A bivariate hierarchical Bayesian model for estimating cropland cash rental rates at the county level
Section 4. Results for non-irrigated cropland in Iowa, Kansas, and Texas

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The model of Section 3 was fit to the non-irrigated cropland cash rental rates reported on the 2009 and 2010 Cash Rent Surveys for Iowa, Kansas, and Texas. These three states were chosen to reflect a range of situations. All counties in Iowa have estimates for corn yields, and cash renting is a relatively common way to rent non-irrigated cropland. Kansas is more agriculturally diverse than Iowa. According to agricultural specialists at NASS, share-renting is a more common way to rent land than cash renting in many parts of Texas, which may explain why realized sample sizes for some Texas counties are as small as zero or one report.

4.1  Covariate selection

The potential covariates for Iowa, Kansas, and Texas are listed in Section 2.2. For each state, the covariates include four variables related to the NCCPI, the total value of production for a county based on the 2007 Census of Agriculture, the expected sales for an operation recorded on the NASS list frame, the farm type recorded on the NASS list frame, and the acres rented for non-irrigated cropland recorded on the NASS Cash Rent Survey. For Iowa, an additional covariate is the corn yield for the county. For Kansas, an additional covariate is the non-irrigated yield index.

The covariates for each state were selected according to the following procedure. First, univariate models were fit to the data for 2009 and 2010 separately using maximum likelihood estimation. The univariate model used for covariate selection is of the form

$$y_{ijt}=x_{ijt}^{\prime }{\alpha }_t+{\nu }_{it}+{ϵ}_{ijt}\mathrm{,}(4.1)$$

where ${ϵ}_{ijt}\sim \text{N}\left(\mathrm{0,}{\sigma }_{ϵ\text{}\mathrm{,}t}^2\right),$ and ${\nu }_{it}\sim \text{N}\left(0,{\sigma }_{\nu \text{}\mathrm{,}t}^2\right).$ The data for each farm operator who reported a non-irrigated cropland cash rental rate in year $t$ were used to fit the univariate model for year $t,$ regardless of whether or not the unit also reported a cash rental rate in year $s\left(s\ne t\right).$ The R function lmer in the package nlme is used for maximum likelihood estimation. For each year, step-wise selection using the R function stepAIC is performed using the BIC measure. The selected covariates are the variables that are in the minimum BIC models for both the 2009 and 2010 univariate models. We acknowledge that the minimum BIC model is a local minimum identified by the stepAIC procedure rather than a global minimum. The selected covariates for Iowa, Kansas, and Texas are as follows:

• Iowa: corn yield, expected sales, non-irrigated acres rented for cash.
• Kansas: non-irrigated yield index, expected sales, farm type.
• Texas: max-NCCPI, expected sales, farm type.

4.2  Estimates of correlation parameters

The exploratory analysis of Section 2.1 suggests a substantial correlation between the non-irrigated cropland cash rental rates for 2009 and 2010. Table 4.1 contains summaries of the posterior distributions of the correlations in the bivariate HB model defined in Section 3.1. The columns labeled “Median” are the posterior medians of the correlations, and lower and upper endpoints of the 95% credible intervals are the 2.5 and 97.5 percentiles of the posterior distributions of the correlations. Even though the variances of $e_{ij09}$ and $e_{ij10}$ are proportional to the inverses of the weights, the correlation is a constant because the weights cancel in the definition of the correlation.

The posterior medians of the county-level and unit-level correlations exceed 0.74 and 0.57, respectively. The lower endpoints of the 95% credible intervals exceed 0.61 and 0.54 for the county-level and unit-level correlations, respectively. For each state, the correlations at the level of the county are larger than the correlations for individual units. The significant correlations suggest the potential for an efficiency gain for the predictors relative to a univariate model.

4.3  Comparison of 2010 predictors for bivariate and univariate models

To demonstrate the gain in efficiency due to the use of the bivariate model relative to a univariate model, we compare the posterior mean squared errors of the predictors from the bivariate model to the posterior mean squared errors of the predictors from a corresponding univariate model. The assumptions of the univariate models are the same as the assumptions of the bivariate models except that the covariance parameters in ${\Sigma }_{ee}$ and ${\Sigma }_{\nu \nu }$ are assumed to equal zero. To fit the univariate models, we use inverse-gamma prior distributions for ${\sigma }_{eet}$ and ${\sigma }_{\nu \nu t}$ $\left(t\mathrm{<mo>=</mo>\; 09,<mtext>\hspace{0.17em}</mtext><mtext>\hspace{0.17em}</mtext>10}\right).$

To compare the bivariate and univariate models, we define the relative posterior MSE (RelMSE) for county $i$ by

$${\text{RelMSE}}_{i\text{}\mathrm{,}10}\mathrm{<mo>=</mo>}\frac{{\text{MSE}}_{i10}^{B\text{Bench}}}{{\text{MSE}}_{i10}^{\text{UNIBench}}}\mathrm{,}(4.2)$$

where ${\text{MSE}}_{i10}^{B\text{Bench}}$ is defined in (3.16) and ${\text{MSE}}_{i10}^{\text{UNIBench}}$ is the posterior MSE based on the corresponding univariate model. The average relative MSEs for Iowa, Kansas, and Texas are 88.71%, 97.27%, and 88.65%, respectively, where the average relative mean squared error for a state is $D^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^D}{\text{RelMSE}}_{i\text{}\mathrm{,}10}.$ Note that the effects of both estimating the covariate mean and benchmarking are incorporated in the forms for the posterior MSE for both the bivariate and univariate models. Because of the significant correlations in the model errors for the two time points, the posterior MSE from a bivariate model is smaller than the posterior MSE from the corresponding univariate model, and the average relative efficiencies are less than one.

To assess the effect of estimating the covariate population mean on the MSE of the predictor, we calculate the average of the ratios ${\widehat{\text{MSE}}}_{2i}{\widehat{\text{MSE}}}_{1i}^{-1}$ for $i\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}D,$ where ${\widehat{\text{MSE}}}_{2i}$ and ${\widehat{\text{MSE}}}_{1i}$ are defined following (3.10). The ratios are 18.21%, 28.20%, and 21.07% for Iowa, Kansas, and Texas, respectively. Compared to Iowa and Texas, the contribution to the prediction MSE due to using the sample covariate mean instead of the population covariate mean is higher in Kansas, and this makes sense since Kansas is more agriculturally diverse. The relatively large average relative MSE for Kansas (97.27%) reflects the relatively large increase in posterior MSE due to estimating the covariate mean.

4.4  Model assessment

To assess model fit, we use the posterior predictive $\text{p}-$ value, which measures departures between the observed data and the model. The posterior predictive $\text{p}-$ value compares the posterior predictive distribution of selected summary statistics to the corresponding values obtained using the original sample. For the analysis below, we use only the elements observed in both 2009 and 2010 (set 1).

We consider two summary statistics: the mean for each year and the multivariate skewness. The mean for year $t$ is the mean of the observations in set 1 for year $t$ and is defined

$${\overline{y}}_t\mathrm{<mo>=</mo>}{\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}\left|A_i\right|\right)}^{-1}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\displaystyle \sum _{j\in A_i}}{y}_{ijt}\mathrm{,}$$

where $A_i$ denotes the elements in set 1 for county $i.$ The multivariate skewness is defined by

$${\widehat{\gamma }}_{\mathrm{1,}p}={\left({\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}\left|A_i\right|\right)}^{-1}{\displaystyle \sum _{i\mathrm{<mo>=</mo>\; 1}}^D}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^D}{\displaystyle \sum _{j\in A_i}}{\displaystyle \sum _{l\in A_i}}{m}_{ijkl}^3\mathrm{,}$$

where $m_{ijkl}={\left(y_{ij}-\overline{y}\right)}^{\prime }{S}^{-1}\left(y_{kl}-\overline{y}\right),$ $y_{ij}\mathrm{<mo>=</mo>}{\left(y_{ij\text{}\mathrm{,}09}\mathrm{,}{y}_{ij\text{}\mathrm{,}10}\right)}^{\prime }\text{},$ $\overline{y}\mathrm{<mo>=</mo>}{\left({\overline{y}}_{09}\mathrm{,}{\overline{y}}_{10}\right)}^{\prime }\text{},$ and $S\mathrm{<mo>=</mo>}{\left({\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^D}\left|A_i\right|-1\right)}^{-1}{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>1}}^D}{\displaystyle {\sum }_{j\in A_i}}\left(y_{ij}-\overline{y}\right){\left(y_{ij}-\overline{y}\right)}^{\prime }.$

The posterior predictive $\text{p}-$ value is defined as the proportion of summary statistics calculated with samples generated from the posterior predictive distribution that exceed the corresponding value based on the original sample. To be specific, let $T\left(y^{\left(r\right)}\right)$ be the summary statistic based on the $r^{\text{th}}$ data set generated from the posterior predictive distribution, where the procedure to generate data from the posterior predictive distribution is defined in Appendix C. Let $T\left(y\right)$ be the corresponding statistic based on the original sample. The posterior predictive $\text{p}-$ value is $R^{-1}{\displaystyle {\sum }_{r\mathrm{<mo>=</mo>\; 1}}^R}I\left[T\left(y^{\left(r\right)}\right)\mathrm{<mo>></mo>}T\left(y\right)\right].$ A $\text{p}-$ value close to 0.5 indicates that the model provides a reasonable fit to the sample data.

Table 4.2 contains the posterior predictive $\text{p}-$ values for Iowa, Kansas, and Texas. For Kansas, the posterior predictive values indicate that the model is a good fit to the data. For Iowa and Texas, the posterior predictive $\text{p}-$ values indicate lack of fit. A further analysis of residuals suggests that the lack of fit may result from outliers. The posterior predictive $\text{p}-$ values far from 0.5 may also arise because we only use the observations sampled in both 2009 and 2010 to calculate the posterior predictive $\text{p}-$ values, while we use the full data set to fit the model.

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