Gibbs Sampling in ARX Models with the “Minnesota” Prior

Abstract

In 1980ies Bayesian ARX models became an important instrument in economic forecasting (see e.g. Litterman 1986). This so-called Minnesota or Litterman prior consists of a hierarchical tightness structure, where 2 hyperparameters have to be specified by the user: λ, which controls the tightness of the own past of a time series and θ which controls the tightness of the past of all other time series. The ARX model for the multivariate time series {(y _1t ,…,y _kt ), t = 1,…,T} explains one time series by the own history and the present and the past of the other time series. The ARX model we are analyzing consists of the following equations for a T x 1 vector y ₁:

$$ {y_1} = {X_1}{b_1} + {X_2}{b_2} + \ldots + {X_K}{b_K} + \varepsilon = XvecB + \varepsilon $$

(1)

with the T x p’ K regressor matrix X = [X ₁ : X ₂ :…: X _K ], where p’ = p + 1. The first T x p’ block X ₁ = [1_n: y _1,‒1:…: y _1,‒p] consists of the constant and the past of the left hand variable. Each Tx(p+1) block X _k represents the current and the past of the time series y_k = {y _kt } _{t =1,…,T} . Thus, the multiple tightness model with normal errors can be formulated as a linear regression model of the standard form y ~ N(Xb,σ ² I _T ), and b = vecB = (b ₁:…: b _K ) is the stacked coefficient vector for K time series responses, where B is a p’ x K coefficient matrix. Note that all the K columns of B are b’ _k = (b _0k , b _1k ,…, b _pk ), k = 1,…, K. The first row of B, i.e. α’ = (b ₀₁ , b ₀₂ ,…, b _0K ) contains the constant b₀₁, and all other contemporaneous coefficients starting with the response of y ₂ on y ₁.