Table Of ContentCarlos A. Flores · Xuan Chen
Average Treatment
Effect Bounds with
an Instrumental
Variable: Theory
and Practice
Average Treatment Effect Bounds
with an Instrumental Variable: Theory and Practice
Carlos A. Flores Xuan Chen
(cid:129)
Average Treatment Effect
Bounds with an Instrumental
Variable: Theory and Practice
123
Carlos A.Flores Xuan Chen
Department ofEconomics, Schoolof Labor andHumanResources
Orfalea Collegeof Business Renmin University of China
California Polytechnic State University Beijing,China
SanLuisObispo, CA,USA
ISBN978-981-13-2016-3 ISBN978-981-13-2017-0 (eBook)
https://doi.org/10.1007/978-981-13-2017-0
LibraryofCongressControlNumber:2018950936
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Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Econometric Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Basic Notation and Parameter of Interest . . . . . . . . . . . . . . . . . . . 7
2.2 The Endogeneity Problem and Partial Identification
of the ATE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Additional Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Bounds Under Different Identification Assumptions . . . . . . . . . . . . . 13
3.1 Manski’s Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 Bounds Under Manski’s Assumptions. . . . . . . . . . . . . . . . 14
3.1.2 Bounds Under the Relaxed MTR and MTS
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Bounds Based on Threshold Crossing Models . . . . . . . . . . . . . . . 18
3.2.1 Bounds in Heckman and Vytlacil (1999, 2000) . . . . . . . . . 19
3.2.2 Bounds in Shaikh and Vytlacil (2005, 2011). . . . . . . . . . . 21
3.3 Bounds Based on the Local Average Treatment Effect
Framework. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3.1 Bounds in Chen et al. (2018) . . . . . . . . . . . . . . . . . . . . . . 25
3.3.2 Bounds in Huber et al. (2017) . . . . . . . . . . . . . . . . . . . . . 32
3.3.3 Bounds in Balke and Pearl (1994, 1997). . . . . . . . . . . . . . 36
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
v
vi Contents
4 Comparison of Bounds Across Different Assumptions . . . . . . . . . . . 41
4.1 Manski’s Approach Versus Threshold Crossing Models . . . . . . . . 41
4.1.1 Manski’s Approach Versus Threshold Crossing Model
on the Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2 Manski’s Approach Versus Joint Threshold Crossing
Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.3 Manski’s Approach Versus Threshold Crossing Model
on the Outcome. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 The LATE Framework Versus the Two Other Approaches . . . . . . 48
4.2.1 Kitagawa (2009) and Hahn (2010) . . . . . . . . . . . . . . . . . . 48
4.2.2 Huber et al. (2017) and Chen et al. (2018) . . . . . . . . . . . . 49
4.3 Summary of the Identification Power of Different
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Assessment of Validity of Different Assumptions . . . . . . . . . . . . . . . 55
5.1 Assessment Based on Threshold Crossing Models . . . . . . . . . . . . 56
5.1.1 Assessment of the Threshold Crossing Model
on the Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.2 Assessment of the Rank Similarity Assumption. . . . . . . . . 58
5.2 Assessment Based on the LATE Framework. . . . . . . . . . . . . . . . . 60
5.2.1 Assessment of the LATE Assumptions . . . . . . . . . . . . . . . 61
5.2.2 Assessment of the Mean Dominance Assumption
Across Strata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Estimation and Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.1 Confidence Intervals for the Parameter: Imbens and Manski
(2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2 Intersection Bounds: Chernozhukov et al. (2013) . . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7 Empirical Applications of Bounds. . . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.1 Empirical Applications Using Manski and Pepper’s (2000)
Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.2 Empirical Applications Using Threshold Crossing Models . . . . . . 90
7.3 Empirical Applications Using LATE Framework. . . . . . . . . . . . . . 91
7.4 Other Empirical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Abstract
This book reviews recent approaches for partial identification of average treatment
effects with instrumental variables in the program evaluation literature, including
Manski’sbounds,boundsbasedonthresholdcrossingmodels,andboundsbasedon
the local average treatment effect (LATE) framework. It compares these bounds
acrossdifferentsetsofassumptions,surveysrelevantmethodstoassessthevalidity
of these assumptions, and discusses estimation and inference methods for the
bounds. The book also reviews some empirical applications employing bounds in
theprogramevaluationliterature.Itaimstobridgethegapbetweentheeconometric
theory on which the different bounds are based and their empirical application to
program evaluation.
vii
Chapter 1
Introduction
Abstract Thischapterintroducesthepurposeandmaincontentofthisbook.Italso
presentsthegeneralideabehindusingpartialidentificationorboundsineconomet-
rics.Itendswiththeorganizationofthisbook.
· ·
Keywords Partialidentification Treatmenteffects Instrumentalvariable
·
Monotonicityassumptions Thresholdcrossingmodels
Localaveragetreatmenteffect(LATE)
Evaluation of causal effects has been the central theme in the program evaluation
literature.Inthisliterature,thepopulationaveragetreatmenteffect(ATE)ofatreat-
mentorinterventiononanoutcomehasbeenoneoftheprimaryparametersofinter-
est among researchers and policy makers. To address the widespread endogeneity
issueinevaluatingcausaleffects,onepopularapproachamongappliedresearchers
isinstrumentalvariables.Traditionalinstrumentalvariable(IV)approachesrelyon
parametric assumptions,and usuallyassume aconstant individual treatmenteffect
acrossunitstoestimatetheATE.AninfluentialIVframeworkallowingforhetero-
geneous effects was developed by Imbens and Angrist (1994) and Angrist et al.
(1996). They showed that IV estimators point identify the local average treatment
effect(LATE)forthesubpopulationofcompliers,whichcomprisesindividualswhose
treatmentstatusisaffectedbytheinstrument.Acommoncriticismoftheirapproach,
however,isthefocusontheeffectforasubpopulation(e.g.,Heckman1996;Robins
andGreenland1996;Deaton2010;Heckman2010).Theinstrument-specificinter-
pretationofLATE hasstimulatedagrowingliteratureonIVapproaches inpursuit
of external validity while allowing for heterogeneous effects. For example, Heck-
manandhiscoauthorsdevelopedaseriesofpapersonthemarginaltreatmenteffect
(MTE)usingalocalinstrumentalvariable(LIV)(HeckmanandVytlacil1999,2005;
Heckman2010;Carneiroetal.2010, 2011).1 TheLIV frameworkprovidesauni-
fied framework linking LATE and ATE, as well as the average treatment effect on
the treated (ATT), which is another popular parameter of interest in the program
1Themarginaltreatmenteffectparameterbridgesthegapbetweenstructuralmodelsandtreatment
effects.
©SpringerNatureSingaporePteLtd.2018 1
C.A.FloresandX.Chen,AverageTreatmentEffectBounds
withanInstrumentalVariable:TheoryandPractice,
https://doi.org/10.1007/978-981-13-2017-0_1
2 1 Introduction
evaluationliterature.PointidentificationoftheATE,however,usuallyrequiresvery
strongassumptions.2 Examplesofsuchassumptionsincludeahomogeneoustreat-
menteffect(sothatLATE =ATE),orthattheinstrumentisstrongenoughtodrive
theprobabilityofbeingtreatedfromzerotoone,bothofwhichmaybehardtosatisfy
inpractice.AngristandFernandez-Val(2010)proposeanalternativewaytoextend
theexternalvalidityofIVestimatesbasedontheavailabilityofmultipleinstruments
forthesamecausalrelationshiptoestimatecausaleffectsforothersubpopulations.
Unfortunately, it can be extremely difficult in practice to find multiple IVs for the
samerelationshipofinterest.
Manski(1990)pioneeredpartialidentificationoftheATE underthemeaninde-
pendenceassumptionoftheinstrument.Sincethen,agrowingliteraturehasderived
boundsontheATE usingIVmethods.Insteadofidentifyingasinglevalueforthe
parameter of interest, partial identification approaches obtain the lower and upper
limitsfortheparameterofinterestunderweakerassumptionsthanthoseneededfor
pointidentification.Suchassumptionsmayfailtodeliverpointestimates,butmay
bestrongenoughtoyieldinformativebounds.Themainmotivationoradvantageof
boundinganalysisistomakeplausibleinferencebyabandoningstrongassumptions
thatmaybeuntenableandhardtojustifyinempiricalapplications.3
Inthisbook,weclassifytheliteratureonpartialidentificationoftheATE using
IVsintothreecategoriesforpresentationpurposes.Thefirstcategoryencompasses
work aiming to improve Manski’s (1990) bounds by using different monotonicity
assumptionsontheoutcome.Forexample,ManskiandPepper(2000)employedthe
monotonetreatmentresponse(MTR)assumptioninManski(1997)andintroduced
themonotonetreatmentselection(MTS)assumption.Thesecondcategoryincludes
literatureimposingthresholdcrossingmodelsonthetreatmentortheoutcome,with
anIVsatisfyingthestatisticalindependenceassumption.Forinstance,Heckmanand
Vytlacil(2000)imposedathresholdcrossingmodelonthetreatment,whileShaikh
and Vytlacil (2011) imposed threshold crossing models on both the treatment and
theoutcome(withthelatterpaperfocusingonabinaryoutcome).Theworkinour
thirdcategoryextendstheLATEframeworktopartialidentificationoftheATE.For
example,BalkeandPearl(1997)consideredthesettingofarandomizedexperiment
with noncompliance to bound the ATE on a binary outcome. Huber et al. (2017)
and Chen et al. (2018) introduced additional mean dominance assumptions on the
outcome,withthelatterpaperalsoaddingamonotonicityassumptionofthetreatment
ontheaverageoutcomeofspecificsubpopulations.
Basedontheaboveclassification,wefirstpresent,foreachofthethreegeneral
frameworks,itsbasicsetting,identificationassumptions,andboundingresults.Then,
wecomparetheidentificationpoweracrossdifferentidentificationassumptions.As
wediscusslater,alternativesetsofassumptionsbasedondifferentapproachesmay
yieldthesameboundsontheATE,andsomeassumptionsareparticularlypowerful
2Pointidentficationmeansthatwecantheoreticallylearnthetrueparametervalueininfinitesamples.
Intuitively,itimpliesthatwecanprovideasingle-valuedestimateofourparameterofinterest.
3Partialidentificationapproachesarealsousedinotherfieldsofeconomics,forexample,game
theoryandauctionmodels(see,e.g.,Tamer2010;HoandRosen2015).
1 Introduction 3
in narrowing the bounds to make it more likely to obtain informative bounds in
practice.Wealsodiscussseveralformalandinformalteststoassesstheplausibility
ofthoseassumptionsinpractice.
Thebookalsotouchesonissuesregardingestimationandinferenceforpartially
identifiedmodels.Thisliteraturehasexperiencedarapidgrowthinthepast20years.
Ratherthanprovidingacomprehensivereviewofthevastliteratureoninferencefor
partially identified models, here we focus on only some estimation and inference
methods for the bounds on the ATE presented in this book.4 An issue that arises
in partial identification models is that confidence intervals can be constructed for
the true value of the parameter of interest (Imbens 2004; Stoye 2009) or for the
entireidentifiedset(e.g.,Chernozhukovetal.2007;RomanoandShaikh2010).The
first type would cover the true parameter value (e.g., the true ATE value) with a
given probability (e.g., 95%), while the second would cover the true identified set
(e.g., the interval from the true value of the lower bound to the true value of the
upper bound) with a given probability. In this book, we consider construction of
confidence intervals for the true value of the parameter following Imbens (2004),
whoarguethatsuchnotionismoreconsistentwiththetraditionalviewofconfidence
intervalsandisthuslikelytobeofgreaterinterest.Inparticular,wefirstconsiderthe
confidence intervals proposed by Imbens (2004), which are applicable when there
existestimatorsofthelowerandupperboundsthatareconsistentandasymptotically
normallydistributed.Then,weconsiderestimationandinferenceincaseswherethe
lowerorupperboundsinvolvemaximumorminimum(or,supremumorinfimum)
operators, respectively, which usually occur in the bounds on the ATE using IV
methods.Suchoperatorscausecomplicationsforestimationandinferencebecause
standardasymptotictheoryisnotapplicabletosuchnon-smoothfunctions(Hirano
andPorter2012).Toaddresssuchcomplication,wediscusstheapproachproposed
byChernozhukovetal.(2013),whosecomputationpackageisavailableinStata.
Partial identification of treatment effects using IV methods has been applied to
many subjects in economics, such as education, labor, and health, among many
others. For example, Gundersen et al. 2012 and Kreider et al. (2012) evaluate the
effects of public social welfare programs on children’s health; Huber et al. (2017)
examinetheeffectsofreceivingprivateschoolingonstudents’educationaloutcomes
inColombia;andChenetal.(2018)analyzetheeffectsofatrainingprogramonlabor
marketoutcomes,tomentionafew.Wereviewsomeapplicationsthathaveemployed
the identification assumptions and methods discussed in this book, putting special
emphasisontheassumptionsusedandtheiridentificationpower.
The rest of the book is organized as follows. Chapter 2 presents the general
econometric framework and the identification issue when estimating theATE, and
alsomotivatestheuseofbounds.Chapter3presentsthethreebasicIVapproachesto
deriveboundsontheATE.Chapter4comparestheidentificationpowerofdifferent
assumptions. Chapter 5 introduces several formal and informal test to assess the
plausibility of the assumptions. Chapter 6 discusses estimation and inference in
4Forreviewsoninferenceforpartiallyidentifiedmodelssee,forexample,Tamer(2010)andCanay
andShaikh(2017).
