# 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