# ugarchllf

Log-likelihood objective function of univariate GARCH(P,Q) processes with Gaussian innovations

## Syntax

```LogLikelihood = ugarchllf(Parameters, U, P, Q)
```

## Arguments

`Parameters`

`(1 + P + Q)`-by-`1` column vector of GARCH(P,Q) process parameters. The first element is the scalar constant term [[KAPPA]] of the GARCH process; the next `P` elements are coefficients associated with the `P` lags of the conditional variance terms; the next `Q` elements are coefficients associated with the `Q` lags of the squared innovations terms.

`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

`LogLikelihood = ugarchllf(Parameters, U, P, Q)` computes the log-likelihood objective function of univariate GARCH(P,Q) processes with Gaussian innovations.

`LogLikelihood` is a scalar value of the GARCH(P,Q) log-likelihood objective function given the input arguments. This function is meant to be optimized via the `fmincon` function of the Optimization Toolbox™ software.

`fmincon` is a minimization routine. To maximize the log-likelihood function, the `LogLikelihood` output parameter is actually the negative of what is formally presented in most time series or econometrics references.

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`, and β represents `Beta`.

`U` is a vector of residuals or innovations (ɛt) representing a mean-zero, discrete time stochastic process. Although ${\sigma }_{t}^{2}$ is generated via 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).

Since `ugarchllf` is really just a helper function, no argument checking is performed. This function is not meant to be called directly from the command line.

 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/`.