Supplemental Material to IDENTIFICATION- AND SINGULARITY-ROBUST INFERENCE FOR MOMENT CONDITION MODELS. Donald W. K. Andrews and Patrik Guggenberger

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1 Supplemental Material to IDENTIFICATION- AND SINGULARITY-ROBUST INFERENCE FOR MOMENT CONDITION MODELS By Donald W. K. Andrews and Patrik Guggenberger January 25 Revised October 28 COWLES FOUNDATION DISCUSSION PAPER NO. 978RS COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 2828 New Haven, Connecticut

2 Supplemental Material for Identi cation- and Singularity-Robust Inference for Moment Condition Models Donald W. K. Andrews Cowles Foundation for Research in Economics Yale University Patrik Guggenberger Department of Economics Pennsylvania State University First Version: March 25, 2 Revised: October 7, 28

3 Contents Outline 4 Further Discussion of the Related Literature 5 2 Subvector SR Tests for Potentially Singular Moments Variance Matrices 7 3 Miscellanei 3. Moore-Penrose Expression for the SR-AR Statistic Computation Implementation SR-CQLR P Test 4. SR-CQLR P Parameter Space De nition of the SR-CQLR P Test Asymptotic Size of the SR-CQLR P Test Asymptotic E ciency of the SR-CQLR P Test under Strong Identi cation Summary Comparison of CLR-type Tests in Kleibergen (25) and AG Tests without the Singularity-Robust Extension 9 5. Asymptotic Results for Tests without the SR Extension Uniformity Framework General Weight Matrices W c n and U b n Uniformity Reparametrization Assumption WU Asymptotic Distributions Singularity-Robust Tests 33 7 Time Series Observations 39 8 SR-CQLR, SR-CQLR P, and Kleibergen s Nonlinear CLR Tests in the Homoskedastic Linear IV Model Normal Linear IV Model with p Endogenous Variables Homoskedastic Linear IV Model SR-CQLR P Test SR-CQLR Test

4 8.5 Kleibergen s Nonlinear CLR Tests De nitions of the Tests p = Case p 2 Case Simulation Results for Singular and Near-Singular Variance Matrices 53 2 Simulation Results for Kleibergen s MVW-CLR Test 55 2 Eigenvalue-Adjustment Procedure Singularity-Robust LM Test Proofs of Lemmas 5.2, 5., and Proofs of Lemma 5.4 and Proposition Proof of Theorem Proof of the Asymptotic Size Results Statement of Results Proof of Theorem Proof of Lemma Proof of Lemma Proof of Lemma Proof of Theorem 5. for the Anderson-Rubin Test and CS Proofs of Theorems 7. and Proofs of Lemmas 8., 8.2, and Proof of Lemma Proof of Lemma Proof of Lemma Proof of Theorem Proof of Theorems 8., 2., and Proof of Theorem Proof of Lemma

5 3.3 Proof of Theorem Proof of Theorem

6 Outline We let AG2 abbreviate the main paper Identi cation- and Singularity-Robust Inference for Moment Condition Models. References to sections with section numbers less than refer to sections of AG2. All theorems, lemmas, and equations with section numbers less than refer to results and equations in AG2. We let SM abbreviate Supplemental Material. We let AG abbreviate the paper Andrews and Guggenberger (27). The SM to AG is given in Andrews and Guggenberger (24). Section provides further discussion of the literature related to AG2. Section 2 extend the subvector tests in Section 8 to allow for the possibility that F = E F g i gi is singular. Section 3 provides some miscellaneous backup material for AG2. Section 4 introduces the SR-CQLR P test that applies when the moment functions are of a multiplicative form, u i ()Z i ; where u i () is a scalar residual and Z i is a k-vector of instrumental variables. Sections 5 and 6 provide parts of the proofs of the asymptotic size results given in Sections 6 and 4. Section 7 generalizes the SR-AR, SR-CQLR, and SR-CQLR P tests from i.i.d. observations to strictly stationary strong mixing observations. Section 8 compares the test statistics and conditioning statistics of the SR-CQLR, SR-CQLR P ; and Kleibergen s (25, 27) CLR tests to those of Moreira s (23) LR statistic and conditioning statistic in the homoskedastic linear IV model with xed (i.e., nonrandom) IV s. Section 9 provides nite-sample null rejection probability simulation results for the SR-AR and SR-CQLR tests for cases where the variance matrix of the moment functions is singular and near singular. Section 2 provides nite-sample simulation results that illustrate that Kleibergen s CLR test with moment-variance weighting can have low power in certain linear IV models with a single right-hand side (rhs) endogenous variable, as the theoretical results in Section 8 suggest. Section 2 establishes some properties of the eigenvalue-adjustment procedure de ned in Section 5. and used in the de nitions of the SR-CQLR and SR-CQLR P tests. Section 22 de nes a new SR-LM test. The remainder of the SM provides the rest of the proofs of the results stated in AG2 and the SM. Section 23 proves Lemmas 5.2, 5., and 4.. Section 24 proves Lemma 5.4 and Proposition 5.5. Section 25 proves Theorem 5.6. Section 26 proves Theorem 5. (using Theorem 5.6). Section 27 proves Theorems 7. and 4.3. Section 28 proves Lemmas 8., 8.2, and 8.3. Section 4

7 29 proves Theorem 7., which concerns the time series results. Section 3 proves Theorems 8. and 2., which concern the subvector inference results. For notational simplicity, throughout the SM, we often suppress the argument for various quantities that depend on the null value : Further Discussion of the Related Literature The rst paragraph of AG2 lists a number of models in which weak identi cation may arise. Speci c references are as follows. For new Keynesian Phillips curve models, see Dufour, Khalaf, and Kichian (26), Nason and Smith (28), and Kleibergen and Mavroeidis (29). For DSGE models, see Canova and Sala (29), Iskrev (2), Qu and Tkachenko (22), Dufour, Khalaf, and Kichian (23), Guerron-Quintana, Inoue, and Kilian (23), Qu (24), Schorfheide (24), and I. Andrews and Mikusheva (25, 26). For the CCAPM, see Stock and Wright (2), Neely, Roy, and Whiteman (2), Yogo (24), Kleibergen (25), Carroll, Slacalek, and Sommer (2), and Gomes and Paz (23). For interest rate dynamics, see Jegannathan, Skoulakis, and Wang (22) and Grant (23). For the BLP model, see Armstrong (26). For the returns-to-schooling wage equations, see Angrist and Krueger (99, 992) and Cruz and Moreira (25). For the time series models, see Hannan (982), Teräsvirta (994), Nelson and Startz (27), and Andrews and Cheng (22, 23b). For the selection model, see Puhani (2). For the mixing and regime switching models, see Cho and White (27), Chen, Ponomareva, and Tamer (24), and references therein. For the nuisance parameter only under the alternative models, see Davies (977) and Andrews and Ploberger (994). Some asymptotic size results in the linear IV regression model with a single right-hand-side endogenous variable (i.e., p = ) include the following. Mikusheva (2) establishes the correct asymptotic size of LM and CLR tests in the linear IV model when the errors are homoskedastic. Guggenberger (22) establishes the correct asymptotic size of heteroskedasticity-robust LM and CLR tests in a heteroskedastic linear IV model. Subvector inference via the Bonferroni or Sche é projection method, is discussed in see Cavanagh, Elliott, and Stock (995), Chaudhuri, Richardson, Robins, and Zivot (2), Chaudhuri and Zivot (2), and McCloskey (2) for Bonferroni s method, and Dufour (989) and Dufour and Jasiak (2) for the projection method. Both methods are conservative, but Bonferroni s method is found to work quite well by Chaudhuri, Richardson, Robins, and Zivot (2) and Chaudhuri and Zivot (2). 27 Andrews (27) provides subvector methods that are closely related to 27 Cavanagh, Elliott, and Stock (995) provide a re nement of Bonferroni s method that is not conservative, but it is much more intensive computationally. McCloskey (2) also considers a re nement of Bonferroni s method. 5

8 the Bonferroni method but are not conservative asymptotically. Other results in the literature on subvector inference include the following. Subvector inference in which nuisance parameters are pro led out is possible in the linear IV regression model with homoskedastic errors using the AR test, but not the LM or CLR tests, see Guggenberger, Kleibergen, Mavroeidis, and Chen (22). Andrews and Cheng (22, 23a,b) provide subvector tests with correct asymptotic size based on extremum estimator objective functions. These subvector methods depend on the following: (i) one has knowledge of the source of the potential lack of identi cation (i.e., which subvectors play the roles of ; ; and in their notation), (ii) there is only one source of lack of identi cation, and (iii) the estimator objective function does not depend on the weakly identi ed parameters (in their notation) when = ; which rules out some weak IV s models. 28 Cheng (25) provides subvector inference in a nonlinear regression model with multiple nonlinear regressors and, hence, multiple potential sources of lack of identi cation. I. Andrews and Mikusheva (26) develop subvector inference methods in a minimum distance context based on Anderson-Rubin-type statistics. Cox (27) provides subvector methods in a class of models that allows for multiple sources of weak identi cation and includes factor models. I. Andrews and Mikusheva (25) provide conditions under which subvector inference is possible in exponential family models (but the requisite conditions seem to be quite restrictive). I. Andrews (25) considers subvector inference in the context of a two-step procedure that determines rst whether one should use an identi cation-robust method or not. Phillips (989) and Choi and Phillips (992) provide asymptotic and nite-sample results for estimators and classical tests in simultaneous equations models that may be unidenti ed or partially identi ed when p : However, their results do not cover weak identi cation (of standard or nonstandard form) or identi cation-robust inference. Hillier (29) provides exact nite-sample results for CLR tests in the linear model under the assumption of homoskedastic normal errors and known covariance matrix. Antoine and Renault (29, 2) consider GMM estimation under semi-strong and strong identi cation, but do not consider tests or CS s that are robust to weak identi cation. Armstrong, Hong, and Nekipelov (22) show that standard Wald tests for multiple restrictions in some nonlinear IV models can exhibit size distortions when some IV s are strongly identi ed and others are semi-strongly identi ed not weakly identi ed. These results indicate that identi cation issues can be more severe in nonlinear models than in linear models, which provides 28 Montiel Olea (22) also provides some subvector analysis in the extremum estimator context of Andrews and Cheng (22). His e cient conditionally similar tests apply to the subvector (; ) of (; ; ) (in Andrews and Cheng s (22) notation), where is a parameter that determines the strength of identi cation and is known to be strongly identi ed. The scope of this subvector analysis is analogous to that of Stock and Wright (2) and Kleibergen (24). 6

9 further motivation for the development of identi cation-robust tests for nonlinear models. 2 Subvector SR Tests for Potentially Singular Moments Variance Matrices Figure SM- provides additional power comparisons to those given in Section 8.4 for the subvector null hypothesis in the endogenous probit model. Figure SM- provides results for = ; whereas Figure in Section 8.4 provides results for = :9: See Section 8.4 for a discussion of the results. In the remainder of this section, we extend the subvector tests in Section 8 to allow for the possibility that F = E F g i gi is singular. We employ the de nitions in (4.3) (4.4) with in place of : That is, br n (; ) := rk( b n (; )) and b n (; ) := b A n (; ) b n (; ) b A n (; ) ; where b n (; ) is the k k diagonal matrix with the eigenvalues of b n (; ) on the diagonal in nonincreasing order, and b A n (; ) is a k k orthogonal matrix of eigenvectors corresponding to the eigenvalues in b n (; ): We partition b A n (; ) according to whether the corresponding eigenvalues are positive or zero: b A n (; ) = [ b A n (; ); b A? n (; )]; where b A n (; ) 2 R kbrn(;) and b A? n (; ) 2 R k(k brn(;)) : The columns of b An (; ) are eigenvectors of b n (; ) that correspond to positive eigenvalues of b n (; ): Analogously, consider the spectral decomposition for the population quantity, de ned in (3.4) with in place of ; i.e., F (; ) = A F (; ) F (; )A F (; ) ; and de ne r F (; ) := rk( F (; )): We partition A F (; ) as A F (; ) = [A F (; ); A? F (; )]; where A F (; ) 2 R kr F (;) ; A? F (; ) 2 R k(k r F (;)) ; (2.) and the columns of A F (; ) are eigenvectors of F (; ) that correspond to positive eigenvalues of F (; ): Let F (; ) denote the upper left r F (; )r F (; ) submatrix of F (; ): The matrix F (; ) is diagonal with the positive eigenvalues of F (; ) on its diagonal in nonincreasing order. As above, we sometimes leave out the argument and denote by b n () the matrix b n ( ; ) and similarly for other expressions. Recall the de nition following (8.6) of e n ; the null-restricted rst-stage GMM estimator. Analogously to the full vector SR test, the subvector SR test is de ned using the nonredundant moment functions. That is, rather than using the moment function g i (; ); the test of the hypothesis in (8.2) is based on g bai (; ) = A b n ( ; e n ) g i (; ) 2 R brn( ; e ) n : (2.2) 7

10 From now on, whenever a subindex b A appears on an object de ned in Section 8.2, it means that it is de ned as in Section 8.2 but resulting from a moment condition model de ned in terms of the nonredundant moment conditions g bai (; ): In particular, b ban (; ) := n bg ban (; ) := n n P i= P n i= g bai (; )g bai (; ) bg ban (; )bg ban (; ) 2 R brn( ; e n )br n( ; e n ) ; g bai (; ); and b ban := arg minjjb' ban bg ban ( ; )jj 2 ; (2.3) 2B where b' ban 2 R brn( ; e n )br n( ; e n ) satis es b' b An b' ban = b ban ( ; e n ): (2.4) The subvector SR-AR and SR-CQLR test statistics, denoted by SR-ARn( S ; b ban ) and SR- QLRn( S ; b ban ); respectively, are de ned as the nonrobust tests are de ned, but based on the moment functions g bai (; ) in place of g i (; ) and using the GMM estimator b ban rather than b n to estimate the nuisance parameter : When br n ( ; e n ) > ; the subvector SR-AR test at nominal size 2 (; ) rejects if SR-ARn( S ; b ban ) > 2 br n( ; e : (2.5) n ); The subvector SR-CQLR test at nominal size 2 (; ) rejects if SR-QLR S n( ; b ban ) > c brn( ; e n );p (n=2 b D ban ( ; b ban ); e J ban ( ; b ban ); ): (2.6) If br n ( ; e n ) = ; then SR-ARn( S ; b ban ) and SR-QLRn( S ; b ban ) := and 2 br n( ; e n ); c brn( ; e n );p (n=2 D b ban ( ; b ban ); J e ban ( ; b ban ); ) := and the two tests do not reject H : and Next, we de ne the parameter spaces for the subvector SR-AR and SR-CQLR tests. We denote the column and null spaces of a matrix by col() and N(); respectively. We impose the conditions in F S AR; de ned in (8.4) which guarantee consistency of the preliminary estimator e n : The parameter space F S AR;2 de ned in (8.5) is modi ed in four ways: (i) the condition min(e F g i g i ) is dropped, (ii) the condition E F sup 2B( ;) jj =2 F ()A F () (g i () E F g i ())jj 2 M is added, (iii) all of the remaining conditions are formulated in terms of the moment functions =2 F ( ; )A F ( ; ) g i ( ; ); rather than g i ( ; ); and (iv) the condition, for some y > ; N( F ( ; )) = N( F ( ; )) for all 2 B( ; y ); where denotes the true value of ; is added. Call the resulting space F S;SR AR;2 : We de ne the null parameter space for the subvector SR 8

11 AR test to be F S;SR AR := F S AR; \ F S;SR AR;2 : (2.7) The null parameter space for the subvector SR-CQLR test, denoted by F S;SR ; is de ned as F S is de ned in (8.7) with the following modi cations. First, FAR S S;SR is replaced by FAR ; and second, all of the remaining conditions are formulated in terms of the moment functions =2 F ( ; )A F ( ; ) g i ( ; ); rather than g i ( ; ): We can also construct con dence regions for with correct asymptotic con dence size by inversion of the subvector SR-AR and SR-CQLR tests. by The relevant parameter spaces are given F S;SR ;AR := f(f; ; ) : (F; ) 2 F S;SR AR ( ); 2 g and F S;SR := f(f; ; ) : (F; ) 2 F S;SR ( ); 2 g; (2.8) respectively, where F S;SR AR ( ) and F S;SR ( ) denote F S;SR AR and F S;SR with the latter sets dependence on made explicit. Note that condition (iv) of F S;SR AR;2 can be restrictive. We now discuss a scenario in which it holds. Consider the case where the moment functions are of the form g i (; ) = u i (; )Z i ; (2.9) where Z i is a vector of instrument variables, the residual u i (; ) is scalar, E F u 2 i ( ; ) > ; and E F u 2 i ( ; )Z i Zi factors into E F u 2 i ( ; )E F Z i Zi : (Note that the latter condition is implied by conditional homoskedasticity: E F (u 2 i ( ; )jz i ) = 2 a.s. for some constant 2 > :) Under these conditions, F ( ; ) = E F u 2 i ( ; )Z i Z i E F u i ( ; )Z i E F u i ( ; )Z i and F ( ; ) = E F u 2 i ( ; )E F Z i Z i : If A F F A F denotes a singular value decomposition of E F Z i Z i with F = Diag( F ; F ); where F 2 R r contains the nonzero eigenvalues and F 2 R k zero eigenvalues and A F r contains the = (A F ; A F ) is a decomposition of the matrix of eigenvectors corresponding to the nonzero/zero eigenvalues, respectively, then A F = N( F ( ; )): It follows that A F E F Z i Zi A F = Diag( F ; F ) and thus, in particular, E F (A F Z i) 2 j = for j = r+; :::; k: Therefore, (A F Z i) j = a.s. for j = r + ; :::; k: But then A F F ( ; )A F = E F u 2 i ( ; )A F Z iz i A F E F u i ( ; )A F Z i E F u i ( ; )Z i A F ; for any 2 B; equals a block diagonal matrix with lower right block equal to (k r)(k r) : This implies F ( ; )A F = k(k r) ; which implies that N( F ( ; )) N( F ( ; )): Thus, in the setup of (2.9), condition (iv) of F S;SR AR;2 holds provided N( F ( ; )) is not a strict subset of N( F ( ; )): 9

12 Note that condition (iv) of F S;SR AR;2 implies that r F () is constant for all 2 B( ; y ): Furthermore, it implies that col( F ( ; )) = col( F ( ; )) for all 2 B( ; y ); i.e., that col(a F ( )) = col(a F ()) for all 2 B( ; y ): Therefore, without loss of generality, under condition (iv) of F S;SR AR;2 ; we can take A F () = A F ( ) for all 2 B( ; y ); i.e., A F () does not depend on for all 2 B( ; y ): The asymptotic size and similarity results for the subvector SR-AR and SR-CQLR tests are as follows. Theorem 2. Suppose Assumption gb holds. The asymptotic sizes of the subvector SR-AR and SR-CQLR tests de ned in (2.5) and (2.6), respectively, equal their nominal size 2 (; ) for the null parameter spaces F S;SR AR and F S;SR ; respectively. These tests are asymptotically similar (in a uniform sense) for the subsets of these parameter spaces that exclude distributions F under which g i = k a.s. Analogous results hold for the corresponding subvector SR-AR and SR-CQLR CS s for the parameter spaces F S;SR S;SR ;AR and F : Comment: Theorem 2. is proved in Section 3 below. 3 Miscellanei 3. Moore-Penrose Expression for the SR-AR Statistic The expression for the SR-AR statistic given in (4.8) of AG2 holds by the following calculations. For notational simplicity, we suppress the dependence of quantities on : We have SR-AR n = nbg n b A n ( b A n b n b An ) b A n bg n = nbg n A b n ( A b n[ A b n ; A b? n ] b n [ A b n ; A b? n ] An b ) A b n bg n = nbg n A b n b b n A nbg n and 2 3 b n brn(k brn) nbg n b+ n bg n = nbg n[ A b n ; A b? n ] 4 (k brn)brn (k brn)(k brn) 5 [ b A n ; b A? n ] bg n = nbg n b A n b n b A nbg n ;(3.) where the spectral decomposition of b n given in (4.3) and (4.4) is used once in each equation above. It is not the case that SR-AR n () equals the rhs expression in (4.8) with probability one when b + n () is replaced by an arbitrary generalized inverse of b n (): The expression for the SR-AR statistic given in (4.6) is preferable to the Moore-Penrose expression in (4.8) for the derivation of the asymptotic results for the SR-AR test.

13 3.2 Computation Implementation The computation times given in Section 5.3 are for the model in Section 9 for the country Australia, although the choice of country has very little e ect on the times. The computation times for the PI-CLC, MM-SU, and MM2-SU tests depend greatly on the choice of implementation parameters. For the PI-CLC test, these include (i) the number of linear combination coe cients "a" considered in the search over [; ]; which we take to be ; (ii) the number of simulation repetitions used to determine the best choice of "a;" which we take to be 2; and (iii) the number of alternative parameter values considered in the search for the best "a;" which we take to be 4 for p = : For the MM-SU and MM2-SU tests, the implementation parameters include (i) the number of variables in the discretization of the maximization problem, which we take to be ; and (ii) the number of points used in the numerical approximations of the integrals h and h2 that appear in the de nitions of these tests, which we take to be : The run-times for the PI-CLC, MM-SU, and MM2-SU tests exclude some items, such as a critical value look up table for the PI-CLC test, that only need to be computed once when carrying out multiple tests. The computations are done in GAUSS using the lmpt application to do the linear programming required by the MM-SU and MM2-SU tests. Note that the computation time for the SR-CQLR test could be reduced by using a look up table for the data-dependent critical values, which depend on p singular values. This would be most useful when p = 2: 4 SR-CQLR P Test In this section, we de ne the SR-CQLR P test, which is quite similar to the SR-CQLR test, but relies on g i () having a product form. This form is g i () = u i ()Z i ; (4.) where Z i is a k vector of IV s, u i () is a scalar residual, and the (random) function u i () is known. This is the case considered in Stock and Wright (2). It covers many GMM situations, but can be restrictive. For example, it rules out Hansen and Scheinkman s (995) moment conditions for continuous-time Markov processes, the moment conditions often used with dynamic panel models, e.g., see Ahn and Schmidt (995), Arellano and Bover (995), and Blundell and Bond (995), and moment conditions of the form g i () = u i () Z i ; where u i () is a vector. The SR-CQLR P test reduces asymptotically to Moreira s (23) CLR test in the homoskedastic linear IV regression model with xed IV s for sequences of distributions in all identi cation

14 categories. In contrast, the SR-CQLR test does so only under sequences in the standard weak, semi-strong, and strong identi cation categories, see Section 6.2 for the de nitions of these identi- cation categories. 4. SR-CQLR P Parameter Space When (4.) holds, we de ne u i u i() 2 R p and u i () u i() u i () The null parameter space for the SR-CQLR P test is A 2 R p+ ; and we have G i () = Z i u i () : 29 (4.2) F SR P := ff 2 F SR : E F jj =2 A F Z i jj 4+ M; E F jju i jj 2+ M; and F E F jj =2 F A F Z i jj 2 u 2 i (u 2 i > c) =2g; (4.3) for some > and some M; c < ; where F and A F are de ned in Section 3.2. By de nition, F SR P F SR F SR AR : The conditions in F SR P su cient condition for the last condition in F SR P are only marginally stronger than those in F SR ; de ned in (3.6). A su ciently large M < (using the rst condition in F SR P inequality). to hold for some c < is E F u 4 i M for some and the Cauchy-Bunyakovsky-Schwarz The conditions in F SR P place no restrictions on the column rank or singular values of E F G i : The conditions in F SR P also place no restrictions on the variance matrix F := E F g i g i of g i; such as min ( F ) for some > or min ( F ) > : Hence, F can be singular. In Section 3.2, it is noted that identi cation failure yields singularity of F in likelihood scenarios. It also does so in all quasi-likelihood scenarios when the quasi-likelihood does not depend on some element(s) of (or some transformation(s) of ) for in a neighborhood of : 3 Another example where F may be singular is the following homoskedastic linear IV model: y i = Y 2i + U i and Y 2i = Z i + V 2i; where all quantities are scalars except Z i ; 2 R d Z and = (; ) 2 R +d Z : 29 As with G(W i; ) de ned in (3.2), u i () need not be a vector of partial derivatives of u i() for all sample realizations of the observations. It could be the vector of partial derivatives of u i() almost surely, rather than for all W i; which allows u i() to have kinks, or a vector of nite di erences of u i(): For the asymptotic size results for the SR-CQLR 2 test given below to hold, u i () can be any random p vector that satis es the conditions in F2 SR (de ned in (4.3)). 3 In this case, the moment functions equal the quasi-score and some element(s) or linear combination(s) of elements of moment functions, equal zero a.s. at (because the quasi-score is of the form g i() = (@=@) log f(w i; ) for some density or conditional density f(w i; )). This yields singularity of the variance matrix of the moment functions and of the expected Jacobian of the moment functions. 2

15 The corresponding reduced-form equations are y i = Z i + V i and Y 2i = Z i + V i; where V i = U i + V 2i : We assume EU i = EV 2i = ; EU i Z i = EV 2i Z i = d Z ; and E(V i V i jz i) = V a.s. for V i := (V i ; V 2i ) and some 2 2 constant matrix V : The moment conditions for are g i () = ((y i Z i )Z i ; (Y 2i Z i )Z i ) 2 R k ; where k = 2d Z : The variance matrix V EZ i Z i of g i ( ) = (V i Z i ; V 2iZ i ) is singular whenever the covariance between the reduced-form errors V i and V 2i is one (or minus one) or EZ i Z i is singular. In this model, we are interested in joint inference concerning and : This is of interest when one wants to see how the magnitude of a ects the range of plausible values. Section 3.2 and Grant (23) note that F can be singular in the model for interest rate dynamics in Jegannathan, Skoulakis, and Wang (22, Sec. 6.2) (JSW). JSW consider ve moment conditions and a four dimensional parameter : The rst four moment functions in JSW are (a(b r i )r 2 i 2 ri ; a(b r i )r 2+ i ( =2) 2 ; (b r i )ri a (=2) 2 r 2 a i ; a(b r i )ri (=2) 3 r 2 i ) ; where = (a; b; ; ) and r i is the interest rate. The second and third functions are equivalent if = (a + )=2; the second and fourth functions are equivalent if = ( + )=2; and the third and fourth functions are equivalent if = a: Hence, the variance matrix of the moment functions is singular when one or more of these three restrictions on the parameters holds. When any two of these restrictions hold, the parameter also is unidenti ed. Next, we specify the parameter space for (F; ) that is used with the SR-CQLR P CS. It is denoted by F;P SR : For notational simplicity, the dependence of the parameter space F SR P in (4.3) on is suppressed. When dealing with the SR-CQLR P CS, rather than test, we make the dependence explicit and write it as FP SR( ): We de ne F SR ;P := f(f; ) : F 2 F SR P ( ); 2 g: (4.4) 4.2 De nition of the SR-CQLR P Test First, we de ne the CQLR P test without the SR extension. It uses the statistics bg n (); b n (); AR n (); and b D n () (de ned in (4.), (4.2), and (5.2)). The CQLR P test also uses analogues e R n () 3

16 and e V n () of b R n () and b V n () (de ned in (5.3)), respectively, which are de ned as follows: er n () := B() I evn k () (B() I k ) 2 R (p+)k(p+)k ; where nx ev n () := n (u i () bu in()) (u i () bu in()) Z i Zi 2 R (p+)k(p+)k ; i= bu in() := e n () Z i 2 R p+ ; e n () := (Z nk Z nk) Z nk U () 2 R k(p+) ; Z nk := (Z ; :::; Z n ) 2 R nk ; U () := (u (); :::; u n()) 2 R n(p+) ; and B() p I p A 2 R (p+)(p+) ; (4.5) where u i () := (u i(); u i () ) is de ned in (4.2). Note that (i) e V n () is an estimator of the variance matrix of the moment functions and their vectorized derivatives, (ii) e V n () exploits the functional form of the moment conditions given in (4.), (iii) e V n () typically is not of a Kronecker product form (because of the average over i = ; :::; n); and (iv) bu in () is the best linear predictor of u i () based on fz i : n g: The estimators e R n (); e Vn (); and e n () (de ned immediately below) are de ned so that the SR-CQLR P test, which employs them, is asymptotically equivalent to Moreira s (23) CLR test under all strengths of identi cation in the homoskedastic linear IV model with xed IV s and p rhs endogenous variables for any p ; see Section 8 for details. The SR-CQLR P test di ers from the SR-CQLR test because e V n () (and the statistics that depend on it) di ers from b V n () (and the statistics that depend on it). We de ne e n () 2 R (p+)(p+) just as b n () is de ned in (5.4) and (5.5), but with e R n () in place of b R n (): We de ne e D n() just as b D n() is de ned in (5.7), but with e n () in place of b n (): That is, ed n() := b n () =2 b Dn () e L =2 n () 2 R kp ; where e L n () := (; I p )( e " n()) (; I p ) : (4.6) The estimator e n () is an estimator of a matrix that could be singular or nearly singular in some cases. For example, in the homoskedastic linear IV model, see Section 8. below, e n () is an estimator of the variance matrix V of the reduced-form errors when is the true parameter, and V could be singular or nearly singular. In the de nition of L e n () above, we use an eigenvalue-adjusted version of e n (); denoted by e" n(); whose condition number (i.e., max ( b n ())= min ( b n ())) is bounded above by construction. Based on the nite-sample simulations, we recommend using " = :: The QLR P statistic without the SR extension, denoted by QLR P n (); is de ned just as QLR n () 4

17 is de ned in (5.7), but with e D n() in place of b D n(): For 2 (; ); the nominal size CQLR P test (without the SR extension) rejects H : = if QLR P n ( ) > c k;p (n =2 e D n ( ); ); (4.7) where c k;p (; ) is de ned in (5.8). The nominal size ( )% CQLR P CS is CS CQLRP ;n := f 2 : QLR P n ( ) c k;p (n =2 e D n ( ); )g: The CQLR P test statistic and critical value satisfy the following invariance properties. Lemma 4. The statistics QLR P n ; c k;p (n =2 e D n ; ); e D n e D n; AR n ; bu in ; e n ; and e L n are invariant to the transformation (Z i ; u i ) (MZ i; u i ) 8i n for any k k nonsingular matrix M: This transformation induces the following transformations: g i Mg i 8i n; G i MG i 8i n; bg n Mbg n ; b Gn M b G n ; b n M b n M ; b jn M b jnm 8j p; b Dn M b D n ; Z nk Z nk M ; e n M e n ; e V n (I p+ M) e V n (I p+ M ) ; and e R n (I p+ M) e R n (I p+ M ) : Comment: This Lemma is important because it implies that one can obtain the correct asymptotic size of the CQLR P test de ned above without assuming that min ( F ) is bounded away from zero. It su ces that F is nonsingular. The reason is that (in the proofs) one can transform the moments by g i M F g i ; where M F F MF = I k; such that the transformed moments have a variance matrix whose eigenvalues are bounded away from zero for some > (since V ar F (M F g i ) = I k ) even if the original moments g i do not. For the CQLR P test with the SR extension, we de ne D b An () as in (5.). We let Z Ai () := ba n () Z i 2 R brn() and Z Ank () := Z nkan b () 2 R nbrn() : We de ne nx ev An () := n (u i () bu Ain()) (u i () bu Ain()) Z Ai ()Z Ai () i= 2 R (p+)brn()(p+)brn() ; where bu Ain() := e An () Z Ai () 2 R p+ ; e An () := (Z Ank () Z Ank ()) Z Ank () U () 2 R brn()(p+) ; (4.8) and br n () and A b n () are de ned in (4.3) and (4.4), respectively. In addition, we de ne R e An (); e An (); L e An (); D e An (); and Q e An () as R b An (); b An (); L b An (); D b An (); and Q b An () are de ned, respectively, in (5.) and (5.2), but with V e An () in place of V b An () in the de nition of R e An (); with R e An () in place of R b An () in the de nition of e An (); and so on in the de nitions of L e An (); ed An (); and Q e An (): We de ne the test statistic SR-QLR P n () as SR-QLR n () is de ned in (5.2), but with Q e An () in place of Q b An (): 5

18 Given these de nitions, the nominal size SR-CQLR P test rejects H : = if SR-QLR P n ( ) > c brn( );p(n =2 e D An ( ); ) or b A? n ( ) bg n ( ) 6= k brn( ) : 3 (4.9) The nominal size ( )% SR-CQLR P CS is CS SR-CQLRP ;n := f 2 : SR-QLR P n ( ) c brn( );p(n =2 e D An ( ); ) and b A? n ( ) bg n ( ) = k brn( ) g: 32 Two simple examples where the extra rejection condition in (4.9) for the SR-CQLR P test (and in (4.7) and (5.3) for the SR-AR and SR-CQLR tests, respectively) improves the power of these tests are the following. First, suppose (X i ; X 2i ) i.i.d. N(; F ); where = ( ; 2 ) 2 R 2 ; F is a 2 2 matrix of ones, and the moment functions are g i () = (X i ; X 2i 2 ) : In this case, F is singular, b An ( ) = (; ) a.s., b A? n ( ) = (; ) a.s., the SR-AR statistic is a quadratic form in b A n ( ) bg n ( ) = X n + X 2n ( + 2 ); where X mn = n P n i= X mi for m = ; 2; and A? n ( ) bg n ( ) = X n X 2n ( 2 ) a.s. If one does not use the extra rejection condition, then the SR-AR test has no power against alternatives = ( ; 2 ) (6= ) for which + 2 = + 2 : The same is true for the SR-CQLR and SR-CQLR P tests (because the SR-QLR n and SR-QLR P n test statistics depend on the SR-AR n test statistic). However, when the extra rejection condition is utilized, all 2 R 2 except those on the line 2 = 2 are rejected with probability one (because X n X 2n = E F X i E F X 2i = 2 a.s.) and this includes all of the alternative values for which + 2 = + 2 : Second, suppose X i i.i.d. N( ; 2 ); = ( ; 2 ) 2 R 2 ; the moment functions are g i () = (X i ; X 2 i 2 2 ) ; and the null hypothesis is H : = ( ; 2 ) : Consider alternative parameters of the form = ( ; ) : Under ; X i has variance zero, X i = X n = a.s., X 2 i = X 2 n = 2 a.s., where X 2 n := n P n i= X2 i ; bg n( ) = ( ; ) a.s., b n ( ) = bg n ( )bg n ( ) bg n ( )bg n ( ) = 22 a.s. (provided b n ( ) is de ned as in (4.) with the sample means subtracted o ), and br n ( ) = a.s. In consequence, if one does not use the extra rejection condition, then the SR-AR, SR-CQLR, and SR-CQLR P tests have no power against alternatives of the form = ( ; ) (because, by de nition, the test statistics and critical values equal zero when br n ( ) = ): However, when the extra rejection condition is utilized, all alternatives of the form 3 By de nition, b A? n ( ) bg n( ) 6= k brn( ) does not hold if br n( ) = k: If br n( ) = ; then SR-QLR P n( ) := and 2 br n( ); := : In this case, b A? n ( ) = I k and the SR-CQLR P test rejects H if bg n( ) 6= k : 32 By de nition, if br n( ) = k; the condition b A? n ( ) bg n( ) = k brn( ) holds. 6

19 = ( ; ) are rejected with probability one. 33;34;35;36 When the sample variance matrix is singular, an alternative to using the SR-AR n ( ) statistic is to arbitrarily delete some moment conditions. However, this typically leads to di erent test results given the same data and can yield substantially di erent power properties of the test depending on which moment conditions are deleted, which is highly undesirable. The following simple example illustrates this. Suppose W i = (W i ; W 2i ; W 3i ) has a normal distribution with mean vector ( ; 2 ; 2 ) ; all variances are equal to one, the covariance between W i and W 2i equals one, (W i ; W 2i ) and W 3i are independent, g(w i ; ) = (W i ; W 2i 2 ; W 3i 2 ) ; and the null hypothesis is H : = for some = ( ; 2 ) 2 R 2 : The sample variance matrix is singular with probability one. A nonsingular sample variance matrix can be obtained by deleting the rst moment condition or the second. If the rst moment condition is deleted, the sample moments evaluated at are (W n2 2 ; W n3 2 ) : If the second moment condition is deleted, they are (W n ; W n3 2 ) : When and 2 2 are not equal (where and 2 denote the true values), these two sets of moment conditions are not the same. Furthermore, it is clear that the power of the two tests based on these two sets of moment conditions is quite di erent because the rst set of sample moments contains no information about ; whereas the second set does. 33 This holds because the extra rejection condition in this case leads one to reject H if X n 6= or Xn = ; which is equivalent a.s. to rejecting if 6= or = (because X n = a.s. and Xn 2 = 2 a.s. under ), which in turn is equivalent to rejecting if 6= (because if 2 > one or both of the two conditions is violated when 6= and if 2 = ; then 6= only if 6= since we are considering the case where 2 = ): 34 In this second example, suppose the null hypothesis is H : = ( ; ) : That is, 2 = : Then, the SR- AR test rejects with probability zero under H and the test is not asymptotically similar. This holds because bg n( ) = (X n ; Xn 2 2 ) = (; ) a.s., br n( ) = a.s., SR-AR n( ) = 2 br n( ); = a.s. (because brn() = a.s.), and the extra rejection condition leads one to reject H if X n 6= or Xn = ; which is equivalent to 6= or = (because X i = a.s.), which holds with probability zero. As shown in Theorem 6., the SR-AR test is asymptotically similar (in a uniform sense) if one excludes null distributions F for which the g i( ) = k a.s. under F; such as in the present example, from the parameter space of null distributions. But, the SR-AR test still has correct asymptotic size without such exclusions. 35 We thank Kirill Evdokimov for bringing these two examples to our attention. 36 An alternative de nition of the SR-AR test is obtained by altering its de nition given in Section 4 as follows. One omits the extra rejection condition given in (4.7), one de nes the SR-AR statistic using a weight matrix that is nonsingular by construction when b n( ) is singular, and one determines the critical value by simulation of the appropriate quadratic form in mean zero normal variates when b n( ) is singular. For example, such a weight matrix can be constructed by adjusting the eigenvalues of b n( ) to be bounded away from zero, and using its inverse. However, this method has two drawbacks. First, it sacri ces power relative to the de nition of the SR-AR test in (4.7). The reason is that it does not reject H with probability one when a violation of the nonstochastic part of the moment conditions occurs. This can be seen in the example with identities in Section 4 and the two examples given here. Second, it cannot be used with the SR-CQLR and SR-CQLR 2 tests introduced in Sections 5 and 4. The reason is that these tests rely on the statistic D b n( ); de ned in (5.2), that employs b n ( ) and if b n ( ) is replaced by a matrix that is nonsingular by construction, such as the eigenvalue-adjusted matrix suggested above, then one does not obtain asymptotic independence of bg n( ) and D b n( ) after suitable normalization, which is needed to obtain the correct asymptotic size of the SR-CQLR and SR-CQLR 2 tests. 7

20 4.3 Asymptotic Size of the SR-CQLR P Test The correct asymptotic size and similarity results for the SR-CQLR P test are as follows. Theorem 4.2 The asymptotic size of the SR-CQLR P test de ned in (4.9) equals its nominal size 2 (; ) for the null parameter spaces FP SR : Furthermore, this test is asymptotically similar (in a uniform sense) for the subset of this parameter space that excludes distributions F under which g i = k a.s. Analogous results hold for the corresponding SR-CQLR P CS for the parameter space F;P SR ; de ned in (4.4). Comments: (i) For distributions F under which g i = k a.s., the SR-CQLR P test rejects the null hypothesis with probability zero when the null is true. Hence, asymptotic similarity only holds when these distributions are excluded from the null parameter spaces. (ii) The proof of Theorem 4.2 is given in Sections 5, 6, and below. 4.4 Asymptotic E ciency of the SR-CQLR P Test under Strong Identi cation Here we show that the SR-CQLR P test is asymptotically e cient in a GMM sense under strong and semi-strong identi cation (when the variance matrix of the moments is nonsingular and the null parameter value is not on the boundary of the parameter space). Suppose k p: Let A F and F be de ned as in (3.4) and (3.5) and the paragraph following these equations with = : De ne F ; P ; and f n;h : n g as F ; W U;P ; and f n;h : n g; respectively, are de ned in (5.6)-(5.8), but with g i and G i replaced by gf =2 i := F A F g i and G =2 F i := F A F G i; with F P replaced by FP SR in the de nition of F W U ; and with W F (:= W (W 2F )) and U F (:= U (U 2F )) de ned as in (5.) with g i and G i replaced by gf i and G F i : In addition, we restrict f n;h : n g to be a sequence for which min(e Fn g i gi ) > for all n : By de nition, a sequence f n;h : n g is said to exhibit strong or semi-strong identi cation if n=2 s pf n! ; where s pf denotes the smallest singular value of E F G F i :37 The LM n and LMn GMM statistics are de ned in (7.). Let 2 p; denote the quantile of the 2 p distribution. The critical value for the LM n and LM GMM n tests is 2 p; : Theorem 4.3 Suppose k p: For any sequence f n;h : n g that exhibits strong or semi-strong identi cation (i.e., for which n =2 s pf n! ) and for which n;h 2 P 8n ; we have 37 The singular value s pf ; de ned here, equals s pf ; de ned in Section 6.2, for all F with min( F ) > ; because in this case F = A F F A F ; =2 F = A F =2 F A F ; =2 F E F G i = A F =2 F A F E F G i = A F E F G F i; and A F is an orthogonal k k matrix. Since we consider sequences here with min( Fn ) = min(e Fn g igi) > for all n ; the de nitions of strong and semi-strong identi cation used here and in Section 6.2 are equivalent. 8

21 (a) SR-QLR P n = QLR P n + o p () = LM n + o p () = LMn GMM + o p () and (b) c k;p (n =2 e D n ; )! p 2 p; : Comments: (i) Theorem 4.3 establishes the asymptotic e ciency (in a GMM sense) of the SR-CQLR P test under strong and semi-strong identi cation. Theorem 4.3 provides asymptotic equivalence results under the null hypothesis, but, by the de nition of contiguity, these asymptotic equivalence results also hold under contiguous local alternatives. (ii) The proof of Theorem 4.3 is given in Section Summary Comparison of CLR-type Tests in Kleibergen (25) and AG2 We brie y summarize some of the results in AG and AG2 concerning Kleibergen s (25) moment-variance-weighted CLR (MVW-CLR) and Jacobian-variance-weighted CLR (JVW-CLR) tests, the SR-CQLR test in AG2, and the SR-CQLR P test introduced above. (i) The MVW-CLR test has correct asymptotic size for all p (for the parameter space in AG, which imposes non-singularity of the variance matrix and some other conditions). (ii) The JVW-CLR test has correct asymptotic size for p = (under similar conditions to the MVW-CLR test). (iii) For p 2; AG provides an expression for the asymptotic size of the JWV-CLR test that depends on a vector of localization parameters. It is unknown whether the asymptotic size exceeds the nominal size. (iv) The MVW-CLR test is not asymptotically equivalent to Moreira s (23) CLR test in the homoskedastic linear IV (HLIV) model for any p : (v) The JVW-CLR test is asymptotically equivalent to Moreira s (23) CLR test in the HLIV model for p = ; but not for p 2: (vi) The SR-CQLR test has correct asymptotic size for the parameter space F SR in Section 3.2, which is larger than the parameter space in (i) and (ii). (vii) The SR-CQLR P test has correct asymptotic size for the parameter space F SR P ( F SR ). (viii) The SR-CQLR test is asymptotically equivalent to Moreira s (23) CLR test in the HLIV model for p = ; but not for p 2; although the di erence for p 2 is only due to the di erence between treating the IV s as random, rather than xed. (ix) The SR-CQLR P test is asymptotically equivalent to Moreira s (23) CLR test in the HLIV model for all p : 5 Tests without the Singularity-Robust Extension The next two sections and Sections below are devoted to the proof of Theorems 6. and 4.2. The proof proceeds in two steps. First, in this section, we establish the correct asymptotic size and asymptotic similarity of the tests and CS s without the SR extension for parameter spaces of distributions that bound min ( F ) away from zero. (These tests are de ned in (4.2), (5.9), and 9

22 (4.7).) We provide parts of the proof of this result in this section and other parts in Sections below. Second, we extend the proof to the case of the SR tests and CS s. We provide the proof of this extension in Section 6 below. 5. Asymptotic Results for Tests without the SR Extension For the AR, CQLR, and CQLR P tests without the SR extension, we consider the following parameter spaces for the distribution F that generates the data under H : = : F AR := ff : E F g i = k ; E F jjg i jj 2+ M; and min (E F g i g i) g; F := ff 2 F AR : E F jjvec(g i )jj 2+ Mg; and F P := ff 2 F : E F jjz i jj 4+ M; E F jju i jj 2+ M; min (E F Z i Z i) g (5.) for some ; > and M < : By de nition, F P F F AR : The parameter spaces F AR ; F; and F P are used for the AR, CQLR, and CQLR P tests, respectively. For the corresponding CS s, we use the parameter spaces: F ;AR := f(f; ) : F 2 F AR ( ); 2 g; F := f(f; ) : F 2 F( ); 2 g; and F ;P := f(f; ) : F 2 F P ( ); 2 g; where F AR ( ); F( ); and F P ( ) equal F AR ; F; and F P ; respectively, with their dependence on made explicit. Theorem 5. The AR, CQLR; and CQLR P tests (without the SR extensions), de ned in (4.2), (5.9), and (4.7), respectively, have asymptotic size equal to their nominal size 2 (; ) and are asymptotically similar (in a uniform sense) for the parameter spaces F AR ; F; and F P ; respectively. Analogous results hold for the corresponding AR; CQLR; and CQLR P CS s for the parameter spaces F ;AR ; F ; and F ;P ; respectively. Comments: (i) The rst step of the proof of Theorems 6. and 4.2 is to prove Theorem 5.. (ii) Theorem 5. holds for both k p and k < p: Both cases are needed in the proof of Theorems 6. and 4.2 (even if k p in Theorems 6. and 4.2). (iii) In Theorem 5., as in Theorems 6. and 4.2, we assume that the parameter space being considered is non-empty. (iv) The results of Theorem 6. still hold if the moment bounds in F AR ; F; and F P are weakened very slightly by, e.g., replacing E F jjg i jj 2+ M in F AR by E F jjg i jj 2 (jjg i jj > j) " j for all integers j for some " j > (that does not depend on F ) for which " j! as j! : The latter conditions are weaker because, for any random variable X and constants ; j > ; EX 2 (jx _ j > j) EjXj 2+ =j : The latter conditions allow for the application of Lindeberg s triangular array central limit theorem for independent random variables, e.g., see Billingsley (979, 2

23 Thm. 27.2, p. 3), in scenarios where the distribution F depends on n: For simplicity, we de ne the parameter spaces as is. Analogously, the results in Theorems 6. and 4.2 still hold if the moment bounds in F SR AR ; F SR ; and FP SR are weakened very slightly by, e.g., replacing E F jj =2 F A F g ijj 2+ M in FAR SR by E F jj =2 F A F g ijj 2 (jj =2 A F g ijj > j) " j for all integers j for some " j > (that does not depend on F ) for which " j! as j! : F The following lemma shows that the critical value function c k;p (D; through its singular values. ) depends on D only Lemma 5.2 Let D be a k p matrix with the singular value decomposition D = CB ; where C is a k k orthogonal matrix of eigenvectors of DD ; B is a p p orthogonal matrix of eigenvectors of D D; and is the k p matrix with the minfk; pg singular values f j : j minfk; pgg of D as its rst minfk; pg diagonal elements and zeros elsewhere, where j is nonincreasing in j: Then, c k;p (D; ) = c k;p (; ): Comment: A consequence of Lemma 5.2 is that the critical value c k;p (n =2 D b n ( ); ) of the CQLR test depends on D b n( ) only through D b n( ) D b n ( ) (because, when k p; the p singular values of n =2 D b n ( ) equal the square roots of the eigenvalues of nd b n( ) D b n ( ) and, when k < p; c k;p (D; ) is the quantile of the 2 k distribution which does not depend on D): 5.2 Uniformity Framework The proofs of Theorems 6., 4.2, and 5. use Corollary 2.(c) in Andrews, Cheng, and Guggenberger (29) (ACG), which provides general su cient conditions for the correct asymptotic size and (uniform) asymptotic similarity of a sequence of tests. Now we state Corollary 2.(c) of ACG. Let f n : n g be a sequence of tests of some null hypothesis whose null distributions are indexed by a parameter with parameter space : Let RP n () denote the null rejection probability of n under : For a nite nonnegative integer J; let fh n () = (h n (); :::; h Jn ()) 2 R J : n g be a sequence of functions on : De ne H := fh 2 (R [ fg) J : h wn ( wn )! h for some subsequence fw n g of fng and some sequence f wn 2 : n gg: (5.2) Assumption B : For any subsequence fw n g of fng and any sequence f wn 2 : n g for which h wn ( wn )! h 2 H; RP wn ( wn )! for some 2 (; ): 2

24 Proposition 5.3 (ACG, Corollary 2.(c)) Under Assumption B ; the tests f n : n g have asymptotic size and are asymptotically similar (in a uniform sense). That is, AsySz := lim sup sup 2 RP n () = and lim inf inf 2 RP n () = lim sup sup n! 2 RP n (): Comments: (i) By Comment 4 to Theorem 2. of ACG, Proposition 5.3 provides asymptotic size and similarity results for nominal CS s, rather than tests, by de ning as one would for a test, but having it depend also on the parameter that is restricted by the null hypothesis, by enlarging the parameter space correspondingly (so it includes all possible values of the parameter that is restricted by the null hypothesis), and by replacing (a) n by a CS based on a sample of size n; (b) by ; (c) RP n () by CP n (); where CP n () denotes the coverage probability of the CS under when the sample size is n; and (d) the rst lim sup n! sup 2 that appears by lim inf n! inf 2 : In the present case, where the null hypotheses are of the form H : = for some 2 ; to establish the asymptotic size of CS s, the parameter is taken to be a subvector of and is speci ed so that the value of this subvector ranges over : (ii) In the application of Proposition 5.3 to prove Theorems 6., 4.2, and 5., one takes to be a one-to-one transformation of F AR ; F; or F P for tests, and one takes to be a one-to-one transformation of F ;AR ; F ; or F ;P for CS s. With these changes, the proofs for tests and CS s are the same. In consequence, we provide explicit proofs for tests only and obtain the proofs for CS s by analogous applications of Proposition 5.3. (iii) We prove the test results in Theorems 5. and 4.2 using Proposition 5.3 by verifying Assumption B for a suitable choice of ; h n (); and : The veri cation of Assumption B is quite easy for the AR test. It is given in Section The veri cations of Assumption B for the CQLR and CQLR P tests are much more di cult. In the remainder of this Section 5, we provide some key results that are used in doing so. (These results are used only for the CQLR and CQLR P tests, not the AR test.) The complete veri cations for the CQLR and CQLR P tests are given in Section 26. n! n! 5.3 General Weight Matrices W c n and U b n As above, for notational simplicity, we suppress the dependence on of many quantities, such as g i ; G i ; u i ; B; and f i ; as well as the quantities V F ; R F ; F ; V e F ; and R e F ; that are introduced below. To provide asymptotic results for the CQLR and CQLR P tests simultaneously, we prove asymptotic results for a QLR test statistic and a conditioning statistic that depend on general random weight matrices W c n 2 R kk and U b n 2 R pp : In particular, we consider statistics of the 22