  This paper proposes two distinct contributions to econometric analysis of
large information sets and structural instabilities. First, it treats a
regression model with time-varying coefficients, stochastic volatility and
exogenous predictors, as an equivalent high-dimensional static regression
problem with thousands of covariates. Inference in this specification proceeds
using Bayesian hierarchical priors that shrink the high-dimensional vector of
coefficients either towards zero or time-invariance. Second, it introduces the
frameworks of factor graphs and message passing as a means of designing
efficient Bayesian estimation algorithms. In particular, a Generalized
Approximate Message Passing (GAMP) algorithm is derived that has low
algorithmic complexity and is trivially parallelizable. The result is a
comprehensive methodology that can be used to estimate time-varying parameter
regressions with arbitrarily large number of exogenous predictors. In a
forecasting exercise for U.S. price inflation this methodology is shown to work
very well.
