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# standard normal distribution

Let b =  b 1 ; b 2  be the OLS estimator and let e =  e 1 ; e 2  be the 2SLS estimator. Under H0 (and homoskedasticity) the OLS estimator is Gauss-Markov e¢ cient, so by the Hausman equality var  b 2 e 2  = var  e 2  var  b 2  =  X0 2 (P Z P 1) X2 1 X0 2M1X2 1   2 where P Z = Z (Z 0Z) 1 Z 0 , P 1 = X1 (X0 1X1) 1 X0 1 , and M1 = In P 1. Thus a valid test statistic for H0 is T =  b 2 e 2 0  (X0 2 (P Z P 1) X2) 1 (X0 2M1X2) 1 1  b 2 e 2  b 2 (12.66) for some estimate b 2 of  2 . Durbin (1954) Örst proposed T as a test for endogeneity in the context of IV estimation, setting b 2 to be the least-squares estimate of  2 . Wu (1973) proposed T as a test for endogeneity in the context of 2SLS estimation, considering a set of possible estimates b 2 , including the regression estimate from (12.65). Hausman (1978) proposed a version of T based on the full contrast b e, and observed that it equals the regression Wald statistic W0 described earlier. In fact, when b 2 is the regression estimate from (12.65), the statistic (12.66) algebraically equals both W0 and the version of (12.66) based on the full contrast b e . We show these equalities below. Thus these three approaches yield exactly the same statistic except for possible di§erences regarding the choice of b 2 . Since the regression F test described earlier has an exact F distribution in the normal sampling model, and thus can exactly control test size, this is the preferred version of the test. The general class of tests are called Durbin-Wu-Hausman tests, Wu-Hausman tests, or Hausman tests, depending on the author. When k2 = 1 (there is one right-hand-side endogenous variable) which is quite common in applications, the endogeneity test can be equivalently expressed at the t-statistic for b in the estimated control function. Thus it is su¢ cient to estimate the control function regression and check the t-statistic for b. If j bj > 2 then we can reject the hypothesis that x2i is exogenous for . We illustrate using the Card proximity example using the two instruments public and private. We Örst estimate the reduced form for education, obtain the residual, and then estimate the control function regression. The residual has a coe¢ cient 0:088 with a standard error of 0.037 and a t-statistic of 2.4. Since the latter exceeds the 5% critical value (its p-value is 0.017) we reject exogeneity. This means that the 2SLS estimates are statistically di§erent from the least-squares estimates of the structural equation and supports our decision to treat education as an endogenous variable. (Alternatively, the F statistic is 2:4 2 = 5:7 with the same p-value). We now show the equality of t