## Implement Box-Jenkins Model Selection and Estimation Using Econometric Modeler App

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.

### Prepare Data for Econometric Modeler

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

data
set.

`load Data_JAustralian`

### Import Data into Econometric Modeler

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 variables, including `PAU`

, appear in the
**Time Series** pane, 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 **Time Series**
pane.

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

### Plot Sample ACF and PACF of Series

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

In the

**Time Series**pane, 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.

### Difference the Series

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

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

The transformed variable `PAUDiff`

appears in the
**Time Series** pane, 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 Sample ACF and PACF of Differenced Series

Plot the sample ACF and PACF of `PAUDiff`

. With
`PAUDiff`

selected in the **Time
Series** pane:

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.

### Specify and Estimate ARIMA Model

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

**Time Series**pane, 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** pane, its value appears in the
**Preview** pane, and its estimation summary appears in the
**Model Summary(ARIMA_PAU)** document.

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

### Check Goodness of Fit

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**Models**pane, 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 Model to Workspace

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

In the

**Time Series**pane, 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 at Command Line

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

## 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.

## See Also

### Apps

### Objects

### Functions

## Related Topics

- Analyze Time Series Data Using Econometric Modeler
- Perform ARIMA Model Residual Diagnostics Using Econometric Modeler App
- Select ARIMA Model for Time Series Using Box-Jenkins Methodology
- Box-Jenkins Methodology
- Detect Serial Correlation Using Econometric Modeler App
- Share Results of Econometric Modeler App Session
- Creating ARIMA Models Using Econometric Modeler App