Bonferroni-Based Size-Correction for Nonstandard Testing Problems

Transcription

1 Bonferroni-Based Size-Correction for Nonstandard Testing Problems Adam McCloskey Brown University October 2011; Tis Version: October 2012 Abstract We develop powerful new size-correction procedures for nonstandard ypotesis testing environments in wic te asymptotic distribution of a test statistic is discontinuous in a parameter under te null ypotesis. Examples of tis form of testing problem are pervasive in econometrics and complicate inference by making size difficult to control. Tis paper introduces two sets of new size-correction metods tat correspond to two different general ypotesis testing frameworks. Te new metods are designed to maximize te power of te underlying test wile maintaining correct asymptotic size uniformly over te parameter space specified by te null ypotesis. Tey involve te construction of critical values tat make use of reasoning derived from Bonferroni bounds. Te first set of new metods provides complementary alternatives to existing size-correction metods, entailing substantially iger power for many testing problems. Te second set of new metods provides te first available asymptotically size-correct tests for te general class of testing problems to wic it applies. Tis class includes ypotesis tests on parameters after consistent model selection and tests on super-efficient/ard-tresolding estimators. We detail te construction and performance of te new tests in tree specific examples: testing after conservative model selection, testing wen a nuisance parameter may be on a boundary and testing after consistent model selection. Keywords: Hypotesis testing, uniform inference, asymptotic size, exact size, power, size-correction, model selection, boundary problems, local asymptotics A previous version of tis paper was circulated under te title Powerful Procedures wit Correct Size for Tests Statistics wit Limit Distributions tat are Discontinuous in Some Parameters. Te autor tanks Donald Andrews, Federico Bugni, Xu Ceng, Kirill Evdokimov, Iván Fernández-Val, Patrik Guggenberger, Hiroaki Kaido, Frank Kleibergen, Hannes Leeb, Blaise Melly, Ulric Müller, Serena Ng, Pierre Perron, Benedikt Pötscer, Zongjun Qu, Eric Renault and Josep Romano for elpful comments. I am especially grateful to Donald Andrews for suggesting a solution to a mistake in a previous draft. I also wis to tank Bruce Hansen for saring some preliminary results. Department of Economics, Brown University, Box B, 64 Waterman St., Providence, RI, (adam [email protected], ttp:// mccloskey/home.tml).

2 1 Introduction Nonstandard econometric testing problems ave gained substantial attention in recent years. In tis paper, we focus on a very broad class of tese problems: tose for wic te null limit distribution of a test statistic is discontinuous in a parameter. Te problems falling into tis class range from tests in te potential presence of identification failure (e.g., Staiger and Stock, 1997 and Andrews and Ceng, 2012) to tests after pretesting or model selection (e.g., Leeb and Pötscer, 2005 and Guggenberger, 2010) to tests wen a parameter may lie on te boundary of its parameter space (e.g., Andrews, 1999, 2001). Toug test statistics tat do not exibit tis type of discontinuity exist for some problems (e.g., Kleibergen, 2002), tey do not for oters. Moreover, suc test statistics may not necessarily be preferred to parameter-discontinuous ones wen good size-correction procedures are available, as tey can ave low power. However, we sidestep te important issue of coosing a test statistic in tis paper, taking it as given. Te usual approximation to te size of a test is te asymptotic probability of rejecting a true null ypotesis (null rejection probability) at a fixed parameter value. For te types of problems studied in tis paper, suc an approximation is grossly misleading. In fact, te discrepancy between tis point-wise null rejection probability (NRP) and te (asymptotic) size of a test can reac unity less te nominal level. Tis problem does not disappear, and often worsens, as te sample size grows. Te inadequacy of point-wise asymptotic approximations and te resulting pitfalls for inference ave been studied extensively in te literature. See, for example, Dufour (1997) in te context of inference on te two-stage least squares estimator, Leeb and Pötscer (2005) in te context of inference after model selection, Stoye (2009) in te context of inference on partially identified parameters and Andrews and Guggenberger (2009a) (AG encefort) in te context of inference on te autoregressive parameter in a first-order autoregressive model. In te parameter-discontinuous testing framework of tis paper, one must examine te maximal NRP uniformly, over te entire parameter space, as te sample size grows in order to determine te asymptotic size of te test (see e.g., te work of Mikuseva, 2007 and Andrews and Guggenberger, 2009b, 2010b). Wen using te test statistics considered in tis paper, one typically takes a conservative approac to control size, leading to igly non-similar tests, i.e., tests for wic te pointwise NRP differs substantially across parameter values. Tis often results in very poor power. In tis paper, we develop novel size-correction metods wit te goal of minimizing te 1

3 degree of conservativeness of te test, and ence maximizing its power, wile maintaining correct asymptotic size. We do so under two different frameworks tat allow for different null limiting beavior of a given test statistic. For te first, termed te single localized limit distribution (SLLD) framework, we adopt te framework studied by AG as it is quite broad in scope. For te second, termed te multiple localized limit distributions (MLLDs) framework, we generalize te SLLD framework in order to accommodate certain complicated asymptotic beaviors of test statistics. To our knowledge, tis latter, more general framework as not yet been studied. It includes examples of testing after consistent model selection and testing on super-efficient/ard-tresolding estimators. Te basic idea beind te size-corrections we introduce is to adaptively learn from te data ow far te true parameters are from te point tat causes te discontinuity in te asymptotic beavior of te test statistic in order to construct critical values (CVs) tat control te size of te test but are not overly conservative. We do tis under a drifting sequence framework by embedding te true parameter values in a sequence indexed by te sample size and a localization parameter. Witin tis framework, we estimate a corresponding localization parameter to find a set of drifting sequences of parameters relevant to te testing problem at and. We ten examine te CVs corresponding to te null limiting quantiles of te test statistic tat obtain under te drifting sequences witin tis set. Toug te localization parameter cannot be consistently estimated under tese drifting sequences, it is often possible to obtain estimators tat are asymptotically centered about teir true values and ence to construct asymptotically valid confidence sets for te true localization parameter. Based upon tis estimator and corresponding confidence sets, we examine tree different size-correction metods in increasing order of computational complexity. For te first, we searc for te maximal CV over a confidence set, rater tan te maximal CV over te entire space of localization parameters, to reduce te degree of conservativeness of a given ypotesis test. Inerent in tis construction are two levels of uncertainty: one for te localization parameter and one for te test statistic itself. We use procedures based on Bonferroni bounds to account for bot. For te second, we also searc for a maximal CV over a confidence set but, instead of using Bonferroni bounds, we account for te two levels of uncertainty by adjusting CV levels according to te asymptotic distributions tat arise under drifting parameter sequences. Tis metod compensates for te asymptotic dependence structure between te test statistic and te CV, leading to more powerful tests. For te tird, we find te smallest CVs over sets of tose justified by te first and second, 2

4 leading to tests wit ig power over most of te parameter space. For testing problems witin te scope of te SLLD framework, our new size-correction metods can be constructed to be eiter uniformly more powerful asymptotically tan existing least-favorable (LF) metods or are more powerful over most of te relevant parameter space. In te latter case, te portions of te parameter space for wic LF metods dominate tend to be very small and suc dominance tends to be nearly undetectable even witin tese portions. Te finite-sample power dominance of our new metods can be very pronounced, sometimes reacing nearly 100% over most of te parameter space. Our size-corrections can also be constructed to direct power toward different regions of te parameter space wile sacrificing very little in oters. We also develop te first size-correction procedures we are aware of to provide tests wit correct asymptotic size for all testing problems falling witin te MLLDs framework. Since tey are adapted from te size-correction procedures used in te SLLD framework to tis generalized framework, tey also adopt many desirable power properties. Te scope of problems to wic our size-correction metods may be applied is quite wide. For illustration, we provide detailed applications to tree nonstandard testing problems. Two of tese examples concern testing after model selection/pretesting. Te first, taken from AG, involves testing after conservative model selection and falls witin te scope of te SLLD framework. Te second, taken from Leeb and Pötscer (2005), considers testing after consistent model selection and falls witin te scope of te MLLDs framework. Te oter example we detail concerns testing wen a nuisance parameter may be on a boundary of its parameter space, taken from Andrews and Guggenberger (2010b). We also briefly discuss a subset of te numerous oter examples for wic our size-corrections can be used. We focus on testing after model selection in te examples because at present, available uniformly valid inference metods tend to be extremely conservative. Inference after model selection is an important issue tat is all too frequently ignored, sometimes being referred to as te quiet scandal of statistics. See, for example, Hansen (2005a) for a discussion of te importance of tis issue. Moreover, excluding te results of AG and Kabaila (1998), te literature as been quite negative wit regards to solving tis inference problem (e.g., Andrews and Guggenberger, 2009b, Leeb and Pötscer, 2005, 2006 and 2008), especially wit regards to inference after consistent model selection. Te recently developed metods for uniform inference of AG are closely related to te metods developed ere. AG also study a given test statistic and adjust CVs according to a drifting parameter sequence framework. However, our metods tend to be (oftentimes, 3

5 muc) less conservative tan teirs. Existing inference procedures tat make use of Bonferroni bounds are also related to tose developed ere. Some of te CVs we employ can also be interpreted as smooted versions of tose based on binary decision rules tat use an inconsistent estimator of te localization parameter. Our CVs are also related to tose tat use a transition function approac to interpolate between LF and standard CVs. In contrast to tese approaces, ours do not necessitate an ad oc coice of transition function. Rater, tey use te data and te limiting beavior of te test statistic to adaptively transition between CVs. See Sections 3.1 and 3.3 for details and references on tese related procedures. Finally, te recent work of Elliott et al. (2012) takes a somewat different approac to some of te problems discussed ere by attempting to numerically determine tests tat approac an asymptotic power bound. Te remainder of tis paper is composed as follows. Section 2 describes te general class of nonstandard ypotesis testing problems we study, subsequently detailing te SLLD and MLLDs frameworks and providing examples. Section 3 goes on to specify te size-correction metods of tis paper under te two localized limit distribution frameworks and provides te conditions under wic some of tese size-corrections yield correct asymptotic size. To conserve space, some of te conditions used to sow size-correctness of procedures in te MLLDs framework are relegated to Appendix III, wic also contains some auxiliary sufficient conditions. Specifics on ow to construct some of te size-corrected CVs are provided for tree econometric examples in Sections 4, 5 and 6. Te finite sample performance of two of te examples, corresponding to testing after model selection, is also analyzed tere. Section 7 concludes. Appendix I contains proofs of te main results of tis paper wile Appendix II is composed of derivations used to sow ow te example testing problems fit te assumptions of te paper. All tables and figures can be found at te end of te document. To simplify notation, we will occasionally abuse it by letting (a 1, a 2 ) denote te vector (a 1, a 2). Te sample size is denoted by n and all limits are taken to mean as n. Let R + = {x R : x 0}, R = {x R : x 0}, R, = R { }, R +, = R + { } and R = R {, }. 1( ) denotes te indicator function. Φ( ) and φ( ) are te usual notation for te distribution and density functions of te standard normal distribution. d and p denote weak convergence and convergence in probability wile O( ), o( ), O p ( ) and o p ( ) denote te usual (stocasitc) orders of magnitude. 4

6 2 Parameter-Discontinuous Asymptotic Distributions In tis paper, we are interested in performing ypotesis tests wen te asymptotic distribution of te test statistic is discontinuous in a parameter under te null ypotesis. We take te test statistic as given and examine te tasks of controlling size and maximizing power for te given statistic. Te important separate issue of coosing a test statistic depends on te specific testing problem at and and is not te focus of tis paper. In order to analyze tis problem, we adopt te same general testing framework as AG. Consider some generic test statistic T n (θ 0 ) used for testing H 0 : θ = θ 0 for some finite-dimensional parameter θ R d. Under H 0, T n (θ 0 ) and its asymptotic distribution depend on some parameter γ Γ. Refer to tis limit distribution as F γ. We decompose γ into tree components, viz. γ = (γ 1, γ 2, γ 3 ), depending on ow eac component affects F γ as follows. Te distribution F γ is discontinuous in γ 1, a parameter in Γ 1 R p, wen one or more of te elements of γ 1 is equal to zero. It also depends on γ 2, a parameter in Γ 2 R q, but γ 2 does not affect te distance of γ to te point of discontinuity in F γ. Te tird component γ 3 may be finite- or infinite-dimensional, lying in some general parameter space Γ 3 (γ 1, γ 2 ) tat may depend on γ 1 and γ 2. Te component γ 3 does not affect te limit distribution F γ but may affect te properties of T n (θ 0 ) in finite samples. Formally, te parameter space for γ is given by Γ = {(γ 1, γ 2, γ 3 ) : γ 1 Γ 1, γ 2 Γ 2, γ 3 Γ 3 (γ 1, γ 2 )}. (1) To complete te preliminary setup, we impose te following product space assumption on Γ 1. Tis assumption is identical to Assumption A of AG. Let ( ) denote te left (rigt) endpoint of an interval tat may be open or closed. Assumption D. (i) Γ satisfies (1) and (ii) Γ 1 = p m=1 Γ 1,m, were Γ 1,m = γ l 1,m, γ u 1,m for some γ l 1,m < γ u 1,m tat satisfy γ l 1,m 0 γ u 1,m for m = 1,..., p. Tis paper introduces testing metods tat are asymptotically size-controlled. Asymptotic size control requires one to asymptotically bound te NRP uniformly over te parameter space admissible under te null ypotesis. In order to assess te uniform limiting beavior of a test, one must examine its beavior along drifting sequences of parameters (see e.g., Andrews and Guggenberger, 2010b and Andrews et al., 2011). In tis vein, we allow γ to depend on te sample size, and empasize tis dependence by denoting it γ n, = (γ n,,1, γ n,,2, γ n,,3 ), were = ( 1, 2 ) H H 1 H 2 is a localization parameter tat describes te limiting 5

7 beavior of te sequence. Te sets H 1 and H 2 depend on Γ 1 and Γ 2 as follows: R p +,, if γ1,m l = 0, H 1 = R,, if γ1,m u = 0, m=1 R, if γ1,m l < 0 and γ1,m u > 0, H 2 = cl(γ 2 ), were cl(γ 2 ) is te closure of Γ 2 wit respect to R q. Given r > 0 and H, define {γ n, } as te sequence of parameters in Γ for wic n r γ n,,1 1 and γ n,,2 2. In tis paper, we consider two broad classes of testing problems: one for wic te limiting beavior of te test statistic is fully caracterized by under any drifting sequence of parameters {γ n, } and te oter for wic tis limiting beavior depends upon bot and te limiting beavior of {γ n,,1 } relative to anoter sequence. 2.1 Single Localized Limit Distribution Framework We begin te analysis wit te simpler of te two cases just described. Tis class of testing problems can be broadly caracterized by Assumption D and te following assumption. Assumption S-B.1. Tere exists a single fixed r > 0 suc tat for all H and corresponding sequences {γ n, }, T n (θ 0 ) W under H 0 and {γ n, d }. Denote te limit distribution function for a given as J, i.e., P (W x) = J (x) and te (1 α) t quantile of W by c (1 α). We refer to J as a localized limit distribution as it obtains under a drifting sequence of parameters indexed by te localization parameter. Assumption S-B.1 is identical to Assumption B of AG. For every sequence of parameters {γ n, } indexed by te same localization parameter, te same limit distribution J obtains, ence te term single in SLLD. We now introduce some new assumptions, noting tat tey are applicable to most of te same econometric applications tat satisfy te assumptions imposed by AG. Assumption S-B.2. Consider some fixed δ (0, 1). (i) As a function in from H into R, c (1 δ) is continuous. (ii) For any H, J ( ) is continuous at c (1 δ). Assumption S-B.2 is a mild continuity assumption. strengten part (i) as follows. To obtain stronger results, we 6

8 Assumption S-BM.1. For some fixed α (0, 1) and pair (δ, δ) [0, α δ] [0, α δ ], as a function of and δ, c (1 δ) is continuous over H and [δ, α δ]. Te quantity δ serves as a lower bound and α δ serves as an upper bound on te points δ for wic c (1 δ) must be continuous. In many examples of interest, W is a continuous random variable wit infinite support so tat c (1) =. For suc examples δ can be set arbitrarily close, but not equal, to zero. Assumptions D, S-B.1, S-B.2 and S-BM.1 and te oter assumptions for tis framework introduced later in Sections 3.1 troug 3.3 old in many nonstandard econometric testing problems of interest. Te following are simple, illustrative examples of suc problems Testing After Conservative Model Selection Various forms of ypotesis tests after model selection exemplify testing problems wit parameter-discontinuous null limit distributions. Conducting a t-test on a parameter of interest after conservative model selection falls witin te framework of Section 2.1, aving a SLLD. Conservative model selection includes, among oters, metods based on te Akaike information criterion (AIC) and standard pre-testing tecniques. As an illustrative example, consider te following problem described by AG. We ave a model given by y i = x 1iθ + x 2iβ 2 + x 3iβ 3 + σε i, (2) for i = 1,..., n, were x i (x 1i, x 2i, x 3i) R k, β (θ, β 2, β 3) R k, x 1i, x 2i, θ, β 2, σ, ε i R, x 3i, β 3 R k 2, te observations {(y i, x i )} are i.i.d. and ε i as mean zero and unit variance conditional on x i. We are interested in testing H 0 : θ = θ 0 after determining weter to include x 2i in te regression model (2), tat is, after determining weter to impose te restriction β 2 = 0. Tis decision is based on weter te absolute value of te pretest t-statistic n 1/2 ˆβ2 T n,2 ˆσ(n 1 X2 M [X 1 :X3 ] X2) 1/2 exceeds a pretest CV c > 0, were c is fixed (i.e., does not depend on n), ˆβ 2 is te standard unrestricted OLS estimator of β 2 in te regression (2) and ˆσ 2 (n k) 1 Y M [X 1 :X 2 :X 3 ] Y wit Y (y 1,..., y n ), Xj [x j1 :... : x jn] for j = 1, 2, 3 and M A I A(A A) 1 A for some generic full-rank matrix A and conformable identity matrix I. Te model selection pretest rejects β 2 = 0 wen T n,2 > c and te subsequent t-statistic for testing H 0 is based 7

9 on te unrestricted version of (2) and is given by ˆT n,1 (θ 0 ) n 1/2 (ˆθ θ 0 ) ˆσ(n 1 X 1 M [X 2 :X 3 ] X 1) 1/2, were ˆθ is te unrestricted OLS estiamtor from regression (2). Conversely, te model selection pretest selects te model witout x 2i, or equivalently restricts β 2 = 0, wen T n,2 c and te resulting t-statistic for H 0 is given by T n,1 (θ 0 ) = n 1/2 ( θ θ 0 ) ˆσ(n 1 X 1 M X 3 X 1) 1/2, were θ is te restricted OLS estimator from regression (2) wit β 2 restricted to equal zero. Hence, for a two-sided test, te post-conservative model selection test statistic for testing H 0 is given by T n (θ 0 ) = T n,1 (θ 0 ) 1( T n,2 c) + ˆT n,1 (θ 0 ) 1( T n,2 > c). Wit straigtforward modification, te results described below also apply to one-sided testing for tis problem. See AG and Andrews and Guggenberger (2009c) for more details. 1 Results in AG sow ow tis testing problem satisfies Assumptions D and S-B.1. Specifically, let G denote te distribution of (ε i, x i ) and define te following quantities x i = x 1i x 3i(E G x 3ix 3i) 1 x 3ix 1i x 2i x 3i(E G x 3ix 3i) 1 x 3ix 2i Q = E G x i x i, and Q 1 = Q11 Q 12. Q 12 Q 22 Ten for tis example, γ 1 = β 2 /σ(q 22 ) 1/2, γ 2 = Q 12 /(Q 11 Q 22 ) 1/2 and γ 3 = (β 2, β 3, σ, G) and d Z 1 1( Z 2 c) + Ẑ1 1( Z 2 > c), if γ 1 = 0 T n (θ 0 ) Ẑ1, if γ 1 0, were Z 1, Ẑ1 and Z 2 are standard normal random variables wit Z 1 independent of Z 2 and Corr(Ẑ1, Z 2 ) = γ 2. Te parameter spaces in (1) are given by Γ 1 = R, Γ 2 = [ 1 + ω, 1 ω] for some ω > 0 and Γ 3 (γ 1, γ 2 ) = {(β 2, β 3, σ, G) : β 2 R, β 3 R k 2, σ (0, ), γ 1 = β 2 /σ(q 22 ) 1/2, 1 Te testing problem described ere also applies to testing a linear combination of regression coefficients after conservative model selection. Tis involves a reparameterization described in Andrews and Guggenberger (2009c). 8

10 γ 2 = Q 12 /(Q 11 Q 22 ) 1/2, λ min (Q) κ, λ min (E G x 3ix 3i) κ, E G x i 2+δ M, E G ε i x i 2+δ M, E G (ε i x i ) = 0 a.s., E G (ε 2 i x i ) = 1 a.s.} for some κ, δ > 0 and M <, were λ min (A) is te smallest eigenvalue of generic matrix A. From tis parameter space, it is clear tat Assumption D olds for tis example. Moreover, Andrews and Guggenberger (2009c) sow tat Assumption S-B.1 olds wit r = 1/2 and x ( ( )) J (x) = ( 1 2 (1 2 2) 1/ t, x) ( 1, c) + 1 (1 2 2), c φ(t)dt, 1/2 (1 2 2) 1/2 x were (a, b) = Φ(a+b) Φ(a b). As defined, H 1 = R and H 2 = [ 1+ω, 1 ω]. Turning to te new assumptions, te lower bound δ of Assumption S-BM.1 can be set arbitrarily close to zero and δ can be set to any quantity in its admissible range [0, α δ ] since W is a continuous random variable wit support over te entire real line for any H, wic can be seen by examining J ( ). Tis fact similarly implies Assumption S-B.2(ii) olds. An assumption imposed later will necessitate a restriction on δ tat we will discuss in Section 4. Continuity of c (1 δ) in follows from te facts tat c (1 δ) = J 1 (1 δ) and J (x) is continuous in H. Te SLLD framework applies to many more complex examples of testing after conservative model selection. For example, results in Leeb (2006) and Leeb and Pötscer (2008) can be used to verify te assumptions of tis paper for a sequential general-to-specific model selection procedure wit multiple potential control variables Testing wen a Nuisance Parameter may be on a Boundary We now explore an illustrative example of a testing problem in wic a nuisance parameter may be on te boundary of its parameter space under te null ypotesis and sow ow it also falls witin te framework of Section 2.1. Tis problem is considered by Andrews and Guggenberger (2010b) and can be described as follows. We ave a sample of size n of an i.i.d. bivariate random vector X i = (X i1, X i2 ) wit distribution F. Under F, te first two moments of X i exist and are given by E F (X i ) = θ and Var F (X i ) = µ σ2 1 σ 1 σ 2 ρ σ 1 σ 2 ρ σ2 2 Say we are interested in te null ypotesis H 0 : θ = θ 0 and we know tat µ 0. Now suppose we use te Gaussian quasi-maximum likeliood estimator of (θ, µ, σ 1, σ 2, ρ) under. 9

11 te restriction µ 0, denoted by (ˆθ n, ˆµ n, ˆσ n1, ˆσ n2, ˆρ n ), to construct a lower one-sided t-test of H 0. 2 Tat is, T n (θ 0 ) = n 1/2 (ˆθ n θ 0 )/ˆσ n1, were ˆθ n = X n1 (ˆρ nˆσ n1 ) min(0, X n2 /ˆσ n2 ) wit X nj = n 1 n i=1 X ij for j = 1, 2. As in te previous example, results for upper one-sided and two-sided tests are quite similar. Results in Andrews and Guggenberger (2010b) provide tat tis testing problem also satisfies Assumptions D and S-B.1. Here, γ 1 = µ/σ 2, γ 2 = ρ and γ 3 = (σ 1, σ 2, F ) and d Z 1 + γ 2 min{0, Z 2 }, if γ 1 = 0 T n (θ 0 ) Z 1, if γ 1 0, were Z 1 and Z 2 are standard normal random variables wit Corr(Z 1, Z 2 ) = γ 2. Te corresponding parameter spaces are Γ 1 = R +, Γ 2 = [ 1 + ω, 1 ω] for some ω > 0 and Γ 3 (γ 1, γ 2 ) = {(σ 1, σ 2, F ) : σ 1, σ 2 (0, ), E f X i 2+δ M, θ = 0, γ 1 = µ/σ 2, γ 2 = ρ} for some M < and δ > 0. 3 From tese definitions, it is immediate tat Assumption D olds. Assumption S-B.1 olds for tis example wit r = 1/2 and W d Z 2,1 + 2 min{0, Z 2,2 + 1 }, were (3) Z 2 = Z 2,1 Z 2,2 d N 0, In order to verify te new assumptions, it is instructive to examine te distribution function J ( ), wic is given by ( ) x 2 1 J (x) = Φ Φ( 1 ) x ( 1 Φ ( 1 2 z (1 2 2) 1/2 )) φ(z)dz (4) (see Appendix II for its derivation). By definition, H 1 = R +, and H 2 = [ 1+ω, 1 ω]. Now, looking at te form of J, we can see tat, as in te previous example, W is a continuous random variable wit support over te entire real line for any H so tat δ of Assumption S-BM.1 can again be set arbitrarily close to zero and δ is left unrestricted over its admissible 2 Te results below also allow for different estimators in tis construction. See Andrews and Guggenberger (2010b) for details. 3 For te purposes of tis paper, we make a small departure from te exact setup used by Andrews and Guggenberger (2010b) in our definition of Γ 2, wic tey define as [ 1, 1]. Tat is, we bound te possible correlation between X i1 and X i2 to be less tan perfect. We do tis in order to employ te size-corrections described later in tis paper. Note tat te analogous assumption is imposed in te above post-conservative model selection example. 10

12 range. We discuss a restriction imposed later on δ in Section 5. For te same reasons given in te previous example, Assumption S-B.2(ii) olds as well. More complicated testing problems wen a nuisance parameter may be on a boundary can also be sown to fit te SLLD framework and later assumptions of tis paper. For example, Andrews (1999, 2001) provides results for more complicated boundary examples tat fit tis framework Oter Examples Tere are many examples in te econometrics literature of testing problems tat fit te SLLD framework. Apart from tose discussed above, tese include, but are not limited to, tests after pretests wit fixed CVs (e.g., Guggenberger, 2010), testing wen te parameter of interest may lie on te boundary of its parameter space (e.g., Andrews and Guggenberger, 2010a), tests on model-averaged estimators (e.g., Hansen, 2007), tests on certain types of srinkage estimators (e.g., Hansen, 2012), tests in autoregressive models tat may contain a unit root (e.g., AG and Mikuseva, 2007), Vuong tests for nonnested model selection (e.g., Si, 2011), subvector tests allowing for weak identification (e.g., Guggenberger et al., 2012) and tests on break dates and coefficients in structural cange models (e.g., Elliott et al., 2012 and Elliott and Müller, 2012). Te SLLD assumptions tecnically preclude certain testing problems wit parameterdiscontinuous null limit distributions suc as testing in moment inequality models (e.g., Andrews and Soares, 2010) and certain tests allowing for weak identification (e.g., Staiger and Stock, 1997). Neverteless, te SLLD framework of tis paper can be modified in a problem-specific manner to incorporate some of tese problems and apply te testing metods introduced later to tem. For example, Assumption D does not allow for testing in te moment inequality context wen te condition of one moment binding depends upon weter anoter moment binds. Yet te results of Andrews and Soares (2010), Andrews and Barwick (2011) and Romano et al. (2012) suggest tat tailoring te assumptions to tis context would permit analogous results to tose presented later. 2.2 Multiple Localized Limit Distributions Framework Te MLLDs framework generalizes te SLLD framework. Te motivation for tis generalization comes from an important class of ypotesis testing problems wit parameterdiscontinuous null limit distributions tat do not satisfy Assumption S-B.1 because under H 0 and a given {γ n, }, te asymptotic beavior of T n (θ 0 ) is not fully caracterized by. 11

13 In tis more general framework, we retain te description of te parameter space given in Assumption D as well as te description of H following it but weaken Assumption S-B.1 to te following. Assumption M-B.1. Tere is a sequence {k n }, a set K H 1 and a single fixed r > 0 suc tat for all H and corresponding sequences {γ n, }, under H 0 : d (i) if lim γ n,,1 /k n K, T n (θ 0 ) W (1) ; (ii) if lim γ n,,1 /k n L K c d, T n (θ 0 ) W (2) (iii) if lim γ n,,1 /k n L c K c, te asymptotic distribution of T n (θ 0 ) is stocastically dominated by W (1) or W (2). Assumption M-B.1 allows for different -dependent localized limit distributions tat are relevant to te different possible limiting beaviors of γ n,,1 /k n. It collapses to Assumption S-B.1 wen k n = n r and K = H 1 or L = H 1. Te auxiliary sequence {k n } may depend upon te elements of {γ n, } toug tis potential dependence is suppressed in te notation. Denote te limit distributions corresponding to (i) and (ii) as J (1) and J (2), wic are te two localized limit distributions tat obtain under te corresponding sequences of γ n,,1 /k n. Similarly, c (1) and c (2) denote te corresponding quantile functions. Finally, we denote te limit random variable under (iii) as W (3). Tis is a sligt abuse of notation because tere may be MLLDs tat obtain under (iii) alone. Distinction between tese distributions is not ; necessary ere because of te imposed stocastic dominance. We will also make use of te following definition: ζ({γ n, }) lim γ n,,1 /k n. Te form tat {k n }, K and L take are specific to te testing problem at and. However, we make a few general remarks in order to provide some intuition. Te MLLDs frameowrk incorporates testing problems after a decision rule tat compares some statistic to a samplesize-dependent quantity, say c n, decides te form T n (θ 0 ) takes. Wen using suc a decision rule, under te drifting sequence of parameters {γ n, }, te null limit distribution of T n (θ 0 ) not only depends on te limit of n r γ n,,1 (and γ n,,2 ), but it also depends on ow fast n r γ n,,1 grows relative to c n. Te sequence {k n } is tus some (scaled) ratio of c n to n r and te sets K and L describe te limiting beavior of n r γ n,,1 relative to c n. Tis setting can clearly be furter generalized to allow for oter sequences like {γ n,,1 /k n } to also determine te limiting beavior of T n (θ 0 ) under H 0. For example, one additional sequence of tis sort could allow for two additional localized limit distributions tat are not necessarily stocastically dominated by any of te oters. In tis case, bot te space containing te limit of γ n,,1 /k n and te limit of tis additional sequence could determine te 12

14 null limiting beavior of T n (θ 0 ) under any given {γ n, }. We conjecture tat a more general case like tis obtains for ypotesis tests after more complicated consistent model selection procedures tan tose for wic asymptotic results under drifting sequences of parameters are presently available. (Te intuition for tis is given in Section below). In tis framework, we will also accommodate certain types of discontinuities in te localized limit distribution J. One form of tese discontinuities occurs wen 1 is on te boundary of its parameter space H 1, entailing infinite values. In order to accommodate tis type of discontinuity, define te following subset of H: H int(h1 ) H 2. Ten te set of corresponding to 1 on te boundary of H 1 is equal to H c. Assumption M-B.2. Consider some fixed δ (0, 1). (i) As a function from H into R, c (i) (1 δ) is continuous for i = 1, 2. (ii) For i = 1, 2 and any H, tere is some finite ε i 0 for wic J (i) ( ) is continuous at c (i) (1 δ) + ε i. Assumption M-B.2 is a continuity assumption tat is a relaxed version of a direct adaptation of Assumption S-B.2 to te MLLDs framework. For te problems in tis class, te localized quantiles can be infinite for H c. Tis is wy (i) is only required as a function from H. Similarly, te localized limit distribution functions can be discontinuous at te teir localized quantiles but continuous in a neigborood near tem. Part (ii) provides te flexibility to allow for tis feature. A sufficient condition for Assumption M-B.2 to old for i = 1 or 2 is tat te W (i) is a continuous random variable wit infinite support and a distribution function tat is continuous in. In tis case ε i = 0. As in te previous framework, we strengten part (i) of tis assumption to obtain stronger results. Assumption M-BM.1. For some fixed α (0, 1) and pairs (δ (i), δ (i) ) [0, α δ (i) ] [0, α δ (i) ] for i = 1, 2, as a function of and δ, c (i) (1 δ) is continuous over H and [δ (i), α δ (i) ]. Analogous remarks to tose on Assumption S-BM.1 can be made ere wit te exception tat continuity in is no longer required at H c Testing After Consistent Model Selection Unlike ypotesis testing after conservative model selection, testing after consistent model selection entails substantially more complicated limiting beavior of a test statistic under te null ypotesis. Te essential difference between conservative and consistent model selection in te context of our examples is tat in consistent model selection, te comparison/critical 13

15 value used in te model selection criterion grows wit te sample size. Tis is te case for example, wit te popular Bayesian information criterion (BIC) and te Hannan-Quinn information criterion. Te simple post-consistent model selection testing framework provided by Leeb and Pötscer (2005) provides an illustrative example of a testing problem tat fits te MLLDs framework. Hence, we sall consider it ere. We now consider te regression model y i = θx 1i + β 2 x 2i + ɛ i (5) for i = 1,..., n, were ɛ i d i.i.d.n(0, σ 2 ) wit σ 2 > 0, X (x 1,..., x n) wit x i (x 1i, x 2i ) is nonstocastic, full-rank and satisfies X X/n Q > 0. For simplicity, assume tat σ 2 is known toug te unknown σ 2 case can be andled similarly. We are interested in testing H 0 : θ = θ 0 after deciding weter or not to include x 2i in te regression model (5) via a consistent model selection rule. As in te conservative model selection framework of Section 2.1.1, tis decision is based on comparing te t-statistic for β 2 wit some CV except now tis CV c n grows in te sample size suc tat c n but c n / n 0. Formally, let σ 2 (X X/n) 1 = σ2 θ,n σ θ,β2,n σ θ,β2,n σβ 2 2,n and ρ n = σ θ,β2,n/(σ θ,n σ β2,n). Ten te model selection procedure cooses to include x 2i in te regression if n ˆβ 2 /σ β2,n > c n, were ˆβ 2 is te (unrestricted) OLS estimator of β 2 in te regression (5), and cooses to restrict β 2 = 0 oterwise. For tis example, we will test H 0 by examining te non-studentized quantity n( θ θ 0 ), were θ is equal to te unrestricted OLS estimator of θ in regression (5) wen n ˆβ 2 /σ β2,n > c n and te restricted OLS estimator of θ in (5) wit β 2 restricted to equal zero wen n ˆβ 2 /σ β2,n c n. 4 Tat is for an upper one-sided test, te post-consistent model selection test statistic for testing H 0 is given by T n (θ 0 ) = n( θ θ 0 )1( n ˆβ 2 /σ β2,n c n ) + n(ˆθ θ 0 )1( n ˆβ 2 /σ β2,n > c n ), were θ and ˆθ are te restricted and unrestricted estimators. Examining T n (θ 0 ) and T n (θ 0 ) and teir corresponding localized null limit distributions would allow us to perform te same analysis for lower one-sided and two-sided tests of H 0. 4 Following Leeb and Pötscer (2005), we examine te non-studentized quantity rater tan te t-statistic because use of te latter will not satisfy Assumption D and is terefore not amenable to te procedures put fort in tis paper. Note tat altoug te studentized quantity does not display a parameter-discontinuous null limit distribution, it suffers te same size-distortion problem wen standard CVs are used. 14

16 Let te limits of all finite sample quantities be denoted by a subscript, e.g., σ 2 θ,. Ten for tis example, γ 1 = β 2 ρ /σ β2,, γ 2 = (γ 2,1, γ 2,2 ) = (σ θ,, ρ ) and γ 3 = (β 2, σ 2, σ 2 θ,n, σ2 β 2,n, ρ n ) and T n (θ 0 ) d N(0, (1 γ 2 2,2)γ 2 2,1), if γ 1 = 0 N(0, γ 2 2,1), if γ 1 0. Te parameter spaces in (1) are given by Γ 1 = R, Γ 2 = [η, M] [ 1 + ω, 1 ω] for some η (0, M], ω (0, 1] and M (0, ) and Γ 3 (γ 1, γ 2 ) = {(β 2, σ 2, σ 2 θ,n, σ 2 β 2,n, ρ n ) : β 2 R, σ 2 (0, ), γ 1 = β 2 ρ /σ β2,, γ 2 = (σ θ,, ρ ) lim σ2 θ,n = γ 2,1, lim σβ 2 2,n = β 2 γ 2,2 /γ 1, lim ρ n = γ 2,2 }. Clearly, Γ satisfies Assumption D. Using te results of Proposition A.2 in Leeb and Pötscer (2005), we can establis tat Assumption M-B.1 is satisfied wit k n = c n ρ n / n, r = 1/2, K = ( 1, 1), J (1) (x) = Φ((1 2 2,2) 1/2 (x/ 2,1 + 1 )), L = [, 1) (1, ] and J (2) (x) = Φ(x/ 2,1 ). See Appendix II for details. Te intuition for wy different null limit distributions for T n (θ 0 ) obtain under {γ n, } depending on ow γ n,,1 = β 2,n ρ n /σ β2,n beaves relative to k n = c n ρ n / n as te sample size grows lies in te fact tat wen γ n,,1 /k n = nβ 2,n /σ β2,nc n < 1, T n (θ 0 ) equals n( θ θ 0 ) asymptotically. Conversely, wen γ n,,1 /k n = nβ 2,n /σ β2,nc n > 1, T n (θ 0 ) equals n(ˆθ θ 0 ) asymptotically. Under H 0 and te drifting sequence {γ n, }, te statistics n( θ θ 0 ) and n(ˆθ θ0 ) ave different limit distributions, corresponding to J (1) and J (2), respectively. In te knife-edge case for wic lim γ n,,1 /k n = 1, te limit of T n (θ 0 ) ten also depends on te limiting beavior of c n ± nβ 2,n /σ β2,n (see Leeb and Pötscer, 2005). However, no matter te limit of tis latter quantity, te limit of T n (θ 0 ) in tis case is always stocastically dominated by te limit tat pertains under eiter lim γ n,,1 /k n < 1 or lim γ n,,1 /k n > 1 (see Appendix II). Hence, under a given drifting sequence {γ n, }, te limiting beavior of T n (θ 0 ) is not fully caracterized by, in violation of Assumption S-B.1. By definition, H 1 = R and H 2 = [η, M] [ 1 + ω, 1 ω]. Turning now to Assumption M-BM.1, δ (1) can be set arbitrarily close to, but strictly greater tan, zero. Since c (1) (1 δ) is te (1 δ) t quantile of a normal distribution wit mean 1 2,1 and variance 2 2,1(1 2 2,2) and H = R [η, M] [ 1 + ω, 1 ω], c (1) (1 δ) is continuous in over H for any δ (0, 1). Continuity in δ over [δ (1), α δ (1) ] also follows for any δ (1) [0, α δ (1) ]. Similar reasoning sows tat, δ (2) can be set arbitrarily close to zero and δ (2) can be anywere in its admissible range for Assumption M-BM.1 to old (we discuss restrictions imposed on δ (i) for i = 1, 2 15

17 by later assumptions in Section 6). For H and i = 1, 2, J (i) is te distribution function of a continuous random variable so tat Assumption M-B.2(ii) olds wit ε i = 0. As wit te oter examples studied in tis paper, tis simple illustrative example is not te most general of its kind to fall into tis framework. Many of te assumptions in te above example can be relaxed. However, uniform asymptotic distributional results for more complex consistent model selection procedures are not readily available in te literature. Tis may be in part due to te very negative results put fort regarding attempts to conduct inference after even te simplest of suc procedures (e.g., Leeb and Pötscer, 2005 and Andrews and Guggenberger, 2009b). As alluded to above, more complicated procedures, suc as a consistent version of te sequential general-to-specific model selection approac of Leeb (2006) and Leeb and Pötscer (2008) or standard BIC approaces to more complicated models, likely require a straigtforward extension of te MLLDs framework and te corresponding CVs introduced below. Te intuition for tis essentially follows from te same intuition as tat used for te simple post-consistent model selection example provided above. As a simplification, suppose now tat anoter regressor enters te potential model (5) wit associated coefficient β 3. Using te obvious notation, we may also wis to determine weter β 3 sould enter te model prior to testing H 0 by comparing n ˆβ 3 /σ β3 to c n. In tis case T n (θ 0 ) would take one of four, rater tan two, possible values depending on bot te value of n ˆβ 3 /σ β3 and n ˆβ 2 /σ β2 relative to c n (ignoring knife-edge cases) Oter Examples Te class of super-efficient/ard-tresolding estimators studied by Andrews and Guggenberger (2009b) and Pötscer and Leeb (2009), including Hodges estimator, also fit te MLLDs framework. A certain subclass of tese estimators in fact requires ε i > 0 in Assumption M-B.2(ii) for i = 1 or 2, unlike te problem considered immediately above. Related problems of testing after pretests using pretest CVs tat grow in te sample size fit te MLLDs framework as well. Toug some very recent work as explored te properties of uniformly valid confidence intervals for some of te problems tat fall into tis framework (e.g., Pötscer and Scneider, 2010), to te autor s knowledge, tis is te first time suc a framework and corresponding uniformly valid inference procedure as been presented at tis level of generality. 16

18 3 Bonferroni-Based Critical Values For test statistics wit parameter-discontinuous null limit distributions, te asymptotic NRP of te test, evaluated at a given parameter value permissible under H 0 can provide a very poor approximation to te true NRP and size of te test, even for large samples. In order to be more precise about tis terminology, we introduce te following definitions for a test of H 0 : θ = θ 0, working under te framework described in Section 2. Let κ n be te (possibly random or sample-size-dependent) CV being used. Te NRP evaluated at γ Γ is given by P θ0,γ(t n (θ 0 ) > κ n ), were P θ0,γ(e) denotes te probability of event E given tat (θ 0, γ) are te true parameters describing te data-generating process (DGP). Te asymptotic NRP of a test statistic T n (θ 0 ) and cv κ n evaluated at γ Γ is given by lim sup P θ0,γ(t n (θ 0 ) > κ n ). Te exact and asymptotic sizes are defined as ExSZ n (θ 0, κ n ) sup P θ0,γ(t n (θ 0 ) > κ n ) γ Γ AsySz(θ 0, κ n ) lim sup ExSZ n (θ 0, κ n ). Note tat te exact and asymptotic sizes of a test ave te concept of uniformity built into teir definitions in tat ExSZ n (θ 0, κ n ) is te largest NRP uniformly over te parameter space Γ and AsySz(θ 0, κ n ) is its limit. In order to ave a test wit approximately controlled exact size, and terefore controlled NRP at any γ Γ, we must control AsySz(θ 0, κ n ). Under te frameworks of tis paper, te primary teoretical step in controlling te asymptotic size of a test is to control te asymptotic NRP under all drifting sequences of parameters {γ n, }. Tat is, if we can find a (sequence of) CV(s) { κ n } suc tat lim sup P θ0,γ n, (T n (θ 0 ) > κ n ) α for all {γ n, } described in Section 2, we can construct a ypotesis test wose asymptotic size is bounded by α (see Andrews and Guggenberger, 2010b, Andrews et al., 2011 or te subsequencing arguments used in Appendix I for details). Since c (1 α) (or c (i) (1 α), i = 1, 2) is te (1 α)t CV of te limit distribution of T n (θ 0 ) under H 0 and te drifting sequence of parameters {γ n, }, we would ideally like to use a CV tat is equal to c (1 α) wenever {γ n, } caracterizes te true DGP in order to maximize te power of te resulting test wile controlling its asymptotic size. Unfortunately, cannot be consistently estimated under all drifting sequence DGPs. Tis as led to te construction of te so-called LF CV sup H c (1 α) and variants tereof (e.g., AG). Guarding against te worst-case drifting sequence DGP, tis CV is often quite large, substantially reducing te power of te resulting test. Toug cannot be consistently estimated under {γ n, }, in typical applications one can 17

19 find an estimator of tat converges in distribution to a random variable centered around te true value under H 0 and tis drifting sequence DGP. Tis allows one to form asymptotically valid confidence sets for and subsequently restrict attention to data-dependent regions inside of H relevant to te testing problem at and, rater tan guarding against te worstcase scenario, leading to smaller CVs and resulting tests wit iger power. However, te additional uncertainty associated wit te estimation of must be taken into account for one to control te asymptotic NRP under all drifting sequences. Tis is were Bonferroni bounds become useful. We now introduce two sets of tree types of robust CVs based upon Bonferroni approaces. Eac set corresponds to CVs to be used witin eiter te SLLD or MLLDs framework. Witin eac set, te CVs are presented in increasing order of computational complexity. As te types of CV grow from least to most computationally complex, appropriately constructed tests using tem tend to gain in power. 3.1 S-Bonf Robust Critical Values We begin by examining Bonferroni-based size-corrected CVs for problems tat are caracterized by a SLLD, as described in Section 2.1. Te first, most conservative but most computationally simple Bonferroni-based CV is defined as follows: c S B(α, δ, ĥn) sup c (1 δ), I α δ (ĥn) were δ [0, α], ĥn is some random vector taking value in an auxiliary space H and I α δ ( ) is a correspondence from H into H. In applications, te space H will typically be equal to H or a space containing H but tis is not necessary for te ensuing assumptions to old. Te random vector ĥn is an estimator of under H 0 and te DGP caracterized by {γ n, } and I α δ (ĥn) serves as a (α δ)-level confidence set for. Construction of an estimator of is typically apparent from te context of te testing problem, given tat 1 = n r γ n,,1 + o(1) and 2 = γ n,,2 +o(1). Te S-Bonf CV generalizes te LF CV: wen I 0 (x) = H for all x H, c S B (α, α, ĥn) = c LF (α) sup H c (1 α). Te tuning parameter δ can be used to direct te power of te test towards different regions of te parameter space H. Procedures using Bonferroni bounds in inference problems involving nuisance parameters and/or composite ypoteses ave appeared in various contexts trougout te econometrics and statistics literature. Examples include Lo (1985), Stock (1991), Berger and Boos (1994), Silvapulle (1996), Staiger and Stock (1997), Romano and Wolf (2000), Hansen (2005b), 18

20 Moon and Scorfeide (2009), Cauduri and Zivot (2011) and Romano et al. (2012). 5 Romano et al. (2012), wic applies a specific form of te S-Bonf CV to ypotesis testing in partially identified moment inequality models, was developed concurrently wit tis work. Te metods of Lo (1985) and Hansen (2005b) are analogous to letting δ α as n in te confidence set I α δ (ĥn), but simply using te maximand c (1 α) in te construction of te CV. Toug tis approac is asymptotically size-correct since it leads to te use of c LF (α) in te limit, it can ave poor finite sample size control since it fails to fully account for te additional uncertainty involved in estimating. We now impose furter assumptions to ensure tat tests utilizing c S B in te current context exibit asymptotic size control. Assumption S-B.3. Consider some fixed β [0, 1]. Under H 0 and wen te drifting sequence of parameters {γ n, } caracterizes te true DGP for any fixed H, tere exists an estimator ĥn taking values in some space H and a (nonrandom) continuous, compactvalued correspondence I β : H H suc tat ĥn d, a random vector taking values in H for wic P ( I β ( )) 1 β. Assumption S-B.3 assures tat ĥn is a well-beaved estimator of under H 0 and {γ n, } and imposes basic continuity assumptions on te correspondence I β ( ) used to construct te confidence set for. It allows I β ( ) to take a variety of forms depending upon te context of te testing problem. For a given β = α δ, tis flexibility can be used to direct te power of te test towards different regions of H or to increase te computational tractability of constructing te S-Bonf-Min CVs (see, e.g., Romano et al., 2012). In a typical testing problem, under H 0 and {γ n, }, ĥn,1 converges weakly to a normally distributed random variable wit mean 1, te true localization parameter, making te construction of I β very straigtforward. Similarly, depending upon te testing problem, different coices of ĥn may lead to tests wit different power properties. For example, it may be advantageous to use ĥ n tat imposes H 0 if tis leads to smaller CVs. Assumption S-B.4. Under H 0 and wen te drifting sequence of parameters {γ n, } caracterizes te true DGP, (T n (θ 0 ), ĥn) d (W, ) for all H. d Assumptions S-B.1 and S-B.3 already provide tat T n (θ 0 ) W and ĥn d under H 0 and {γ n, } so tat Assumption S-B.4 only ensures tat tis weak convergence occurs jointly. Assumptions S-B.3 and S-B.4 also allow for muc flexibility in te estimation of. Since 5 I tank Hannes Leeb and Benedikt Pötscer for alerting me to some of te early references in te statistics literature troug te note Leeb and Pötscer (2012). 19

21 weak convergence is only required under H 0, ĥn may be constructed to be an asymptotically biased estimator of under some alternatives. We may now state te first result. Teorem S-B. Under Assumptions D and S-B.1 troug S-B.4 for β = α δ, δ AsySz(θ 0, c S B(α, δ, ĥn)) α. Teorem S-B establises te asymptotic size control of tests using S-Bonf CVs as well as providing a lower bound for te asymptotic size. If a consistent estimator of γ 2 is available, we can plug it into te CV to reduce its magnitude and increase te power of te test. Define te plug-in (PI) S-Bonf CV c S B P I (α, δ, ĥn) as above but wit I α δ (ĥn) j = {ˆγ n,2,j p } for j = p + 1,..., p + q. Tis generalizes te PI LF CV (e.g., AG): wen I 0 (x) = H 1 {(x p+1,..., x p+q )} for all x H, c S B P I (α, α, ĥn) = c LF P I (α, γ n,2 ˆ ) sup 1 H 1 c (1,ˆγ n,2 )(1 α). Similar comments to tose regarding te coice of δ, ĥn and I α δ ( ) apply to tis CV as well. We impose te following consistency assumption ten state te analogous result for tests using PI S-Bonf CVs. Assumption PI. ˆγ n,2 γ n,2 p 0 for all sequences {γ n } wit γ n Γ for all n. Corollary S-B-PI. Under Assumptions D, S-B.1 troug S-B.4 and PI for β = α δ, δ AsySz(θ 0, c S B P I(α, δ, ĥn) α. 3.2 S-Bonf-Adj Robust Critical Values Toug tey are size-controlled, te S-Bonf CVs may be conservative, even in a uniform sense. Tis can be seen from te fact tat we can only establis a lower bound of δ α on te asymptotic size of tests using tese CVs. Instead of relying on Bonferroni bounds, for a given confidence set level, we can directly adjust te level of te localized quantile function according to te limit distribution of (T n (θ 0, ĥn)) to improve power as follows: c S B A(α, β, ĥn) sup c (1 ᾱ), I β (ĥn) were β [0, 1] and ᾱ = inf H α() wit α() [α β, α] solving P (W sup c (1 α())) = α I β ( ) or α() = α if P (W sup Iβ ( ) c (1 α)) < α. Unlike te S-Bonf CV, te level adjustment in tis S-Bonf-Adj CV compensates for te limiting dependence between T n (θ 0 ) 20

22 and sup Iβ (ĥn) c ( ). Te level ᾱ is an implicit function of te tuning parameter β, automatically adjusting to te user s coice. Clearly, for any given coice of δ [0, α], c S B A (α, α δ, ĥn) c S B (α, δ, ĥn) so tat a test using c S B A (α, α δ, ĥn) necessarily as iger power. Note tat, like te S-Bonf CVs, te S-Bonf-Adj CVs also generalize te LF CV: wen I 0 (x) = H for all x H, P (W c LF (α)) = P (W sup I0 ( ) c (1 α)) α for all H so tat α() = α for all H and c S B A (α, α, ĥn) = c LF (α). To sow correct asymptotic size of te S-Bonf-Adj CVs, we impose one additional condition wic is essentially a continuity assumption. Assumption S-BA. (i) P (W = c S B A (α, β, )) = 0 for all H. (ii) P (W c S B A (α, β, )) = α for some H. Assumption S-BA is analogous to Assumption Rob2 of Andrews and Ceng (2012) in te current general context. Part (ii) is not required for te asymptotic size to be bounded above by α; it is only used to sow tat te lower bound is also equal to α. Part (i) olds if W c S B A (α, ) is a continuous random variable for all H, as is typical in applications. Teorem S-BA. Under Assumptions D, S-B.1, S-B.2(i) evaluated at δ = ᾱ, S-B.3, S-B.4 and S-BA, AsySz(θ 0, c S B A(α, β, ĥn)) = α. We can also define a PI version of tese CVs to furter improve power as follows: c S B A P I(α, β, ĥn) sup c (1 ā(ˆγ n,2 )), I β (ĥn) were I β (ĥn) j = {ˆγ n,2,j p } for j = p + 1,..., p + q, β [0, α] and ā : H 2 [δ, α δ] [α β, α] is a level-adjustment function. For similar reasons to tose given above, a test using c S B A P I (α, α δ, ĥn) tat satisfies te following assumption necessarily as iger power tan one using c S B P I (α, δ, ĥn) and generalizes c LF P I (α, ˆγ n,2 ). To sow correct asymptotic size of tests using te PI S-Bonf-Adj CVs, we introduce a different continuity assumption. Assumption S-BA-PI. (i) ā : H 2 [δ, α δ] is a continuous function. (ii) P (W = c S B A P I (α, β, )) = 0 for all H. (iii) P (W sup Iβ ( ) c (1 ā( 2 ))) α for all H and tere is some H suc tat P (W c S B A P I (α, )) = α. 21

23 Te second statement of part (iii) is not required for te asymptotic size to be bounded above by α; it is only used to sow tat te lower bound is also equal to α. As before, part (ii) olds if W c B A P I (α, ) is a continuous random variable, as is typical. Parts (i) and (iii) will old if ā( ) is constructed appropriately. In practice, for a single given testing problem, one does not need to determine te entire function ā( ) since it will only be evaluated at a single point, ˆγ n,2. Supposing ā(ˆγ n,2 ) comes from evaluating a function satisfying Assumption S-BA-PI provides one wit te asymptotic teoretical justification needed for correct asymptotic size and is consistent wit practical implementation. For a given testing problem, practical determination of ā( ) tat is consistent wit part (iii) and yields te igest power proceeds by finding te largest ā [δ, α δ] suc tat sup P (W (1, γ n,2 ˆ ) sup c (1 ā)) α. (6) 1 H 1 I β (( 1,ˆγ n,2 )) Corollary S-BA-PI. Under Assumptions D, S-B.1, S-B.3, S-B.4, PI, S-BM.1 and S-BA- PI, AsySz(θ 0, c S B A P I(α, β, ĥn)) = α. Te coice of β allows te practitioner te flexibility to direct power toward regions of te localization parameter space H. One euristic way to direct power toward a given (e.g., identification strengt) is to use te PI S-Bonf-Adj CV wit te β tat minimizes te distance between c (1 α) and c S B A P I (α, β, ) (assuming E[ĥn] ). Tis will yield a S- Bonf-Adj CV tat is close to te true localized quantile evaluated at wit ig probability under drifting sequences {γ n, }. More generally, one may coose β to direct power towards regions of H according to a weigting sceme in analogy to maximizing weigted average power. For example, let F H be a probability measure wit support on H. One could select β to minimize c (1 α) c S B A P I (α, β, ) df H() S-Bonf-Min Robust Critical Values Toug β can be cosen to direct power in te construction of te S-Bonf-Adj CVs, different coices of β trade off power over different regions of te parameter space. (Tis is also true of δ in te S-Bonf CV.) Depending upon te coice of β, te S-Bonf-Adj CVs can lack power over large portions of te parameter space. In order to overcome tis obstacle and produce tests tat can simultaneously direct power wile maintaining ig power over most of te 6 Recalling Jensen s inequality, one may prefer to coose β to minimize c (1 α) E[c S B A P I (α, β, )] or c (1 α) E[c S B A P I (α, β, )] df H (). 22

24 parameter space, we introduce te following CV tat minimizes over a set of S-Bonf and S-Bonf-Adj CVs: were c S BM(α, ĥn) c min B (α, ĥn) +, { c min B (α, ĥn) = min c S B A(α, β 1, ĥn),..., c S B A(α, β r, ĥn), and = sup H () wit () 0 solving or () = 0 if P (W c min B (α, )) < α. P (W c min B (α, ) + ()) = α inf δ [δ,α δ] } c S B(α, δ, ĥn) Here we refer to te input S-Bonf-Adj CVs c S B A (α, β i, ĥn) as te ancors and te corresponding values in te parameter space H for wic te β i are cosen to direct power toward as te ancor points. Toug computation of c S BM is more expensive tan tat of c S B A, te ancors allow one to direct power wile te minimum over S-Bonf-Min CVs component allows te test to acieve relatively ig power over te entire parameter space H. Te former is acieved by bringing te value of c min B A (α, ĥn) close to tat of c (1 α) wit ig probability at te ancor points. Te latter is acieved by enabling c min B A (α, ) to (conservatively) mimic te beavior of c (1 α) over te entire parameter space H. A simple coice of ancor is te LF CV. Te size-correction factor (SCF) automatically adapts to te user s coices of confidence set correspondence I β ( ), localization parameter estimate ĥn and te parameters δ, δ and β 1,..., β r since it is constructed so tat te test as correct asymptotic size under H 0 for te given coice of tese objects. Given te conservativeness already built into te Bonferroni approac, will typically be quite small wen te number of ancors is small, often being numerically indistinguisable from zero, so tat c S BM (α, ĥn) c S B (α, β i, ĥn), c S B (α, δ, ĥn) for δ [δ, α δ] and i = 1,..., r, resulting in tests wit iger power. In fact, in many contexts, simply setting = 0 will result in a test witout perfect (asymptotic) size control, but very little size distortion and necessarily iger power. For some testing problems, size-control can be attained wit set exactly equal to zero. See Corollary SCF in Appendix III for a sufficient condition for tis to occur. Inerent in te construction of c S BM (α, ĥn) are te parameters δ and δ. It sould be empasized tat tese parameters are not tuning parameters. Tey serve te role of restricting te localized quantile function c ( ) and te confidence set I α δ to regions of continuity (see 23

25 te discussion following Assumptions S-BM.1 and S-BM.2). Given tat te aim of using S-Bonf-Min CVs is to maximize power, te minimization space [δ, α δ] sould be set as large as possible in order to obtain te smallest CV. Tis means tat δ and δ sould be set close to zero wile satisfying Assumptions S-BM.1 and S-BM.2. Beyond computational complexity, tere is a tradeoff involved in using too many ancor points (large r): te more ancor points one uses, te larger te size-correction factor will be. Tis will lower power in portions of te parameter space tat are far from te ancor points. It is wort pointing out ere te similarities tat te use of S-Bonf-Min CVs sares wit existing metods. In te specific contexts of inference in moment inequality models and inference robust to identification strengt, Andrews and Soares (2010), Andrews and Barwick (2011) and Andrews and Ceng (2012) also use CVs tat are functions of an inconsistent estimator of te localization parameter. Some of te size-correction tecniques in tese papers use a binary decision rule tat uses as CV te localized quantile wit 1 = once ĥn,1 crosses a sample-size dependent tresold and c LF (α) oterwise. In contrast, te S-Bonf-Min CVs confine te range of used to construct te test statistic to I α δ (ĥn) and subsequently searc for te smallest of tese among admissible δ values (and ancors), witout requiring an ad oc tresold specification. In some sense tis can be tougt of as a smooted version of tese autors metods. Ceng (2008) and Kabaila (1998) also employ CV selection metods based on binary decision rules for inference in te specific contexts of a weakly identified nonlinear regression model and conservative model selection. Peraps te most similar metodologies to tose using S-Bonf-Min CVs are tose tat use a transition function to smoot between te LF CV and te one tat obtains under infinite values of te localization parameter in a manner tat depends upon an inconsistent estimate of te localization parameter. Tese metods are advocated in specific contexts by Andrews and Soares (2010) and Andrews and Barwick (2011) (using certain coices of teir ψ function) and Andrews and Ceng (2012) (teir type 2 robust CV). Te transition function metods of Andrews and Barwick (2011) and Andrews and Ceng (2012) also necessitate te use of SCFs similar to in te definition of c S BM. In contrast to tese metods, te S-Bonf- Min CVs do not require an ad oc coice of transition function. Rater, tey adaptively use te full localized quantile function (rater tan two points of it) and te data to coose wic points along te localized quantile curves are relevant to te finite-sample testing problem at and. Tis allows c S BM (α, ) to closely mimic te true (1 α)t localized quantile function. See Section 4.1 and Figures 1-2 for illustrations. Tis leads to asymptotic NRPs tat are close to te asymptotic size of te test over wide ranges of so tat te tests are 24

26 nearly similar and consequently attain ig power. To sow tat tests utilizing c S BM (α, ĥn) ave correct (non-conservative) asymptotic size, we strengten Assumption S-B.3 as follows. Assumption S-BM.2. Consider some fixed α (0, 1) and some pair (δ, δ) [0, α δ] [0, α δ ]. (i) Assumption S-B.3 olds for all β [ δ, α δ ]. (ii) I β () is continuous in β over [ δ, α δ ] for all H. Te quantity δ serves as a lower bound and α δ serves as an upper bound on te points β for wic te correspondence I β () must be continuous. In many applications, te confidence set I β corresponds to a confidence set for a normal random variable. Hence, in order to satisfy Assumption S-B.3, we would need I 0 (x) = H. However, tis will typically involve a discontinuity of I β () at β = 0 so tat we must bound β from below by some value δ greater tan zero. To sow correct asymptotic size of tests using te S-Bonf-Min CVs, we introduce a new continuity assumption similar to S-BA. Assumption S-BM.3. (i) P (W = c S BM (α, )) = 0 for all H. (ii) P (W c S BM (α, )) = α for some H. Similar comments to tose following Assumption S-BA apply ere. We provide an easyto-verify sufficient condition for part (ii) to old in Proposition S-BM of Appendix III. Teorem S-BM. Under Assumptions D, S-B.1, S-B.2(i) evaluated at te δ i = ᾱ i tat corresponds to β i, S-B.3 evaluated at β i, S-B.4 and S-BM.1 troug S-BM.3 for i = 1,..., r, AsySz(θ 0, c S BM(α, ĥn)) = α. Depending upon te values of δ, δ and β i for i = 1,..., r used in te construction of c S BM (α, ĥn), some of te assumptions imposed in Teorem S-BM may be redundant. For example, Assumption S-B.2(i) evaluated at δ i = ᾱ i for i = 1,..., r is implied by Assumption S-BM.1 if ᾱ i [δ, α δ] for all i = 1,..., r. Similarly, Assumption S-B.3 evaluated at β i for i = 1,..., r is implied by Assumption S-BM.2 if β i [ δ, α δ ] for all i = 1,..., r. As wit all of te CVs examined in tis paper, if a consistent estimator of γ 2 is available, we may furter improve power by using a PI version of te S-Bonf-Min CV: c S BM P I(α, ĥn) c min B P I (α, ĥn) + η(ˆγ n,2 ), 25

27 were { c min B P I (α, ĥn) = min c B A P I (α, β 1, ĥn),..., c B A P I (α, β r, ĥn), inf δ [δ m,α δ m ] } c S B P I(α, δ, ĥn) and η : H 2 R. Note tat, for eac i = 1,..., r in te construction of ancors c B A P I (α, β i, ĥn), a different level-adjustment function ā i : H 2 [δ i, α δ i ] [α β i, α] may be used. Apart from using a PI estimator in te minimum-of-bonferroni approac, te magnitude of te CV is furter reduced and te power of te test is furter increased by te use of a SCF tat depends upon te PI estimator. Tis SCF function, satisfying te following assumption, is often numerically indistinguisable from zero in applications, making its practical relevance minimal as simply using c min B P I (α, ĥn) often leads to minimal size-distortion. Tis again results from te conservativeness inerent in te Bonferroni approac. Similarly to te analogous SCF function used by Andrews and Barwick (2011), in order to maximize power one sould select te smallest SCF function tat satisfies te following assumption. Assumption S-BM-PI. (i) η : H 2 R is a continuous function. (ii) P (W = c min B P I (α, ) + η( 2 )) = 0 for all H. (iii) P (W c min B P I (α, ) + η( 2 )) α for all H and tere is some H suc tat P (W c S BM P I (α, )) = α. Part (i) is analogous to Assumption η1 and parts (ii) and (iii) are analogous to Assumption η3 of Andrews and Barwick (2011). As wit te oter PI CVs introduced in tis paper, te SCF function η and level-adjustment functions ā i only need to be computed at te single value of 2 = ˆγ n,2 in practice. Similar comments to tose following Assumption S-BA-PI apply. We may now present te result establising te correct asymptotic size of tests utilizing tis PI CV. Corollary S-BM-PI. Under Assumptions D, PI, S-B.1, S-B.3 evaluated at β i, S-B.4, PI, S-BA-PI(i) and (iii) evaluated at β i and corresponding ā i ( ), S-BM.1 and S-BM.2 evaluated at pairs (δ m, δ m ) and (δ i, δ i ) and S-BM-PI for i = 1,..., r, AsySz(θ 0, c S BM P I(α, ĥn)) = α. Depending upon te values of (δ m, δ m ), (δ i, δ i ) and β i for i = 1,..., r used in te construction of c BM A P I (α, ĥn), (part of) Assumption S-B.3 evaluated at β i for i = 1,..., r may be redundant. For example, it is implied by Assumption S-BM.2 evaluated at pairs 26

28 (δ m, δ m ) and (δ i, δ i ) for i = 1,..., r if β i r j=1 [δ j, δ j ] [δ m, δ m ] for i = 1,..., r. It is also wort noting tat te full force of Assumption S-BA-PI(iii) is not necessary for te result of Corollary S-BM-A-PI to old. Te assumption is imposed as a guide for te ancoring procedure to direct power appropriately. As was te case for te SCF above, te addition of η(ˆγ n,2 ) is not always necessary to obtain correct asymptotic size. Corollary SCF in Appendix III provides a precise condition under wic it may be dispensed wit. 3.4 M-Bonf Robust Critical Values Te S-Bonf robust CVs tat apply to problems wit a SLLD can be adapted to andle tose wit MLLDs. In order to do tis we need a metod to coose wic localized limit distribution J (1) or J (2) to construct te CVs from. For tis, we utilize an estimator of ζ({γ n, }) tat tells us wit increasing precision wic region of H 1 tat ζ({γ n, }) lies witin. Construction of tis estimator can be deduced from te form of γ n,,1 /k n. In applications, ˆζ is typically equal to n r ĥ n,1 scaled by kn 1 since γ n,,1 n r 1 and ζ({γ n, }) γ n,,1 /k n. Te M-Bonf CV for te MLLDs framework is tus defined as follows: were c M B (α, δ, ĥn) c (1) B (α, δ, ĥn), if c (2) B (α, δ, ĥn), if ˆζ K ˆζ L max{c (1) B (α, δ, ĥn), c (2) B (α, δ, ĥn)}, if ˆζ K c L c, c (i) B (α, δ, ĥn) sup c (i) (1 δ) + ε i I α δ (ĥn) for i = 1, 2, K int(k) is closed and L int(l) is closed. Asymptotic power is igest for sets K and L tat are very close to K and L. Tis makes te region using te most conservative CV in te construction, K c L c, relatively small. Similarly, ε i for i = 1, 2 sould be cosen to be te smallest feasible value tat satisfies Assumption M-B.2(ii) in order to maximize power Unlike existing approaces, adapting te Bonferroni approac to te MLLDs framework allows one to size correct te CVs tat pertain to te typical type of testing problem tat falls into tis framework witout killing te power of te test. Tis type of problem entails (at least) one of te localized distribution quantiles being equal to for some H so tat approaces putting positive weigt on LF CVs (suc as tose of AG, Andrews and Soares, 2010, Andrews and Ceng, 2012 and Andrews and Barwick, 2011) will lead to infinite CVs and zero power over muc or all of te parameter space. 27

29 We now impose additional assumptions under wic tests utilizing M-Bonf CVs ave correct asymptotic size. Some of tese are obvious counterparts to tose imposed for S-Bonf CVs. Assumption M-B.3. Consider some fixed β [0, 1]. Under H 0 and wen te drifting sequence of parameters {γ n, } caracterizes te true DGP for any fixed H, tere exists an estimator ĥn taking values in some space H and a (nonrandom) continuous, compactvalued correspondence I β : H H suc tat ĥn d, a random vector taking values in H for wic P ( I β ( )) 1 β and I β (ĥn), I β ( ) H wp 1. Te condition tat I β (ĥn), I β ( ) H wp 1 if H is not restrictive and can be ensured by te proper construction of te confidence set for. In te typical application, it states tat if te entries of are finite, te confidence set for does not include infinite values. Similar remarks to tose following Assumption S-B.3 apply ere as well. Assumption M-B.4. Under H 0 and wen te drifting sequence of parameters {γ n, } caracterizes te true DGP for any H, (i) if ζ({γ n, }) K, (T n (θ 0 ), ĥn) d (W (1), ) (ii) if ζ({γ n, }) L, (T n (θ 0 ), ĥn) d (W (2), ). Joint convergence does not need to be establised for ζ({γ n, }) L c K c. remarks to tose following Assumption S-B.4 apply ere as well. Similar Assumption M-B.5. ˆζ {γ n, }. p ζ({γ n, }) under H 0 and te drifting sequence of parameters Under Assumption M-B.5, te estimator ˆζ yields te decision rule tat cooses wic Bonferroni CV is applicable under any given sequence {γ n, } as te sample size increases. Recall tat ˆζ is typically equal to kn 1 n r ĥ n,1. Furtermore, ĥn,1 is typically asymptotically centered about 1 under H 0 so tat for finite 1, ĥn,1 = 1 + O p (1) = n r γ n,,1 + O p (1) and since typically k n n r, ˆζ = kn 1 n r (n r γ n,,1 + O p (1)) = kn 1 γ n,,1 + o p (1) = ζ({γ n, }) + o p (1). Tis wy ζ({γ n, }) consistently estimable even toug 1 is not. As in te SLLD case, we can use a consistent estimator of γ 2 to increase te power of tests using M-Bonf robust CVs. Define te PI M-Bonf CV c M B P I (α, δ, ĥn) te same as c M B (α, δ, ĥn) 28

30 but wit I α δ (ĥn) j = {ˆγ n,2,j p } for j = p+1,..., p+q. For i = 1, 2, also define c (i) B P I (α, δ, ĥn) identically to c (i) B (α, δ, ĥn) but wit I α δ (ĥn) j = {ˆγ n,2,j p } for j = p + 1,..., p + q. To conserve space, we relegate and additional ig-level assumption used to provide an upper bound on te NRP under drifting sequences of parameters for wic H c and assumptions used to establis a lower bound on te size of a test using M-Bonf and PI M-Bonf CVs to Appendix III. We furtermore refer te reader to Lemmas M-B and M-B-PI in Appendix I for te formal statements and proofs tat AsySz(θ 0, c M B (α, δ, ĥn)), AsySz(θ 0, c M B P I(α, δ, ĥn)) α under appropriate conditions, as well as a brief discussion of a sufficient condition. Te size-corrected CVs c M B (α, δ, ĥn) and c M B P I (α, δ, ĥn) simultaneously accommodate problems of non-uniformity in te point-wise asymptotics of te distribution of te test statistic (Assumption D), localized null limit distributions tat depend on te localization parameter discontinuously (Assumption M-B.1), localized null limit distributions tat are discontinuous (Assumption M-B.2(ii)) and localized null limit distributions tat escape to ± (Assumption M-B.6 in Appendix III). 3.5 M-Bonf-Adj Robust Critical Values Like teir S-Bonf counterparts, te M-Bonf CVs may lead to conservative testing in a uniform sense and can be level-adjusted to improve power. Specifically, define te M-Bonf-Adj CV, c M B A (α, β, ĥn), identically to te M-Bonf CV but replace c (i) B (α, δ, ĥn) in te definition wit c (i) B A (α, δ, ĥn) for i = 1, 2, were β [0, 1], c (i) B A (α, β, ĥn) = sup c (i) (1 ᾱ(i) ) I β (ĥn) and ᾱ (i) = inf H α (i) () wit α (i) () [0, α] solving P (W (i) sup c (i) (1 α(i) ())) = α I β ( ) or α() = α if P (W (i) sup Iβ ( ) c(i) (1 α)) < α for i = 1, 2. Analogous to te S-Bonf-Adj CV case, we can use a PI version of te M-Bonf-Adj CV to improve power wen a consistent estimator of γ 2 is available. Define te PI M-Bonf-Adj CV, c M B A P I (α, β, ĥn), identically to te PI M-Bonf CV but replace c (i) B P I (α, δ, ĥn) wit c (i) B A P I (α, δ, ĥn) for i = 1, 2, were β [0, 1], I β (ĥn) j = {ˆγ n,2,j p } for j = p + 1,..., p + q, c (i) B A P I (α, β, ĥn) = sup c (i) (1 ā(i) (ˆγ n,2 )) I β (ĥn) 29

31 and ā (i) : H 2 [δ (i), α δ (i) ] for i = 1, 2. Also in analogy wit te S-Bonf-Adj CV, computation of c M B A (α, α δ, ĥn) (c M B A P I (α, α δ, ĥn)) is costlier tan tat of c M B (α, δ, ĥn) (c M B P I (α, δ, ĥn)) but results in tests wit iger power. As is te case for (PI) S-Bonf-Adj CVs, te coice of β in te construction of te (PI) M-Bonf-Adj CVs provides te practitioner wit te ability to direct te power of te resulting tests toward different regions of H. Straigtforward generalization allows for different β s to be cosen to correspond to te different localized limit distributions used in te construction of te CVs, furter enancing tis flexibility. Te formal statements of te conditions under wic AsySz(θ 0, c M B A(α, β, ĥn)), AsySz(θ 0, c M B A P I(α, β, ĥn)) = α may be found in Corollaries M-BA and M-BA-PI located in Appendix III. Additional assumptions imposed in tese corollaries may also be found tere. 3.6 M-Bonf-Min Robust Critical Values As is te case for S-Bonf and S-Bonf-Adj CVs in te SLLD framework, te M-Bonf and M-Bonf-Adj CVs can be appropriately minimized to maintain ig power over most of te parameter space wile retaining correct asymptotic size and directing power. For tis, consider te Bonf-Min CVs adapted to te MLLDs framework: c M BM(α, ĥn) c (1) min B (α, ĥn) + 1, c (2) min B (α, ĥn) + 2, if ˆζ K if ˆζ L max{c (1) min B (α, ĥn) + 1, c (2) min B (α, ĥn) + 2 }, if ˆζ K c L c, were for i = 1, 2, { c (i) min B (α, ĥn) = min c (i) B A (α, β 1, ĥn),..., c (i) B A (α, β r, ĥn), inf δ [δ (i),α δ (i) ] } c (i) B (α, δ, ĥn), and i = sup H i () wit i () 0 solving P (W (i) c (i) min B (α, ) + i ()) = α or i () = 0 if P (W (i) c (i) min B (α, ) + i ()) < α for H. Aside from a couple of generalizing adjustments, c M BM (α, ĥn) is essentially a version of c S BM (α, ĥn) tat is selected by ˆζ. Hence, very similar comments to tose made about c S BM (α, ĥn) and te objects used in its construction apply for c M BM (α, ĥn) (see Section 3.3). We also impose some analogous assumptions to tose used in te SLLD framework. 30

32 Assumption M-BM.2. Consider some fixed α (0, 1) and some pairs (δ (i), δ (i) ) [0, α δ (i) ] [0, α δ (i) ] for i = 1, 2. (i) Assumption M-B.3 olds for all β [ δ (1), α δ (1) ] [ δ (2), α δ (2) ]. (ii) I β () is continuous in β over [ δ (1), α δ (1) ] [ δ (2), α δ (2) ] for all H. Analogous comments to tose following Assumption S-BM.2 apply ere. Assumption M-BM.3. (i) P (W (i) = c (i) min B (α, ) + i ) = 0 for all H and i = 1, 2. (ii) Eiter (a) tere is some (1) H and {γ n, (1)} Γ wit ζ({γ n, (1)}) int( K) suc tat lim inf P θ0,γ n, (1) (T n(θ 0 ) > c (1) min B (α, ĥn) + 1 ) = α or (b) tere is some (2) H and {γ n, (2)} Γ wit ζ({γ n, (2)}) int( L) suc tat lim inf P θ0,γ n, (2) (T n(θ 0 ) > c (2) min B (α, ĥn) + 2 ) = α. Similar comments to tose following Assumption S-BM.3 apply ere as well. Part (ii) is not required for te asymptotic size to be bounded above by α; it is only used to sow tat te lower bound is also equal to α. Proposition M-BM in Appendix III contains a sufficient condition for part (ii) to old. Assumption M-BM.4. Consider some fixed α (0, 1) and some pairs (δ (i), δ (i) ) [0, α δ (i) ] [0, α δ (i) ] for i = 1, 2. Consider any H c. (i) For any finite n and i = 1, 2, P θ0,γ n, ( c (i) min B (α, ĥn) + i < ) = 1. (ii) If ζ({γ n, }) K, ten lim sup P θ0,γ n, (T n (θ 0 ) > c (1) min B (α, ĥn) + 1 ) α. If ζ({γ n, }) L, ten lim sup P θ0,γ n, (T n (θ 0 ) > c (2) min B (α, ĥn) + 2 ) α. (iii) If ζ({γ n, }) L c K c, tere are some { γ (1) n, and ζ({ γ (2) n, }) L and }, { γ(2) n, c (i) (i) min B (α, ĥn( γ n, plim )) c (i) (i) min B (α, ĥn(γ n, )) 1 } Γ suc tat ζ({ γ(1) n, }) K wp 1 for i = 1, 2, were ĥn(γ n, ) (ĥn( γ (i) n, )) denotes te estimator of Assumption B.3 wen H 0 and te drifting sequence of parameters {γ n, } ({ γ (i) n, }) caracterize te true DGP. Assumption M-BM.4 consists of ig-level conditions tat are not difficult to verify in te typical application. It is used to ensure an upper bound on te NRP under drifting sequences of parameters for wic H c, tat is, cases for wic entries of 1 are infinite. By properly constructing te confidence set I α δ ( ), part (i) can be sown to old since in applications ĥn as finite entries in finite samples and te localized quantiles are finite for H. Part (ii) must be verified directly. In te applications we ave encountered, 31

33 tis is not difficult since te M-Bonf-Min CVs ave closed-form solutions. In applications, te sequences { γ (i) n, } in part (iii) can be obtained by simply scaling {γ n,} appropriately for i = 1, 2. Te probability limit condition is typically not difficult to verify because te CVs ave closed form solutions and/or tey are invariant to differences in drifting sequences wit te same localization parameter (leading to a ratio of exactly one in te limit). See te verification of te similar assumption, Assumption M-BM-PI.2 for te consistent model selection example in Appendix II for illustrative details. We may now establis te correct size of tests using M-Bonf-Min CVs. Teorem M-BM. Under Assumptions D and M-B.1, M-B.2(i) evaluated at δ (1) i = ᾱ (1) i corresponding to β i and c (1) and δ (2) i = ᾱ (2) i corresponding to β i and c (2), eiter (a) M-B.3 evaluated at β i or (b) c (j) B A (α, β i, ) is invariant to for j = 1, 2, M-B.4, M-B.5 and M-BM.1 troug M-BM.4 for i = 1,..., r, AsySz(θ 0, c M BM(α, ĥn)) = α. A similar sufficient condition to te one discussed following Lemma M-B in Appendix I applies ere as well. We at last consider te PI version of te M-Bonf-Min robust CVs to furter increase power wile retaining correct asymptotic size in tis context: c M BM P I(α, ĥn) c (1) min B P I (α, ĥn) + η 1 (ˆγ n,2 ), c (2) min B P I (α, ĥn) + η 2 (ˆγ n,2 ), if ˆζ K if ˆζ L max{c (1) min B P I (α, ĥn) + η 1 (ˆγ n,2 ), c (2) min B P I (α, ĥn) + η 2 (ˆγ n,2 )}, if ˆζ K c L c, were { c (i) min B P I (α, ĥn) = min c (i) B A P I (α, β 1, ĥn),..., c (i) B A P I (α, β r, ĥn), inf δ [δ (i),α δ (i) ] } c (i) B P I (α, δ, ĥn), and η i : H 2 R for i = 1, 2. Analogous remarks to tose made about c S BM P I (α, ĥn) apply. We now adapt some of te previous assumptions to te present framework and establis te correct asymptotic size of tests using tis PI CV. Assumption M-BM-PI.1. (i) η i : H 2 R is a continuous function for i = 1, 2. (ii) P (W (i) = c (i) min B P I (α, ) + η i ( 2 )) = 0 for all H and i = 1, 2. (iii) P (W (i) c min B P I (α, ) + η i ( 2 )) α for all H and i = 1, 2. 32

34 (iv) Eiter (a) tere is some (1) H and {γ n, (1)} Γ wit ζ({γ n, (1)}) int( K) suc tat lim inf P θ0,γ n, (1) (T n(θ 0 ) > c (1) min B P I (α, ĥn) + η 1 (ˆγ n,2 )) = α or (b) tere is some (2) H and {γ n, (2)} Γ wit ζ({γ n, (2)}) int( L) suc tat lim inf P θ0,γ n, (2) (T n(θ 0 ) > c (2) min B P I (α, ĥn) + η 2 (ˆγ n,2 )) = α. Parts (i)-(iii) are te MLLDs counterparts to Assumption S-BM-PI(i)-(iii) and tus may be considered analogously. As in te case of Assumption M-BM.3(ii), we can provide a sufficient condition for part (iv) to old tat may be easier to verify. See Proposition M- BM-PI in Appendix III for details. Assumption M-BM-PI.2. Consider some fixed α (0, 1) and some pairs (δ (i), δ (i) ) [0, α δ (i) ] [0, α δ (i) ] for i = 1, 2. Consider any H c. (i) For any finite n and i = 1, 2, P θ0,γ n, ( c (i) min B P I (α, ĥn) < ) = 1. (ii) If ζ({γ n, }) K, ten lim sup P θ0,γ n, (T n (θ 0 ) > c (1) min B P I (α, ĥn)+η 1 (ˆγ n,2 )) α. If ζ({γ n, }) L, ten lim sup P θ0,γ n, (T n (θ 0 ) > c (2) min B P I (α, ĥn) + η 2 (ˆγ n,2 )) α. (iii) If ζ({γ n, }) L c K c, tere are some { γ (1) n, and ζ({ γ (2) n, }) L and }, { γ(2) n, c (i) (i) min B P I (α, ĥn( γ n, plim )) c (i) (i) min B P I (α, ĥn(γ n, )) 1 } Γ suc tat ζ({ γ(1) n, }) K wp 1 for i = 1, 2, were ĥn(γ n, ) (ĥn( γ (i) n, )) denotes te estimator of Assumption B.3 wen H 0 and te drifting sequence of parameters {γ n, } ({ γ (i) n, }) caracterize te true DGP. Tis assumption is te direct adaption of Assumption M-BM.4 to te PI version of te M-Bonf-Min CVs. We may now establis te correct asymptotic size of te PI test. Corollary M-BM-PI. Under Assumptions D, PI, M-B.1, eiter (a) M-B.3 evaluated at β i or (b) c (j) B A (α, β i, ) is invariant to 1 for j = 1, 2 and i = 1,..., r, M-B.4, M-B.5, M-BA-PI(i) and (iii) evaluated at β i and corresponding ā (1) i ( ), ā (2) i ( ), M-BM.1, M-BM.2, M-BM-PI.1 and M-BM-PI.2, AsySz(θ 0, c M BM P I(α, ĥn)) = α. Assumption M-BA-PI is given in Appendix III. A straigtforward adaptation of te sufficiency conditions provided in Corollary SCF (in Appendix III) to te MLLDs context caracterizes a class of problems for wic setting i (η i (ˆγ n,2 )) to zero for i = 1 and/or 2 yields a test wit correct asymptotic size. 33

35 4 Testing After Conservative Model Selection We now illustrate a way to construct te objects used in c S BM P I (α, ĥn) for te ypotesis testing problem after conservative model selection introduced in Section We also sow ow te remainder of te assumptions imposed in Corollary S-BM-PI are satisfied for tis problem, enabling us to use a PI S-Bonf-Min CV to conduct a test wit correct asymptotic size. First, define ĥn = (ĥn,1, ˆγ n,2 ), were ĥ n,1 = n β2 ˆσ(n 1 X2 M [X 1 :X3 ] X2) wit β 1/2 2 (X2 M X 3 X2) 1 X2 M X 3 (Y X1θ 0 ) ( β 2 is te restricted least squares estimator of β 2, imposing H 0 ) and ˆγ n,2 can be defined as in AG as ˆγ n,2 = n 1 n i=1 x 1ix 2i (n 1 n i=1 x2 1i n 1 n i=1 x2 2i )1/2 wit {(x 1i, x 2i )} being te residuals from te regressions of x ji on x 3i for j = 1, 2. 7 let H = H and define te correspondence I β : H H as follows: ( 1, 2 ), if 1 = ± I β () = ([ z ξ, z 1 β+ξ ], 2 ), if 1 R, Second, were ξ = ξ(β) (0, β) is a continuous function of β, 1 H 1, 2 H 2 and z b denotes te b t quantile of te standard normal distribution. 8 Te function ξ(β) can be cosen to direct power against certain regions of H 1 (at te expense of oters). Te agnostic, and peraps intuitive, coice of ξ(β) would be simply ξ(β) = β/2. Tis is not te only configuration of I β tat will satisfy Assumption S-BM.2. For example, if te practitioner as a priori knowledge on te sign tat β 2 takes, tis can be incorporated into te analysis to improve power (see te next example). Wit tese definitions in and, we may verify te remaining assumptions for some cosen significance level α (0, 1). To see te proofs tat Assumptions S-BM.2 and S-B.4 are satisfied for δ tat can be set arbitrarily close, but not equal to zero, ceck Appendix II. A byproduct of tese proofs is tat Assumption S-B.3 is also satisfied for any β (0, 1). 7 Tis is clearly not te only construction of ĥn tat satisfies te relevant assumptions. 8 In tis and te following examples, I β () is defined at infinite values of 1 for teoretical completeness only. For te estimator ĥn we consider, it is in fact not possible to ave ĥn,1 = wit positive probability for any finite n. 34

36 Assumption PI follows from te te weak law of large numbers for L 1+d -bounded independent random variables. Now, te fact tat under H 0 and wen te drifting sequence {γ n, } wit 1 < caracterizes te true DGP, T n,1 (θ 0 ) ĥ n,1 T n,2 ˆT n,1 (θ 0 ) ĥ n,1 T n,2 d N d N 1 2 (1 2 2) 1/ , 1, (1 2 2) 1/2 0 2 (1 2 2) 1/ (7), (see Appendix II) makes it straigtforward to calculate functions ā i ( ) and η( ) satisfying Assumptions S-BA-PI(i)+(iii) and S-BM-PI via computer simulation. For example, working wit te parameter space wit H 2 = [ 0.99, 0.99] (limiting te absolute maximum asymptotic correlation between OLS estimators to 0.99) and setting α = 0.05, δ m = δ m = 0.005, c = 1.96 (standard pretesting), ξ(β) = β/2, r = 1 and β 1 = 0, ā 1 ( ) = α, wic clearly satisfies Assumption S-BA-PI(i) for δ 1 = 0. Furtermore, P (W sup c (1 ā 1 ( 2 ))) = P (W sup c (1, 2 )(1 α)) α I 0 ( ) 1 H 1 for all H and tere is some H suc tat P (W sup 1 H 1 c (1, 2 )(1 α)) = α, by te continuity properties of W and c (1 α) discussed in Section 2.1.1, so tat Assumption S- BA-PI(iii) is satisfied for β 1 = 0 and ā 1 ( ) = α. Finally, a function η( ) satisfying Assumption S-BM-PI for tis coice of ancor is simply given by η( ) = 0. Clearly η( ) is continuous over H 2 so tat Assumption S-BM-PI(i) is satisfied. Examination of te random variables W and and teir distribution functions reveals tat W c min B P I (α, ) is an absolutely continuous random variable so tat Assumption S-BM-PI(ii) is satisfied. Te function η( ) was constructed so tat Assumption B3-PI(iii) olds and can be set to (7.3,0), for example. Alternatively, we can use te ancor to direct power toward a point in H. Using te same values as above, but finding β 1 to minimize te distance between c 0,0.6 (0.95) and c S B A P I (0.05, β, (0, 0.6)) (in order to direct power toward = (0, 0.6)), we find β 1 = 0.89 wit corresponding ā 1 (x) = (0.99 x 0.91) + ( x )1(0.91 > x > 0.9) (0.9 x 0.61) + ( x )1(0.61 > x > 0.6) (0.6 x 0.21) + ( x )1(0.21 x 0.2) 35

37 (0.2 x 0). Tis function was constructed to satisfy Assumption S-BA-PI(i)+(iii) wile remaining as large as possible. Finally, a very small η function satisfying Assumption S-BM-PI for tis coice of ancor is given by η(x) = (0.61 x )1(0.61 > x > 0.58) (0.58 x 0.57) + ( x 0.54)1(0.57 > x > 0.55) (0.55 x 0.48) + ( x 0.47)1(0.48 > x > 0.47). Assumption S-BM-PI olds for identical reasons to tose given for te case of β 1 = 0 above. Here, can be set to (2.45,0.6), for example. Note tat, for eiter coice of ancor, η( ) is eiter numerically indistinguisable from zero or sup γ2 Γ 2 η(γ 2 ) = At te same time, c min B P I (α, ) ranges from about 2 to 23.7 so tat, even wen it is not equal to zero, te SCF is numerically dwarfed by c min B P I (α, ). In fact, upon artificially setting η( ) = 0 for te β 1 = 0.89 coice of ancor, te asymptotic size of te test is 5.1%, entailing minimal size distortion. 4.1 Critical Value Graps: An Illustration As an illustration of ow te Bonf-Min CVs operate, we grap c S BM P I (α, ) for te two different ancor coices corresponding to β 1 = 0 and β 1 = 0.89, discussed above. In te construction of te graps, α, δ m, δ m, c, ξ(β), ā 1 ( ) and η( ) were set to te same values as above. Figure 1 graps four CV functions for 2 = 0.9 as a function of 1 H 1 : te true localized CV function (c ), te PI LF CV (c LF P I ), te PI S-Bonf-Min CV function (c S BM P I ) using te PI LF CV as its ancor (β 1 = 0) and te PI S-Bonf-Min CV function using c S B A P I (0.05, 0.89, ) as its ancor. We examine te two different PI S-Bonf-Min CVs to compare te differences tat te coice of ancor makes in te construction of te CVs. We can see tat te PI S-Bonf-Min CV functions differentiate between different regions of 1. Toug bot S-Bonf-Min CVs conservatively mimic te underlying localized quantile function c (1 α), te CV using c S B A P I (0.05, 0.89, ) as its ancor appears to more closely track it tan does tat using c LF P I (0.05, 2 ). For all but a small portion of te parameter space H 1, te CV using c S B A P I (0.05, 0.89, ) as its ancor also lies below tat 9 Strictly speaking, for eiter coice of ancor, η(γ 2 ) > 0 for all γ 2 0 in tis problem. However, for practical purposes, for values of η( ) less tan 0.005, numerical approximation error dominates any miniscule size distortion tat may arise from setting η( ) exactly equal to zero. 36

38 using c LF P I (0.05, 2 ) so tat we can expect tests using te former to tend to ave iger power tan tests using te latter. Bot te S-Bonf-Min CVs lie above te true localized quantile function in order to account for te asymptotic uncertainty in estimating 1. At teir peaks, te S-Bonf-Min CVs are eiter sligtly larger or no larger tan te PI LF CV, entailing corresponding tests wit very minimal power loss over any portion of te parameter space. Wen 1 falls outside of te range of rougly [ 6, 6], te PI S-Bonf-Min CVs collapse to approximate te standard normal asymptotic CV wile te PI LF CV is constant and very ig for all values of 1. Tis enables tests using te PI S-Bonf-Min CVs to obtain substantially iger power wen ĥn,1 lies outside of tis range. Tis is also te case for large portions witin [ 6, 6]. Taking tese facts togeter, we can expect large power gains and minimal power loss from using S-Bonf-Min CVs. We can also see from tis grap ow te S- Bonf-Min CVs smoot between teir ancors and te standard normal CVs. In comparison, a tecnique using a binary decision rule (in te spirit of e.g., Andrews and Soares, 2010 and Andrews and Ceng, 2012) would coose te PI LF CV for ĥn,1 witin some range and te standard normal CV for ĥn,1 outside of tis range, rater tan adaptively using te data to transition between tem. We also examine te above CV functions for 2 = 0.6 to ascertain te generality of te above statements. We again see tat te S-Bonf-Min CV using c S B A P I (0.05, 0.89, ) as its ancor is smaller tan tat using c LF P I (0.05, 2 ) over most of H 1 and tat wen 1 lies outside of a given range, te S-Bonf-Min CVs are substantially smaller tan teir LF counterparts. We also again see te smooting described above. Te general features of tese two graps are sared for all values of 2 altoug te differences between te PI LF and S-Bonf-Min CVs srink as 2 approaces zero (compare te vertical axes in Figures 1 and 2). 4.2 Finite Sample Properties We now analyze ow te S-Bonf-Min CVs examined trougout tis section beave in a realistic sample size. To tis end, let us consider (an approximation to) te exact size and power of te two-sided post-conservative model selection t-test using tese PI S-Bonf-Min CVs at a sample size of n = 120. Te null ypotesis for te size analysis and te subsequent power analysis (to follow) is H 0 : θ = 0. Let us consider te simplest model for wic only te two regressors x 1 and x 2 enter. In te formation of te CVs, we used te exact same constructions as tose used in te above CV grap illustration. In order to keep te scale of te t-statistic similar across values of γ 2, we fixed te variance of te OLS estimators of θ and 37

39 β 2 to unity. Additionally, we fixed σ = 1 and used normally distributed data so tat x 1i and x 2i are normally distributed wit mean zero, variance 1/(1 γ 2 2) and covariance γ 2 /(1 γ 2 2) and ε i is drawn from an independent standard normal distribution. For te size analysis, we follow AG and, for a fixed value of γ 2, report te maximum NRP over te parameter space of Γ 1. We do tis by examining te point-wise NRP of te test for a wide range of values for β 2 and take te maximum over β 2 for eac γ 2 under study. 10 Te size values correspond to a 5% nominal level and are computed from 10,000 Monte Carlo replications, reported in Table 1. We examine te exact sizes of tests using c S P I BM (0.05, ĥn) wit ancor c LF P I (0.05, ˆγ n,2 ), c S P I BM (0.05, ĥn) wit ancor c S B A P I (0.05, 0.89, ĥn) and tose using te latter CV but ignoring te SCF function, tat is using c min B P I (0.05, ĥn), in order to assess te practical importance of te SCF function. 11 Table 1 indicates tat in tis finitesample scenario, t-tests using te first type of CV ave excellent size properties, wit a maximal NRP of over te entire grid of parameter values considered. Tests using te CV wit ancor c S B A P I (0.05, 0.89, ĥn) exibit a 3.6% size distortion. Te practical relevance of tis size distortion may be questionable as it occurs at extreme values of te correlation parameter, γ 2 = ±0.99. If we were to restrict te correlation parameter space to, say H 2 = [ 0.95, 0.95], we would not see tese size distortions. Moreover, tey disappear as te sample size grows: for n = 1200, te maximal NRP for γ 2 = 0.99 is equal to 4.8%. It is quite interesting to note tat ignoring te SCF (or artificially setting η( ) = 0) in tis problem does not introduce any additional size distortion. Moving on to te property te new CVs are designed to enance, we examine te power of te two-sided post-conservative model selection t-testing procedure using c S BM P I (α, ĥn) wit te two different ancors for various values of β 2, θ and γ 2 and compare it to te power corresponding to c LF P I (α, ĥn). Figures 3-7 are te finite sample (n = 120) power curves for tese tree testing procedures at a 5% level. Eac grap is plotted against te range -5 to 5 for β 2 at various fixed values of γ 2 and θ and based on 10,000 Monte Carlo replications (examining a range of β 2 is te finite sample counterpart to examining a range of 1 ). Due to te symmetry properties of te localized null limit distribution, results for γ 2 are quite similar: te corresponding power graps are a reflection of te graps for γ 2 across β 2 = 0. Te power graps display some interesting features. First, te power of te tests using c S BM P I (α, ĥn) wit eiter ancor is very good over most of te parameter space. 10 Specifically, we searc over a grid of values for β 2 in te interval [0, 10] using step sizes.0025,.025, and.25 on te intervals [0, 0.8], [0.8, 3] and [3, 10], respectively. We also examine te value of β 2 = 999, Note tat te SCF function for te CV wit ancor c LF P I (0.05, ˆγ n,2 ) is already equal to zero. Te 38

40 tests using c S BM P I (α, ĥn) acieve tis, wile retaining asymptotic validity, in part by using standard normal CVs wen β 2 is far enoug away from zero, wic is determined by ĥ n,1. Tis is a consequence of te minimum of Bonferroni CVs component of te S-Bonf- Min CVs. Second, it is evident tat te testing procedures using PI S-Bonf-Min CVs ave iger power tan tat using te PI LF procedure in all cases and often to a very large extent. For example, Figure 7 sows a power difference of nearly 100% between te testing procedures over almost te entire parameter space for β 2. For te S-Bonf-Min CV using te c S B A P I (0.05, 0.89, ĥn) ancor, altoug tere is a small region of te parameter space for wic c LF P I (α, ĥn) < c S BM P I (α, ĥn), tis region, and te difference between te CVs witin it, are so small tat any power dominance by te use of c LF P I (α, ĥn) is indiscernible. Tird, te tests using S-Bonf-Min CVs tend to maintain power at or above te maximum power of te tests using PI LF CVs over most of te parameter space of β 2. Fourt, te ranges in β 2 of low power are very small for tests using S-Bonf-Min CVs, in contrast to tose using c LF P I (α, ĥn). Fift, te envelope were te power of te tree tests coincide tends to be a very small portion of β 2 s parameter space. Sixt, te differences in power values over β 2 for a particular test is increasing in γ 2, as sould be expected. Tis brings us to a related feature: te differences in power between te testing procedures is larger for larger values of γ 2. Finally, we can see tat te S-Bonf-Min CV test using ancor c S B A P I (0.05, 0.89, ) as te best power performance. As displayed in Figures 1 and 2, tis CV is not always te smallest. However, te differences between te two S-Bonf-Min CVs at points for wic tat using te c LF P I (0.05, ĥn) ancor is smaller, are so small tat tey ave no perceptible effect on power. 5 Hypotesis Testing wen a Nuisance Parameter may be on a Boundary In tis section we sow ow to construct a PI S-Bonf-Min CV for te testing problem introduced in Section 2.1.2, testing wen a nuisance parameter may be on te boundary of its parameter space. Our construction of te confidence set I β is somewat different in te context of tis problem tan te previous. It is instructive to notice tis difference as it exemplifies te general feature tat te confidence set I β sould be tailored to te testing problem at and and provides some guidance on ow to construct tis object in different testing scenarios. First, ĥn (n 1/2 Xn,2 /ˆσ n,2, ˆγ n,2 ), were ˆγ n,2 = ˆρ n. Second, let H = R [ 1 + ω, 1 ω] 39

41 (note te difference wit H in tis context) and define I β : H H as follows: ( 1, 2 ), if 1 = I β () = I β ( 1, 2 ) = ([max{0, 1 + z ξ }, max{0, 1 + z 1 β+ξ }], 2 ), if 1 <, using te same notation as in te previous example. As mentioned in te previous example, ξ can be cosen to direct power against particular regions of te alternative ypotesis. Also, in te context of tis example, it makes sense to judiciously coose ξ. For example, Andrews and Guggenberger (2010b) illustrate tat wen examining lower one-sided tests, te limit of te test statistic W is stocastically decreasing (increasing) in 1 for 2 < 0 ( 2 0). Wen W is stocastically decreasing (increasing), large (small) values of ξ may increase te power of te test as te searc in over te CVs of te test statistic will be limited to a set corresponding to smaller values. Assumption PI follows immediately from te consistency of ˆρ n. Assumption S-B.4 follows from te definition of T n (θ 0 ), te continuous mapping teorem and te facts tat ˆρ n consistent and n1/2 Xn,1 /ˆσ n,1 n 1/2 Xn,2 /ˆσ n,2 d 0 + Z 2 1 by te central limit teorem. Fixing a significance level α (0, 1), it can be sown tat Assumption S-B.3 is satisfied for any β (0, 1) and Assumption S-BM.2 is satisfied for δ tat can be set arbitrarily close to zero (see Appendix II). For te sake of brevity, we omit te details on appropriate ā i ( ) and η( ) functions tat satisfy Assumptions S-BA-PI(i)+(iii) and S-BM-PI, simply noting tat teir general features are very similar to tose in te previous example. 6 Testing After Consistent Model Selection In tis section we sow ow to construct te objects used in c M BM P I (α, ĥn) as well as sowing ow te remainder of te assumptions of Corollary M-BM-PI are satisfied in te ypotesis testing after consistent model selection example introduced in Section First, let ĥn = ( n ˆβ 2 ρ n /σ β2,n, ˆγ n,2 ), were ˆγ n,2 = (σ θ,n, ρ n ). Second, let H = H and, using te same notation as in previous examples, ( 1, 2,1, 2,2 ), if 1 = ± I β () = I β ( 1, 2, 3 ) = ([ 1 + ρ n z ξ, 1 + ρ n z 1 β+ξ ], 2,1, 2,2 ), if 1 R, 40 is

42 were 1 H 1 and 2 = ( 2,1, 2,2 ) H 2. Te remarks made in Section 4 concerning te coice of ξ and configuration of I β also apply ere. For te additional estimator not present in te SLLD framework, let ˆζ = n ˆβ 2 /σ β2,nc n. Fix some α (0, 1). To see tat Assumption M-BM.2 is satisfied, note tat ĥ d n,1 nγ 1 + N(0, ρ 2 n) (8) so we ave, = ( 1, 2 ), if 1 = ± ( 1 + Z ρ, 2 ), if 1 R, were Z d ρ N(0, ρ 2 ). To satisfy tis assumption, δ (1) and δ (2) need only be set greater tan zero. Te proof tat Assumption M-BM.2 is satisfied for δ (i) (0, 1) is essentially te same as te proof of Assumption S-BM.2 for te post-conservative model selection example except for te additional condition of I β (ĥn), I β ( ) H wp 1 if H. Te same is true for sowing tat Assumption M-B.3 olds wen evaluated at any β (0, 1). It is clear tat tis additional condition is satisfied from te definition of I β ( ) and te expressions for ĥn and above. To see ow Assumption M-B.4 is satisfied, ceck Appendix II. Regarding Assumption M-B.5, under H 0 and {γ n, }, since c n, ˆζ = nβ2,n σ β2,nc n + 1 c n Z p ζ, were Z d N(0, 1). For Assumption PI, simply note tat ˆγ n,2 = (σ θ,n, ρ n ) = γ n,2 and te remainder of te assumption olds by construction. Now, te fact tat under H 0 and wen te drifting sequence {γ n, } wit 1 < caracterizes te true DGP, n( θ θ0 ) d N 1 2,1, 2 2,1(1 2 2,2) 0, ĥ n, ,2 n(ˆθ θ0 ) d N 0, 2 2,1 2,1 2 2,2 (9) ĥ n,1 1 2,1 2 2,2 2 2,2 (see Appendix II), makes it straigtforward to calculate functions ā (1) i ( ), ā (2) i ( ) and η i ( ) satisfying Assumption M-BA-PI(i)+(iii) and M-BM-PI.1(i)-(iii). For example, upon setting δ (i), δ (i) > 0 for i = 1, 2, r = 1 and β 1 = 0, ā (1) 1 = ā (2) 1 = α satisfy Assumption M-BA-PI(i)+(iii) in analogy wit tis coice of ancor in te post- 41

43 conservative model selection example. Furtermore, c (i) B A (α, 0, ) = c(i) LF P I (α, 2) sup c (i) ( 1, 2 )(1 α) 1 H 1 is invariant to 1 for i = 1, 2, satisfying condition (b) of Corollary M-BM-PI. For tis coice of ancor, we can simply set η i ( ) = 0 for i = 1, 2 witout te need to calculate via simulation because te MLLDs analog of te sufficient condition in Corollary SCF (in Appendix III) olds. To see tis and ow Assumptions M-BM-PI.1 and M-BM-PI.2 are satisfied in tis example, ceck Appendix II. 6.1 Finite Sample Properties As in te case for two-sided post-conservative model selection testing, we examine te finite sample beavior of te upper one-sided post-consistent model selection test but now based upon te PI M-Bonf-Min CVs using c (i) LF P I (α, ĥn) as te ancor in eac localized limit distribution corresponding to i = 1, 2. We use te exact same DGP as tat used in te postconservative example to conduct te analysis. Of course since te regressors are stocastic, tis violates te assumptions of Leeb and Pötscer (2005). Neverteless, as we sall see, size is well controlled for tis DGP, consistent wit te claim tat Leeb and Pötscer s (2005) analysis can be extended to cases of stocastic regressors. Wit te same DGP, te only differences wit te post-conservative example are (i) instead of te studentized t-statistic, we use te non-studentized version wile providing a PI estimator for te variance of ˆθ and (ii) we examine a model selection CV tat is growing in te sample size, namely, te BIC coice of c n = log n. We also use te same coices for ξ(β) as in te finite sample analysis of te post-conservative example as well as δ (i) and δ (i) values corresponding to te values of δ and δ in tat example. For te size analysis, we approximate te exact size using te same procedure. Table 2 reports te maximum over β 2 finite sample null rejection frequencies at a 5% nominal level. Te table sows tat te new test again as excellent size properties wit a maximal NRP of 5.2% over te entire grid of parameters considered. In tis example, construction of te PI M-Bonf-Min CV involves very little computation since η i ( ) = 0 and c M BM P I (α, ĥn) is a closed-form a function of ĥn and standard normal CVs. Moving on to te power properties of te test, we computed power functions for te 5% test corresponding to four representative values of γ 2,2, te correlation between te ˆθ and ˆβ 2 : 0.9, 0.6, 0.3 and 0. Te power functions are graped against β 2 in Figures 8-11 and are based on 10,000 Monte Carlo replications. Eac grap sows power curves as θ moves 42

44 away from te null value of zero for tree different values: θ = 0.1, 0.2 and 0.3. As in te post-conservative example, te results for γ 2,2 are quite similar. Te power curves in tis one-sided post-consistent model selection test are similar to tose in te two-sided post-conservative model selection test wit one important difference: tey do not seem to exibit as complex power beavior near β 2 = 0. As in te post-conservative example, power is very good over most of te parameter space. Power losses occur only over very small portions of te parameter space and, even at teir lowest points, tey are well above te nominal size of te text except wen θ is very close to te null value of zero. Te power losses occur at points surrounding β 2 = 0 since it is te area of te parameter space were c (1) min B P I (α, ĥn) is more likely to be selected by ˆζ. In tis particular problem, c (1) min B P I (α, ĥn) is a more conservative CV tan c (2) min B P I (α, ĥn), as te former is constructed by taking a maximum over an interval centered about ĥn,1 wile te latter is strictly a PI CV tat does not depend upon ĥn,1 (see Appendix II for details). 7 Conclusion Tis study provides new metods of size-correction for tests wen te null limit distribution of a test statistic is discontinuous in a parameter. Te CVs utilized by tese size-corrections entail power gains in a variety of circumstances over existing size-correction metods. Tey also enable one to conduct tests wit correct asymptotic size in a general class of testing problems for wic uniformly valid metods were previously unavailable. 43

45 8 Appendix I: Proofs of Main Results Tis appendix is composed of te proofs of te main results, followed by auxiliary lemmas used in tese proofs, some of wic are of independent interest. Proof of Teorem S-B: By te Teorem of te Maximum, Assumptions S-B.2 and S-B.3, c S B (α, δ, ) is continuous over H. Now, take any H and suppose te drifting sequence of parameters {γ n, } caracterizes te true DGP. Ten, Assumptions S-B.2 troug S-B.4 imply lim sup P θ0,γ n, (T n (θ 0 ) > c S B(α, δ, ĥn)) P (W c S B(α, δ, )) = P (W c S B(α, δ, ) c (1 δ)) + P (W c (1 δ) > c S B(α, δ, )) + P (c (1 δ) > W c S B(α, δ, )) P (W c (1 δ)) + P (c (1 δ) > c S B(α, δ, )) α (A.1) since c (1 δ) is te (1 δ) t quantile of W and P ( / I α δ ( )) α δ. Hence, we ave a uniform upper bound on te asymptotic NRP under all {γ n, } sequences. To establis full uniformity over Γ, let { γ n } be a sequence in Γ suc tat AsySz(θ 0, c S B(α, δ, ĥn)) lim sup sup P θ0,γ(t n (θ 0 ) > c S B(α, δ, ĥn)) γ Γ = lim sup P θ0, γ n (T n (θ 0 ) > c S B(α, δ, ĥn)). Suc a sequence always exists. Let { γ kn } be a subsequence of { γ n } suc tat lim sup P θ0, γ n (T n (θ 0 ) > c S B(α, δ, ĥn)) = lim P θ0, γ kn (T kn (θ 0 ) > c S B(α, δ, ĥk n )). Suc a subsequence always exists. Since H is compact, tere exists a subsequence of { γ kn }, call it { γ (kn) j }, for wic lim ((k n) j ) r γ (kn) j,1 H 1 and k Tat is, {γ (kn) j } = {γ (kn) j,} for some H. Hence, lim γ (k n) j,2 H 2. (k n) j AsySz(θ 0, c S B(α, δ, ĥn)) = lim P θ0, γ (kn)j,(t (kn) j (θ 0 ) > c S B(α, δ, ĥ(k n) j )) α, were te inequality follows from (A.1). Finally, δ = AsySz(θ 0, sup c (1 δ)) AsySz(θ 0, c S B(α, δ, ĥn)), H A.1

46 were te equality is te direct result of Teorem 2 in AG (Assumptions D, S-B.1 and S-B.2 imply Assumptions A, B, L and M(a) of teir paper old wit α replaced by δ ) and te inequality follows from te fact tat for all γ Γ and n. c S B(α, δ, ĥn) sup c (1 δ) sup c (1 δ) I β (ĥn) H Proof of Corollary S-B-PI: Te upper bound follows from identical arguments to tose made in te proof of Teorem S-B. For te lower bound, simply note tat for eac 2 H 2, tere is some 1 H 1 suc tat sup 1 H 1 c (1, 2 )(1 δ) = c ( 1, 2 )(1 δ) by virtue of te extreme value teorem and Assumption S-B.2(i). Hence, for some H wit sup 1 H 1 c (1, 2 ) (1 δ) = c (1 δ), AsySz(θ 0, c S B P I(α, δ, ĥn)) AsySz(θ 0, c LF P I (δ, ˆγ n,2 )) lim inf P θ 0,γ n, (T n (θ 0 ) > sup 1 H 1 c (1,ˆγ n,2 )(1 δ)) P (W > c (1 δ)) = δ, were te tird inequality follows from Assumptions S-B.1, PI and te continuity of sup 1 H 1 c (1, )(1 δ), te latter of wic is implied by Assumption S-B.2(i). Te equality follows from Assumption S-B.2(ii). Te sequence {γ n, } Γ exists by, cf., Lemma 7 of Andrews and Guggenberger (2010b). Proof of Teorem S-BA: By te same arguments as tose used in te proof of Teorem S-B, for any H, lim sup P θ0,γ n, (T n (θ 0 ) > c S B A(α, β, ĥn)) P (W sup c (1 ᾱ)). I β ( ) Tus, te same subsequencing argument as tat used in te proof of Teorem S-B provides tat tere is some H suc tat AsySz(θ 0, c S B A(α, β, ĥn)) P (W sup c (1 ᾱ)) α, I β ( ) were te latter inequality olds by S-BA(i) and te definition of ᾱ. On te oter and, AsySz(θ 0, c S B A(α, β, ĥn)) lim sup P θ0,γ n, (T n (θ 0 ) > c S B A(α, β, ĥn)) P (W c S B A(α, β, ) = α. Proof of Corollary S-BA-PI: By te Teorem of te Maximum, Assumptions S-BM.1, S- BA-PI(i) and S-B.3 imply tat c S B A P I (α, β, ) is continuous. Hence, by te same arguments as tose used in te proof of Teorem S-B, for any H, lim sup P θ0,γ n, (T n (θ 0 ) > c S B A P I(α, β, ĥn)) P (W sup c (1 ā( 2 ))). I β ( ) A.2

47 Te same subsequencing argument provides tat tere is some H suc tat AsySz(θ 0, c S B A P I(α, β, ĥn)) P (W by Assumptions PI and S-BA-PI(ii)-(iii). On te oter and, sup (1 ā( 2 ))) α I β ( ) AsySz(θ 0, c S B A P I(α, β, ĥn)) lim sup P θ0,γ (T n, n(θ 0 > c S B A P I(α, β, ĥn))) P (W c S B A P I(α, β, )) = α. Proof of Teorem S-BM: Lemma BM provides tat c min B (α, ) is continuous over H by Assumptions S-B.2(i) evaluated at δ i = ᾱ i, S-B.3 evaluated at β i, S-BM.1 and S-BM.2 for i = 1,..., r. Hence, using Assumptions S-B.1 and S-B.4, for any H, lim sup P θ0,γ n, (T n (θ 0 ) > c S BM(α, ĥn)) P (W c min B (α, ) + ). Tus, te same subsequencing argument as tat used in te proof of Teorem S-B provides tat tere is some H suc tat AsySz(θ 0, c S BM(α, ĥn)) P (W c min B (α, ) + ) α, were te latter inequality olds by Assumption S-BM.3(i) and te definition of. On te oter and, from Assumption S-BM.3(ii), AsySz(θ 0, c S BM(α, ĥn)) lim sup P θ0,γ n, (T n (θ 0 ) > c S BM(α, ĥn)) P (W c S BM(α, )) = α. Proof of Corollary S-BM-PI: Lemma BM provides tat c min B P I (α, ) is continuous over H by Assumptions S-B.3 evaluated at β i and S-BM.1 and S-BM.2 at pairs (δ m, δ m ) and (δ i, δ i ) for i = 1,..., r. Hence, using Assumptions S-B.1, S-B.4, PI and S-BM-PI(i), lim sup P θ0,γ n, (T n (θ 0 ) > c S BM P I(α, ĥn)) P (W c min B P I (α, ) + η( 2 )) for any H. Tus, te same subsequencing argument used in te proof of Teorem S-B provides tat tere is some H suc tat AsySz(θ 0, c S BM P I(α, ĥn)) P (W c min B P I (α, ) + η( 2 )) α, wic follows from Assumption S-BM-PI(ii)-(iii). On te oter and, using Assumption S-BM-PI(iii), te same argument used in te proof of Teorem S-BM provides AsySz(θ 0, c S BM P I(α, ĥn)) α. A.3

48 Proof of Teorem M-BM: Many parts of te proof are quite similar to te corresponding parts in te proof of Lemma M-B, so details are omitted. We now study lim sup P θ0,γ n, (T n (θ 0 ) > c M BM (α, ĥn)) for te same tree cases corresponding to Assumption M-B.1. Case 1: ζ({γ n, }) K. Using very similar arguments to tose used in te proof of Lemma M-B, we can sow tat for any H, lim sup P θ0,γ n, (T n (θ 0 ) > c M BM(α, ĥn)) lim sup P θ0,γ n, (T n (θ 0 ) c (1) min B (α, ĥn) + 1 ), (A.2) ere using Assumptions M-B.2(i) evaluated at δ (1) i = ᾱ (1) i corresponding to β i and c (1) and δ (2) i = ᾱ (2) i corresponding to β i and c (2), M-B.3 evaluated at β i, M-BM.1, M-BM.2(i) and M-BM.4(i) for i = 1,..., r to establis te finiteness of c (2) min B (α, ĥn) + 2 for any finite n and H. Now, for H c, Assumption M-BM.4(ii) immediately provides tat (A.2) is bounded above by α. For H, Assumptions M-B.2(i) evaluated at δ (1) i = ᾱ (1) i corresponding to β i and c (1) and δ (2) i = ᾱ (2) i corresponding to β i and c (2), M-B.3 evaluated at β i, for i = 1,..., r, M-BM.1 and M-BM.2, in conjunction wit Lemma BM, imply tat c (1) min B (α, ) + 1 is continuous so tat by Assumption M-B.4(i), (A.2) is bounded above by P (W (1) c (1) min B (α, ) + 1 ) α, were te inequality results from Assumption M-BM.3(i) and te definition of 1. Case 2: ζ({γ n, }) L. Te proof tat lim sup P θ0,γ n, (T n (θ 0 ) > c M BM (α, ĥn)) α for all H tat fall into tis category is again symmetric to te proof for Case 1. Case 3: ζ({γ n, }) K c L c. Assumptions M-B.2(i) evaluated at δ (1) i = ᾱ (1) i corresponding to β i and c (1) and δ (2) i = ᾱ (2) i corresponding to β i and c (2), M-B.3 evaluated at β i, for i = 1,..., r, M-BM.1, M-BM.2(i) and M-BM.4(i) and similar arguments to tose in te proof of Lemma M-B provide tat for any H, lim sup P θ0,γ n, (T n (θ 0 ) > c M BM(α, ĥn)) lim sup P θ0,γ n, (T n (θ 0 ) max{c (1) min B (α, ĥn) + 1, c (2) min B (α, ĥn) + 2 }), (A.3) in tis case. Assumptions M-B.1(iii) and M-BM.4(ii)-(iii) and a similar cange of measure to tat used in te proof of Teorem B-M provide tat (A.3) is bounded above by α for H c. For H, if W (3) is stocastically dominated by W (1), (A.3) is bounded above by lim sup P θ0,γ n, (W (1) c (1) min B (α, ĥn) + 1 ) P (W (1) c (1) min B (α, ) + 1 ) α, were we ave again use te continuity of c (1) min B (α, ) + 1 and te final inequality was establised above. If W (3) is stocastically dominated by W (2), te argument is symmetric. We can now establis tat AsySz(θ 0, c M BM (α, ĥn)) α using te same type of subsequencing argument as tat used in te proofs of all te main results. A.4

49 Finally, if (a) of Assumption M-BM.3(ii) olds, similar arguments to tose used to establis (A.4) yield AsySz(θ 0, c M BM(α, ĥn)) lim sup P θ0,γ (T n, (1) n(θ 0 ) > c M BM(α, ĥn)) lim inf P θ 0,γ (T n, (1) n(θ 0 ) > c (1) min B (α, ĥn) + 1 ) = α. If (b) of Assumption M-BM.3(ii) olds, te argument is symmetric. Proof of Corollary M-BM-PI: Te proof is very similar to te proof of Teorem M- BM so most details are omitted. Assumptions M-B.3 evaluated at β i for i = 1,..., r, M- BM.1, M-BM.2(i), M-BM-PI.1(i) and M-BM-PI.2(i) can be used to establis te finiteness of c (i) min B P I (α, ĥn) + η i (ˆγ n,2 ) for i = 1, 2. In tis case, Assumption M-BM-PI.1(iii) is used directly to establis te upper bound on limiting null rejection probabilities wen H. For te lower bound, Assumption M-BM-PI.1(iv) plays te role of M-BM.3(ii) in Teorem M-BM. Te following is an auxiliary Lemma used in many of te proofs. Lemma BM. Let H, D, H and B be metric spaces, f : H D R be a function tat is continuous in bot of its arguments and G : H B H be a compact-valued correspondence tat is continuous in bot of its arguments. If te set {(d, b) D B : d + b = a} is nonempty and compact, ten is continuous in y H. inf sup {(d,b) D B:d+b=a} x G(y,b) f(x, d) Proof : Te maximum teorem provides tat sup x G(y,b) f(x, d) is continuous in (y, b, d) H B D. A second application of te maximum teorem yields te lemma s claim since, as a correspondence from H into D B, {(d, b) D B : d + b = a} is compact-valued and trivially continuous. Te following lemma establises te correct asymptotic size of tests using M-Bonf CVs. Assumptions M-B.6 and M-B-LB can be found in Appendix III. Lemma M-B. Under Assumptions D and M-B.1 troug M-B.6 for β = α δ, If Assumption M-B-LB also olds, AsySz(θ 0, c M B (α, δ, ĥn)) α. P (W (i) (i) > c (i) (i) (1 δ) + ε i ) AsySz(θ 0, c M B (α, δ, ĥn)), were i = 1 if (a) olds and i = 2 if (b) olds. A.5

50 Proof: Take any H. We break down te proof tat lim sup P θ0,γ n, (T n (θ 0 ) > c M B (α, δ, ĥn)) α for any H into te tree cases corresponding to Assumption M-B.1. Case 1: ζ({γ n, }) K. Assumptions M-B.2(i), M-B.3 and M-B.6(i) provide tat for any H and finite n, P θ0,γ n, ( sup (1 δ) < ) = 1. Hence, for any H and ɛ > 0, P θ0,γ n, ( 1(ˆζ c (2) I α δ (ĥn) L)c (2) B (α, δ, ĥn) ɛ) P θ0,γ n, (ˆζ L) 0 by Assumption M-B.5 and te fact tat L int(k c ). Hence, using te fact tat L c = K ( K c L c ), lim sup + 1(ˆζ P θ0,γ n, (T n (θ 0 ) > c M B (α, δ, ĥn)) = lim sup P θ0,γ n, (T n (θ 0 ) > 1(ˆζ (2) L)c B (α, δ, ĥn) + 1(ˆζ K c L c ) max{c (1) B (α, δ, ĥn), c (2) B lim sup P θ0,γ n, (T n (θ 0 ) > 1(ˆζ L c )c (1) B (α, δ, ĥn) + 1(ˆζ = lim sup P θ0,γ n, (T n (θ 0 ) c (1) B (α, δ, ĥn)), (A.4) (α, δ, ĥn)}) (2) L)c B (α, δ, ĥn)) (1) K)c B (α, δ, ĥn) were te final equality follows from Assumption M-B.5 and te fact tat 1(x L c ) is continuous at any x K. If H c, Assumption M-B.6(ii) immediately implies tat (A.4) is bounded above by α. Suppose ten tat H. Given Assumptions M-B.2(i) and M- B.3, te Maximum Teorem implies tat c (1) B (α, δ, ) is continuous wen I β( ) H. Hence, Assumptions M-B.1(i), M-B.3 and M-B.4(i) imply tat (A.4) is bounded above by P (W (1) c (1) B (α, δ, )) P (W (1) c (1) (1 δ) + ε 1) + P ( / I α δ ( )) α, (A.5) were te first inequality is te result of te same type of Bonferroni arguments made in te proof of Teorem S-B and te second inequality follows from Assumptions M-B.2(ii) and M-B.3. Case 2: ζ({γ n, }) K. Te proof tat lim sup P θ0,γ n, (T n (θ 0 ) > c M B (α, δ, ĥn)) α for all H in tis case is symmetric to te proof for Case 1. Case 3: ζ({γ n, }) K c L c. Assumptions M-B.2(i), M-B.3 and M-B.6(i) provide tat for i = 1, 2, any H and finite n, so tat for any ɛ > 0, we ave P θ0,γ n, ( 1(ˆζ P θ0,γ n, ( sup c (i) (1 δ) < ) = 1 I α δ (ĥn) K)c (1) B (α, δ, ĥn) ɛ), P θ0,γ n, ( 1(ˆζ L)c (2) B (α, δ, ĥn) ɛ) 0 by Assumption M-B.5. Hence, similar expressions to tose leading to (A.4) yield lim sup P θ0,γ n, (T n (θ 0 ) > c M B (α, δ, ĥn)) A.6

51 lim sup P θ0,γ n, (T n (θ 0 ) max{c (1) B (α, δ, ĥn), c (2) B (α, δ, ĥn)}). (A.6) Suppose H c. If W (3) is stocastically dominated by W (1), (A.6) is bounded above by lim sup P θ0,γ n, (T n (θ 0 ) > c (1) B (α, δ, ĥn)) lim sup P θ0 (T, γ (1) n (θ 0 ) > c (1) B (α, δ, ĥn)) α, n, for some { γ (1) n, } Γ wit ζ({ γ(1) n, }) K satisfying Assumption M-B.6(iii), were te first inequality follows from Assumption M-B.6(iii) and te second from Assumption M-B.6(ii). If W (3) is stocastically dominated by W (2), te argument is symmetric. Now suppose H. If W (3) is stocastically dominated by W (1), (A.6) is bounded above by lim sup P θ0,γ n, (W (1) c (1) B (α, δ, ĥn)) P (W (1) c (1) B (α, δ, )) α, were te first inequality results from te continuity of c (1) B (α, δ, ) wen I β H and Assumption B.3 and te second inequality as been establised in (A.5). If W (3) is stocastically dominated by W (2), te argument is symmetric. Now, since we ave establised tat lim sup P θ0,γ n, (T n (θ 0 ) > c M B (α, δ, ĥn)) α for any H, AsySz(θ 0, c M B (α, δ, ĥn)) α follows from te same type of subsequencing argument as tat used in te proof of Teorem S-B. Finally, if (a) of Assumption M-B-LB olds, AsySz(θ 0, c M B (α, δ, ĥn)) lim sup P θ0,γ (T n, (1) n(θ 0 ) > c M B (α, δ, ĥn)) lim inf P θ 0,γ (T n, (1) n(θ 0 ) > c (1) B (α, δ, ĥn)) lim inf P θ 0,γ (T n, (1) n(θ 0 ) > sup c (1) (1 δ) + ε 1) H P (W (1) (1) > c (1) (1) (1 δ) + ε 1 ), were te second inequality follows from very similar arguments to tose used to establis (A.4) and te final inequality follows from Assumption M-B.1(i). If (b) of Assumption M-B-LB olds, te argument is symmetric. Clearly, if ε i = 0 and J (i) ( ) is continuous at c (i) (i)(1 δ), ten te lower bound is equal to δ. We now remark on an important, easy to verify, sufficient condition tat implies some of te imposed assumptions old: as a function from H in R, c (i) (1 δ) is continuous for i = 1 or 2. Clearly tis a strengtening of Assumption M-B.2(i). In conjunction wit te oter assumptions of Lemma M-B, it also implies tat one of te s in Assumption M-B-LB exists (but it does not imply te existence of corresponding γ sequences) and M-B.6(i)-(ii) olds. To see wy Assumptions M-B.6(ii) olds in tis case, notice tat application of te Maximum Teorem provides tat c (i) B (α, δ, ) is continuous and Bonferroni arguments yield P (W (i) c (i) B (α, δ, )) α. See te arguments surrounding (A.5) in te proof of Lemma M- B for details. Finally, if tis condition olds, te statements made in Assumption M-B.6(iii) A.7

52 wit regard to { γ (j) n, } may be dropped since ten te same arguments made in Case 3 of te proof for H may be generalized to all H. Te following lemma establises te correct asymptotic size of tests using PI M-Bonf CVs. Assumption M-B-PI-LB can be found in Appendix III. Lemma M-B-PI. Under Assumptions D, PI and M-B.1 troug M-B.6 for β = α δ, If Assumption M-B-PI-LB also olds, P (W (i) ( (i) > 1, 2 ) c(i) ( (i) AsySz(θ 0, c M B P I(α, δ, ĥn)) α. were i = 1 if (a) olds and i = 2 if (b) olds. 1, 2 )(1 δ) + ε i) AsySz(θ 0, c M B P I(α, δ, ĥn)), Proof: Te upper bound follows from identical arguments to tose made in Lemma M-B. For te lower bound, if (a) of Assumption M-B-PI-LB olds, AsySz(θ 0, c M B P I(α, δ, ĥn)) lim sup lim inf lim inf P (W (1) ( (1) P θ0,γ (T (1) n (θ 0 ) > c M n,( 1, B P I(α, δ, ĥn)) 2 ) P θ 0,γ (T (1) n (θ 0 ) > c (1) n,( 1, 2 ) B1 P I (α, δ, ĥn)) P θ 0,γ (T (1) n (θ 0 ) > sup c (1) n,( 1, 2 ) ( 1,ˆγ n,2 ) (1 δ) + ε 1) 1 H 1 > 1, 2 ) c(1) ( (1) 1, 2 )(1 δ) + ε 1), were te second inequality follows from very similar arguments to tose used to establis (A.4) and te final inequality follows from Assumptions M-B.1(i), PI and M-B-PI-LB. If (b) of Assumption M-B-PI-LB olds, te argument is symmetric. 9 Appendix II: Derivation of Results for Examples 9.1 Testing After Conservative Model Selection Te following is a verification tat Assumption S-BM.2 is satisfied in te context of tis example for δ tat can be set arbitrarily close to zero. Assume tat H 0 olds and te drifting sequence of parameters {γ n, } caracterizes te true DGP. (i) First, te weak law of large numbers for L 1+d -bounded independent random variables p provides tat ˆγ n,2 2. Results used to sow (S11.10) in Andrews and Guggenberger (2009c) provide tat for 1 <, ĥn,1 d Z, were Z d N(0, 1). Furtermore, wen 1 =, ĥn,1 p 1 since ĥn,1 = n 1/2 β2 /(σq 22 + o p (1)) and Hence, n 1/2 β2 = σ(q 22 ) 1/2 n 1/2 γ n,,1 + σn 1/2 (X 2X 2 ) 1 X 2ε = σ(q 22 ) 1/2 n 1/2 γ n,,1 + O p (1). = ( 1, 2 ), if 1 = ± ( Z, 2 ), if 1 R. A.8

53 Second, take any β (0, 1). Using te definition of I β, if 1 = ±, ten If 1 R, P θ0,γ n, ( I β ( )) = P (( 1, 2 ) ([ 1 + = P (Z [z ξ, z 1 β+ξ ]) = 1 β. P θ0,γ n, ( I β ( )) = P ( = ) = Z z ξ, Z z 1 β+ξ ], 2 )) Tird, as defined, I β is clearly compact-valued for any β (0, 1). Finally, to sow continuity of I β over H, first note tat for β > 0, I β () = ([ z ξ, z 1 β+ξ ], 2 ). Now, take any H = R [ 1 + ω, 1 ω] and let { k } be a sequence in H suc tat k. For eac k, take any b k I β ( k ) suc tat b k b. Ten, b k = (a k ( 1,k ,k z ξ) + (1 a k ) ( 1,k ,k z 1 β+ξ), 2,k ), for some a k [0, 1], were is a binary operator defined for any pair (a, b) [0, 1] R as follows: ab, if a 0 a b = 0, if a = 0. Now note tat for (a, b, c) [0, 1] R [ 1 + ω, 1 ω], f(a, b, c) a (b + 1 c 2 z ξ ) + (1 a) (b + 1 c 2 z 1 β+ξ ) a(b + 1 c 2 z ξ ) + (1 a)(b + 1 c 2 z 1 β+ξ ), if a (0, 1) = b + 1 c 2 z 1 β+ξ, if a = 0 b + 1 c 2 z ξ, if a = 1 is continuous in all tree arguments. Ten, since bot b k and k converge and [0, 1] is compact, a k a [0, 1]. Hence, b = (a ( z ξ )+(1 a) ( z 1 β+ξ ), 2 ) I β () so tat I β is upper emicontinuous. Now, take any H and let { m } be a sequence in H suc tat m. For eac b I β (), b = (a ( z 1+ξ ) + (1 a) ( z 1 β+ξ ), 2 ) for some a [0, 1]. Ten, for b m = (a ( 1,m ,mz ξ ) + (1 a) ( 1,m ,mz 1 β+ξ ), 2,m ) I β ( m ), b m b so tat I β is also lower emicontinuous and terefore continuous for any β (0, 1). (ii) First note tat 2 is trivially continuous in β. Similarly for β > 0, wen 1 =, [ z ξ, z 1 β+ξ ] = [ 1, 1 ] is trivially continuous in β. Very similar arguments to tose used in (i) above sow tat [ z 1, z 2 ] is continuous in (z 1, z 2 ) {( z 1, z 2 ) R 2 : z 1 z 2 } wen 1 <. Since z ξ = z ξ(β) and z 1 β+ξ = z 1 β+ξ(β) are continuous in β (0, 1), we ave te continuity of I β () in β over (0, 1) for all H. A.9

54 To see tat Assumption S-B.4 is satisfied, assume tat H 0 olds and te drifting sequence of parameters {γ n, } caracterizes te true DGP. Assume 1 <. Using Lemma S3 of Andrews and Guggenberger (2009c), ĥ n,1 = n1/2 β 2 /σ n + (n 1 X 2X 2 ) 1 n 1/2 X 2ε (ˆσ/σ n )(n 1 X 2M X1 X 2 ) 1/2 = 1 + (Q 22 n ) 1/2 (Q n,22 ) 1 e 2n 1/2 X ε + o p (1) were X = [x 1,..., x n ] and te second equality follows analogously to (S11.18) in Andrews and Guggenberger (2009c). Combining tis wit expressions (S11.18) and (S11.25) of Andrews and Guggenberger (2009c), T n,1 (θ 0 ) ĥ n,1 T n,2 = 1 2 (1 2 ) 1/2 + Q 1/2 n,11 n 1/2 e 1X ε 1 + (Q 22 n ) 1/2 Q 1 n,22e 2n 1/2 X ε 1 + (Q 22 n ) 1/2 (e 2 Q n,12 Q 1 n,11e 1 ) n 1/2 X ε + o p(1) by te Lindberg Central Limit Teorem and te Cramér-Wold device, were te limiting random vector is a multivariate normal. Similarly, ( ˆT n,1 (θ 0 ), ĥn,1, T n,2 ) d (Ẑ,1, 1, Z,2 ), a multivariate normal random vector. If 1 =, ĥn,1 p 1 so tat we again obtain joint convergence of te above random tree-vectors. Now, take any (t 1, t 2 ) R 2 and fixed x R, d P θ0,γ n, (t 1 T n (θ 0 ) + t 2 ĥ n,1 x) = P θ0,γ n, (t 1 T n,1 (θ 0 ) + t 2 ĥ n,1 x, T n,2 c) + P θ0,γ n, (t 1 ˆT n,1 (θ 0 ) + t 2 ĥ n,1 x, T n,2 > c) P (t 1 Z,1 + t 2 1 x, Z,2 c) + P (t 1 Ẑ,1 + t 2 1 x, Z,2 > c) = P (t 1 W + t 2 1 x), since W d Z,1 I( Z,2 c) + Ẑ,1 I( Z,2 > c) by (S11.10) of Andrews and Guggenberger (2009c). Hence, te Cramér-Wold device yields te joint convergence of T n (θ 0 ) and ĥn,1. p Finally, since ˆγ n,2 2, we ave te joint convergence of T n (θ 0 ) and ĥn = (ĥn,1, ˆγ n,2 ). Here we sow tat (7) olds. Assume H 0 olds and te drifting sequence of parameters {γ n, } caracterizes te true DGP wit 1 <. We ave already sown te joint convergence of te two random vectors to multivariate normal random vectors above. Te asymptotic covariance matrix is all tat remains to be sown. Andrews and Guggenberger (2009c) sow Var( Z,1 ) = Var(Ẑ,1) = Var(Z,2 ) = 1, Cov( Z,1, Z,2 ) = 0 and Cov(Ẑ,1, Z,2 ) = 2. Using te representation above, we obtain te following results in a similar fasion: Cov( Z,1, 1 ) = lim E Gn (Q 22 n ) 1/2 Q 1 n,22e 2n 1 X X e 1 Q 1/2 n,11 Z,1 1 Z,2 = lim (Q 22 n ) 1/2 Q 1 n,22e 2Q n e 1 Q 1/2 n,11 = lim (Q 22 n ) 1/2 Q 1 n,22q n,21 Q 1/2 n,11 A.10,

55 = lim γ n,,2 (1 γ 2 n,,2) 1/2 = 2 (1 2 2) 1/2, Cov(Z,2, 1 ) = lim E Gn (Q 22 n ) 1/2 Q 1 n,22e 2n 1 X X (e 2 Q n,12 Q 1 n,11e 1 )(Q 22 n ) 1/2 = lim (Q 22 n ) 1/2 Q 1 n,22e 2Q n (e 2 Q n,12 Q 1 n,11e 1 )(Q 22 n ) 1/2 = lim (Q 22 n ) 1/2 Q 1 n,22(q n,22 Q 2 n,12q 1 n,11)(q 22 n ) 1/2 = lim 1 γn,,2 2 = 1 2 2, Cov(Ẑ,1, 1 ) = lim E Gn (Q 22 n ) 1/2 Q 1 n,22e 2n 1 X X (e 1 Q n,12 Q 1 n,22e 2 )(Q 11 n ) 1/2 = lim (Q 22 n ) 1/2 Q 1 n,22e 2Q n (e 1 Q n,12 Q 1 n,22e 2 )(Q 11 n ) 1/2 = lim (Q 22 n ) 1/2 Q 1 n,22(q n,12 Q n,12 Q 1 n,22q n,22 )(Q 11 n ) 1/2 = Testing wen a Nuisance Parameter may be on a Boundary To verify tat te distribution function of W is given by (4), note tat by (3), W d Z 2,1, if Z 2,2 1 Z 2,1 + 2 (Z 2,2 + 1 ), if Z 2,2 < 1 so tat Since Z 2,1 is equal to P (W x) = P ( Z 2,1 x, Z 2,2 1 ) (A.7) + P ( Z 2,1 + 2 (Z 2,2 + 1 ) x, Z 2,2 < 1 ). (A.8) d 2 Z 2, Z, were Z d N(0, 1) and E[Z 2,2 Z] = 0, we ave tat (A.8) P ( ) ( ) 1 22 Z x x, Z 2,2 < 1 = Φ Φ( 1 ) For similar reasons, Z 2,2 conditional on Z 2,1 = z 1 is distributed N( 2 z 1, 1 2 2) so tat, letting f(z 2 z 1 ) denote te conditional density, (A.7) is equal to ( 1 ( ) ) f(z 2 z 1 )φ(z 1 )d z2 d z1 = 1 (1 2 2) 1/2 z2 2 z 1 φ dz x 1 x (1 2 ) 1/2 2 φ(z 1 )dz 1 ( 1 (1 ( ) ) 2 2 ) 1/2 = 1 φ z 2 d z 2 yielding te desired result. = x x ( 1 Φ ( 1 2 z (1 2 2) 1/2 A.11 )) φ(z)dz, 2 z 1 (1 2 2) 1/2 φ(z 1 )dz 1

56 Here we verify Assumption S-BM.2 and te claim tat δ can be set arbitrarily close to zero in tis example. Assume tat H 0 olds and te drifting sequence of parameters {γ n, } caracterizes te true DGP. p (i) First, ˆγ n,2 = ˆρ n 2 and te central limit teorem provides tat for 1 <, d n 1/2 Xn,2 /ˆσ n,2 1 + Z 2,2, were Z 2,2 p since ˆσ n,2 σ 2, n 1/2 γ n,,1 and d N(0, 1). Wen 1 =, n 1/2 Xn,2 /ˆσ n,2 p 1 n 1/2 Xn,2 = 1 n n i=1 [σ 2 γ n,,1 + e i2 ] = σ 2 n n n 1/2 γ n,,1 + O p (1) i=1 were e i2 is iid wit finite variance σ 2 2. Hence, = ( 1, 2 ) if 1 = ( 1 + Z 2,2, 2 ) if 1 R +. Second, take any β (0, 1). 1 =. If 1 R +, As in te previous example, P θ0,γ n, ( I β ( )) = 1 wen P ( I β ( )) = P ( 1 [max{0, 1 + Z 2,2 + z ξ }, max{0, 1 + Z 2,2 + z 1 β+ξ }]) P ( 1 [max{0, 1 + Z 2,2 + z ξ }, 1 + Z 2,2 + z 1 β+ξ ]) = P ( 1 [ 1 + Z 2,2 + z ξ, 1 + Z 2,2 + z 1 β+ξ ]) = P (Z 2,2 [z ξ, z 1 β+ξ ]) = 1 β since P ( 1 < 0) = 0. Tird, as defined, I β () is clearly compact-valued. Finally, to sow continuity of I β () over H, note tat for any β (0, 1), I β () = ([max{0, 1 +z ξ }, max{0, 1 + z 1 β+ξ }], 2 ). Now, take any H = R and let { k } be a sequence in H suc tat k. For eac k, take any b k I β ( k ) suc tat b k b. Ten we ave four cases: (i) if 1,k =, b k = k = ( 1,k + z ξ, 2,k ) = ( 1,k + z 1 β+ξ, 2,k ); (ii) if 1,k + z 1 β+ξ 0, b k = (0, 2,k ); (iii) if 1,k + z ξ 0 and 1,k + z 1 β+ξ > 0, b k = ((1 a k )( 1,k + z 1 β+ξ ), 2,k ) for some a k [0, 1]; (iv) if ( 1,k + z ξ ) (0, ), ten b k = (a k ( 1,k + z ξ ) + (1 a k )( 1,k + z 1 β+ξ ), 2,k ) for some a k [0, 1]. In oter terms, b k = (a k ( 1,k + z ξ ) + (1 a k ) ( 1,k + z 1 β+ξ ), 2,k ) for some a k [0, 1], were is a binary operator defined for any pair (a, b) [0, 1] H 1 as follows: ab, if b R + a b = 0, if b < 0 or bot b = and a = 0 b, if b = and a > 0. Now note tat for (a, b) [0, 1] H 1, f(a, b) a (b + z ξ ) + (1 a) (b + z 1 β+ξ ) A.12

57 = a(b + z ξ ) + (1 a)(b + z 1 β+ξ ), if (b + z ξ ) R + (1 a)(b + z 1 β+ξ ), if (b + z 1 β+ξ ) R + and (b + z ξ ) < 0 0, if (b + z 1 β+ξ ) < 0 b + z 1 β+ξ, if (b + z ξ ) = is clearly continuous in a. It is also continuous in b since lim b zξ f(a, b) = (1 a)(z 1 β+ξ z ξ ), lim b z1 β+ξ f(a, b) = 0 and lim b f(a, b) = b + z 1 β+ξ. Ten, te same arguments used for te previous example provide tat I β is upper emicontinuous. Similar reasoning yields lower emicontinuity so tat I β is continuous for all β (0, 1). Very similar arguments to tose used in te previous example establis te continuity of I β () in β over (0, 1) for all H. 9.3 Testing After Consistent Model Selection To prove tat Assumption M-B.1 olds for tis example wit te quantities as described in Section 2.2.1, note te following. As defined, part (i) of Assumption M-B.1 is implied by 1.(i) of Proposition A.2 of Leeb and Pötscer (2005) wen we treat te domain of Φ( ) as R. Part (ii) of Assumption M-B.1 follows similarly from 2.(i) of Proposition A.2 in Leeb and Pötscer (2005). To sow tat part (iii) of Assumption M-B.1 olds, we must examine te cases for wic γ n,,1 /k n = nβ 2,n /σ β2,nc n ±1. First, if nβ 2,n /σ β2,nc n 1 and c n nβ 2,n /σ β2,n or nβ 2,n /σ β2,nc n 1 and c n + nβ 2,n /σ β2,n, ten, by 1.(ii)-(iii) of Proposition A.2 in Leeb and Pötscer (2005), te null limit distribution of T n (θ 0 ) is given by J (1). Second, if nβ 2,n /σ β2,nc n 1 and c n nβ 2,n /σ β2,n or nβ2,n /σ β2,nc n 1 and c n nβ 2,n /σ β2,n, ten, by 2.(ii)-(iii) of Proposition A.2 in Leeb and Pötscer (2005), te null limit distribution of T n (θ 0 ) is given by J (2). Tird, suppose nβ2,n /σ β2,nc n 1 and c n nβ 2,n /σ β2,n r for some r R. If 1 lim nγn,,1 = lim nβ2,n ρ n /σ β2,n =, te null limit distribution of T n (θ 0 ), evaluated at x is given by x ( ) ( ) 1 u r + 2,2 1 2,1u Φ(r) + φ Φ du 2,1 2,1 (1 2 2,2) 1/2 (see 3. of Proposition A.2 in Leeb and Pötscer, 2005). Tis distribution function is increasing in r and its limit as r is equal to J (2) (x) so tat te null limit distribution of T n (θ 0 ) is stocastically dominated by J (2). If 1 =, te null limit distribution of T n (θ 0 ), evaluated at x is given by x ( ) ( ) 1 u r + 2,2 1 2,1u φ Φ du 2,1 2,1 (1 2 2,2) 1/2 (again see 3. of Proposition A.2 in Leeb and Pötscer, 2005), wile J (1) (x) = 0 for all x R, i.e., it is te distribution function of a pointmass at, and ence stocastically dominates. If 1 R, 2,2 lim ρ n = 0 since c n so tat nβ 2,n /σ β2,n. Hence, by 3. of A.13

58 Proposition A.2 in Leeb and Pötscer (2005), te null limit distribution of T n (θ 0 ), evaluated at x is given by Φ(r)Φ(x/ 2,1 + 1 ) + Φ( r)φ(x/ 2,1 ), wic is stocastically dominated by J (1) (x) if 1 0 and by J (2) (x) if 1 > 0. Te proof of te fourt and final case for wic nβ2,n /σ β2,nc n 1 and c n + nβ 2,n /σ β2,n s for some s R is quite similar to te previous case and is ence omitted. To see tat Assumption M-B.4 is satisfied, begin by noting tat under H 0 and {γ n, }, n( θ θ0 ) = n(x 1X 1 ) 1 X 1X 2 β 2,n + n(x 1X 1 ) 1 X 1ɛ n(ˆθ θ0 ) = n(x 1M X2 X 1 ) 1 X 1M X2 ɛ ĥ n,1 = nρ n β 2,n /σ β2,n + n(ρ n /σ β2,n)(x 2M X1 X 2 ) 1 X 2M X1 ɛ, were X i is te i t column of X for i = 1, 2 and ɛ = (ɛ 1,..., ɛ n ), so tat n( θ θ0 ) d N σ θ,n nγn,,1, σ2 θ,n (1 ρ2 n) 0 d ĥ n,1 nγn,,1 0 ρ 2 n n(ˆθ θ0 ) d 0 σ N, θ,n 2 σ θ,β2,nρ n /σ β2,n ĥ n,1 nγn,,1 σ θ,β2,nρ n /σ β2,n ρ 2 n W (1) 1 d, W (2) 1. Ten, for any fixed (t 1, t 2 ) R 2 and x R and {γ n, } suc tat ζ({γ n, }) / { 1, 1}, P θ0,γ n, (t 1 T n (θ 0 ) + t 2 ĥ n,1 x) = P θ0,γ n, (t 1 n( θ θ0 ) + t 2 ĥ n,1 x, ˆζ 1) + P θ0,γ n, (t 1 n(ˆθ θ0 ) + t 2 ĥ n,1 x, ˆζ > 1) P (t 1 W (1) + t 2 1 x), if ζ({γ n, }) < 1 P (t 1 W (2) + t 2 1 x), if ζ({γ n, }) > 1. Hence, te Cramér-Wold device, along wit te facts tat γ n,,2,1 = σ θ,n σ θ, = 2,1 and γ n,,2,2 = ρ n ρ = 2,2, yields te desired result. To see tat (9) olds, simply note tat te limiting random vectors in (9) are te limits of ( n( θ θ 0 ), ĥn,1) and ( n(ˆθ θ 0 ), ĥn,1) above. Assumption M-BM-PI.1(i) is trivially satisfied since η i ( ) = 0 for i = 1, 2. Examining te distribution functions of and W (i) for i = 1, 2 (see Sections 6 and 2.2.1, respectively), we can see tat for H: sup c (1) (1 δ) = 1 2,1 2,1 2,2 z ξ(α δ) + 2,1 (1 2 2,2) 1/2 z 1 δ, I α δ ( ) c (1) B A P I (α, 0, ) = c (1) LF P I (α, 2) = sup 1 H 1 { 1 2,1 + 2,1 (1 2 2,2) 1/2 z 1 α } =, A.14

59 so tat c (1) min B P I (α, ) = 1 2,1 + inf W (1) d 1 2,1, 1 1 δ [δ (1),α δ (1) ] { 2,1 2,2 z ξ(α δ) + 2,1 (1 2 2,2) 1/2 z 1 δ }, 2 2,1(1 2 2,2) ,2 W (1) c (1) min B P I (α, ) d N(0, 2 2,1) + inf { 2,1 2,2 z ξ(α δ) + 2,1 (1 2 2,2) 1/2 z 1 δ }, δ [δ (1),α δ (1) ] wic is an absolutely continuous random variable. Similarly, for H, W (2) c(2) min B P I (α, ) d N(0, 2 2,1)+min{ 2,1 z 1 α, inf δ [δ (2),α δ (2) ] 2,1 z 1 δ } d N(0, 2 2,1)+ 2,1 z 1 α so tat Assumption M-BM-PI.1(ii) is satisfied. We can see from te above expressions tat for H, δ (1) argmin δ [δ (1),α δ (1) ] c(1) B P I (α, δ, ) = argmin δ [δ (1),α δ (1) ] { 2,1 2,2 z ξ(α δ) + 2,1 (1 2 2,2) 1/2 z 1 δ }, wic is nonrandom. Similarly, for H, δ (2) argmin δ [δ (2),α δ (2) ] c(2) B P I (α, δ, ) = argmin δ [δ (2),α δ (2) ] 2,1z 1 δ = 2,1 z 1 α+ δ(2), wic is nonrandom, so tat te MLLDs analog of Corollary SCF olds. Tis implies tat Assumption M-BM-PI.1(iii) is satisfied wit η i ( ) = 0 for i = 1, 2. Tis latter result also guarantees tat part (b) of Assumption M-BM-PI.1(iv) is satisfied since lim inf P θ 0,γ n, (T n (θ 0 ) > c (2) min B P I (α, ĥn)) = P (W (2) > 2,1 z 1 α ) = α for all H wit ζ({γ n, }) L by Assumption M-B.4 and te continuity given by Lemma BM. To find te sequence {γ n, (2)}, take any sequence {γ n } wit lim nγn,1 int( L) H 1 = (, 1) (1, ) and γ n,2,2 = ρ n = 1/c n (γ n,2,1 need only converge to some 2,1 [η, M]) so tat lim γ n,1 /k n = lim nγn,1 int( L). For tis sequence, (2) = (lim nγn,1, 2,1, 0). Moving on to Assumption M-BM-PI.2, note tat c (1) min B P I (α, ĥn) sup Iα δ (ĥn) c(1) (1 δ) for any δ [δ (1), α δ (1) ]. Suppose P θ0,γ n, ( sup Iα δ (ĥn) c(1) (1 δ) = ) > 0 for some suc δ and H c. Tis means ) P θ0,γ n, ( sup { 1ˆγ n,2,1 + z 1 δˆγ n,2,1 (1 ˆγ n,2,2) 2 1/2 } = 1 [ĥn,1+ˆγ n,2,2 z ξ,ĥn,1+ˆγ n,2,2 z 1 α+δ+ξ ] A.15 > 0

60 and consequently P θ0,γ n, ( or [ĥn,1 + ˆγ n,2,2 z ξ, ĥn,1 + ˆγ n,2,2 z 1 β+ξ ]) > 0. d But tis is at odds wit te fact tat ĥn nγ n,,1 + N(0, ˆγ n,2,2) 2 under P θ0,γ n, and te definition of Γ for tis problem, and we arrive at a contradiction. Also, we clearly ave P θ0,γ n, ( c (2) min B P I (α, ĥn) = ) = 0 since c (2) min B P I (α, ĥn) = ˆγ n,2,1 z 1 α. Hence, Assumption M-BM-PI.2(i) is satisfied. Now, consider any H c and {γ n, } wit ζ({γ n, }) K. Using te facts tat, under γ n,, c (1) B A P I (α, 0, ĥn) = c (1) LF P I (α, ĥn)) = sup 1 H 1 { 1ˆγ n,2,1 + ˆγ n,2,1 (1 ˆγ n,2,2 ) 1/2 } =, sup c (1) (1 δ) = nγ n,,1 n ˆγ n,2,1ˆγ n,2,2 X 2M X1 ɛ I α δ (ĥn) σ β2,nx 2M X1 X 2 for any δ [δ (1), α δ (1) ], we ave We also ave so tat ˆγ n,2,1ˆγ n,2,2 z ξ + ˆγ n,2,1 (1 ˆγ 2 n,2,2) 1/2 z 1 δ c (1) min B P I (α, ĥn) = nγ n,,1 n ˆγ n,2,1ˆγ n,2,2 X 2M X1 ɛ σ β2,nx 2M X1 X 2 + inf { ˆγ n,2,1ˆγ n,2,2 z ξ + ˆγ n,2,1 (1 ˆγ n,2,2) 2 1/2 z 1 δ }. δ [δ (1),α δ (1) ] ( T n (θ 0 ) = nγ n,,1 + ) n X 1ɛ 1(ζ({γ X 1X n, }) + o p (1) 1) 1 + n X 1M X2 ɛ 1(ζ({γ X 1M n, }) + o p (1) > 1) X2 X 1 = nγ n,,1 + n X 1ɛ X 1X 1 + o p (1) lim sup P θ0,γ n, (T n (θ 0 ) > c (1) min B P I (α, ĥn)) = lim sup P θ0,γ n, ( n X 1ɛ + n ˆγ n,2,1ˆγ n,2,2 X 2M X1 ɛ X 1X 1 σ β2,nx 2M X1 X 2 inf { ˆγ n,2,1ˆγ n,2,2 z ξ + ˆγ n,2,1 (1 ˆγ n,2,2) 2 1/2 z 1 δ }) δ [δ (1),α δ (1) ] = P ( 2,1 (1 2 2,2) 1/2 Z 1 + 2,1 2,2 Z 2 inf { 2,1 2,2 z ξ(α δ) + 2,1 (1 2 2,2) 1/2 z 1 δ }) δ [δ (1),α δ (1) ] = P ( 2,1 (1 2 2,2) 1/2 Z 1 + 2,1 2,2 Z 2 2,1 2,2 z ξ(α δ) + 2,1 (1 2 2,2) 1/2 z 1 δ) P ( 2,1 (1 2 2,2) 1/2 Z 1 2,1 (1 2 2,2) 1/2 z 1 δ) + P ( 2,1 2,2 Z 2 2,1 2,2 z 1 ξ(α δ) ) A.16

61 = δ + ξ(α δ) < δ + α δ = α, were Z 1, Z 2 d N(0, 1) and δ = argmin δ [δ (1),α δ (1) ] { 2,1 2,2 z ξ(α δ) + 2,1 (1 2 2,2) 1/2 z 1 δ }. Now consider any H c and {γ n, } wit ζ({γ n, }) L. Ten, P θ0,γ n, (T n (θ 0 ) > c (2) min B P I (α, ĥn)) = P θ0,γ n, (T n (θ 0 ) > ˆγ n,2,1 z 1 α ) P (W (2) > 2,1 z 1 α ) = α. Hence, Assumption M-BM-PI.2(ii) is satisfied. Finally, suppose ζ({γ n, }) L c K c for some H c. Let γ (1) n,,1 = χ (1)γ n,,1 and γ (1) n,,2 = γ n,,2 for all n wit χ (1) (0, 1) so tat (1) lim n γ n,,1 = χ (1) lim nγn,,1 = 1 = ± and, recalling tat L c K c = { 1, 1}, (1) lim n γ n,,1 /k n = χ (1) lim nγn,,1 /k n { χ (1), χ (1) } ( 1, 1) = K. Under {γ n, }, by results derived above, c (1) min B P I (α, ĥn) = nγ n,,1 γ n,,2,1 + O p (1) so tat c (1) (1) min B P I (α, ĥn( γ n, plim )) c (1) min B P I (α, ĥn(γ n, )) = plim nχ (1) γ n,,1 γ n,,2 + O p (1) nγ n,,1 γ n,,2 + O p (1) = χ (1) 1. Now let γ (2) n,,1 = χ (2)γ n,,1 and γ (1) n,,2 = γ n,,2 for all n wit χ (2) (1, ) so tat lim n γ (2) n,,1 = χ (2) lim nγn,,1 = 1 and (2) lim n γ n,,1 /k n = χ (2) lim nγn,,1 /k n { χ (2), χ (2) } [, 1) (1, ] = L. Under {γ n, }, c (2) min B P I (α, ĥn) = ˆγ n,2,1 z 1 α so tat c (2) (2) min B P I (α, ĥn( γ n, plim )) c (2) min B P I (α, ĥn(γ n, )) = plim 1 = 1. Hence, Assumption M-BM-PI.2(iii) olds. 10 Appendix III: Auxiliary Assumptions and Results In tis appendix, we provide some auxiliary results mentioned in te main text as well as some auxiliary assumptions enforced in te lemmas of Appendix I. Corollary SCF. Suppose Assumptions D, S-B.1, S-B.2(i) evaluated at te δ i = ᾱ i tat corresponds to β i, S-B.3 evaluated at β i, S-B.4, S-BM.1 and S-BM.2 old for i = 1,..., r. If tere are some H and ĩ {1,..., r} suc tat P (W sup Iβĩ( ) c (1 ᾱĩ)) = α and, for every H, eiter (a) P (c S BM (α, ) = c S B (α, δ, )) = 1 for some nonrandom δ [δ, α δ] or (b) P (c S BM (α, ) = c B A (α, βî, )) = 1 for some nonrandom î {1,..., r}, ten AsySz(θ 0, c S BM(α, ĥn)) = α A.17

62 wen = 0. Suppose instead tat Assumptions D, PI, S-B.1, S-B.3 evaluated at β i, S-B.4, PI, S- BA-PI(i) and (iii) evaluated at β i and corresponding ā i ( ), S-BM.1 and S-BM.2 evaluated at pairs (δ m, δ m ) and (δ i, δ i ) old for i = 1,..., r. If tere are some H and ĩ {1,..., r} suc tat P (W sup Iβĩ( ) c (1 āĩ( 2))) = α and, for every H eiter (a) P (c S BM P I (α, ) = c S B P I (α, δ, )) = 1 for some nonrandom δ [δ m, α δ m ] or (b) P (c S BM P I (α, ) = c B A P I (α, βî, )) = 1 for some nonrandom î {1,..., r}, ten wen η( ) = 0. AsySz(θ 0, c S BM P I(α, ĥn)) = α Proof: We provide te proof for te non-pi case as tat for te PI case is quite similar. By te same arguments as tose used in te proof of Teorem S-BM, for any H, If (a) olds, lim sup P θ0,γ n, (T n (θ 0 ) > c S min B(α, ĥn)) P (W c min B (α, )). P (W c min B (α, )) = P (W c S B(α, δ, )) α, were te inequality follows from (A.1). If (b) olds, P (W c min B (α, )) = P (W c B A (α, βî, )) α. We ten obtain AsySz(θ 0, c S BM (α, ĥn)) α by identical arguments to tose used in te proof of Teorem S-B. Finally, since c S BM (α, ĥn) c B A (α, βĩ, ĥn) for all γ Γ and n, α = P (W sup c (1 ᾱĩ)) AsySz(θ 0, c B A (α, βĩ, ĥn)) AsySz(θ 0, c S BM(α, ĥn)), I βĩ( ) similarly to te proof of Teorem S-BA. Te enforcement of Assumption S-B.2 for δ = α in te above corrolary is only used to establis te lower bound of α on te asymptotic size. Te new condition tat ensures size control in te absence of a SCF olds, for example, in te non-pi context wen δ is invariant to and in te PI context wen δ is invariant to 1. Te following proposition provides a sufficient condition for part (ii) of Assumption S- BM.3 to old. Proposition S-BM. (i) For > 0, Assumption S-BM.3(ii) olds if te distribution function of W W c min B (α, ), J ( ) say, is continuous over H and strictly increasing at () for all Ĥ { H : J (0) 1 α}. (ii) For = 0, Assumption S-BM.3(ii) olds if tere are some H and ĩ {1,..., r} suc tat P (W sup Iβĩ( ) c (1 ᾱĩ)) = α. A.18

63 1 Proof: (i) Note tat for Ĥ, () = J (1 α) is continuous over Ĥ H (by te continuity of J in ) and tat Ĥ is compact (being te preimage of a closed set of a continuous function). Hence, te extreme value teorem provides tat = sup Ĥ () = ( ) for some Ĥ so tat P (W cs BM (α, )) = P (W c min B (α, )+ ( )) = α since Ĥ. (ii) P (W c S BM (α, )) α by construction and P (W c S BM (α, )) P (W sup Iβĩ( ) c (1 ᾱĩ)) = α. Te following two assumptions are used in Lemma M-B in Appendix I. Assumption M-B.6. Consider some fixed (α, δ) (0, 1) [0, α] and any H c. (i) For any finite n and i = 1, 2, P θ0,γ n, ( sup c (i) (1 δ) < ) = 1. I α δ (ĥn) (ii) If ζ({γ n, }) K, ten lim sup P θ0,γ n, (T n (θ 0 ) > c (1) B (α, δ, ĥn)) α. If ζ({γ n, }) L, ten lim sup P θ0,γ n, (T n (θ 0 ) > c (2) B (α, δ, ĥn)) α. (iii) If ζ({γ n, }) L c K c, tere are some { γ (1) n, and ζ({ γ (2) n,,1 }) L and sup Iα δ plim (ĥn( γ(i) n, sup Iα δ (ĥn(γ n,)) c(i) }, { γ(2) n, c(i) )) (1 δ) } Γ suc tat ζ({ γ(1) n,,1 }) K (1 δ) 1 wp 1 for i = 1, 2, were ĥn(γ n, ) (ĥn( γ (i) n, )) denotes te estimator of Assumption B.3 wen H 0 and te drifting sequence of parameters {γ n, } ({ γ (i) n, }) caracterize te true DGP. Similar comments to tose following Assumption M-BM.4 apply to te above assumption. Te following assumption is used only to establis a lower bound on te size of a test using M-Bonf CVs. It is not required to sow size control. Assumption M-B-LB. Eiter (a) sup H c (1) (1 δ) = c(1) (1 δ) for some (1) H and (1) tere is some {γ n, (1)} Γ wit ζ({γ n, (1)}) int( K) or (b) sup H c (2) (1 δ) = c(2) (1 δ) (2) for some (2) H and tere is some {γ n, (2)} Γ wit ζ({γ n, (2)}) int( L). Like Assumption M-B.2, tis assumption is also essentially a continuity condition. An appeal to te continuity in of one of te localized quantiles and te extreme value teorem can be used to verify it so long as te corresponding condition on te normalized drifting sequence is sown to old. We replace Assumption M-B-LB by te following to establis te lower bound on te asymptotic size of te corresponding test using PI M-Bonf CVs. A.19

64 Assumption M-B-PI-LB. Consider some fixed δ (0, 1). For some 2 H 2, eiter (a) sup 1 H 1 c (1) ( 1, 2 )(1 δ) = c(1) δ) for some ( (1) 1, (1) 2 )(1 1 H 1, sup 1 H 1 c (1) ( 1, )(1 δ) is continuous at 2 as a function into R and tere is some {γ n,( (1) 1, 2 )} Γ wit ζ({γ n,( (1) 1, 2 )}) int( K) or (b) sup 1 H 1 c (2) ( 1, 2 )(1 δ) = c(2) ( (2) 1, 2 )(1 δ) for some (2) 1 H 1, sup 1 H 1 c (2) ( 1, ) (1 δ) is continuous at 2 as a function into R and tere is some {γ n,( (2) 1, 2 ζ({γ (2) n,( 1, 2 )}) int( L). )} Γ wit Similar comments wit regard to continuity to tose following Assumption M-B-LB apply ere as well. We now present formal statements regarding te asymptotic size of tests using (PI) M- Bonf-Adj CVs as well as additional imposed assumptions. Te proofs of te following two corollaries are very similar to tose found in Appendix I and are ence omitted. Assumption M-BA. (i) P (W (i) = c (i) B A (α, β, )) = 0 for all H and i = 1, 2. (ii) Eiter (a) P (W (1) c (1) B A (α, β, (1) )) = α for some (1) H and tere is some {γ n, (1)} Γ wit ζ({γ n, (1)}) int( K) or (b) P (W (2) c (2) B A (α, β, (2) )) = α for some (2) H and tere is some {γ n, (2)} Γ wit ζ({γ n, (2)}) int( L). Similar remarks to tose following Assumption S-BA apply ere. Corollary M-BA. Under Assumptions D, M-B.1, M-B.2(i) evaluated at δ = ᾱ (1) for i = 1 and δ = ᾱ (2) for i = 2, M-B.3 troug M-B.6, replacing α δ by β and δ by ᾱ (1) for i = 1 and ᾱ (2) for i = 2 in parts (i) and (iii) of Assumption M-B.6 and c (i) B by c(i) B A for i = 1, 2 in part (ii) of Assumption M-B.6, and M-BA, AsySz(θ 0, c M B A(α, β, ĥn)) = α. Assumption M-BA-PI. (i) For i = 1, 2, ā (i) : H 2 [δ (i), α δ (i) ] is a continuous function. (ii) For i = 1, 2, P (W (i) = c (i) B A P I (α, β, )) = 0 for all H. (iii) For i = 1, 2, P (W (i) sup Iβ ( ) c(i) (1 ā(i) ( 2 ))) α for all H. (iv) Eiter (a) P (W (1) c (1) B A P I (α, β, (1) )) = α for some (1) H and tere is some {γ n, (1)} Γ wit ζ({γ n, (1)}) int( K) or (b) P (W (2) c (2) B A P I (α, β, (2) )) = α for some (2) H and tere is some {γ n, (2)} Γ wit ζ({γ n, (2)}) int( L). Similar remarks to tose following Assumption S-BA-PI apply ere. Corollary M-BA-PI. Under Assumptions D, PI, M-B.1, M-BM.1, M-B.3 troug M-B.6, replacing α δ by β and δ by ā (1) (ˆγ n,2 ) for i = 1 and ā (2) (ˆγ n,2 ) for i = 2 in parts (i) and (iii) of Assumption M-B.6 and c (i) B by c(i) B A P I for i = 1, 2 in part (ii) of Assumption M-B.6, and M-BA-PI, AsySz(θ 0, c M B A P I(α, β, ĥn)) = α. We now present te final two sufficient conditions, useful for verifying Assumptions imposed in Teorem M-BM and Corollary M-BM-PI. A.20

65 Proposition M-BM. Suppose Assumptions D, M-B.1, M-B.2(i) evaluated at δ (1) i = ᾱ (1) i i corresponding to β i and c (2), eiter (a) M- corresponding to β i and c (1) and δ (2) i = ᾱ (2) B.3 evaluated at β i or (b) c (1) B A (α, β i, ) and c (2) B A (α, β i, ) are invariant to, M-B.4, M- B.5, M-BM.1, M-BM.2 and M-BM.3(i) old for i = 1,..., r. If eiter (a) P (W (1) (1) c (1) min B (α, (1) ) + 1 ) = α for some (1) H and tere is some {γ n, (1)} Γ wit ζ({γ n, (1)}) int( K) or (b) P (W (2) c (2) (2) min B (α, (2) ) + 2 ) = α for some (2) H and tere is some {γ n, (2)} Γ wit ζ({γ n, (2)}) int( L), ten Assumption M-BM.3(ii) olds. Proof: Suppose (a) olds. Ten, α lim sup P θ0,γ (T n, (1) n(θ 0 ) > c (1) min B (α, ĥn) + 1 ) lim inf P θ 0,γ (T n, (1) n(θ 0 ) > c (1) min B (α, ĥn) + 1 ) P (W (1) (1) > c (1) min B (α, (1) ) + 1 ) = α, were te first inequality is sown in te proof of Teorem M-BM. Te proof wen (b) olds is very similar. Proposition M-BM-PI. Suppose Assumptions D, PI, M-B.1, eiter (a) M-B.3 evaluated at β i or (b) c (1) B A (α, β i, ) and c (2) B A (α, β i, ) are invariant to 1 for i = 1,..., r, M-B.4, M-B.5, M-BM.1, M-BM.2 and M-BM-PI.1(i)-(iii) old. If eiter (a) P (W (1) (1) 2 )) = α for some (1) H and tere is some {γ n, (1)} Γ c (1) min B P I (α, (1) ) + η 1 ( (1) wit ζ({γ n, (1)) int( K) or (b) P (W (2) (2) c (2) min B P I (α, (2) ) + η 2 ( (2) 2 )) = α for some (2) H and tere is some {γ n, (2)} Γ wit ζ({γ n, (2)) int( L), ten Assumption M-BM-PI.1(iv) olds. Proof: Suppose (a) olds. Ten, α lim sup P θ0,γ (T n, (1) n(θ 0 ) > c (1) min B P I (α, ĥn) + η 1 (ˆγ n,2 )) lim inf P θ 0,γ (T n, (1) n(θ 0 ) > c (1) min B P I (α, ĥn) + η 1 (ˆγ n,2 )) P (W (1) (1) > c (1) min B P I (α, (1) ) + η 1 ( (2) 2 )) = α, were te first inequality is sown in te proof of Corollary M-BM-PI. Te proof wen (b) olds is very similar. Tese sufficient conditions are useful because tey are typically te direct byproducts of te construction of te SCFs i (or SCF functions η i ( )) so tat te only part tat needs to be verified is te existence of te corresponding parameter sequences. A.21

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69 Table 1: Maximum (Over β 2 ) Null Rejection Probabilities for Different Values of γ 2 in te Post-Conservative Model Selection Example γ 2 = Max β = 0, w/o SCF β = 0.89, w/o SCF β = 0.89, w/ SCF Table 2: Maximum (Over β 2 ) Null Rejection Probabilities for Different Values of γ 2,2 in te Post Consistent Model Selection Example γ 2,2 = Max β = 0, w/o SCF =

70 _1 c_(0.95) c_{bm-pi}(0.05,), beta=0 c_{bm-pi}(0.05,), beta=0.89 c_{lf-pi}(0.05,_2) Figure 1: Post-Conservative MS Critical Values, 2 = _1 c_(0.95) c_{bm-pi}(0.05,), beta=0 c_{bm-pi}(0.05,), beta=0.89 c_{lf-pi}(0.05,_2) Figure 2: Post-Conservative MS Critical Values, 2 = 0.6

71 beta_2 PI LF PI S-Bonf-Min, beta=0 PI S-Bonf-Min, beta=0.89 Figure 3: Post-Conservative MS Power, γ 2 = 0.9, θ = beta_2 PI LF PI S-Bonf-Min, beta=0 PI S-Bonf-Min, beta=0.89 Figure 4: Post-Conservative MS Power, γ 2 = 0.6, θ = 0.2

72 beta_2 PI LF PI S-Bonf-Min, beta=0 PI S-Bonf-Min, beta=0.89 Figure 5: Post-Conservative MS Power, γ 2 = 0.3, θ = beta_2 PI LF PI S-Bonf-Min, beta=0 PI S-Bonf-Min, beta=0.89 Figure 6: Post-Conservative MS Power, γ 2 = 0, θ = 0.2

73 beta_2 PI LF PI S-Bonf-Min, beta=0 PI S-Bonf-Min, beta=0.89 Figure 7: Post-Conservative MS Power, γ 2 = 0.9, θ = beta_2 teta=0.1 teta=0.2 teta=0.3 Figure 8: Post-Consistent MS Power Using PI M-Bonf-Min CVs, γ 2,2 = 0.9

74 beta_2 teta=0.1 teta=0.2 teta=0.3 Figure 9: Post-Consistent MS Power Using PI M-Bonf-Min CVs, γ 2,2 = beta_2 teta=0.1 teta=0.2 teta=0.3 Figure 10: Post-Consistent MS Power Using PI M-Bonf-Min CVs, γ 2,2 = 0.3

75 beta_2 teta=0.1 teta=0.2 teta=0.3 Figure 11: Post-Consistent MS Power Using PI M-Bonf-Min CVs, γ 2,2 = 0