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# 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, , of a GARCH(P,Q) process is modeled as

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

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

Although is generated using the equation above, ɛt and are related as

where is an independent, identically distributed (iid) sequence ~ N(0,1).

 Note   ugarch corresponds generally to the Econometrics Toolbox™ function garchfit. 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