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.

The effect of round-off error on long memory processes

“…Despite the fact that, as we have demonstrated in the previous sections, the Hurst exponent of the discretized process is the same as the one of the original process, we will show here that both methods give estimates of the Hurst exponent which are systematically and significantly negatively biased. These results are consistent with the literature on long-memory signal plus noise processes (Arteche (2004), and Hurvich, Moulines and Soulier (2005)) and on general non-linear transformations of long-memory processes (Dalla, Giraitis and Hidalgo (2006)).…”

The effect of round-off error on long memory processes

“…Despite the fact that, as we have demonstrated in the previous sections, the Hurst exponent of the discretized process is the same as the one of the original process, we will show here that both methods give estimates of the Hurst exponent which are systematically and significantly negatively biased. These results are consistent with the literature on long-memory signal plus noise processes (Arteche (2004), and Hurvich, Moulines and Soulier (2005)) and on general non-linear transformations of long-memory processes (Dalla, Giraitis and Hidalgo (2006)).…”

Consistent estimation of the memory parameter for nonlinear time series

“…However, the noise can significantly increase the finite-sample bias of the estimator of a Y , so that larger samples are required to achieve the same precision as in estimation without the noise (Z t ). In the case of Gaussian or linear signal, similar observations were made by Hurvich et al (2005), Arteche (2004), and by Deo and Hurvich (2001) and Sun and Phillips (2003) for the log-periodogram estimator, assuming that the signal and the noise are independent processes. Our approach does not assume that the signal is a Gaussian or linear process and the noise is an i.i.d.…”

“…In the case of Gaussian or linear signal, similar observations were made by Hurvich et al. (2005), Arteche (2004), and by Deo and Hurvich (2001) and Sun and Phillips (2003) for the log‐periodogram estimator, assuming that the signal and the noise are independent processes.…”

The memory of beta

“…4 This enables us to obtain more precise and less noisy beta estimates (see also Bollerslev et al, 2016;Bollerslev et al, 2020;Hollstein et al, 2020). Noise in the beta series of Andersen et al (2006) could potentially lead to a downward bias in memory estimates, as found by Deo & Hurvich (2001) and Arteche (2004). In contrast, we find that changing the bandwidth in the 2ELW estimator leads to similar estimates of the memory parameter.…”

The effect of round-off error on long memory processes

“…Despite the fact that, as we have demonstrated in the previous sections, the Hurst exponent of the discretized process is the same as the one of the original process, we will show here that both methods give estimates of the Hurst exponent which are systematically and significantly negatively biased. These results are consistent with the literature on long-memory signal plus noise processes (Arteche (2004), and Hurvich, Moulines and Soulier (2005)) and on general non-linear transformations of long-memory processes (Dalla, Giraitis and Hidalgo (2006)).…”

Consistent estimation of the memory parameter for nonlinear time series

“…However, the noise can significantly increase the finite-sample bias of the estimator of a Y , so that larger samples are required to achieve the same precision as in estimation without the noise (Z t ). In the case of Gaussian or linear signal, similar observations were made by Hurvich et al (2005), Arteche (2004), and by Deo and Hurvich (2001) and Sun and Phillips (2003) for the log-periodogram estimator, assuming that the signal and the noise are independent processes. Our approach does not assume that the signal is a Gaussian or linear process and the noise is an i.i.d.…”

“…In the case of Gaussian or linear signal, similar observations were made by Hurvich et al. (2005), Arteche (2004), and by Deo and Hurvich (2001) and Sun and Phillips (2003) for the log‐periodogram estimator, assuming that the signal and the noise are independent processes.…”

The memory of beta

“…4 This enables us to obtain more precise and less noisy beta estimates (see also Bollerslev et al, 2016;Bollerslev et al, 2020;Hollstein et al, 2020). Noise in the beta series of Andersen et al (2006) could potentially lead to a downward bias in memory estimates, as found by Deo & Hurvich (2001) and Arteche (2004). In contrast, we find that changing the bandwidth in the 2ELW estimator leads to similar estimates of the memory parameter.…”

The effect of round-off error on long memory processes

“…Despite the fact that, as we have demonstrated in the previous sections, the Hurst exponent of the discretized process is the same as the one of the original process, we will show here that both methods give estimates of the Hurst exponent which are systematically and significantly negatively biased. These results are consistent with the literature on long-memory signal plus noise processes (Arteche (2004), and Hurvich, Moulines and Soulier (2005)) and on general non-linear transformations of long-memory processes (Dalla, Giraitis and Hidalgo (2006)).…”

Consistent estimation of the memory parameter for nonlinear time series

“…However, the noise can significantly increase the finite-sample bias of the estimator of a Y , so that larger samples are required to achieve the same precision as in estimation without the noise (Z t ). In the case of Gaussian or linear signal, similar observations were made by Hurvich et al (2005), Arteche (2004), and by Deo and Hurvich (2001) and Sun and Phillips (2003) for the log-periodogram estimator, assuming that the signal and the noise are independent processes. Our approach does not assume that the signal is a Gaussian or linear process and the noise is an i.i.d.…”

“…In the case of Gaussian or linear signal, similar observations were made by Hurvich et al. (2005), Arteche (2004), and by Deo and Hurvich (2001) and Sun and Phillips (2003) for the log‐periodogram estimator, assuming that the signal and the noise are independent processes.…”

The memory of beta

“…4 This enables us to obtain more precise and less noisy beta estimates (see also Bollerslev et al, 2016;Bollerslev et al, 2020;Hollstein et al, 2020). Noise in the beta series of Andersen et al (2006) could potentially lead to a downward bias in memory estimates, as found by Deo & Hurvich (2001) and Arteche (2004). In contrast, we find that changing the bandwidth in the 2ELW estimator leads to similar estimates of the memory parameter.…”