ugarchllf (Financial Toolbox)

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 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, i.e., the residuals or innovations (_t), 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.

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.

`Parameters`	`(1 + P + Q)`- by-1 column vector of GARCH(P,Q) process parameters. The first element is the scalar constant term 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, i.e., the residuals or innovations (_t), 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.
`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.

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,

_t², of a GARCH(P,Q) process is modeled as

where

represents the argument Alpha, and

represents Beta.

U is a vector of residuals or innovations (

_t) representing a mean-zero, discrete time stochastic process. Although

_t² is generated via the equation above,

_t and

_t² are related as

where {v_t} is an independent, identically distributed (i.i.d.) 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: ugarchllf corresponds generally to the GARCH Toolbox function garchllfn. The GARCH Toolbox provides a comprehensive and integrated computing environment for the analysis of volatility in time series. For information see the GARCH Toolbox User's Guide or the financial products Web page at http://www.mathworks.com/products/finprod/.

See Also
ugarch, ugarchpred, ugarchsim, and the GARCH Toolbox function garchllfn

ugarch ugarchpred