In the single equation setting, EViews provides views that perform Engle and Granger (1987) and Phillips and Ouliaris (1990) residual-based tests, Hansen’s instability test (Hansen 1992b), and Park’s added variables test (Park 1992).

The Engle-Granger and Phillips-Ouliaris residual-based tests for cointegration are simply unit root tests applied to the residuals obtained from SOLS estimation of Equation (28.1). Under the assumption that the series are not cointegrated, all linear combinations of , including the residuals from SOLS, are unit root nonstationary. Therefore, a test of the null hypothesis of no cointegration against the alternative of cointegration corresponds to a unit root test of the null of nonstationarity against the alternative of stationarity.

The Engle-Granger test estimates a -lag augmented regression of the form

The number of lagged differences should increase to infinity with the (zero-lag) sample size but at a rate slower than .

We consider the two standard ADF test statistics, one based on the t-statistic for testing the null hypothesis of nonstationarity and the other based directly on the normalized autocorrelation coefficient :

where is the usual OLS estimator of the standard error of the estimated

(Stock 1986, Hayashi 2000). There is a practical question as to whether the standard error estimate in Equation (28.19) should employ a degree-of-freedom correction. Following common usage, EViews standalone unit root tests and the Engle-Granger cointegration tests both use the d.f.-corrected estimated standard error , with the latter test offering an option to turn off the correction.

In contrast to the Engle-Granger test, the Phillips-Ouliaris test obtains an estimate of by running the unaugmented Dickey-Fuller regression

and using the results to compute estimates of the long-run variance and the strict one-sided long-run variance of the residuals. By default, EViews d.f.-corrects the estimates of both long-run variances, but the correction may be turned off. (The d.f. correction employed in the Phillips-Ouliaris test differs slightly from the ones in FMOLS and CCR estimation since the former applies to the estimators of both long-run variances, while the latter apply only to the estimate of the conditional long-run variance).

As with ADF and PP statistics, the asymptotic distributions of the Engle-Granger and Phillips-Ouliaris and statistics are non-standard and depend on the deterministic regressors specification, so that critical values for the statistics are obtained from simulation results. Note that the dependence on the deterministics occurs despite the fact that the auxiliary regressions themselves exclude the deterministics (since those terms have already been removed from the residuals). In addition, the critical values for the ADF and PP test statistics must account for the fact that the residuals used in the tests depend upon estimated coefficients.

MacKinnon (1996) provides response surface regression results for obtaining critical values for four different assumptions about the deterministic regressors in the cointegrating equation (none, constant (level), linear trend, quadratic trend) and values of from 1 to 12. (Recall that is the number of cointegrating regressors less the number of deterministic trend regressors excluded from the cointegrating equation.) When computing critical values, EViews will ignore the presence of any user-specified deterministic regressors since corresponding simulation results are not available. Furthermore, results for will be used for cases that exceed that value.

To perform the Engle-Granger test, open an estimated equation and select View/Cointegration and select Engle-Granger in the Test Method dropdown. The dialog will change to display the options for this specifying the number of augmenting lags in the ADF regression.

Most of the entries are self-explanatory, though a few deserve a bit of discussion. First, the “Rho S.E.” and “Residual variance” are the (possibly) d.f. corrected coefficient standard error and the squared standard error of the regression. Next, the “Long-run residual variance” is the estimate of the long-run variance of the residual based on the estimated parametric model. The estimator is obtained by taking the residual variance and dividing it by the square of 1 minus the sum of the lag difference coefficients. These residual variance and long-run variances are used to obtain the denominator of the z-statistic ( Equation (28.18)). Lastly, the “Number of stochastic trends” entry reports the value used to obtain the p-values. In the leading case, is simply the number of cointegrating variables (including the dependent) in the system, but the value must generally account for deterministic trend terms in the system that are excluded from the cointegrating equation.

The dialog changes to show a single Options button for controlling the estimation of the long-run variance and the strict one-sided long-run variance . The default settings instruct EViews to compute these long-run variances using a non-prewhitened Bartlett kernel estimator with a fixed Newey-West bandwidth. To change these settings, click on the Options button and fill out the dialog. Since the default settings are sufficient for our needs, simply click on the OK button to compute the test statistics.

There are a couple of new results. The “Bias corrected Rho - 1” reports the estimated value of Equation (28.21) and the “Rho* S.E.” corresponds to Equation (28.23). The “Long-run residual variance” and “Long-run residual autocovariance” are the estimates of and , respectively. It is worth noting that the ratio of to the S.E. of the regression, which is a measure of the amount of residual autocorrelation in the long-run variance, is the scaling factor used in adjusting the raw t-statistic to form tau.

Hansen (1992) outlines a test of the null hypothesis of cointegration against the alternative of no cointegration. He notes that under the alternative hypothesis of no cointegration, one should expect to see evidence of parameter instability. He proposes (among others) use of the test statistic, which arises from the theory of Lagrange Multiplier tests for parameter instability, to evaluate the stability of the parameters.

The statistic examines time-variation in the scores from the estimated equation. Let be the vector of estimated individual score contributions from the estimated equation, and define the partial sums,

where by construction. For FMOLS, we have

where is the residual for the transformed regression. Then Hansen chooses a constant measure of the parameter instability and forms the statistic

For FMOLS, the natural estimator for is

The and may be defined analogously to least squares for CCR using the transformed data. For DOLS is defined for the subset of original regressors , and may be computed using the method employed in computing the original coefficient standard errors.

The distribution of is nonstandard and depends on , the number of cointegrating regressors less the number of deterministic trend regressors excluded from the cointegrating equation, and the number of trending regressors in the system. Hansen (1992) has tabulated simulation results and provided polynomial functions allowing for computation of p-values for various values of and . When computing p-values, EViews ignores the presence of user-specified deterministic regressors in your equation.

Park’s test is an added variable test. The test is computed by testing for the significance of spurious time trends in a cointegrating equation estimated using one of the methods described above.

Suppose we estimate equation Equation (28.1) where, to simplify, we let consist solely of powers of trend up to order . Then the Park test estimates the spurious regression model including from to spurious powers of trend

and tests for the joint significance of the coefficients . Under the null hypothesis of cointegration, the spurious trend coefficients should be insignificant since the residual is stationary, while under the alternative, the spurious trend terms will mimic the remaining stochastic trend in the residual. Note that unless you wish to treat the constant as one of your spurious regressors, it should be included in the original equation specification.

Since the additional variables are simply deterministic regressors, we may apply a joint Wald test of significance to . Under the maintained hypothesis that the original specification of the cointegrating equation is correct, the resulting test statistic is asymptotically .