Persistence and Kurtosis in GARCH and Stochastic Volatility Models

Abstract

This article shows that the relationship between kurtosis, persistence of shocks to volatility, and first-order autocorrelation of squares is different in GARCH and ARSV models. This difference can explain why, when these models are fitted to the same series, the persistence estimated is usually higher in GARCH than in ARSV models, and, why gaussian ARSV models seem to be adequate, whereas GARCH models often require leptokurtic conditional distributions. We also show that introducing the asymmetric response of volatility to positive and negative returns does not change the conclusions. These results are illustrated with the analysis of daily financial returns.

Predicting Time-Varying Parameters with Parameter-Driven and Observation-Driven Models

“…It is typically the case that higher order conditional moments of y t , such as kurtosis, are at least as high for p(y t |F t−1 ; ψ) as for p(y t |θ t ; ψ). For example, Carnero, Peña, and Ruiz (2004), among others, show that the Gaussian stochastic volatility model of Table 1 and (2) is conditionally leptokurtic. Similarly, the stochastic count and duration models we study below display conditional over-dispersion.…”

Properties of the Sample Autocorrelations of Nonlinear Transformations in Long-Memory Stochastic Volatility Models

“…Although the assumption of Gaussianity for t can seem ad hoc, Andersen et al (2001aAndersen et al ( , 2001b show that the distribution of the log volatility can be well approximated by a normal distribution. On the other hand, the assumption of gaussianity for 4 t in SV models is not as restrictive as it is in GARCH processes [see, Carnero, Peña, and Ruiz (2004)]. Finally, note that Model (1) is stationary if j0j`1 and d`0.5 and nests the short memory ARSV model when d 0.…”

An Introduction to Univariate GARCH Models

“…Typically in applications, p = q = 1 and g 1 (s t 1 ) = s t 1 where is a parameter to be estimated: This generalization is called the autoregressive stochastic volatility (SV) model, and it substantially increases the ‡exibility of the EGARCH parameterization. For evidence of this, see Malmsten and Teräsvirta (2004) and Carnero, Peña and Ruiz (2004). A disadvantage is that model evaluation becomes more complicated than that of EGARCH models because the estimation does not yield residuals.…”