GARCH Toolbox™

Maximum Likelihood Estimation

This section explains how the garchfit estimation engine uses maximum likelihood to estimate the parameters needed to fit the specified models to a given univariate return series.

Given an observed univariate time series and the conditional mean and variance models described in Conditional Mean and Variance Models, garchfit does the following:

• Infers the innovations (residuals) from the return series.

• Estimates, by maximum likelihood, the parameters needed to fit the specified models to the return series.

Given a vector of initial parameter estimates, as described in Initial Parameter Estimates, the garchfit function calls the Optimization Toolbox™ fmincon function to perform constrained optimization of a scalar function of several variables; that is, the log-likelihood function. This technique is called constrained nonlinear optimization or nonlinear programming. In turn, fmincon calls the appropriate log-likelihood objective function to estimate the model parameters using maximum likelihood estimation (MLE).

The chosen log-likelihood objective function proceeds as follows:

• Given the vector of current parameter values and the observed data Series, the log-likelihood function infers the process innovations (residuals) by inverse filtering. This inference operation rearranges the conditional mean equation to solve for the current innovation ε_t:

  

  This equation is a whitening filter, transforming a (possibly) correlated process y_t into an uncorrelated white noise process ε_t.

• The log-likelihood function then uses the inferred innovations ε_t to infer the corresponding conditional variances via recursive substitution into the previous model-dependent conditional variance equations Equation 2-4, Equation 2-5, and Equation 2-6.

• Finally, the function uses the inferred innovations and conditional variances to evaluate the appropriate log-likelihood objective function. If ε_t is Gaussian, the log-likelihood function is

  (6-1)

  If ε_t is Student's t, the log-likelihood function is

  (6-2)

  where T is the sample size, that is, the number of rows in the series {y_t}. The degrees of freedom ν must be greater than 2.

The conditional mean equation, Equation 2-2, and the conditional variance equations, Equation 2-4, Equation 2-5, and Equation 2-6, are recursive, and generally require presample observations to initiate inverse filtering. For this reason, the objective functions shown here are referred to as conditional log-likelihood functions. Evaluation of the log-likelihood function is conditioned, or based, on a set of presample observations. For more information about the methods used to specify these presample observations, see Presample Observations.

The iterative numerical optimization repeats the previous three steps until it satisfies suitable termination criteria. For more information, see Termination Criteria and Optimization Results .

Estimation

Initial Parameter Estimates