Optimal predictions of powers of conditionally heteroskedastic processes

Francq, Christian and Zakoian, Jean-Michel (2010): Optimal predictions of powers of conditionally heteroskedastic processes.

Abstract

In conditionally heteroskedastic models, the optimal prediction of powers, or logarithms, of the absolute process has a simple expression in terms of the volatility process and an expectation involving the independent process. A standard procedure for estimating this prediction is to estimate the volatility by gaussian quasi-maximum likelihood (QML) in a first step, and to use empirical means based on rescaled innovations to estimate the expectation in a second step. This paper proposes an alternative one-step procedure, based on an appropriate non-gaussian QML estimation of the model, and establishes the asymptotic properties of the two approaches. Their performances are compared for finite-order GARCH models and for the infinite ARCH. For the standard GARCH(p, q) and the Asymmetric Power GARCH(p,q), it is shown that the ARE of the estimators only depends on the prediction problem and some moments of the independent process. An application to indexes of major stock exchanges is proposed.

• Item Type: MPRA Paper
• Language: English
• Keywords: APARCH; Infinite ARCH; Conditional Heteroskedasticity; Efficiency of estimators; GARCH; Prediction; Quasi Maximum Likelihood Estimation
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

Andersen, T.G. and T. Bollerslev (1998) Answering the skeptics: yes, standard volatility models do provide accurate forecasts. International Economic Review 39, 885--906.

Baillie, R. and T. Bollerslev (1992) Prediction in dynamic models with time-dependent conditional variance. Journal of Econometrics 52, 91--113.

Bardet, J-M. and O. Wintenberger (2009) Asymptotic normality of the Quasi-maximum likelihood estimator for multidimensional causal processes. The Annals of Statistics 37, 2730--2759.

Berkes, I. and L. Horvath (2004) The efficiency of the estimators of the parameters in GARCH processes. The Annals of Statistics 32, 633--655.

Berkes, I., Horvath, L. and P. Kokoszka (2003) GARCH processes: structure and estimation. Bernoulli 9, 201--227.

Billingsley, P.(1961) The Lindeberg-Levy theorem for martingales. Proceedings of the American Mathematical Society 12, 788--792.

Billingsley, P. (1995) Probability and Measure. John Wiley \& Sons, New York.

Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307--327.

Bougerol, P. and N. Picard (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115--127.

Davis, J.-P. Kreiss and T. Mikosch. New York: Springer. Zakoïan, J-M. (1994) Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931--955.

Ding, Z., Granger, C. and R.F. Engle (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83--106.

Douc, R., Roueff, F. and P. Soulier (2008) On the existence of some ARCH($\infty$) processes. Stochastic Processes and their Applications 118, 755--761.

Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50, 987--1007.

Engle, R.F. and T. Bollerslev (1986) Modeling the persistence of conditional variances. Econometric Reviews 5, 1--50.

Engle, R.F. and D.F. Kraft (1983) Multiperiod forecast error variances of inflation estimated from ARCH models. Applied Time Series Analysis of Economic Data. Washington, DC: Bureau of the Census.

Engle, R.F. and J.R. Russell (1998) Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data. Econometrica 66, 1127--1162.

Escanciano, J.C. (2009) Quasi-maximum likelihood estimation of semi-strong GARCH models. Econometric Theory 25, 561--570.

Francq, C. and J-M. Zakoïan (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605--637.

Giraitis, L., Kokoszka, P. and R. Leipus (2000) Stationary ARCH models: dependence structure and central limit theorem. Econometric Theory 16, 3--22.

Giraitis, L., Leipus, R., and D. Surgailis (2008) ARCH($\infty$) and long memory properties. In Handbook of Financial Time Series. Eds. T. G. Andersen, R. A. Davis, J-P.

Kreiss, T. Mikosch. Springer, Berlin Heidelberg New York. Hall, P. and Q. Yao (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285--317.

Hamadeh, T. and J-M. Zakoian (2009) Asymptotic properties of least-squares and quasi-maximum likelihood estimators for a class of nonlinear GARCH Processes. Unpublished Document, University Lille 3.

Higgins, M.L. and A.K. Bera (1992) A class of nonlinear ARCH models. International Economic Review, 33, 137--158.

Karanasos, M. (1999) Prediction in ARMA models with GARCH in mean effects. Journal of Time Series Analysis 22, 555--576.

Kazakevicius, V. and R. Leipus (2003) A new theorem on existence of invariant distributions with applications to ARCH processes. Journal of Applied Probability 40, 147--162.

Loève, M. (1977) Probability Theory I, 4th edition Springer, New-York.

Nelson, D. B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318--334.

Newey, W.K. and D.G. Steigerwald (1997) Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroskedasticity models. Econometrica 65, 587--599.

Pan, J., Wang, H., and H. Tong (2008) Estimation and tests for power-transformed and threshold GARCH models. Journal of Econometrics, 142, 352--378.

Pascual, L., Romo, J., and E. Ruiz (2005) Bootstrap prediction for returns and volatilities in GARCH models. Computational Statistics \& Data Analysis 50, 2293--2312.

Robinson, P.M. (1991) Testing for strong correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics, 47, 67--84.

Robinson, P.M. and P. Zaffaroni (2006) Pseudo-Maximum Likelihood Estimation of Arch($\infty$) Models. The Annals of Statistics, 34, 1049--1074.

Straumann, D. and T. Mikosch (2006) Quasi-maximum likelihood estimation in conditionally heteroscedastic Time Series: a stochastic recurrence equations approach. The Annals of Statistics 5, 2449--2495.

Teräsvirta, T. (2007) An introduction to univariate GARCH models. Forthcoming in Handbook of Financial Time Series, ed. by T.G. Andersen, R.A.