Table 1 Comparison of the predictive power of OLS and Bayesian yield models.

Model specifications
OLS 1 OLS 2 OLS 3 OLS 4 Bayes 1 Bayes 2
Intercepts Uniform County Interacted County Partial Partial
Coefficients Uniform Uniform Interacted Interacted Partial Partial
Error variance Uniform Uniform Uniform Uniform County Uniform
R2 by model: estimated and evaluated on all years
OLS 1 OLS 2 OLS 3 OLS 4 Bayes 1 Bayes 2
Barley 0.36 0.71 0.57 0.75 0.74 0.75
Corn 0.48 0.76 0.65 0.78 0.81 0.82
Cotton 0.32 0.64 0.55 0.70 0.68 0.69
Rice 0.75 0.84 0.81 0.84 0.85 0.85
Soybeans 0.47 0.72 0.65 0.76 0.78 0.79
Wheat 0.42 0.71 0.56 0.73 0.76 0.76
R2 by model: estimated on 1949–1994, evaluated on 1995–2009
OLS 1 OLS 2 OLS 3 OLS 4 Bayes 1 Bayes 2
Barley −0.11 0.43 0.20 0.45 0.48 0.46
Corn −0.09 0.20 0.07 −1.05 0.27 0.17
Cotton 0.07 0.31 0.14 −37.50 0.21 0.12
Rice 0.20 0.37 0.12 −1.59 0.19 0.14
Soybeans 0.26 0.47 0.39 −16.27 0.53 0.48
Wheat 0.16 0.49 0.31 0.47 0.51 0.50
1. Table cells show R2 by crop and model specification, using all data (top) and under cross-validation on 1995–2009 (bottom). The first four columns are ordinary least-squares (OLS) specifications, variously including region-specific intercepts and covariate interactions. The last two columns are for the Bayesian model, with partially pooled intercepts and coefficients, either allowing each county to have a different variance (Bayes 1) or constraining all to have the same variance (Bayes 2). In all cases, R$${\,}^{2}=1-\frac{\sum {\left({y}_{i}-{\hat{y}}_{i}\right)}^{2}}{\sum {\left({y}_{i}-{\bar{y}}_{i}\right)}^{2}}$$, where yi is the observed log yield for county-year i. $${\hat{y}}_{i}$$ is the point estimate for OLS and the posterior prediction for the mean MCMC parameter draw for the Bayesian model, and $${\bar{y}}_{i}$$ is the average across all observations of yi.