Loglikelihood objective function of univariate GARCH(P,Q) processes with Gaussian innovations
LogLikelihood = ugarchllf(Parameters, U, P, Q)

 
 Single column vector of random disturbances, that is,
the residuals or innovations (ɛ_{t}), of
an econometric model representing a meanzero, discretetime stochastic
process. The innovations time series
 
 Nonnegative, scalar integer representing a model order
of the GARCH process.  
 Positive, scalar integer representing a model order of
the GARCH process. 
LogLikelihood = ugarchllf(Parameters, U, P, Q)
computes the loglikelihood objective function of univariate GARCH(P,Q)
processes with Gaussian innovations.
LogLikelihood
is a scalar value of the GARCH(P,Q)
loglikelihood 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 loglikelihood function, the LogLikelihood
output
parameter is actually the negative of what is formally presented in
most time series or econometrics references.
The timeconditional variance, $${\sigma}_{t}^{2}$$, of a GARCH(P,Q) process is modeled as
$${\sigma}_{t}^{2}=K+{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}{\sigma}_{ti}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}{\epsilon}_{tj}^{2}},$$
where α represents the argument Alpha
,
and β represents Beta
.
U
is a vector of residuals or innovations
(ɛ_{t})
representing a meanzero, discrete time stochastic process. Although $${\sigma}_{t}^{2}$$ is generated via the equation
above, ɛ_{t} 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 