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, 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`	Non-negative, 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 MATLAB 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, H(t), of a GARCH(P,Q) process is modeled as

H(t) = Kappa + Alpha(1)*H(t-1) + Alpha(2)*H(t-2) +...+ 
Alpha(P)*H(t-P)+ Beta(1)*U^2(t-1)+ Beta(2)*U^2(t-2)+...+ Beta(Q)*U^2(t-Q)

U is vector of innovations representing a mean-zero, discrete time stochastic process. Although H is generated via the equation above, U and H are related as

           U(t) = sqrt(H(t))*v(t)

where {v(t)} is an 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.