Let's say I go long US equity futures (for example, S&P 500) and short European equity futures (for example, Eurostoxx or Dax) so that dollar value of notional exposure is equal (for example, if I buy 100 E-mini S&P 500 futures that represents 100*$50*1,195=$5,975,000 and I need to calculate the number of contracts of european equity futures that represents that same amount taking into consideration exchange rate and notional value that a contract represents). Intuitively I know that such spread can still have a directional risk exposure because 'betas' are not matched. There are a few challenges in approaching verifying that empirically because settlement times are different, etc. Has anybody worked on a similar project? I'd appreciate if you have any suggestions for me.

I guess technically you should try to match the betas only. for that multiply the daily xchange rate to Eusrostaxx to convert to dollar and get beta with S&P . Divide this by the currency and contract size of Eurotoxx future . Just remember that you have exposure to both currency and equity movement here .