This thesis explores the development and novel application of linear panel data methods that use latent grouping variables in the modelling of time-varying unobservable heterogeneity. The methods are tailored for use in microeconomic applications with observational panel data by: a) controlling for individual-specific intercepts; b) focusing on the economic interpretability of the time-varying heterogeneity component of the models; c) addressing the problem of estimating unknown group memberships across an unknown number of latent groups. The most general model studied also allows for latent group structures in the partial effects of observed covariates, where groups in the covariate effects can be independent from groups in the unobservable heterogeneity. Classical and Bayesian statistical methodologies are considered, with the main methodological contributions being in the development of Bayesian approaches. For the kinds of applications studied, the Bayesian methods are shown to have more favourable properties, both in principle and in practice. Empirical applications to retirement decumulation and smoking policy in Australia demonstrate how the methods developed in this thesis may be used to learn about economically meaningful latent behavioural patterns across a range of applications.