# ugarch

(To be removed) Univariate GARCH(P,Q) parameter estimation with Gaussian innovations

As an alternative to `ugarch`, use the `garch` object to create conditional variance models. For more information, see Specify GARCH Models Using garch.

`ugarch` is removed. Use the `garch` object to create conditional variance models and the `estimate` function to fit conditional variance models to data. For more information, see Likelihood Ratio Test for Conditional Variance Models.

## Syntax

```[Kappa, Alpha, Beta] = ugarch(U, P, Q)
```

## 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`.

`P`

Nonnegative, scalar integer representing a model order of the GARCH process. `P` is the number of lags of the conditional variance. `P` can be zero; when `P = 0`, a GARCH(0,Q) process is actually an ARCH(Q) process.

`Q `

Positive, scalar integer representing a model order of the GARCH process. `Q` is the number of lags of the squared innovations.

## Description

`[Kappa, Alpha, Beta] = ugarch(U, P, Q)` computes estimated univariate GARCH(P,Q) parameters with Gaussian innovations.

`Kappa` is the estimated scalar constant term ([[KAPPA]]) of the GARCH process.

`Alpha` is a `P`-by-`1` vector of estimated coefficients, where `P` is the number of lags of the conditional variance included in the GARCH process.

`Beta` is a `Q`-by-`1` vector of estimated coefficients, where `Q` is the number of lags of the squared innovations included in the GARCH process.

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 a GARCH(P,Q) process.

## References

James D. Hamilton, Time Series Analysis, Princeton University Press, 1994