Load and Import Data into Econometric Modeler

At the command line, load the Data_Canada.mat data set.

load Data_Canada

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:

The Canadian interest and inflation rate variables appear in the Time Series pane, and a time series plot of all the series appears in the Time Series Plot(INF_C) figure window.

Plot only the three interest rate series INT_L, INT_M, and INT_S together. In the Time Series pane, click INT_L and Ctrl+click INT_M and INT_S. Then, on the Plots tab, in the Plots section, click Time Series. The time series plot appears in the Time Series(INT_L) document.

The interest rate series each appear nonstationary, and they appear to move together with mean-reverting spread. In other words, they exhibit cointegration. To establish these properties, this example conducts statistical tests.

Conduct Stationarity Test

Assess whether each interest rate series is stationary by conducting Phillips-Perron unit root tests. For each series, assume a stationary AR(1) process with drift for the alternative hypothesis. You can confirm this property by viewing the autocorrelation and partial autocorrelation function plots.

Close all plots in the right pane, and perform the following procedure for each series INT_L, INT_M, and INT_S.

Position the test results to view them simultaneously.

All tests fail to reject the null hypothesis that the series contains a unit root process.

Conduct Cointegration Test

To create and estimate a VEC model of unit root series, the series need to exhibit cointegration. Conduct a Johansen test. Because the raw series do not contain a linear trend, assume that the only deterministic term in the model is an intercept in the cointegrating relation (H1* Johansen form), and include 1 lagged difference term in the model.

Econometric Modeler conducts a separate test for each cointegration rank 0 through 2 (the number of series – 1). The test rejects the null hypothesis of no cointegration (Cointegration rank = 0), but fails to reject the null hypothesis of Cointegration rank ≤ 1. The conclusion is to set the cointegration rank of the VEC model to 1.

Estimate VEC Models

Estimate 3-D VEC(p) models of the interest rate series, with a cointegration rank of 1 and p = 1 and 2.

Select Model with Best In-Sample Fit

The estimation summary in each Model Summary(VECp) tab contains a plot of fitted values and residuals, with respect to the time series in the Time Series list, a standard statistical table of estimates and inferences, and a table of information criteria.

Compare the information criteria of each estimated model simultaneously by positioning the estimation summary documents so that they occupy the left and right sections of the right pane. The model with the lowest value has the best, parsimonious fit.

The VEC(1) model VEC produces the lowest AIC and BIC values. Choose this model for further analysis.

Check Goodness of Fit

Inspect the following VEC(1) plots of each residual series:

This example diagnoses the residuals visually. Alternatively, you can conduct statistical tests to diagnose the residuals.

Dismiss the Model Summary(VEC2) document by clicking on its tab.

On the Models pane, click VEC.

Plot separate residual histograms. On the Econometric Modeler tab, in the Diagnostics section, click Residual Diagnostics > Residual Histogram. Histograms of the each residual series appear in the Histogram(VEC) document.

Each residual series appears approximately centered around 0 and approximately normal.

Plot separate residual quantile-quantile plots. With VEC selected in the Time Series pane, on the Econometric Modeler tab, in the Diagnostics section, click Residual Diagnostics > Residual Q-Q Plot. Quantile-quantile plots of each residual series appear in the QQPlot(VEC) document.

The residual series are slightly skewed left and have lighter tails than the normal distribution. This example proceeds without addressing possible skewness and light tails.

Plot separate ACF plots of each residual series. With VEC selected in the Time Series pane, on the Econometric Modeler tab, in the Diagnostics section, click Residual Diagnostics > Autocorrelation Function. ACF plots of each residual series appear in the ACF(VEC) document.

The residuals do not exhibit significant autocorrelation.

Plot separate ACF plots of each squared residual series. With VEC selected in the Time Series pane, on the Econometric Modeler tab, in the Diagnostics section, click Residual Diagnostics > Squared Residual Autocorrelation. ACF plots of each squared residual series appear in the ACF(VEC)2 document.

Each squared residual series has significant autocorrelations at early lags, which suggests that heteroscedasticity is present in all series. This example proceeds without addressing possible heteroscedasticity.

Export Model to Workspace

Export the model to the workspace.

The variables INT_L, INT_M, INT_S, and VEC appear in the workspace.

Generate Forecasts at Command Line

Generate forecasts and approximate 95% forecast intervals from the estimated VEC(1) model for the next five years. For convenience, use the entire series as a presample for the forecasts. The forecast function discards all specified presample observations except for the required final observation.

Y0 = [INT_L INT_M INT_S]; 
[YF,YFMSE] = forecast(VEC,5,Y0);
YFSE = cell2mat(cellfun(@(x)sqrt(diag(x)'),YFMSE,UniformOutput=false));
UB = YF + 1.96*YFSE;
LB = YF - 1.96*YFSE;
datesF = DataTimeTable.Time(end) + calyears(1:5);

figure
tiledlayout(3,1)
for j = 1:VEC.NumSeries
    nexttile
    h1 = plot(DataTimeTable.Time,Y0(:,j),Color=[.75,.75,.75]);
    hold on
    h2 = plot(datesF,YF(:,j),"r",LineWidth=2);
    h3 = plot(datesF,UB(:,j),"k--",LineWidth=1.5);
    plot(datesF,LB(:,j),"k--",LineWidth=1.5);
    ct = [Y0(end,j) YF(1,j); Y0(end,j) LB(1,j); Y0(end,j) UB(1,j);];
    plot([DataTimeTable.Time(end); datesF(1)],ct,Color=[.75,.75,.75])
    legend([h1 h2 h3],VEC.SeriesNames(j),"Forecast", ...
       "Forecast interval",Location="northwest")
    hold off
end