MATLAB Examples

Regression Model with AR Errors and t Innovations

This example shows how to set the innovation distribution of a regression model with AR errors to a $t$ distribution.

Specify the regression model with AR(4) errors:

$$\begin{array}{l}{y_t} = {X_t}\left[ \begin{array}{l} - 2\\0.5\end{array} \right] + {u_t}\\{u_t} = 0.2{u_{t - 1}} + 0.1{u_{t - 4}} + {\varepsilon _t},\end{array}$$

where $\varepsilon_t$ has a $t$ distribution with the default degrees of freedom and unit variance.

Mdl = regARIMA('AR',{0.2,0.1},'ARLags',[1,4],...
    'Constant',0,'Beta',[-2;0.5],'Variance',1,...
    'Distribution','t')
Mdl = 

    Regression with ARIMA(4,0,0) Error Model:
    ------------------------------------------
    Distribution: Name = 't', DoF = NaN
       Intercept: 0
            Beta: [-2 0.5]
               P: 4
               D: 0
               Q: 0
              AR: {0.2 0.1} at Lags [1 4]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: 1

The default degrees of freedom is NaN. If you don't know the degrees of freedom, then you can estimate it by passing Mdl and the data to estimate.

Specify a $t_{10}$ distribution.

Mdl.Distribution = struct('Name','t','DoF',10)
Mdl = 

    Regression with ARIMA(4,0,0) Error Model:
    ------------------------------------------
    Distribution: Name = 't', DoF = 10
       Intercept: 0
            Beta: [-2 0.5]
               P: 4
               D: 0
               Q: 0
              AR: {0.2 0.1} at Lags [1 4]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: 1

You can simulate or forecast responses using simulate or forecast because Mdl is completely specified.

In applications, such as simulation, the software normalizes the random $t$ innovations. In other words, Variance overrides the theoretical variance of the $t$ random variable (which is DoF/(DoF - 2)), but preserves the kurtosis of the distribution.