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## Maximum Likelihood Estimation of regARIMA Models

### Innovation Distribution

For regression models with ARIMA time series errors in Econometrics Toolbox™, εt = σzt, where:

• εt is the innovation corresponding to observation t.

• σ is the constant variance of the innovations. You can set its value using the `Variance` property of a `regARIMA` model.

• zt is the innovation distribution. You can set the distribution using the `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 εt has a Student’s t distribution, then

`${z}_{t}={T}_{\nu }\sqrt{\frac{\nu -2}{\nu }},$`

where Tν is a Student’s t random variable with ν > 2 degrees of freedom. Subsequently, zt is t-distributed with mean 0 and variance 1, but has the same kurtosis as Tν. Therefore, εt is t-distributed with mean 0, variance σ, and has the same kurtosis as Tν.

`estimate` builds and optimizes the likelihood objective function based on εt by:

1. Estimating c and β using MLR

2. Inferring the unconditional disturbances from the estimated regression model, ${\stackrel{^}{u}}_{t}={y}_{t}-\stackrel{^}{c}-{X}_{t}\stackrel{^}{\beta }$

3. Estimating the ARIMA error model, ${\stackrel{^}{u}}_{t}={Η}^{-1}\left(L\right)Ν\left(L\right){\epsilon }_{t},$ where H(L) is the compound autoregressive polynomial and N(L) is the compound moving average polynomial

4. Inferring the innovations from the ARIMA error model, ${\stackrel{^}{\epsilon }}_{t}={\stackrel{^}{Η}}^{-1}\left(L\right)\stackrel{^}{Ν}\left(L\right){\stackrel{^}{u}}_{t}$

5. 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.

### Loglikelihood Functions

Given its history, the innovations are conditionally independent. Let Ht denote the history of the process available at time t, where t = 1,...,T. The likelihood function of the innovations is

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

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 zt is standard Gaussian, then the loglikelihood objective function is

`$logL=-\frac{T}{2}\mathrm{log}\left(2\pi \right)-\frac{T}{2}\mathrm{log}{\sigma }^{2}-\frac{1}{2{\sigma }^{2}}\sum _{t=1}^{T}{\epsilon }_{t}^{2}.$`

• If zt is a standardized Student’s t, then the loglikelihood objective function is

`$logL=T\mathrm{log}\left[\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\Gamma \left(\frac{\nu }{2}\right)\sqrt{\pi \left(\nu -2\right)}}\right]-\frac{T}{2}{\sigma }^{2}-\frac{\nu +1}{2}\sum _{t=1}^{T}log\left[1+\frac{{\epsilon }_{t}^{2}}{{\sigma }^{2}\left(\nu -2\right)}\right].$`

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