The Hausman Test and Weak Instruments y

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1 The Hausman Test and Weak Instruments y Jinyong Hahn z John C. Ham x Hyungsik Roger Moon { September 7, Abstract We consider the following problem. There is a structural equation of interest that contains an explanatory variable that theory predicts is endogenous. There are one or more instrumental variables that credibly are exogenous with regard to this structural equation, but which have limited explanatory power for the endogenous variable. Further, there is one or more potentially strong instruments, which has much more explanatory power but which may not be exogenous. Hausman (978) provided a test for the exogeneity of the second instrument when none of the instruments are weak. Here we focus on how the standard Hausman test does in the presence of weak instruments using the Staiger-Stock asymptotics. It is natural to conjecture that the standard version of the Hausman test would be invalid in the weak instruments case, which we con rm. However, we provide a version of the Hausman test that is valid even in the presence of weak IV, and illustrate how to implement the test in the presence of heteroskedasticity. We show that the situation we analyze occurs in several important economic examples. Our Monte Carlo experiments show that our procedure works relatively well in nite samples. We should note that our test is not consistent, although we believe that it is impossible to construct a consistent test with weak instruments. JEL Classi cation: C Key Words: Hausman Test, Weak Instruments. Introduction The weak instruments problem has led to the development to two strands of research, each of which is characterized by a di erent asymptotic approximation. The rst of these, which we will call the We thank Takeshi Amemiya, an associate editor, two referees, and Joris Pinkse for helpful comments and suggestions, and Martin Weidner for proofreading. Hahn, Ham, and Moon acknowledge supports from the National Science Foundation. Any opinions, ndings, conclusions, or recommendations in this material are those of the authors and do not necessarily re ect the views of the National Science Foundation. y All the omitted proofs and derivations are available in our Online Appeneix (Hahn, Ham, and Moon ()) which is available at www-rcf.usc.edu/~moonr or upon request by to the authors. z Department of Economics, University of California, Los Angeles. x Department of Economics, University of Maryland, IZA, IRP (UW-Madison). { Department of Economics, University of Southern California

2 many-instrument asymptotics, emphasizes the nite sample distortion which can be explained by the approximation where the number of instruments grows to in nity as a function of the sample size. This literature often concludes that the IV estimators are still approximately normal, but that the asymptotic variance estimators need to address the nite sample issue. Because the many-instrument asymptotics still produces a normal approximation for the estimators, the implication for practitioners is more or less a simple message that the standard error calculations need to be re ned. On the other hand, there is the concern that the many-instrument asymptotics may not be relevant for situations where the degree of overidenti cation is mild. When the model is just or only mildly overidenti ed, and the explanatory power of the instruments is small, the alternative approximation due to Staiger and Stock (997) is intuitively appealing. This approximation is characterized by alternative asymptotics where the rst stage coe cient shrinks to zero as a function of the square root of the sample size. We will refer to this approximation as the weak-instruments asymptotics or Staiger and Stock asymptotics. Under Staiger and Stock s asymptotic approximation, many usual statistics have nonstandard asymptotic distributions. For example, it is well-known that IV estimators and t-statistics have nonstandard distributions. Staiger and Stock (997) also considered tests of overidenti cation under their asymptotics, and established that standard tests of overidenti cation do not have chi-square ( hereafter) distributions either. On the other hand, they showed that in the context of comparing the weak IV against OLS, a version of Hausman test statistic as usually employed has a correct asymptotic size, although they did observe that the test is not consistent. This nding is important because there is no standard test in the literature to determine whether conventional asymptotics or Staiger and Stock s alternative asymptotics is more appropriate for a given nite sample. The version of Hausman test has the identical asymptotic distribution under both asymptotics, and is thus a exception to the rule of thumb that test statistics tend to have nonstandard distribution under weak instrument asymptotics. It is a useful exception in that practitioners do not need to worry about the weakness of the IV and its potentially complicated consequences. In this paper, we extend Staiger and Stock s (997) analysis and document further exceptional cases. We next consider the standard Hausman test that examines the di erence of two IV estimators based on two di erent sets of instruments, and show that it possesses a certain robustness property in that its asymptotic distribution is invariant to whether conventional or weak instrument asymptotics are adopted. We consider a Hausman test that compares weak IV against strong IV. It is wellknown that the test statistic has a distribution under conventional asymptotics. We establish that a version of Hausman test continues to have the distribution even under the weak instrument asymptotics. We then show that a version of the overidenti cation test, which we interpret to be a natural generalization of the Hausman test, has such robustness. Finally, we also provide empirical researchers with a version of the Hausman test that can be used with heteroscedasticity under both See Bekker (994), Donald and Newey (), and Hahn and Hausman (), among others. See also Kleibergen (), Moreira (3), and Andrews, Moreira, and Stock (4).

3 conventional and Staiger-Stock asymptotics; although quite straightforward theoretically, neither case is currently available in the literature. Besides being of theoretical interest, our result has substantial practical implications because empirical researchers often face the following problem. 3 They have a structural equation of interest that contains an explanatory variable that theory predicts is endogenous. They want to obtain a con dence interval for the estimated coe cient on the structural parameter, or for a set of coe cients from the structural equation. On the one hand they have one or more instrumental variables that credibly are exogenous with regard to this structural equation, but which have limited explanatory power for the endogenous variable. On the other hand they have one or more strong instruments, which have much more explanatory power but which may not be exogenous. Researchers currently can take one of two tacks. First, if the researcher only uses the weak instruments, the standard errors on the structural equation calculated by standard methods may be very large. Moreover, it may be the case that the standard asymptotic distribution for IV estimators is invalid because of the weak instrument problem. Second, in the vast majority of cases, empirical researchers use the strong instrument since it is simple to use and likely to produce statistically signi cant results. Thus it has obvious appeal to the researcher, but also has the obvious disadvantage that the researcher may obtain inconsistent results if the strong instrument is not a valid instrument. We would propose that researchers take a third approach in their work: use the strong instrument but provide a diagnostic via a Hausman test comparing the results using the strong and weak instrument. However, this approach raises the concern of whether the Hausman test is valid when some of the instruments are weak, which our paper naturally addresses. The outline of the paper is as follows. We outline our model and assumptions in Section. In Section 3 we motivate the paper by showing that the situation we analyze arises in several important economic examples: i) estimating models of life cycle labor supply behavior; ii) estimating dynamic models such as a health production function for individuals in a developing country and iii) estimating the return to schooling. In Section 4 we consider the conventional Hausman test under weak IV asymptotics and show that, in general, it will not have the standard distribution, but if the model is exactly identi ed given the weak instruments, one of the standard tests can be used without modi- cation. In section 5 we provide a modi cation of the Hausman test when the model is overidenti ed given the weak instruments that has a standard distribution. In Section 6 we rst consider the case where it is the weak instrument, and not the strong instrument, that may not be valid. then extend our test to the case where the errors are heteroskedastic. 4 We Finally we extend the test 3 See Section 3 below. 4 To carry out this extension, we must rst derive the standard Hausman test to for the case where both the exogenous and potentially endogenous instruments are strong and the errors are heteroskedastic. Although this derivation is quite straightforward from the perspective of econometric theory, it should be helpful to applied researchers since this test is not currently available in the literature. 3

4 statistic from Section 5 to improve its power while maintaining its good size properties. The results of our Monte Carlo experiments are presented in Section 7. They show that there is indeed a problem with the standard tests when there are weak instruments which overidentify the model, and that our general procedure works relatively well in this case in nite samples. Model and Assumptions We consider a simultaneous equation linear regression model y = Y + " and Y = Z + V; where " and V are mean zero unobserved error matrices, y is an n vector of dependent variables, Y is an n K matrix of regressors that are correlated with ", Z is an n L matrix of IV s with L K that are independent of V: 5 paper are based on n!. The sample size is denoted by n and all the asymptotic results of the We assume that the IV s consist of two components, Z = [W; S], where W is an n L w matrix that contains weak IV s and S is an n L s matrix that contains strong, but potentially invalid, IV s. Further, e S is the residual when S is projected on W in the population, S = W w + e S: () We also denote y i ; Y i ; w i ; s i ; es i ; " i; and v i to be the ith row of y ; Y ; W; S; e S; "; and V; respectively. We assume that L w K: Throughout this section, we will assume that W is orthogonal to the regression error ", that is, E [w i " i ] =. The main object of interest in this paper is a test for the validity of the IV s in S. In this case, the hypotheses that we are testing are H : E [s i " i ] = (or E [es i " i ] = ), () H : E [s i " i ] 6= (or E [es i " i ] 6= ). (3) Thus, if the exclusion restriction is violated, then it is only through the possible correlation between es i and " i. 6 Let s = [E (es i es i )] E (es i " i ) denote the coe cient of the projection of " on e S. We write that " = e S s + V v + e; (4) 5 We follow the standard approach, and assume that included exogenous variables are partialled out - see the online appendix for more details. 6 The main focus of the paper is not to obtain a post-speci cation test inference, but rather to investigate the validity of various versions of the Hausman tests themselves with weak IV s. See Guggenberger(8), e.g., for issues on postspeci cation test inference. 4

5 where v = [E (v i v i )] E (v i " i ) denotes the coe cient of projection of " i on v i. We will assume that e i is uncorrelated with es i and v i and has mean zero and variance e. Our null and alternative hypotheses can then be rewritten as H : s = ; H : s 6= : The basic idea of the Hausman test statistics for the null hypothesis () is based on the di erence of the following two estimators 7 : b w = YP W Y Y P W y ; = YP Z Y Y P Z y : When the conventional asymptotic approximation is valid, b z is an e cient but non-robust estimator, while b w is a less e cient, but robust, estimator. Then the conventional Hausman test statistics measure the di erence b w z b using various weight matrices. We rst consider three versions of the Hausman test that are used widely in the literature: H = bw H = b ";w bw H 3 = b ";z bw h z b b ";w YP W Y b ";z YP i Z Y bw h z b h Y P W Y Y P Z Y i bw Y P W Y Y P Z Y i bw ; ; ; where b ";w = n y Y b w y Y b w (5) and b ";z = n y Y y Y : (6) Under conventional asymptotics these test statistics all converge to K ; a (central) chi-square distribution with d.f. K under the null. Therefore, the conventional asymptotics suggest that we compare these test statistics with the critical value from K : Our contribution is to consider the properties of the test statistics under the assumption that W is weak, and S is strong but potentially invalid, under the asymptotics developed by Staiger and Stock (997). 3 Economic Examples The problem we analyze arises in many empirical studies; here we show this for three important cases. 7 Given a matrix A; we use notation P A = A (A A) A and M A = I P A throughout. 5

6 3. Life Cycle Labor Supply Models Researchers often consider the following model to describe the (annual) intertemporal labor supply function for prime-aged males 8 ln(h it ) = ln(w it ) + + X it + e it + it ; (7) where denotes the rst di erence. In (7) h it are hours of work in year t for individual i, w it is his real hourly wage rate in that year, X it are time changing demographic variables, and e it is an idiosyncratic error term. Further, it is a rational expectations error term which is orthogonal to all variables known in period t ; thus ln(w it ) is correlated with it : We also expect that ln(w it ) is correlated with e it since variable w it is formed by dividing annual earnings by h it, and the latter is thought to contain substantial measurement error. MaCurdy (98) used polynomials in age as IV for ln(w it ); but Altonji (986) argued that MaCurdy s instruments were weak in the sense of not being jointly signi cant in the rst stage equation. Instead Altonji considered a direct measure wm it of the wage which is obtained from a question put to individuals in the sample what is your hourly wage rate? He assumes that the measurement error in w it and wm it are independent, and thus only considering the error term e it, the variable ln(wm it ) is a valid IV for ln(w it ). However, as Altonji noted, this potential instrument will not be independent of it unless wm it is known in period t. He next considered ln(wm it ) as an IV for ln(w it ); since it will be orthogonal to it, but nds that the correlation between ln(w it ) and ln(wm it ) is too weak to be empirically useful. Instead he assumes that the wage is known one period in advance so that ln(wm it ) is indeed an appropriate IV for ln(w it ). Thus our procedure could be used to o er readers a diagnostic test whether ln(wm it ) is indeed a valid IV, using either (or both) MaCurdy s polynomial in age or ln(wm it ) as the weak instruments. 3. Dynamic Models Researchers often consider dynamic panel data regressions of the form y it = y it + X it + u it : (8) In (8) y it is a scaler dependent variable for individual i in year t, X it is a vector of exogenous explanatory variables, and u it is an error term. Since it is unreasonable to assume that the error term u it is independent over time for the same person, y it must be treated as endogenous. Natural instruments are lagged values of X it, but researchers often nd that these lagged values of X it do a poor job of explaining y it. Instead they often assume a MA (k) structure for u it ; which implies that y it k is a valid IV for y it. However, the choice of k is usually arbitrary, since economic theory does not 8 See MaCurdy 98, Altoni 986, Ham 986, Ham and Reilly. Corner solutions at zero hours are not important for this group and thus a regression framework is appropriate. 6

7 provide any guidance on this issue. Again our test can be used here, where y it k is the strong instrument and the lags of X it are the weak instruments. An example of such an equation is given in Strauss and Thomas (995), where (8) is a health production function for individuals in a developing country, and the X it represents variables such as distance to the village health clinic. They used the strong instruments (lagged y it ), but could have used our procedure below to obtain a diagnostic for their approach. 3.3 Estimating the Return to Schooling Consider the wage equation ln(w i ) = S i + A i + X i + e i : (9) In (9) the variables w i, S i and A i represent the hourly wage, years of schooling, and ability (as measured by a test such as the AFQT in the case of the NLS data), while X i represents variables such as race, experience, and experienced squared. The problem here is that even conditional on ability A i, S i and e i may be correlated. For example, an increase in ambition may increase both S i and e i ; leading to a positive correlation between these variables. One possible instrument for S i is the father s education F E i, which Willis and Rosen (979) use to identify a more complicated version of (9). They argue that children from wealthier families have a lower discount rate than poorer children, since wealthy parents are more likely to help nance their children s education. In practice F E i will be an important determinant of S i conditional on A i and X i : However, it may be an invalid IV since it can also re ect the father s ambition, which he may pass on to his children; if so, F E i will be correlated with e i and thus will be an invalid IV. An alternative IV is the father s age, F A i ; in the year that the individual turned eighteen, since this will also a ect the family s ability to help its children pay for college, but is unlikely to be correlated with e i : Unfortunately F A i may have little predictive power for S i conditional on A i and X i ; and again out test provides a diagnostic for Willis and Rosen s assumption that F E i is a valid IV. 4 Hausman Tests under Weak IV Asymptotics We now investigate the theoretical properties of the conventional Hausman test for the hypothesis of () under the assumption that W is weak. More speci cally, we assume that the coe cient of the population projection of Y on W shrinks to zero at the rate p n, while the coe cient of population projection of Y on S does not. For this purpose, we adopt the following parameterization: Y = W C p n + e S s + V: () We assume that L s K. Suppose that we consider H, H, or H 3 ; but adopt Staiger and Stock s (997) alternative asymptotic approximation. Because their asymptotics implies that the asymptotic 7

8 distribution of b w is not normal, it is natural to conjecture that the asymptotic size of the conventional procedure would be distorted. Not surprisingly, H, H, and H 3 are usually not distributed as K under the weak-instrument asymptotic approximation see Theorem 4 in Appendix C.. Our rst main contribution is to recognize that there is an important exception. We show that the H 3 is asymptotically K under the null despite the presence of weak instruments if the model is exactly identi ed with only the weak IV: Theorem Assume Conditions and in Appendix A. 9 Suppose that L w = K and Y W has full rank K. Then, (a) under the null hypothesis (), (b) under the alternative hypothesis (3), H 3 ) Z Z K; H 3 ) (Z + ) (Z + ) K () ; where Z N (; I K ) and the noncentrality parameter is de ned in (6) in the Appendix. Proof. In Appendix C.. Theorem indicates that, as long as the weak instrument W exactly identi es (L w = K), the standard practice of using H 3 along with a critical value from K is asymptotically valid even under the weak-instrument asymptotics. The weak-instrument asymptotic distribution under the null is identical to the standard asymptotic distribution. The weak-instrument asymptotic distribution under the alternative is K () ; a noncentral chi-square distribution with d.f. K; which dominates the asymptotic distribution under the null K, and therefore, the test is unbiased under the weakinstrument asymptotics. The invalidity of H and H (even under exact identi cation) can be attributed to the failure to estimate " consistently under the null. On the other hand, even H 3, which is based on a consistent estimator of " under the null, is invalid without exact identi cation. This is one of the main di erences between our results and the results in Staiger and Stock (997), who test for exogeneity of the regressors (there the strong IV s are the regressors). In the next section, we develop a modi cation of the Hausman test that does not require exact identi cation. This modi cation requires " to be estimated consistently under the null. 5 Generalized Hausman Test The results in Section 4 imply that a version of Hausman test, i.e., H 3 (but not H and H ); combined with a critical value from K, is asymptotically valid even under the weak-instrument asymptotics as 9 We impose standard regularity conditions, which are discussed in the Appendix. 8

9 long as is exactly identi ed by weak instruments (L w = K). On the other hand, Theorem 4 in the Appendix C. shows that neither H, H, nor H 3 ; along with a critical value from K, is valid under the weak-instrument asymptotics if the weak IV s overidentify, that is, when L w > K. One might argue that the overidenti ed case is not of practical concern because a practitioner can always choose a subset of weak instruments from W that exactly identi es and thus H 3 can be used: Although one can resolve the situation in this fashion, it is not clear which of the K weak instruments should be chosen out of L w. We show in this section that there is a version of the speci cation test which can be used with a critical value from a chi-square distribution even under the weak-instrument asymptotics when the model is overidenti ed with the weak IV s. Suppose that the model is in fact overidenti ed with the weak instruments ( L w K). Under the null that the strong IV s are valid, we have the moment condition E hw i y i Yi plim b i z = : However, under the alternative that the strong IV s are not valid, we have E hw i y i Yi plim b i z 6= because the probability limit of b z will be di erent from under the alternative. From these observations, we might consider a test statistic based on With some algebra, it can be shown that p n W y Y : Lemma Assume Conditions and in Appendix A. Under the null () and conventional asymptotics, p n W y Y ) N ; " ; where is the probability limit of n b and Proof. In the Online Appendix. Therefore, the test statistic is equal to b = W W W Y Y P Z Y Y W : H b " = b " y Y W b W y Y ; () where b = W W (W Y ) (Y P ZY ) (Y W ) and b " denotes some consistent estimator for ". In light of Lemma, it is straightforward to conclude that the (conventional) asymptotic distribution of H b " is Lw. In other words, researchers can use H b " to obtain a -test with standard critical values even when the model is overidenti ed with the weak IV s. Proposition gives an interpretation of the new statistic H b " : 9

10 Proposition When L w = K, H b " = bw b " h Y P W Y Y P Z Y i bw : Proof. In the Online Appendix. From Proposition, we can conclude that H b " can be understood as a version of the Hausman test in a special case where L w = K. Depending on the estimator b " used, the statistic H b " can be understood to be an extension of H or H 3. Recall b ";z in (6) : It turns out that H b ";z ; which is comparable to H3, has desirable asymptotic properties. Theorem Assume Conditions and in Appendix A. Assume that L w K: (a) Under the null hypothesis, H b ";z ) Lw ; (b) under the alternative hypothesis, H b ";z ) ( + Z) ( + Z) ; where Z N (; I Lw ) and is the same noncentrality parameter as in Theorem. Proof. In Appendix D.. Theorem indicates that using H b ";z along with a critical value from is asymptotically valid even under the weak-instrument asymptotics, and the weak-instrument asymptotic distribution under the null is identical to the standard asymptotic distribution. The weak-instrument asymptotic distribution under the alternative dominates the asymptotic distribution under the null, and therefore, the test is unbiased under the weak-instrument asymptotics. On the other hand, the test statistic does not diverge to in nity under the alternative, as is the case with standard asymptotics, and therefore the test is not consistent under weak-instrument asymptotics regardless of whether the model is exactly identi ed or over-identi ed by the weak IV s. 6 Discussion We rst consider two deviations from our assumptions. First, we consider the case where the strong IV is valid under both null and alternative hypotheses, while the weak IV is valid only under the null. Second, we examine the consequences of heteroscedasticity. After this, we consider the issue of improving power. When L w = K, we have H b ";w = H and H b ";z = H3. We thank an anonymous referee for suggesting this question.

11 6. When the Weak IV are Valid Only Under the Alternative Hypothesis We may want to consider an alternative scenario, where the strong IV are valid both under the null and the alternative, and the weak IV are valid only under the null. Although this scenario is unlikely to be common in practice, we address this situation for its theoretical interest. In this case, the model could be modi ed as y = Y + " and Y = W ~ p C n + S s + V, where the alternative hypothesis is now written " = W f w +V v +e. Here W f is the population projection residual of W on S: W = S s + W f. Here we consider the properties of (generalized) Hausman test statistic H b ";z. It can be shown that H b ";z is distributed as Ls under the null, but diverges to under the alternative. 3 In other words, the H b ";z has the identical properties as under the conventional asymptotics! 6. Heteroscedasticity It is well-known that SLS is not e cient under heteroscedasticity, and the usual form of the Hausman test would no longer be valid even under the null. This implies that, even with conventional asymptotics, the Hausman test has to be modi ed. We note that there does not exist a standard modi cation of Hausman test to accommodate heteroscedasticity. We consider one possible modi cation here. Since the size of the Hausman test is valid only when L w = K even under homoscedasticity, we assume that the dimension of the weak IV s is the same as the dimension of ; that is, L w = K: Here for expositional simplicity we assume that is a scalar (that is, K = ). The extension to a general K is straightforward and has been placed in the Online appendix. Given that the Hausman test has an interpretation of a comparison between b w and b z, a natural modi cation of the Hausman test statistic would take the form n bw z b pn () dvar bw z b where Var d pn bw denotes a consistent estimator of the asymptotic variance of p n bw under conventional asymptotics. To see this in more detail, by de nition, under the null we have p n bw z b = p p n bw n bz W Y W " Y = p P ZY Y P p Z" ; n n n n Given that the role of w and s is switched, we note that our test would be based on p n S y Y z b. 3 The proof is available in our Online Appendix.

12 and its asymptotic variance under conventional asymptotics is E wi " i (E [w i Y i ]) E (Y i zi ) (E (z izi )) E z i w i " i h i E (Y i zi ) (E (z izi )) E (z i Y i ) (E (zi zi )) E (z i Y i ) i : E [w i Y i ] + E (Y izi ) (E (z izi )) E z i zi " i h E (Y i zi ) (E (z izi )) E (z i Y i ) One can use the idea behind White s heteroscedasticity corrected standard errors, using the standard IV estimator s residuals ^" z = y Y z b. A natural choice for a consistent estimator of pn Var bw z b is pn dvar bw z b n i= = w i b" i n i= n i= w Y iz i n i= z iz i n i= z iw i b" i iy i n i= w h iy i n i= Y izi n i= z izi n i= z i iy i n i= + Y izi n i= z izi n i= z iz i^" i n i= z izi n i= z iy i h n i= Y izi n i= z izi n i= z i : iy i Although the above derivation is quite straightforward from the perspective of econometric theory, it should be helpful to applied researchers since the standard Hausman term where both the exogenous and potentially endogenous instruments are strong is not currently available in the literature when there is heteroskedasticity. In the Online appendix we show that under the Staiger-Stock asymptotics H hetero (^" z ) = n bw z b ) (Z + hetero ) ; (3) dvar bw z b where Z N (; ) and hetero = ww C P h lim n n n i= E wi (~s i ( s s ) + v i ( v ) + e i ) i = ; where = ( s eses s ) s eses s : Since hetero = ( = ) under the null, H hetero (^" z ) ) : Therefore, the test is valid under the null. On the other hand, under the alternative, H hetero (^" z ) ) ( hetero ) ; and thus the test H hetero (^" z ) becomes asymptotically unbiased.

13 6.3 Improving the Power of H 3 In Section 4, we noted that H 3 is asymptotically unbiased for the case where L = K. On the other hand, the test statistic does not diverge to in nity under the alternative, as is the case with standard asymptotics, and therefore the test is not consistent under weak-instrument asymptotics. Given that the consistency of a test is usually understood to be a necessity, a researcher may conclude that a test using H 3 is de cient. We should note, though, that a consistent test is probably impossible to construct given the nature of weak instruments. Many other tests are inconsistent with weak IV s, and a lack of consistency is not limited to the weak IV literature. (For example, a recent test by Andrews (3) exhibits similar properties.) The lack of consistency suggests that it would be a useful endeavor to try to improve the power of the test while maintaining its good size properties. For this purpose, we propose the following version of the Hausman test: H 4 = e ";z bw h where e ";z is obtained by the following algorithm: Y P W Y Y P Z Y i bw. Using the IV estimator b z, we get the IV residual b" z = y Y. ;. Regress the IV residual b" z on Z = [S; W ] to get the residual M Z b" z and de ne e ";z = n b" zm Z b" z = b ";z n b" zp Z b" z : (4) From (4), one can see that the proposed estimator e ";z modi es b ";z by subtracting n b" zp Z b" z. Although it is generally preferable to use the modi ed estimator e ";z, there are two special cases where such modi cation is unnecessary and b ";z does not need to be modi ed. The rst case is when = eses s belongs to the space spanned by the columns of = eses s (or s ). Recall the de nition = ( s eses s ) s eses s : We then have = eses s = = eses s ( s eses s ) s eses s, so that s = s. This coincidence depends on the alternative, which is not known to the practitioner, so it probably has little practical importance. The second case is of more practical signi cance. Suppose that the model is exactly identi ed by the strong instrument s i, that is, L s = K, and s is invertible. We then have s = s and =, and there is no need for the second step modi cation above we can use b ";z in place of b ". We believe that the second case is empirically more relevant than the rst case, because in many applications the endogenous regressor Y i and the strong IV s i are scalars. Theorem 3 below shows that H 4 thus de ned has the usual K under the null, and its asymptotic distribution under the null stochastically dominates K. 3

14 Theorem 3 Assume Conditions and in Appendix A. Suppose that L w = K. Then H 4 ) K under the null. Under the alternative hypothesis (3), H 4 ) (Z + ) (Z + ), where Z N (; I K ), is the same noncentrality parameter as in Theorem. Also, = p lim b ";z under the alternative and = p lim e ";z under the alternative. Proof. Omitted because Theorem 3 is an immediate consequence of Theorem in Section 5. In the Appendix we show that under the alternative, e ";z! p = ( v ) vv ( v ) + e and b ";z! p = ( s s ) eses ( s s ) +, where = plim bz = ( s eses s ) s eses s. Here it is obvious to see =, which implies that the asymptotic power of the modi ed test H 4 is larger than H 3. In Section 5, we proposed a generalized Hausman test statistic H () for the case where L w > K. A natural question is whether the power of H b ";z is dominated by H e ";z. Using Theorem, it is easy to see that H e ";z is asymptotically unbiased and its power dominates that of H b ";z. As such, H e ";z is a desirable test. We note that, if L w = K; H e ";z simpli es to H e ";z = e bw ";z h Y P W Y Y P Z Y i bw = H 4 (5) by Proposition. Based on this equality, we will, without too much loss of generality, de ne H 4 = H e ";z even for the overidenti ed case. 4 7 Monte Carlo Simulations The data generating process used in the Monte Carlo simulations is y i = Y i + s i s + " i Y i = w i w + s i s + v i ; i = ; : : : ; n " #! where (wi ; s i ) iid N (; I) ; (" i ; v i ) iid N ; ; and y i and Y i are scalars. Further, = and w and s are proportional to vectors consisting of ones. They are related to the (partial) rst stage R by R w = w w w w + ; R s = s s s s + : We xed R s = : throughout the simulation, and we considered R w = :; :. We set s = under the null, and s = ( s ; ; : : : ; ) under the alternative. We consider n = ; 5, = 4 ; ; 3 4, R w = :; : :3; :5; :; :, and s =. The dimensions of w i and s i are L w = ; 5 and 4 From our Online Appendix one sees that a test based on H 4 is quite easy to construct in Stata or similar programs. 4

15 L s = ; ; 5 respectively. The nominal size of the test is 5%. (Additional cases are considered in our Online Appendix.) All of the simulation results are based on 5 runs. Table A looks at the size of the test for H H 4 when there is one weak IV and = 4 for di erent sample sizes and numbers of strong instruments. Thus we rst consider the case where the model is exactly identi ed using the weak instruments and the degree of endogeneity is relatively small. The rst section of the table considers this speci cation for the three di erent sample sizes and the six values of R w. Ideally each entry should be.5, so that we see that in each case the size with low R w is much too small for H and H but is dead on for H 3 and H 4. As R w and the sample size increase, the size distortion of both H and H deceases. The bottom two sections of Table A consider the case of two and ve strong instruments respectively. Now the size of H 3 and H 4 are still equal to.5 or.6 in the rest of the cases, while H and H continue to be under-sized. Table B looks at the size of the test for H H 4 when there are ve weak IV and = 3 4 ; i.e. a case where the model is overidenti ed under the weak instruments and the degree of endogeneity is considerably higher. In this case, for tests H 3 and H 4 we consider H 3 = H b ";z and H4 = H e ";z ; where the test statistic H () is de ned in () : Now the size of each test is biased upwards - this is especially true for H and H with low R w. However, it is interesting to note that H 4 substantially outperforms H 3 for most of the cases, which is intuitively plausible since H 4 was developed for the case where the model is overidenti ed using the weak instruments. Comparing the results in Tables A and B does raise an interesting dilemma. On the one hand, researchers can improve the size of the test by using only one of the weak IV s when the model is overidenti ed under the weak IV. On the other hand, since di erent researchers are likely to make di erent choices in terms of which weak IV to use, they will obtain di erent test results for identical models. Further, there is also the potential problem of researchers running all ve regressions when there are ve weak instruments and one endogenous variable, and choosing the results they like the best. In Table C we consider the power properties of H 3 and H 4 when there is one weak IV, ve strong IV and s = (; ; : : : ; ); i.e. only one of the strong instruments is invalid. Note that this is a conservative example in that it will be harder to reject the null when it is false than if all the strong IV were invalid. Given that we have weak instruments, it is unrealistic to expect the entries in Table C to be close to one. Not surprisingly, the power of each test rises with the sample size and the explanatory power of the weak instruments. It is also interesting to note that when R w is low, the power of H 4 is often more than double that of H 3 when n =, and about 5% greater than that of H 3 when n = 5. Thus in terms of power with low R w, H 4 substantially out performs H 3 for all sample sizes in our example. This is to be expected as the model is overidenti ed under the strong IV, and H 4 was developed with power considerations in mind. 5 (Recall that there is no need to use the modi ed 5 The corresponding power statistics for H 4 when the model is overidenti ed under the weak instruments are somewhat higher than those in Table C. 5

16 version of e ";z when the model is exactly identi ed under the strong instruments.) When R w is high, the power gain decreases. However, in this case, the power itself is much higher than that for the case with low R w. 8 Conclusion Hausman (978) provides a test for whether an instrument(s) is valid given that the model is identi ed by other instruments which can be treated as exogenous. However, as we show in a series of examples, researchers often face the problem that the most acceptable instruments are also quite weak, while most strong instruments are potentially endogenous. Using Staiger-Stock asymptotics, we show that the standard Hausman test for this case may have a size distortion under the null in the presence of weak instruments unless the model is exactly identi ed using the weak instruments. We then provide a form of the Hausman test that eliminates the problem for the overidenti ed case. Finally, we show in our Online Appendix that this test is easy for empirical researchers to implement using a program like Stata. Our Monte Carlo results suggest that there is indeed a problem with the standard tests, and that our general procedure works relatively well in nite samples. 6

17 Appendix A Regularity Conditions Condition We assume the following. (i) n Z Z! p zz > ; n e S e S!p eses > ; n V V! p vv > ; n Y Y! p ; (ii) n Z V! p ; n Z e! p ; e ns V! p ; e ns e! p ; and (iii) n " "! p " > ; " # ww n e e! p e ws > ; where zz = and the notation > in (i) signi es positive de niteness of the matrices. sw ss Remark Condition assumes the weak law of large numbers of the variables in Z; e S; V; and Y. The asymptotic orthogonalities in Condition (ii) re ect the de nitions of the parameterizations in (), (), and (4). In Condition (iii) we assume homoscedasticity of the errors, as is common in the literature. We discuss heteroscedasticity in Section 6. Condition We also assume that 3 3 p n vec W S e vec (Z wes ) 6 p 4 n vec (W V ) 7 5 ) 6 4 vec (Z wv ) 7 5 N ; diag eses ww ; vv ww ; e ww ; p n W e Z we where diag eses ww ; vv ww ; e ww is a block diagonal matrix consisting of eses ww ; vv ww ; and e ww as blocks. B Preliminaries Proofs of the lemmas below are available in our Online Appendix. Lemma Assume Condition holds. The following hold both under the null and under the alternative. " (a) ww ww w n Z Z! p w ww w ww w + eses " # (b) n Z Y! p : eses s (c) n Y Z n Z Z n Z Y! p s eses s : # : Lemma 3 Assume Conditions and hold. The following hold under the null hypothesis in Section 4. (a) pn W " ) Z wv v + Z we : 7

18 " Y (b) P # W Y Y P ) W " (c) n Z "! p : (d) n Y "! p vv v : (e) b z! p : (f) b ";z! p ": (g) e ";z! p ": " (ww C + Z wes s + Z wv ) ww ( ww C + Z wes s + Z wv ) ( ww C + Z wes s + Z wv ) ww (Z wv v + Z we ) Lemma 4 Assume Conditions and hold. The following hold under the alternative hypothesis in Section 4. (a) n Z "! p " eses s # : (b) b z! p +, where = ( s eses s ) s eses s : (c) pn W (" Y ) ) Z wes ( s s ) ww C + Z wv ( v ) + Z we : (d) e ";z! p ( v ) vv ( v ) + e: (e) b ";z! p ( s s ) eses ( s s ) + ( v ) vv ( v ) + e: # : C Proofs of the Results in Section 4 We begin by presenting a rather natural result on the properties of the Hausman test with weak IV. In Theorem 4 below, we show that the Hausman test does not have the usual distribution under the null. C. The Asymptotic Distribution of Hausman Test Before presenting Theorem 4, we introduce the following de nitions: de ne " # " # Dw (ww C + Z wes s + Z wv ) ww ( ww C + Z wes s + Z wv ) = N w ( ww C + Z wes s + Z wv ) ; ww (Z wv v + Z we ) (Z wes ; Z wv ; Z we ) = + " = D w N w = vv v = D w N w = vv v " v vv vv v ; (Z wes ; Z wv ) = e (ww C + Z wes s + Z wv ) ww ( ww C + Z wes s + Z wv ) = ( ww C + Z wes s + Z wv ) wwz wv v ; 8

19 H = bw H = b ";w bw and H 3 = b ";z bw h z b b ";w YP W Y b ";z YP i Z Y bw h z b h Y P W Y Y P Z Y i bw Y P W Y Y P Z Y i bw Theorem 4 Assume Conditions and hold, and suppose that Z denotes a random vector of N (; I K ) that is independent of (Z wes ; Z wv ). Under the null hypothesis (), ; : ; (a) H ; H ) e (Z " wes ; Z wv ; Z we ) ( (Z wes ; Z wv ) + Z) ( (Z wes ; Z wv ) + Z). (b) H 3 ) e ( (Z " wes ; Z wv ) + Z) ( (Z wes ; Z wv ) + Z) : Suppose that L w = K; that is, W exactly identi es. Then, under the null hypothesis (), (c) H ; H ) (Z wes ; Z wv ; Z we ) Z Z: (d) H 3 ) Z Z K : Proof Part (a): Here we show only the limit of H. The limit of H can be derived by similar fashion and we omit the proof. By Lemma 3(b), we have " Y P W Y Y P W " # ) " Dw From this, we can deduce that b w ) Dw N w. Also, by Lemma 3(d), (e), and (f), and Lemma (c), we have = + o p (), b ";z = " + o p (), n Y " = vv v + o p (), and Y P ZY n = O p (). Therefore, we obtain N w b ";w = n " " bw n Y " + bw bw n Y Y ) " NwD w vv v + NwD w Dw N w = " + = D w N w = vv v = D w N w = vv v v vv vv v : # : Now note that H = bw z b b ";z YP Z Y b ";z YP Z Y n = bw + o p () b ";z YP Z Y b b ";w YP W Y w b + o p () ( = bw Y + o p () b ";z P ZY n b ";w YP W Y w b + o p () ";z Y Y b ";z = b ";w " P W Y Y P W Y Y P W " + o p () ; b ";w YP W Y b ";w YP W Y w b P Z Y b ";w YP o W Y P ZY n Y b ";w P ) W Y n 9

20 where the last equality follows from Y P W Y n = o p (). We therefore obtain H ) " + = D w N w = vv v = D w N w = vv v v vv vv v N wd w N w : Let Z = e (ww C + Z wes s + Z wv ) ww ( ww C + Z wes s + Z wv ) = (ww C + Z wes s + Z wv ) wwz we : Then, Z N (; I K ) and Z is independent of (Z wes ; Z wv ) : Recalling the de nitions of (Z wes ; Z wv ; Z we ) and (Z wes ; Z wv ), the limit of H is e " (Z wes ; Z wv ; Z we ) ( (Z wes ; Z wv ) + Z) ( (Z wes ; Z wv ) + Z) : Part (b): Notice that under the null hypothesis, by Lemma 3(f), b ";z! p ". Using similar arguments to those used in the proof of Part (a), we can show that H 3 = " bw Y P W Y bw + o p () = " " P W Y Y P W Y Y P W " + o p () ) " N wd w N w = e " ( (Z wes ; Z wv ) + Z) ( (Z wes ; Z wv ) + Z) : Part (c): When W exactly identi es, N wd w N w = (Z wv v + Z we ) ww (Z wv v + Z we ). In this case, de ne Z = " = ww (Z wv v + Z we ) N (; I K ) : Then, the limit distribution of H (and H ) is now (Z wes ; Z wv ; Z we ) Z Z as required. C. Proof of Theorem Part (a): We proceed as in Part (c) in the proof of Theorem 4. De ne Z = " = ww (Z wv v + Z we ) N (; I K ) : Then, the limit distribution of H 3 is then Z Z K. Part (b): Using similar argument in the proof of Theorem 4 Part (a), we have H 3 = b ";z bw z b YP Z Y Y P Z Y YP W Y Y P W Y w b = b ";z bw Y + o p () P ZY Y P ZY Y P W Y Y n n n P W Y w b + o p () = b ";z bw + o p () YP W Y + o p () w b + o p () = b ";z (" Y ) P W Y Y P W Y Y P W (" Y ) + o p () = b ";z (" Y ) P W (" Y ) + o p () ;

21 where the second line holds since = + + o p () by Lemma 4(b), the third line holds since Y P W Y n = o p () ; and the last line follows since the dimension of W and dimension of Y are the same and Y W is full rank. By Lemmas 4 (c) and (e) and Condition, we can write p n W (" Y ) ) Z wes ( s s ) + Z wv ( v ) + Z we ww C and b ";z! p ( s s ) eses ( s s ) + ( v ) vv ( v ) + e = ; say. De ne and Then, Z = = ww (Z wes ( s s ) + Z wv ( v ) + Z we ) = = wwc: (6) H 3 ) (Z+) (Z+) ; where Z N (; I K ) : D Proofs of the Results in Section 5 D. Some Useful Lemmas We introduce a few lemmas below that are helpful in proving Theorem. Lemmas 5 and 6 assume that the estimator b " is consistent under the null even when we adopt the weak IV asymptotics, and nd the limit of the test statistic H b " under the null and under the alternative, respectively. In Lemma 7 we provide an estimator b " that is consistent under the null even when we adopt the weak IV asymptotics. Lemma 5 Assume Conditions and hold. Suppose that b " is consistent for " under the null using weak instrument asymptotics. Then H b " ) Lw. Proof The result easily follows from the proof of Lemma 6 below by noting that s =, =, and " = e under the null. In Lemma 5, we make the additional assumption that b " is consistent for " under the weak instrument asymptotics in order to isolate the properties of the numerator. This is inspired by the

22 discussion in the previous section, where we saw that H failed to converge to a central chi-square distribution (see Proposition 4) because the estimator b ";w in (5) is inconsistent for ". It turns out that the properties of b " have implications for the the power property of H b " under the weak instrument asymptotics. De ne = ( v ) vv ( v ) + e and where (Z wes ) = = ww (Z wes ( s s ) ww C) ; = plim bz = s eses s s eses s denotes the asymptotic bias of b z under the alternative hypothesis. Lemma 6 Assume Conditions and hold. Under the alternative hypothesis (3), H b " ) ( + Z) ( + Z) ; where Z N (; I Lw ) and is the same noncentrality parameter as in Theorem (b): Proof Recall the de nition We start with the analysis of Note that H b " = y b " Y z b W b W y Y z b = N b " ; say. N = y Y W b W y Y : p n W y Y = p n W " p n W Y! n Y Z n ZZ n Z Y n Y Z n Z Z n Z ": Using Lemmas, 3, and 4, we can write p n W y Y = p n W (" Y ) + o p () ) Z wes ( s s ) + Z wv ( v ) + Z we ww C: Because we have n b = n W W n Y P Z Y = O p () ; n W Y = O p pn ; n W Y n Y P Z Y n Y W = ww + o p ()

23 under both the null and the alternative. We may therefore write that N = (Z wes ( s s ) + Z wv ( v ) + Z we ww C) ww (Z wes ( s s ) + Z wv ( v ) + Z we ww C) + o p () : Also, under the alternative, by Lemma 4(e), b "! p ; where Now let and = ( s s ) ~s~s ( s s ) + ( v ) vv ( v ) + e: Z = = ww (Z wes ( s s ) + Z wv ( v ) + Z we ) ; = = wwc: Then, it is easy to see that Z N (; I K ). By writing we obtain the desired conclusion. H b " N = b " = ( + Z) ( + Z) + o p () ; Lemmas 5 and 6 imply that it is important to choose b " such that it is consistent for " under the null, and consistent for under the alternative. To see this, suppose that we use b ";z in (6) for b ". It can be shown 6 that b ";z! p " under the null, but b ";z! p ( s s ) eses ( s s ) + under the alternative. In other words, we have = if we use b ";z. This implies that the asymptotic distribution of H b ";z under the alternative is the mixture of distributions ( (Z wes ) + Z) ( (Z wes ) + Z) multiplied by a constant less than or equal to. Therefore, the test H b ";z may be asymptotically biased. 7 Below, we present asymptotic properties of e ";z developed in (4). It turns out that if we use e ";z as an estimate of ", then the ratio converges to under the alternative: Lemma 7 Assume Conditions and. Under the null, e ";z! p ", and under the alternative, e ";z! p = ( v ) vv ( v ) + e. Proof The required results follow by Lemma 3(g) and Lemma 4(d) in Appendix B. 6 See Lemma 3(f) and Lemma 4(e) in the appendix. 7 Recall that H b ";z = H3 when L w = K. The upshot is that unless L w = L s = K; H 4 will be more powerful than H 3; hence our focus on calculating H 4 in the Online Appendix. 3

24 D. Proof of Theorem Proof Part(a) follows by Lemmas 5 and 7 above in Section D.. Part (b) follows by Lemmas 6 and 7 from Section D.. E Computational Issue For convenience to practitioners, we provide below an alternative algorithm to compute H 4 in the special case when the weak instruments W exactly identify the coe cient. The algorithm is based on the characterization (5) in Section 5: For a general over-identi ed case, see our Online Appendix. Compute the SLS b z by using the instruments Z = [W; S]. Let b V z = b ";z (Y P ZY ) denote the standard variance estimator of b z, where b ";z = n b" zb" z denotes the standard estimator of ". Computation of e ";z:. Using the IV estimator b z, get the IV residual b" z = y Y.. Regress the IV residual b" z on Z = [S; W ], and get the residual e" z = M Z b" z : 3. Calculate e ";z = n e" ze" z. Let ev z = e ";z bv b ";z z : Compute the SLS estimate b w by using the instruments W. Let b V w = b ";w (Y P W Y ) denote the standard variance estimator of b w, where b ";w = n b" wb" w denotes the standard estimator of ". Let ev w = e ";z bv b ";w w : H 4 can now be calculated as H 4 = bw h evw e Vz i bw : 4

25 References [] Altonji, Joseph G. (986): Intertemporal Substitution in Labor Supply: Evidence from Micro Data, Journal of Political Economy, 94, S76-S5. [] Andrews, Donald W.K. (3): End-of-Sample Instability Tests, Econometrica, 7, [3] Andrews, Donald W.K., Marcelo Moreira, and James H. Stock (6): Optimal Two-Sided Invariant Similar Tests for Instrumental Variables Regression, Econometrica, 74, [4] Bekker, Paul A. (994): Alternative Approximations to the Distributions of Instrumental Variable Estimators, Econometrica, 6, [5] Donald, Stephen D. and Whitney K. Newey (): Choosing the Number of Instruments, Econometrica, 69, 6-9. [6] Guggenberger, Patrik (8): The Impact of a Hausman Pretest on the Asymptotic Size of a Hypothesis Test, Working Paper, UCLA. [7] Hahn, Jinyong, John C. Ham, and Hyungsik R. Moon (): The Hausman Test and Weak Instruments: Online Appendix, available at www-rcf.usc.edu/~moonr. [8] Hahn, Jinyong and Jerry A. Hausman (): A New Speci cation Test for the Validity of Instrumental Variables, Econometrica, 7, [9] Ham, John C. (986): Testing Whether Unemployment Represents Life-Cycle Labor Supply, Review of Economic Studies, 53, [] Ham, John C. and Kevin T. Reilly (): Testing Intertemporal Substitution, Implicit Contracts, and Hours Restriction Models of the Labor Market Using Micro Data, American Economic Review, 9, [] Hausman, Jerry A. (978): Speci cation Tests in Econometrics, Econometrica, 46, 5-7. [] Kleibergen, Frank (): Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression, Econometrica, 7, [3] MaCurdy, Thomas E. (98): An Empirical Model of Labour Supply in a Life-Cycle Setting, Journal of Political Economy, 89, [4] Moreira, Marcelo (3): A Conditional Likelihood Ratio Test for Structural Models, Econometrica, 7,

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