Arch Models: Properties, Estimation and Testing

Abstract

The aim of this survey paper is to provide an account of some of the important developments in the autoregressive conditional heteroskedasticity (ARCH) model since its inception in a seminal paper by Engle (1982). This model takes account of many observed properties of asset prices, and therefore, various interpretations can be attributed to it. We start with the basic ARCH models and discuss their different interpretations. ARCH models have been generalized in different directions to accommodate more and more features of the real world. We provide a comprehensive treatment of many of the extensions of the original ARCH model. Next we discuss estimation and testing for ARCH models and note that these models lead to some interesting and unique problems. There have been numerous applications and we mention some of these as we present different models. The paper includes a glossary of the acronyms for the models we describe.

Practical Issues in the Analysis of Univariate GARCH Models

“…Also, the EGARCH is covariance stationary provided P q j=1 b j < 1. Another GARCH variant that is capable of modeling leverage effects is the threshold GARCH (TGARCH) model, 9 which has the following form…”

“…See, for example, [9], [14], [74], [76], [27] and [83]. There are also several comprehensive surveys that focus on the forecasting performance of GARCH models including [78], [77], and [3].…”

Persistence and Kurtosis in GARCH and Stochastic Volatility Models

“…1.1 The GARCH(1,1) Model GARCH models are very popular for representing the dynamic evolution of the volatility of financial returns and have been extensively analyzed in the literature [see, e.g., Bollerslev, Engle, and Nelson (1994), Engle (1994), Bera and Higgins (1995), Ló pez (1995), andMcAleer andOxley (2003), among many others]. If y t follows a GARCH(1,1) model, then the volatility is given by…”

Estimation of stochastic volatility models via Monte Carlo maximum likelihood

“…R . Econometric specifications of this form are known as ARCH models and have achieved widespread popularity in applied empirical research (Bollerslev et al, 1992;Bollerslev et al, 1993;Bera and Higgins, 1993). Alternatively, volatility may be modelled as an unobserved component following some latent stochastic process, such as an autoregression.…”