## Estimate ARIMAX Model Using Econometric Modeler App

This example shows how to specify and estimate an ARIMAX model
using the Econometric Modeler app. The data set, which is stored in
`Data_CreditDefaults.mat`

, contains annual investment-grade
corporate bond default rates, among other predictors, from 1984 through 2004.
Consider modeling corporate bond default rates as a linear, dynamic function of the
other time series in the data set.

### Import Data into Econometric Modeler

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

data
set.

`load Data_CreditDefaults`

For more details on the data set, enter `Description`

at the
command line.

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 `IGD`

, appear in the
**Time Series** pane, and a time series plot containing all
the series appears in the **Time Series Plot(AGE)** figure
window.

### Assess Stationarity of Dependent Variable

In the **Time Series** pane, double-click
`IGD`

. The value of `IGD`

appears in the **Preview** pane, and a time series plot for
`IGD`

appears in the **Time Series
Plot(IGD)** figure window.

`IGD`

appears to be stationary.

Assess whether `IGD`

has a unit root by conducting a
Phillips-Perron test:

On the

**Econometric Modeler**tab, in the**Tests**section, click**New Test**>**Phillips-Perron Test**.On the

**PP**tab, in the**Parameters**section, set**Number of Lags**to`1`

.In the

**Tests**section, click**Run Test**.

The test results in the **Results** table of the
**PP(IGD)** document.

The test rejects the null hypothesis that `IGD`

contains a unit root.

### Inspect Correlation and Collinearity Among Variables

Plot the pairwise correlations between variables.

Select all variables in the

**Time Series**pane by clicking`AGE`

, then press**Shift**and click`SPR`

.Click the

**Plots**tab, then click**Correlations**.

A correlations plot appears in the
**Correlations(AGE)** figure window.

All predictors appear weakly associated with `IGD`

.
You can test whether the correlation coefficients are significant by using
`corrplot`

at the command line.

Assess whether any variables are collinear by performing Belsley collinearity diagnostics:

In the

**Time Series**pane, select all variables.Click the

**Econometric Modeler**tab. Then, in the**Tests**section, click**New Test**>**Belsley Collinearity Diagnostics**.

Tabular results appear in the **Collinearity(AGE)**
document.

None of the condition indices are greater than the condition-index tolerance
(`30`

). Therefore, the variables do not exhibit
multicollinearity.

### Specify and Estimate ARIMAX Model

Consider an ARIMAX(0,0,1) model for `IGD`

containing
all predictors. Specify and estimate the model.

In the

**Time Series**pane, click`IGD`

.Click the

**Econometric Modeler**tab. Then, in the**Models**section, click the arrow to display the models gallery.In the models gallery, in the

**ARMA/ARIMA Models**section, click**ARIMAX**.In the ARIMAX Model Parameters dialog box, on the

**Lag Order**tab, set**Moving Average Order**to`1`

.In the

**Predictors**section, select the**Include?**check box for each time series.Click

**Estimate**. The model variable`ARIMAX_IGD`

appears in the**Models**pane, its value appears in the**Preview**pane, and its estimation summary appears in the**Model Summary(ARIMAX_IGD)**document.

At a 0.10 significance level, all predictors and the MA coefficient are significant.

Close all figure windows and documents.

### 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.

In the

**Models**pane, select`ARIMAX_IGD`

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

The residual histogram and quantile-quantile plots suggest that the residuals might not be normally distributed. According to the ACF plot, the residuals do not exhibit serial correlation. Standard inferences rely on the normality of the residuals. To remedy nonnormality, you can try transforming the response, then estimating the model using the transformed response.