The GARCH model for time series contains several highly constrained parameters. This example presents estimates and confidence limits for a GARCH(1,1) model applied to 20 years of monthly observations on the capitilization weighted returns of the Wilshire 5000 index.
Define the time series
where t = 1, 2, ... T, and an observed time series with expected value .
where , , and
The log-likelihood conditional on max(p,q) initial estimates of the conditional variances is, for
The customary constraints applied to the parameters to enforce stationarity and a positive conditional variance are
To be able to construct a uniform prior, it is necessary to bound from above and from above and below. For the example,
The prior, then, is constructed from the above constraints:
where A is the hypervolume inscribed by the inequalities, and V = 400 for this example.
The parameter estimates are presented in Table 1 along with confidence limits computed by four different methods. It can be observed there that the confidence limits of the structural parameter, , is very consistent across the methods. is not itself constrained. However, it's confidence limits can be affected by estimates of the remaining parameters occuring in the region of their constraint boundaries. This, however, is provided that the distribution of is correlated more than about .7 with these parameters. Table 2 presents the estimate of the correlation matrix of the parameters calculated from the maximum likelihood estimate. We see there that the largest correlation is quite small, -.05, and thus we should expect that the constraints on the remaining parameters will have little effect on the structural parameter.
Figure 6: Kernel density plots of the parameters of a GARCH(1,1) model of the Wilshire 5000 capitalization weighted returns
Kernel density plots were constructed for each of the parameters from the weighted likelihood bootstrap results (Figure 6). The posterior distribution of appears to be slightly skewed resulting in a slight difference between the mean and the mode. The mean of the distribution, 1.1167, is very close to the maximum likelihood estimate, 1.0947. The mode, however, appears to be about 1.2.