Autoregressive Conditional Skewness

Fuqua School of Business Working Paper No. 9604

34 Pages Posted: 8 May 2000

See all articles by Akhtar R. Siddique

Akhtar R. Siddique

Government of the United States of America - Risk Analysis Division

Campbell R. Harvey

Duke University - Fuqua School of Business; National Bureau of Economic Research (NBER); Duke Innovation & Entrepreneurship Initiative

Date Written: April 6, 1999


We present a framework for modeling and estimating dynamics of variance and skewness from time-series data using a maximum likelihood approach assuming that the errors from the mean have a non-central conditional t distribution. We parameterize conditional variance and conditional skewness in an autoregressive framework similar to that of GARCH models and estimate the parameters in a conditional noncentral t distribution. The likelihood function has two time-varying parameters, the degrees of freedom and the noncentrality parameter. We apply this methodology to daily and monthly equity returns data from the U.S., Germany and Japan, concurrently estimating conditional mean, variance and skewness. We find that there is significant conditional skewness.

We then use this model to understand how the inclusion of conditional skewness affects some of the well-known stylized facts about conditional variance and the relation between returns and conditional variance. Two important stylized facts about conditional variance we examine are persistence and asymmetry in variance. Persistence refers to the tendency where high conditional variance is followed by high conditional variance. Asymmetry in variance, i.e., the observation that conditional variance depends on the sign of the innovation to the conditional mean has been documented in asymmetric variance models used in Glosten, Jagannathan, and Runkle (1993) and Engle and Ng (1993). We find that the evidence of asymmetric variance is really just conditional skewness. Inclusion of conditional skewness also impacts the persistence in conditional variance. However, we also find that there are significant seasonalities in variance and the results also depend on how the seasonal effects are accommodated in the estimation methodology. We also examine the relation between expected returns and volatility.

Keywords: Time-varying skewness, GARCH, non-central t-distribution, leverage effect

JEL Classification: C22, C51, C40, G12

Suggested Citation

Siddique, Akhtar R. and Harvey, Campbell R., Autoregressive Conditional Skewness (April 6, 1999). Fuqua School of Business Working Paper No. 9604. Available at SSRN: or

Akhtar R. Siddique

Government of the United States of America - Risk Analysis Division ( email )

400 7th Ave
Washington, DC 20219
United States
202-649-5526 (Phone)

Campbell R. Harvey (Contact Author)

Duke University - Fuqua School of Business ( email )

Box 90120
Durham, NC 27708-0120
United States
919-660-7768 (Phone)
919-660-8030 (Fax)

National Bureau of Economic Research (NBER)

1050 Massachusetts Avenue
Cambridge, MA 02138
United States

Duke Innovation & Entrepreneurship Initiative ( email )

215 Morris St., Suite 300
Durham, NC 27701
United States

Register to save articles to
your library


Paper statistics

Abstract Views