LOG-PERIODOGRAM ESTIMATION OF LONG MEMORY VOLATILITY DEPENDENCIES WITH CONDITIONALLY HEAVY TAILED RETURNS

ABSTRACT

Many recent papers have used semiparametric methods, especially the log-periodogram regression, to detect and estimate long memory in the volatility of asset returns. In these papers, the volatility is proxied by measures such as squared, log-squared, and absolute returns. While the evidence for the existence of long memory is strong using any of these measures, the actual long memory parameter estimates can be sensitive to which measure is used. In Monte-Carlo simulations, I find that if the data is conditionally leptokurtic, the log-periodogram regression estimator using squared returns has a large downward bias, which is avoided by using other volatility measures. In United States stock return data, I find that squared returns give much lower estimates of the long memory parameter than the alternative volatility measures, which is consistent with the simulation results. I conclude that researchers should avoid using the squared returns in the semiparametric estimation of long memory volatility dependencies.

Keywords:

• Semiparametric methods
• Fractional integration
• Stochastic volatility
• Stock returns
• Heavy tails
• JEL Classification: C22, G10

ACKNOWLEDGMENTS

I am grateful to an associate editor and two anonymous referees for their extremely helpful comments. All errors are my sole responsibility. The views in this paper should not be interpreted as reflecting the views of the Board of Governors of the Federal Reserve System, or of other members of its staff.

Notes

Throughout this paper, the ∼ notation means that the limit of the ratio of the quantities on the left and right hand sides of the symbol is a finite positive constant.

Strictly, the proofs in Robinson[25] required some of the very lowest frequencies to be omitted from the regression in Eq. 9. Subsequently, Hurvich et al.[26] provided a version of this proof which does not require any such trimming.

Changing σ_ϵ ² amounts to multiplying y _t ² or | y _t | by a constant, or adding a constant to log (y _t ²); the log-periodogram regression is numerically invariant to these transformations.

For example, Fridman and Harris[37] and Liesenfeld and Jung[39] both consider the parametric estimation of a short memory autoregressive stochastic volatility model, by simulated maximum likelihood, allowing for conditional nonnormality. Both papers find evidence of significant conditional nonnormality in stock returns.