L2SLS are consistent and asymptotically normal (Davidson and MacKinnon, 1993, p. 641). _Stata_ very easily estimates both L2SLS and LOLS with a single simple command but it does not estimate the sum of these two log-likelihoods (except using Roodman’s _cmp_ command). That limitation in _Stata_ does not make this model complex.

The RM likelihood is (2) LRM = L2SLS(y0-yfmδ- **x** β0, yfm- **x** βfm- **z** π) for everyone in the sample including those without choice where **z** is a set of identifying instruments having parameters π. The **z** instruments, as RM explain, are interactions of **x** with a dummy variable for having the choice to participate. Both PK and RM estimate their models by maximizing their respective likelihoods.

There are two differences between the PK log-likelihood (equation 1) and the RM log-likelihood (equation 2). First, for those without credit program choice, RM estimate the determinants of the outcome y0 (household expenditure) as part of the 2SLS likelihood even though credit is deterministically and uniformly zero for every observation so that ordinary least squares is unbiased and fully efficient. It is not clear what it means to do 2SLS over a subsample in which the variable that is treated as endogenous only takes the value of deterministic zero. In contrast, PK treat the determinants of y0 for those exogenously excluded from credit as exogenous (in the OLS part of likelihood (1)). Second, in seeming contradiction to point #1, RM add instruments **z** based upon the exogeneity of choice. Given that they have treated exogenous observations as endogenous, they must do this in order for the impact of credit to be identified. In contrast, PK do not need to add instruments **z** because they treat exogenous observations as exogenous. There is no _a priori_ reason to believe that maximum likelihood estimates of credit impacts based upon RM’s approach using the 2SLS likelihood for all observations is any more robust to deviations from normality, as claimed by RM (p. 19), than PK’s estimates that use OLS for some of the observations. Furthermore, PK’s log-likelihood is much simpler because it does not need the artifice of constructing instruments **z,** and because it uses OLS rather than 2SLS for the exogenous observations.

Consider the implications of RM’s estimating the parameters β and π from a first-stage credit equation sample that includes observations lacking the choice to participate. Since credit is deterministically zero (that is, credit is zero with a sample and theoretical variance of zero) for those without choice, how can these observations provide any information to the estimation of the first-stage parameters?<sup>4</sup> 4 Another issue that arises from RM’s inclusion of nonchoice households in the first-stage sample is that having wealthier, landed households in the first-stage RM can lead to bias because “Even with villagelevel fixed effects, bias can emerge when programs base their placement decisions on the characteristics of sub-village groups. ...The village-level fixed effects will only control for the average characteristics of the village sample (RM, fn. 7).” This critique by RM is not valid for the PK model, but is valid for the RM setup. PK do not include landed, ineligible households in the first-stage equation where the village fixed effects are estimated -- PK only estimate the village fixed effects with the functionally landless subsample of the villages with programs. RM do include wealthy, landed household in their first-stage.

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