Food Stamps

I was reading this piece by Ed Glaeser (the danger of following links posted at Noahpinion) and came across this rather interesting sentance:

The childhood obesity problem should also make us wonder whether food stamps are really good for kids.

My question is rather simple: how do we know that these two factors are causally related?  Chuldhood obesity is a complicated problem, but one possible driver is low quality food (such as potato chips) that is cheap, easily stored and (per calorie) relatively inexpensive.  Is it not plausible that reducing food budgets could increase obesity by focusing food intake even more on these foods? 

I worry when we attribute a complex phenomonon (seen at all sorts of socio-economic levels) with a single government program.  I am not saying that this statement is incorrect (and it is phrased as speculative), but it seems like too important of a proposition to be confinded to a single sentance.  In particular, I would be interested in the counter-factuals:

1. Food Stamps
2. Cash Transfers
3. No Assistence

And a comparison of childhood outcomes (obesity but also starvation) under these three different scenarios. Or am I missing the relevant research and we already know the answer?