These examples show how to conduct statistical hypothesis tests for assessing whether a time series is a unit root process by using the Econometric Modeler app. The test you use depends on your assumptions about the nature of the nonstationarity of an underlying model.

This example uses the Augmented Dickey-Fuller and
Phillips-Perron tests to assess whether a time series is a unit root process.
The null hypothesis for both tests is that the time series is a unit root
process. The data set, stored in `Data_USEconModel.mat`

,
contains the US gross domestic product (GDP) measured quarterly, among other
series.

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

data
set.

`load Data_USEconModel`

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, among `GDP`

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

In the **Data Browser**, double-click
`GDP`

. A time series plot of
`GDP`

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

The series appears to grow without bound.

Apply the log transformation to `GDP`

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

In the **Data Browser**, a variable representing the
logged GDP (`GDPLog`

) appears. A time series plot
of the logged GDP appears in the **Time Series
Plot(GDPLog)** figure window.

The logged GDP series appears to have a time trend or drift term.

Using the Augmented Dickey-Fuller test, test the null hypothesis that the
logged GDP series has a unit root against a trend stationary AR(1) model
alternative. Conduct a separate test for an AR(1) model with drift
alternative. For the null hypothesis of both tests, include the restriction
that the trend and drift terms, respectively, are zero by conducting
*F* tests.

With

`GDPLog`

selected in the**Data Browser**, on the**Econometric Modeler**tab, in the**Tests**section, click**New Test**>**Augmented Dickey-Fuller Test**.On the

**ADF**tab, in the**Parameters**section:Set

**Number of Lags**to`1`

.Select

**Model**>**Trend Stationary**.Select

**Test Statistic**>**F statistic**.

In the

**Tests**section, click**Run Test**.Repeat steps 2 and 3, but select

**Model**>**Autoregressive with Drift**instead.

The test results appear in the **Results** table of the
**ADF(GDPLog)** document.

For the test supposing a trend stationary AR(1) model alternative, the null hypothesis is not rejected. For the test assuming an AR(1) model with drift, the null hypothesis is rejected.

Apply the Phillips-Perron test using the same assumptions as in the Augmented Dickey-Fuller tests, except the trend and drift terms in the null model cannot be zero.

With

`GDPLog`

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

**PP**tab, in the**Parameters**section:Set

**Number of Lags**to`1`

.Select

**Model**>**Trend Stationary**.

In the

**Tests**section, click**Run Test**.Repeat steps 2 and 3, but select

**Model**>**Autoregressive with Drift**instead.

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

The null is not rejected for both tests. These results suggest that the logged GDP possibly has a unit root.

The difference in the null models can account for the differences between the Augmented Dickey-Fuller and Phillips-Perron test results.

This example uses the Kwiatkowski, Phillips, Schmidt, and
Shin (KPSS) test to assess whether a time series is a unit root process. The
null hypothesis is that the time series is stationary. The data set, stored in
`Data_NelsonPlosser.mat`

, contains annual nominal wages,
among other US macroeconomic series.

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

data set.

`load Data_NelsonPlosser`

Convert the table `DataTable`

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

dates = datetime(dates,12,31,'Format','yyyy'); % 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, including the nominal wages `WN`

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

In the **Data Browser**, double-click
`WN`

. A time series plot of
`WN`

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

The series appears to grow without bound, and wage measurements are missing before 1900. To zoom into values occurring after 1900, pause on the plot, click , and enclose the time series in the box produced by dragging the cross hair.

Apply the log transformation to `WN`

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

In the **Data Browser**, a variable representing the
logged wages (`WNLog`

) appears. The logged series
appears in the **Time Series Plot(WNLog)** figure
window.

The logged wages appear to have a linear trend.

Using the KPSS test, test the null hypothesis that the logged wages are trend stationary against the unit root alternative. As suggested in [1], conduct three separate tests by specifying 7, 9, and 11 lags in the autoregressive model.

With

`WNLog`

selected in the**Data Browser**, on the**Econometric Modeler**tab, in the**Tests**section, click**New Test**>**KPSS Test**.On the

**KPSS**tab, in the**Parameters**section, set**Number of Lags**to`7`

.In the

**Tests**section, click**Run Test**.Repeat steps 2 and 3, but set

**Number of Lags**to`9`

instead.Repeat steps 2 and 3, but set

**Number of Lags**to`11`

instead.

The test results appear in the **Results** table of the
**KPSS(WNLog)** document.

All tests fail to reject the null hypothesis that the logged wages are trend stationary.

This example uses the variance ratio test to assess the null
hypothesis that a time series is a random walk. The data set, stored in
`CAPMuniverse.mat`

, contains market data for daily returns
of stocks and cash (money market) from the period January 1, 2000 to November 7,
2005.

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

data
set.

`load CAPMuniverse`

The series are in the timetable `AssetsTimeTable`

. The
first column of data (`AAPL`

) is the daily return of a
technology stock. The last column is the daily return for cash (the daily
money market rate, `CASH`

).

Accumulate the daily technology stock and cash returns.

AssetsTimeTable.AAPLcumsum = cumsum(AssetsTimeTable.AAPL); AssetsTimeTable.CASHcumsum = cumsum(AssetsTimeTable.CASH);

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

econometricModeler

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

Import `AssetsTimeTable`

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`AssetsTimeTable`

variable.Click

**Import**.

The variables, including stock and cash prices
(`AAPLcumsum`

and
`CASHcumsum`

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

In the **Data Browser**, double-click
`AAPLcumsum`

. A time series plot of
`AAPLcumsum`

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

The accumulated returns of the stock appear to wander at first, with high variability, and then grow without bound after 2004.

Using the variance ratio test, test the null hypothesis that the series of accumulated stock returns is a random walk. First, test without assuming IID innovations for the alternative model, then test assuming IID innovations.

With

`AAPLcumsum`

selected in the**Data Browser**, on the**Econometric Modeler**tab, in the**Tests**section, click**New Test**>**Variance Ratio Test**.On the

**VRatio**tab, in the**Tests**section, click**Run Test**.On the

**VRatio**tab, in the**Parameters**section, select the**IID Innovations**check box.In the

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

The test results appear in the **Results** table of the
**VRatio(AAPLcumsum)** document.

Without assuming IID innovations for the alternative model, the test fails to reject the random walk null model. However, assuming IID innovations, the test rejects the null hypothesis. This result might be due to heteroscedasticity in the series, that is, the series might be a heteroscedastic random walk.

In the **Data Browser**, double-click
`CASHcumsum`

. A time series plot of
`CASHcumsum`

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

The series of accumulated cash returns exhibits low variability and appears to have long-term trends.

Test the null hypothesis that the series of accumulated cash returns is a random walk:

With

`CASHcumsum`

selected in the**Data Browser**, on the**Econometric Modeler**tab, in the**Tests**section, click**New Test**>**Variance Ratio Test**.On the

**VRatio**tab, in the**Parameters**section, clear the**IID Innovations**box.In the

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

The test results appear in the **Results** tab of the
**VRatio(CASHcumsum)** document.

The test rejects the null hypothesis that the series of accumulated cash returns is a random walk.

[1]
Kwiatkowski, D., P. C.
B. Phillips, P. Schmidt, and Y. Shin. “Testing the Null Hypothesis of Stationarity against the
Alternative of a Unit Root.” *Journal of Econometrics*. Vol. 54, 1992, pp.
159–178.

`adftest`

| `kpsstest`

| `lmctest`

| `vratiotest`