Hierarchical Bayesian Model with Inequality Constraints for US County Estimates

Abstract

In the production of US agricultural official statistics, certain inequality and benchmarking constraints must be satisfied. For example, available administrative data provide an accurate lower bound for the county-level estimates of planted acres, produced by the U.S. Department of Agriculture’s (USDA) National Agricultural statistics Services (NASS). In addition, the county-level estimates within a state need to add to the state-level estimates. A sub-area hierarchical Bayesian model with inequality constraints to produce county-level estimates that satisfy these important relationships is discussed, along with associated measures of uncertainty. This model combines the County Agricultural Production Survey (CAPS) data with administrative data. Inequality constraints add complexity to fitting the model and present a computational challenge to a full Bayesian approach. To evaluate the inclusion of these constraints, the models with and without inequality constraints were compared using 2014 corn planted acres estimates for three states. The performance of the model with inequality constraints illustrates the improvement of county-level estimates in accuracy and precision while preserving required relationships.

Smoothing County-Level Sampling Variances to Improve Small Area Models’ Outputs

“…Then, as described in Section 3.2, the new set of values was drawn from the empirical distribution to update the unreliable variances for the "anomalous" counties within each state. The updated sampling variances obtained via the bootstrap method satisfied the inequalities in model ( 3) and provided more reasonable values for the extreme sampling variances, which were further used as inputs in the model in Equation (4).…”

“…Yield is defined as the ratio of total production to the harvested acreage. Starting in 2020, several HB subarea-level (small area) models have been implemented as extensions of the FH model to improve the precision of the estimates at the county level (see [2][3][4][5][6]). The sampling variances of the yield estimates are produced using a second-order Taylor series approximation and, due to various reasons (e.g., sparseness in data), could result in zero, very small or very large estimated variances for several counties.…”

Using Small Area Estimation to Produce Official Statistics

“…Developing the methodology to enforce this lower bound within (1) was technically challenging. Nandram et al [20] and Chen et al [21] proposed and implemented the constrained model (4) for planted acres.…”

“…For instance, Nandram et al [20] showed how to incorporate the area-specific inequality constraints and benchmarking into the Fay-Herriot model using simulated datasets with properties resembling an Illinois corn crop. Chen et al [21] examined the performance of the model with inequality constraints and, through a case study, illustrated the improvement in the county-level estimates in terms of accuracy and precision while preserving the required relationships. Erciulescu et al [19] discussed the yield model and different methods of applying benchmarking constraints to a triplet (numerator, denominator, ratio) and illustrated results for 2014 for corn and soybeans in Indiana, Iowa, and Illinois.…”

“…The convergence was monitored using trace plots, the multiple potential scale reduction factors ( R close to 1), and the Geweke test of stationarity for each chain (see [32,33]). More details can be found in [15,[19][20][21].…”