GARCH Toolbox

Introducing GARCH

GARCH stands for Generalized Autoregressive Conditional Heteroscedasticity. Loosely speaking, you can think of heteroscedasticity as time-varying variance (i.e., volatility). Conditional implies a dependence on the observations of the immediate past, and autoregressive describes a feedback mechanism that incorporates past observations into the present. GARCH then is a mechanism that includes past variances in the explanation of future variances. More specifically, GARCH is a time series modeling technique that uses past variances and past variance forecasts to forecast future variances.

In this manual, whenever a time series is said to have GARCH effects, the series is heteroskedastic, i.e., its variances vary with time. If its variances remain constant with time, the series is homoskedastic.

Why Use GARCH?

GARCH modeling builds on advances in the understanding and modeling of volatility in the last decade. It takes into account excess kurtosis (i.e. fat tail behavior) and volatility clustering, two important characteristics of financial time series. It provides accurate forecasts of variances and covariances of asset returns through its ability to model time-varying conditional variances. As a consequence, you can apply GARCH models to such diverse fields as risk management, portfolio management and asset allocation, option pricing, foreign exchange, and the term structure of interest rates.

You can find highly significant GARCH effects in equity markets, not only for individual stocks, but for stock portfolios and indices, and equity futures markets as well [5]. These effects are important in such areas as value-at-risk (VaR) and other risk management applications that concern the efficient allocation of capital. You can use GARCH models to examine the relationship between long- and short-term interest rates. As the uncertainty for rates over various horizons changes through time, you can also apply GARCH models in the analysis of time-varying risk premiums [5]. Foreign exchange markets, which couple highly persistent periods of volatility and tranquility with significant fat tail behavior [5], are particularly well suited for GARCH modeling.

Note    Bollerslev [4] developed GARCH as a generalization of Engle's [8] original ARCH volatility modeling technique. Bollerslev designed GARCH to offer a more parsimonious model (i.e., using fewer parameters) that lessens the computational burden.

GARCH Limitations

Although GARCH models are useful across a wide range of applications, they do have limitations:

• GARCH models are only part of a solution. Although GARCH models are usually applied to return series, financial decisions are rarely based solely on expected returns and volatilities.
• GARCH models are parametric specifications that operate best under relatively stable market conditions [11]. Although GARCH is explicitly designed to model time-varying conditional variances, GARCH models often fail to capture highly irregular phenomena, including wild market fluctuations (e.g., crashes and subsequent rebounds), and other highly unanticipated events that can lead to significant structural change.
• GARCH models often fail to fully capture the fat tails observed in asset return series. Heteroscedasticity explains some of the fat tail behavior, but typically not all of it. Fat tail distributions, such as student-t, have been applied in GARCH modeling, but often the choice of distribution is a matter of trial and error.

GARCH Overview

Using GARCH to Model Financial Time Series