For conditional mean models in Econometrics Toolbox™, the
form of the innovation process is where zt can
be standardized Gaussian or Student's t with degrees of freedom. Specify
your distribution choice in the
arima model object
The innovation variance, can
be a positive scalar constant, or characterized by a conditional variance
model. Specify the form of the conditional variance using the
If you specify a conditional variance model, the parameters of that
model are estimated with the conditional mean model parameters simultaneously.
Given a stationary model,
applying an inverse filter yields a solution for the innovation
For example, for an AR(p) process,
where is the degree p AR operator polynomial.
estimate uses maximum likelihood to estimate
the parameters of an
fitted values for any parameters in the input model object equal to
any equality constraints in the input model object, and does not return
estimates for parameters with equality constraints.
Given the history of a process, innovations are conditionally independent. Let Ht denote the history of a process available at time t, t = 1,...,N. The likelihood function for the innovation series is given by
where f is a standardized Gaussian or t density function.
The exact form of the loglikelihood objective function depends on the parametric form of the innovation distribution.
If zt has a standard Gaussian distribution, then the loglikelihood function is
If zt has a standardized Student's t distribution with degrees of freedom, then the loglikelihood function is
estimate performs covariance matrix estimation for
maximum likelihood estimates using the outer product of gradients