Yes, there were a few analysts who saw the Great Recession coming. And, no, so far as I can tell, they didn't acquire this insight aided by DSGE models. They looked at the vulnerable structures and pulled the fire alarm. They didn't, so far as I can tell, run any fancy stat pack routines, Bayesian or otherwise.
Here's one post mortem for your reading pleasure.
Of particular interest ('He' is Raghuram Rajan):
He said the rollout of complicated instruments such as credit-default swaps and mortgage-backed securities made the global financial system a riskier place.
If you follow the link in his section, you find this bit:
He then focused on the "credit default swaps" which promised to repay delinquent loans in exchange for moderate insurance premiums. Noting that nobody really knew how realistically these swaps were priced, Rajan said that the banks were probably taking excessive risks because they trusted that the insurer would repay them. In these circumstances, a sudden increase in defaulting loans could exceed the reserves of the insurer, leading to a financial crisis. This is exactly what happened two years later, leading to the 2008 financial crisis.
The "It's all Blythe Masters fault!" quote. Not mentioned in either reporting is the key fact of CDS: it's exactly the same as betting on someone else's throw of the dice at a casino. The CDS is a pure bet by an entity not attached to the asset being bet on. And, crucially, there was no limit to the betting. Now you know why AIG crashed. Or,
as reported
Or you could just sell GE credit default swaps. You get money from other banks, and all you have to give is the promise to pay if something bad happens. That's zero money down and a profit limited only by how many you can sell.
That was from 2005, long before the cracks began to appear. But he noted that the CDS, et al, had changed the rules of the game in a negative way. Alas, such insight comes not from running fancy quant models, but from reading the newspaper; and having eyes and ears inside financial institutions. What data
can tell you: how are the gross money flows changing? But you don't need regression to see such. As the Great Recession proved, time series analysis, of any variety, only works when the rules of the game are stable. Change the rules...?