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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

`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.

Given the history of a process, innovations are conditionally
independent. Let *H _{t}* denote
the history of a process available at time

$$f({\epsilon}_{1},{\epsilon}_{2},\dots ,{\epsilon}_{N}|{H}_{N-1})={\displaystyle \prod _{t=1}^{N}f(}{\epsilon}_{t}|{H}_{t-1}),$$

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*has a standard Gaussian distribution, then the loglikelihood function is_{t}$$LLF=-\frac{N}{2}\mathrm{log}(2\pi )-\frac{1}{2}{\displaystyle \sum _{t=1}^{N}\mathrm{log}{\sigma}_{t}^{2}-}\frac{1}{2}{\displaystyle \sum _{t=1}^{N}\frac{{\epsilon}_{t}^{2}}{{\sigma}_{t}^{2}}}.$$

If

*z*has a standardized Student’s_{t}*t*distribution with $$\nu >2$$ degrees of freedom, then the loglikelihood function is$$LLF=N\mathrm{log}\left[\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sqrt{\pi (\nu -2)}\Gamma \left(\frac{\nu}{2}\right)}\right]-\frac{1}{2}{\displaystyle \sum _{t=1}^{N}\mathrm{log}{\sigma}_{t}^{2}}-\frac{\nu +1}{2}{\displaystyle \sum _{t=1}^{N}\mathrm{log}\left[1+\frac{{\epsilon}_{t}^{2}}{{\sigma}_{t}^{2}(\nu -2)}\right]}.$$

`estimate`

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