Gary A, Wagner & Erick M. Elder (both of University of Arkansas-Little Rock, Department of Economics & Finance) have published Revenue Cycles and the Distribution of Shortfalls in U.S. States: Implications for an "Optimal" Rainy Day Fund, 60 Nat’l Tax J. 727 (2007). Here is the abstract:

Slowdowns in economic activity often leave state policymakers facing severe budget shortfalls and the prospects of reducing services. In this paper we apply a Markov switching regression to monthly state–level data to model the distribution of expansions and contractions. This allows us not only to construct distributions of the revenue shortfalls states are likely to confront during recessions, but also to construct savings rate rules that depend on the uncertain duration in both expansions and contractions. Our results have important implications for policymakers who may wish to smooth cyclical fluctuations in the budget via a rainy day fund.