# ugarchpred

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, that is, the residuals or innovations (ɛt), 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.

 Note:   The latest value of residuals is the last element of vector `U`.

`Kappa`

Scalar constant term [[KAPPA]] 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` (that is, ɛt). 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 vector of the conditional variances (σt2) 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 pre-sample values of σt2 to the unconditional variance of the {ɛt} process. `H` is a single column vector of the same length as the input innovations vector `U`.

The time-conditional variance, ${\sigma }_{t}^{2}$, of a GARCH(P,Q) process is modeled as

${\sigma }_{t}^{2}=K+\sum _{i=1}^{P}{\alpha }_{i}{\sigma }_{t-i}^{2}+\sum _{j=1}^{Q}{\beta }_{j}{\epsilon }_{t-j}^{2},$

where α represents the argument `Alpha`, β represents `Beta`, and the GARCH(P,Q) coefficients {Κ, α, β} are subject to the following constraints.

$\begin{array}{l}\sum _{i=1}^{P}{\alpha }_{i}+\sum _{j=1}^{Q}{\beta }_{j}<1\\ K>0\\ \begin{array}{cc}{\alpha }_{i}\ge 0& i=1,2,\dots ,P\\ {\beta }_{j}\ge 0& j=1,2,\dots ,Q.\end{array}\end{array}$

Note that `U` is a vector of residuals or innovations (ɛt) of an econometric model, representing a mean-zero, discrete-time stochastic process.

Although ${\sigma }_{t}^{2}$ is generated using the equation above, ɛt 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 `http://www.mathworks.com/products/finprod/`.

## Examples

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