Main Content

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.

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.

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.

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 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`

.

[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.