This dissertation is comprised of three essays. In the first chapter, I quantify the effect of government intervention in the U.S. housing market to determine whether it can explain the nonexistence of mortgage contracts that are contingent on house prices, which are found to be optimal in the mortgage design literature. In the model, contract and down payment choices, and the corresponding mortgage interest rates, are endogenous for heterogeneous households that are subject to idiosyncratic income and house price shocks, as well as to an aggregate house price shock. I find that the implicit subsidy to government-sponsored enterprises leads to the dominance of fixed-rate mortgages in the U.S. housing market, and in a world without government intervention, mortgage contracts that are contingent on house prices emerge endogenously. In this world, contingent contracts decrease the cyclicality of foreclosure rate by adjusting the value of debt during a housing crisis. However, they increase the average foreclosure rate in normal times due to endogenously decreasing the down payment of households, since contingent contracts are relatively cheaper for low down payment options in equilibrium. In the second chapter, Luis Miguel Diez Catalan, Simone Civale and I show the limitations of normal mixtures. We document that normal mixtures are flexible enough to capture the empirical patterns documented in the literature. In the third chapter, we develop and test a discretization method to calibrate a Markov chain that features non-zero skewness and high kurtosis. The proposed method applies the logic of Tauchen (1986) to a first-order autoregressive process with normal mixture innovations, which, as we discuss, can be calibrated to feature non-zero skewness and high kurtosis. We then illustrate an application of our method in an Aiyagari economy. We find that an idiosyncratic shock with higher kurtosis decreases the equilibrium interest rate, whereas higher left skewness increases it.