Spatial VAR

Definition

A Spatial Bayesian VAR (Spatial VAR or SVAR) is a Bayesian VAR (BVAR) model in which the Minnesota Prior's cross-variable shrinkage is differentiated by geographic proximity. Variables from spatially contiguous units receive prior means and variances calibrated to reflect greater expected influence than variables from distant units, encoding the belief that neighboring regions' economies are more interdependent than distant ones.

Key Ideas

The standard Minnesota prior uses a single cross-variable weight $\theta = 0.2$ for all cross-variable lags, regardless of geography. The spatial prior replaces $\theta$ with equation-specific spatial weights from a contiguity matrix.
The prior mean structure encodes a spatial random walk: the best single-step forecast of region $i$ is a weighted average of own current value (weight 0.5) and current values of contiguous neighbors (total weight 0.5, split equally).
Non-contiguous variables are shrunk more aggressively than under standard Minnesota.
Standard deviations retain the Minnesota harmonic lag decay but are differentiated spatially.
Estimation uses Theil-Goldberger mixed estimation — identical mechanics to standard BVAR, no Markov Chain Monte Carlo (MCMC) required.

How It Works

Contiguity Matrix W

Construct the binary contiguity matrix $C$ where $C_{ij} = 1$ if regions $i$ and $j$ share a border and $C_{ii} = 0$ . Row-standardize:

$W = \mathrm{diag}(C \mathbf{1})^{-1} C$

Each row of $W$ sums to 1. Entry $w_{ij} = 1/m_i$ where $m_i$ is the number of first-order neighbors of region $i$ . The spatial lag $Wy$ is a weighted average of neighboring values at each time point.

Spatial Prior Means

For equation $i$ (region $i$ ), the prior mean on the first lag of variable $j$ is:

$\mu_{ij,1} = \begin{cases} 0.5 & \text{if } j = i \quad \text{(own)} \\ 0.5 \cdot w_{ij} & \text{if } j \in \mathcal{N}(i) \quad \text{(contiguous neighbor)} \\ 0 & \text{otherwise} \end{cases}$

Prior means on lags $\ell > 1$ are 0 for all variables. This is the spatial random walk belief: the prior says the best forecast of $y_{i,t+1}$ is $0.5\,y_{i,t} + 0.5\sum_{j \in \mathcal{N}(i)} w_{ij} y_{j,t}$ .

Spatial Prior Standard Deviations

Retain the Minnesota harmonic decay form, but differentiate by spatial relationship:

Own variable ( $j = i$ ): standard Minnesota tightness, $\sigma_{ii,\ell} \propto \lambda/\ell$
Contiguous neighbor ( $j \in \mathcal{N}(i)$ ): same tightness as own, scaled by $w_{ij} \cdot (s_i/s_j)$
Non-contiguous ( $j \notin \mathcal{N}(i)$ , $j \neq i$ ): tighter shrinkage toward zero

The overall tightness hyperparameter $\lambda$ (and lag-decay $\gamma$ ) remain as in standard Minnesota.

Alternative: Spatial Lag as Explicit Variable

For regions at the boundary of the study area (fewer neighbors than average), an alternative is to include $Wy$ as an explicit variable in the VAR system and assign it a random-walk prior mean of 1 on its first lag, 0 on all others. This avoids distorting the implicit spatial prior for edge equations and is preferred when the spatial averaging involves fewer than three neighbors.

Estimation

Identical to standard Minnesota BVAR via Theil-Goldberger mixed estimation (ridge-type posterior mean). The spatial weights change only the prior hyperparameters fed into the standard formula:

$\hat{\beta} = (X'X + \kappa R'R)^{-1}(X'Y + \kappa R'r)$

where $R$ and $r$ now encode the spatial prior precision and means.

Why It Matters

Provides a principled, computationally free way to incorporate geographic structure into regional BVAR models — no extra computation beyond constructing W.
Outperforms standard Minnesota BVAR at medium-to-long forecast horizons (4–12 steps) for regional employment data (LeSage-Krivelyova 1999: SVAR wins mean absolute percentage error (MAPE) in 138/240 comparisons vs. BVAR, dominates at horizons 4–12).
SVAR also has smaller forecast error dispersion — more consistent across variables and states — not just better on average.
The contiguity matrix W is transparent, interpretable, and can be replaced by distance-based, trade-flow, or economic-similarity matrices.

Open Questions

Optimal spatial weight matrix: first-order contiguity ignores distance decay, trade flows, and higher-order proximity.
Combining spatial priors with cointegration structure (spatial vector error correction model (VECM)) remains unexplored.
Whether spatial priors improve performance in non-regional settings with network structure (e.g., interbank markets, global supply chains).