priors.minnesota_prior()

Return the Minnesota (Litterman) prior moments for VAR coefficients.

Usage

Source

priors.minnesota_prior(
    n_lags,
    n_obs,
    tightness,
    *,
    cross_shrinkage=0.5,
    decay="harmonic",
    own_lag_mean=1.0
)

The prior shrinks the coefficient of variable j at lag l in the equation of variable i toward a mean that is nonzero only for the first own lag, with a standard deviation that tightens on longer lags and on cross-variable lags:

m_{l, ij} = \begin{cases} m_{\text{own}} & l = 1,\ i = j \\ 0 & \text{otherwise} \end{cases}, \qquad s_{l, ij} = \lambda \, d(l) \, \begin{cases} 1 & i = j \\ \kappa & i \neq j \end{cases},

with d(l) = 1/l (decay="harmonic") or d(l) = 1/l^2 (decay="geometric", the label Impulso uses; in Doan, Litterman and Sims it is the harmonic decay with exponent two). m_{\text{own}} = 1 is the random-walk belief for series in levels; use own_lag_mean=0.0 for differenced or otherwise stationary series. The parameterization follows Impulso’s MinnesotaPrior (fixed tightness rather than an estimated one, no residual-scale ratios), with own_lag_mean as the one addition. The classic Litterman scale ratio for the coefficient of variable j in equation i is \sigma_i / \sigma_j; apply it, when the series are on different scales, as scale * (sigma[:, None] / sigma[None, :]).

Parameters

n_lags: int

Number of lags p \ge 1.

n_obs: int

Number of series k \ge 1.

tightness: ArrayLike

Overall shrinkage \lambda > 0 (the standard deviation of the first own lag). A modeling choice, so it has no default; a jax scalar is accepted, which lets a model sample it.

cross_shrinkage: ArrayLike = 0.5

Relative shrinkage \kappa \in [0, 1] of cross-variable lags versus own lags (1.0 treats them alike, 0.0 pins them to the mean).

decay: Literal["harmonic", "geometric"] = "harmonic"

Lag decay d(l): "harmonic" for 1/l, "geometric" for 1/l^2.

own_lag_mean: ArrayLike = 1.0

Prior mean m_{\text{own}} of the first own lag.

Returns

loc: Float[Array, "lags obs obs"]

Prior means in the phi layout of var_step().

scale: Float[Array, "lags obs obs"]

Prior standard deviations in the same layout.

Raises

ValueError

If n_lags or n_obs is below one, or a Python-number tightness is not positive or cross_shrinkage is outside [0, 1] (an unknown decay is a type error under the package’s runtime type checking).

Examples

loc, scale = minnesota_prior(n_lags=2, n_obs=3, tightness=0.5, own_lag_mean=0.0)
phi = numpyro.sample("phi", dist.Normal(loc, scale).to_event(3))

References

Litterman, R. B. (1986). Forecasting with Bayesian vector autoregressions: five years of experience. Journal of Business & Economic Statistics, 4(1).

Doan, T., Litterman, R. B. and Sims, C. A. (1984). Forecasting and conditional projection using realistic prior distributions. Econometric Reviews, 3(1).

Pinder, T. Impulso, MinnesotaPrior (documentation, repository).