Estimate Multiplicative ARIMA Model Using Econometric Modeler App

This example uses the Box-Jenkins method [1] to determine a multiplicative seasonal ARIMA (SARIMA) model for a univariate series by using the Econometric Modeler app. Specifically, the example shows this procedure:

Visually determine whether the series has an exponential trend, and then remove the trend if one is present.
The remaining steps determine the structure of the conditional mean model to fit to the series without the exponential trend.
Visually determine whether the transformed series has seasonal integration, and then remove the seasonal integration, if it is present, by taking the seasonal difference. Record the degrees of seasonal integration removed.
Determine whether the transformed series has a unit root by conducting a statistical test, and then apply the first difference to the series to stabilize it, if a unit root is present. Record whether a unit root exists.
Determine the nonseasonal AR and MA degrees of the conditional mean model for the stationary series by inspecting correlograms of the stabilized series. Record the polynomial degrees.
Use the results of steps 2 through 4 to create and estimate a conditional mean model for the nonstationary series with the exponential trend removed.

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 DataTimeTable into the app:

On the Econometric Modeler tab, in the Import section, click the Import button .
In the Import Data dialog box, in the Import? column, select the check box for the DataTimeTable 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.

This screen shot shows a time series plot of the variable PSSG where the x axis shows a time period from the late 1940's through the early 1960's.

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

To properly determine the lag operator polynomial orders for the conditional mean model, correlograms require a stationary 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.

This screen shot shows a time series plot of the variable PSSGLog where the x axis shows a time period from the late 1940's through the early 1960's.

The exponential growth appears to be removed from the series.

The rest of the example determines the parameters of conditional mean model for the series PSSGLog by several visual and statistical diagnoses and transformations.

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.

This screen shot shows a time series plot of the variable PSSGLogSeasonalDiff where the x axis shows a time period from the late 1940's through the early 1960's, but the line for PSSGLogSeasonalDiff starts at 1950.

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.

A Results table showing "Augmented Dickey-Fuller Test (PSSGLogSeasonalDiff Log); Null Hypothesis: PSSGLogSeasonalDiff Log contains a unit root". The table shows columns entitled select, null rejected, P-value, test statistic, Critical Value, Lags, model, test statistic, and Significance Level. There are two rows below the headings.

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.

This screen shot shows a time series plot of the variable PSSGStable where the x axis shows a time period from the late 1940's through the early 1960's, but the line for PSSGStable starts at 1950.

Identify Model for Series

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

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.

This set of time series plots compare the differences between the Sample Autocorrelation Function of PSSGStable in the ACF tab and the Sample Partial Autocorrelation Function of PSSGStable in the PACF tab. Lag is shown on the x axis and blue horizontal lines indicate confidence bounds.

According to [1], the autocorrelations in the ACF and PACF, the results in previous steps, and the balance of model parsimony with complexity, suggest that the following SARIMA(0,1,1)×(0,1,1)₁₂ model is appropriate for the nonstationary series with the exponential trend removed PSSGLog.

$(1 - L) (1 - L^{12}) y_{t} = (1 + θ_{1} L) (1 + Θ_{12} L^{12}) ε_{t} .$

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 section
  1. Set Degrees of Integration to 1.
  2. Set Moving Average Order to 1.
  3. Clear the Include Constant Term check box.
- Seasonal section
  1. Set Period to 12 to indicate monthly data.
  2. Set Moving Average Order to 1.
  3. 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.

Model Summary (SARIMA_PSSGLog) tab shows time series plots of Model Fit for PSSGLog and SARIMA_PSSGLog and Residual Plot for SARIMA_PSSGLog. To the right are two tables, one for Parameters on top and one for Goodness of Fit below.

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.