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Generalized Eigenvalues

ractice. 12.24 The Peril of Bootstrap 2SLS Standard Errors It is tempting to use the bootstrap algorithm to estimate variance matrices and standard errors for the 2SLS estimator. In fact this is one of the most common use of bootstrap methods in current econometric practice. Unfortunately this is an unjustiÖed and ill-conceived idea and should not be done. In Önite samples the 2SLS estimator may not have a Önite second moment, meaning that bootstrap variance estimates are unstable and unreliable. Theorem 12.7 shows that under jointly normality the 2SLS estimator will have a Önite variance if and only if the number of overidentifying restrictions is two or larger. Thus for just-identiÖed IV, and 2SLS with one degree of overidentiÖcation, the Önite sample variance is inÖnite. The bootstrap will be attempting to estimate this value ñinÖnity ñand will yield nonsensical answers. When the observations are not jointly normal there is no Önite sample theory (so it is possible that the Önite sample variance is actually Önite) but this is unknown and unveriÖable. In overidentiÖed settings when the number of overidentifying restrictions is two or larger the bootstrap can be applied for standard error estimation. However this is not the most common application of IV methods in econometric practice and thus should be viewed as the exception rather than the norm. To understand what is going on, consider the simplest case of a just-identiÖed model with a single endogeneous regressor and no included exogeneous regressors. In this case the estimator can be written as a ratio of means biv = Pn i=1 P ziei n i=1 zixi : Under joint normality of (ei ; xi), this has a Cauchy-like distribution which does not possess any Önite integer moments. The trouble is that the denominator can be either positive or negative, and arbitrarily close to zero. This means that the ratio can take arbitrar CHAPTER 12. INSTRUMENTAL VARIABLES 435 To illustrate let us return to the basic Card IV wage regression from column 2 of Table 12.1 which uses college as an instrument for education. Estimate this equation for the subsample of black men, which has n = 703 observations. We focus on the coe¢ cient for the return to education. The coe¢ cient estimate is reported in Table 12.3, along with asymptotic, jackknife, and two bootstrap standard errors each calculated with 10,000 bootstrap replications.