13.4 Testing endogeneity and instrument validity
The Hausman test
If \(X\) is in fact exogenous, both least squares and IV are consistent, and least squares is more efficient. If \(X\) is endogenous, only IV is consistent. Hausman (1978) proposed testing exogeneity by comparing the two.
Lemma 13.4.1. Let \(\hat \beta _E\) be asymptotically efficient and \(\hat \beta _C\) consistent, both asymptotically normal under \(H_0\). Then the difference \(q = \hat \beta _C - \hat \beta _E\) is asymptotically uncorrelated with \(\hat \beta _E\), and \(\Var (q) = \Var (\hat \beta _C) - \Var (\hat \beta _E)\) asymptotically.
Proof. Suppose \(\Cov (\hat \beta _E, q) = C \neq 0\). For any matrix \(A\) the estimator \(\hat \beta _E + Aq\) is consistent, with variance \(\Var (\hat \beta _E) + AC' + CA' + A\Var (q)A'\). Taking \(A = -tC\Var (q)^{-1}\) with a small \(t > 0\) gives \(\Var (\hat \beta _E) - 2tC\Var (q)^{-1}C' + t^2C\Var (q)^{-1}C'\), which for small \(t\) is smaller than \(\Var (\hat \beta _E)\) by a non-zero psd matrix, contradicting efficiency. So \(C = 0\), and then \(\Var (\hat \beta _C) = \Var (\hat \beta _E + q) = \Var (\hat \beta _E) + \Var (q)\). □
Theorem 13.4.2 (Hausman). Under \(H_0\) that \(X\) is exogenous, with homoscedastic disturbances, \[ H = (b_{\text {IV}} - b)'\bigl [\widehat \Var (b_{\text {IV}}) - \widehat \Var (b)\bigr ]^{-1}(b_{\text {IV}} - b) \xrightarrow {d} \chi ^2_q, \] where \(q\) is the number of endogenous regressors being tested (a generalised inverse is used if the matrix is singular). Equivalently, let \(\hat V = M_ZX_1\) be the first-stage residuals for the suspect regressors \(X_1\); then the least squares coefficient of \(X\) in the regression of \(Y\) on \(X\) and \(\hat V\) equals \(b_{\text {2SLS}}\), and the Wald test of the coefficients of \(\hat V\) is asymptotically equivalent to \(H\).
Proof. The \(\chi ^2\) limit follows from Lemma 13.4.1, with \(b\) efficient under \(H_0\) by the Gauss–Markov theorem, and the quadratic-form result of Theorem 5.4.2. For the regression form, take one suspect regressor and write \(X = [X_1\ X_2]\), with \(X_2\) exogenous and included in \(Z\). By the Frisch–Waugh–Lovell Theorem 6.4.1, the coefficients of \(X\) in the regression of \(Y\) on \(X\) and \(\hat V\) are those of the regression of \(M_{\hat V}Y\) on \(M_{\hat V}X\). Now \(\hat V'X_1 = X_1'M_ZX_1 = \hat V'\hat V\) and \(\hat V'X_2 = 0\), so \(M_{\hat V}X_1 = X_1 - \hat V = P_ZX_1\) and \(M_{\hat V}X_2 = X_2 = P_ZX_2\): \(M_{\hat V}X = \hat X\). Since \(\hat X'\hat V = 0\), \(\hat X'M_{\hat V}Y = \hat X'Y\), and the coefficient is \((\hat X'\hat X)^{-1}\hat X'Y = b_{\text {2SLS}}\). The coefficient of \(\hat V\) measures how far least squares departs from 2SLS, and a zero coefficient means that the two coincide. □
The regression form, often called the Durbin–Wu–Hausman or control function test, is easy to run and to make robust to heteroscedasticity, by using robust standard errors for the coefficient of \(\hat V\). The intuition is attractive: \(\hat V\) is the part of \(X_1\) not explained by the instruments, the part suspected of being correlated with \(U\). If it has no explanatory power for \(Y\), there is no evidence of endogeneity.
The Sargan test of overidentifying restrictions
Exogeneity of the instruments cannot be tested when the equation is exactly identified: the IV residuals are orthogonal to \(Z\) by construction. With \(\ell > k\) there are \(\ell - k\) surplus restrictions, and they can be tested.
Theorem 13.4.3 (Sargan). Let \(\hat u\) be the 2SLS residuals. Under the hypothesis that all the instruments are exogenous, with homoscedastic disturbances, \[ S = nR^2 \xrightarrow {d} \chi ^2_{\ell - k}, \] where \(R^2\) is from the regression of \(\hat u\) on \(Z\).
Sketch of proof. \(S = \hat u'P_Z\hat u/(\hat u'\hat u/n)\). Writing \(\hat u = U - X(b_{\text {2SLS}} - \beta )\) and substituting the expression for \(b_{\text {2SLS}} - \beta \) shows that, asymptotically, \(P_Z\hat u/\sigma \) behaves like \((P_Z - P_{\hat X})U/\sigma \). Since \(\hat X\) lies in the column space of \(Z\), \(P_Z - P_{\hat X}\) is a projection of rank \(\ell - k\), and the quadratic form in \(U/\sigma \) is asymptotically \(\chi ^2_{\ell - k}\) by Theorem 5.4.2, the asymptotic normality of \(\frac {1}{\sqrt n}Z'U\) taking the place of exact normality. □
The test has a serious limitation: it can only detect instruments that disagree with each other. If all the instruments are invalid in the same way, they agree, the test does not reject, and all the estimates are wrong together. A rejection says that at least one instrument is invalid, without saying which. In the end the case for an instrument rests on economic argument: why should it affect \(Y\) only through \(X\)?
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