## Estimate Multiplicative ARIMA Model Using Econometric Modeler App

This example shows how to estimate a multiplicative seasonal ARIMA model by using the Econometric Modeler app. The data set `Data_Airline.mat`

contains monthly counts of airline passengers.

### Import Data into Econometric Modeler

At the command line, load the `Data_Airline.mat`

data set.

`load Data_Airline`

At the command line, open the **Econometric Modeler** app.

econometricModeler

Alternatively, open the app from the apps gallery (see **Econometric
Modeler**).

Import `DataTable`

into the app:

On the

**Econometric Modeler**tab, in the**Import**section, click .In the

**Import Data**dialog box, in the**Import?**column, select the check box for the`DataTable`

variable.Click

**Import**.

The variable `PSSG`

appears in the **Time Series** pane, its value appears in the **Preview** pane, and its time series plot appears in the **Time Series Plot(PSSG)** figure window.

The series exhibits a seasonal trend, serial correlation, and possible exponential growth. For an interactive analysis of serial correlation, see Detect Serial Correlation Using Econometric Modeler App.

### Stabilize Series

Address the exponential trend by applying the log transform to `PSSG`

.

In the

**Time Series**pane, select`PSSG`

.On the

**Econometric Modeler**tab, in the**Transforms**section, click**Log**.

The transformed variable `PSSGLog`

appears in the **Time Series** pane, its value appears in the **Preview** pane, and its time series plot appears in the **Time Series Plot(PSSGLog)** figure window.

The exponential growth appears to be removed from the series.

Address the seasonal trend by applying the 12th order seasonal difference. With `PSSGLog`

selected in the **Time Series** pane, on the **Econometric Modeler** tab, in the **Transforms** section, set **Seasonal** to `12`

. Then, click **Seasonal**.

The transformed variable `PSSGLogSeasonalDiff`

appears in the **Time Series** pane, and its time series plot appears in the **Time Series Plot(PSSGLogSeasonalDiff)** figure window.

The transformed series appears to have a unit root.

Test the null hypothesis that `PSSGLogSeasonalDiff`

has a unit root by using the Augmented Dickey-Fuller test. Specify that the alternative is an AR(0) model, then test again specifying an AR(1) model. Adjust the significance level to 0.025 to maintain a total significance level of 0.05.

With

`PSSGLogSeasonalDiff`

selected in the**Time Series**pane, on the**Econometric Modeler**tab, in the**Tests**section, click**New Test**>**Augmented Dickey-Fuller Test**.On the

**ADF**tab, in the**Parameters**section, set**Significance Level**to`0.025`

.In the

**Tests**section, click**Run Test**.In the

**Parameters**section, set**Number of Lags**to`1`

.In the

**Tests**section, click**Run Test**.

The test results appear in the **Results** table of the **ADF(PSSGLogSeasonalDiff)** document.

Both tests fail to reject the null hypothesis that the series is a unit root process.

Address the unit root by applying the first difference to `PSSGLogSeasonalDiff`

. With `PSSGLogSeasonalDiff`

selected in the **Time Series** pane, click the **Econometric Modeler** tab. Then, in the **Transforms** section, click **Difference**.

The transformed variable `PSSGLogSeasonalDiffDiff`

appears in the **Time Series** pane, and its time series plot appears in the **Time Series Plot(PSSGLogSeasonalDiffDiff)** figure window.

In the **Time Series** pane, rename the `PSSGLogSeasonalDiffDiff`

variable by clicking it twice to select its name and entering `PSSGStable`

.

The app updates the names of all documents associated with the transformed series.

### Identify Model for Series

Determine the lag structure for a conditional mean model of the data by plotting the sample autocorrelation function (ACF) and partial autocorrelation function (PACF).

With

`PSSGStable`

selected in the**Time Series**pane, click the**Plots**tab, then click**ACF**.Show the first 50 lags of the ACF. On the

**ACF**tab, set**Number of Lags**to`50`

.Click the

**Plots**tab, then click**PACF**.Show the first 50 lags of the PACF. On the

**PACF**tab, set**Number of Lags**to`50`

.Drag the

**ACF(PSSGStable)**figure window above the**PACF(PSSGStable)**figure window.

According to [1], the autocorrelations in the ACF and PACF suggest that the following SARIMA(0,1,1)×(0,1,1)_{12} model is appropriate for PSSGLog.

$$(1-L)\left(1-{L}^{12}\right){y}_{t}=\left(1+{\theta}_{1}L\right)\left(1+{\Theta}_{12}{L}^{12}\right){\epsilon}_{t}.$$

Close all figure windows.

### Specify and Estimate SARIMA Model

Specify the SARIMA(0,1,1)×(0,1,1)_{12} model.

In the

**Time Series**pane, select the`PSSGLog`

time series.On the

**Econometric Modeler**tab, in the**Models**section, click the arrow >**SARIMA**.In the

**SARIMA Model Parameters**dialog box, on the**Lag Order**tab:**Nonseasonal**sectionSet

**Degrees of Integration**to`1`

.Set

**Moving Average Order**to`1`

.Clear the

**Include Constant Term**check box.

**Seasonal**sectionSet

**Period**to`12`

to indicate monthly data.Set

**Moving Average Order**to`1`

.Select the

**Include Seasonal Difference**check box.

Click

**Estimate**.

The model variable `SARIMA_PSSGLog`

appears in the **Models** pane, its value appears in the **Preview** pane, and its estimation summary appears in the **Model Summary(SARIMA_PSSGLog)** document.

The results include:

**Model Fit**— A time series plot of`PSSGLog`

and the fitted values from`SARIMA_PSSGLog`

.**Residual Plot**— A time series plot of the residuals of`SARIMA_PSSGLog`

.**Parameters**— A table of estimated parameters of`SARIMA_PSSGLog`

. Because the constant term was held fixed to 0 during estimation, its value and standard error are 0.**Goodness of Fit**— The AIC and BIC fit statistics of`SARIMA_PSSGLog`

.

## References

[1] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. *Time Series Analysis: Forecasting and Control*. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.