# Do Engle-Granger residuals need to be normally distributed?

Do EngleGranger residuals need to be normally distributed?

• Does the Engle-Granger paper say anything about this? Have you checked it? (It will probably stay a rhetorical question before the OP gets enough points to comment.) – Richard Hardy Mar 21 '17 at 20:12

The Engle and Granger test for cointegration is basically an ADF test in the estimated error of the long run equation, say $\hat{u}$. The ADF test is looking for stationarity of this process (or trend-stationary). It does not require normality, as there are many non-normal processes which are stationary (see here or here).

For instance, this paper compares the power of different unit root tests (including ADF) for a whole range of distribution of the variable in question. Distributions include normal and non-normal like Chi-squared, Beta, and Laplace (double exponential in their terminology). Their interest is to propose more powerful unit root tests, but an aside conclusion is that, even under non-normality, the ADF test performs "well" (their proposed tests perform better though).

Having this in mind, the next question is: does the error in a cointegrated, long run equation has to be normal? This is akin to the question of whether errors in a regression have to be normal. There are tons of related posts in CV about this. The short answer seems to be: hopefully yes, but might not be. Examples here, here, here, and more generally, here.

(Richard Hardy's summary of this normality issue in the comments is certainly more precise on this than my summary. Basically, if you have a small sample, you want normality, so that standard tests have the theoretical distribution the should have (so, tests are valid). If you have a large sample, the Central Limit theorem will do that job for you.)

• Three points: (1) minor quibble: predicted error? I would say residual (=estimated error); (2) the existence of nonnormal yet stationary processes cannot by itself imply anything about the assumptions of the ADF test, so the first paragraph suggests a misleading connection; (3) the short answer is, normality of errors is required only for the validity of small sample $t$ and $F$ tests (while consistency, asymptotic normality, Gauss-Markov theorem does not rely on normality). – Richard Hardy Mar 22 '17 at 7:41
• Thanks. Added the comments you mention. Let me know if I was right in (3). Feel free to edit if not. – luchonacho Mar 22 '17 at 10:18
• That's better, although the first paragraph is unchanged and still suggests a relation between the assumptions of ADF test and general requirements for stationary processes. – Richard Hardy Mar 22 '17 at 10:33
• Mmm, what I am saying is that ADF tests for stationarity and that normality is not necessary, as there are many stationary but non-normal processes. The point of the new second paragraph was to show an example that indeed the ADF test remains valid even with non-normal but stationary processes, confirming the first paragraph claim. – luchonacho Mar 22 '17 at 11:02
• The problem is that ADF is not a general test for any type of stationarity, and therefore it may or may not have specific assumptions beyond what is required for stationarity in general -- just to stress the logic of the argument. Now we know that ADF test does not require normality -- but it is not because stationarity does not require normality, but rather because such is the construction/design of the ADF test. – Richard Hardy Mar 22 '17 at 11:10