This example shows how to specify and estimate a regression model
with ARMA errors using the Econometric Modeler app. The data set, which
is stored in `Data_USEconModel.mat`

, contains the US personal
consumption expenditures measured quarterly, among other series.

Consider modeling the US personal consumption expenditures
(`PCEC`

, in $ billions) as a linear function of the effective
federal funds rate (`FEDFUNDS`

), unemployment rate
(`UNRATE`

), and real gross domestic product
(`GDP`

, in $ billions with respect to the year 2000).

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

data
set.

`load Data_USEconModel`

Convert the federal funds and unemployment rates from percents to decimals.

DataTable.UNRATE = 0.01*DataTable.UNRATE; DataTable.FEDFUNDS = 0.01*DataTable.FEDFUNDS;

Convert the nominal GDP to real GDP by dividing all values by the GDP deflator
(`GDPDEF`

) and scaling the result by 100. Create a column
in `DataTable`

for the real GDP series.

DataTable.RealGDP = 100*DataTable.GDP./DataTable.GDPDEF;

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**.

All time series variables in `DataTable`

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

Plot the `PCEC`

, `RealGDP`

,
`FEDFUNDS`

, and `UNRATE`

series on separate plots.

In the

**Data Browser**, double-click`PCEC`

.Repeat step 1 for

`RealGDP`

,`FEDFUNDS`

, and`UNRATE`

.In the right pane, drag the

**Time Series Plot(PCEC)**figure window to the top so that it occupies the first two quadrants.Drag the

**Time Series Plot(RealGDP)**figure window to the first quadrant.Drag the

**Time Series Plot(UNRATE)**figure window to the third quadrant.

The `PCEC`

and `RealGDP`

series appear to have an exponential trend. The
`UNRATE`

and `FEDFUNDS`

series appear to have a stochastic trend.

Right-click the tab for any figure window, then select **Close All
** to close all the figure windows.

Check whether the series are collinear by performing Belsley collinearity diagnostics.

In the

**Data Browser**, select`PCEC`

. Then, press**Ctrl**and click to select`RealGDP`

,`FEDFUNDS`

, and`UNRATE`

.On the

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

The Belsley collinearity diagnostics results appear in the
**Collinearity(FEDFUNDS)** document.

All condition indices are below the default condition-index tolerance, which is 30. The time series do not appear to be collinear.

Specify a linear model in which `PCEC`

is the
response and `RealGDP`

,
`FEDFUNDS`

, and `UNRATE`

are predictors.

In the

**Data Browser**, select`PCEC`

.Click the

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

**Regression Models**section, click**MLR**.In the

**MLR Model Parameters**dialog box, in the**Predictors**section, select the**Include?**check box for the`FEDFUNDS`

,`RealGDP`

, and`UNRATE`

time series.Click

**Estimate**.

The model variable `MLR_PCEC`

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

In the **Model Summary(MLR_PCEC)** figure window, the
residual plot suggests that the standard linear model assumption of uncorrelated
errors is violated. The residuals appear autocorrelated, nonstationary, and
possibly heteroscedastic.

To stabilize the residuals, stabilize the response and predictor series by
converting the `PCEC`

and
`RealGDP`

prices to returns, and by applying the
first difference to `FEDFUNDS`

and
`UNRATE`

.

Convert `PCEC`

and `RealGDP`

prices to returns:

In the

**Data Browser**, select the`PCEC`

time series.On the

**Econometric Modeler**tab, in the**Transforms**section, click**Log**.In the

**Data Browser**, a variable representing the logged`PCEC`

prices (`PCECLog`

) appears.In the

**Data Browser**, select`PCECLog`

.On the

**Econometric Modeler**tab, in the**Transforms**section, click**Difference**.In the

**Data Browser**, a variable representing the first differences of the logged`PCEC`

prices (`PCECLogDiff`

) appears.Rename the

`PCECLogDiff`

variable to`PCECReturns`

.In the

**Data Browser**, right-click`PCECLogDiff`

.In the context menu, select

**Rename**.Enter

`PCECReturns`

.

Repeat steps 1 through 5, but select the

`RealGDP`

time series instead. Rename the real GDP returns to`RealGDPReturns`

.

Apply the first difference to `FEDFUNDS`

and
`UNRATE`

:

In the

**Data Browser**, select the`FEDFUNDS`

time series.On the

**Econometric Modeler**tab, in the**Transforms**section, click**Difference**.In the

**Data Browser**, a variable representing the first difference of`FEDFUNDS`

(`FEDFUNDSDiff`

) appears.Repeat steps 1 and 2, but select the

`UNRATE`

time series instead.Close all figure windows and documents.

Respecify the linear model, but use the stabilized series instead.

In the

**Data Browser**, select`PCECReturns`

.On the

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

**Regression Models**section, click**MLR**.In the

**MLR Model Parameters**dialog box, in the**Predictors**section, select the**Include?**check box for the`FEDFUNDSDiff`

,`RealGDPReturns`

, and`UNRATEDiff`

time series.Click

**Estimate**.

The model variable `MLR_PCECReturns`

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

The residual plot suggests that the residuals are autocorrelated.

Assess whether the residuals are normally distributed and autocorrelated by generating quantile-quantile and ACF plots.

Create a quantile-quantile plot of the
`MLR_PCECReturns`

model residuals:

In the

**Data Browser**, select the`MLR_PCECReturns`

model.On the

**Econometric Modeler**tab, in the**Diagnostics**section, click**Residual Diagnostics**>**Residual Q-Q Plot**.

The residuals are skewed to the right.

Plot the ACF of the residuals:

In the

**Data Browser**, select the`MLR_PCECReturns`

model.On the

**Econometric Modeler**tab, in the**Diagnostics**section, click**Residual Diagnostics**>**Autocorrelation Function**.On the

**ACF**tab, set**Number of Lags**to`40`

.

The plot shows autocorrelation in the first 34 lags.

Attempt to remedy the autocorrelation in the residuals by specifying a
regression model with ARMA(1,1) errors for
`PCECReturns`

.

In the

**Data Browser**, select`PCECReturns`

.Click the

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

**Regression Models**section, click**RegARMA**.In the

**regARMA Model Parameters**dialog box:In the

**Lag Order**tab:Set

**Autoregressive Order**to`1`

.Set

**Moving Average Order**to`1`

.

In the

**Predictors**section, select the**Include?**check box for the`FEDFUNDSDiff`

,`RealGDPReturns`

, and`UNRATEDiff`

time series.Click

**Estimate**.

The model variable `RegARMA_PCECReturns`

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

The *t* statistics suggest that all coefficients are
significant, except for the coefficient of
`UNRATEDiff`

. The residuals appear to fluctuate around
*y* = 0 without autocorrelation.

Assess whether the residuals of the
`RegARMA_PCECReturns`

model are normally
distributed and autocorrelated by generating quantile-quantile and ACF
plots.

Create a quantile-quantile plot of the
`RegARMA_PCECReturns`

model residuals:

In the

**Data Browser**, select the`RegARMA_PCECReturns`

model.On the

**Econometric Modeler**tab, in the**Diagnostics**section, click**Residual Diagnostics**>**Residual Q-Q Plot**.

The residuals appear approximately normally distributed.

Plot the ACF of the residuals:

In the

**Data Browser**, select the`RegARMA_PCECReturns`

model.On the

**Econometric Modeler**tab, in the**Diagnostics**section, click**Residual Diagnostics**>**Autocorrelation Function**.

The first autocorrelation lag is significant.

From here, you can estimate multiple models that differ by the number of autoregressive and moving average polynomial orders in the ARMA error model. Then, choose the model with the lowest fit statistic. Or, you can check the predictive performance of the models by comparing forecasts to out-of-sample data.

`autocorr`

|`collintest`

|`estimate`

|`fitlm`