Economics is a science of partial knowledge. Data are scarce relative to the number of plausible models, structural relationships shift over time, and policy decisions must be made under profound uncertainty. Bayesian econometrics provides a natural language for this setting: prior distributions encode economic theory and institutional knowledge, the posterior quantifies what the data support, and model averaging reflects honest uncertainty about the correct specification.
The field grew from the pioneering work of Arnold Zellner at the University of Chicago in the 1960s and 1970s, gained computational traction with MCMC methods in the 1990s, and is now central to macroeconomic forecasting, monetary policy analysis, and structural estimation.
Bayesian Vector Autoregressions
The Bayesian vector autoregression (BVAR) is perhaps the most influential tool in Bayesian macroeconometrics. A VAR models a vector of macroeconomic time series (GDP growth, inflation, interest rates) as a linear function of its own lagged values. With many variables and lags, the number of parameters explodes, making unrestricted estimation imprecise. The Bayesian solution is to impose shrinkage through the prior.
Minnesota Prior (Litterman, 1986) E[Aₖ] = I for k=1, 0 otherwise (random walk prior)
Var[(Aₖ)ᵢⱼ] = λ² / k² · (σᵢ / σⱼ)² for i ≠ j (cross-variable shrinkage)
Where λ → Overall shrinkage hyperparameter
k → Lag order (more distant lags shrunk more aggressively)
σᵢ → Scale of variable i (normalization factor)
The Minnesota prior, introduced by Robert Litterman at the Federal Reserve Bank of Minneapolis in 1986, embodies the belief that each economic variable approximately follows a random walk — a reasonable baseline for many macro series. The prior shrinks off-diagonal coefficients (cross-variable effects) and higher-order lags toward zero, dramatically reducing effective dimensionality while allowing the data to overcome the prior when evidence is strong.
Before the Minnesota prior, VARs were criticized for overfitting and producing poor forecasts. The Bayesian shrinkage approach resolved this tension: BVARs with the Minnesota prior consistently outperform both unrestricted VARs and many structural models in macroeconomic forecasting comparisons. The Federal Reserve Bank of Minneapolis, the European Central Bank, and numerous other institutions use BVARs as workhorse forecasting tools.
Bayesian Model Averaging
Economists routinely face model uncertainty: should the regression include the exchange rate? The oil price? A nonlinear term? Bayesian model averaging (BMA) provides a principled solution: compute the posterior probability of each model given the data, and average predictions and parameter estimates across models weighted by these probabilities.
p(Mₖ | y) ∝ p(y | Mₖ) · p(Mₖ)
Where Δ → Quantity of interest (forecast, parameter, policy effect)
p(Mₖ | y) → Posterior model probability
p(y | Mₖ) → Marginal likelihood of model Mₖ (integrated likelihood)
The marginal likelihood automatically penalizes complex models through the Bayesian Occam's razor: a model with many parameters spreads its prior predictive probability thinly, so it receives less credit for fitting the data unless the improvement is substantial. BMA has been applied extensively to cross-country growth regressions, where dozens of potential predictors of economic growth have been proposed and no single specification dominates.
Structural Estimation
Modern macroeconomics relies on dynamic stochastic general equilibrium (DSGE) models — systems of equations derived from optimizing behavior of households and firms. These models have many parameters (preferences, technology, policy rules) that must be estimated from aggregate data. Bayesian methods are the standard approach.
The Smets and Wouters (2003, 2007) model, a medium-scale DSGE model estimated with Bayesian methods, demonstrated that DSGE models could forecast as well as unrestricted BVARs. This result was influential at central banks worldwide and cemented Bayesian estimation as the standard methodology for DSGE analysis.
"Bayesian analysis is particularly well suited for economic applications because it naturally handles the small sample sizes, identification issues, and model uncertainty that are endemic to economics." — Arnold Zellner, An Introduction to Bayesian Inference in Econometrics (1971)
Forecasting and Density Prediction
Bayesian econometrics naturally produces density forecasts — full predictive distributions for future observations — rather than point forecasts. The posterior predictive distribution integrates over parameter uncertainty and, when BMA is used, model uncertainty. Central banks increasingly report density forecasts (fan charts) for inflation and GDP growth, reflecting the full range of uncertainty about the economic outlook.
Bayesian Methods in Microeconometrics
Bayesian methods are also used in microeconometrics, particularly for discrete choice models, hierarchical models of consumer behavior, and treatment effect estimation. The Bayesian approach to instrumental variables — which can be numerically unstable in frequentist estimation when instruments are weak — provides well-defined posteriors even in challenging identification settings.
In structural econometrics, priors are not arbitrary — they encode theoretical restrictions and institutional knowledge. A prior that concentrates the price elasticity of demand near negative values reflects basic economic theory. A prior that constrains the Taylor rule coefficient on inflation to be above one reflects the expectation that central banks respond aggressively to inflation. These priors are transparent and testable, making assumptions explicit rather than hidden in model specification choices.
Key Computational Tools
Bayesian econometrics has driven and adopted key computational advances. The Gibbs sampler, popularized by Gelfand and Smith (1990), was quickly adopted for Bayesian VARs and regression models. The Metropolis-Hastings algorithm enabled estimation of nonlinear DSGE models. More recently, sequential Monte Carlo (particle filters) methods are used for estimating time-varying parameter models and for computing marginal likelihoods for model comparison.