## Conduct Cointegration Test Using Econometric Modeler

This example models the annual Canadian inflation and interest rate series by using the Econometric Modeler app. The example performs the following actions in the app:

Test each raw series for stationarity.

Test for cointegration using the Engle-Granger cointegration test.

Test for cointegration among all possible cointegration ranks using the Johansen cointegration test.

The data set, which is stored in `Data_Canada`

, contains annual
Canadian inflation and interest rates from 1954 through 1994.

### Load and Import Data into Econometric Modeler

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

data
set.

`load Data_Canada`

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 Canadian interest and inflation rate variables appear in the
**Time Series** pane, and a time series plot of all the
series appears in the **Time Series Plot(INF_C)** figure
window.

Plot only the three interest rate series `INT_L`

,
`INT_M`

, and `INT_S`

together. In the **Time Series** pane, click
`INT_L`

and **Ctrl** click
`INT_M`

and `INT_S`

. Then,
on the **Plots** tab, in the **Plots**
section, click **Time Series**. The time series plot
appears in the **Time Series(INT_L)** document.

The interest rate series each appear nonstationary, and they appear to move together with mean-reverting spread. In other words, they exhibit cointegration. To establish these properties, this example conducts statistical tests.

### Conduct Stationarity Test

Assess whether each interest rate series is stationary by conducting Phillips-Perron unit-root tests. For each series, assume a stationary AR(1) process with drift for the alternative hypothesis. You can confirm this property by viewing the autocorrelation and partial autocorrelation function plots.

Close all plots in the right pane, and perform the following procedure for
each series `INT_L`

, `INT_M`

,
and `INT_S`

.

In the

**Time Series**pane, click a series.On the

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

**PP**tab, in the**Parameters**section, in the**Number of Lags**box, type`1`

, and in the**Model**list select**Autoregressive with Drift**.In the

**Tests**section, click**Run Test**. Test results for the selected series appear in the**PP(**tab.)`series`

Position the test results to view them simultaneously.

All tests fail to reject the null hypothesis that the series contains a unit root process.

### Conduct Cointegration Tests

Econometric Modeler supports the Engle-Granger and Johansen cointegration tests. Before conducting a cointegration, determine whether there is visual evidence of cointegration by regressing each interest rate on the other interest rates.

In the cointegrating relation, assign

`INT_L`

as the dependent variable and`INT_M`

and`INT_S`

as the independent variables. In the**Time Series**pane, click`INT_L`

.On the

**Econometric Modeler**tab, in the**Models**section, click the arrow >**MLR**.In the MLR Model Parameters dialog box, select the

**Include?**check boxes of the time series`INT_M`

and`INT_S`

.Click

**Estimate**. The model variable`MLR_INT_L`

appears in the**Models**pane, its value appears in the**Preview**pane, and its estimation summary appears in the**Model Summary(MLR_INT_L)**document.Iterate steps 1 through 4 twice. For the first iteration, assign

`INT_M`

as the dependent variable and`INT_L`

and`INT_S`

as the independent variables. For the second iteration, assign`INT_S`

as the dependent variable and`INT_M`

and`INT_L`

as the independent variables.On each

**Model Summary(MLR_**tab, determine whether the residual plot appears stationary.)`seriesName`

The residual series of the regression on
`INT_L`

appears trending, and the other residual
series appear stationary. For the Engle-Granger test, choose
`INT_S`

as the dependent variable.

### Conduct Engle-Granger Test

The Engle-Granger tests for one cointegrating relation by performing two univariate regressions: the cointegrating regression and the subsequent residual regression (for more details, see Identifying Single Cointegrating Relations). Therefore, to perform the cointegrating regression, the test requires:

The response series that takes the role of the dependent variable in the first regression.

Deterministic terms to include in the cointegrating regression.

Then, the test assesses whether the residuals resulting from the
cointegrating regression are a unit root process. Available tests are the
Augmented Dickey-Fuller (`adftest`

) or Phillips-Perron
(`pptest`

) test. Both perform a residual regression to
form the test statistic. Therefore, the test also requires:

The unit root test to conduct, either the

`adftest`

or`pptest`

function. The residual regression model for both tests is an AR model without deterministic terms. For more flexible models, such as AR models with drift, call the functions at the command line.Number of lagged residuals to include in the AR model.

Conduct the Engle-Granger test.

In the

**Time Series**pane, click`INT_L`

and**Ctrl**+click`INT_M`

and`INT_S`

.In the

**Tests**section, click**New Test**>**Engle-Granger Test**.On the

**EGCI**tab, in the**Parameters**section:In the

**Dependent Variable**list, select**INT_S**.In the

**Residual Regression Form**list, select**PP**for the Phillips-Perron unit root test.In the

**Number of Lags**box, type`1`

.

In the

**Tests**section, click**Run Test**. Test results appear in the**EGCI**document.In the

**Tests**section, click**Run Test**again. Test results and a plot of the cointegration relation for the largest rank appear in the**EGCI**tab.

The test rejects the null hypothesis that the series does not exhibit cointegration. Although the test is suited to determine whether series are cointegrated, the results are limited to that. For example, the test does not give insight into the cointegrating rank, which is required to form a VEC model of the series. For more details, see Identifying Single Cointegrating Relations.

### Johansen Test

The Johansen cointegration test runs a separate test for each possible
cointegration rank 0 through *m* – 1, where
*m* is the number of series. This characteristic makes the
Johansen test better suited than the Engle-Granger test to determine the
cointegrating rank for a VEC model of the series. Also, the Johansen test
framework is multivariate, and, therefore, results are not relative to an
arbitrary choice for a univariate regression response variable. For more
details, see Identifying Multiple Cointegrating Relations.

The Johansen test requires:

Deterministic terms to include in the cointegrating relation and the model in levels.

Number of short-run lags in the VEC model of the series.

Because the raw series do not contain a linear trend, assume that
the only deterministic term in the model is an intercept in the cointegrating
relation (`H1*`

Johansen form), and include 1 lagged
difference term in the model.

With

`INT_L`

,`INT_M`

, and`INT_S`

selected in the**Time Series**pane, click**Tests**section, click**New Test**>**Johansen Test**.On the

**JCI**tab, in the**Parameters**section, in the**Number of Lags**box, type`1`

, and in the**Model**list select**H1***.In the

**Tests**section, click**Run Test**. Test results and a plot of the cointegration relation for the largest rank appear in the**JCI**tab.

Econometric Modeler conducts a separate test for each cointegration rank 0
through 2 (the number of series – 1). The test rejects the null hypothesis of no
cointegration (**Cointegration rank** = 0), but fails to reject
the null hypotheses of **Cointegration rank** ≤ 1 and
**Cointegration rank** ≤ 2. The results suggest that the
cointegration rank is 1.