Is there a technique to choose the time-frame for a cointegration test (eg Augmented Dickey-Fueller's)?
I have never read about any formal procedure for this. And, I don't remember this issue is even treated in C.Alexander's book Market Risk Analysis, Practical Financial Econometrics that dedicate a whole part to the cointegration of financial time series. One may well find tests for cointegration succeeding (failing) for a certain time frame and failing (succeeding) for a longer (shorter) time length.
This may be of interest to test for integration for different time lengths. And, one may argue the more often it succeed the better it is to take advantage of the paired-behavior of your time series.
Another way around is to consider first the time frame you wish, depending on the issue at stake (is that pure stats observation? do you want to ground trading decisions on these tests?).
Finally, one may wish to choose time frames avoiding known in advance events likely to cause shocks in prices (company earnings season, ....).
Why not use the entire data. But before you do clean the data by checking for structural breaks (t / F statistic of a dummy variable). If there exist a functional break, then you know you have to perform the co-integration tests separately for each time frames.
If you use a preferable time then the question is what criteria will you be choosing this time-frame on.