Discriminating between (in)valid external instruments and (in)valid exclusion restrictions

Transcription

1 Discussion Paper: 05/04 Discriminating between (in)valid external instruments and (in)valid exclusion restrictions Jan F. Kiviet Amsterdam School of Economics Roetersstraat 08 WB AMSTERDAM The Netherlands

2 Discriminating between (in)valid external instruments and (in)valid exclusion restrictions Jan F. Kiviet Version of 5 May 06 JEL Classi cations: C0, C, C6 Keywords: di erence in Sargan-Hansen tests; exclusion restrictions; maintained hypothesis; instrument validity tests; overidentifying restrictions Abstract In models estimated by (generalized) method of moments a test on coe cient restrictions can either be based on a Wald statistic or on the di erence between evaluated criterion functions. From their correspondence it easily follows that the statistic used for testing instrument validity, the Sargan-Hansen overidentifying restrictions (OR) statistic, is equivalent to an exclusion restrictions test statistic for a nonunique group of regressor variables. We prove that asymptotically this is the case too for incremental OR tests. However, we also demonstrate that, despite this equivalence of test statistics, one can nevertheless distinguish between either the (in)validity of some additional instruments or the (un)tenability of particular exclusion restrictions. This, however, requires to be explicit about the adopted maintained hypothesis. It also highlights that recent warnings in the literature that overidentifying restrictions tests may mislead practitioners should not be directed towards the test, but to practitioners who do not realize that inference based on such tests is unavoidably conditional on the validity of particular just-identifying statistically untestable assumptions. Emeritus Professor of Econometrics, Amsterdam School of Economics, University of Amsterdam, PO Box 5867, 00 NJ Amsterdam, The Netherlands (j.f.kiviet@uva.nl). Most of this paper was written while I was enjoying hospitality as Visting Professor at the Division of Economics, School of Humanities and Social Sciences, Nanyang Technological University, 4 Nanyang Drive, Singapore Helpful comments by João Santos Silva on an earlier version of this paper and further suggestions by JEM co-editor Jason Abrevaya are gratefully acknowledged.

3 . Introduction In empirical econometric models it is often not unlikely that some explanatory variables are in fact endogenous, because their realizations are contemporaneously correlated with the error terms of the model. To overcome inference problems due to such correlation one often exploits external (non-explanatory) variables that should be uncorrelated with these errors. To verify whether these variables (instruments) seem uncorrelated with the disturbances indeed one can use a so-called test for overidentifying restrictions (OR), see Sargan (958), Hansen (98). The mandatory use of OR tests on a routine basis has been advocated at various places in the literature. See, for instance, the list mentioned in Baum et al. (003, footnote ). However, warnings that OR tests may mislead practitioners were recently issued in Parente and Santos Silva (0) and Guggenberger (0), and can also be found in, for instance, Hayashi (000, p.8) and Cameron and Trivedi (005, p.77). In this study we will place these warnings regarding OR tests into perspective and highlight that they only apply if one is hesitant with respect to formulating an explicit maintained hypothesis, involving su cient a priori orthogonality conditions and exclusion restrictions. Also we will expose the algebraic equivalences between OR and CR (coe cient restrictions) test statistics and explain why these tests nevertheless allow distinct inferences. Moreover, we discuss to what degree similar issues arise for incremental OR tests. For the sake of simplicity we will focus here mainly on the linear method of moments context, but our derivations can be extended to more general settings, as we shall indicate. First, while introducing our notation, we review some general results on testing in a linear instrumental variables context, such as illustrating that the classical trinity of test principles established in the context of Maximum Likelihood inference, constituted by Wald (W), Likelihood Ratio (LR) and Lagrange Multiplier (LM), have their counterparts in a semiparametric setting in which models can e ciently and robustly be estimated by (generalized) method of moments, see also Newey and West (987). In a linear method of moments context CR tests based on a W statistic are in fact equivalent to the test obtained by taking the di erence between evaluated criterion functions (CF) for the restricted and the unrestricted model. The latter reminiscences of an LR-type test. By employing an estimate for the disturbance variance obtained from the restricted model both W and CF can be implemented as a kind of LM-type test. A very particular implementation of a CF test gives the Sargan-Hansen statistic for testing overidentifying restrictions. Thus, the very same test statistic can be interpreted either as a CR test or as an OR instrument validity test. However, we will highlight that these two interpretations require di erent maintained hypotheses. This has not always been made

4 very clear in earlier literature. We also prove that in linear models similar equivalences and di erences exist between exclusion restrictions tests and so-called incremental overidentifying restrictions (IOR) tests (also addressed as di erence in Sargan-Hansen tests), which verify the validity of a subset of instruments. From these ndings we conclude that strictly following formal statistical test principles enables to characterize the context in which OR and IOR tests should not mislead and in fact allow to distinguish inference regarding the (in)validity of some additional orthogonality conditions from inference on the (un)tenability of particular exclusion restrictions. Practical problems do emerge only when one is unable or unwilling to condition the analysis on a rm initial maintained hypothesis. The structure of this study is as follows. In Section we introduce the model and various test procedures and concepts. Section 3 demonstrates how the standard Sargan test can either be used for testing instrument validity or under a di erent maintained hypothesis can be interpreted as a CR test. Section 4 demonstrates to what degree these results do apply to incremental Sargan tests as well. Finally Section 5 indicates options for further generalizations and summarizes the conclusions.. Corresponding test principles We focus on the single linear simultaneous regression model y = X + " with " (0; "I); where X = (x ; :::; x n ) 0 is an n K matrix of rank K with unknown coe cient vector R K : To cope with E(x i " i ) = x" 6= 0 for i = ; :::; n we will employ an n L instrumental variables matrix Z = (z ; :::; z n ) 0 of rank L K. The instrumental variables (IV) or two-stage least-squares (SLS) estimator is given by ^ = (X 0 P Z X) X 0 P Z y; (.) where P A = A(A 0 A) A 0 for any full column rank matrix A. When E(z i " i ) = 0 (the instruments are valid) and standard regularity conditions are ful lled too (including su cient relevance of the instruments) then the IV estimator is consistent and asymptotically normal, whereas its variance can be estimated by V d ar( ^) = ^ "(X 0 P Z X) ; where ^ " = ^" 0^"=n with residuals ^" = y X ^: A test on any set of K < K linear restrictions on the coe cients can after a simple transformation of the model be represented as testing exclusion restrictions H 0 : = 0 in y = X + X + " with " (0; "I); (.) where X j is n K j and j is K j for j = ; : Making use of standard results on partitioned regression applied to the second-stage regression, in which y is regressed on 3

5 P Z X = (P Z X ; P Z X ) = ( ^X ; ^X ); one easily nds that ^ = ( ^ ; 0 ^ ) 0 0 ; with ^ = ( ^X M 0 ^X ^X ) ^X0 M ^X y (.3) and V d ar( ^ ) = ^ "( ^X M 0 ^X ^X ) ; where M A = I P A : The W statistic for testing H 0 : = 0 against alternative H : 6= 0 readily follows and is given by W = ^ 0 [ d V ar( ^ )] ^ : (.4) Under H 0 it is asymptotically (K ) distributed, provided the maintained hypothesis does hold indeed. The maintained hypothesis (which is the union of H 0 and H and all underlying assumptions) can here be characterized loosely as M : (i) the actual data generating process (DGP) is nested in model (.) with K regressors; (ii) X and Z show su cient regularity; (iii) n = Z 0 "! d N (0; " plim n Z 0 Z): Substituting (.3) in (.4) and making use of a result on projection matrices for a partitioned full column rank matrix A = (A ; A ) which says P A = P A + P MA A ; we nd W = y 0 P M y=^ ^X ^X " = y 0 (P ^X P ^X )y=^ " = y 0 (M ^X M ^X)y=^ ": (.5) Hence, the test statistic can easily be obtained by taking the di erence between the restricted and the unrestricted sum of squared residuals of second stage regressions, while scaling by a consistent estimator of ": If the latter is obtained from the restricted residuals ~" = y X ~ ; where ~ = ( ^X 0 ^X ) ^X0 y (.6) is the restricted IV estimator (imposing = 0) and ~ " = ~" 0 ~"=n, the test statistic reminds of a Lagrange Multiplier test. Therefore we indicate it as LM = y 0 (M ^X M ^X)y=~ "; (.7) which is asymptotically equivalent with W : A test statistic with similarities to the LR principle is found as follows. Estimator ^ is obtained by minimizing with respect to a quadratic form in the vector Z 0 (y X) in which a weighting matrix is used proportional to (Z 0 Z) : This yields a consistent estimator with optimal variance under the maintained hypothesis. De ning the criterion function as Q(; y; X; Z; ") = (y X) 0 Z(Z 0 Z) Z 0 (y X)="; (.8) one can verify that it attains its minimum for ^ of (.), giving Q( ^; y; X; Z; ") = (y X ^) 0 P Z (y X ^)= " = (y 0 P Z y ^ 0 ^Xy + ^0 ^X0 ^X ^)= " = (y 0 P Z y y 0 P ^Xy)= " = (y 0 M ^Xy y 0 M Z y)= "; 4

6 which has in its numerator the di erence between the sum of the squares of the secondstage residuals and the reduced form residuals. In the model under = 0; upon using the criterion function Q( ; y; X ; Z; "); one nds ~ of (.6) with Q( ~ ; y; X ; Z; ") = (y 0 M ^X y y 0 M Z y)=": The di erence between these two minimized criterion functions is Q( ~ ; y; X ; Z; ") Q( ^; y; X; Z; ") = (y 0 M ^X y y 0 M ^Xy)= ": Now substituting either ^ " or ~ "; we nd that taking the di erence between these two evaluated criterion functions yields either CF (^ ") = Q( ~ ; y; X ; Z; ^ ") Q( ^; y; X; Z; ^ ") = W (.9) or CF (~ ") = LM : (.0) These equivalences are algebraic, thus hold irrespective of the validity of M : 3. The Sargan test and exclusion restrictions A special situation occurs in case L = K: This specializes M to what we will indicate by M L=K : Under this maintained hypothesis the model is just identi ed, but overidenti ed under the K coe cient restrictions = 0: When the model is just identi ed then, when minimizing Q(; y; X; Z; "); the L = K orthogonality conditions Z 0 (y X ^) = 0 will all be satis ed in the sample, giving Q( ^; y; X; Z; ") = 0: So, in this particular situation, (.9) specializes into W L=K = CF L=K (^ ") = Q( ~ ; y; X ; Z; ^ ") = ~" 0 P Z ~"=^ "; (3.) and (.0) into the asymptotically equivalent statistic LM L=K = CF L=K (~ ") = ~" 0 P Z ~"=~ ": (3.) The latter expression is the well-known Sargan test statistic which is used to check the validity of the instruments by testing the L which has just the K regressors X ; and can be expressed as K overidentifying restrictions in the model y = X + " with " (0; " I): (3.3) From its algebraic equivalence with a CR test it is immediately obvious that testing for OR has similarities with testing the validity of L K exclusion restrictions. However, there are dissimilarities as well. Validity of the L orthogonality conditions E(z i " i ) = 0 is part of the maintained hypothesis M L=K for the test of L K exclusion 5

7 restrictions = 0 in model (.), whereas the Sargan test is used to verify validity of the orthogonality conditions E(z i " i ) = 0 in model (3.3). Although using exactly the same test statistic and critical values, we will explicate that the two tests have intrinsically di erent H 0 and H and build on di erent maintained hypotheses. We are not aware of literature where these di erences and their consequences have been spelled out in full detail. Below we aim to ll this gap. Note that expression (3.) is invariant regarding X : So, replacing X by any alternative n K matrix X a would yield equivalent results for CF L=K (~ "), provided P Z (X ; X a ) has full column rank K = L: Thus, (X ; X a ) should have full column rank and the variables X a should be su ciently related to the instruments Z; but at the same time they may also be endogenous regarding ": Due to the projection of the regressors on the space spanned by the instruments Z in the second stage regression, and the presence of the regressors X ; the CR test actually does not test the explanatory power of X a as such, but just that of M ^X P Z X a = (P Z P ^X P Z )X a = (P Z P ^X )X a ; which is its projection on the space spanned by Z; as far as this is orthogonal to P Z X : A viable choice for X a would be the following. Let F = (F ; F ) be a full rank L L matrix, where F is L L ; F is L L ; with L = L + L and L = K : Now consider the transformed instruments Z f = (Z f ; Z f ) = ZF = (ZF ; ZF ); and let F be such that square matrix Z f0 X has full rank. Obviously, F is nonunique. Note that if the transformed instruments Z f are not just relevant for X ; but were valid for the errors " of model (3.3) as well, then the coe cients would be just identi ed by them. When taking X a = M Z f Z f ; then P Z (X ; X a ) has full column rank as required, and statistic CF L=K (~ ") for testing H 0 : = 0 in model y = X + M Z f Z f + " (3.4) is therefore equivalent to the Sargan test statistic CF L=K (~ "): Estimating of (3.4) by IV using the instruments Z or Z f (which is equivalent because P Z = P Z f ) yields because P Z f M MZ f ^ = (X 0 P Z f M MZ f = (X 0 P Z f Z P f Z f X ) XP 0 Z f M MZ f Z P f Z f y (3.5) X ) X 0 P Z f y = (Z f0 X ) Z f0 y = ^ ; = P Z f : Hence, augmenting model (3.3) by Z P f Z f = P Z f P MZ f Z f Z f and using instruments Z or Z f yields coe cient estimates ^ for K regressors M Z f the regressors X which are equivalent to the simple IV estimator ^ obtained in model (3.3) when just using the K instruments Z f ; irrespective of the validity of any of the instruments. Apart from the interpretation of CF L=K (~ ") as testing K = L K exclusion restrictions = 0 in model (.) under maintained hypothesis M L=K ; which presupposes 6

8 the K orthogonality conditions E(z i " i ) = 0; yet another coherent interpretation of the numerically equivalent test statistic CF L=K (~ ") is the following. It tests whether the L K variables Z f are additional valid external instruments for model (3.3) under the maintained hypothesis given by M L=K : (i) the DGP is nested in model (3.3) which has K < K = L regressors; (ii) X and Z (and thus Z f ) show su cient regularity; (iii) n = Z f0 "! d N (0; " plim n Z f0 Z f ): Note that for adopting M L=K in practice it might be desirable to specify matrix F ; this issue will be addressed below. Under M L=K the value of E(z f i " i ) = z f is unspeci ed. Obviously, if next to " Z f the variables Z f are uncorrelated with " then M Z f Z f should not have explanatory power in model (3.4). Hence, when adopting M L=K ; then under the null hypothesis = 0 in model (3.4) all instruments Z are valid for model (3.3), implying z f = 0 " so that ~ is consistent, asymptotically normal and e cient for : Below we will simply use the acronym AE (asymptotically e cient) to indicate for an estimator if, under the assumptions made, it is consistent with asymptotically normal distribution and smallest asymptotic variance (in a matrix sense). The alternative 6= 0 implies z f 6= 0: So, " in the context of model (3.4) test statistic CF L=K (~ ") tests the null hypothesis given by the L K orthogonality conditions E(z f i " i ) = 0 or overidentifying restrictions = 0: When these conditions/restrictions are invalid then, given validity of M L=K ; estimator ^ = (Z f0 X ) Z f0 y for is AE in model (3.3). In the extended regression model (3.4) all variables in both Z f and Z f constitute valid instruments regarding " under M L=K ; because inclusion of the regressors M Z f Z f purges the disturbances from their correlation with Z f : That this regression too produces the AE estimator ^ for the coe cients ; as is proved by (3.5), is because it constitutes a special case of the so-called "partialling out" result which will be reviewed at the end of the next section. It is the case that truth of the null = 0 under M L=K implies E(z i " i ) = 0; and so does truth of the null E(z f i " i ) = 0 under M L=K : Under both null hypotheses estimator ~ is AE for : However, imposing the respective null hypotheses implies either accepting coe cient restrictions or accepting extra orthogonality assumptions. Invalidity of = 0 under M L=K renders ^ the AE estimator for ; but invalidity of E(z i" i ) = 0 under M L=K renders ^ AE for the coe cients of just the regressors X ; since in this setup the DGP is supposed to be nested in model (3.3) which has just K regressors. So, on the basis of the single test statistic LM L=K (~ ") = CF L=K (~ "); or the as- (^ "); one can discriminate between either (in)validity of ymptotically equivalent W L=K L K exclusion restrictions in model (.) or (in)validity of L K external instruments in model (3.3). This just hinges upon which of the nonnested maintained hypotheses has actually been adopted: M L=K or M L=K : In both contexts the tests concern a possible 7

9 loss/gain in the degree of overidenti cation by L K : Namely, either due to rejecting/accepting exclusion restrictions on the coe cients of potential explanatory variables X in model (.), or on accepting/rejecting exclusion restrictions on the coe cients of the auxiliary regressors M Z f Z f in model (3.4). The latter implies accepting/rejecting validity of L K external instruments Z f : The warnings of Parente and Santos Silva (0) and Gugenberger (0) concern the problems that emerge when using the Sargan statistic for testing jointly the L orthogonality conditions E(z i " i ) = 0 in model (3.3). However, using this L dimensional null hypothesis, is asking too much from this L K degrees of freedom test statistic, leading to inconsistency of the test, meaning: existence of nonlocal alternatives to the null that are not rejected with probability one asymptotically. The classic Sargan test on instrument validity is only consistent in case one is willing to adopt maintained hypothesis M L=K : It is uncomfortable but also unavoidable that this maintained hypothesis embraces the statistically untestable orthogonality conditions expressed by E(z f i " i ) = 0 (i = ; :::; n): These untestable conditions entail: It is possible to nd K linearly independent linear combinations of the K variables in matrix Z which establish valid instruments for model (3.3) and ensure just identi cation of : It is the researcher who may decide whether (s)he leaves these K orthogonality conditions implicit, or makes them explicit by actually specifying the matrix F ; for instance by choosing F = (I; O) 0 and then thus simply assuming that the rst K instruments in the matrix Z are valid in model (3.3). In that special case the explicit null hypothesis of the Sargan test is that the instruments M Z Z are valid too, which would imply E(z i " i ) = 0 in model (3.3). 4. The incremental Sargan test and exclusion restrictions The validity of a subset of the instruments Z f for model (3.3) can be tested by a socalled incremental Sargan or IOR test. For that we will examine too in what way such a procedure has (dis)similarities with testing exclusion restrictions. However, for this analysis we will take as our starting point the usual model with K regressors and L > K instruments and therefore have to introduce some further notation. We consider a partition of the matrix of L instruments, denoted as Z = (Z m ; Z a ); where for j fm; ag matrix Z j = (z j ; :::; z jn ) 0 is n L j : Matrix Z m ; with L m K; contains the instruments which are maintained to be valid, whereas the validity of the additional L a > 0 instruments Z a will be examined by testing. So the maintained hypothesis is now given by M L>K : (i) the DGP is nested in model y = X + "; which has K regressors; (ii) X 8

10 and Z = (Z m ; Z a ) with L > L m K show su cient regularity; (iii) n = Zm" 0! d N (0; " plim n ZmZ 0 m ): Just using the instruments Z m yields estimator and residuals ^ m = (X 0 P Zm X) X 0 P Zm y; ^" m = y X ^ m : (4.) From the foregoing section it follows that the Sargan statistic S m = n ^" 0 mp Zm ^" m =^" 0 m^" m (4.) is asymptotically (L m K) distributed under M L>K when L m > K; whereas it is zero for L m = K: Using the additional instruments as well yields ^ given in (.) and ^" = y X ^ and enables to calculate S ma = n ^" 0 P Z ^"=^" 0^": (4.3) Now the null hypothesis E(z ia " i ) = 0; which involves L a orthogonality conditions for model (.) additional to the L m orthogonality conditions already implied by M L>K ; can be tested by the incremental Sargan statistic IS a = S ma S m ; (4.4) which under M L>K has asymptotic null distribution (L a ): Note that for L m = K statistic IS a corresponds to the Sargan test examined in the foregoing section, but now the role of X is plaid by X; so K is now K; whereas K is given here by L a and what earlier was called ~" is given by ^" now. Most published derivations of the null distribution of IS a start o from a GMM (generalized method of moments) context. Our simple linear framework allows a very concise derivation, which goes as follows. Without loss of generality we assume Z 0 mz a = O; which may have been achieved by replacing the original Z a by M Zm Z a : P Z is not a ected by this partial orthogonalization, but now P Z = P Zm + P Za : From P Z ^" = P Z [I X(X 0 P Z X) X 0 P Z ]" = (P Z P PZ X)" we nd ^" 0 P Z ^" = " 0 P Z " " 0 P PZ X" = " 0 (P Zm + P Za P PZ X)" and ^" 0 mp Zm ^" m = " 0 (P Zm P PZm X)"; so ^" 0 P Z ^" ^" 0 mp Zm ^" m = " 0 (P Za +P PZm X P PZ X)": The matrix in this quadratic form is symmetric and idempotent. The latter follows from P Za P PZm X = O and (P Za +P PZm X)P PZ X = (P Za +P Zm )X(X 0 P Z Z) X 0 P Z = P PZ X: Since tr(p Za + P PZm X P PZ X) = L a + K K = L a this means that under M L>K and the null hypothesis E(z ia " i ) = 0 (i = ; :::; n); so when all instrument in Z are valid, IS d a! " 0 (P Za + P PZm X P PZ X)"=" d! (L a ); because under the null both plim ^" 0 m^" m =n = " and plim ^" 0^"=n = ": 9

11 We shall now examine tests for omitted regressors (or exclusion restrictions) and seek cases which show correspondences with statistic IS a. Consider the extended model with K a L K additional regressors X a given by and the maintained hypothesis de ned by y = X + X a a + " a ; with " a (0; " a I); (4.5) M L>K : (i) the DGP is nested in model y = X + X a a + " a ; which has K + K a L regressors; (ii) (X; X a ) and Z show su cient regularity; (iii) n = Z 0 " d a! N (0; " a plim n Z 0 Z): Let us indicate the IV results when estimating model (4.5) using instruments Z as y = X ^ + X a ^a + ^" a : Then the LM-like test statistic for H 0 : a = 0 is given by CF a (^ ") = (^" 0 P Z ^" ^" 0 ap Z ^" a )=^ "; (4.6) which under M L>K has asymptotic null distribution (K a ): Still assuming without loss of generality that Z 0 mz a = O; we shall rst examine the speci c case X a = Z a ; for which expression (4.6) can be specialized by using the following results. Substituting X a = Z a we obtain ^ = (X 0 P Z M Za P Z X) X 0 P Z M Za y = (X 0 P Zm X) X 0 P Zm y = ^ m ; (4.7) because M Za P Z = P Z P Za P Z = P Z P Za = P Zm : And, since in IV estimation the residuals are orthogonal to the second-stage regressors, we have Z 0 a^" a = 0: This yields P Z ^" a = (P Zm + P Za )^" a = P Zm (y X ^ Z a ^a ) = P Zm (y X ^ ) = P Zm (y X ^ m ) = P Zm ^" m ; where we used (4.7) for the last equality. From this we nd CF Xa=Za a (^ ") = (^" 0 P Z ^" ^" 0 ap Z ^" a )=^ " = (^" 0 P Z ^" ^" 0 mp Zm ^" m )=^ " ^" 0 = S ma S m^" m m : (4.8) ^" 0^" Under M L>K and H 0 : a = 0 both ^" 0 m^" m =n and ^" 0^"=n converge to "; thus the ratio ^" 0 m^" m =^" 0^" equals asymptotically. Therefore, under their respective null hypotheses, the incremental Sargan test statistic IS a = S ma S m is asymptotically equivalent with the exclusion restrictions test statistic CF Xa=Za (^ a "): In fact, this also holds for more general matrices than just X a = Z a : Consider instead X a = Z a C + XC + M Z C 3 + "C 4 ; (4.9) where the nite matrices C ; C ; C 3 ; C 4 are L a K a ; K K a ; n K a and K a respectively. Now writing ^X a = P Z Xa and ^X = P Z X it follows that ^X a = Z a C + ^XC + P Z "C 4 : Under the null hypothesis a = 0; we have plim n Z 0 " = 0; hence 0

12 P Z "C 4! O; thus M ^X ^X a! M ^XZ a C : The regularity assumption of M L>K requires this to have full column rank, hence C should have rank K a L a : If K a = L a and C has full rank we obtain P M ^X ^X a! P M ^XZ a : Thus, tests for the omission of regressors X a which belong to group (4.9) with C of full rank K a = L a will yield test statistics which under the null are all asymptotically equivalent with statistic IS a. Estimator ^ is AE both under M L>K with valid exclusion restrictions a = 0 and under M L>K with valid additional instruments Z a : However, under M L>K while instruments Z a are invalid ^ m is AE, whereas under M L>K with a 6= 0 the estimators ^ and ^ a are AE in model (4.5), yielding an estimator for asymptotically equivalent to ^ m only if X a belongs to group (4.9) with K a = L a ; C of full rank and C = O: The algebraic estimator equivalence for X a = Z a given in (4.7) reestablishes the so-called partialling out result, which says that IV coe cient estimates are invariant when (putative) exogenous regressors are deliberately omitted, provided they are also removed from and ltered out of the remaining instruments. In Hendry (0) this analytic result is established by simulation, which leads to the unsatisfactory conclusion that (in our notation) ^ and ^ m "hardly di er". Moreover, in the simulation design used the instruments are all valid, L = 3; K = ; L a = and all variables are normally distributed, whereas neither of these is required for the algebraic result to hold exactly. 5. Interpretation and conclusion The above results lead to clear guidelines only when a researcher is willing to adopt unambiguously a particular maintained hypothesis, which (s)he should according to sound Neyman-Pearson statistical test methodology. In practice, however, a more exible though opportunistic approach may often seem more appealing. The two juxtaposed maintained hypotheses M L=K and M L=K in Section 3, and their generalizations M L>K and M L>K in Section 4, both suppose that knowledge is available already of a model speci cation in which the DGP is nested, whereas endeavors to reach that stage form usually the major challenge in an empirical modelling exercise. This explains why researchers when obtaining a signi cant OR test may either conclude: (a) some instruments are invalid, (b) the model speci cation is de cient (so its implied errors may correlate even with instruments which would be valid for an adequate model speci cation), or (c) both model and instruments are unsound. We have shown that this shilly-shally is a consequence of not building on a rm maintained hypothesis. Under M L>K an insigni cant OR statistic is either due to validity of the tested additional moment conditions, or to lack of power of the test under (a), but not to (b) or (c), whereas a signi cant OR statistic is either due to a Type I error or to invalid instruments. Prac- See, for instance, Hayashi (000, p.8) and Cameron and Trivedi (005, p.77).

13 titioners who interpret an insigni cant OR statistic as approval of the chosen model speci cation and the employed instruments should always realize that this presupposes validity of the untested hypothesis that this instrument set contains a subset which already just-identi es the coe cients of this chosen model speci cation. The presented results on instrument validity and coe cient restriction tests can rather straight-forwardly be generalized to linear models with nonspherical disturbances estimated by generalized method of moments, because GMM conforms to IV/SLS after proper transformations. When " (0; ") this transformation requires (as in GLS) to premultiply the model by an n n matrix ; where 0 = ; so that " = " (0; "I); but employ now as instruments Z y = ( 0 ) Z; see Kiviet and Feng (06). Hence, all above results can easily be reformulated for the linear GMM context, where (incremental) OR tests are usually addressed as Sargan-Hansen tests and indicated as (di erences) in J tests. In case of unknown heteroskedasticity (hence is diagonal but not available), rather than robustifying ine cient IV, its ndings ^ and ^" = y X ^ can be employed in a second step to obtain asymptotically e cient feasible GMM inference. This can be represented 3 as follows. Let y i = y i =^" i ; X = (x ; ::; x n) 0 with x i = x i =^" i and Z y = (z y ; :::; z y n) with z y i = z i ^" i : Next regressing y on X using instruments Z y yields ^ y = (X 0 P Z yx ) X 0 P Z yy = [X 0 Z(Z y0 Z y ) Z 0 X] X 0 Z(Z y0 Z y ) Z 0 y; with Z y0 Z y = n i=^" i z i z 0 i: Its variance could be estimated consistently by d V ar( ^ y ) = [X 0 Z(Z y0 Z y ) Z 0 X] ; and writing ^" y = y X ^y the Sargan-Hansen statistic is S y = ^" y0 P Z y ^" y = (y X ^ y ) 0 Z(Z y0 Z y ) Z 0 (y X ^ y ); which is asymptotically equivalent to the GMM maximized criterion function. Hence, also for feasible GMM all earlier ndings on CR and IOR tests have their (asymptotically equivalent) counterparts. Note, however, that a robusti ed variance estimator for ^ would be ( ^X 0 ^X) n i=^" i ^x i^x 0 i( ^X 0 ^X) with P Z X = ^X = (^x ; :::; ^x n ) 0 : Employing this in a robusti ed version of the CR test W of (.4) does no longer seem to lead to a statistic which can be expressed equivalently as the di erence between maximized robusti ed criterion functions, as in (.9). Note that a robusti ed Sargan statistic for IV is given by ^" 0 ( n i=^" i z i z 0 i) ^" = (y X ^) 0 Z(Z y0 Z y ) Z 0 (y X ^): Due to the consistency of ^ this is asymptotically equivalent 4 to the -step statistic S y ; and thus a robusti ed CF test has correspondences with a CR test based on -step feasible GMM estimator ^ y and no longer with one based on the ine cient ^. Also in the extended contexts indicated in the preceding paragraph the practical relevance of the here presented ndings is the following. An (incremental) Sargan-Hansen When embedded into a properly extended framework generalizations for nonlinear models can be obtained as well; see, for instance, Newey (985). 3 See Newey and West (987, p.786). 4 Baum et al (003, p.8) claim that S and S y are even numerically equivalent, which does not seem right.

14 test on the validity of (particular) orthogonality conditions uses exactly the same (or at least an under the null asymptotically equivalent) test statistic with the same asymptotic null distribution as a test for zero restrictions on the coe cients of additional regressor variables which may stem from a particular fairly wide group. However, in essence these two test procedures build on di erent nonnested maintained hypotheses which enables them to produce discriminative inferences. References Baum, C., Scha er, M., Stillman, S., 003. Instrumental variables and GMM: estimation and testing. Stata Journal 3, -3. Cameron, A.C., Trivedi, P.K., 005. Microeconometrics: Methods and Applications. Cambridge University Press. Guggenberger, P., 0. A note on the (in)consistency of the test of overidentifying restrictions and the concepts of true and pseudo-true parameters. Economics Letters 7, Hayashi, F., 000. Econometrics. Princeton, Princeton University Press. Hansen, P.L., 98. Large sample properties of generalized method of moments estimators. Econometrica 50, Hendry, D.F., 0. On adding over-identifying instrumental variables to simultaneous equations. Economics Letters, Kiviet, J.F., Feng, Q., 06. E ciency gains by modifying GMM estimation in linear models under heteroskedasticity. UvA-Econometrics working paper 04/06. Newey, W., 985. Generalized method of moments speci cation testing. Journal of Econometrics 9, Newey, W.K., West, K.D., 987. Hypothesis testing with e cient method of moments estimation. International Economic Review 8, Parente, P.M.D.C, Santos Silva, J.M.C., 0. A cautionary note on tests of overidentifying restrictions. Economics Letters 5, Sargan, J.D., 958. Estimation of economic relationships using instrumental variables. Econometrica 6,