by a factor of 10. Of course, as we have shown above, the RM estimates are not interpretable, as they do not correspond either to the PK model or to any formal econometric model of program evaluation that RM or anyone else has ever set out, not to mention that the linear LIML estimation strategy RM adopt just makes the estimates even more strange.

Can one show that, in general, the linear LIML does not “work” in the PK model as RM have formulated it, or is it simply a reflection of some defect in the PK data? In our Table 8 below, we reproduce the simulation results of our Table 1 (above) but add the linear LIML estimates to the set of estimates reported. These are exactly the same simulation datasets described earlier. Recall that the errors are normally distributed and the instruments are not “weak” in these simulations. The linear LIML estimates of column (4) of our Table 8, the only new column in this table as compared to Table 1, have clearly bizarre point estimates (means of 200 simulations are presented) and huge standard deviations. It is no surprise that RM get outlandish parameter values and large standard errors in their Table 6 when they estimate the PK model with their linear LIML setup. Our Table 9 reports the estimates from one of the simulation datasets. The model estimates an effect of female credit from group 1 to be -244.388 when the true value is 0.75, and the male credit effect of group 3 to be 177.334 when the true value is 0.25. This should not be surprising given the large standard deviations reported in column (4) of our Table 9. Almost anything can result when an erroneous econometric model is applied to an artificially altered data matrix.

# **7.  Sources of identification in Pitt and Khandker:  Some new results**

In this section, we further examine two aspects of our instrumental variable approach that have been attacked by RM. The first is the validity of the exclusion restriction underlying their use of interactions between program choice and the set of exogenous variables (including the village fixed effects) as instruments. The second is the application of the “one-half acre” program eligibility rule. We will show that identification does not require both of these, and present new results dropping each assumption in turn.

The PK model is identified even after dropping all of the exogenous variables including the village fixed effects from the first-stage equations given by equation (4). All that is required is a constant term. Consequently, the identifying instrument is whether or not a household had choice to join a female credit program and/or a male credit program. Estimating a model with only a constant term in the first-stage is of interest because it suggests the extent to which the set of exogenous variables X (and village fixed effects), which have been subject to pointed attack by RM, actually drive the results. Note that even if program placement is nonrandom - that is, even if the program availability is correlated with the village effects - dropping village fixed effects from the first-stage does not cause bias in the second-stage as long as the second-stage equation includes the village fixed effects, as it does in PK.

24