## Select ARCH Lags for GARCH Model Using Econometric Modeler App

This example shows how to select the appropriate number of ARCH
and GARCH lags for a GARCH model by using the Econometric Modeler app. The data set,
stored in `Data_MarkPound`

, contains daily Deutschmark/British
pound bilateral spot exchange rates from 1984 through 1991.

### Import Data into Econometric Modeler

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

data
set.

`load Data_MarkPound`

At the command line, open the **Econometric Modeler** app.

econometricModeler

Alternatively, open the app from the apps gallery (see **Econometric
Modeler**).

Import `Data`

to 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**Data**variable.Click

**Import**.

The variable `Data1`

appears in the **Time
Series** pane, and its time series plot appears in the
**Time Series Plot(Data1)** figure window.

The exchange rate looks nonstationary (it does not appear to fluctuate around a fixed level).

### Transform Data

Convert the exchange rates to returns.

With

`Data1`

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

**Time Series**pane, a variable representing the logged exchange rates (`Data1Log`

) appears , and its time series plot appears in the**Time Series Plot(Data1Log)**figure window.In the

**Time Series**pane, select`Data1Log`

.On the

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

In the **Time Series** pane, a variable representing the
returns (`Data1LogDiff`

) appears. A time series plot of
the differenced series appears in the **Time Series
Plot(Data1LogDiff)** figure window.

### Check for Autocorrelation

In the **Time Series** pane, rename the
`Data1LogDiff`

variable by clicking it twice to
select its name and entering `Returns`

.

The app updates the names of all documents associated with the returns.

The returns series fluctuates around a common level, but exhibits volatility clustering. Large changes in the returns tend to cluster together, and small changes tend to cluster together. That is, the series exhibits conditional heteroscedasticity.

Visually assess whether the returns have serial correlation by plotting the sample ACF and PACF:

Close all figure windows in the right pane.

In the

**Time Series**pane, select the`Returns`

time series.Click the

**Plots**tab, then click**ACF**.Click the

**Plots**tab, then click**PACF**.Drag the

**PACF(Returns)**figure window below the**ACF(Returns)**figure window so that you can view them simultaneously.

The sample ACF and PACF show virtually no significant autocorrelation.

Conduct the Ljung-Box Q-test to assess whether there is significant serial correlation in the returns for at most 5, 10, and 15 lags. To maintain a false-discovery rate of approximately 0.05, specify a significance level of 0.05/3 = 0.0167 for each test.

Close the

**ACF(Returns)**and**PACF(Returns)**figure windows.With

`Returns`

selected in the**Time Series**pane, on the**Econometric Modeler**tab, in the**Tests**section, click**New Test**>**Ljung-Box Q-Test**.On the

**LBQ**tab, in the**Parameters**section, set**Number of Lags**to`5`

.Set

**Significance Level**to`0.0167`

.In the

**Tests**section, click**Run Test**.Repeat steps 3 through 5 twice, with these changes.

Set

**Number of Lags**to 10 and the**DOF**to 10.Set

**Number of Lags**to 15 and the**DOF**to 15.

The test results appear in the **Results** table of the
**LBQ(Returns)** document.

The Ljung-Box Q-test null hypothesis that all autocorrelations up to the tested lags are zero is not rejected for tests at lags 5, 10, and 15. These results, and the ACF and PACF, suggest that a conditional mean model is not needed for this returns series.

### Check for Conditional Heteroscedasticity

To check the returns for conditional heteroscedasticity, **Econometric
Modeler** requires a series of squared residuals. After importing the
squared residuals into the app, visually assess whether there is conditional
heteroscedasticity by plotting the ACF and PACF of the squared residuals. Then,
determine the appropriate number of lags for a GARCH model of the returns by
conducting Engle's ARCH test.

Compute the series of squared residuals at the command line by demeaning the returns, then squaring each element of the result.

Export `Returns`

to the command line:

In the

**Time Series**pane, right-click`Returns`

.In the context menu, select

`Export`

.

`Returns`

appears in the MATLAB^{®} Workspace.

Remove the mean from the returns, then square each element of the result. To
ensure all series in the **Time Series** pane are synchronized,
Econometric Modeler prepends first-differenced series with a
`NaN`

value. Therefore, to estimate the sample mean, use
`mean(Returns,'omitnan')`

.

```
Residuals = Returns - mean(Returns,'omitnan');
Residuals2 = Residuals.^2;
```

Create a table containing the `Returns`

and
`Residuals2`

variables.

Tbl = table(Returns,Residuals,Residuals2);

Import `Tbl`

into Econometric Modeler:

On the

**Econometric Modeler**tab, in the**Import**section, click .The app must clear the right pane and all documents before importing new data. Therefore, after clicking

**Import**, in the**Econometric Modeler**dialog box, click**OK**.In the

**Import Data**dialog box, in the**Import?**column, select the check box for the**Tbl**variable.Click

**Import**.

The variables appear in the **Time Series** pane, and a time
series plot of all the series appears in the **Time Series
Plot(Residuals)** figure window.

Plot the ACF and PACF of the squared residuals.

Close the

**Time Series Plot(Residuals)**figure window.In the

**Time Series**pane, select the`Residuals2`

time series.Click the

**Plots**tab, then click**ACF**.Click the

**Plots**tab, then click**PACF**.Drag the

**PACF(Residuals2)**figure window below the**ACF(Residuals2)**figure window so that you can view them simultaneously.

The sample ACF and PACF of the squared returns show significant autocorrelation. This result suggests that a GARCH model with lagged variances and lagged squared innovations might be appropriate for modeling the returns.

Conduct Engle's ARCH test on the residuals series. Specify a two-lag ARCH model alternative hypothesis.

Close all figure windows.

In the

**Time Series**pane, select the`Residuals`

time series.On the

**Econometric Modeler**tab, in the**Tests**section, click**New Test**>**Engle's ARCH Test**.On the

**ARCH**tab, in the**Parameters**section, set**Number of Lags**to`2`

.In the

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

The test results appear in the **Results** table of the
**ARCH(Residuals)** document.

Engle's ARCH test rejects the null hypothesis of no ARCH effects in favor of the alternative ARCH model with two lagged squared innovations. An ARCH model with two lagged innovations is locally equivalent to a GARCH(1,1) model.

### Create and Fit GARCH Model

Fit a GARCH(1,1) model to the returns series.

In the

**Time Series**pane, select the`Returns`

time series.Click the

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

**GARCH Models**section, click**GARCH**.In the

**GARCH Model Parameters**dialog box, on the**Lag Order**tab:Set

**GARCH Degree**to`1`

.Set

**ARCH Degree**to`1`

.Because the returns required demeaning, include an offset by selecting the

**Include Offset**check box.

Click

**Estimate**.

The model variable `GARCH_Returns`

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

An alternative way to select lags for a GARCH model is by fitting several models containing different lag polynomial degrees. Then, choose the model yielding the minimal AIC.