Forecasting under the MCMC framework can be done easily. The procedure is simply to use the fitted model in each Gibbs iteration to generate samples for the forecasting period. In a sense, forecasting here is done by using the fitted model to simulate realizations for the forecasting period. We use the univariate stochastic volatility model to illustrate the procedure; forecasts of other models can be obtained by the same method.
Consider the stochastic volatility model in Eqs. (12.20) and (12.21). Suppose that there are n returns available and we are interested in predicting the return rn+i and volatility hn+i for i = 1, … , ℓ, where ℓ > 0. Assume that the explanatory variables xjt in Eq. (12.20) are either available or can be predicted sequentially during the forecasting period. Recall that estimation of the model under the MCMC framework is done by Gibbs sampling, which draws parameter values from their conditional posterior distributions iteratively. Denote the parameters by 
, 
, and 
for the jth Gibbs iteration. In other words, at the jth Gibbs iteration, the model is
12.58 
We can use this model to generate a realization of rn+i and hn+i for i = 1, … , ℓ. Denote the simulated realizations by rn+i, j and hn+i, j, respectively. These realizations are generated as follows:
- Draw a random sample vn+1 from
and use Eq. (12.59) to compute hn+1, j. - Draw a random sample ϵn+1 from N(0, 1) to obtain
and use Eq. (12.58) to compute rn+1, j. - Repeat the prior two steps sequentially for n + i with i = 2, … , ℓ.
If we run a Gibbs sampling for M + N iterations in model estimation, we only need to compute the forecasts for the last N iterations. This results in a random sample for rn+i and hn+i. More specifically, we obtain
These two random samples can be used to make inference. For example, point forecasts of the return rn+i and volatility hn+i are simply the sample means of the two random samples. Similarly, the sample standard deviations can be used as the variances of forecast errors. To improve the computational efficiency in volatility forecast, importance sampling can be used; see Gelman, Carlin, Stern, and Rubin (2003).
Example 12.7
(Example 12.3 continued) As a demonstration, we consider the monthly log return series of the S&P 500 index from 1962 to 1999. Table 12.6 gives the point forecasts of the return and its volatility for five forecast horizons starting with December 1999. Both the GARCH model in Eq. (12.26) and the stochastic volatility model in Eq. (12.27) are used in the forecasting. The volatility forecasts of the GARCH(1,1) model increase gradually with the forecast horizon to the unconditional variance 3.349/(1 − 0.086 − 0.735) = 18.78. The volatility forecasts of the stochastic volatility model are higher than those of the GARCH model. This is understandable because the stochastic volatility model takes into consideration the parameter uncertainty in producing forecasts. In contrast, the GARCH model assumes that the parameters are fixed and given in Eq. (12.26). This is an important difference and is one of the reasons that GARCH models tend to underestimate the volatility in comparison with the implied volatility obtained from derivative pricing.
Table 12.6 Volatility Forecasts for Monthly Log Return of S&P 500 Indexa
a
The data span is from January 1962 to December 1999 and the forecast origin is December 1999. Forecasts of the stochastic volatility model are obtained by a Gibbs sampling with 2000 + 2000 iterations.
Remark
Besides the advantage of taking into consideration parameter uncertainty in forecast, the MCMC method produces in effect a predictive distribution of the volatility of interest. The predictive distribution is more informative than a simple point forecast. It can be used, for instance, to obtain the quantiles needed in value at risk calculation. □