Create Regression Models with SARIMA Errors

SARMA Error Model Without an Intercept

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

Specify the default regression model with $$SARMA(1,1)\times (2,1,1{)}_4$$ errors:

$$\begin{array}{c}{y}_t=X_t\beta +u_t\\ \left(1-a_{1}L\right)\left(1-A_4{L}^4-A_8{L}^8\right)\left(1-L^4\right)u_t=\left(1+b_{1}L\right)\left(1+B_4{L}^4\right){\epsilon }_t.\end{array}$$

Mdl = regARIMA('ARLags',1,'SARLags',[4, 8],...
    'Seasonality',4,'MALags',1,'SMALags',4,'Intercept',0)

Mdl = 
  regARIMA with properties:

     Description: "ARMA(1,1) Error Model Seasonally Integrated with Seasonal AR(8) and MA(4) (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: 0
            Beta: [1×0]
               P: 13
               Q: 5
              AR: {NaN} at lag [1]
             SAR: {NaN NaN} at lags [4 8]
              MA: {NaN} at lag [1]
             SMA: {NaN} at lag [4]
     Seasonality: 4
        Variance: NaN

The name-value pair argument:

• 'ARLags',1 specifies which lags have nonzero coefficients in the nonseasonal autoregressive polynomial, so $$a(L)=(1-a_{1}L)$$.

• 'SARLags',[4 8] specifies which lags have nonzero coefficients in the seasonal autoregressive polynomial, so $$A(L)=(1-A_4{L}^4-A_8{L}^8)$$.

• 'MALags',1 specifies which lags have nonzero coefficients in the nonseasonal moving average polynomial, so $$b(L)=(1+b_{1}L)$$.

• 'SMALags',4 specifies which lags have nonzero coefficients in the seasonal moving average polynomial, so $$B(L)=(1+B_4{L}^4)$$.

• 'Seasonality',4 specifies the degree of seasonal integration and corresponds to $$(1-L^4)$$.

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

Property P = p + D + $$p_s$$ + s = 1 + 0 + 8 + 4 = 13, and property Q = q + $$q_s$$ = 1 + 4 = 5. Therefore, the software requires at least 13 presample observation to initialize Mdl.

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.

Be aware that the regression model intercept (Intercept) is not identifiable in regression models with ARIMA errors. If you want to estimate Mdl, then you must set Intercept to a value using, for example, dot notation. Otherwise, estimate might return a spurious estimate of Intercept.

Known Parameter Values for a Regression Model with SARIMA Errors

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

Specify the regression model with $$SARIMA(1,1,1)\times (2,1,0{)}_{12}$$ errors:

$$\begin{array}{c}{y}_t=X_t\beta +u_t\\ \left(1-0.2L\right)\left(1-L\right)\left(1-0.25L^{12}-0.1L^{24}\right)\left(1-L^{12}\right)u_t=\left(1+0.15L\right){\epsilon }_t,\end{array}$$

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

Mdl = regARIMA('AR',0.2,'SAR',{0.25, 0.1},'SARLags',[12 24],...
    'D',1,'Seasonality',12,'MA',0.15,'Intercept',0,'Variance',1)

Mdl = 
  regARIMA with properties:

     Description: "ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(24) (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: 0
            Beta: [1×0]
               P: 38
               D: 1
               Q: 1
              AR: {0.2} at lag [1]
             SAR: {0.25 0.1} at lags [12 24]
              MA: {0.15} at lag [1]
             SMA: {}
     Seasonality: 12
        Variance: 1

The parameters in Mdl do not contain NaN values, and therefore there is no need to estimate Mdl. However, you can simulate or forecast responses by passing Mdl to simulate or forecast.

Regression Model with SARIMA Errors and t Innovations

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

Specify the regression model with $$SARIMA(1,1,1)\times (2,1,0{)}_{12}$$ errors:

$$\begin{array}{c}{y}_t=X_t\beta +u_t\\ \left(1-0.2L\right)\left(1-L\right)\left(1-0.25L^{12}-0.1L^{24}\right)\left(1-L^{12}\right)u_t=\left(1+0.15L\right){\epsilon }_t,\end{array}$$

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

Mdl = regARIMA('AR',0.2,'SAR',{0.25, 0.1},'SARLags',[12 24],...
    'D',1,'Seasonality',12,'MA',0.15,'Intercept',0,...
		'Variance',1,'Distribution','t')

Mdl = 
  regARIMA with properties:

     Description: "ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(24) (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = NaN
       Intercept: 0
            Beta: [1×0]
               P: 38
               D: 1
               Q: 1
              AR: {0.2} at lag [1]
             SAR: {0.25 0.1} at lags [12 24]
              MA: {0.15} at lag [1]
             SMA: {}
     Seasonality: 12
        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: "ARIMA(1,1,1) Error Model Seasonally Integrated with Seasonal AR(24) (t Distribution)"
      SeriesName: "Y"
    Distribution: Name = "t", DoF = 10
       Intercept: 0
            Beta: [1×0]
               P: 38
               D: 1
               Q: 1
              AR: {0.2} at lag [1]
             SAR: {0.25 0.1} at lags [12 24]
              MA: {0.15} at lag [1]
             SMA: {}
     Seasonality: 12
        Variance: 1

You can simulate or forecast responses by passing Mdl to 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.