Not really but we do know what did happen.
Before you post it, your come back will be, it prevented a depression. Which is just a guess, just as it would be a guess to say what would happen without the stimulus.
The entire theory for enacting the stimulus was what happened in 1929 when the stock market crash similar to 2008. The (GOP) government did nothing and just let things work themselves out on their own ("the market will adjust"). There was no stimulus, no government actions...nothing.
Guess what happened? The depression got much...MUCH....worse...the economy went from being bad to totally falling off a cliff. It led to the 1932 elections with Democrats taking over the government and the peak of the depression didn't occur until 1933...a full four years after it started.
So yeah....we kinda have a good idea what would've happened without stimulus. The central government was the ONLY creditable name out there to get money flowing...it had to take action, and it did.