Cointegration: ADF Test Critical Values

The second step in the Engle-Granger cointegration approach is to test if the residuals from the regression have unit root via ADF test. When we apply the ADF test on residuals (estimates) instead on actual time-series we can not use the Dickey and Fuller critical values and p-values that are reported. The critical values when we use the ADF test on residuals are stricter than the original critical values (this means that the critical values are lower and thus it is less likely to reject null hypothesis of unit root). There are several sets of critical values – as Engle and Yoo (1987), MacKinnon (1991), Phillips and Ouliaris (1990).

Phillips and Ouliaris critical values are available here: http://finpko.faculty.ku.edu/myssi/FIN938/Phillips%20%26%20Ouliaris_Asymp%20Props%20of%20Resid%20Based%20Tests%20for%20Coint_Econometrica_1990.pdf – Table IIa (no intercept and no trend), IIb (intercept but no trend) and IIc  (both intercept and trend).

We can express formally the three equations as follows:

From the table above, if we have one explanatory variable, constant only at 5% level of significance, the critical value is -3.37 (regression b case); in the case of both constant and trend, the critical value is -3.8.

This is for Phillips and Ouliaris critical values. MacKinnon (http://qed.econ.queensu.ca/working_papers/papers/qed_wp_1227.pdf) approaches the situation differently – he estimates response surface regression and the function used to calculate the critical values is:

Here is a table with MacKinnon critical values, corresponding to the second case regression – intercept, no trend (when comparing the critical values, it should be noted that in MacKinnon approach N is the number of cointegrating variables, while in Phillips and Ouliaris N is number of explanatory variables):

Based on the same procedure I interpolated the critical values for the third case – intercept and trend:

From the table with MacKinnon critical values at 5% level of significance if we have two variables with 200 observations and we have only a constant included in the ADF-regression, the critical value is -3.368; in case of 500 observations, the critical value is -3.350.

If we have both constant and trend, at 5% level of significance and 200 observations, the MacKinnon critical value is -3.828.

It is generally accepted that if intercept is included in the cointegrating regression, it is omitted in the ADF equation.