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Matrix Decompositions

 b 2sls b 2sls . By standard calculations we can also show that bootstrap t-ratios are asymptotically normal. Theorem 12.8 Under Assumption 12.2, as n ! 1 p n  b 2sls b 2sls d ! N (0;V ) where V is the 2SLS asymptotic variance from Theorem 12.2. Furthermore, T = p n  b 2sls b 2sls s CHAPTER 12. INSTRUMENTAL VARIABLES 434 This shows that percentile-type and percentile-t conÖdence intervals are asymptotically valid. One might expect that the asymptotic reÖnement arguments extend to the BCa and percentile-t methods, but this does not appear to be the case. While p n  b 2sls b 2sls and p n  b 2sls  have the same asymptotic distribution, they di§er in Önite samples by an Op n 1=2  term. This means that they have distinct Edgeworth expansions. Consequently, unadjusted bootstrap methods will not achieve an asymptotic reÖnement. An alternative suggested by Hall and Horowitz (1996) is to recenter the bootstrap 2SLS estimator so that it satisÖes the correct orthogonality condition. DeÖne b 2sls =  X0Z Z 0Z 1 Z 0X 1 X0Z Z 0Z 1 Z 0y Z 0be  : We can see that p n  b 2sls b 2sls = 1 n X0Z  1 n Z 0Z 1 1 n Z 0X !1   1 n X0Z   1 n Z 0Z 1 1 p n Xn i=1 (z i e i E (z i e i ))! which directly converges to the N (0;V ) distribution without special handling. Hall and Horowitz (1996) show that percentile-t methods applied to b 2sls achieve an asymptotic reÖnement and are thus preferred to the unadjusted bootstrap estimator. This recentered estimator, however, is not the standard implementation of the bootstrap for 2SLS as used in empi