2SLS estimates leave out that option. Exactly what LIML means in two-stage least squares estimation is described in detail below. A comparison of the linear LIML estimates of columns (2) and (3) and the standard 2SLS estimates in columns (3) and (4) suggest that they have no relationship to each other at all. For example, the coefficient on the variable _nontarpk_ in the 2SLS is 0.146 with a t-statistic of 3.39, but in the linear LIML it is -2.27 with (of course) a t- statistic of -0.01. RM estimated this model by standard 2SLS and posted it online as part of their response to Pitt (2011a), titled “Response to Pitt’s Response to Roodman and Morduch’s Replication of..., etc.” (available at http://blogs.cgdev.org/open_book/2011/03/response-to-pitts- <u>response-to-roodman-and-morduchs-replication-of-etc.php), and thus should be aware of this</u> discrepancy in the t-statistics. Our Table 6 provides their 2SLS from that March 2011 response to Pitt as well as the linear LIML estimates that appear in this paper, which was distributed as a working paper in December 2011.

Consider as well the second column of parameter estimates in Table 6 of RM. Stata once again warns that the estimated parameter covariance matrix is not of full rank, and 105 out of 106 t-values in the second-stage are less than 0.5 in absolute value.<sup>31</sup> The Stata output of coefficient and t-statistics are copied and pasted into our Table 7. The (centered) R-squared of this regression is -7.26. RM once again ignore this evidence of serious issues with the estimates, and proceed to report parameters, t-ratios, and Kleibergen-Paap underidentification and weak identification tests.

The likely source of the clearly unidentified linear LIML models that RM present in their Table 6 is the extraordinarily artificial nature of the matrix of exogenous variables that they construct for the first-stage equations. To see why this matters, consider the nature of the linear LIML estimator. Leaving aside both sample weights and clustering so as to leave the notation uncluttered, the κ-class estimator of the vector of second-stage parameters **b** = [ **β** δ] is where **Z** is the full set of exogenous regressors in the first-stage, **X** is the set of included exogenous regressors **X1** plus the endogenous credit variables **C** , **X** = [ **X1 C** ], in the secondstage, and **MZ** = **I – Z(Z ́Z)**<sup>**-1**</sup> **Z ́** . The linear LIML estimator sets κ equal to the minimum eigenvalue of **(C ́MZC)**<sup>**-1/2**</sup> **C ́MX1C ́MX1C(C ́MZC)**<sup>**-1/2**</sup> , where **Mx1 = I – X1(X1 ́X1)**<sup>**-1**</sup> **X1 ́** . Exactly what this minimum eigenvalue represents in the RM formulation of the PK model is unclear because (i) the X and Z matrices have been by swelled by a factor of 58; (ii) most of the observations for **C** are deterministically zero but not treated as such; and (iii) 102 exogenous variables are dropped by Stata because of perfect collinearity. Which variables are kept by Stata and which are dropped is likely to be arbitrary, at least from point of view of statistical theory, so that dropping some columns from **X1** rather than from **X2** ( **Z = [X1 X2]** ), the matrix of excluded exogenous variables, may arbitrarily alter the mininum eigenvalue.

> 31 The only exception to this invariance of the t-ratios is for a survey round dummy variable with a t-ratio of 6.2.

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