dfuller - MacKinnon critical values change when you specify drift option

[Sean Fiedler]

I am using the dfuller command to test for a unit root, and am looking at the critical values returned for comparison with my test statistic. Whether I include the option "drift" or do not include it, the estimated regression used to create the test statistic is the same, so I am assuming that having "drift" is just the default option for the command. I can also use the "reg" option to see the reported regression output.

My problem is that the critical values that get reported do change (and change greatly!), based on whether I specify the drift option or not, even though the estimated regression is the same. The reported test statistic is the same, the regression is the same, but the critical values (and corresponding p-value) are very different. Is this a bug, or operating as intended?

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dfuller input , reg
dfuller input, drift reg

[Sebastian Kripfganz]

That is intended. The option drift affects the null hypothesis of the test, assuming that the coefficient of L1.input is zero but allowing the intercept term to be nonzero. Without the drift option, it is assumed under the null hypothesis that the intercept is zero as well. Yet, because of the super-consistency of the autocorrelation parameter (faster convergence rate than the estimator for the intercept and the coefficient of additional first-differenced terms, if any) we have the odd result that also without the drift assumption the test statistic itself does not depend on whether we include an intercept or not but the distribution of the test statistic under the null hypothesis is affected. The intercept is often included to get a better power of the test even when assuming that there is no drift under the null hypothesis.

[Sean Fiedler]

Ok I think I understand. In the regression for delta y, specifying the drift option makes the null hypothesis only about the lag term coefficient, whereas not including the drift option makes the null hypothesis about both the lag term coefficient and the intercept. And depending on which case, the distribution of the test statistic can be different. This is news to me as I thought the claim of "y has a unit root" was entirely dependent on just the lag term coefficient. But I guess the claim of "y has a unit root with drift" is a different claim/different hypothesis. Thank you very much for the explanation.