errors and check if (i) the log-likelihood is bimodal, and (ii) the frequency with which the maximum of the log-likelihood corresponds to the true value of the parameter of interest -- the credit effect, δ. The results of this simulation (Table 4) demonstrate that (i) bimodality was found in all 100 simulations, and is likely to be a “common” feature of PK-type models; (ii) all 100 simulations used data that were constructed with normal errors, so that non-normality is not an implication of the finding of bimodality, as RM suggest; (iii) the global maximum was tightly centered around the true value of δ, so that bimodality is innocuous in the simulations because the true estimate is recovered in every case; (iv) “weak” instruments are not necessary for bimodality, as RM suggest, as the instruments in the simulations were sufficiently “strong” to identify δ with tight bounds; and (v) the inferior mode was always in the direction of the least squares bias, that is, the mode had a smaller credit effect δ. In no case did the mode associated with the smaller value of δ have a higher likelihood. RM do not dispute that the PK estimates are the global maximum. Although the data-generating process of the PK data is, of course, unknown, this is suggestive evidence that the global maximum in PK, the one with a significant and positive credit effect, is the “true” maximum.

The simulated data-generating process has two genders and one program. Forty variables Xi were generated from independent normal (0,1) random variables. The treatment chosen by those who are offered treatment is cf = 0.5∑x - 7μ + 5εf for females, and cm = 0.5∑x - 7μ + 5εm for males, where μ , εf and εm, are independent normal (0,1) random variables. The outcome is y = 1.0*cf + 1.0*cm + 0.5∑x + 7μ + 5ν, where ν is an independent normal (0,1) random variable. Half of the generated sample is female and half male. Treatment was made available to half of each gender-defined sub-sample. One hundred data sets with 5000 observations each were independently generated and the models estimated with maximum likelihood using Roodman’s _cmp_ estimation package to trace the converged likelihood over a range of the credit effect parameters δ = δf = δm starting at -0.3 and incrementing by 0.1 until δf =1.5 . The gender breakdown is not really relevant in this exercise, since the point is to examine the shape of the log-likelihood surface with respect to δ, the parameter on credit.

In Table 4, as noted, the log-likelihood with respect to δ is bimodal in every single simulated dataset. The global maximum was tightly clustered around the true value of δ = 1, and the local maximum (inferior mode) was clustered around δ = -0.1.<sup>23</sup> Figure 1 presents the log-likelihood surface for the 1<sup>st</sup> , 25<sup>th</sup> , 75<sup>th</sup> , and 100<sup>th</sup> simulated data set. The resemblance to the PK log-likelihood surface presented by RM (see also our Figure 2) is striking. One attribute of a bimodal log-likelihood is that selectively dropping observations can “cut off” the mode corresponding to the global maximum, resulting in a discrete (and negative, in this case) jump to the local mode. Indeed, it was not difficult to find data-generating

> 23 When iterated to convergence, the mean value of the estimated δ was 0.998 with a standard deviation of 0.032, and the asymptotic t-ratios on δ were in excess of 30.0 in all simulated datasets.  The data-generating process thus seems to result in a very well-identified model; that is, the instruments are not weak.

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