Asymptotics for Duration-Driven Long Range Dependent Processes

TL;DR

This paper studies processes with long-range dependence caused by heavy-tailed durations, deriving their asymptotic distributions for Fourier transforms and autocovariances, and comparing these to classical models.

Contribution

It provides the first detailed asymptotic analysis of duration-driven long-range dependent processes and compares their behavior to traditional fractional models.

Findings

At low frequencies, DFTs converge to a stable law.

Sample autocovariances at fixed lags converge to a degenerate distribution.

High-frequency DFTs converge to a Gaussian law.

Abstract

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.