This example shows how to use the Box-Jenkins methodology to select and estimate an ARIMA model by using the Econometric Modeler app. Then, it shows how to export the estimated model to generate forecasts. The data set, which is stored in `Data_JAustralian.mat`

, contains the log quarterly Australian Consumer Price Index (CPI) measured from 1972 and 1991, among other time series.

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

data set.

`load Data_JAustralian`

Convert the table `DataTable`

to a timetable:

Clear the row names of

`DataTable`

.Convert the sampling times to a

`datetime`

vector.Convert the table to a timetable by associating the rows with the sampling times in

`dates`

.

DataTable.Properties.RowNames = {}; dates = datetime(dates,'ConvertFrom','datenum',... 'Format','ddMMMyyyy','Locale','en_US'); DataTable = table2timetable(DataTable,'RowTimes',dates);

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 variables, including `PAU`

, appear in the **Data Browser**, and a time series plot of all the series appears in the **Time Series Plot(EXCH)** figure window.

Create a time series plot of `PAU`

by double-clicking `PAU`

in the **Data Browser**.

The series appears nonstationary because it has a clear upward trend.

Plot the sample autocorrelation function (ACF) and partial autocorrelation function (PACF).

In the

**Data Browser**, select the`PAU`

time series.Click the

**Plots**tab, then click**ACF**.Click the

**Plots**tab, then click**PACF**.Close all figure windows except for the correlograms. Then, drag the

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

The significant, linearly decaying sample ACF indicates a nonstationary process.

Close the **ACF(PAU)** and **PACF(PAU)** figure windows.

Take a first difference of the data. With `PAU`

selected in the **Data Browser**, on the **Econometric Modeler** tab, in the **Transforms** section, click **Difference**.

The transformed variable `PAUDiff`

appears in the **Data Browser**, and its time series plot appears in the **Time Series Plot(PAUDiff)** figure window.

Differencing removes the linear trend. The differenced series appears more stationary.

Plot the sample ACF and PACF of `PAUDiff`

. With `PAUDiff`

selected in the **Data Browser**:

Click the

**Plots**tab, then click**ACF**.Click the

**Plots**tab, then click**PACF**.Close the

**Time Series Plot(PAUDiff)**figure window. Then, drag the**ACF(PAUDiff)**figure window above the**PACF(PAUDiff)**figure window.

The sample ACF of the differenced series decays more quickly. The sample PACF cuts off after lag 2. This behavior is consistent with a second-degree autoregressive (AR(2)) model for the differenced series.

Close the **ACF(PAUDiff)** and **PACF(PAUDiff)** figure windows.

Estimate an ARIMA(2,1,0) model for the log quarterly Australian CPI. This model has one degree of nonseasonal differencing and two AR lags.

In the

**Data Browser**, select the`PAU`

time series.On the

**Econometric Modeler**tab, in the**Models**section, click**ARIMA**.In the

**ARIMA Model Parameters**dialog box, on the**Lag Order**tab:Set

**Degree of Integration**to`1`

.Set

**Autoregressive Order**to`2`

.

Click

**Estimate**.

The model variable `ARIMA_PAU`

appears in the **Models** section of the **Data Browser**, and its estimation summary appears in the **Model Summary(ARIMA_PAU)** document.

Both AR coefficients are significant at a 5% significance level.

Check that the residuals are normally distributed and uncorrelated by plotting a histogram, quantile-quantile plot, and ACF of the residuals.

Close the

**Model Summary(ARIMA_PAU)**document.With

`ARIMA_PAU`

selected in the**Data Browser**, on the**Econometric Modeler**tab, in the**Diagnostics**section, click**Residual Diagnostics**>**Residual Histogram**.Click

**Residual Diagnostics**>**Residual Q-Q Plot**.Click

**Residual Diagnostics**>**Autocorrelation Function**.In the right pane, drag the

**Histogram(ARIMA_PAU)**and**QQPlot(ARIMA_PAU)**figure windows so that they occupy the upper two quadrants, and drag the ACF so that it occupies the lower two quadrants.

The residual plots suggest that the residuals are approximately normally distributed and uncorrelated. However, there is some indication of an excess of large residuals. This behavior suggests that a *t* innovation distribution might be appropriate.

Export the model to the MATLAB^{®} Workspace.

In the

**Data Browser**, select the`PAU`

time series.On the

**Econometric Modeler**tab, in the**Export**section, click**Export**>**Export Variables**.In the

**Export Variables**dialog box, select the**Select**check box for the**ARIMA_PAU**model.Click

**Export**. The check box for the**PAU**time series is already selected.

The variables `PAU`

and `ARIMA_PAU`

appear in the workspace.

Generate forecasts and approximate 95% forecast intervals from the estimated ARIMA(2,1,0) model for the next four years (16 quarters). Use the entire series as a presample for the forecasts.

[PAUF,PAUMSE] = forecast(ARIMA_PAU,16,'Y0',PAU); UB = PAUF + 1.96*sqrt(PAUMSE); LB = PAUF - 1.96*sqrt(PAUMSE); datesF = dates(end) + calquarters(1:16); figure h4 = plot(dates,PAU,'Color',[.75,.75,.75]); hold on h5 = plot(datesF,PAUF,'r','LineWidth',2); h6 = plot(datesF,UB,'k--','LineWidth',1.5); plot(datesF,LB,'k--','LineWidth',1.5); legend([h4,h5,h6],'Log CPI','Forecast',... 'Forecast Interval','Location','Northwest') title('Log Australian CPI Forecast') hold off

[1] 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.

- Econometric Modeler App Overview
- Perform ARIMA Model Residual Diagnostics Using Econometric Modeler App
- Box-Jenkins Model Selection
- Box-Jenkins Methodology
- Detect Serial Correlation Using Econometric Modeler App
- Share Results of Econometric Modeler App Session
- Creating ARIMA Models Using Econometric Modeler App