Create Regression Models with AR Errors

These examples show how to create regression models with AR errors using regARIMA. For details on specifying regression models with AR errors using the Econometric Modeler app, see Specify Regression Model with ARMA Errors Using Econometric Modeler App.

Default Regression Model with AR Errors

This example shows how to apply the shorthand regARIMA(p,D,q) syntax to specify a regression model with AR errors.

Specify the default regression model with AR(3) errors:

$$\begin{array}{l}{y}_t=c+X_t\beta +u_t\\ u_t=a_1{u}_{t-1}+a_2{u}_{t-2}+a_3{u}_{t-3}+{\epsilon }_t.\end{array}$$

Mdl = regARIMA(3,0,0)

Mdl = 
  regARIMA with properties:

     Description: "ARMA(3,0) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 3
               Q: 0
              AR: {NaN NaN NaN} at lags [1 2 3]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: NaN

The software sets the innovation distribution to Gaussian, and each parameter to NaN. The AR coefficients are at lags 1 through 3.

Pass Mdl into estimate with data to estimate the parameters set to NaN. Though Beta is not in the display, if you pass a matrix of predictors ($$X_t$$) into estimate, then estimate estimates Beta. The estimate function infers the number of regression coefficients in Beta from the number of columns in $$X_t$$.

Tasks such as simulation and forecasting using simulate and forecast do not accept models with at least one NaN for a parameter value. Use dot notation to modify parameter values.

AR Error Model Without an Intercept

This example shows how to specify a regression model with AR errors without a regression intercept.

Specify the default regression model with AR(3) errors:

$$\begin{array}{l}{y}_t=X_t\beta +u_t\\ u_t=a_1{u}_{t-1}+a_2{u}_{t-2}+a_3{u}_{t-3}+{\epsilon }_t.\end{array}$$

 Mdl = regARIMA('ARLags',1:3,'Intercept',0)

Mdl = 
  regARIMA with properties:

     Description: "ARMA(3,0) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: 0
            Beta: [1×0]
               P: 3
               Q: 0
              AR: {NaN NaN NaN} at lags [1 2 3]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: NaN

The software sets Intercept to 0, but all other estimable parameters in Mdl are NaN values by default.

Since Intercept is not a NaN, it is an equality constraint during estimation. In other words, if you pass Mdl and data into estimate, then estimate sets Intercept to 0 during estimation.

You can modify the properties of Mdl using dot notation.

AR Error Model with Nonconsecutive Lags

This example shows how to specify a regression model with AR errors, where the nonzero AR terms are at nonconsecutive lags.

Specify the regression model with AR(4) errors:

$$\begin{array}{l}{y}_t=c+X_t\beta +u_t\\ u_t=a_1{u}_{t-1}+a_4{u}_{t-4}+{\epsilon }_t.\end{array}$$

Mdl = regARIMA('ARLags',[1,4])

Mdl = 
  regARIMA with properties:

     Description: "ARMA(4,0) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 4
               Q: 0
              AR: {NaN NaN} at lags [1 4]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: NaN

The AR coefficients are at lags 1 and 4.

Verify that the AR coefficients at lags 2 and 3 are 0.

Mdl.AR

ans=1×4 cell array
    {[NaN]}    {[0]}    {[0]}    {[NaN]}

The software displays a 1-by-4 cell array. Each consecutive cell contains the corresponding AR coefficient value.

Pass Mdl and data into estimate. The software estimates all parameters that have the value NaN. Then, estimate holds $$a_2$$ = 0 and $$a_3$$ = 0 during estimation.

Known Parameter Values for a Regression Model with AR Errors

This example shows how to specify values for all parameters of a regression model with AR errors.

Specify the regression model with AR(4) errors:

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

where $${\epsilon }_t$$ is Gaussian with unit variance.

Mdl = regARIMA('AR',{0.2,0.1},'ARLags',[1,4], ...
    'Intercept',0,'Beta',[-2;0.5],'Variance',1)

Mdl = 
  regARIMA with properties:

     Description: "Regression with ARMA(4,0) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: 0
            Beta: [-2 0.5]
               P: 4
               Q: 0
              AR: {0.2 0.1} at lags [1 4]
             SAR: {}
              MA: {}
             SMA: {}
        Variance: 1

There are no NaN values in any Mdl properties, and therefore there is no need to estimate Mdl using estimate. However, you can simulate or forecast responses from Mdl using simulate or forecast.

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:

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

where $${\epsilon }_t$$ has a $$t$$ distribution with the default degrees of freedom and unit variance.

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

Mdl = 
  regARIMA with properties:

     Description: "Regression with ARMA(4,0) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = NaN
       Intercept: 0
            Beta: [-2 0.5]
               P: 4
               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 = 
  regARIMA with properties:

     Description: "Regression with ARMA(4,0) Error Model (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = 10
       Intercept: 0
            Beta: [-2 0.5]
               P: 4
               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.