LeSage-Krivelyova (1999) A Spatial Prior for Bayesian Vector Autoregressive Models

Summary

LeSage and Krivelyova (1999) extend the Minnesota prior to exploit spatial contiguity between regional units. The cross-variable shrinkage parameters of the standard prior are replaced with equation-specific weights derived from a row-standardized contiguity matrix W: contiguous neighbors receive prior means of 0.5/m (split from the own-variable's 0.5), while non-contiguous variables are shrunk more aggressively. Estimation is standard Theil-Goldberger mixed estimation — no Markov chain Monte Carlo (MCMC) required. In a 20-industry × 8-state Midwest employment forecasting experiment, the spatial Bayesian vector autoregression (SVAR) outperforms the standard Minnesota Bayesian VAR (BVAR) in 138 of 240 mean absolute percentage error (MAPE) comparisons, with dominance concentrated at forecast horizons 4–12.

Key Claims

• The spatial prior encodes a "spatial random walk": the prior mean on the first own lag is 0.5 (vs. 1.0 in standard Minnesota), with the remaining 0.5 distributed equally across contiguous neighbors' first lags via weights $w_{ij} = 1/m_i$.
• Contiguous-neighbor lags receive the same prior standard deviations as own lags (scaled by $w_{ij} \cdot s_i/s_j$); non-contiguous lags are tightly shrunk toward zero.
• The spatial weighting matrix $W = \mathrm{diag}(C\mathbf{1})^{-1}C$ is constructed from a row-standardized first-order contiguity matrix (shared-border criterion).
• For edge regions (fewer neighbors than average), an alternative embeds the spatial lag $Wy$ as an explicit VAR variable with a random-walk prior mean of 1.
• SVAR wins MAPE in 138/240 comparisons vs. BVAR; standard BVAR wins 102; the error correction model (ECM) performs worst overall.
• SVAR has smaller forecast error dispersion across states and industries (Table 5) — more consistent than BVAR, not just better on average.

Concepts Introduced or Extended

• Spatial VAR
• Minnesota Prior
• Vector Autoregression

Entities Mentioned

• James P. LeSage
• Anna Krivelyova

Quotes

"The spatial prior produced more accurate forecasts in more than 70 percent of cases for forecast horizons 4 through 12." (p. 311)

My Take

The paper's key contribution is computationally free: swap the scalar $\theta = 0.2$ for a vector of spatially calibrated weights drawn from a contiguity matrix. The "spatial random walk" motivation is intuitive for regional labor markets. The main limitation is that first-order contiguity is a coarse proxy — it ignores distance decay, trade flows, or economic similarity. The finding that gains concentrate at horizons 4–12 is expected: spatial information is most valuable when own-variable autoregressive dynamics weaken. The paper does not address cointegrated regional systems; LeSage (1990) showed those cases may still be better handled by ECM.