ugarchpred (Financial Toolbox)

Forecast conditional variance of univariate GARCH(P,Q) processes.

Syntax

[VarianceForecast, H] = ugarchpred(U, Kappa, Alpha, Beta, 
NumPeriods)

Arguments


`U`	Single column vector of random disturbances, i.e., the residuals, or innovations, of an econometric model representing a mean-zero, discrete-time stochastic process. The innovations time series `U` is assumed to follow a GARCH(P,Q) process.
`Kappa`	Scalar constant term of the GARCH process.
`Alpha`	`P`-by-`1` vector of coefficients, where `P` is the number of lags of the conditional variance included in the GARCH process. `Alpha` can be an empty matrix, in which case `P` is assumed 0; when `P = 0`, a GARCH(0,Q) process is actually an ARCH(Q) process.
Beta	`Q`-by-`1` vector of coefficients, where `Q` is the number of lags of the squared innovations included in the GARCH process.
`NumPeriods`	Positive, scalar integer representing the forecast horizon of interest, expressed in periods compatible with the sampling frequency of the input innovations column vector `U`.

Description

[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)-by-1 vector of the minimum mean-square error forecast of the conditional variance of the innovations time series vector U. The first element contains the 1-period-ahead forecast, the second element contains the 2-period-ahead 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 single column vector of the same length as the input innovations vector U. To model the GARCH(P,Q) process, you must construct the conditional variance time series, H(t), (see below). This represents the time series 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 pre-sample values of H(t) to the unconditional variance of the U(t) process.

GARCH(P,Q) coefficients {Kappa, Alpha, Beta} are subject to constraints

Kappa > 0
Alpha(i) >= 0 for i = 1,2,...P
Beta(i)  >= 0 for i = 1,2,...Q
sum(Alpha(i) + Beta(j)) < 1 for i = 1,2,...P and j = 1,2,...Q

The time-conditional variance, H(t), of a GARCH(P,Q) process is modeled as:

H(t) = Kappa + Alpha(1)*H(t-1) + Alpha(2)*H(t-2) +...+ Alpha(P)*H(t-P)+ Beta(1)*U^2(t-1)+ Beta(2)*U^2(t-2)+...+ Beta(Q)*U^2(t-Q)

Note that U is a vector of innovations, or regression residuals of an econometric model, representing a mean-zero, discrete-time stochastic process. That is, it is assumed that a regression model has already been run, and that U(t) = y(t) - F(X(t),B) is the time series of innovations derived from the model.

Although H is generated via the equation above, U and H are related as:

U(t) = sqrt(H(t))*v(t)

where v(t) is an i.i.d. sequence.

Example

See ugarchsim for an example of forecasting the conditional variance of a univariate GARCH(P,Q) process.