and contrast their performance on simulated Fractional Gaussian Noises and fractionally integrated series with lengths between 100 and 10,000 data points and H values between 0.55 and 0.90 or d values between 0.05 and 0.40. We apply all 12 estimators to the Campito Mountain data and estimate the accuracy of their estimates using the Beran goodness of fit test for long memory time series.
WaveletBased Estimation Procedures for Seasonal LongMemory Models
, 2000
"... We introduce the multiscale analysis of seasonal persistent processes; i.e., time series models with a singularity in their spectral density function at a or multiple frequencies in [0; 1=2]. The discrete wavelet packet transform (DWPT) and an nondecimated version of it known as the maximal over ..."
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Cited by 1 (0 self)
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We introduce the multiscale analysis of seasonal persistent processes; i.e., time series models with a singularity in their spectral density function at a or multiple frequencies in [0; 1=2]. The discrete wavelet packet transform (DWPT) and an nondecimated version of it known as the maximal overlap DWPT (MODWPT) are introduced as an alternative method to spectral techniques for analyzing time series that exhibit seasonal longmemory. Approximate maximum likelihood estimation is performed by replacing the variance/covariance matrix with diagonalized matrix based on the ability of the DWPT to approximately decorrelate a seasonal persistent process. Simulations are performed using this waveletbased maximum likelihood technique on a variety of time series models. An application of this methodology to atmospheric CO 2 measurements is also presented. Keywords. Discrete wavelet packet transform, Gegenbauer process, long memory, multitaper spectral estimation, periodogram, wavelet...
Preface Acknowledgments
, 2002
"... at the division of Design, Manufacture, and Industrial Innovation. This supplement and the S+Wavelets module are products of the efforts made by the Insightful Corporation. Many of the mathematical and statistical ideas behind this software are described more thoroughly in the book coauthored by on ..."
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at the division of Design, Manufacture, and Industrial Innovation. This supplement and the S+Wavelets module are products of the efforts made by the Insightful Corporation. Many of the mathematical and statistical ideas behind this software are described more thoroughly in the book coauthored by one of us (Percival) and Andrew Walden entitled, Wavelet Methods for Time Series Analysis,