Instrument endogeneity and identification-robust tests: some analytical results

Transcription

1 MPRA Munich Personal RePEc Archive Instrument endogeneity and identification-robust tests: some analytical results Firmin Sabro Doko chatoka and Jean-Marie Dufour 3. May 2008 Online at htt://mra.ub.uni-muenchen.de/2963/ MPRA Paer No. 2963, osted 9. March 20 2:28 UC

2 Instrument endogeneity and identification-robust tests: some analytical results Firmin Doko Université de Montréal Jean-Marie Dufour McGill University First version: November 2007 Revised: December 2007, February 2008 his version: February 2008 Comiled: May 3, 2008, :36m his aer has been ublished in the Journal of Statistical Planning and Inference, 38 (2008), he authors thank ze Leung Lai and Ingram Olkin for several useful comments. his work was suorted by the William Dow Chair in Political Economy (McGill University), the Canada Research Chair Program (Chair in Econometrics, Université de Montréal), the Bank of Canada (Research Fellowshi), a Guggenheim Fellowshi, a Konrad-Adenauer Fellowshi (Alexander-von-Humboldt Foundation, Germany), the Institut de finance mathématique de Montréal (IFM2), the Canadian Network of Centres of Excellence [rogram on Mathematics of Information echnology and Comlex Systems (MIACS)], the Natural Sciences and Engineering Research Council of Canada, the Social Sciences and Humanities Research Council of Canada, the Fonds de recherche sur la société et la culture (Québec), and the Fonds de recherche sur la nature et les technologies (Québec), and a Killam Fellowshi (Canada Council for the Arts). Université de Montréal, Déartement de Sciences Économiques, C.P. 628, succ. Centre-ville, Montréal, QC H3C 3J7. EL: () ext. 5283; Fax: () ; f.doko.tchatoka@umontreal.ca. William Dow Professor of Economics, McGill University, Centre interuniversitaire de recherche en analyse des organisations (CIRANO), and Centre interuniversitaire de recherche en économie quantitative (CIREQ). Mailing address: Deartment of Economics, McGill University, Leacock Building, Room 59, 855 Sherbrooke Street West, Montréal, Québec H3A 27, Canada. EL: () ; FAX: () ; jean-marie.dufour@mcgill.ca. Web age: htt://

3 ABSRAC When some exlanatory variables in a regression are correlated with the disturbance term, instrumental variable methods are tyically emloyed to make reliable inferences. Furthermore, to avoid difficulties associated with weak instruments, identification robust methods are often roosed. However, it is hard to assess whether an instrumental variable is valid in ractice because instrument validity is based on the questionable assumtion that some of them are exogenous. In this aer, we focus on structural models and analyze the effects of instrument endogeneity on two identification-robust rocedures, the Anderson-Rubin (949, AR) and the Kleibergen (2002, K) tests, with or without weak instruments. wo main setus are considered: () the level of instrument endogeneity is fixed (does not deend on the samle size), and (2) the instruments are locally exogenous, i.e. the arameter which controls instrument endogeneity aroaches zero as the samle size increases. In the first setu, we show that both test rocedures are in general consistent against the resence of invalid instruments (hence asymtotically invalid for the hyothesis of interest), whether the instruments are strong or weak. We also describe cases where test consistency may not hold, but the asymtotic distribution is modified in a way that would lead to size distortions in large samles. hese include, in articular, cases where the 2SLS estimator remains consistent, but the AR and K tests are asymtotically invalid. In the second setu, we find (non-degenerate) asymtotic non-central chi-square distributions in all cases, and describe cases where the non-centrality arameter is zero and the asymtotic distribution remains the same as in the case of valid instruments (desite the resence of invalid instruments). Overall, our results underscore the imortance of checking for the resence of ossibly invalid instruments when alying identification-robust tests. Key words: simultaneous equations; instrumental variables; locally weak instruments; invalid instruments; locally exogenous instruments. i

4 Contents. Introduction 2. Framework 2 3. est statistics 5 4. Asymtotic theory with invalid and weak instruments Possibly invalid instruments Locally exogenous instruments Conclusion 0 A. Aendix: Proofs List of Assumtions, Proositions and heorems 4. heorem : Asymtotic distribution of the AR statistic heorem : Asymtotic distribution of the K statistic heorem : Asymtotic distributions with locally exogenous instruments heorem : Asymtotic distributions with weak locally exogenous instruments Proof of heorem Proof of heorem Proof of heorem Proof of heorem ii

5 . Introduction he last decade shows growing interest for so-called weak instruments roblems in the econometric literature, i.e. situations where instruments are oorly correlated with endogenous exlanatory variables; see the reviews of Dufour (2003) and Stock, Wright and Yogo (2002). More generally, these can be viewed as situations where model arameters are not identified or close not to being identifiable, as meant in the econometric literature [see Dufour and Hsiao (2008)]. When instruments are weak, the limiting distributions of standard test statistics like Student, Wald, likelihood ratio and Lagrange multilier criteria in structural models often deend heavily on nuisance arameters; see e.g. Phillis (989), Bekker (994), Dufour (997), Staiger and Stock (997) and Wang and Zivot (998). In articular, standard Wald-tye rocedures based on the use of asymtotic standard errors are very unreliable in the resence of weak identification. As a result, several authors have worked on roosing more reliable statistical rocedures that would be alicable in such contexts. Interestingly, in the early days of simultaneous-equations econometrics, Anderson and Rubin (949, AR) roosed a rocedure which is comletely robust to weak instruments as well as to other difficulties such as missing instruments [see Dufour (2003), Dufour and aamouti (2005, 2007)]. But the AR rocedure may suffer from ower losses when too many instruments are used. So alternative methods largely try to alliate this difficulty, for examle: seudo-ivotal LM-tye and LR-tye statistics [Wang and Zivot (998), Kleibergen (2002), Moreira (2003)], samle-slitting methods [Dufour and Jasiak (200)], aroximately otimal instruments [Dufour and aamouti (2003)], systematic search methods for identifying relevant instruments and excluding unimortant instruments [Hall, Rudebusch and Wilcox (996), Hall and Peixe (2003), Dufour and aamouti (2003), Donald and Newey (200)]. However, all these rocedures including the AR method rely on the availability on valid (exogenous) instruments. his raises the question: what haens to these rocedures when some of the instruments are endogenous? In articular, what haens if an invalid instrument is added to a set of valid instruments? How robust are these inference rocedures to instrument endogeneity? Do alternative inference rocedures behave differently? If yes, what is their relative erformance in the resence of instrument endogeneity? We view the roblem of instrument endogeneity as imortant because it is hard in ractice to assess whether an instrumental variable is valid, i.e. whether it is uncorrelated with the disturbance term. Instrument validity or orthogonality tests are built on the availability of a number of undisuted valid instruments, at least as great as the number of coefficients to be estimated, whereas the validity of those initial instruments is not testable. In the econometric literature, little is known about test rocedures when some instruments are both invalid and weak. Hahn and Hausman (2003) deal with both instrument endogeneity and weakness, but they focus on estimation. Ashley (2006) roosed a sensitivity analysis of IV estimators when instruments are imerfect, his results however are only alicable if the covariance between the structural error term and some instruments is known, which is not necessary the case as it is showed in this aer. Analyzing the effect of instrument invalidity on the limiting and emirical distribution of IV estimators, Kivet and Niemczyk (2006) conclude that for the accuracy of asym-

6 totic aroximations, instrument weakness is much more detrimental than instrument invalidity and that the realizations of IV estimators based on strong but ossibly invalid instruments seem usually much closer to the true arameter values than those obtained from valid but weak instruments. However, this finding of Kiviet and Niemczyk leaves oen crucial questions: is it really ossible to make reliable inference with endogenous instruments? Is instrument endogeneity really more detrimental than its weakness on inference rocedures like a general family of Anderson-Rubintye rocedures? Swanson and Chao (2005) roosed a weak-instrument unified framework, but they do not take into account ossible invalidity of some instruments. Finally, Small (2007) has recently studied the roerties of tests for identifying restrictions [Sargan (958), Kadane and Anderson (977)], which can be sensitive to the use of endogenous instruments, and he roosed a sensitivity analysis to assess the imortance of the issue. hese results, however, do not allow for weak identification. In this aer, we focus on structural models and analyze the effects of instrument endogeneity on the Anderson and Rubin (949) and Kleibergen (2002) tests, in the resence of ossibly weak instruments. After formulating a general asymtotic framework which allows one to study these issues in a convenient way, we consider two main setus: () the one where the level of instrument endogeneity is fixed (i.e., it does not deend on the samle size), and (2) the one where the instruments are locally exogenous, i.e. the arameter which controls instrument endogeneity aroaches zero (at rate /2 ) as the samle size increases. In the first setu, we show that both test rocedures studied are in general consistent against the resence of invalid instruments (hence asymtotically invalid for the hyothesis of interest), whether the instruments are strong or weak. We also observe there are cases where consistency may not hold, but the asymtotic distribution is modified in a way that would lead to size distortions in large samles. In the second setu, asymtotic noncentral chi-square distributions are derived, and we give conditions under which the non-centrality arameter is zero and the asymtotic distribution remains the same as in the case of valid instruments (desite the resence of invalid instruments). Overall, our results underscore the imortance of checking for the resence of ossibly invalid instruments when alying identification-robust tests. he aer is organized as follows. Section 2 formulates the model considered. Section 3 describes briefly the statistics. Section 4 studies the asymtotic distribution of the statistics (under the null hyothesis) when some instruments are invalid. We conclude in section 5. Proofs are resented in the Aendix. 2. Framework We consider the following standard simultaneous equation framework, which has been the basis of much work on inference in model with ossibly weak instruments [see the reviews of Dufour (2003) and Stock et al. (2002)]: y = Y β + Zγ + u, (2.) Y = XΠ + ZΓ + V, (2.2) 2

7 where y is a vector of observations on the deendent variable, Y = [Y,..., Y ] is a G matrix of observations on exlanatory (ossibly) endogenous variables (G ), Z is a r matrix of observations on the included exogenous variables, X = [X,..., X ] is a k (k G) fullcolumn-rank matrix of observations on (suosedly) exogenous variables (instruments) excluded from the structural equation (2.), u = [u,..., u ] and V = [V,..., V ] = [v,..., v G ] are resectively vector and G disturbance matrices, β and γ are G and r vectors of unknown coefficients, Π and Γ are k G and r G matrices of unknown coefficients. he usual necessary and sufficient condition for identification of this model is rank(π) = G. Since we focus on the arameter β in our analysis, we can simlify the resentation of the results without notable loss of generality by setting γ = 0 and Γ = 0, so that Z dros from the model. With this simlification, model (2.)-(2.2) reduces to We also assume that y = Y β + u, (2.3) Y = XΠ + V. (2.4) u t = V t a + ε t, t =,...,, (2.5) X t = X 0t + W t, t =,...,, (2.6) u t = W t b + e t, t =,...,, (2.7) where X 0 = [X 0,..., X 0 ] is a k matrix of exogenous variables, ε t is uncorrelated with V t, and e t are uncorrelated with W t. V t and W t have mean zero and covariance matrices Σ V and Σ W,ε t and e t have mean zero and variances σ 2 ε and σ2 e resectively, while a and b are G and k vectors of unknown coefficients. (2.5)-(2.7) can be rewritten in matrix form as: u = V a + ε, (2.8) X = X 0 + W, (2.9) u = Wb + e, (2.0) where X 0 is uncorrelated with W, V, ε and e, while W = [W,..., W ] is uncorrelated with e but may be correlated with u (when b 0). So a controls the endogeneity of the variable Y, whereas b reresents the ossible endogeneity of the instruments X. If b = 0, the instruments X are valid; otherwise, they are invalid (endogenous). More recisely, if b 0, i.e., there exists at least one i such that b i 0, i =,..., k, and the corresonding variable X i does not constitute a valid instrument. We also make the following generic assumtions on the asymtotic behaviour of model variables [where A > 0 for a matrix A means that A is ositive definite (.d.), and refers to limits as ]: [ [ ] [ ] ΣV 0 ] V ε V ε 0 σ 2 > 0, (2.) ε 3

8 [ X0 W ] [ X0 W ] [ Σ0 0 0 Σ W [ ], Σ 0 > 0, (2.2) [ ] X 0 V ε e 0, (2.3) X X Σ X, (2.4) [ [ ] [ ] ΣW 0 ] W e W e 0 σ 2, (2.5) e X e (X W Σ W )b W V Σ WV, (2.6) ] [ ] Se N [0, Σ S ], (2.7) L S b S e N [ 0, σ 2 eσ X ], Sb N [ 0, σ 2 eσ b ], (2.8) where Σ V is G G fixed matrix, Σ 0 and Σ W are k k fixed matrices, S e and S b are k random vectors. Note that Σ W may be singular, and S b may not be indeendent of S e. From the above assumtions, it is easy to see that: where X 0u 0, X 0e 0, (2.9) X u X V ϕ = Σ W b, Σ WV, (2.20) [ [ ] [ ] σ 2 u V u V Σ = u δ ] > 0, (2.2) δ Σ V [ [ ] [ ] σ 2 u W u W u ϕ ], ϕ Σ W (2.22) [ ] [ ] [ ] δwε 0 W V ε e, 0 δ V e (2.23) δ = Σ V a, σ 2 u = a Σ V a + σ 2 ε = σ 2 e + b Σ W b, (2.24) Σ V a = Σ WV b + δ V e, Σ W b = Σ WV a + δ Wε, (2.25) Σ X = Σ 0 + Σ W > 0, Σ XY = Σ X Π + Σ WV, (2.26) Σ Y = Π Σ X Π + Σ V + Σ WV Π + Π Σ WV. (2.27) Finally, we denote by N(Σ W ) the null set of the linear ma on R k characterized by the matrix Σ W : N(Σ W ) = {x R k : Σ W x = 0}. (2.28) 4

9 If Σ W is a full-column-rank matrix, then N(Σ W ) = {0}; otherwise, there is at least one x 0 0 such that Σ W x 0 = 0. he setu described above is quite wide and does allow one to study several questions associated with the ossible resence of invalid instruments. In articular, an imortant ractical roblem consists in studying the effect on inference of adding an invalid instrument to a list of valid (ossibly identifying) instruments. Note that this roblem is distinct from studying the effect of imosing incorrect overidentifying restrictions [as done by Small (2007)]. o better see the issues studied here, it will be useful to consider a simle examle. Examle 2. Consider a model with one endogenous exlanatory variable (G = ) and two candidate instruments (k = 2). hen Y and V are vectors, X = [X, X 2 ] and W = [W, W 2 ] are 2 matrices, Π = [π, π 2 ] and b = [b, b 2 ] are vectors of dimension 2, and Y = XΠ + V = X π + X 2 π 2 + V, (2.29) u = Wb + e = W b + W 2 b 2 + e. (2.30) Let us further assume that X is a valid instrument (with W = 0), E[u X ] = 0, X 2 = W 2, π 2 = 0 and b = 0, where e is indeendent of X and X 2 (with finite mean zero), so that Y = XΠ + V = X π + V, (2.3) u = Wb + e = W 2 b 2 + e. (2.32) Here W 2 is not a valid instrument when b 2 0. But the structural equation (2.3) may in rincile be estimated using only X as an instrument, because E[u X ] = 0; if X is not a weak instrument (π 0) and satisfies usual regularity conditions, a consistent estimate of β can be obtained. Among other things, we study below the effect (on some identification-robust tests) of taking X 2 as an instrument when b 2 0, i.e. when X 2 is correlated with u. Note that the condition E[u X ] = 0 does not entail E[e X, X 2 ] = 0, which is a maintained hyothesis used by Small (2007). So the roblem considered here is distinct from the roblem of testing overidentifying restrictions [studied, for examle, by Sargan (958), Kadane and Anderson (977) and Small (2007)]. 3. est statistics We consider in this aer the roblem of testing H 0 : β = β 0 (3.) where some of the instruments used are in fact endogenous (b 0). We analyze the behavior of the Anderson-Rubin and Kleibergen statistics. he Anderson and Rubin (949) test for H 0 in equation (2.3) involves considering the transformed equation y Y β 0 = X + ε (3.2) 5

10 where = Π(β β 0 ) and ε = u + V (β β 0 ). H 0 can then be assessed by testing H 0 : = 0. he AR-statistic for H 0 is given by AR(β 0 ) = k (y Y β 0 ) P X (y Y β 0 ) (y Y β 0 ) M X (y Y β 0 )/( k) (3.3) where M B = I P B and P B = B(B B) B is the rojection matrix on the sace sanned by the columns of B. If b = 0, the asymtotic distribution of AR(β 0 ) is a χ 2 (k)/k under H 0. If furthermore u N[0,σ 2 I ] and X is indeendent of u, then AR(β 0 ) F(k, k) under H 0 irresective of whether the instruments are strong or weak. However, when some instruments are invalid, the distribution of the AR statistic may be affected. Kleibergen (2002) roosed a modification of the AR statistic to take into account the fact that this statistic may have low ower when there are too many instruments in the model. he modified statistic for testing H 0 can be written K(β 0 ) = (y Y β 0 ) PỸ (β0 ) (y Y β 0) (y Y β 0 ) M X (y Y β 0 )/( k) (3.4) where [ Ỹ (β 0 ) = X Π(β 0 ), Π(β0 ) = (X X) X Y (y Y β 0 ) S ] uv (β 0 ), (3.5) S uu (β 0 ) S uu (β 0 ) = k (y Y β 0) M X (y Y β 0 ), S uv (β 0 ) = k (y Y β 0) M X Y. (3.6) Unlike the AR statistic which rojects y Y β 0 on the k columns of X, the K statistic rojects y Y β 0 on the G columns of X Π(β 0 ). If the instruments X are exogenous, Π(β0 ) is both a consistent estimator of Π and asymtotically indeendent of X (y Y β 0 ) under H 0, and K(β 0 ) converges to a χ 2 (G). However, if some instruments are invalid (b 0), Π(β0 ) may not be asymtotically indeendent of X (y Y β 0 ) and the asymtotic distribution of the K statistic may not be a χ 2 (G). If the model contains only one instrument and one endogenous variable (G = k = ), the AR and K statistics are equivalent and ivotal even in finite samles whenever b = 0. When k >, even if b = 0, the K statistic is not ivotal in finite samles but is asymtotically ivotal, whereas the AR statistic is ivotal even in finite samles (when X is indeendent of u). Following Staiger and Stock (997), we refer to the locally weak-instrument asymtotic setu by considering a limiting sequence of Π where Π is local-to-zero. We also consider a limiting sequence of b where b is local-to-zero. We refer to this later limiting sequence as locally exogenous instruments asymtotic. We do not study this aer conditional tests such as those roosed by Moreira (2003), because the distributional theory for such tests is considerably more comlex and would go beyond the scoe of a short aer like the resent one. 6

11 4. Asymtotic theory with invalid and weak instruments In this section, we study the large-samle roerties of the statistics described above when some of the instruments used are invalid. wo setus are considered. he first is the ossibly invalid instrument setu, i.e., the endogeneity arameter b is a fixed vector. he second is the locally exogenous instrument setu, i.e., b is local-to-zero. 4.. Possibly invalid instruments We consider first the case where the endogeneity arameter b is a constant vector and we analyze the asymtotic distributions of the statistics. Our results cover both strong and weak-instrument asymtotic. heorem 4. below summarizes the asymtotic behavior of the AR statistic when some instruments may be endogenous. For a random variable S whose distribution deends on the samle size, the notation S L + means that P[S > x] as, for any x. heorem 4. ASYMPOIC DISRIBUION OF HE AR SAISIC. Suose that the assumtions (2.3)-(2.8) hold, with b = b 0 and β = β 0, where b 0 and β 0 are given vectors. If b 0 / N(Σ W ), then If b 0 N(Σ W ), then AR(β 0 ) L +. (4.) AR(β 0 ) L kσ 2 (S e + S b ) Σ X (S e + S b ) (4.2) u where S e and S b are defined in (2.7)-(2.8). If b 0 = 0, then AR(β 0 ) L k χ2 (k). (4.3) In the above theorem, no restriction is imosed on the rank of Π. In articular, the result holds even if Π is not a full-column rank matrix. When b 0 / N(Σ W ), the AR statistic diverges under the null hyothesis H 0. When b 0 N(Σ W ), the limiting distribution of the AR statistic does not diverge, but the AR test is not valid unless S b = 0. Of course, when b 0 = 0 which is the classical exogenous instrument setu S b = 0 and the AR test is asymtotically valid. heorem 4.2 below summarizes the asymtotic behavior of the K statistic when some instruments are ossibly invalid. heorem 4.2 ASYMPOIC DISRIBUION OF HE K SAISIC. Suose that the assumtions (2.3)-(2.8) hold, with b = b 0 and β = β 0, where b 0 and β 0 are given vectors. (A) If b 0 / N(Σ W ) then K(β 0 ) L + (4.4) when at least one of the following two conditions holds: (i) Π = Π 0 0 with rank( Σ XY ) = G, or (ii) Π = Π 0 / with rank(σxy ) = G, where Σ XY = Σ XY Σ W b 0 (q uv / σ 2 u), Σ XY = Σ WV Σ W b 0 (q uv / σ 2 u), 7

12 q uv = δ b 0 Σ WΣ X Σ WV, σ 2 u = σ2 u b 0 Σ WΣ X Σ Wb 0. (B) If b 0 N(Σ W ), then K(β 0 ) L σ 2 (S e + S b ) Σ X Σ XY (Σ XY Σ X Σ XY ) Σ XY Σ X (S e + S b ) (4.5) u when Π = Π 0 0 and rank(σ XY ) = G, and K(β 0 ) L σ 2 (S e + S b ) Σ X Σ WV (Σ WV Σ X Σ WV ) Σ WV Σ X (S e + S b ) (4.6) u when Π = Π 0 / and rank(σ WV ) = G. (C) If b 0 = 0, then K(β 0 ) L χ 2 (G) (4.7) when at least one of the following two conditions holds: (i) Π = Π 0 0 with rank(σ XY ) = G, or (ii) Π = Π 0 / with rank(σ WV ) = G. Unlike heorem 4. for the AR statistic, heorem 4.2 requires an additional rank assumtion. When b 0 / N(Σ W ), the null limiting distribution of the K statistic diverges. his means that the K test often rejects H 0 asymtotically when b 0 / N(Σ W ). Furthermore, when b 0 N(Σ W ), the K test is not asymtotically valid unless S b = 0. As exected, if b 0 = 0 (i.e., S b = 0), the K statistic converges to a χ 2 (G). It is worthwhile to note that the case where the rank assumtion fails [e.g., the artial identification of β] is not covered in this aer. Finally, it is interesting to observe that the limiting value of the two-stage least-squares (2SLS) estimator of β, is given by β = ( Ŷ Ŷ ) Ŷ y = [ Y X(X X) X Y ] Y X(X X) X y, (4.8) lim β = β + [ Σ XY Σ X Σ ] Σ XY XY Σ X Σ Wb (4.9) rovided rank(σ XY ) = G, so that β is consistent when b 0 N(Σ W ) and Σ XY has full column rank (even if some instruments are invalid). If b 0 N(Σ W ) but b 0 0, the asymtotic level of the Anderson-Rubin and Kleibergen tests can be affected Locally exogenous instruments We consider now the case where the endogeneity arameter b is local-to-zero. As in the revious subsection, we analyze the limiting distributions of the statistics. he results also cover two setus: locally exogenous instruments [Π = Π 0 0, b = b 0 / ], and weak locally exogenous instruments [Π = Π 0 /, b = b 0 / ]. heorem 4.3 and heorem 4.4 below derive the distributions of the statistics for both setus. 8

13 heorem 4.3 ASYMPOIC DISRIBUIONS WIH LOCALLY EXOGENOUS INSRUMENS. Suose that the assumtions (2.3)-(2.8) hold, with b = b 0 /, Π = Π 0 0 and β = β 0, where b 0 and β 0 are given vectors, and Π 0 is a given matrix. If b 0 / N(Σ W ), then AR(β 0 ) L k χ2 (k,µ ), (4.0) where K(β 0 ) L χ 2 (G,m m) if rank(σ XY ) = G, (4.) µ = σ 2 b 0 Σ WΣ X Σ Wb 0, m = (Σ XY e σ Σ X Σ XY ) /2 Σ XY Σ X Σ Wb 0, (4.2) e and Σ X, Σ XY, and Σ W are given in (2.)-(2.27). If b 0 N(Σ W ), then AR(β 0 ) L k χ2 (k), (4.3) K(β 0 ) L χ 2 (G) if rank(σ XY ) = G. (4.4) heorem 4.4 ASYMPOIC DISRIBUIONS WIH WEAK LOCALLY EXOGENOUS INSRU- MENS. Suose that the assumtions (2.3)-(2.8) hold, with b = b 0 /, Π = Π 0 / and β = β 0, where b 0 and β 0 are given vectors, and Π 0 is a given matrix (Π 0 = 0 is allowed). If b 0 / N(Σ W ), then AR(β 0 ) L k χ2 (k,µ ), (4.5) K(β 0 ) L χ 2 (G, m m) if rank(σ WV ) = G, (4.6) where m = σ e (Σ WV Σ X Σ WV ) /2 Σ WV Σ X Σ Wb 0, (4.7) and Σ X, Σ WV, Σ W and µ are defined in heorem 4.3. If b 0 N(Σ W ), then AR(β 0 ) L k χ2 (k), (4.8) K(β 0 ) L χ 2 (G) if rank(σ WV ) = G. (4.9) We make the following remarks concerning heorem 4.3 and heorem 4.4. First, the endogeneity arameter b is local-to-zero, and for b 0 N(Σ W ) the AR and K tests are asymtotically valid. However, unlike the AR test, note that the validity of the K test is established under an additional rank assumtion (the case where this additional rank assumtion fails is not covered in this aer). So, when b 0 N(Σ W ), the inference with locally exogenous instruments using the AR and K tests is feasible (at least in large samles). Second, if b 0 / N(Σ W ), the results in both theorems are different from those of heorems 4. and 4.2 because the limiting distributions of both statistics 9

14 do not diverge. hird, even though the AR and K statistics have non-central chi-square limiting distributions when b 0 / N(Σ W ), they are not ivotal since the non-centrality arameters deend on nuisance arameters. In addition, the limiting distributions of both statistics cannot be bounded by any ivotal distribution. It will be useful to see how the above theorems aly in a simle examle. Examle 4. Consider again model (2.29)-(2.30), which involves one endogenous exlanatory variable and two instruments. If the matrix Σ W is invertible, then N(Σ W ) = {0}, and heorem 4. entails that AR(β 0 ) L + under the null hyothesis β = β 0. Similarly, if Σ XY 0, then rank( Σ XY ) = G = and heorem 4.2 entails that K(β 0 ) L + when β = β 0. If X is a valid instrument (with W = 0) and X 2 = W 2 with W 2 W 2/ σ 2 W 2 > 0, we have [ ] 0 0 Σ W = 0 σ 2 W 2 (4.20) which is a matrix of rank one, and N(Σ W ) = {(x, x 2 ) : x 2 = 0}. If b 2 = 0, then b 0 N(Σ W ) and heorem 4. entails that the asymtotic distribution given by (4.2) holds for AR(β 0 ), while for K(β 0 ) art B of heorem 4.2 is alicable. Of course, when b 0 = 0, AR(β 0 ) follows the usual χ 2 (2)/2 asymtotic distribution, while K(β 0 ) follows a χ 2 () distribution. For locally exogenous instruments, theorems 4.3 and 4.4 can be alied in a similar way. 5. Conclusion In this aer, we have established conditions under which the AR and K tests are asymtotically valid even if some instruments used are endogenous. We have also showed that when these conditions fail, the limiting distributions of both statistics may diverge. Furthermore, when these conditions fail, under locally exogenous instruments setu, the limiting distributions of the statistics deend on nuisance arameters and cannot be bounded by any ivotal distribution. In consequence, the weak-instrument rocedure roosed by Wang and Zivot (998), the unified weak instruments framework of Swanson and Chao (2005) and the inference with imerfect instruments suggested by Ashley (2006) are not alicable. Overall, our results underscore the imortance of checking for the resence of ossibly invalid instruments when alying identification-robust tests. hey also suggest that sensitivity analyses where different sets of instruments are considered [Ashley (2006), Small (2007)] can be quite useful for the interretation of emirical results based on instrumental variables. 0

15 A. Aendix: Proofs PROOF OF HEOREM 4. (y Y β 0 ) M X (y Y β 0 ) k Note first that = u u k k ( u )( X X X ) ( X ) u (A.) where, by the assumtions (2.3)-(2.8), u u k σ 2 u > 0, X X Σ X > 0, ( u )( X X X (y Y β 0 ) M X (y Y β 0 ) k X u ) ( X ) u = X 0 u + W W b 0 + W e b 0Σ W Σ X Σ Wb 0, σ 2 u = σ 2 u b 0Σ W Σ X Σ Wb 0 0. Σ W b 0, (A.2) (A.3) (A.4) (A) Suose now that b 0 / N(Σ W ). hen b 0 Σ WΣ X Σ Wb 0 > 0 and the numerator of the AR statistic diverges: ( u (y Y β 0 ) )( X X X P X (y Y β 0 ) = ) ( X ) u L +, (A.5) hence AR(β 0 ) L +. (B) If b 0 N(Σ W ), we have Σ W b 0 = 0 and σ 2 u = σ2 u. Further, hen, hence X u = X (e + Wb 0 ) = X e + X Wb 0, X u = [X u Σ W b 0 ] = X e + (X W Σ W )b 0 L S = Se + S b. ( u (y Y β 0 ) X P X (y Y β 0 ) = )( X X (y Y β 0 ) M X (y Y β 0 ) k AR(β 0 ) L kσ 2 S Σ X S. u ) ( X ) u σ 2 u, L S Σ X S, (C) Finally, if b 0 = 0, we have b 0 N(Σ W ), with the extra restrictions u = e, σ 2 u = σ 2 e, (A.6) (A.7) (A.8) (A.9) (A.0) (A.) S = X u = X e L N [ 0, σ 2 e Σ X],

16 hence AR(β 0 ) L kσ 2 S eσ X S e e k χ2 (k). (A.2) PROOF OF HEOREM 4.2 We note first, as in (A.)-(A.4), that S uu (β 0 ) = (y Y β 0) M X (y Y β 0 ) k σ 2 u, X X Σ X > 0, (A) Suose that b 0 / N(Σ W ). (i) Let Π = Π 0 0. hen, we have X u Σ W b 0. (A.3) S uv (β 0 ) = k (y Y β 0) M X Y q uv = δ b 0 Σ WΣ X Σ WV, ( X Π(β ) X X ( Y X ) 0 ) = X X u S uv (β 0 ) Σ X S uu (β 0 ) Σ XY, where Σ XY = Σ XY Σ W b 0 (q uv / σ 2 u), and (A.4) (A.5) Ỹ (β 0 ) u = Π(β 0 ) X u Σ XY Σ X Σ Wb 0, (A.6) Ỹ (β 0 ) Ỹ (β 0 ) Σ XY Σ X Σ XY. If rank( Σ XY ) = G, then Σ XY Σ Σ X XY > 0 and Σ Σ X XY Σ W b 0 0 for b 0 / N(Σ W ), hence ] [Ỹ (β0 ) Ỹ (β 0 ) Ỹ (β0 ) u u Ỹ (β 0 ) Consequently, the numerator of the K statistic diverges: (A.7) b 0 Σ WΣ X Σ XY ( Σ XY Σ X Σ XY ) Σ XY Σ X Σ Wb 0 > 0. (y Y β 0 ) PỸ (β0 ) (y Y β 0) = u Ỹ (β 0 ) ] [Ỹ (β0 ) Ỹ (β 0 ) Ỹ (β0 ) u + (A.8) and (ii) Let Π = Π 0 /. hen K(β 0 ) L +. (A.9) (y Y β 0 ) PỸ (β0 ) (y Y β 0) = u Ỹ (β 0 ) ] [Ỹ (β0 ) Ỹ (β 0 ) Ỹ (β0 ) u, (A.20) 2

17 where Ỹ (β 0 ) Ỹ (β 0 ) L Σ XY Σ X Σ XY, Ỹ (β 0 ) u Σ X Σ XY Σ W b 0, with ΣXY = Σ WV Σ W b 0 (q uv / σ 2 u ). If rank(σ XY ) = G, then the numerator of the K statistic diverges, and K(β 0 ) L +. (B) If b 0 N(Σ W ), we have Σ W b 0 = 0, σ 2 u = σ2 u and X u L S = S e + S b as in (A.7)-(A.8). (i) If Π = Π 0 0, we have when b 0 N(Σ W ), the denominator of the K statistic satisfies while the denominator can be written (y Y β 0) M X (y Y β 0 ) σ 2 u, (y Y β 0 ) PỸ (β0 ) (y Y β 0) = u X Π(β0 ) [Ỹ (β0 ) Ỹ (β 0 ) ] Π(β0 ) X u (A.2) (A.22) where Π(β 0 ) Σ X Σ XY, Ỹ (β 0 ) Ỹ (β 0 ) Σ XY Σ X Σ XY, Ỹ (β 0 ) u Σ X Σ XY S. (A.23) If rank(σ XY ) = G, we have Σ XY Σ X Σ XY > 0, hence K(β 0 ) L σ 2 S Σ X Σ XY (Σ XY Σ X Σ XY ) Σ XY Σ X S. u (A.24) (ii) If Π = Π 0 /, the numerator of the K statistic is (y Y β 0 ) PỸ (β0 ) (y Y β 0) = u X Π(β0 ) [Ỹ (β0 ) Ỹ (β 0 ) ] Π(β0 ) X u, (A.25) hence Π(β 0 ) Σ X Σ WV, Ỹ (β 0 ) Ỹ (β 0 ) Σ WV Σ X Σ WV, Ỹ (β 0 ) u Σ X Σ WV S. (A.26) If rank(σ WV ) = G, then K(β 0 ) L σ 2 S Σ X Σ WV (Σ WV Σ X Σ WV ) Σ WV Σ X S. u (A.27) (C) Finally, if b 0 = 0, we have b 0 N(Σ W ), with the extra restrictions u = e, σ 2 u = σ 2 e, S = X u L N [ 0, σ 2 e Σ X], 3

18 hence, if Π = Π 0 0, K(β 0 ) L σ 2 S eσ X Σ XY (Σ XY Σ X Σ XY ) Σ XY Σ X S e χ 2 (G), e (A.28) and if Π = Π 0 / (where Π 0 = 0 is allowed), K(β 0 ) L σ 2 S eσ X Σ WV (Σ WV ΣX Σ WV ) Σ WV ΣX S e χ 2 (G). e (A.29) PROOF OF HEOREM 4.3 Since b is now local-to-zero, we have X u L Se + Σ W b 0, X X Further, we have Σ X, X u 0, (y Y β 0) M X (y Y β 0 ) k σ 2 u > 0. (A.30) u u k = (e + W b 0 ) (e + W b 0 ) k = e e k + b 0 W e ( k) + e Wb 0 ( k) + b 0 e Wb 0 ( k) σ 2 e = σ 2 u. (A.3) (A) Let b 0 / N(Σ W ). hen, AR(β 0 ) L kσ 2 (S e + Σ W b 0 ) Σ X (S e + Σ W b 0 ) e k χ2 (k,µ ) (A.32) where µ = Ỹ (β 0 ) u b σ 2 0 Σ WΣ e X Σ Wb 0 0. Similarly, we have Ỹ (β 0 ) Ỹ (β 0 ) L Σ X Σ XY (S e + Σ W b 0 ). So, if rank(σ XY ) = G, we have Σ XY Σ X Σ XY and K(β 0 ) L σ 2 (S e + Σ W b 0 ) Σ X Σ XY (Σ XY Σ X Σ XY ) Σ XY Σ X (S e + Σ W b 0 ) χ 2 (G,m m) e (A.33) where m = σ e (Σ XY Σ X Σ XY ) /2 Σ XY Σ X Σ Wb 0 0. (B) If b 0 N(Σ W ), we have Σ W b 0 = 0. hen µ = 0 and m = 0, hence AR(β 0 ) L k χ2 (k) and K(β 0 ) L χ 2 (G). PROOF OF HEOREM 4.4 he roof of heorem 4.3 for the AR statistic covers heorem 4.4. he roof for the K statistic is similar to the one in heorem

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