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

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

##### نویسنده
• V. Omelchenko
##### چکیده

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

برای دسترسی به متن کامل این مقاله و 23 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

## One-factor-Garch models for German stocks: Estimation and forecasting

This paper presents theoretical models and their empirical results for the return and variance dynamics of German stocks. A factor structure is used in order to allow for a parsimonious modeling of the rst two moments of returns. Dynamic factor models with GARCH dynamics (GARCH(1,1)-M, IGARCH(1,1)-M, Nonlinear Asymmetric GARCH(1,1)-M and Glosten-Jagannathan-Runkle GARCH(1,1)-M) and three di ere...

متن کامل

## The Mean Variance Mixing Garch (1,1) Process a New Approach to Quantify Conditional Skewness

We present a general framework for a GARCH (1,1) type of process with innovations using a probability law of the mean-variance mixing type. We call the process the mean variance mixing GARCH (1,1) or MVM GARCH (1,1). One implication of this particular specification is a GARCH process with skewed innovations and constant mean dynamics. This is achieved without using a location parameter to compe...

متن کامل

متن کامل

## Estimating and forecasting volatility of stock indices using asymmetric GARCH models and (Skewed) Student-t densities

This paper examines the forecasting performance of four GARCH(1,1) models (GARCH, EGARCH, GJR and APARCH) used with three distributions (Normal, Student-t and Skewed Student-t). We explore and compare different possible sources of forecasts improvements: asymmetry in the conditional variance, fat-tailed distributions and skewed distributions. Two major European stock indices (FTSE 100 and DAX 3...

متن کامل

## The Residual Cusum Test for the Constancy of Parameters in GARCH(1,1) Models

In this paper we consider the problem of testing for a parameter change in GARCH(1,1) models based on the residual cusum test. The test appears to circumvent the drawbacks, such as size distortions and low powers of the cusum of squares test proposed by Kim, Cho and Lee (2000). It is shown that the proposed test statistic has a limiting distribution of the sup of a standard Brownian bridge. A s...

متن کامل

## Stable Limits of Martingale Transforms with Application to the Estimation of Garch Parameters By

In this paper we study the asymptotic behavior of the Gaussian quasi maximum likelihood estimator of a stationary GARCH process with heavytailed innovations. This means that the innovations are regularly varying with index α ∈ (2,4). Then, in particular, the marginal distribution of the GARCH process has infinite fourth moment and standard asymptotic theory with normal limits and √ n-rates brea...

متن کامل

ذخیره در منابع من

با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دسترسی به متن کامل این مقاله و 23 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

##### عنوان ژورنال:

دوره   شماره

صفحات  -

تاریخ انتشار 2010