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

TESTING FOR LONG MEMORY IN VOLATILITY

We consider the asymptotic behavior of log-periodogram regression estimators of the memory parameter in long-memory stochastic volatility models, under the null hypothesis of short memory in volatility. We show that in this situation, if the periodogram is computed from the log squared returns, then the estimator is asymptotically normal, with the same asymptotic mean and variance that would hold if the series were Gaussian. In particular, for the widely used GPH estimator dGPH under the null hypothesis, the asymptotic mean of mýdGPH is zero and the asymptotic variance is piò/24 where m is the number of Fourier frequencies used in the regression. This justifies an ordinary Wald test for long memory in volatility based on the log periodogram of the log squared returns.Statistics Working Papers Serie

Frequency Domain Bootstrap Methods For Time Series

Two frequency domain bootstrap methods for weakly stationary time series will be proposed. The motivations for the proposed methods will be discussed, and the performance of the first method will be compared with that of a recently proposed method of Swanpoel and van Wyk, in a Monte Carol study. It is found that, when applied to the problem of estimating the variance of a log spectrum estimate, all methods under consideration can sometimes perform poorly. Overall, the frequency domain method used in conjunction with automatic spectrum estimate choice criterion developed by Hurvich, is found to perform best

Estimating fractional cointegration in the presence of polynomial trends

We propose and derive the asymptotic distribution of a tapered narrow-band least squares estimator (NBLSE) of the cointegration parameter ÃÂò in the framework of fractional cointegration. This tapered estimator is invariant to deterministic polynomial trends. In particular, we allow for arbitrary linear time trends that often occur in practice. Our simulations show that, in the case of no deterministic trends, the estimator is superior to ordinary least squares (OLS) and the nontapered NBLSE proposed by P.M. Robinson when the levels have a unit root and the cointegrating relationship between the series is weak. In terms of rate of convergence, our estimator converges faster under certain circumstances, and never slower, than either OLS or the nontapered NBLSE. In a data analysis of interest rates, we find stronger evidence of cointegration if the tapered NBLSE is used for the cointegration parameter than if OLS is used.Statistics Working Papers Serie

The Averaged Periodogram Estimator for a Power Law in Coherency

We prove the consistency of the averaged periodogram estimator (APE) in two new cases. First, we prove that the APE is consistent for negative memory parameters, after suitable tapering. Second, we prove that the APE is consistent for a power law in the cross-spectrum and therefore for a power law in the coherency, provided that sufficiently many frequencies are used in estimation. Simulation evidence suggests that the lower bound on the number of frequencies is a necessary condition for consistency. For a Taylor series approximation to the estimator of the power law in the cross-spectrum, we consider the rate of convergence, and obtain a central limit theorem under suitable regularity conditions.J.P. Morgan Chase and Co. and New York UniversityStatistics 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 yields more accurate confidence intervals than the widely-used GPH estimator. In an empirical analysis of the daily Deutschemark/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

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