Modeling stock markets’ volatility using GARCH models with Normal, Student’s t and stable Paretian distributions

José Dias Curto,José Castro Pinto,Gonçalo Nuno Tavares

doi:10.1007/s00362-007-0080-5

Abstract

As GARCH models and stable Paretian distributions have been revisited in the recent past with the papers of Hansen and Lunde (J Appl Econom 20: 873–889, 2005) and Bidarkota and McCulloch (Quant Finance 4: 256–265, 2004), respectively, in this paper we discuss alternative conditional distributional models for the daily returns of the US, German and Portuguese main stock market indexes, considering ARMA-GARCH models driven by Normal, Student’s t and stable Paretian distributed innovations. We find that a GARCH model with stable Paretian innovations fits returns clearly better than the more popular Normal distribution and slightly better than the Student’s t distribution. However, the Student’s t outperforms the Normal and stable Paretian distributions when the out-of-sample density forecasts are considered.

Highlights

Since the birth of the modern empirical finance two main approaches have been considered to model the empirical distribution of financial assets returns
As the empirical findings suggest the presence of volatility clusters, one might represent this kind of returns behavior using a model where the conditional variance is serially correlated and since the seminal paper of Engle (1982) was published, the second one, that we name the conditional approach, became common in empirical finance
In this paper we examine the conditional distribution of daily returns in the US, the German and the Portuguese equity markets, comparing the stable Paretian distribution to the Gaussian and the Student’s t distributions for innovations

Summary

Introduction

Since the birth of the modern empirical finance two main approaches have been considered to model the empirical distribution of financial assets returns. The first one, that we name the unconditional approach, admits that stock prices follow a random walk and several models have been proposed to describe the unconditional distribution of financial returns. The Gaussian distribution was the first to be considered and the normality became one of the most important assumptions in the classical financial models, namely the Portfolio Theory, the Capital Asset Pricing Model (CAPM) and the. Given the homoskedastic nature of the conditional distribution implicit in these models, they are unable to capture the volatility clustering that is common in financial asset returns. In this paper we examine the conditional distribution of daily returns in the US, the German and the Portuguese equity markets, comparing the stable Paretian distribution to the Gaussian and the Student’s t distributions for innovations.

The models

Statistical properties of returns

Modeling the empirical distribution of returns

Out-of-sample density forecasts

Conclusions