Maximum Likelihood Estimation for Conditional Mean Models

Innovation Distribution

For conditional mean models in Econometrics Toolbox™, the form of the innovation process is$${\epsilon }_t={\sigma }_t{z}_t,$$ where z_t can be standardized Gaussian or Student’s t with $$\nu >2$$ degrees of freedom. Specify your distribution choice in the arima model object Distribution property.

The innovation variance, $${\sigma }_t^2,$$ can be a positive scalar constant, or characterized by a conditional variance model. Specify the form of the conditional variance using the Variance property. 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,

$$y_t=\mu +\psi (L){\epsilon }_t,$$

applying an inverse filter yields a solution for the innovation $${\epsilon }_t$$

$${\epsilon }_t={\psi }^{-1}(L)(y_t-\mu ).$$

For example, for an AR(p) process,

$${\epsilon }_t=-c+\varphi (L)y_t,$$

where $$\varphi (L)=(1-{\varphi }_{1}L-\cdots -{\varphi }_p{L}^p)$$ is the degree p AR operator polynomial.

estimate uses maximum likelihood to estimate the parameters of an arima model. estimate returns fitted values for any parameters in the input model object equal to NaN. estimate honors any equality constraints in the input model object, and does not return estimates for parameters with equality constraints.

Loglikelihood Functions

Given the history of a process, innovations are conditionally independent. Let H_t denote the history of a process available at time t, t = 1,...,N. The likelihood function for the innovation series is given by

$$f({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_N|H_{N-1})={\displaystyle \prod _{t=1}^{N}f(}{\epsilon }_t|H_{t-1}),$$

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 z_t has a standard Gaussian distribution, then the loglikelihood function is

  $$LLF=-\frac{N}{2}\log (2\pi )-\frac{1}{2}{\displaystyle \sum _{t=1}^N\log {\sigma }_t^2-}\frac{1}{2}{\displaystyle \sum _{t=1}^N\frac{{\epsilon }_t^2}{{\sigma }_t^2}}.$$

• If z_t has a standardized Student’s t distribution with $$\nu >2$$ degrees of freedom, then the loglikelihood function is

  $$LLF=N\log \left[\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sqrt{\pi (\nu -2)}\Gamma \left(\frac{\nu }{2}\right)}\right]-\frac{1}{2}{\displaystyle \sum _{t=1}^N\log {\sigma }_t^2}-\frac{\nu +1}{2}{\displaystyle \sum _{t=1}^N\log \left[1+\frac{{\epsilon }_t^2}{{\sigma }_t^2(\nu -2)}\right]}.$$

estimate performs covariance matrix estimation for maximum likelihood estimates using the outer product of gradients (OPG) method.