The Minnesota Prior¶

The Minnesota prior (Impulso, 1986) is the most widely used prior for Bayesian VARs. It encodes the belief that each variable follows a random walk, with coefficients on other variables' lags shrunk toward zero.

Key hyperparameters¶

Parameter	Default	Meaning
`tightness`	0.1	Overall shrinkage. Smaller = more shrinkage toward prior.
`decay`	`"harmonic"`	How fast coefficients shrink on longer lags. `"harmonic"`: \(1/l\). `"geometric"`: \(1/l^2\).
`cross_shrinkage`	0.5	Relative shrinkage on other variables' lags vs own lags. 0 = only own lags matter, 1 = equal treatment.

Intuition¶

The prior mean for the coefficient on a variable's own first lag is 1.0 (random walk). All other coefficients have prior mean 0.0. The prior standard deviation controls how far the posterior can move from these defaults.

Usage in Impulso¶

from impulso import VAR
from impulso.priors import MinnesotaPrior

# Use defaults
spec = VAR(lags=4, prior="minnesota")

# Customize hyperparameters
prior = MinnesotaPrior(tightness=0.2, decay="geometric", cross_shrinkage=0.3)
spec = VAR(lags=4, prior=prior)