Forecast conditional variance of univariate GARCH(P,Q) processes
[VarianceForecast, H] = ugarchpred(U, Kappa, Alpha, Beta,
NumPeriods)
 Single column vector of random disturbances, that is,
the residuals or innovations (ɛ_{t}), of
an econometric model representing a meanzero, discretetime stochastic
process. The innovations time series
 
 Scalar constant term [[KAPPA]] of the GARCH process.  

 

 
 Positive, scalar integer representing the forecast horizon
of interest, expressed in periods compatible with the sampling frequency
of the input innovations column vector 
[VarianceForecast, H] = ugarchpred(U, Kappa, Alpha,
Beta, NumPeriods)
forecasts the conditional variance of
univariate GARCH(P,Q) processes.
VarianceForecast
is a number of periods (NUMPERIODS
by1
)
vector of the minimum meansquare error forecast of the conditional
variance of the innovations time series vector U
(that
is, ɛ_{t}). The first element contains the
1periodahead forecast, the second element contains the 2periodahead
forecast, and so on. Thus, if a forecast horizon greater than 1 is
specified (NUMPERIODS > 1
), the forecasts of
all intermediate horizons are returned as well. In this case, the
last element contains the variance forecast of the specified horizon, NumPeriods
from
the most recent observation in U
.
H
is a vector of the conditional variances
(σ_{t}^{2}) corresponding
to the innovations vector U
. It is inferred from
the innovations U
, and is a reconstruction of the
"past" conditional variances, whereas the VarianceForecast
output
represents the projection of conditional variances into the "future."
This sequence is based on setting presample values of σ_{t}^{2} to
the unconditional variance of the {ɛ_{t}}
process. H
is a single column vector of the same
length as the input innovations vector U
.
The timeconditional variance, $${\sigma}_{t}^{2}$$, of a GARCH(P,Q) process is modeled as
$${\sigma}_{t}^{2}=K+{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}{\sigma}_{ti}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}{\epsilon}_{tj}^{2}},$$
where α represents the argument Alpha
, β represents Beta
,
and the GARCH(P,Q) coefficients {Κ, α, β}
are subject to the following constraints.
$$\begin{array}{l}{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}}<1\\ K>0\\ \begin{array}{cc}{\alpha}_{i}\ge 0& i=1,2,\dots ,P\\ {\beta}_{j}\ge 0& j=1,2,\dots ,Q.\end{array}\end{array}$$
Note that U
is a vector of residuals or innovations
(ɛ_{t})
of an econometric model, representing a meanzero, discretetime stochastic
process.
Although $${\sigma}_{t}^{2}$$ is generated using the equation above, ɛ_{t} and $${\sigma}_{t}^{2}$$ are related as
$${\epsilon}_{t}={\sigma}_{t}{\upsilon}_{t},$$
where $$\left\{{\upsilon}_{t}\right\}$$ is an independent, identically distributed (iid) sequence ~ N(0,1).
Note
The Econometrics Toolbox™ software provides a comprehensive
and integrated computing environment for the analysis of volatility
in time series. For information, see the Econometrics Toolbox documentation
or the financial products Web page at 
See ugarchsim
for an
example of forecasting the conditional variance of a univariate GARCH(P,Q)
process.