694—Chapter 38. Cointegration Testing

there is one cointegrating relation. Conditional on there being only one cointegrating relation, the LR test does not reject the imposed restriction at conventional levels.

The output also reports the estimated b and a imposing the restrictions. Since the cointegration test does not specify the number of cointegrating relations, results for all ranks that are consistent with the specified restrictions will be displayed. For example, suppose the restriction is:

B(2,1) = 1

Since this is a restriction on the second cointegrating vector, EViews will display results for ranks r = 2, 3, º, k – 1 (if the VAR has only k = 2 variables, EViews will return an error message pointing out that the “implied rank from restrictions must be of reduced order”).

For each rank, the output reports whether convergence was achieved and the number of iterations. The output also reports whether the restrictions identify all cointegrating parameters under the assumed rank. If the cointegrating vectors are identified, asymptotic standard errors will be reported together with the parameters b .

Single-Equation Cointegration Tests

You may use a group or an equation object estimated using cointreg to perform Engle and Granger (1987) or Phillips and Ouliaris (1990) single-equation residual-based cointegration tests. A description of the single-equation model underlying these tests is provided in “Background” on page 219. Details on the computation of the tests and the associated options may be found in “Residual-based Tests,” on page 234.

Briefly, the Engle-Granger and Phillips-Ouliaris residual-based tests for cointegration are simply unit root tests applied to the residuals obtained from a static OLS cointegrating regression. Under the assumption that the series are not cointegrated, the residuals are unit root nonstationary. Therefore, a test of the null hypothesis of no cointegration against the alternative of cointegration may be constructed by computing a unit root test of the null of residual nonstationarity against the alternative of residual stationarity.

How to Perform a Residual-Based Cointegration Test

We illustrate the single-equation cointegration tests using Hamilton’s (1994) purchasing power parity example (p. 598) for the log of the U.S. price level (P_T), log of the Dollar-Lira exchange rate (S_T), and the log of the Italian price level (PSTAR_T) from 1973m1 to 1989m10. We will use these data, which are provided in “Hamilton_rates.WF1”, to construct Engle-Granger and Phillips-Ouliaris tests assuming the constant is the only deterministic regressor in the cointegrating equation.

To carry out the Engle-Granger of Phillips-Ouliaris cointegration tests, first create a group, say G1, containing the series P_T, S_T, and PSTAR_T, then select View/Cointegration

696—Chapter 38. Cointegration Testing

Date: 05/11/09 Time: 15:52

Series: P_T S_T PSTAR_T

Sample: 1973M01 1989M10

Included observations: 202

Null hypothesis: Series are not cointegrated

Cointegrating equation deterministics: C

Additional regressor deterministics: @TREND

Fixed lag specification (lag=12)

**Number of stochastic trends in asymptotic distribution

The top two portions of the output describe the test setup and summarize the test results. Regarding the test results, note that EViews computes both the Engle-Granger tau-statistic (t-statistic) and normalized autocorrelation coefficient (which we term the z-statistic) for residuals obtained using each series in the group as the dependent variable in a cointegrating regression. Here we see that the test results are broadly similar for different dependent variables, with the tau-statistic uniformly failing to reject the null of no cointegration at conventional levels. The results for the z-statistics are mixed, with the residuals from the P_T equation rejecting the unit root null at the 5% level. On balance, however, the test statistics suggest that we cannot reject the null hypothesis of no cointegration.

The bottom portion of the results show intermediate calculations for the test corresponding to each dependent variable. “Residual-based Tests,” on page 234 offers a discussion of these statistics. We do note that there are only 2 stochastic trends in the asymptotic distribution (instead of the 3 corresponding to the number of variables in the group) as a result of our assumption of a non-zero drift in the regressors.

Alternately, you may compute the Phillips-Ouliaris test statistic. Once again select View/ Cointegration Test/Single-Equation Cointegration Test from the Group toolbar or main menu, but this time choose Phillips-Ouliaris in the Test Method combo.