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80
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 214 (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...
Estimation in Conditionally Heteroscedastic Time Series Models
 Lecture Notes in Statist. 181
, 2005
"... This paper studies the quasimaximumlikelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt = σt Zt, where the unobservable volatility σt is a parametric function of (Xt−1,...,Xt−p,σt−1,...,σt−q) for some p,q ≥ 0, and (Zt) is standardized i ..."
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Cited by 83 (2 self)
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This paper studies the quasimaximumlikelihood estimator (QMLE) in a general conditionally heteroscedastic time series model of multiplicative form Xt = σt Zt, where the unobservable volatility σt is a parametric function of (Xt−1,...,Xt−p,σt−1,...,σt−q) for some p,q ≥ 0, and (Zt) is standardized i.i.d. noise. We assume that these models are solutions to stochastic recurrence equations which satisfy a contraction (random Lipschitz coefficient) property. These assumptions are satisfied for the popular GARCH, asymmetric GARCH and exponential GARCH processes. Exploiting the contraction property, we give conditions for the existence and uniqueness of a strictly stationary solution (Xt) to the stochastic recurrence equation and establish consistency and asymptotic normality of the QMLE. We also discuss the problem of invertibility of such time series models.
NeymanPearson detection of GaussMarkov signals in noise: Closedform error exponent and properties,” preprint
, 2004
"... Abstract — The performance of NeymanPearson detection of correlated stochastic signals using noisy observations is investigated via the error exponent for the miss probability with a fixed level. Using the statespace structure of the signal and observation model, a closedform expression for the er ..."
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Cited by 34 (16 self)
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Abstract — The performance of NeymanPearson detection of correlated stochastic signals using noisy observations is investigated via the error exponent for the miss probability with a fixed level. Using the statespace structure of the signal and observation model, a closedform expression for the error exponent is derived, and the connection between the asymptotic behavior of the optimal detector and that of the Kalman filter is established. The properties of the error exponent are investigated for the scalar case. It is shown that the error exponent has distinct characteristics with respect to correlation strength: for signaltonoise ratio (SNR)> 1 the error exponent decreases monotonically as the correlation becomes stronger, whereas for SNR < 1 there is an optimal correlation that maximizes the error exponent for a given SNR. I.
Cointegration in fractional systems with unknown integration orders. Econometrica 1727–1766
, 2003
"... Cointegrated bivariate nonstationary time series are considered in a fractional context, without allowance for deterministic trends. Both the observable series and the cointegrating error can be fractional processes. The familiar situation in which the respective integration orders are 1 and 0 is ne ..."
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Cited by 33 (7 self)
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Cointegrated bivariate nonstationary time series are considered in a fractional context, without allowance for deterministic trends. Both the observable series and the cointegrating error can be fractional processes. The familiar situation in which the respective integration orders are 1 and 0 is nested, but these values have typically been assumed known. We allow one or more of them to be unknown real values, in which case Robinson and Marinucci (1997,2001) have justified least squares estimates of the cointegrating vector, as well as narrowband frequencydomain estimates, which may be less biased. While consistent, these estimates do not always have optimal convergence rates, and they have nonstandard limit distributional behaviour. We consider estimates formulated in the frequency domain, that consequently allow for a wide variety of (parametric) autocorrelation in the short memory input series, as well as timedomain estimates based on autoregressive transformation. Both can be interpreted as approximating generalized least squares and Gaussian maximum likelihood estimates. The estimates share the same limiting distribution, having mixed normal asymptotics (yielding Wald test statistics with χ 2 null limit distributions), irrespective of whether the integration orders are known or unknown, subject in the latter case to their estimation with adequate rates of convergence. The parameters describing the short memory stationary input series are √ nconsistently estimable, but the assumptions imposed on these series are much more general than ones of autoregressive moving average type. A Monte Carlo study of finitesample performance is included. JEL Classification: C22
Parameter Estimation for Infinite Variance Fractional ARIMA
 ARIMA, Annals of Statistics
, 1996
"... Consider the fractional ARIMA time series with innovations that have infinite variance. This is a finite parameter model which exhibits both longrange dependence (long memory) and high variability. We prove the consistency of an estimator of the unknown parameters which is based on the periodogram ..."
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Cited by 29 (5 self)
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Consider the fractional ARIMA time series with innovations that have infinite variance. This is a finite parameter model which exhibits both longrange 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 longrange 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...
Estimating semiparametric ARCH(∞) models by kernel smoothing methods
 Econometrica
, 2005
"... Contents: ..."
Measles metapopulation dynamics: A gravity model for epidemiological coupling and dynamics. The American Naturalist
, 2004
"... Abstract: Local oscillatory dynamics, fadeout rates and regional phase differences are the most important spatialtemporal manifestations of acute epidemic dynamics. However, few systems are well enough understood and documented to detect all of these properties and to explain their interaction wit ..."
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Cited by 28 (4 self)
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Abstract: Local oscillatory dynamics, fadeout rates and regional phase differences are the most important spatialtemporal manifestations of acute epidemic dynamics. However, few systems are well enough understood and documented to detect all of these properties and to explain their interaction with spatiotemporal variations in population structure and demography. Based on a gravity coupling model and a time series susceptibleinfectedrecovered (TSIR) model for local dynamics, we propose a new model for regional measles epidemic dynamics. The model can capture all the major spatiotemporal properties in prevaccination epidemics of measles in England and Wales.
Semiparametric Bayesian inference for time series with mixed spectra
 J. Royal Statist. Soc. Ser. B
, 1996
"... This paper provides a Bayesian analysis of such a model. The main contribution of our paper is that different features of the datasuch as the spectral density of the stationary term, the regression parameters, unknown frequencies and missing observationsare combined in a hierarchical Bayesian fr ..."
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Cited by 26 (1 self)
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This paper provides a Bayesian analysis of such a model. The main contribution of our paper is that different features of the datasuch as the spectral density of the stationary term, the regression parameters, unknown frequencies and missing observationsare combined in a hierarchical Bayesian framework and estimated simultaneously. A Bayesian test to detect the presence of deterministic components in the data is also constructed. Applications of our methods to simulated and real data suggest that they perform well. We place a smoothness prior, similar to that in Wahba (1980), on the logarithm of the spectral density. To make the estimation of the spectral density computationally tractable, Whittle's (1957) approximation to the Gaussian likelihood is used. This results in a nonparametric regression problem with the logarithm of the periodogram as the dependent variable, the logarithm of the spectral density as the unknown regression curve, and observation errors having log chisquared distributions. By approximating the logarithm of a chisquared distribution as a mixture of normals, the approximate log likelihood together with the prior for the spectral density can be expressed as a state space model with errors that are mixtures of normals. The computation is carried out efficiently by Markov chain Monte Carlo using the sampling approach in Carter and Kohn (1994). To make the paper easier to read the full model is introduced in a number of steps. Section 2 shows how to estimate the spectral density of a stationary process in the absence of deterministic components. Section 3 extends the estimation to the signal plus noise model with missing observations. Section 4 shows by example how the results in Sections 2 and 3 can be combined to analyze data and studies emp...
SemiParametric Graphical Estimation Techniques for LongMemory Data.
, 1996
"... This paper reviews several periodogrambased methods for estimating the longmemory 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 t ..."
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Cited by 19 (4 self)
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This paper reviews several periodogrambased methods for estimating the longmemory 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 longmemory 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 (longrange dependence) parameter H in a given d...
Edgeworth expansions for semiparametric Whittle estimation of long memory
 Ann. Statist
, 2003
"... The semiparametric local Whittle or Gaussian estimate of the long memory parameter is known to have especially nice limiting distributional properties, being asymptotically normal with a limiting variance that is completely known. However in moderate samples the normal approximation may notbevery go ..."
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Cited by 16 (3 self)
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The semiparametric local Whittle or Gaussian estimate of the long memory parameter is known to have especially nice limiting distributional properties, being asymptotically normal with a limiting variance that is completely known. However in moderate samples the normal approximation may notbevery good, so we consider a re ned, Edgeworth, approximation, for both a tapered estimate, and the original untapered one. For the tapered estimate, our higherorder correction involves two terms, one of order m;1=2 (where m is the bandwidth number in the estimation), the other a bias term, which increases in m � depending on the relative magnitude of the terms, one or the other may dominate, or they may balance. For the untapered estimate we obtain an expansion in which, for m increasing fast enough, the correction consists only of abiasterm. We discuss applications of our expansions to improved statistical inference and bandwidth choice. We assume Gaussianity, but in other respects our assumptions seem mild. AMS Subject classi cation. Primary 62G20 � secondary 62M10