In equation (3), α1 measures the elasticity of the value of output with respect to operated land area. If this is estimated to be less than one and significant, it suggests that the value of output rises less quickly than operated land area and therefore the inverse productivity hypothesis holds (Lamb, 2003). If constant returns to scale are assumed, all production factors, including yields and inputs, can be converted to per acre terms and equation (4) estimated. For this equation, the inverse productivity relationship holds if β1 is found to be significantly less than zero.

Researchers that estimate this relationship using self-reported land areas assume that land has been adequately measured or that mismeasurement can be controlled for in some manner within the specification. If this is not the case, the estimate may be biased. The direction of bias depends largely on the manner in which self-reported areas deviate from the actual land size. For example, it may be the case that farmers consistently over or under report land areas across the land distribution or, alternatively, that over or under reporting varies with land size. The specific issues in reporting will influence the direction of bias associated with the inverse productivity relationship. The previous analysis of the deviation between self-reported and GPS-measured areas indicates that farmers over report small land areas and under report larger areas suggesting estimates of the inverse productivity relationship will vary between the two measures.

To test this hypothesis, we estimate equations (3) and (4) with GPS and with self-reported land areas to identify the difference. Estimating both equations allows verification of the consistency in the relationship regardless of the approach. In running these equations we are testing two distinct hypotheses. First, whether the inverse productivity relationship holds even when using a GPS land measure. Second, whether the estimate using the GPS land measure differs in a statistical sense from the estimate using self-reported land area.

Along with providing a basic estimate of the productivity-land relationship, as seen in equations (3) and (4) additional variables are included (e.g. Pi, Hi, Xi, θr) in the specification to check whether any estimated discrepancy in the productivity-land relationship between the two land measures can be controlled for through the inclusion of these additional variables. Towards this end, a stepwise approach is taken to see how the difference between the two measures varies with the addition of controls. Each step includes the addition of key sets of controls noted in the equations. The basic model includes only the land variables with locational (enumeration area) fixed effects to control for differences across location (particularly agroecological differences). The initial analysis focuses on the metadata, which is subsequently confirmed using the individual country data.

Table 6 reports the results for the analysis of value of output (equation 3) using the metadata. Tests of difference (p-values) between the estimates using GPS measured land area and selfreported land area are reported at the bottom of the table. The first estimations, which include no control variables but include enumeration area fixed effects, show that for both land measures the estimated elasticities are substantially lower than one, supporting the inverse productivity relationship. The elasticities for the self-reported areas are lower than for GPS areas—tests show that they are significantly lower—indicating that while the results confirm an inverse productivity relationship, the self-reported area results suggest it is stronger than when using GPS areas. The subsequent regressions incorporate plot level variables, household factors and finally labor and nonlabor inputs. The inclusion of these factors, while lowering the elasticities for both measures, reduces the estimate more for self-reported areas. The final results provide support for the inverse productivity relationship and indicate that self-reported area indicates a significantly stronger inverse productivity relationship than estimated by the GPS measure.

Table 7 follows a similar analysis estimating in a stepwise fashion the value of output per acre (equation 4) using the metadata. Again, tests of difference between the estimates using GPS and self-reported areas are reported at the bottom of the table. The results are consistent with Table 6. The estimates for both self-reported and GPS land measures suggest an inverse productivity 11