Asymptotics for Duration-Driven Long Range Dependent Processes

We consider processes with second order long range dependence resulting from heavy tailed durations. We refer to this phenomenon as duration-driven long range dependence (DDLRD), as opposed to the more widely studied linear long range dependence based on fractional differencing of an $iid$ process. We consider in detail two specific processes having DDLRD, originally presented in Taqqu and Levy (1986), and Parke (1999). For these processes, we obtain the limiting distribution of suitably standardized discrete Fourier transforms (DFTs) and sample autocovariances. At low frequencies, the standardized DFTs converge to a stable law, as do the standardized sample autocovariances at fixed lags. Finite collections of standardized sample autocovariances at a fixed set of lags converge to a degenerate distribution. The standardized DFTs at high frequencies converge to a Gaussian law. Our asymptotic results are strikingly similar for the two DDLRD processes studied. We calibrate our asymptotic results with a simulation study which also investigates the properties of the semiparametric log periodogram regression estimator of the memory parameter

Multistep forecasting of long memory series using fractional exponential models

We develop forecasting methodology for the fractional exponential (FEXP) model. First, we devise algorithms for fast exact computation of the coefficients in the infinite order autoregressive and moving average representations of a FEXP process. We also describe an algorithm to accurately approximate the autocovariances and to simulate realizations of the process. Next, we present a fast frequency-domain cross validation method for selecting the order of the model. This model selection method is designed to yield the model which provides the best multistep forecast for the given lead time, without assuming that the process actually obeys a FEXP model. Finally, we use the infinite order autoregressive coefficients of a fitted FEXP model to construct multistep forecasts of inflation in the United Kingdom. These forecasts are substantially different than those from a fitted ARFIMA model.Statistics Working Papers Serie

A Pure-Jump Transaction-Level Price Model Yielding Cointegration, Leverage, and Nonsynchronous Trading Effects

We propose a new transaction-level bivariate log-price model, which yields fractional or standard cointegration. Most existing models for cointegration require the choice of a fixed sampling interval ¢t. By contrast, our proposed model is constructed at the transaction level, thus determining the properties of returns at all sampling frequencies. The two ingredients of our model are a Long Memory Stochastic Duration process for the waiting times f¿kg between trades, and a pair of stationary noise processes (fekg and f´kg) which determine the jump sizes in the pure-jump log-price process. The fekg, assumed to be i:i:d: Gaussian, produce a Martingale component in log prices. We assume that the microstructure noise f´kg obeys a certain model with memory parameter d´ 2 (¡1=2; 0) (fractional cointegration case) or d´ = ¡1 (standard cointegration case). Our log-price model includes feedback between the disturbances of the two log-price series. This feedback yields cointegration, in that there exists a linear combination of the two series that reduces the memory parameter from 1 to 1 + d´ 2 (0:5; 1) [ f0g. Returns at sampling interval ¢t are asymptotically uncorrelated at any fixed lag as ¢t increases. We prove that the cointegrating parameter can be consistently estimated by the ordinary least-squares estimator, and obtain a lower bound on the rate of convergence. We propose transaction-level method-of-moments estimators of several of the other parameters in our model. We present a data analysis, which provides evidence of fractional cointegration. We then consider special cases and generalizations of our model, mostly in simulation studies, to argue that the suitably-modified model is able to capture a variety of additional properties and stylized facts, including leverage, portfolio return autocorrelation due to nonsynchronous trading, Granger causality, and volatility feedback. The ability of the model to capture these effects stems in most cases from the fact that the model treats the (stochastic) intertrade durations in a fully endogenous way.Statistics Working Papers Serie

The Local Whittle Estimator of Long Memory Stochastic Volatility

We propose a new semiparametric estimator of the degree of persistence in volatility for long memory stochastic volatility (LMSV) models. The estimator uses the periodogram of the log squared returns in a local Whittle criterion which explicitly accounts for the noise term in the LMSV model. Finite-sample and asymptotic standard errors for the estimator are provided. An extensive simulation study reveals that the local Whittle estimator is much less biased and that the finite-sample standard errors yield more accurate confidence intervals than the widely-used GPH estimator. The estimator is also found to be robust against possible leverage effects. In an empirical analysis of the daily Deutsche Mark/US Dollar exchange rate, the new estimator indicates stronger persistence in volatility than the GPH estimator, provided that a large number of frequencies is used.Statistics Working Papers Serie

ON THE LOG PERIODOGRAM REGRESSION ESTIMATOR OF THE MEMORY PARAMETER IN LONG MEMORY STOCHASTIC VOLATILITY MODELS

We consider semiparametric estimation of the memory parameter in a long memory stochastic volatility model. We study the estimator based on a log periodogram regression as originally proposed by Geweke and Porter-Hudak (1983, Journal of Time Series Analysis 4, 221âÃÂÃÂ238). Expressions for the asymptotic bias and variance of the estimator are obtained, and the asymptotic distribution is shown to be the same as that obtained in recent literature for a Gaussian long memory series. The theoretical result does not require omission of a block of frequencies near the origin. We show that this ability to use the lowest frequencies is particularly desirable in the context of the long memory stochastic volatility model.Statistics Working Papers Serie

Semiparametric Estimation of Multivariate Fractional

We consider the semiparametric estimation of fractional cointegration in a multivariate process of cointegrating rank r > 0. We estimate the cointegrating relationships by the eigenvectors corresponding to the r smallest eigenvalues of an averaged periodogram matrix of tapered, differenced observations. The number of frequencies m used in the periodogram average is held fixed as the sample size grows. We first show that the averaged periodogram matrix converges in distribution to a singular matrix whose null eigenvectors span the space of cointegrating vectors. We then show that the angle between the estimated cointegrating vectors and the space of true cointegrating vectors is Op(nduôd) where d and du are the memory parameters of the observations and cointegrating errors, respectively. The proposed estimator is invariant to the labeling of the component series, and therefore does not require one of the variables to be specified as a dependent variable. We determine the rate of convergence of the r smallest eigenvalues of the periodogram matrix, and present a criterion which allows for consistent estimation of r. Finally, we apply our methodology to the analysis of fractional cointegration in interest rates.Statistics Working Papers Serie

Computationally Efficient Gaussian Maximum Likelihood Methods for Vector ARFIMA Models

In this paper, we discuss two distinct multivariate time series models that extend the univariate ARFIMA model. We describe algorithms for computing the covariances of each model, for computing the quadratic form and approximating the determinant for maximum likelihood estimation, and for simulating from each model. We compare the speed and accuracy of each algorithm to existing methods and measure the performance of the maximum likelihood estimator compared to existing methods. We also fit models to data on unemployment and inflation in the United States, to data on goods and services inflation in the United States, and to data about precipitation in the Great Lakes.Statistics Working Papers Serie

Estimation of Long Memory in Volatility

We discuss some of the issues pertaining to modelling and estimating long memory in volatility. The main focus is on semi parametric estimation of the memory parameter in the long memory stochastic volatility model. We present the asymptotic properties of the log periodogram regression estimator of the memory parameter in this model. A modest simulation study of the estimator is also presented to study its behaviour when the volatility possesses only short memory. We conclude with a discussion of the appropriate choice of transformation of returns to measure persistence in volatility.Statistics Working Papers Serie

On the Correlation Matrix of the Discrete Fourier Transform and the Fast Solution of Large Toeplitz Systems For Long-Memory Time Series

For long-memory time series, we show that the Toeplitz system §n(f)x = b can be solved in O(n log5=2 n) operations using a well-known version of the preconditioned conjugate gradient method, where §n(f) is the n£n covariance matrix, f is the spectral density and b is a known vector. Solutions of such systems are needed for optimal linear prediction and interpolation. We establish connections between this preconditioning method and the frequency domain analysis of time series. Indeed, the running time of the algorithm is determined by rate of increase of the condition number of the correlation matrix of the discrete Fourier transform vector, as the sample size tends to 1. We derive an upper bound for this condition number. The bound is of interest in its own right, as it sheds some light on the widely-used but heuristic approximation that the standardized DFT coefficients are uncorrelated with equal variances. We present applications of the preconditioning methodology to the forecasting and smoothing of volatility in a long memory stochastic volatility model, and to the evaluation of the Gaussian likelihood function of a long-memory model.Statistics Working Papers Serie