This section illustrates how to estimate a GARCH model. Assuming that ut follows normal or Gaussian distribution conditional on past history, the prediction error

Figure 5.3: Histogram of the simulated GARCH process

decomposition of the log-likelihood function of the GARCH model conditional on initial values is:

The unknown model parameters c, a (i = 0 p) and ffj, (j = 1 q) can be estimated using conditional maximum likelihood estimation (MLE). Details of the maximization are given in Hamilton (1994). Once the MLE estimates of the parameters are found, estimates of the time varying volatility a2 (t =1 T) are also obtained as a side product.

5.5. GARCH Model Extensions

In many cases, the basic GARCH model (5.3.2) provides a reasonably good model for analyzing financial time series and estimating conditional volatility. However, there are some aspects of the model which can be improved so that it can better capture the characteristics and dynamics of a particular time series.

In the basic GARCH model, since only squared residuals u2^ enter the equation, the signs of the residuals or shocks have no effects on conditional volatility. However, a stylized fact of financial volatility is that bad news (negative shocks) tends to have a larger impact on volatility than good news (positive shocks).

5.5.1. EGARCH Model

Nelson (1991) proposed the following exponential GARCH (EGARCH) model to allow for leverage effects:

where ht = log of. Note that wiles ut-i is positive, the total effect of ut-Y is (1 + Bi) ut-1 ; in contrast, when ut-r is negative, the total effect of ut-1 is (1 — 7i) ut-b . Bad news can have a larger impact on volatility, and the value of y would be expected to be negative.

5.5.2. TGARCH Model

Another GARCH variant that is capable of modeling leverage effects is the threshold GARCH (TGARCH) model, which has the following form:

That is, depending on whether ut-i is above or below the threshold value of zero, has different effects on the conditional variance a2: when ut-i is positive, the total effects are given by aiu^-i; when ut-i is negative, the total effects are given by (ai + Yi)uut-i. So one would expect Yi to be positive for bad news to have larger impacts.

5.5.3. PGARCH Model

The basic GARCH model can be also extended to allow for leverage effects. This is made possible by treating the basic GARCH model as a special case of the power GARCH (PGARCH) model proposed by Ding and (1993) (1993):

where d is a positive exponent, and Yi denotes the coefficient of leverage effects. Note that when d = 2, (5.5.1) reduces to the basic GARCH model with leverage effects.

Two Components Model The GARCH model can be used to model mean reversion in conditional volatility. Recall the mean reverting form of the basic GARCH(1 1) model:

where o2 = a0 (1 — a — f{) is the unconditional long run level of volatility which is constant over time. Engle and Lee (1999) propose a model with time varying long run volatility level. The general form of the two components model is:

The long run component qt follows a highly persistent PGARCH(1 1) model, and the transitory component $t follows another PGARCH(1 1) model.

GARCH-in-the-Mean Model In financial investment, high risk is often expected to lead to high returns. Although modern capital asset pricing theory does not imply such a simple relationship, it does suggest there are some interactions between expected returns and risk as measured by volatility. Engle, Lilien and Robins (1987) propose to extend the basic GARCH model so that the conditional volatility can generate a risk premium which is part of the expected returns. This extended GARCH model is often referred to as GARCH-in-the-mean (GARCH-M) model.

The GARCH-M model extends the conditional mean equation (5.2.1) as follows:

where g(-) can be an arbitrary function of volatility at.

Exogenous Variables in Conditional Mean So far the conditional mean equation has been restricted to a constant when considering GARCH models, except for the GARCH-M model where volatility was allowed to enter the mean equation as an explanatory variable. It is possible to add ARMA terms as well as exogenous explanatory variables in the conditional mean equation. A more general form for the conditional mean equation is

where Xt is a k x 1 vector of regressors and 8 is a vector of coefficients.

Also, one can add explanatory variables into the conditional variance formula which may have impacts on conditional volatility.

Error Distributions In all the examples illustrated so far, a normal error distribution has been exclusively used. However, given the well known fat tails in financial time series, it may be more desirable to use a distribution which has fatter tails than the normal distribution.

EViews allows two fat-tailed error distributions for fitting GARCH models: the Student t distribution and the generalized error distribution.

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