Gaussian semiparametric estimation in long memory in stochastic volatility and signal plus noise models

Abstract

This paper considers the persistence found in the volatility of many financial time series by means of a local Long Memory in Stochastic Volatility model and analyzes the performance of the Gaussian semiparametric or local Whittle estimator of the memory parameter in a long memory signal plus noise model which includes the Long Memory in Stochastic Volatility as a particular case. It is proved that this estimate preserves the consistency and asymptotic normality encountered in observable long memory series and under milder conditions it is more efficient than the estimator based on a log-periodogram regression. Although the asymptotic properties do not depend on the signal-to-noise ratio the finite sample performance rely upon this magnitude and an appropriate choice of the bandwidth is important to minimize the influence of the added noise. I analyze the effect of the bandwidth via Monte Carlo. An application to a Spanish stock index is finally included.JEL classification: C13; C22.

Local polynomial Whittle estimation of perturbed fractional processes

“…0 and m 4d 0 +1 n 4d 0 ! 0, respectively, see Hurvich & Ray (2003) and Arteche (2004). Therefore, Theorem 2 provides an improvement in the rate of convergence relative to existing estimators of the memory parameter for perturbed fractional processes.…”

“…This is in contrast to the orders O((m=n) 2 ) and O((m=n) 2d 0 ) (assuming su¢ cient smoothness) for the LWN and LW estimators, respectively, see Hurvich & Ray (2003) and Arteche (2004). Thus, as in Andrews & Sun (2004) for the pure long memory case, the order of magnitude of the asymptotic bias is smaller when modeling the (smooth) spectral density of the short-memory component locally by a polynomial instead of a constant.…”

“…Indeed, for non-perturbed processes (where w ( ) = 0) the bias of the standard semiparametric frequency domain estimators is of order O( 2 m ), whereas the leading bias term when w ( ) 6 = 0 is of order O( 2d m ) (in both cases assuming su¢ cient smoothness of the spectral density). As shown in Deo & Hurvich (2001) and Arteche (2004), this bias is typically negative and can be very large (note that d < 1). Therefore, estimating long memory in perturbed time series can be challenging, and calls for an estimator which explicitly accounts for the perturbation.…”

“…This assumption is common in the random walk plus noise unobserved components models, and has also been imposed by Breidt et al (1998), Deo & Hurvich (2001), and Arteche (2004), among others, in the LMSV model. To accommodate the leverage e¤ect, we could allow contemporaneous correlation, while the return process remains a martingale di¤erence sequence by replacing y t with y t 1 in (3).…”

“…0. Although the popular log-periodogram regression estimator of Geweke & Porter-Hudak (1983) and Robinson (1995b) and the local Whittle (LW) estimator of Künsch (1987) and Robinson (1995a) both preserve consistency and asymptotic normality when applied to perturbed fractional processes, as shown recently by Deo & Hurvich (2001) and Arteche (2004), these estimators can be severely biased since they do not take the perturbation into account. Indeed, for non-perturbed processes (where w ( ) = 0) the bias of the standard semiparametric frequency domain estimators is of order O( 2 m ), whereas the leading bias term when w ( ) 6 = 0 is of order O( 2d m ) (in both cases assuming su¢ cient smoothness of the spectral density).…”