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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

**Note**

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.