Bayesian spatial analysis of US agricultural land values

ABSTRACT

This study analyses the spatial patterns in agricultural land values using a Bayesian spatial econometrics approach. The model is motivated by land capitalisation theory, which captures the current value of a parcel of farmland as a discounted, future stream of returns associated with its agricultural use and pressures to convert the land. This study extends the land capitalisation model to incorporate variations in neighbouring agricultural lands. The results suggest that farmland prices are subject to spatial connectivities, implying that markets for cropland are highly localised, whereas markets for agricultural commodities are generally global in nature. The estimates imply that changes to neighbouring cropland rental rates and land enrolled in conservation explain about seven percent and eleven percent, respectively, of the variation in local land values.

KEYWORDS:

• Bayesian spatial model
• farmland rental rates
• spatial model
• agricultural land markets

JEL:

• C11
• Q15
• R32

ACKNOWLEDGEMENTS

This research was supported [in part] by the US Department of Agriculture, Economic Research Service.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s). The findings and conclusions in this publication are those of the author(s) and should not be construed to represent any official USDA or U.S. Government determination or policy.

Notes

1 There is a large amount of literature on the use of non-geographically based weights matrices or combinations of geographic structures (e.g., Elhorst et al. (2012)) within spatial econometric models (Debarsy & LeSage, 2018).

2 For this study, we estimate the spatial econometric models in a Bayesian framework, so all the models will be denoted with the letter ‘B’ in front of the acronym to represent the Bayesian approach. For example, the Bayesian SLX model will be denoted by the acronym ‘BSLX’.

3 The contiguity matrix is defined such that any county that shares a border with a local county is assigned a value of unity in the weight matrix – this is sometimes referred to as queen design. The distanced-based matrix assigns a value to unity to any county that is within a 100-mile distance of a local county. The nearest-neighbour design assigns a value of unity to the six nearest neighbours within a particular distance of a local county. The assignment of the six-nearest-neighbours matrix is calculated prior to row-normalisation. The maximum calculated Euclidean distance between county pairs in the six-nearest-neighbour specification was about 230 miles, so we prespecified 100 miles in the distance-based specification to provide for a variety of defined distances among the separate spatial weight matrices. We chose the six nearest neighbours because the five or less neighbours yielded an average distance between county centroids that was too close to the 100-mile distance-based matrix specification, and any numbers above six states yielded an average county-pair distance that made it incomparable to the 100-mile specification.

4 A six-nearest-neighbour matrix is generally not symmetric.

5 The other models are specified in similar manner, but we discuss this BSAR here for ease of illustration.

6 The hyperparameter r is for the draws on the chi-squared distribution, which estimates the variance terms in the matrix V. We specified the models such that the V matrix accounts for heterogeneity within the variance or heteroskedasticity.

7 Although simple in principle, the marginal likelihood must be approximated if there is no closed-formed solution to the expression of the Bayesian model. In the current study, we estimated the marginal likelihood using an MCMC approach offered within LeSage (2023).

8 These effects estimates are only for the spatial lag of X model, but the effects for the spatial autoregressive model are estimated in a similar manner. The coefficients on the spatial error model have an interpretation that is similar to a linear model.

9 To generate the full data set, we imputed the following approximate amount of observations for these variables: cash rental rate – 8000; cash receipts – 78; price per acre – 5300; and CRP – 4100. The multiple imputation algorithm followed a ‘predictive mean matching’ method (van Buuren, 2018). This method generates imputed values for own county as well as neighbouring county values (i.e., if neighbouring counties are part of matching set based on mean values). That way, one particular county’s imputed values are less sensitive to under or over prediction.

10 We also compared estimates between a non-spatial, Bayesian fixed effects estimator and a non-spatial, non-Bayesian fixed effects estimator. The results, which are offered in the online appendix, between the two models were nearly identical with differences at the third or fourth decimal place of estimates.

11 As a sensitivity analysis for the spatial weight matrix, we also estimated a model in which the weights were defined according to an exponential distance decay function. The estimates for the Bayesian spatial econometric models, based on the exponential distance decay weight matrix, are offered in the online appendix. The results were nearly identical to the findings offered in Table 3.

12 LeSage and Parent (2007) developed an approach to evaluate the marginal likelihood and produce posterior model probabilities that can be applied to the SAR, SLX, SEM and SDM models. The approach relies upon the stylised normal-inverse gamma conjugate prior (outlined above) in the Bayesian model comparison literature (Fernández et al., 2001; Koop, 2003). This allows for an analytical integration of all the model parameters except for the spatial dependence parameter, which is handled using univariate numerical integration (LeSage, 2023).

13 The Bayes factor analyses use flat priors for each of the models. Using uninformative priors can be problematic as Bayes factors only measures evidence in the data, but it must be interpreted relative to the prior evidence (O’Hagan, 2006). Thus, any findings from the Bayes factor analyses must be interpreted with caution.

Additional information

Funding

This research was supported, in part, by cooperative agreement 58-6000-1-0127 offered by the US Department of Agriculture, Economic Research Service.