Of course we may never find anything thats fully integrated but if they are close to being "closely integrated" then it works. Correlation is a must. Correlated stocks tend to give better returns and are more predictable in nature. I've held some trades for over 4 months at a time with no sign of making profits. Ive exited such trades due to sheer boredom of watching them do nothing. Of course some day they did revert to mean and had i held on to them i would have made money. The question I ask then is " what is my opportunity cost of holding these positions?".
There are multiple reason due to which i dont use correlation or cointegration. I must confess in my earlier years i used to obsess over both of them but over the years i have grown to realize that its not of much use.
Out of correlation and cointegration, the latter is of course the superior statstical tool. however, there are very very few pairs which are fully or "closely integrated". I mean if you will run the test you will find very very very few pairs which are cointegrated at 90 percent confidence interval.
Even if we do find such pairs, they very rarely diverge from each other hence we dont get much trading opportunities. ( once a year types )
Correlation is a useless metric in pair trading. Firstly it does not tell you the causality. Secondly , historical correlation does not point out to future correlation. Thirdly, the coefficient of correlation will vary greatly across time frames ( monthly, 3 monthly , 6 monthly and yearly ).
Apart from their individual drawbacks, what is a common drawback with both of them is that they dont tell us about
"Structural Breaks" in pairs. Structural breaks may happen in between a fully cointegrated pair as well as a fully correlated pair ( correl coefficient =1 ). It may happen due to a variety of reasons like Earnings Surprise , M&A activity, Company Specific factors like fabrication, fraud, governance, promoters selling their stake etc. etc.
About the opportunity cost of holding a pair, we break down our trading system to trade 4 types of signals.
1. Very Short Term ( 1-2 Days or even Intra Day)
2. Short Term ( 3-7 Days )
3. Positional ( 5-20 Days )
4. Tracking the Structural Break
The days mentioned are working days, obviously. Now the time frame which is the highest is obviously the 4th one when a pair undergoes structural break. We obviously dont know the reason behind the structural break until very late ( when we read about it in media ). So the opportunity cost is highest in the 4th category.