IntroductionConditional Moment Inequalities/EqualitiesModelConfidence SetsTest StatisticsGeneral Form of the Test StatisticFunction SInstrumentsWeight Function QComputation of Sums, Integrals, and SupremaGMS Confidence SetsGMS Critical ValuesGMS Critical Values for Approximate Test StatisticsBootstrap GMS Critical ValuesDefinition of 0=x"0127n(0=x"0112)“Plug-in Asymptotic” Confidence SetsUniform Asymptotic Coverage ProbabilitiesNotationUniform Asymptotic Distribution of the Test StatisticAn Additional GMS AssumptionUniform Asymptotic Coverage Probability ResultsPower Against Fixed AlternativesPower Against n-1/2-Local AlternativesPreliminary Consistent Estimation ofIdentified Parameters and Time SeriesMonte Carlo SimulationsDescription of the TestsQuantile Selection ModelDescription of the Modelg FunctionsSimulation ResultsInterval Outcome Regression ModelDescription of Modelg FunctionsSimulation ResultsEntry Game ModelDescription of the Modelg FunctionsSimulation ResultsAppendix AOutlineAppendix BKolmogorov-Smirnov and ApproximateCvM Tests and CS'sInstruments and Weight FunctionsExample: Verification of AssumptionsLA1-LA3 and LA3 Uniformity Issues with Infinite-DimensionalNuisance ParametersProblems with Pointwise AsymptoticsSubsampling Critical ValuesDefinitionAsymptotic Coverage Probabilitiesof Subsampling Confidence SetsAppendix CProofs of Lemmas 2 and 3 and Theorem 2(b)Proofs of Results for Fixed AlternativesProofs of Results for n-1/2-Local Alternatives Proofs Concerning the Verificationof Assumptions S1-S4Appendix DProofs of Kolmogorov-Smirnov and ApproximateCramér von Mises ResultsProof of Lemma B2 Regarding GB-spline, Gbox,dd, and Gc/d Proofs of Theorems B4 and B5 RegardingUniformity IssuesProofs of Subsampling ResultsAppendix EPreliminary Lemmas E1-E3Proof of Lemma A1(a)Proof of Lemma A1(b)Proof of Lemma E1Proof of Lemma E2Proof of Lemma E3Appendix FQuantile Selection ModelInterval Outcome Regression ModelEntry Game ModelSupplement to INFERENCE BASED ON CONDITIONAL MOMENT INEQUALITIES By Donald W.K. Andrews and Xiaoxia Shi June 2010 COWLES FOUNDATION DISCUSSION PAPER NO. 1761S COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/Supplemental MaterialforInference Based onConditional Moment InequalitiesDonald W. K. AndrewsCowles Foundation for Research in EconomicsYale UniversityXiaoxia ShiDepartment of EconomicsYale UniversityNovember 2007Revised: June 201011 OutlineThis Supplement includes …ve appendices.The …rst appendix, Appendix B, provides a number of supplemental results to themain paper. These include:(i) results for Kolmogorov-Smirnov (KS) and approximate Cramér von Mises (A-CvM) tests and CS’s in Section 12.1,(ii) three additional examples of collections G and probability measures Q that satisfyAssumptions CI, M, FA(e), and Q in Section 12.2,(iii) an illustration of the veri…cation of Assumptions LA1-LA3 in Section 12.3,(iv) an illustration of some uniformity issues that arise with in…nite-dimensionalnuisance parameters in Section 12.4,(v) an illustration of problems with pointwise asymptotics in Section 12.5, and(vi) coverage probability results for subsampling tests and CS’s under drifting se-quences of distributions in Section 12.6.Appendix C provides proofs of the results that are stated in the main paper but arenot proved in Appendix A. These include:(i) the proofs of Lemmas 2 and 3 and Theorem 2(b) in Section 13.1,(ii) the proofs of Lemma 4 and Theorem 3 concerning …xed alternatives in Section13.2,(iii) the proof of Theorem 4 concerning local power in Section 13.3, and(iv) the proof of Lemma 1 concerning the veri…cation of Assumptions S1-S4 in Section13.4.Appendix D provides proofs of the results stated in Appendix B. These include:(i) the proofs of Kolmogorov-Smirnov and approximate Cramér von Mises results inSection 14.1,(ii) the proof of Lemma B2 in Section 14.2,(iii) the proofs of Theorems B4 and B5 regarding uniformity issues in Section 14.3,and(iv) the proofs of the subsampling results in Section 14.4.Appendix E proves Lemma A1 which is stated in Appendix A of the main pap er.Appendix F provides some additional material concerning the Monte Carlo simula-tion results.112 Appendix B12.1 Kolmogorov-Smirnov and ApproximateCvM Tests and CS’sIn this Section, we provide results for Kolmogorov-Smirnov (KS) and approximateCvM (A-CvM) tests and CS’s de…ned in Sections 3.1 and 4.2, respectively. A-CvM testsare Cramér-von Mises-type tests in which the test statistic is an in…nite sum that istruncated to include only the …rst snfunctions fg1; :::; gsng or the test statistic is anintegral with respect to the measure Q and the integral is approximated by a (possiblyweighted) average over the functions fg1; :::; gsng; which are obtained by simulation or bya quasi-Monte Carlo (QMC) method. The same functions fg1; :::; gsng are used for thetest statistic and the critical value. In the case of simulated functions, the probabilisticresults given here are for …xed (i.e., non-random) functions fg1; :::; gsng: If fg1; :::; gsngare obtained via i.i.d. draws from Q; then the probability results are made conditionalon the observed functions fg1; :::; gsng for n  1:We show that (i) KS and A-CvM CS’s have uniform asymptotic coverage probabilitiesthat are greater than or equal to their nominal level 1; (ii) KS and A-CvM tests haveasymptotic power equal to one for all …xed alternatives, and (iii) KS and A-CvM testshave asymptotic power that is arbitrarily close to one for a broad array of n1=2-localalternatives whose localization parameter is arbitrarily large.We consider a slightly more general KS statistic than that de…ned in (3.7):Tn() = supg2GnS(n1=2mn(; g); n(; g)); (12.1)where Gn G:For KS tests and CS’s, we make use of the following assumptions.Assumption KS. Gn" G as n ! 1:Let Wbddenote a subset of W (the set of k k positive de…nite matrices) containingmatrices whose eigenvalues are bounded away from zero and in…nity.Assumption S20. S(m; ) is uniformly continuous in the sense that for all bounded2sets M in Rkand all sets Wbdsup2Rp+f0gvsupm;m02M:jjmm0jjsup;02Wbd:jj0jjjS(m + ; )  S(m0+ ; 0)j ! 0 as  ! 0:The following Lemma shows that Assumption S20is not restrictive.Lemma B1. The functions S1; S2; and S3satisfy Assumption S20.The