Consider the following GARCH(1,1) model

(9.58)

(9.59)

Suppose that the researcher had estimated the above GARCH model for a series of returns on a stock index and obtained the following parameter estimates: If the researcher has data available up to and including time T, write down a set of equations in and and their lagged values,which could be employed to produce one-, two-, and three-step-ahead forecasts for the conditional variance of yt

.

What is needed is to generate forecasts of σT+1 2

|ΩT

, σT+2 2

|ΩT

, …,

σT+s 2

|ΩT where ΩT denotes all information available up to and including observation T. For time T, the conditional variance equation is given by equation (9.59). Adding one to each of the time subscripts of this equation, and then two, and then three would yield equations

(9.60)–(9.62)

(9.60)

(9.61)

(9.62)

Let be the one-step-ahead forecast for σ

2 made at time T. This is easy to calculate since, at time T, the values of all the terms on the RHS are known. would be obtained by taking the conditional expectation of equation (9.60).
Given how is the two-step-ahead forecast for σ
2 made at time T, calculated?
(9.63)
From equation (9.61), it is possible to write (9.64)
where is the expectation, made at time T, of squared disturbance term. It is necessary to find using the expression for the variance of a random variable ut . The model assumes that the series ut has zero mean, so that the variance can be written (9.65)
The conditional variance of ut is so (9.66)
Turning this argument around, and applying it to the problem at hand (9.67)
but is not known at time T, so it is replaced with the forecast for it, so that equation (9.64) becomes (9.68)
(9.69)
What about the three-step-ahead forecast?
By similar arguments, (9.70)
(9.71)
(9.72)

Any s-step-ahead forecasts would be produced by (9.74)
for any value of s ≥ 2.
It is worth noting at this point that variances, and therefore variance forecasts, are additive over time. This is a very useful property.
Suppose, for example, that using daily foreign exchange returns, one-, two-, three-, four-, and five-step-ahead variance forecasts have been produced, i.e., a forecast has been constructed for each day of the next trading week. The forecasted variance for the whole week would simply be the sum of the five daily variance forecasts. If the standard deviation is the required volatility estimate rather than the variance, simply take the square root of the variance forecasts. Note also, however, that standard deviations are not additive. Hence, if daily standard deviations are the required volatility measure, they must be squared to turn them to variances. Then the variances would be added and the square root taken to obtain a weekly standard deviation.