Log-likelihood objective function of univariate GARCH(P,Q) processes with Gaussian innovations
LogLikelihood = ugarchllf(Parameters, U, P, Q)
(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.
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.
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.
Positive, scalar integer representing a model order of the GARCH process. Q is the number of lags of the squared innovations.
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, , of a GARCH(P,Q) process is modeled as
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 is generated via the equation above, ɛt and are related as
where 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/.