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Log-likelihood objective function of univariate GARCH(P,Q) processes with Gaussian innovations
LogLikelihood = ugarchllf(Parameters, U, P, Q)
Parameters | (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. |
U | Single column vector of random disturbances, that is, the residuals or innovations (ɛ_{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. |
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, $${\sigma}_{t}^{2}$$, of a GARCH(P,Q) process is modeled as
$${\sigma}_{t}^{2}=K+{\displaystyle \sum _{i=1}^{P}{\alpha}_{i}{\sigma}_{t-i}^{2}}+{\displaystyle \sum _{j=1}^{Q}{\beta}_{j}{\epsilon}_{t-j}^{2}},$$
where α represents the argument Alpha, and β represents Beta.
U is a vector of residuals or innovations (ɛ_{t}) representing a mean-zero, 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 http://www.mathworks.com/products/finprod/. |