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## Specify the 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)" 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.