# ugarch

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

## 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