ugarchllf (Financial Toolbox)

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