A bivariate hierarchical Bayesian model for estimating cropland cash rental rates at the county level
Section 1. Introduction
The National Agricultural Statistics Service (NASS) of the United States Department of Agriculture (USDA) conducts hundreds of surveys each year to obtain estimates related to diverse aspects of US agriculture. Examples of parameters that NASS estimates include total production, harvested area, and crop yield. Estimation for sub-state domains, such as counties, is difficult due to small sample sizes. Our interest is in estimation of the county-level cash rental rate, the market value of land rented on a per acre basis for cash only.
Estimates of county-level cash rental rates serve many purposes. Farmers use the estimates for guidance in determining rental agreements (Dhuyvetter and Kastens, 2009). Agronomists use the estimates to study research questions related to the interplay between cash rental rates and other economic characteristics such as commodity prices and fuel costs (Woodard, Paulson, Baylis and Woddard, 2010). NASS’s published estimates of mean cash rental rates at the county level have implications for the Conservation Reserve Program, a policy that encourages agricultural landowners to conserve their land. The 2008 and 2014 Farm Bills require NASS to collect data on cash rental rates for three land use categories – non-irrigated cropland, irrigated cropland, and permanent pasture – for counties with at least 20,000 acres of cropland or pastureland.
To satisfy the requirements of the 2008 and 2014 Farm Bills, NASS conducts a Cash Rent Survey. A concern is that direct estimators of county means from the Cash Rent Surveys may be unstable due to small realized sample sizes. We investigate the use of mixed models (Rao and Molina, 2015) to stabilize the estimators of average cash rental rates at the county level. NASS publishes estimates of average cash rental rates at the state level before county level estimation from the Cash Rent Survey is complete. To maintain internal consistency, the county predictors must satisfy a benchmarking restriction.
In a frequentist framework, Berg, Cecere and Ghosh (2014) use area-level models to predict county-level cash rental rates for all states and for the three land use categories of non-irrigated cropland, irrigated cropland, and permanent pasture. For each combination of land use category and state, the method of Berg et al. (2014) uses data from two years. An assumption that the variances for the two years are the same motivates the Pitman-Morgan transformation, which converts the vector of observations for the two time points into an average and a difference. After separate univariate models are applied to the average and the difference, the predictor for each time point is obtained by adding the predictor of the average to half of the predictor of the difference. The method of Berg et al. (2014) is demonstrated to provide a practical approach to obtaining reasonable predictions across a diverse range of conditions. Nonetheless, the effects of simplifying assumptions warrant additional investigation. If the variances for the two time-points differ, then, as discussed in Berg et al. (2014), the mean squared error (MSE) estimator based on the Pitman-Morgan transformtion can have a negative bias. Further, the Berg et al. (2014) method does not account for the effect of benchmarking when estimating the MSE.
This study addresses the issues of non-constant variances across time and the effect of benchmarking on efficiency in the context of the NASS Cash Rent Surveys through the use of a bivariate hierarchical Bayesian (HB) model for the unit-level data. The model is sufficiently flexible to allow the variances to differ between the two time-points. The use of Bayesian methods for inference facilitates estimation of the increase in posterior MSE due to benchmarking. Another innovation of the bivariate HB approach is that it incorporates the survey weights in the variance model. We also aim to improve the efficiency of the predictors for particular situations, relative to Berg et al. (2014), by allowing the covariates to differ across states. Datta, Day and Maiti (1998) examine HB bivariate models for the county crop acreage data of Battese, Harter and Fuller (1988). Our model extends the Datta et al. (1998) model to account for a relationship between the weight and the variance as well as an unbalanced data structure.
We focus on prediction of county level cash rental rates for non-irrigated cropland using the responses to the 2009 and 2010 Cash Rent Surveys as well as external sources of auxiliary information. In Section 2, we discuss the survey data and the auxiliary information in detail. We describe the bivariate HB model in Section 3. In Section 4, we summarize results for non-irrigated cropland in Iowa, Kansas, and Texas. In Section 5, we summarize and discuss possible future research applicable to both estimation of cropland cash rental rates and small area estimation more generally.
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