This dissertation consists of three projects which focus on methods, with direct applications to American and comparative politics, and are intended to either advance researchers' understanding of existing methods or to help develop new methods for political science research. These works focus generally on two methodological areas: discrete choice modeling and time-series cross-section (TSCS) methodology. In chapter 1 I conduct Monte Carlo simulations to compare multinomial logit (MNL), multinomial probit (MNP), and mixed/random parameters logit (MXL) on 7 criteria, including the accuracy of coefficient point estimates. Simulated data represent a range of violations of the independence of irrelevant alternatives (IIA) assumption, and show that MNL nearly always provides more accurate coefficients than MNP and MXL, even when the IIA assumption is severely violated. In chapter 2 a new method, called the between effects estimation routine (BEER), is developed to maximize information from TSCS data to model the cross-sectional effects while allowing these effects to change over time. This method is applied to two examples. First, it is used to analyze the variation in regional authority and federalism in 21 countries over 54 years. Second, it is used to reconsider the effect of income on state voting in US presidential elections. In chapter 3 I develop a method to estimate a non-linear logistic regression with survey weights which uses Markov-Chain Monte Carlo (MCMC) estimation. To demonstrate the utility of the method, I consider a voting model for U.S. presidential general elections which is non-linear, and addresses a long-standing theoretical debate in the study of American elections regarding the moderating role of personal importance on the effect of issue evaluations on the vote.