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These examples show how to assess whether a series has volatility clustering by using
the **Econometric Modeler** app. Methods include inspecting correlograms of
squared residuals and testing for significant ARCH lags. The data set, stored in
`Data_EquityIdx.mat`

, contains a series of daily NASDAQ closing
prices from 1990 through 2001.

This example shows how to visually determine whether a series has significant ARCH effects by plotting the autocorrelation function (ACF) and partial autocorrelation function (PACF) of a series of squared residuals.

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

data
set.

`load Data_EquityIdx`

The data set contains a table of NASDAQ and NYSE closing prices, among
other variables. For more details about the data set, enter
`Description`

at the command line.

Convert the table `DataTable`

to a timetable (for
details, see Prepare Time Series Data for Econometric Modeler App).

dates = datetime(dates,'ConvertFrom','datenum',... 'Format','ddMMMyyyy'); % Convert dates to datetimes DataTable.Properties.RowNames = {}; % Clear row names DataTable = table2timetable(DataTable,'RowTimes',dates); % Convert table to timetable

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**.

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

Convert the daily close NASDAQ index series to a percentage return series by taking the log of the series, then taking the first difference of the logged series:

In the

**Data Browser**, select`NASDAQ`

.On the

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

`NASDAQLog`

selected, in the**Transforms**section, click**Difference**.Change the name of

`NASDAQLogDiff`

to`NASDAQReturns`

:In the

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

.In the context menu, select

**Rename**.Enter

`NASDAQReturns`

.

The time series plot of the NASDAQ returns appears in the **Time
Series Plot(NASDAQReturns)** figure window.

The returns appear to fluctuate around a constant level, but exhibit 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.

Compute squared residuals:

Export

`NASDAQReturns`

to the MATLAB^{®}Workspace:In the

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

.In the context menu, select

**Export**.

`NASDAQReturns`

appears in the MATLAB Workspace.At the command line:

For numerical stability, scale the returns by a factor of 100.

Create a residual series by removing the mean from the scaled returns series. Because you took the first difference of the NASDAQ prices to create the returns, the first element of the returns is missing. Therefore, to estimate the sample mean of the series, you must use

`nanmean`

.Square the residuals.

Add the squared residuals as a new variable to the

`DataTable`

timetable.

NASDAQReturns = 100*NASDAQReturns; NASDAQResiduals = NASDAQReturns - nanmean(NASDAQReturns); NASDAQResiduals2 = NASDAQResiduals.^2; DataTable.NASDAQResiduals2 = NASDAQResiduals2;

In Econometric Modeler, import `DataTable`

:

On the

**Econometric Modeler**tab, in the**Import**section, click .In the

**Econometric Modeler**dialog box, click**OK**to clear all variables and documents in the app.In the

**Import Data**dialog box, in the**Import?**column, select the check box for`DataTable`

.Click

**Import**.

Plot the ACF and PACF:

In the

**Data Browser**, select the`NASDAQResiduals2`

time series.Click the

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

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

**Time Series Plot(NASDAQ)**figure window. Then, position the**ACF(NASDAQResiduals2)**figure window above the**PACF(NASDAQResiduals2)**figure window.

The sample ACF and PACF show significant autocorrelation in the squared residuals. This result indicates that volatility clustering is present.

This example shows how to test squared residuals for significant ARCH effects using the Ljung-Box Q-test.

At the command line:

Load the

`Data_EquityIdx.mat`

data set.Convert the NASDAQ prices to returns. To maintain the correct time base, prepend the resulting returns with a

`NaN`

value.Scale the NASDAQ returns.

Compute residuals by removing the mean from the scaled returns.

Square the residuals.

Add the vector of squared residuals as a variable to

`DataTable`

.Convert

`DataTable`

from a table to a timetable.

For more details on the steps, see Inspect Correlograms of Squared Residuals for ARCH Effects.

load Data_EquityIdx NASDAQReturns = 100*price2ret(DataTable.NASDAQ); NASDAQReturns = [NaN; NASDAQReturns]; NASDAQResiduals2 = (NASDAQReturns - nanmean(NASDAQReturns)).^2; DataTable.NASDAQResiduals2 = NASDAQResiduals2; dates = datetime(dates,'ConvertFrom','datenum'); DataTable.Properties.RowNames = {}; DataTable = table2timetable(DataTable,'RowTimes',dates);

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**.

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

Test the null hypothesis that the first *m* = 5
autocorrelation lags of the squared residuals are jointly zero by using the
Ljung-Box Q-test. Then, test the null hypothesis that the first
*m* = 10 autocorrelation lags of the squared residuals
are jointly zero.

In the

**Data Browser**, select the`NASDAQResiduals2`

time series.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 both the**Number of Lags**and**DOF**to`5`

. To maintain a significance level of 0.05 for the two tests, set**Significance Level**to 0.025.In the

**Tests**section, click**Run Test**.Repeat steps 3 and 4, but set both the

**Number of Lags**and**DOF**to`10`

instead.

The test results appear in the **Results** table of the
**LBQ(NASDAQResiduals2)** document.

The null hypothesis is rejected for the two tests. The
*p*-value for each test is 0. The results show that not
every autocorrelation up to lag 5 (or 10) is zero, indicating volatility
clustering in the squared residuals.

This example shows how to test residuals for significant ARCH effects using the Engle's ARCH Test.

At the command line:

Load the

`Data_EquityIdx.mat`

data set.Convert the NASDAQ prices to returns. To maintain the correct time base, prepend the resulting returns with a

`NaN`

value.Scale the NASDAQ returns.

Compute residuals by removing the mean from the scaled returns.

Add the vector of residuals as a variable to

`DataTable`

.Convert

`DataTable`

from a table to a timetable.

For more details on the steps, see Inspect Correlograms of Squared Residuals for ARCH Effects.

load Data_EquityIdx NASDAQReturns = 100*price2ret(DataTable.NASDAQ); NASDAQReturns = [NaN; NASDAQReturns]; NASDAQResiduals = NASDAQReturns - nanmean(NASDAQReturns); DataTable.NASDAQResiduals = NASDAQResiduals; dates = datetime(dates,'ConvertFrom','datenum'); DataTable.Properties.RowNames = {}; DataTable = table2timetable(DataTable,'RowTimes',dates);

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**.

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

Test the null hypothesis that the NASDAQ residuals series exhibits no ARCH effects by using Engle's ARCH test. Specify that the residuals series is an ARCH(2) model.

In the

**Data Browser**, select the`NASDAQResiduals`

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(NASDAQResiduals)** document.

The null hypothesis is rejected in favor of the ARCH(2) alternative. The test result indicates significant volatility clustering in the residuals.