Dirk Krueger and Hanno Lustig destroy another literature I had thought was promising...
In retrospect, I suppose it should have been obvious that uninsurable idiosyncratic risk that doesn't widen the distribution of consumption in downturns cannot help account for a large equity premium. To account for an equity return premium, you do need to make those states of the world in which stocks lose money extremely unpleasant for the marginal agent--you have to produce extremely high marginal utilities of consumption in those states of the world. And, given the correlations, you have to do so without moving aggregate consumption spending very much.
Idiosyncratic risk uncorrelated with systematic risk simply doesn't, to first order, move marginal utility away from average consumption. So it simply cannot do the job.
Hanno Lustig and Dirk Krueger (2009), "When is Market Incompleteness Irrelevant for the Price of Aggregate Risk?"
In a model with a large number of agents who have constant relative risk aversion (CRRA) preferences, market incompleteness has no effect on the premium for aggregate risk if the distribution of idiosyncratic risk is independent of aggregate shocks and aggregate consumption growth is distributed independently over time. In the equilibrium, which features trade and binding solvency constraints, as opposed to Constantinides and Duffie (1996), households only use the stock market to smooth consumption; there is no trade in bond markets. Furthermore, we show that the cross-sectional wealth and consumption distributions are not affected by aggregate shocks. These results hold regardless of the persistence of idiosyncratic shocks. A weaker irrelevance result survives when we allow for predictability in aggregate consumption growth.