For regression models with ARIMA time series errors in Econometrics
Toolbox™, *ε _{t}* =

*ε*is the innovation corresponding to observation_{t}*t*.*σ*is the constant variance of the innovations. You can set its value using the`Variance`

property of a`regARIMA`

model.*z*is the innovation distribution. You can set the distribution using the_{t}`Distribution`

property of a`regARIMA`

model. Specify either a standard Gaussian (the default) or standardized Student’s*t*with*ν*> 2 or`NaN`

degrees of freedom.### Note

If

*ε*has a Student’s_{t}*t*distribution, then$${z}_{t}={T}_{\nu}\sqrt{\frac{\nu -2}{\nu}},$$

where

*T*is a Student’s_{ν}*t*random variable with*ν*> 2 degrees of freedom. Subsequently,*z*is_{t}*t*-distributed with mean 0 and variance 1, but has the same kurtosis as*T*. Therefore,_{ν}*ε*is_{t}*t*-distributed with mean 0, variance*σ*, and has the same kurtosis as*T*._{ν}

`estimate`

builds and optimizes the likelihood objective function based on *ε _{t}* by:

Estimating

*c*and*β*using MLRInferring the unconditional disturbances from the estimated regression model, $${\widehat{u}}_{t}={y}_{t}-\widehat{c}-{X}_{t}\widehat{\beta}$$

Estimating the ARIMA error model, $${\widehat{u}}_{t}={{\rm H}}^{-1}(L){\rm N}(L){\epsilon}_{t},$$ where

*H*(*L*) is the compound autoregressive polynomial and*N*(*L*) is the compound moving average polynomialInferring the innovations from the ARIMA error model, $${\widehat{\epsilon}}_{t}={\widehat{{\rm H}}}^{-1}(L)\widehat{{\rm N}}(L){\widehat{u}}_{t}$$

Maximizing the loglikelihood objective function with respect to the free parameters

If the unconditional disturbance process is nonstationary (i.e., the nonseasonal or seasonal integration degree is greater than 0), then the regression intercept, *c*, is not identifiable. `estimate`

returns a `NaN`

for *c* when it fits integrated models. For details, see Intercept Identifiability in Regression Models with ARIMA Errors.

`estimate`

estimates all parameters in the `regARIMA`

model set to `NaN`

. `estimate`

honors any equality constraints in the `regARIMA`

model, i.e., `estimate`

fixes the parameters at the values that you set during estimation.

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

$$f({\epsilon}_{1},\mathrm{...},{\epsilon}_{T}\text{|}{H}_{T-1})={\displaystyle \prod}_{t=1}^{T}f({\epsilon}_{t}{\text{|H}}_{t-1}),$$

where *f* is the standard Gaussian or *t* probability density function.

The exact form of the loglikelihood objective function depends on the parametric form of the innovation distribution.

If

*z*is standard Gaussian, then the loglikelihood objective function is_{t}$$logL=-\frac{T}{2}\mathrm{log}(2\pi )-\frac{T}{2}\mathrm{log}{\sigma}^{2}-\frac{1}{2{\sigma}^{2}}{\displaystyle \sum _{t=1}^{T}{\epsilon}_{t}^{2}}.$$

If

*z*is a standardized Student’s_{t}*t*, then the loglikelihood objective function is$$logL=T\mathrm{log}\left[\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu}{2}\right)\sqrt{\pi (\nu -2)}}\right]-\frac{T}{2}{\sigma}^{2}-\frac{\nu +1}{2}{\displaystyle \sum _{t=1}^{T}l}og\left[1+\frac{{\epsilon}_{t}^{2}}{{\sigma}^{2}(\nu -2)}\right].$$

`estimate`

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