Journal of Financial Econometrics Advance Access published online on February 21, 2008
Journal of Financial Econometrics, doi:10.1093/jjfinec/nbn002
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Parameterizing Unconditional Skewness in Models for Financial Time Series
South Western University of Finance and Economics, China and Dalarna University, Sweden
University of Technology Sydney, Australia
University of Aarhus, Denmark and Stockholm School of Economics, Sweden
Address correspondence to: Annastiina Silvennoinen, School of Finance and Economics, University of Technology Sydney, PO Box 123, Broadway NSW 2007, Australia, or e-mail: annastiina.silvennoinen{at}uts.edu.au.
JEL Classification: C22
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Abstract |
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In this paper we consider the third-moment structure of a class of time series models. It is often argued that the marginal distribution of financial time series such as returns is skewed. Therefore it is of importance to know what properties a model should possess if it is to accommodate unconditional skewness. We consider modeling the unconditional mean and variance using models that respond nonlinearly or asymmetrically to shocks. We investigate the implications of these models on the third-moment structure of the marginal distribution as well as conditions under which the unconditional distribution exhibits skewness and nonzero third-order autocovariance structure. In this respect, an asymmetric or nonlinear specification of the conditional mean is found to be of greater importance than the properties of the conditional variance. Several examples are discussed and, whenever possible, explicit analytical expressions provided for all third-order moments and cross-moments. Finally, we introduce a new tool, the shock impact curve, for investigating the impact of shocks on the conditional mean squared error of return series.
KEYWORDS: asymmetry, GARCH, nonlinearity, shock impact curve, time series, unconditional skewness
This research has been supported by the Danish National Research Foundation and the Jan Wallander and Tom Hedelius Foundation, Grant No. J03–41. We would like to thank Pentti Saikkonen, Markku Lanne, Tony Hall, Peter Phillips, and Mika Meitz for useful discussions. Part of the research was done when the second and the third author were visiting the School of Finance and Economics, University of Technology, Sydney, and when the second author was visiting CREATES, University of Aarhus. The kind hospitality of both institutions is gratefully acknowledged. Our special thanks go to Tony Hall for making the visit to UTS possible. Material from this paper has been presented at the workshop "Econometrics and Computational Economics " Helsinki, November 2004; 14th meeting of the New Zealand Econometric Study Group, Christchurch, March 2005; RUESG Workshop on Financial Econometrics, Helsinki, August 2005; International Conference on Finance, Copenhagen, September 2005; Far Eastern Meeting of the Econometric Society, Beijing, July 2006; Econometric Society European Meeting, Vienna, August 2006; and in seminars at Stockholm School of Economics, and the University of Technology, Sydney. We would like to thank the participants in these occasions for their comments. Comments by two anonymous referees, an associate editor, and in particular those by the Editor (Eric Renault) have greatly improved the presentation. The responsibility for any errors and shortcomings in this paper remains ours.
Received December 15, 2005; revised July 11, 2007; accepted December 18, 2007