Kadiyala-Karlsson (1993) Forecasting with Generalized Bayesian Vector Autoregressions

Summary

Kadiyala and Karlsson compare five prior distributions for Bayesian vector autoregression (VAR) forecasting — the Minnesota prior, Normal-Wishart, Diffuse (Jeffreys'), Normal-Diffuse, and the Extended Natural Conjugate (ENC) prior — across three forecasting experiments on Canadian GNP/M2 data and US wheat export data. The central critique of the Minnesota prior is that it forces equation-independence and treats the residual variance-covariance matrix $\Psi$ as known and diagonal — the wrong tradeoff, since practitioners can form prior beliefs about regression coefficients more easily than about $\Psi$. Priors that allow inter-equation dependence consistently match or beat the Minnesota prior; OLS is worst in every experiment.

Key Claims

Model and Notation

• VAR: $y_t = \sum_{i=1}^p y_{t-i}A_i + x_tC + u_t$, $u_t \sim N(0,\Psi)$; stacked as $\mathbf{Y} = \mathbf{Z}\Gamma + \mathbf{U}$.
• OLS estimator $\hat\Gamma$; residual standard error from univariate AR on variable $i$: $s_i$.
• Three tightness hyperparameters: $\pi_1$ (own lags), $\pi_2$ (cross lags), $\pi_3$ (exogenous). Calibrated values (Table I): $\pi_1 = 0.07$, $\pi_2 = 0.007$, $\pi_3 = 1.4\times10^5$.

The Five Priors

Minnesota prior (Litterman 1986a): equations treated independently; $\gamma_i \sim N(\tilde\gamma_i, \tilde\Sigma_i)$ with $\tilde\Sigma_i$ diagonal and $\psi_{ii}$ fixed at $0.81s_i^2$. Prior variances (eq. 2):

$$\tilde\Sigma_i = \text{diag}\!\left\{\frac{\pi_1}{k},\quad \frac{\pi_2 s_i^2}{k s_j^2},\quad 0.81\pi_3 s_i^2\right\}$$

for own-lag $k$, cross-lag $k$ of variable $j\neq i$, and exogenous coefficients respectively. Key limitation: treats $\Psi$ as known and diagonal — the opposite of what practitioners know.

Normal-Wishart prior: $\gamma|\Psi \sim N(\tilde\gamma, \Psi\otimes\tilde\Omega)$, $\Psi \sim IW(\tilde\Psi, \alpha)$ (inverse-Wishart, IW). Natural conjugate — posterior $\gamma|\Psi,y \sim N(\bar\gamma, \Psi\otimes\bar\Omega)$, $\Psi|y \sim IW(\bar\Psi, T+\alpha)$; marginal $\Gamma \sim MT(\tilde\Omega^{-1}, \tilde\Psi, \bar\Gamma, \alpha)$ (matric-variate $t$, MT). Allows non-diagonal $\Psi$ and yields closed-form posterior moments. Limitation: $\Psi\otimes\tilde\Omega$ structure forces symmetric treatment of all equations, so own-lag prior variances differ from Minnesota.

Diffuse (Jeffreys') prior: $p(\gamma,\Psi) \propto |\Psi|^{-(q+1)/2}$. Posterior centered on OLS: $E(\Gamma|y) = \hat\Gamma$; moments analogous to Zellner seemingly unrelated regressions (SUR). Same $\Psi\otimes(Z'Z)^{-1}$ restriction as Normal-Wishart.

Normal-Diffuse prior (new): $\gamma \sim N(\tilde\gamma,\tilde\Sigma)$, $p(\Psi)\propto|\Psi|^{-(q+1)/2}$. Combines a Minnesota-type prior on regression parameters with a diffuse prior on $\Psi$; allows non-diagonal $\Psi$ and inter-equation dependence. Marginal posterior of $\gamma$ is product of marginal prior and a matricvariate $t$ factor — can be bimodal when prior and likelihood strongly conflict.

ENC prior (new, Drèze-Morales 1976): reparameterizes the VAR with block-diagonal $\Delta$ (columns $\gamma_i$ on diagonal). Prior: $p(\Delta)\propto|\tilde\Psi+(\Delta-\tilde\Delta)'\bar{M}(\Delta-\tilde\Delta)|^{-\alpha/2}$, $\Psi|\Delta\sim IW(\cdot)$. Overcomes the $\Psi\otimes\Omega$ restriction on $\text{Var}(\gamma)$ that constrains the Normal-Wishart. Marginal posterior of $\Delta$ is not matricvariate $t$; no closed-form moments — numerical evaluation required.

Forecasting

• Minnesota: chain rule using posterior means.
• Normal-Wishart, Diffuse: draw directly from matricvariate $t$ posterior; evaluate forecast integral as sample mean.
• ENC, Normal-Diffuse: no closed-form posterior — importance sampling (Kloek-van Dijk 1978) with antithetic variates. Importance functions: product of multivariate $t$ distributions (ENC); 2-0 poly $t$-distribution (Normal-Diffuse, Drèze 1977).

Empirical Results

• Exp. 1 — Canadian small sample (log GNP + log M2, 17–26 obs, 12-step): ENC and Normal-Diffuse best for log GNP; Minnesota significantly worse than ENC/Normal-Diffuse in 4 cases; OLS worst.
• Exp. 2 — Canadian large sample (56–75 obs): Normal-Diffuse best for log GNP; Diffuse best for log M2; differences small as likelihood dominates prior.
• Exp. 3 — US wheat export (4-variable: SHIP, EXCH, PRICE, SALE; monthly Jan 1974–Mar 1985; 30 sets × 24-step): ENC and Normal-Wishart best overall; Normal-Diffuse and ENC dominate for SHIP and PRICE.
• General finding: Minnesota prior never provides forecasts significantly better than the more general priors; in small samples it is significantly worse.

Concepts Introduced or Extended

• Minnesota Prior — critique of equation-independence and fixed-diagonal-$\Psi$ limitation
• Noninformative Prior for VAR — Diffuse and Normal-Diffuse priors; bimodality caveat for Normal-Diffuse
• Vector Autoregression — prior sensitivity; Normal-Wishart and ENC posterior derivations

Entities Mentioned

• K. Rao Kadiyala
• Sune Karlsson

Quotes

"The prior distributions that allow for dependencies between the equations of the VAR give rise to better forecasts." (abstract)

"In no case does the Minnesota prior provide forecasts that are significantly better than the more general prior distributions."

"The Normal-Wishart and Diffuse priors are especially suitable in this context, since analytic expressions for the first and second posterior moments of the parameters are available."

My Take

The paper's central contribution is empirical: the Minnesota prior's forced equation-independence is costly in small samples, and more flexible priors consistently match or beat it. The theoretical critique — treating $\Psi$ as known diagonal is the wrong tradeoff — is well-argued and anticipates the later Sims-Zha framework. The ENC prior is technically elegant (the Drèze-Morales reparameterization is clean) but computationally demanding; the paper notes Gibbs sampling as an alternative, which Kadiyala and Karlsson (1997) later pursued. The Normal-Wishart prior is the practical choice for applications requiring closed-form posterior moments (impulse response functions, IRFs, and variance decompositions), as the authors note in the conclusion.