  In all areas of human knowledge, datasets are increasing in both size and
complexity, creating the need for richer statistical models. This trend is also
true for economic data, where high-dimensional and nonlinear/nonparametric
inference is the norm in several fields of applied econometric work. The
purpose of this paper is to introduce the reader to the world of Bayesian model
determination, by surveying modern shrinkage and variable selection algorithms
and methodologies. Bayesian inference is a natural probabilistic framework for
quantifying uncertainty and learning about model parameters, and this feature
is particularly important for inference in modern models of high dimensions and
increased complexity.
  We begin with a linear regression setting in order to introduce various
classes of priors that lead to shrinkage/sparse estimators of comparable value
to popular penalized likelihood estimators (e.g.\ ridge, lasso). We explore
various methods of exact and approximate inference, and discuss their pros and
cons. Finally, we explore how priors developed for the simple regression
setting can be extended in a straightforward way to various classes of
interesting econometric models. In particular, the following case-studies are
considered, that demonstrate application of Bayesian shrinkage and variable
selection strategies to popular econometric contexts: i) vector autoregressive
models; ii) factor models; iii) time-varying parameter regressions; iv)
confounder selection in treatment effects models; and v) quantile regression
models. A MATLAB package and an accompanying technical manual allow the reader
to replicate many of the algorithms described in this review.
