This paper addresses an existing gap in the developing literature on conditional skewness. We develop a simple procedure to evaluate parametric conditional skewness models. This procedure is based on regressing the realized skewness measures on model-implied conditional skewness values. We find that an asymmetric generalized autoregressive conditional heteroscedasticity specification on shape parameters with a skewed generalized error distribution provides the best in-sample fit for the data, as well as reasonable predictions of the realized skewness measure. Our empirical findings imply significant asymmetry with respect to positive and negative news in both conditional asymmetry and kurtosis processes.
We thank two anonymous referees, Chris Adcock (the editor), Torben Andersen, Bruce Hansen, Scott Hendry, Glen Keenleyside, Stanislav Khrapov, Nicola Loperfido, Dilip K. Patro, Kazuhiko Shinki, Akhtar Siddique, seminar participants at Joint Statistical Meetings 2011, Midwest Econometric Group meeting 2011, the Bank of Canada, Wayne State University (mathematics department), Office of the Comptroller of the Currency, and the MAF 2012 conference for many useful comments. An earlier version of this paper has been circulated and presented at various seminars and conferences under the title ‘Regime Switching in the Conditional Skewness of S&P500 Returns’. The remaining errors are ours. The analysis and the conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Federal Reserve Board of Governors or the Bank of Canada.
1. By traditional, we mean standardized third and fourth moments of a random variable.
2. We define the ‘upside variance’ as the variance of the returns conditional upon their realization above a certain threshold. Their variance conditional upon their realization below the same threshold is called ‘downside variance’. Based on these two definitions, we define the difference between upside variance and downside variance as the ‘RSV’.
3. We find that the evidence for asymmetry-in-asymmetry itself is relatively weak. But the flexibility offered by separating the contributions of positive and negative shocks improves the model's performance significantly.
4. Suitability of γ(x) as a skewness measure critically depends on the measure of volatility used in modeling the returns process. Later in the paper, we show that our results are based on the Engle and Ng (Citation1993) NGARCH volatility model. Based on empirical results, we argue that NGARCH is a perfectly adequate volatility measure and, thus, our conditional skewness measures are well specified.
5. This result means that RSV, , satisfies (P1) up to a multiplicative constant.
6. In practice, we put less weight on the first condition, with β0 statistically indistinguishable from 0. As Chernov (Citation2007) documents this issue in a parallel literature, pinning down the correct functional form of the statistical relationship between latent variables is difficult. Thus, we focus on the more robust and theoretically more important relationship between our skewness measures through slope parameters. That said, we report results for joint tests for in our discussion of empirical findings.
7. See Bangert et al. (Citation1986), Kimber and Jeynes (Citation1987) and Toth and Szentimrey (Citation1990), among others, for examples of using the binormal distribution in data modeling, statistical analysis and robustness studies.
8. To keep our notation consistent with the RV literature, our timing convention differs slightly from the familiar GARCH notation. Throughout the paper, the subscript t on any variable means that it is observed exactly at time t. In the traditional GARCH notation, the subscript t in the conditional variance means that it is the variance of the time t returns. Hence, the variance is observed at time t−1.
9. This means that, in our tables, the reported LR test statistics are computed by using log-likelihood values from model Mx and model Mx−1. That is, the LR statistic for model M2 is based on log-likelihood values for models M1 and M2. When necessary, we indicate that we have used log-likelihood values from non-sequential but nested models to construct a test statistic.
10. Our testing procedure is similar in spirit to the forward selection method used in econometrics for model selection through sequential significance testing. Thus, our procedure does not account for global significance level of tests. For nested models, this procedure is set up to reject restricted (null) against the unrestricted (alternative) models. Our goal is to provide a simple rule for model selection. One may consider a different rule which leads to a different model choice. For example, backward elimination, forward selection and stepwise selection rules do not necessarily pick the same model. Thus, using a different rule leads to a different set of results. While we acknowledge that the global significance level may be lower than the level of the test for each single pair of models, this does not influence our choice of model, since our model selection rule does not account for this contingency.
11. Please note that in Tables 2, 3, 5, 6, and 8, we abuse the notation to save space. That is, κ1+ and γ1+ represent both the coefficients for the ARCH-like terms and the coefficients for positive innovation shocks.
12. critical values at 5% and 1% confidence levels are 5.99 and 9.21, respectively, while the computed LR test statistic is over 11.