A note on the relationship between GARCH and symmetric stable processes

This note provides some explanations and extensions for the interesting results in Ghose and Kroner (1995). Specifically, we address the following points: (1) It is shown that the stable distribution and the stationary ARCH distributions are partially nested with respect to their tail shapes; (2) A novel interpretation of the McCulloch estimator is developed from the vantage point of extreme value theory; (3) This interpretation not only explains the apparent bias in some of the reported estimates, but it also helps in remedying the problem. Taken together, all three points reinforce the main conclusion of Ghose and Kroner.

[1]  J. McCulloch,et al.  Simple consistent estimators of stable distribution parameters , 1986 .

[2]  H. Kesten.  Random difference equations and Renewal theory for products of random matrices , 1973 .

[3]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[4]  Sidney I. Resnick,et al.  A Simple Asymptotic Estimate for the Index of a Stable Distribution , 1980 .

[5]  Sidney I. Resnick,et al.  Extremal behaviour of solutions to a stochastic difference equation with applications to arch processes , 1989 .

[6]  W. Vervaat.  On a stochastic difference equation and a representation of non–negative infinitely divisible random variables , 1979, Advances in Applied Probability.

[7]  Adrian Pagan,et al.  The econometrics of financial markets , 1996 .

[8]  Daniel B. Nelson.  Stationarity and Persistence in the GARCH(1,1) Model , 1990, Econometric Theory.

[9]  R. Baillie,et al.  Fractionally integrated generalized autoregressive conditional heteroskedasticity , 1996 .

[10]  P. Hall.  On Some Simple Estimates of an Exponent of Regular Variation , 1982 .

[11]  N. Bingham.  INDEPENDENT AND STATIONARY SEQUENCES OF RANDOM VARIABLES , 1973 .

[12]  Charles M. Goldie,et al.  SLOW VARIATION WITH REMAINDER: THEORY AND APPLICATIONS , 1987 .

[13]  Charles M. Grinstead,et al.  Introduction to probability , 1999, Statistics for the Behavioural Sciences.

[14]  Russell P. Robins,et al.  Estimating Time Varying Risk Premia in the Term Structure: The Arch-M Model , 1987 .

[15]  A. Dekkers,et al.  Optimal choice of sample fraction in extreme-value estimation , 1993 .

[16]  K. Kroner,et al.  The relationship between GARCH and symmetric stable processes: Finding the source of fat tails in financial data , 1995 .

[17]  E. Fama,et al.  Parameter Estimates for Symmetric Stable Distributions , 1971 .