Parameter Estimation of the Stable GARCH(1,1)-Model

The paper aims to show methodology of parameter estimation of the stable GARCH(1,1) model. There are represented and compared 3 methods of finding estimates of their parameters. We assume that we have a stable GARCH(1,1) model with the stable symmetric innovation. We search for the estimates of parameters of the stable GARCH model under assumption that we don’t know anything about parameters, that is neither α nor parameters of the model are known, and all the information must be extracted from the observations. We develop the methodology of parameter estimation by simulating a sample from the stable GARCH model and estimating its parameters and comparing estimates with theoretical values. It has been shown by simulation that we can obtain appropriate estimates of the parameters having only 1000 observations. Introduction The stable distributions are characterized by the same convolution and limit properties as the normal distributions and they can be an extension of it. The normal distribution also belongs to the family of the stable distributions as a special case and the financial models such as stable GARCH(p,q) can serve as an extension of the classical GARCH(p,q) model with the normal innovation. The classical GARCH models with the stable innovation are quiet popular in economic practice but sometimes the behavior of prices of many assets is not typical for the normal models therefore, it is reasonable to replace the normal distribution by a family of the distributions extending the normal one. There are developed 3 methodologies of parameter estimation that will be described below. We can use Two Sample Kolmogorov Smirnov test to verify if the estimates are correct. The second sample will be obtained by the simulation. Simulation will be the central point of this methodology. Definition of the Stable Distributions There are four equivalent definitions of the stable distributions. We will mention only one of them which we need for our purposes. The rest of them specifies convolution and limit properties of the stable distributions and can be derived from the definition specifying its characteristic function. Definition A random variable X has the stable distribution if its characteristic function is of the form: E exp(i · u ·X) = exp ( −σα|u| [ 1− iβ ( tan πα 2 ) sign(u) ] + iμu ) , for α 6= 1 and E exp(i · u ·X) = exp ( −σ|u| [ 1 + iβ ( 2 π ) sign(u) ln |u| ] + iμu ) , for α = 1 This definition is the most important for our purposes because it gives us the explicit form of the characteristic function. The density and the distribution functions of the stable distributions do not have any explicit form with only 3 exceptions: normal distribution, Cauchy distribution and Levy distribution. The constants contained in the formula of the characteristic function are parameters of the stable distributions which uniquely determine them. The most important parameter is α which is called the tail index. α ∈ (0, 2] and if α = 2 then we deal with the normal distribution. If α < 2 then for any a ≥ α EX =∞ and EX <∞ if a < α. The parameter σ is called the scale parameter which has the properties of the standard deviation. The parameter μ is called the location parameter, it equals the mean, i.e. EX = μ if α > 1. The parameter β is called the index of asymmetry which equals zero if α = 2 and belongs to the interval [−1, 1] if α < 2. The stable distribution with parameters α, σ, β, and μ is denoted by Sα(σ, β, μ). The centralized stable distribution Sα(σ, 0, 0) will be denoted by SαS. E.g 137 WDS'09 Proceedings of Contributed Papers, Part I, 137–142, 2009. ISBN 978-80-7378-101-9 © MATFYZPRESS