A State-Dependent BART VAR Connectedness Operator for Agricultural Commodity Markets: A Finite-Difference Construction

This paper introduces the BART-based State-Dependent Finite-Step Connectedness (BART-SDFC) operator for nonlinear vector autoregressions estimated with Bayesian additive regression trees (BART). Because a BART response surface is piecewise constant, the operator is a finite-step symmetric secant matrix, not a classical derivative. Evaluated at observed states, it is stabilized by a spectral-radius cap and mapped through a finite-horizon generalized forecast-error variance decomposition (GFEVD). We state normalization and net-flow balance properties and test sensitivity to step size, stability cap, VAR order, and forecast horizon. The application uses monthly World Bank prices for maize, U.S. hard red winter wheat, Thai 5% rice, soybeans, and soybean meal from January 1990 to December 2025, with lagged oil, fertilizer, and Oceanic Niño Index conditions. At h = 0.05 and cap = 0.98, the posterior mean BART-SDFC TCI is 39.22% (time-averaged 95% interval conditional on fixed covariance: 37.10%–41.55%); across the tested h-grid and caps, mean TCI ranges from 37.57% to 39.57%. BART has lower RMSE than linear VAR for all five commodities in a common 129-month holdout, but no Diebold–Mariano test is significant at 5%. All five equations meet the selected rank-normalized R-hat < 1.01 criterion, with minimum bulk ESS 1,056; connectedness diagnostics also meet the reported thresholds. The network intervals condition on fixed residual covariance. The operator summarizes predictive transmission, not structural causality.