Sylvia Kaufmann University of Basel Master/Doctoral level FS25 Advanced Time Series Analysis The lecture introduces Bayesian econometrics, with a particular focus on time series analysis, from univariate to multivariate high-dimensional. The primary goal in Bayesian inference is to derive the posterior distribution of an object of interest, being usually parameters or some latent variables. Therefore, in a first part we define the basic components specifying the Bayesian setup, the prior and the likelihood, and discuss principles of posterior updating. As for most econometric models the posterior distribution is not of a known standard form nor available in analytical form, the posterior distribution is approximated or estimated by sampling methods. We introduce two generic samplers based on Markov chain Monte Carlo (MCMC) simulation methods to estimate the posterior distribution: Metropolis-Hastings and Gibbs sampling. Bayesian inference inherently lends itself to a probabilistic interpretation or discussion of model estimates. To quantify uncertainty, we derive procedures to obtain credible intervals, for parameters as well as (non)linear transformations of parameters. Finally, we also discuss approaches to perform model choice or (forecast) evaluation, like MCMC-based estimation of the marginal likelihood or K-fold cross-validation. The Bayesian approach circumvents estimation difficulties when either data is scarce or high-dimensional. To deal with these issues, we discuss ways of specifying informative prior distributions and prior distributions that induce shrinkage into parameters. In a last part, we introduce latent variables which allow extending models to regime-switching parameters or extracting a small number of common factors from high-dimensional datasets. The lecture also includes the analytical discussion of time series models. We derive properties of the time series process, discuss stationarity and invertibility conditions, derive conditional and unconditional moments. As single parameters are not of prime interest, tools like impulse responses and variance decomposition are used to interpret multivariate time series models. We discuss various strategies of structural identification. The lecture includes exercise sessions with applications in time series modelling. Syllabus Slides and other material available for download in ADAM