RM simulate. The only variables in the first-stage are intercepts, and as argued below, those intercepts should not be identifying instruments. Even with normal errors, the RM model is not consistent and the PK model is consistent if there are sufficient exogenous covariates and deterministic (non-choice) observations to identify the model. It is the existence of exogenous covariates that generates the plethora of interaction terms in the misspecified 2SLS setup of RM that make the RM instrument set weak and the linear LIML estimates invalid. As PK (1998) estimates a model with 254 free parameters, the RM simulation without any exogenous **x** slope variables is a seriously deficient model from which to make claims about the PK and RM approaches, particularly without making any mention of the crucial importance of these covariates.

To examine the claims of RM concerning the consistency of their approach and the inconsistency of PK’s approach, Table 2 presents estimates of credit effects (δf and δm) using Stata code similar to that used by RM (see Appendix 1 of RM) but now with 80 exogenous slope variables (covariates) rather than none.<sup>23</sup> The means of 100 replications estimated by RM’s linear LIML method yield mean parameter estimates that are more than 60 percent larger than the true values of 1.00. The PK estimator applied to the same simulated data results in mean parameter estimates only 1 percent from the true value when the errors are normal or skewed and the first-stage is linear. In the case of a Tobit first-stage, the mean parameters are 1.00 when the errors are normal, and off only slightly when the data are skewed in the way that RM specify. RM’s very odd choice of simulating a model without any exogenous variables masks the inconsistency of their approach and the robustness of that of PK.

The general claim made by RM that having any skew in the second-stage error “violates” the PK model is clearly wrong on its face. As equation (1) above makes clear, the determinants of the outcome (household consumption expenditure) for those without choice is simply estimated by PK using ordinary least squares (OLS) in the LOLS part of the likelihood function. As noted earlier, even though the assumption of normality is made in constructing the OLS likelihood, it is well known that OLS by linear regression provides exactly the same parameter estimates as OLS by maximum likelihood no matter what the distribution of the errors. Consequently, having skewed errors for those without credit program choice is of no consequence. Adding the level of skewness suggested by RM only to the nonchoice observations in the simulation model reported in Table 3 (nonchoice observations constitute about 80 percent of the sample), yields no evidence of bias whatsoever (last row of Table 2). Exogenous observations are treated as exogenous in PK, and treating nonchoice observations as if they have choice and are endogenous (as RM do) results in mistaken conclusions such as this.

RM’s failure to understand that non-normal errors can only be an issue for observations with choice matters because RM test for skewness and excess kurtosis from predicted “second-stage” residuals for

> 23 The 80 independent variables are generated as independent N(0,1) random variates. In the PK (1998) paper, the number of independent variables K is about 80.  As the source of inconsistency in the RM approach is their treatment of household observations without choice, this simulation generates samples with a large share of non-choice households, and for which there are three program types, as in the actual Bangladesh case examined in PK (1998).  This simulation thus mirrors the RM model used to estimate column 1 of RM’s Table 5.

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