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Conditional volatility, skewness, and kurtosis: Existence, persistence, and comovements. (English) Zbl 1178.91226
Summary: 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.

MSC:
91G70 Statistical methods; risk measures
62P05 Applications of statistics to actuarial sciences and financial mathematics
90C20 Quadratic programming
Software:
SNOPT
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[1] Baker, G.A.; Graves-Morris, P., Padé approximants, (1996), Cambridge University Press Cambridge
[2] Barone-Adesi, G., Arbitrage equilibrium with skewed asset returns, Journal of financial and quantitative analysis, 20, 3, 299-313, (1985)
[3] Bera, A.K.; Higgins, M.L., ARCH models: properties, estimation and testing, Journal of economic surveys, 7, 4, 305-362, (1993)
[4] Black, F., 1976. Studies in stock price volatility changes. Proceedings of the 1976 Business Meeting of the Business and Economic Statistics Section. American Statistical Association.
[5] Bollerslev, T., Generalized autoregressive conditional heteroskedasticity, Journal of econometrics, 31, 3, 307-327, (1986) · Zbl 0616.62119
[6] Bollerslev, T.; Wooldridge, J.M., Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances, Econometric reviews, 11, 2, 143-172, (1992) · Zbl 0850.62884
[7] Bollerslev, T.; Chou, R.Y.; Kroner, K.F., ARCH modelling in finance: a review of the theory and empirical evidence, Journal of econometrics, 52, 1-2, 5-59, (1992) · Zbl 0825.90057
[8] Bollerslev, T., Engle, R.F., Nelson, D.B., 1994. ARCH models. In: Engle, R.F., McFadden, D.L. (Eds.), Handbook of Econometrics, Vol. IV. Elsevier Science, Amsterdam, pp. 2959-3038 (Chapter 49).
[9] Campbell, J.Y.; Hentschel, L., No news is Goos news: an asymmetric model of changing volatility, Journal of financial economics, 31, 3, 281-318, (1992)
[10] Engle, R.F., Auto-regressive conditional heteroskedasticity with estimates of the variance of united kingdom inflation, Econometrica, 50, 4, 987-1007, (1982) · Zbl 0491.62099
[11] Engle, R.F.; Gonzalez-Rivera, G., Semi-parametric ARCH models, Journal of business and economic statistics, 9, 4, 345-359, (1991)
[12] Fama, E., Mandelbrot and the stable Paretian hypothesis, Journal of business, 36, 420-429, (1963)
[13] Fang, H.; Lai, T.-Y., Co-kurtosis and capital asset pricing, Financial review, 32, 2, 293-307, (1997)
[14] Friend, I.; Westerfield, R., Co-skewness and capital asset pricing, Journal of finance, 35, 4, 897-913, (1980)
[15] Gill, Ph.E., Murray, W., Saunders, M.A., 1997. SNOPT: an SQP algorithm of large-scale constrained optimization. Report NA 97-2, Department of Mathematics, University of California, San Diego. · Zbl 1210.90176
[16] Gill, Ph.E., Murray, W., Saunders, M.A., 1999. User’s guide for SNOPT 5.3: a Fortran package for large-scale nonlinear programming. Working paper, Department of Mathematics, UCSD, California.
[17] Glosten, R.T.; Jagannathan, R.; Runkle, D., On the relation between the expected value and the volatility of the nominal excess return on stocks, Journal of finance, 48, 5, 1779-1801, (1993)
[18] Gradshteyn, I.S.; Ryzhik, I.M., Table of integrals, series, and products, (1994), Academic Press New York · Zbl 0918.65002
[19] Hamao, Y.; Masulis, R.W.; Ng, V., Correlations in price changes and volatility across international stock markets, Review of financial studies, 3, 2, 281-307, (1990)
[20] Hamburger, H., Über eine erweiterung des stieltjesschen momentproblems, Mathematische zeitschrift, 7, 235-319, (1920) · JFM 47.0427.04
[21] Hansen, L.P., Large sample properties of generalized method of moments estimator, Econometrica, 50, 4, 1029-1054, (1982) · Zbl 0502.62098
[22] Hansen, B.E., Autoregressive conditional density estimation, International economic review, 35, 3, 705-730, (1994) · Zbl 0807.62090
[23] Harvey, C.R.; Siddique, A., Autoregressive conditional skewness, Journal of financial and quantitative analysis, 34, 4, 465-487, (1999)
[24] Harvey, C.R.; Siddique, A., Conditional skewness in asset pricing tests, Journal of finance, 55, 3, 1263-1295, (2000)
[25] Hausdorff, F., Summationsmethoden und momentfolgen, I, Mathematische zeitschrift, 9, 74-109, (1921) · JFM 48.2005.01
[26] Hausdorff, F., Summationsmethoden und momentfolgen, II, Mathematische zeitschrift, 9, 280-299, (1921) · JFM 48.2005.02
[27] Hwang, S.; Satchell, S.E., Modelling emerging market risk premia using higher moments, Journal of finance and economics, 4, 4, 271-296, (1999)
[28] Jorion, Ph., Predicting volatility in the foreign exchange market, Journal of finance, 50, 2, 507-528, (1995)
[29] Kan, R.; Zhou, G., A critique of the stochastic discount factor methodology, Journal of finance, 54, 4, 1221-1248, (1999)
[30] Korkie, B., Sivakumar, R., Turtle, H.J., 1997. Skewness persistence: it matters, just not how we thought. Working paper, Faculty of Business, University of Alberta, Canada.
[31] Kraus, A.; Litzenberger, R.H., Skewness preference and the valuation of risk assets, Journal of finance, 31, 4, 1085-1100, (1976)
[32] Kroner, K.F.; Ng, V.K., Modeling asymmetric comovements of asset returns, Review of financial studies, 11, 4, 817-844, (1998)
[33] Longin, F.; Solnik, B., Is the correlation in international equity returns constant: 1960-1990?, Journal of international money and finance, 14, 1, 3-26, (1995)
[34] Loretan, M.; Phillips, P.C.B., Testing covariance stationarity under moment condition failure with an application to common stock returns, Journal of empirical finance, 1, 211-248, (1994)
[35] Mandelbrot, B., The variation of certain speculative prices, Journal of business, 35, 394-419, (1963)
[36] Mood, A.M.; Graybill, F.A.; Boes, D.C., Introduction to the theory of statistics, (1982), McGraw-Hill New York
[37] Nelsen, R.B., An introduction to copulas, (1999), Springer, Berlin New York · Zbl 0909.62052
[38] Nelson, D.B., Conditional heteroskedasticity in asset returns: a new approach, Econometrica, 59, 2, 370-397, (1991) · Zbl 0722.62069
[39] Newey, W., Generalized method of moments specification testing, Journal of econometrics, 29, 229-256, (1985) · Zbl 0606.62132
[40] Premaratne, G., Bera, A.K., 1999. Modeling asymmetry and excess kurtosis in stock return data. Working paper, Department of Economics, University of Illinois.
[41] Ramchand, L.; Susmel, R., Volatility and cross correlation across major stock markets, Journal of empirical finance, 5, 4, 397-416, (1998)
[42] Rockinger, M., Jondeau, E., 2001. Conditional dependency of financial series: an application of copulas. Working paper, Banque de France, NER#82.
[43] Rockinger, M.; Jondeau, E., Entropy densities with an application to autoregressive conditional skewness and kurtosis, Journal of econometrics, 106, 1, 119-142, (2002) · Zbl 1043.62110
[44] Rubinstein, M.E., The fundamental theorem of parameter-preference security valuation, Journal of financial and quantitative analysis, 8, 1, 61-69, (1973)
[45] Sears, R.S.; Wei, K.C.J., Asset pricing, higher moments, and the market risk premium: a note, Journal of finance, 40, 4, 1251-1253, (1985)
[46] Sears, R.S.; Wei, K.C.J., The structure of skewness preferences in asset pricing models with higher moments: an empirical test, Financial review, 23, 1, 25-38, (1988)
[47] Sentana, E., Quadratic ARCH models, Review of economic studies, 62, 4, 639-661, (1995) · Zbl 0847.90035
[48] Stieltjes, T.J., Recherches sur LES fractions continues, Annales de la faculté des sciences de Toulouse, 8, 1, 1-22, (1894) · JFM 25.0326.01
[49] Susmel, R.; Engle, R.F., Hourly volatility spillovers between international equity markets, Journal of international money and finance, 13, 1, 3-25, (1994)
[50] Tan, K.-J., Risk return and the three-moment capital asset pricing model: another look, Journal of banking and finance, 15, 2, 449-460, (1991)
[51] White, H., A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity, Econometrica, 48, 4, 817-838, (1980) · Zbl 0459.62051
[52] Widder, D.V., The Laplace transform, (1946), Princeton University Press Princeton, NJ · Zbl 0060.24801
[53] Zakoı̈an, J.M., Threshold heteroskedastic models, Journal of economic dynamics and control, 18, 5, 931-955, (1994) · Zbl 0875.90197
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