Specify Default Regression Model with ARIMA Errors

This example shows how to specify the default regression model with ARIMA errors using the shorthand ARIMA($$p$$, $$D$$, $$q$$) notation corresponding to the following equation:

$$\begin{array}{c}{y}_t=c+u_t\\ \left(1-{\varphi }_{1}L-{\varphi }_2{L}^2-{\varphi }_3{L}^3\right){\left(1-L\right)}^D{u}_t=\left(1+{\theta }_{1}L+{\theta }_2{L}^2\right){\epsilon }_t.\end{array}$$

Specify a regression model with ARIMA(3,1,2) errors.

Mdl = regARIMA(3,1,2)

Mdl = 
  regARIMA with properties:

     Description: "ARIMA(3,1,2) Error Model (Gaussian Distribution)"
      SeriesName: "Y"
    Distribution: Name = "Gaussian"
       Intercept: NaN
            Beta: [1×0]
               P: 4
               D: 1
               Q: 2
              AR: {NaN NaN NaN} at lags [1 2 3]
             SAR: {}
              MA: {NaN NaN} at lags [1 2]
             SMA: {}
        Variance: NaN

The model specification for Mdl appears in the Command Window. By default, regARIMA sets:

• The autoregressive (AR) parameter values to NaN at lags [1 2 3]

• The moving average (MA) parameter values to NaN at lags [1 2]

• The variance (Variance) of the innovation process, $${\epsilon }_t$$, to NaN

• The distribution (Distribution) of $${\epsilon }_t$$ to Gaussian

• The regression model intercept to NaN

There is no regression component (Beta) by default.

The property: 

• P = p + D, which represents the number of presample observations that the software requires to initialize the autoregressive component of the model to perform, for example, estimation.

• D represents the level of nonseasonal integration.

• Q represents the number of presample observations that the software requires to initialize the moving average component of the model to perform, for example, estimation.

Fit Mdl to data by passing it and the data into estimate. If you pass the predictor series into estimate, then estimate estimates Beta by default.

You can modify the properties of Mdl using dot notation.

References:

Box, G. E. P., G. M. Jenkins, and G. C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.