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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 inData_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 theData_JAustralian.matdata set.

loadData_JAustralian

Convert the tableDataTableto a timetable:

  1. Clear the row names ofDataTable.

  2. Convert the sampling times to adatetimevector.

  3. Convert the table to a timetable by associating the rows with the sampling times indates.

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

Import Data into Econometric Modeler

At the command line, open theEconometric Modeler应用程序。

econometricModeler

Alternatively, open the app from the apps gallery (seeEconometric Modeler).

ImportDataTableinto the app:

  1. On theEconometric Modelertab, in theImportsection, click.

  2. In theImport Datadialog box, in theImport?column, select the check box for theDataTablevariable.

  3. ClickImport.

The variables, includingPAU, appear in theTime Seriespane, and a time series plot of all the series appears in theTime Series Plot(EXCH)figure window.

Create a time series plot ofPAUby double-clickingPAUin theTime Seriespane.

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).

  1. In theTime Seriespane, select thePAUtime series.

  2. Click thePlotstab, then clickACF.

  3. Click thePlotstab, then clickPACF.

  4. Close all figure windows except for the correlograms. Then, drag theACF(PAU)figure window above thePACF(PAU)figure window.

重要的,李nearly decaying sample ACF indicates a nonstationary process.

Close theACF(PAU)andPACF(PAU)figure windows.

Difference the Series

Take a first difference of the data. WithPAUselected in theTime Seriespane, on theEconometric Modelertab, in theTransformssection, clickDifference.

The transformed variablePAUDiffappears in theTime Seriespane, and its time series plot appears in theTime 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 ofPAUDiff. WithPAUDiffselected in theTime Seriespane:

  1. Click thePlotstab, then clickACF.

  2. Click thePlotstab, then clickPACF.

  3. Close theTime Series Plot(PAUDiff)figure window. Then, drag theACF(PAUDiff)figure window above thePACF(PAUDiff)figure window.

差的样品ACF系列莫衰减re 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 theACF(PAUDiff)andPACF(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.

  1. In theTime Seriespane, select thePAUtime series.

  2. On theEconometric Modelertab, in theModelssection, clickARIMA.

  3. In theARIMA Model Parametersdialog box, on theLag Ordertab:

    1. SetDegree of Integrationto1.

    2. SetAutoregressive Orderto2.

  4. ClickEstimate.

The model variableARIMA_PAUappears in theModelspane, its value appears in thePreviewpane, and its estimation summary appears in theModel 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.

  1. Close theModel Summary(ARIMA_PAU)document.

  2. WithARIMA_PAUselected in theModelspane, on theEconometric Modelertab, in theDiagnosticssection, clickResidual Diagnostics>Residual Histogram.

  3. ClickResidual Diagnostics>Residual Q-Q Plot.

  4. ClickResidual Diagnostics>Autocorrelation Function.

  5. In the right pane, drag theHistogram(ARIMA_PAU)andQQPlot(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 atinnovation distribution might be appropriate.

模型导出到工作空间

Export the model to the MATLAB®Workspace.

  1. In theTime Seriespane, select thePAUtime series.

  2. On theEconometric Modelertab, in theExportsection, clickExport>Export Variables.

  3. In theExport Variablesdialog box, select theSelectcheck box for theARIMA_PAUmodel.

  4. ClickExport. The check box for thePAUtime series is already selected.

The variablesPAUandARIMA_PAUappear 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]); holdonh5 = 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') holdoff

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

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