Volatility Regressions with Fat Tails

Résumé

Nowadays, a common practice to forecast integrated variance is to do simple OLS auto-regressions of the observed realized variance data. However, non-parametric estimates of the tail index of this realized variance process reveal that its second moment is possibly unbounded. In this case, the behavior of the OLS estimators and the corresponding statistics are unclear. We prove that when the second moment of the spot variance is unbounded, the slope of the spot variance's auto-regression converges to a random variable when the sample size diverges. Likewise, the same result holds when one considers either integrated variance's auto-regression or the realized variance one. We also characterize the connection between these slopes whether the second moment of the spot variance is finite or not. Our theory also allows for a nonstationary spot variance process. We derive the results for the case of several lags in the auto-regressions and multifactor volatility process. A simulation study corroborates our theoretical findings.