This thesis has two central contributions. One is economic; it explains the behavior of sovereign interest rates after a default. The other contribution is principally technical; it defines and provides an accurate closed-form approximation to a common filtering problem. The main steps in the argument are as follows: - Standard economic theory predicts that after conditioning on macroeconomic conditions, the cost of sovereign borrowing is independent of a country's past default history. - Empirical evidence contradicts this prediction. Countries which have recently defaulted pay more to borrow than would be expected given their observable macroeconomic conditions. This extra cost of borrowing is the conditional default premium. - I propose a model of sovereign default where the government's immediate cost of default is governed by a hidden type. - The ability of the model to generate high post-default spreads depends crucially on the continuous nature of the government' type. - The evolution of investors beliefs of the government's type given the past default history is a threshold filtering problem. This is a common problem in many models with persistent hidden types and binary actions. - I propose an approximate solution to the threshold filtering problem, which I term the threshold filter. This provides a closed-form solution to the approximate evolution of the first two moments of the distribution of the hidden state. This is a substantially more accurate solution to the problem than standard filtering methods, such as the unscented Kalman filter or exact Gaussian filter. - I use the threshold filter to model investor's beliefs and so solve for equilibrium in the sovereign default model. - A calibrated version of the model generates quantitatively accurate high post-default spreads. - I show that the threshold filtering problem shows up in a wide variety of other economic problems, including market entry and long-term unemployment. The threshold filter can be applied to these scenarios as well.