Conditional Volatility, Skewness, and Kurtosis:
Existence, Persistence, and Comovements
Eric Jondeau and Michael Rockinger*
Journal of Economic Dynamics and Control, 2003, 27, 1699-1737.
Abstract
Recent portfolio-choice, asset-pricing, value-at-risk, and option-valuation
models highlight the importance of modeling the asymmetry and tail-fatness of
returns. These characteristics are captured by the skewness and the kurtosis.
We characterize the maximal range of skewness and kurtosis for which a density
exists and show that the generalized Student-t distribution spans a large domain
in the maximal set. We use this distribution to model innovations of a GARCH
type model, where parameters are conditional. After demonstrating that an autoregressive
specification of the parameters may yield spurious results, we estimate and
test restrictions of the model, for a set of daily stock-index and foreign-exchange
returns. The estimation is implemented as a constrained optimization via a sequential
quadratic programming algorithm. Adequacy tests demonstrate the importance of
a time-varying distribution for the innovations. In almost all series, we find
time dependency of the asymmetry parameter, whereas the degree-of-freedom parameter
is generally found to be constant over time. We also provide evidence that skewness
is strongly persistent, but kurtosis is much less so. A simulation validates
our estimations and we conjecture that normality holds for the estimates. In
a cross-section setting, we also document covariability of moments beyond volatility,
suggesting that extreme realizations tend to occur simultaneously on different
markets.
Keywords: Volatility, Skewness, Kurtosis, Generalized Student-t distribution,
GARCH, Stock indices, Exchange rates, SNOPT.
JEL classification: C22, C51, G12.
* HEC Lausanne