Feel the Causation
This is why you don't leap to conclusions from one data point.
I'm a little confused on this one. In the last year, Wisconsin's public sector employment has fallen by less than 2 percent. Moreover, a very similar change in economic conditions is happening in Minnesota, too, which hasn't seen the concerted reductions in public employment, despite the shutdown of the state government last summer (that's what that big drop below 400,000 is from). I'm not from the area, but it seems like something regional is going on.
Let's consider Chinn's argument a little more broadly, which is that the shrinking or not of the public-sector can explain the differences in the states' relative output growth. Now the Bureau of Economic Analysis says rightly that state and local governments have been a net drag on growth--0.32 percent in the fourth quarter of 2011--but I don't see the empirical support for Chinn's point.
Now there are certainly confounding third variables here, and the causation can run both ways in the graph above, but it stands to reason that the government employment reductions haven't been causing flagging growth in any state. We should keep in mind that tax revenues force state government spending to be procyclical, so layoffs may not be a cause, so much as a consequence, of a weak economy in that particular state.
For the math nerds: the correlation here is super weak and shallow. A 6 percent reduction in government employment appears needed to bring down output growth by 1 percent. The goodness of fit from linear regression is basically zero.