interest (Schafer and Graham, 2002).<sup>14</sup> Moreover, the mean value and associated standard error for our land productivity measure, namely plot-level gross value of output per acre in local currency, exhibit more drastic changes under complete case analysis following MI in comparison to the incomplete case analysis based on plots with an observed GPS-based area measure. The next section delves deeper into the empirical implications of incomplete vs. multiply-imputed complete case analysis of GPS-based plot areas in the context of the inverse scale-land productivity relationship.

# 5. EMPIRICAL APPLICATION: INVERSE SCALE-LAND PRODUCTIVITY RELATIONSHIP

Starting with the seminal work of Sen (1962, 1966), who observed an inverse relationship (IR) between farm size and output per hectare in Indian agriculture, a large number of empirical studies have presented evidence that appears to corroborate that hypothesis (Lau and Yotopoulos, 1971; Yotopoulos and Lau, 1973; Berry and Cline, 1979; Carter, 1984, Eswaran and Kotwal, 1985, 1986; Barrett, 1996; Benjamin and Brandt, 2002; Larson et al. 2012). A smaller set of empirical studies, however, does not find evidence of such a relationship (Hill, 1972; Kevane, 1996; Zaibet and Dunn, 1998). Binswanger et al. (1995) and Eastwood et al. (2010) provide careful discussions of both the theory and the empirics of the IR debate, a full review of which is beyond the scope of this paper.

Following Barrett et al. (2010), an inverse relation between farm size and land productivity may have three main explanations: (i) imperfect factor markets, (ii) omitted variables and, in particular, omitted controls for land quality, and (iii) statistical issues related to the measurement of plot size. Imperfect factor markets (i.e. labor, land, insurance) are linked to differences in the shadow price of production factors that in turn lead to differences in the application of inputs per unit of land, in ways that are correlated with farm size.

Much of the earlier contributions to the IR debate focused on testing this type of explanation. Assunçao and Ghatak (2003) demonstrate how unobserved heterogeneity in farmer ability may theoretically explain the observed differences in productivity. On one hand, Bhalla and Roy (1988) and Benjamin (1995) have challenged the existence of the IR

> 14 We conducted conditional mean imputation for missing GPS-based plot areas in each setting, replacing missing values by the EA mean, provided that there were at least ten GPS-based plot areas in a given EA. In case, the criterion on the EA-level observation count was not satisfied, a coarser grouping with at least ten GPS-based plot areas is used, in order of district, region, and nation. While the resulting mean for GPS-based plot areas were similar to those obtained under casewise deletion as well as MI, the associated standard errors were consistently lower, in accordance with the expectations. These results, while available upon request, have not been reported given the lack of a statistical basis for choosing any particular sample size (Graham et al., 1997). After all, “one imputed value cannot itself represent any uncertainty about which value to impute: if one value were really adequate, then that value was never missing. Hence, analyses that treat imputed values just like observed values generally systematically underestimate uncertainty, even assuming the precise reasons for nonresponse are known.”  Rub n,  988:79

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