## 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.