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A Note on the Normalized Errors in ARCH and Stochastic Volatility Models

Published online by Cambridge University Press:  11 February 2009

Daniel B. Nelson
Daniel B. Nelson
Affiliation:
University of Chicago and National Bureau of Economic Research
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Abstract

It is well-known that conditional heteroskedasticity thickens the tails of the unconditional distribution of an error term relative to its conditional distribution. To what extent do imperfect forecasts of the conditional variance undo this tail thickening? This note considers the effect of changing the quality of the information embodied in a forecast of a conditional variance. Adding noise of a certain form thickens the tails of the normalized errors, but decreasing the amount of information used in the forecast may or may not thicken the tails. We also explore the relation between tail thickness and various notions of “optimal” volatility forecasts. The relationship is surprisingly complicated.

Type
Research Article
Information
Econometric Theory , Volume 12 , Issue 1 , March 1996 , pp. 113 - 128
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

Akgiray, V. (1989) Conditional heteroskedasticity in time series of stock returns: Evidence and forecasts. Journal of Business 62, 55–80.CrossRefGoogle Scholar
Bawa, V. (1982) Stochastic dominance: A research bibliography. Management Science 28, 698–712.CrossRefGoogle Scholar
Bollerslev, T. (1987) A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics 69, 542–547.CrossRefGoogle Scholar
Bollerslev, T., Engie, R.F., & Nelson, D.B. (1994) ARCH models. In Engle, R.F. & McFad-den, D. (eds.). The Handbook of Econometrics, vol. 4, pp. 2959–3038. New York: North-Holland.CrossRefGoogle Scholar
DeGroot, M.H. (1970) Optimal Statistical Decisions. New York: McGraw-Hill.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1008.CrossRefGoogle Scholar
Engle, R.F., T. Hong, A. Kane, & Noh, J. (1993) Arbitrage valuation of variance forecasts with simulated options. In Chance, D.M. & Trippi, R.R. (eds.), Advances in Futures and Options Research. Greenwich, CT: JAI Press.Google Scholar
Geweke, J. (1988) Exact inference in models with autoregressive conditional heteroskedasticity. In Barnett, W.A.Berndt, E.R., & White, H. (eds.), Dynamic Econometric Modelling. New York: Cambridge University Press.Google Scholar
Hamilton, J.O. (1989) A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384.CrossRefGoogle Scholar
Jacquier, E., Poison, N.G., & Rossi, P.E. (1994) Bayesian analysis of stochastic volatility models. Journal of Business and Economic Statistics 12, 371–389.Google Scholar
Lumsdaine, R.L. (1994) Asymptotic Properties of the Quasi-Maximum Likelihood Estimator in GARCH(l.l) and IGARCH(l.l) Models. Working paper, Princeton University.Google Scholar
McCulIoch, R. & Rossi, P.E. (1990) Posterior, predictive, and utility-based approaches to testing the arbitrage pricing theory. Journal of Financial Economics 28, 7–38.CrossRefGoogle Scholar
Melino, A. & Turnbull, S. (1990) Pricing foreign currency options with stochastic volatility. Journal of Econometrics 45, 239–266.CrossRefGoogle Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347–370.CrossRefGoogle Scholar
Nelson, D.B. & Foster, D.P. (1994) Asymptotic filtering theory for univariate ARCH models. Econometrica 62, 1–41.Google Scholar
Pagan, A.R. & Schwert, G.W. (1990) Alternative models for conditional stock market volatility. Journal of Econometrics 45, 267–290.CrossRefGoogle Scholar
Rothschild, M. & Stiglitz, J.E. (1970) Increasing risk I: A definition. Journal of Economic Theory 2, 225–243.CrossRefGoogle Scholar
Shephard, N. (1994) Local scale models: State space alternatives to integrated GARCH processes. Journal of Econometrics 60, 181–202.CrossRefGoogle Scholar
Taylor, S.J. (1994) Modelling stochastic volatility. Mathematical Finance 4, 183–204.CrossRefGoogle Scholar
Tong, H. (1990) Non-Linear Time Series: A Dynamical System Approach. Oxford, UK: Clarendon Press.CrossRefGoogle Scholar
Weiss, A.A. (1986) Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2, 107–131.Google Scholar
West, K.D., H.J. Edison, & Cho, D. (1993) A utility based comparison of some models of exchange rate volatility. Journal of International Economics 35, 23–45.CrossRefGoogle Scholar
Whitmore, G. & Findlay, M. (1978) Stochastic Dominance: An Approach to Decision Making under Risk. Lexington, Massachusetts: Heath.Google Scholar
Zellner, A. (1971) An Introduction to Bayesian Inference in Econometrics. New York: Wiley.Google Scholar