Garch Models

An important assumption in the ARMA model

\[\phi(B)W_t = \theta_0 + \theta(B)A_t, t \in Z\]

is that the errors \(A_t\) are independent random variables with mean 0 and constant variance, \(\sigma^2\).

For some time series, the assumptions of independent errors and constant variance will not hold for the underlying series. For example, in the stock market, large errors in one direction are often followed by large errors in the opposite direction. Variability over time can increase with a price level or trading volume. To account for heteroscedastic (non-equal) variances, Engle ([1]) introduced the Autoregressive Conditional Heteroscedastic or ARCH, model:

\[A_t = z_t \sigma_t\]

\[\sigma _t^2 = \sigma^2 + \sum\limits_{i = 1}^q {\alpha _i A_{t - i}^2},\]

where \(z_t\) is a sequence of independent and identically distributed standard normal random variables. In the ARCH model, the variance term depends on previous squared errors, up to a given lag q. A generalized ARCH model, called GARCH, was introduced by Bollerslev ([2]) and has the form:

\[A_t = z_t \sigma_t\]

\[\sigma _t^2 = \sigma^2 + \sum\limits_{i = 1}^p {\beta _i \sigma _{t - i}^2} + \sum\limits_{i = 1}^q {\alpha _i A_{t - i}^2},\]

In the GARCH model, the variance has an auto-regressive term in addition to the squared error term. The function imsl.timeseries.garch() estimates ARCH or GARCH models.

[1]	Engle, C. (1982), Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation, Econometrica, 50, 987-1008.

[2]	Bollerslev, T. (1986), Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, 307-327.