treatment. Clearly, the number of observations without choice of treatment critically affects identification – at the limit, if there is only one such observation in the data, the model with both exogenous variables X and treatment c is not identified – and yet the authors proceed as if this were just another two-stage least squares problem. They seem to recognize there might be a problem, but then fail to properly address it: _Because of the unusual design of the PK estimator, no theory is available to measure instrument weakness in terms of its potential for distorting estimates. We find some evidence, however, that credit eligibility is a weak instrument for credit uptake. Differentiating credit by gender exacerbates the weakness_ . (p. 27) If there is no statistical theory, then on what basis is there evidence?

Finally, one can easily simulate PK-like data sets and (i) estimate the model as PK do, by maximum likelihood, and (ii) estimate the model as RM do, using the wrongly augmented set of instruments and 2SLS.<sup>16</sup> The results of generating 200 such datasets are provided in Table 1.

Using the RM approach, the simulated data generating process has a positive bias for males and a negative bias for females<sup>17</sup> . Using PK’s maximum likelihood method there is absolutely no evidence of bias arising from “weak instruments.” The mean value for the F- statistic for the first-stage regressions given by equations (4) and (5) are 3.47 and 3.41, respectively. Using RM’s 2SLS method, however, there is strong evidence of bias in the direction of the least squares bias. The male parameters are almost twice as large as the true values, and the female parameters are one-third too small, so that RM’s method finds that male and female credit effects are about the same while the true female effect is three times as large as the true male credit effect. The Kleibergen-Papp rk rank test F statistic for weak instruments in the 2SLS has a mean value of 0.736, and the null hypothesis of “weak instruments” is never rejected in the 200 simulations. It is no surprise that the instruments are weak in RM’s approach,

> 16 RM actually estimate their model with linear LIML _not standard_ two-stage least squares.  This turns out to be another crucial error on their part that is discussed at length below.  We use standard 2SLS here so as to highlight the effect of the weak instruments bias that arises as a consequence of RM’s padded matrix of instrumental variables.  The further degree of bias arising from the use of linear LIML is estimated with these same simulated datasets below and discussed at length.

> 17 For the simulation, 40 variables Xi were generated from independent normal (0,1) random variables.  The treatment chosen by females who are offered treatment is cf = 0.5∑x + 12μ + 8εf, and the treatment chosen by males who are offered treatment is cm = 0.5∑x - 12μ + 8εm, where μ , εf, and εm are independent normal (0,1) random variables.  The outcome is y =   0.75*cf + 0.25*cm + 0.5∑x  - 12μ + 8ν, where ν is an independent normal (0,1) random variable.  Female treatment was made available to 40 percent of the sample, male treatment to 40 percent, and no treatment to 40 percent.  Half of the observations with female treatment also had male treatment, so when taking into account the overlap the percentages sum to 100 percent.  Those treated were randomly assigned to one of three groups.  Two hundred data sets with 5000 observations each were independently generated and the models estimated with maximum likelihood using Roodman’s _cmp_ estimation package, and the _ivreg2_ (2SLS) package of _Stata_ .

13