Bootstrap methods and applications in econometrics : a brief survey

B ootstrap M ethods andA pplications in Econometrics - aB riefSurvey ^¤

P ålB ergström

January19 9 9

A bstract

T his paperprovides a briefsurvey ofthe bootstrap and its use in econometrics. A s anintroduction, thepapergives adescriptionofthe basics ofthemethod, with aspecialempasis on bootstrap testing. A fairly large amountofspace is devoted to discussingwhy bootstrap tests providere…nements comparedtotoequivalentasymptotictests.

A series ofdi¤ erentrecentapplications intheeconometrics litterature is then surveyed in orderto give a picture ofthis rapidly evolving research …eld.

Keywords: B ootstrap;SampleR euseM ethods;SimulationM ethods JEL Classi…cation: C4

¤T his survey has bene…tted from comments from Eva Johansson and A nders Klev- marken, neitherofwhom can beheld responsibleforremainingobscurities and errors.

yD epartmentofEconomics, U ppsalaU niversity, B ox513, SE 7 51 20 U ppsala, Sweden.

e-mail: P al.B ergstrom@ nek.uu.se

1 Introduction

T his papergives abriefand maybeabitsubjectivesurveyofthebootstrap andits useineconometrics. Sinceresearchinthis …eldhas beenveryactive, especially in the last…ve years orso, I willgive abriefintroduction tothe

”basics” ofthe bootstrap butmainly try tofocus on the ”frontier” ofthis rapidlyevolving…eld. I willdothis by…rstsayingafewwords onthebasic ideas andthentrytoexplainhowthebootstrap canbeapplied inthethree majorcontexts, i.e:

²Estimatingvariance(standarderrors)

²Correctingforbias

²Formingtests

H avingdonethatI willdwelluponthesubjectonastowhythebootstrap actually provides asymptotic re…nements a bit longer. I believe that the understandingoftheseissuesreallyhelpstellingusifitisagoodideatoapply the bootstrap in a speci…c context. H aving gone through these somewhat messy arguments, I will…nally describeaselection ofapplications and also hintatsomequestions openforfutureresearch.

1.1 T he B ootstrap

T he bootstrap as a computationaldevice was invented and introduced by Efron (19 7 9 ) as a quite intuitive and (perhaps deceptively) simple way of

…ndingapproximations ofquantities thatare very hard, oreven impossible tocompute analytically. T he basic idea is totake the sample thatwe are interestedinandthinkofitas ifitwas apopulationandthenbyresampling

create a new sample, a bootstrap sample, which we use to compute some quantitythatweareinterestedin. Ifwerepeatthis severaltimes, obtaining lots ofbootstrap samples, wecan use the mean ofthecomputed quantities as anestimateoftheexpectedvalueofthis bootstrapped quantity.

L et us just considera simple example: Suppose that we, just forthe sake ofthe argument, would want to compute the bootstrap mean fora sampleof, say, heightsof30 economists. T ocalculatethebootstrap mean, we considerthe30 heightswehaveastheentirepopulation. W ethen, usingsome unbiased pseudo random numbergenerator, perform 30 random drawings from our”fake” population, butallthetimedrawingwithreplacement, i.e., we put the heights drawn back into the population all the time. T hen, almostcertainly, someheights willbedrawn severaltimes and somenotat all. T he bootstrap sample thatwe getwillthus di¤ ersomewhatfrom the fake population, i.e. the originalsample. W e nowcompute the quantity of interest, which in this case was the mean, usingthe bootstrap sample and obtain one realisation ofthe bootstrap estimatorforthe mean. W e then repeatthis severaltimes, say athousand, and thus getthousand bootstrap samples and thousand realisations ofthe bootstrap mean. Computing the meanofthesethousandrealisations willgiveusourestimateoftheexpected valueofthebootstrap mean.

N ow, whywouldanyonebotherdoingthis?W ell, doingpreciselywhatwe justdid, we would be suprised indeed ifthe bootstrap mean deviated from themean ofthe originalsample. T he aboveexampleis hence notapartic- ularilyinterestingone. In manycases however, this simpleprinciplecan be used toapproxiamte quantities thatare very hard tocompute analytically.

In addition tothis, bootstrap quantities can, undercertain circumstances, be shown toconverge tothe true values more rapidly than asymptoticap- proximations, and also be used to correctforbias. W e willreturn to this shortly.

T hatis basicallyit. H owever, as weshallsee, thedevilis in thedetails, andevengivingthisverybasicexplanationonwhatthebootstrapisallabout, twoimportantcritisisms couldberaisedagainstthedescriptionabove.

1.T he description too sloppy. Ifwe wanttomake a more rigorous description followinge.g. Shao& T u (19 9 5) the bootstrap is really a combination oftwo techniques: the substitution principle and a nu- mericalapproximation. W ithoutgettingintodetails, the ideais that there exists abootstrap distribution, which is the distribution we get when we do the resampling conditionalon ourpresentsample. T he substitution thatwe make is to replace this unknown distribution of interestbyanempiricaldistribution, e.g. theempiricaldistributionof oursample. T his gives us atheoreticalbootstrap distribution, which mayhaveseveralinterestingcharacteristics, andisthesubjectofstudy intheoreticalworkonbootstrap estimators. T heseestimators however seldom haveclosedform solutions¹, andwehenceneedtoapproximate them numerically, whichis whatwedowhenweactuallycarryoutthe repeatedresamplingandaverageoverthebootstrap samples togetthe expected value.

2.T hede…nitionis tonarrow. T hegeneraltendencyintheliterature is tosaythatthebootstrap is abitmorethan whatwejusthavedis- cussed, and therfore to labelthe above procedure ”a non-parametric bootstrap”. Itis then thoughtofas non-parametricas opposed toan estimatorwhere we would use family ofdistributions ratherthan an empiricalonedoingthebootstrap. Inoursimplisticexperimentabove, itwouldcorrespondtosayingthatN (¹y;s) isthebestapproximationto theheights ofeconomists, andthen drawvalues from this distribution

1Seeforexample Shao& Tu (19 9 5) pp. 10fforan example wheresuch aclosed form solution actually exists. T he boostrap estimatorofthe variance ofa sample median is shown tobeequivalenttotheestimatorofM aritz & Jarret(19 7 8).

ratherthandoingtheresampling, yieldinganevenmorerediculus esti- mator. T hemoregeneralde…nitionofthebootstrap wouldthenbeany proceduredrawingsamples from aD G P, eithergiven byaparametric familyorbyanempiricaldistribution, whichuses thesesocalledboot- strap samples to draw inference aboutquantities ofinterest. In the followingwe willalmostexclusively be discussingthe non-parametric bootstrap andwillhencesurpressthe”non-parametric” epithetifthere is noriskofconfusion.

2 B asicuse ofthebootstrap

H avinglooked brie‡yatthegeneralprinciplewewillnowlookatthe…elds wherethebootstrap has beenusedmostcommonlyup untilpresent.

2.1 StandardErrors

T he …rst use made ofthe bootstrap was to estimate the standard errors fore.g. estimators in cases where there were noanalyticalasymptotic ap- proximations available. (O nceagain, youwould hardlyeverbeinterestedin bootstrappingthevarianceforasamplemean.) T heprocedureofcalculating bootstrap standarderrorsforbasicallyanyestimatorisstraightforwardlyde- scribedinEfron& T ibshirani (19 9 3), andwewillherejustsketchthegeneral ideaandthen sayafewwords ofwarning.

L et’s however…rst…xsomenotation, whichI willtrytosticktothrough- out. W ewillusethesuperscriptb toindicateanybootstrappedquantity;we wille.g. labeltheoriginalsamplesy, thequantitiesofinterestµ (y) andhence theirbootstrap analogues as y^b_jandµ^b_j^³y^b_j^´wherejindicates thej:thoutof B bootstrap realisations. T he expected value ofthe bootstrapped quantity which is obtained by calculating the mean ofthe bootstrap realisations is denotedbyµ^b:

In terms oftheintroduced notation, thegeneralideais nowtoestimate the standard errorofa parameterestimate ofinterest by computing the standarddeviation ofµ^b_j; thatis

se (µ (y)) =c 0

@ 1

B ¡1

XB j= 1

³µ^b_j¡µ^b´2

1 A

1 2

:

A typicalexample where this type ofbootstrap estimatorhas been ad- vocated is the variance ofM anski’s maximum score estimator(see G reene (19 9 7 ) pp 9 02 f). Since there is nolikelihood argumentbehind the M anski (19 7 5) estimator, standard information matrix estimates are notavailable and the bootstrap might seem useful. T here are however, as yet to my knowledge, no theoretical results established on this bootstrap estimator, whichmakes usingitasomewhatriskybusiness, sincewecannotbeassured thattheestimates willconvergetotruevalues atall².

So, arebootstrap standarderrorsuseful?W ell, inhislecturenotesonthe bootstrap, M arcN erlovegivesasimpleexampleofresamplingO L S residuals usingthesetoestimatethestandarderrorsfortheregressioncoe¢cients, …nd- ingtheseseverelydownwardbiased(N erlove(19 9 8)). T hereasonclaimed, as describedbyN erlove(19 9 8), is thatthe…ttedresiduals which heresamples

e = y¡X µ willhaveacovariancematrixof

E (ee⁰) = ¾^{2 ³}I ¡X (X ⁰X )^¡1X ^0´

2FortheH orowitz(19 9 2) smoothedmaximum scoreestimatorbootstrapresultsdoexist (H orowitz (19 9 6)), which however is a direct corrolary ofassymptotic approximations existing forthe originalestimator, making the bootstrap less necessary butstilluseful accordingtotheresults in thelatterstudy.

whichis certainlydi¤ erentfrom theerrorterm heuses inhis D G P, whichis

"»N (0 ;¾²I ):T hiswillnotworkverywell, andthereasonisthedependence ofthebootstrappedquantityofinterest, thevarianceoftheO L S-coe¢cients onunknownparameters. Forreasonsthatsoonwillbecomeevident, wewould howevernotexpectthe bootstrap standard errors toperform substantially worsethantheasymptoticones. T hemainproblem inP rof. N erlovesaplica- tion is probablythatheuses unadjusted residuals. Since O L S residuals are generallysmallerthan theerrorterms oftheregression model, theseshould beadjustedbyscalingthem by^qn= (n ¡k). Ifdoneproperly, thebootstrap estimatevarianceshould, withanincreasingnumberofbootstrap iterations, convergeexactlytotheasymptoticO L S variance-covarianceestimator.

T hereareseldom anye¢ciencygainsfrom applyingthebootstrap toesti- matestandarderrors, forthosereasonsthat, onceagain, willbecomeevident whenwelookatthepropertiesofbootstrap tests, shortly. T heliteraturehas overthelastdecadehencemovedawayfrom varianceestimationbyusingthe bootstrap, andinsteadfocusedonbootstrap tests, testswhichinmostofthe cases are the reasons as towhy we are interested in the variance estimates in the…rstplace.

2.2 B ias correction

B eforeturningtothe…eld ofbootstrap tests, wewillconsideranothercom- mon application ofthebootstrap, i.e. correctingforbias. Even ifwe know thatan estimatoris consistent, itmightsu¤ erfrom bias in …nite samples.

B yapplyingthebootstrap wecantrytocorrectthisbias usingthefollowing simpleprocedure:

1. Estimatetheparameterofinterest, µ; bye.g. O L S, IV orM L toobtain

^µ:

2. ConstructB bootstrap samples and computeµ^bas

µ^b= 1 B

XB j= 1

µ^b_j

whereµ^b_jis thesameestimatorthatwas usedtoobtained^µ applied to thebootstrap sample.

3. Estimatethebias as

b^b³^µ^´= µ^b¡^µ 4. Calculatethebias-correctedestimateas

~µ = ^µ¡b^b³^µ^´= 2 ^µ¡µ^b

T heideabehindtheprocedureaboveishencethatthedi¤erencebetween theestimate^µ andthetruevalueµ; shouldbethesameas thedi¤ erencebe- tween µ^band ^µ; or, loosely, thatthe the relation ofthebootstrap sampleto the originalsampleis the sameas the relation between the originalsample andthetruepopulation. T his forms thebasis forthesimplestform ofboot- strap biasreduction. Furtherdescriptions ontheseprocedures maybefound in Efron& T ibshirani (19 9 3).

T hereis anobvious problem withthis approach, namelythatweassume thatthebias is constantanddoes notvarywiththeparametervalue. T here is normallynogood reason toexpectthis tobethe case. In M acKinnon &

Smith Jr. (19 9 8) the use ofthe bootstrap is explored in settings where the bias function is notassumed to be constant. T heirresults ofgeneralising the bias function is encouraging, though there is a cleartrade-o¤ in terms ofe¢ciencyloss from usingthebias corrected estimator, totheextentthat usingthecorrections mayincreasethemeansquared erroroftheestimator.

InFerrari & Cribari-N eto(19 9 8), theauthors seektounifytheliterature ofbootstrap bias correction with the one ofanalytical ditto. In the pa- per, whichis somewhatinvolved, theequivalenceofanalyticalandbootstrap correction is demonstrated forM L estimeators ofmodels with one parame- ter. Formoregeneralmodels theauthors providesom M onteCarloevidence expressing a weak preference forthe analyticalcorrection, con…rming the results ofM acKinnon & Smith Jr. (19 9 8) in the respectthatthebootstrap corrections mayinduceincreasedM SE.

2.3 B ootstrap tests

T he main reason forusing bootstrap tests ratherthan asymptotic tests is thatthelattermayin…nitesamples bebiased, i.e. theyhaveempiricalsizes thatdi¤erfrom theirnominalones. A main feature ofbootstrap tests is that, undercertainconditionwhichwewilllookintoshortly, theirempirical sizeswillconvergetothetruesizes fasterthanasymptotictests andattimes convergeconsiderablyfaster. B ootstraptestswithcorrectsizescanalsooften be shown to have basically the same powerproperties as theirasymptotic counterparts³. B efore discussingthe issues ofconvergence, wewilldescribe whatabootstrap testis allabout.

T opinpointthedi¤erences, letus…rstbrie‡yconsidertraditionalhypoth- esis testing. Supposethatwehaveasamplefrom whichwehaveobtainedan estimate ^µ ofan unknown parameterµ:T otesta hypothesis on this single parameter, say H 0 :µ = 0 , we simply employ a t-test, which we knowwill haveacertaindistributionatleastasymptotically, giventhatthenullistrue.

U singthis approximationwewillassess whethertheteststatisticis likelyto havebeendrawnfrom thedistribution in question.

3T he results mentioned as wellas those given below in this section are proofed and discussed atfurtherlength in D avidson & M acKinnon (19 9 6b), D avidson & M acKinnon (19 9 6a) and H orowitz (19 9 7 )

W henformingabootstrap testinthis context, wecould insteadusethe factthatwe knowthe sample valueofµ in thesample, and then treatthis sampleas ifitwas apopulationforwhichthetruevalueoftheparameterof interestis in facttheestimated parameter, ^µ.⁴ W e then create abootstrap samplebyrandomly resamplingobservations from ouroriginalsamplewith replacement, thereby obtainingasample with the same size as the original one, butnotwiththesamecomposition. T hebootstrap sampleis thenused toobtain a bootstrap estimate, ^µ^b; which in repeated (re)sampling willbe equalto^µ on average:

T heresamplingprocedureis thenas usualcarried outalargenumberof times andforeachbootstrap estimateateststatisticis formedbasedonthe nullthat^µ^b = ^µ:D oingthatwe willobtain a distribution ofteststatistics whichisgeneratedtakingthecharacteristicsofthedataintheoriginalsample intoconsideration, while explicitly imposingthe restriction thatthe nullis true. Calculatingthe9 5thpercentilefortheabsolutevaluesofthet-statistics obtainedfrom thebootstrap estimates, wegetthebootstrap criticalvaluefor ourt-testatthe5 % signi…cancelevel, withwhichwecancomparethevalue ofthe t-testobtained from the originalsample testinge.g. the hypothesis

^µ = 0 :T he principle behind the bootstrap testis hence toconstructatrue null, e.g. ^µ^b= ^µ; andthensimulatethedistributionoftheteststatisticusing thedataathand.

Eventhoughtheintuitionmightappearstraightforward, therigorous ar- gumentastowhythebootstrap providesre…nementscomparedto…rstorder asymptotics is somewhatinvolved. T hefundamentalpropertywerequireof thetestin orderforthebootstrap toprovidere…nements compared to…rst orderasymptotics, is the one ofpivotalness, i.e. thatthe testdistribution does notdepend onanyunknownparameters.

4It is certainly not necessary to use the set-up suggested here, since all bootstrap schemes thatimposethetruenullwould bevalid. H owever, in cases such as this when µ delimits H0, itseems naturaland straightforward tousethepresentsetting.

T his is certainly true asymptotically forthe t-test, since we knowthat its distribution willconverge toa N (0 ;1); which is evidently independent ofany unknown parameters. Forthe t-testwe may obtain a distribution independentofunknown parameters even in …nite samples⁵, which would suggest that a bootstrap t-test should work quite well in even when the samplesizeiscomparativelysmall. Formosttests, thepropertyofpivotalness is full…lled asymptotically, since theirlimitingdistributions are quite often normaloch chi-squared orwhatever. Ifwe howeverin the linearregression contextarewillingtoassumenormalerrors(underthenull), severaltestssuch astestsforserialcorrelation, heteroscedasticity(includingA R CH ), skewness andkurtosisareexactlypivotalandthebootstraptestwillthenbeexacteven in …nitesamples. T oseewhybootstrap testactuallyprovidere…nements as compared toasymptotictests, and why thepivotalness is importantin this context, weneedargueabitmorerigorously, whichwillbethescopeofnext section.

3 W hydobootstraptestsprovidere…nements?

M yconjecturefrom studyingtheliteratureis thatthe…eld wheretheboot- strap has beenmostcommonlyapplied is theoneofbootstrap testing. O ne reasonforthis maybethatthereis acleartheoreticalsupportfortheboot- straps abilitytoprovidere…nements as comparedtoasymptoticapproxima- tions when bootstrappingpivotaltests. Sincethe importanceofbootstrap- ping”pivotalquantities” is bynowwellunderstood byresearchers activein the …eld, I thinkthatitis usefultospelloutthe mostwidely spread proof on as to why this re…nementoccurs in somewhatgreaterdetail. T he de- scriptionoftheproofis somewhat”sketchy”, butbasicallyfollows H orowitz (19 9 7 ) and H all(19 9 2). T he readerwhodesires more ofrigourshould look

5T hatis, ifwearewillingtoassumenormalresiduals or…xed regressors.

up especiallythelatterreference.

L etusstartoutwithasetofdatawhichisarandomsamplefX ⁱ:i= 1;:::;ng from adistribution, the CD F ofwhich wewilldenoteby F . Ifitwould be possibletodescribethe distribution by some…nitesetofparameters, µ; we wouldwritetheCD F as F (x;µ), which would beequivalenttoadoptingan entirely parametric method. T he proofwe are aboutto go through could thestraightforwardlycarriedoutforaparametricbootstrap, butforthecase mostrelevanttothis survey, thenon-parametricbootstrap, we willhaveto make use ofa more generalEmpiricalD istribution Function (ED F) which wewilldenotebyFn; thegenericestimatorofwhich is

F^n (x) = n^¡1

Xn i= 1

I (X i·x):

T his estimator, ^Fn(x); willundermild regularity conditions converge toF almostsurelyattherateofO ^³n^¡12´. (Seee.g. D avidson (19 9 4) p. 332)

L etusfurthermoreintroduceatestforacertainH 0aboutthedistribution from whichfX ig isdrawn, andlabelthistestTⁿ(X 1;:::;X n); the…nite-sample distribution ofwhich underthe trueH 0 is Gn(z ;F )´ P(Tn ·z ) .(z being thecriticalvalueforrejection). Ifwenowtakethecaseofasymmetrictwo- sided test, werejectH ₀ atthe ® levelifjTnj> z n® wherethe criticalvalue z n ® solves

® = 1¡(G n (zn ®;F )¡Gn (¡zn®;F )) (1) SincewedonotknowF , wecannotobtainz n® rightaway. D ependingon thecircumstances, therearenowatleastthreedi¤erentways toproceed:

1. SupposethatTn is pivotalin …nite samples. T his means thatGn will notdependonF atall, andwewillknowthevalueofz n® exactly. T his isforinstancethecasewithat-testonaregressioncoe¢centifwehave normallydistributed errors. W hatwesimplydois toobtain z n® from

astandard t-distribution.

2. Fewtests are pivotalin …nite samples. M ostofthetests employed in econometrics are howeverasymptotically pivotal. In the case ofour t-test, itwillconverge toastandard normalvariable, which ofcourse is independentofF and any otherparameter, forthatmatter. W hat we then do is to use the standard normaldistribution to obtain an approximation forz n®:

3. T heseasymptoticapproximations can attimes bequitepoor. A third routeishencetoaproximateF byFn andhenceform thetestbasedon Gn (z ;Fn); and thatis essentially whata bootstrap testis allabout, i.e. inourtwo-tailed casesolving

® = 1¡^³Gn

³z _n®^b ;Fn

´¡Gn

³¡zn®^b ;Fn

´´ (2)

where z _{n ®}^b is the bootstrap criticalvalue. N ormally we cannotobtain an analyticalexpression forGn

³z _n®^b ;F_n^´and we musthence resortto numericalsimulations through M onte Carlo resampling, which is the waythebootstrap tests arenormallycarried out⁶.

T he main bene…tofthe third route, the bootstrap, is thatthese tests doconverge fasterthan asymptotic approximation. T oprove this we need the higherorderapproximation known as the Edgeworth expansion which appliedtoourempiricaldistribution forthetesttakes theform of

Gn (z ;F ) = G (z ;F ) + n^¡12g1(z ;F ) + n^¡1g2 (z ;F ) + o^³n^¡1´; (3)

where G (z ;F ) is the the asymptotic CD F ofTn; g₁ an even function ofz

6R ecallthe discussion in the introduction, thatthe bootstrap consists ofcombining twoprinciples. First, the substitution principle;we replace F by Fn:Second, numerical approximation, carried outbyresampling.

foreachF and g2 anoddditto. Furthermoreg2 (z ;Fn) willconvergealmost surelytog2 (z ;F ) uniformlyoverz ⁷ :N owusing(1) and (3) wegetthat

® = P(jTnj> z ) = 1 ¡[G (z ;F ) ¡G (¡z ;F )]¡2 n^¡1g2 (z ;F ) + o^³n^¡1´: N ote that we have used the evenness ofg1 and that o (n^¡1)§o (n^¡1) = o (n^¡1):

Supposethatwenowform abootstrap testreplacingF byFn; toget

®^b= P^b³¯¯¯T_n^b¯¯¯> z ^´= 1¡[G (z ;Fn)¡G (¡z ;Fn)]¡2 n^¡1g2 (z ;Fn)+ o^³n^¡1´: Subtractingnowthe true value ofthe testsize from the bootstrap test sizeweget

P^b³¯¯_¯T_n^b_¯^¯¯> z ^´¡P(jTnj> z ) = [G (z ;F ) ¡G (z ;Fn)] (4) [¡G (¡z ;F ) ¡G (¡z ;Fn)]

+ 2 n^¡1[g2 (z ;F )¡g2 (z ;Fn)]

+ o^³n^¡1´

IfG (¢) is su¢cientlysmooth, whatmatters herewillberate ofconvergence ofFn to F; which as we have previously stated is ofO ^³n^¡12´; and is the leadingorderintheexpressionabove. W ehenceseethatthebootstrap test underquitegeneralconditions havesizes thatconvergetotheirtrueones at the rate ofO ^³n^{¡12 ´}. T his is howeverthe same rate ofconvergence as the standard asymptoticand wewould gain nothingfrom takingthetroubleto usebootstrap tests. B utherecomes thetrick. Ifthetestthatwebootstrap

7 JuststatingtheEdgeworth expansion (orratherits inversion) is aregrettably unin- tuitive way ofpresentingthe proofand certainly a‡awofthis exposition. T hese higher orderexpansions are howeverquite tricky sfu¤ , and mustadmitthatI …nd ithard to conveyany…rm intuition here.

is asymptotically pivotal, its distribution willnotdepend on any unknown parameters, which directly implies thatG (z ;Fn) = G (z ;F ) forallz ; and that(4 ) simpli…es to

P^b³¯¯_¯T_n^b¯¯_¯> z ^´¡P(jTnj> z ) = + 2 n^¡1[g2 (z ;F )¡g2 (z ;Fn)]+ o^³n^¡1´

= 2 O ^³n^¡1´O ^³n^¡21´+ o^³n^¡1´= o^³n^¡1´ havingappliedthementioned convergenceresultforg2 (z ;Fn) tog2 (z ;F ):

W e hence …nally obtain the desired result: Ifthe testwe bootstrap is asymptoticallypivotal, which almostalltests used in econometrics are, the bootstraptestwillconvergefasterbyanorderof(atleast) O ^³n^¡12´compared totheasymptoticapproximation.

4 A fewapplications ofthebootstrap

W ewillnowsurvey afewrecentcontributions tothe bootstrap partofthe econometrics literature. T he bias towards tests and the usage ofpivotal quantities inthis exposition, is claimedtobeamanifestationofthestateof theliterature, ratherthanmyownpreferences.

4.1 SU R -regressions

Itis awell-known property ofthe Z ellnerSU R estimator, thatthe asymp- toticstandard-errorsoftheregressioncoe¢cientsmaybeseverelydownward biased. A lready morethan ten years agoM arais (19 86) and in apublished pieceofworkafewyearslaterA tkinson& W ilson(19 9 2), attempttoaddress the problem by using bootstrap standard errors. T he evidence was mixed butdid indicate some improvement. Itshould be evidentfrom the earlier discussions on pivotalness, thatthe results thatthe authors obtained most

likelywereparameterdependent, andthatabootstrap testwouldhavemore clear-cutresults. InR ilstone& V eall(19 9 6) theapproachofusingpercintile- tcon…dence intervals was adopted. T his approach involves bootstrapping pivotalquantities, and hence yielded results much more clear-cutand en- couragingthanintheprevious studies: thebootstrap con…denceintervals in theirM onteCarlostudyalmostexactlycoveredthenominalones.

W hereas this approachdoes notaddmuchconceptuallycomparedtothe testprocedures describedearlier, ithoweverallows assymetriccon…dencein- tervals whichmightbeapotentiallyimportantfeature, andwewilltherefore lookattheprocedureusedbrie‡y.

T hedescription on as tohowthebootstrap samplewas createdis some- whatvagueinthepaper, butwas presumablydoneresamplingtheresiduals ofthesecondstep, whichshouldbeindependentbyconstruction. T healgo- rith then works brie‡yas follows:

1. Estimateµ bySU R obtaining^µ and ^¾^³^µ^´

2. R esampleandapplySU R tothebootstrapdatasettoget^µ^band^¾

µ^µ^b

¶

3. Createabootstrap t-statisticas

^t^b= ^µ^b¡^µ

^

¾

µ^µ^b

¶

4. R epeatsteps 2-3 afewhundredtimes orso.

5. Sortthedistributionof^t^b andextract^t^b_{1¡® =2} and ^t^b_{® =2}: 6. Form a(1 ¡® ) * 100% bootstrap-tcon…denceintervalas

=^bt= ^h^µ¡^¾^³^µ^´^t^b_{1¡® =2};^µ¡^¾^³^µ^´^t^b_{® =2}ⁱ

4.2 H ausmantests

W ong(19 9 6) provides aniceand straightforward implementation ofaboot- strap test. T he H ausman exogeneity test, which is asymptotically pivotal converging to a ² distribution, is claimed to have a bootstrap equivalent which is notjustconvergingofo^³n^¡12´buto (n^¡1) fasterthan the asymp- toticapproximation⁸. D atais generated as

y = ¯₀ + x¯₁+ u

andunderatruenullofexogeneitythereis nocorrelationbetweenx andu.

Forafalsenullthesearecorrelated, butanadditionalregressorz isgenerated whichisuncorrelatedwithubutcorrelatedwithx. T healternativeestimator toO L S is standard IV . T healgorithm forthebootstrap testis as follows:

1. EstimatetheequationbyO L S andIV andcomputetheH ausmantest Q^

2. R esampleresiduals estimatedas u^b= y¡^¯^{O LS}₀ + x^¯^{O LS}₁

3. Constructy^b= ^¯^{O LS}₀ + x^¯^{O LS}₁ + u^b(N otethatusingtheO L S estimates constructingthebootstrap data-set, thetruenullisexplicitlyimposed) 4. Estimate once again by O L S and IV and compute the H ausman test

Q^^b

5. R epeatsteps 2 - 4 lots oftimes.

6. Sortthedistributionof ^Q^b

7 . R ejectexogeneityatthe® ¤10 0 % -levelif ^Q > ^Q^b_1¡®

8T his is generallytruefortests convergingtoa²:SeeH all(19 9 2).

W hereas the asymptotictestappears size distorted (undersized) in the M onteCarlopresentedinthepaper, thebootstrapworksperfectlywellinthe experiments reported. Italsoappears thattheadvantageofusingbootstrap increasesasthecorrelationbetweentheinstrumentandtheregressorsbecoms low, which shouldhaveimportantimplications forpracticalapplications.

4.3 T ime Series andD ynamicmodels

A n importantissuethatwehavenotyettouched upon, is howtocarryout thebootstrapindynamicmodels. T hereisanextensivesurveyavailableinL i

& M addala(19 9 6), whereanentireissueofEconometricR eviews is devoted tothepaperand …vecommentingnotes byotherleadingresearchers in the

…eld. W ewillherejustdiscuss thegeneralprincipleaccordingtowhich the bootstrap iscarriedoutindynamicmodels, andhowtimeseriesmodelswith non-IID errors couldbehandled.

L etus …rstconsiderasimpledynamicmodelofthetype

y_t= ® y_t¡1+ ¯₀ + ¯₁x_t+ "_t; "_t»I I D^³0 ;¾²_"^´

A s longas the errors areIID , theimplementation ofthesocalled recursive bootstrap is straightforward:

1. Estimate® and¯ bysomeestimatorofpreference.

2. O btainrescaled⁹ residuals

~"t=

µ n

n ¡k

¶¹₂

^"t

where

^

"t= yt¡^® yt¡1+ ^¯₀ + ^¯₁xt

9T he rescalingis necessary due tothe aforementioned factthatO L S residuals under- estimatethetrueerrors.

3. G enerate abootstrap sample by resamplingfrom the ~"t:s, and either usinganactualy0 ordrawingitfrom itsunconditionaldistribution, i.e.

y0 »N

à ¯^₀

1¡^®; ¾^² 1¡^®²

!

andthen creatingthebootstrap datarecursivelyas

y_t^b= ^® y_t¡1+ ^¯₀ + ^¯₁x_t+ ~"^b_t (5)

4. A pply the estimatortothe bootstrap sample and calculate a pivotal statisticofinterest. T hencarryonas usual.

Ifthereis anerrorstructureofmorecomplexform present, thesocalled M oving B locks bootstrap could be used in static time series models. T he generalideais toperservethis errorstructures byresampling(overlapping) blocks ofresiduals ratherthan resamplingthem onebyone. T herearesev- eralproblems with this approach, notthe leastthatthe serialdependence willjustprevailwithin the blocks, makingthe approximation ofthe resid- uals distribution arather”rough” one. T his and otherproblems as wellas di¤ erentways ofimplementing the moving blocks bootstrap is thoroughly discussed in L i & M addala(19 9 6).

4.4 N onstationarityandthe B ootstrap

Firstand foremost, itneeds tobestressed thattheissue as toifand when the bootstrap can be applied in nonstationary contexts appears to be an open question. Severalstudies exist, givingsomewhatmixed evidence. T he standard results ofimprovements compared to asymptotic approximations donotimmediatelycarryoverwhen thequantities webootstrap comefrom none-stationarytimeseries.

T his particularresearch …eld is quiteactiveand someprogress has been achieved. T he earliestreference in the econometrics literature seems tobe H arris (19 9 2), whereabootstrap testforaunitrootin aunivariatecontext (theordinaryD ickey-Fullertest) is showntohavebetterproperties thanthe asymptotic test, once the unitroot is imposed. T here are no theoretical results, atleastnotin the econometrics literature, telling us ifwe should expectimprovements from bootstrap tests inthesecases, andtryingtogen- eralisetheunivariatetesttoabootstrap versionoftheJohansen(19 88)- test, H arris & Judge (19 9 8) …nd thatthe bootstrap does notworkatall. T heir conjecture is thatitis the mix ofstationary and non-stationary series that makesthebootstrap back…re. T hereishowever, asmentioned, notheoretical explanations, as yet.

A s forcointegratingregressions, where the residuals have been brought to stationarity, there howeverseems to be a case forthe bootstrap. L i &

M addala(19 9 7 ) gives evidenceforimproved inferenceusingbootstrap tests on the coe¢cients ofcointegratingvectors. A n importantimplication from theirstudies is thatwhen the series involved are I (1), it is necessary to bootstrap theresiduals(whichundercointegrationareI (0 )). Furtherresults ontheusefulnessofbootstraptestsoncointegratingvectorsestimatedbythe Johansen(19 88)-procedureis alsogiveninG redenho¤ (19 9 8).

4.5 G M M B ootstrap tests

Evenifthebootstrap is quiteabletoimproveinferencefortheseestimators as well, formingbootstrap tests forG M M estimators provides somespeci…c di¢culties. Suppose thatwe have used the G M M to obtain a parameter vectorofinterest, ^µ; and have hence assumed thatthe vectorsatis…es the populationmomentconditions

E [g (x;z ;µ)]= 0 ;

whereg (x;z ;µ) is avectorofmomentconditions, x regressors andz instru- ments. Inordertorestrict^µ^btoequal^µ; thebootstrap samplemomentsmust satisfythesamemomentconditions as theoriginalsample, whichwouldnot bethecaseifwebootstrap observations with probability1=N in theG M M case. T hereasonis thatthesamplemoments

^ gN

³^µ^´= 1 N

XN i= 1

g^³x;z ;^µ^´;

aregenerallynotzerowhenthemodelisoveridenti…ed. Ifwewanttorestrict

^µ^btoequal^µ, wemustthereforerestrictthesamplemomentsusedaccordingly.

T his problem has recently been noted and addressed in two papers, each suggestingadi¤ erentapproachtosolvetheproblem.

4.5.1 B rown& N ewey

B rown & N ewey(19 9 5) consideran approach based on usingan alternative estimatorofthe distribution ofthe data. T he above problem is solved by replacingtheempiricaldistributionwithamomentrestrictedestimatorofthe distributions. B ydoingthis, themomentconditions areexplicitlyimposed.

M oreformally, B rown& N eweyuseadistributionfunctionestimatorthat imposes the momentconditions. Instead ofresampling with probabilities 1=N , each observation is given an individualprobability, pi, ofbeingdrawn.

T hese estimated probabilities re‡ecthowwellthe momentrestrictions are ful…lledineachcase.

T he probabilities are calculated using a so called empirical likelihood approach¹⁰. L etobservationiintheoriginaldatabedrawnwithaprobability

10SeefurtherO wen (19 88) foratreatiseon EmpiricalL ikelihood

pi, wherepisolves thefollowingmaximisation problem:

pmax1:::pN

XN i= 1

ln(pi) s:t:pi > 0 ; (6)

XN i= 1

pi = 1;

XN i= 1

pig^³z i;^µ^´ = 0 :

H ere, g^³z i;^µ^´is obtained from ane¢cientG M M estimation on theorig- inalsample. Instead ofsolvingthe maximisation problem in (6), B rown &

N ewey presentan easierway to calculate the probabilities above without havingtosolve an N -dimensionalmaximisation problem. L et^gi= g^³z i;^µ^´ be aJ £1 vectorofmoments, i= 1;:::;N . Furthermore, let^¸ be a J£1 vectorgivenby

m ax¸

XN i= 1

ln(1 + ¸ ^gi) s:t:1 + ¸ ^gi> 0 : T hen piis givenby

^

pi= N ^¡1³1 + ^¸ ^gi

´_¡1

:

T his empiricallikelihood estimatoris a memberofa class ofdistribution estimators, whicharemomentrestricted. B rown& N eweyshowthat, ifone only has information aboutthe momentconditions, the proposed moment restricted estimatoris theasymptoticallymoste¢cientestimatoravailable.

4.5.2 H all& H orowitz

H all& H orowitz(19 9 6) proposeadi¤erentwayofrecentringtheG M M boot- strap. T hey create bootstrap samples in the traditional way, that is, by drawing each observation from the empiricaldistribution with probability 1=N . Instead ofrecentringthe distribution as B rown & N ewey do, they re- centre the moments around theirempiricalvalues and use these recentred

moments when estimatingthemodelandwhen formingabootstrap version oftheoveridenti…cationtest-statistic.

L etg^³z _;^µ^´be given by a G M M estimation on the originaldata. T he recentredmoments arethen

~

g^b_N (µ) = N ^¡1

XN i= 1

g^³z _;^bµ^´¡¹g;

where¹g is givenby

¹

g = N ^¡1

XN i= 1

g^³z ;^µ^´: 4.5.3 A pplications

M onteCarloevidencefortheperformanceofthetwodi¤ erentapproaches in adynamicpaneldatamodelisprovidedinB ergström, D ahlberg& Johansson (19 9 7 ) and B ergström (19 9 7 ), whereitis demonstrated thateven ifneither oftheapproches canbaarguedtobebetterthantheotheras awhole, they bothprovideniceimprovementsascomparedtoasymptoticapproximations.

Q uitefewapplicationsoftheG M M bootstrap havebeenundertaken. T oour knowledgetheonlyones areD ahlberg& Johansson(19 9 7 ) andB ergström &

L indberg(19 9 8).

5 Finalremarks

T his has beenaroughandreadyexposureofsomeofthebasics andsomeof theongoingresearchinthebootstrap …eld. T herearesurelyimportantparts ofandpaths throughthis …eldthatI haveleftout, buthopefullythis paper couldserveas ashortintroductionintothis fascinating…eld. T hebootstrap is certainlyausefuldevice, ifapplied properly. Iftheproblem ofinterestis designed rigorously, the correct(pivotal) quantity is bootstrapped, and the

resultis evaluatedcarefully, usingthebootstrap couldprobablyprovequite rewardingin manycontexts.

T he principles ofthe bootstrap are easy and intuitive. Y et, in orderto fullyunderstandwhythebootstrap works betterthan …rst-orderasymptot- ics in a speci…c case, itis importantthatthe theoreticalproperties ofthe boostrap estimatorsareinvestigatedthoroughly, somethingthatmightprove veryimportantforestimators appliedtoespeciallynonstationarydata. T his should bean important…eld forfurthertheoreticalresarch. W hen itcomes toempiricalapplications, bootstrap procedureshavebecome, ifnotyetstan- dard, butatleastincreasingly importantto researchers in the time series

…eld. A pplications formicro- and paneldata are really much more scarce, butshould becomemuchmorecommon, sincethereareseveralencouraging theoreticaland M onte Carloresults speakingin favourofthe bootstrap in thesesettings.

R eferences

A tkinson, S. & W ilson, P. W . (19 9 2). T hebias ofbootstrapped versus con- ventionalstandarderrors inthegenerallinearandSU R models, Econo- metricT heory8: 258–27 5.

B ergström, P. (19 9 7 ). O n bootstrap standard errors in dynamicpaneldata models., W orkingpaper23, D epartmentofEconomics U ppsalaU niver- sity.

B ergström, P., D ahlberg, M . & Johansson, E. (19 9 7 ). G M M bootstrapping and testingin dynamicpanels, W orkingpaper10, D epartmentofEco- nomics U ppsalaU niversity.

B ergström, P. & L indberg, S. (19 9 8). Firms’ …nancialpolicy and labour demand: T heoryand evidence, W orkingpaper??, D epartmentofEco- nomics U ppsalaU niversity.

B rown, B . & N ewey, W . (19 9 5). B ootstrappingforG M M , M imeo, M IT . D ahlberg, M .& Johansson, E.(19 9 7 ).A nexaminationofthedynamicbehav-

ioroflocalgovernments usingG M M bootstrappingmethods, Chapter III in "Essays on Estimation M ethods and L ocalP ublic Economics."

D octoralD isseration, D epartmentofEconomics, U ppsalaU niversity.

D avidson, J. (19 9 4). StochasticL imitT heory, O xfordU niversityP ress, O x- ford.

D avidson, R . & M acKinnon, J. G . (19 9 3). Estimation and Inference in Econometrics, O xfordU niversityP ress, O xford.

D avidson, R . & M acKinnon, J. G . (19 9 6a). T he powerofbootstrap tests, M imeo, Q ueen’s InstituteforEconomicR esearch.

D avidson, R . & M acKinnon, J. G . (19 9 6b). T hesizedistortionofbootstrap tests, D iscussion P aper9 37 , Q ueen’s InstituteforEconomicR esearch.

Efron, B . (19 7 9 ). B ootstrap methods: A notherlookatthejackknife, A nnals ofStatistics7 : 1–26.

Efron, B . & T ibshirani, R . J. (19 9 3). A n Introduction to the B ootstrap, Chapman and H all, N ewY ork.

Ferrari, S. & Cribari-N eto, F. (19 9 8). O n bootstrap an analyticalbias cor- rection, EconomicL etters58: 7 –15.

G redenho¤, M . (19 9 8). B ootstrap Inference in T ime Series Econometrics, P hD thesis, Stockholm SchoolofEconomics.

G reene, W . H . (19 9 7 ). Econometric A nalysis, 3d Edition, P rentice H all, U pperSaddleR iver, N ewJersey.

H all, P. (19 9 2). T he B ootstrap and Edgeworth Expansion, Springer-V erlag, N ewY ork.

H all, P. & H orowitz, J. L . (19 9 6). B ootstrap criticalvalues fortests based onG eneralized-M ethod-of-M oments estimators, Econometrica64: 89 1–

9 16.

H arris, R . (19 9 2). Smallsample testing forunitroots, O xford B ulletin of Economics andStatistics54(4): 615–625.

H arris, R . & Judge, G . (19 9 8). Smallsampletestingforcointegrationusing thebootstrap approach, Economics L etters58: 31–37 .

H orowitz, J. (19 9 7 ). B ootstrap methods in econometrics: T heory and nu- mericalperformance, in D . M . Kreps & K. F. W allis (eds), A dvances in Economics andEconometrics: T heoryandA pplications, Cambridge U niversityP ress, Cambridge, G reatB ritain, chapter7 , pp. 188–222.

H orowitz, J.L .(19 9 2). A smoothedmaximum scoreestimatorforthebinary responsemodel,60: 505–531.

H orowitz, J. L . (19 9 6). B ootstrap critical values for tests based on the smoothed maximum score estimator, M imeo, D ptofEconomics, U ni- versityofIowa.

Johansen, S. (19 88). Statisticalanalysis ofcointegration vectors, Journalof EconomicD ynamics andControl12: 231–254.

L i, H .& M addala, G .(19 9 6). B ootstrappingtimeseriesmodels, Econometric R eviews15(2): 115–158.

L i, H .& M addala, G .(19 9 7 ). B ootstrappingcointegratingrelations, Journal ofEconometrics80: 29 7 –318.

M acKinnon, J. G . & Smith Jr., A . A . (19 9 8). A pproximate bias correction in econometrics, JournalofEconometrics85: 205–230.

M anski, C.(19 7 5). T hemaximum scoreestimatorofthestochasticutilityof choice, JournalofEconometrics3: 205–228.

M arais, M .(19 86). O nthe…nitesampleperformanceofestimatedgenarlized leastsquarsinseeminglyunrelatedregression, W orkingpaper, G raduate SchoolofB usiness, U niversityofChicago.

M aritz, J. & Jarret, R . (19 7 8). A note on estimating the variance ofthe samplemedian,7 3: 19 4–19 6.

N erlove, M . L . (19 9 8). B ootstrapping- lecture notes, M imeo, A R EC, U ni- veristyofM aryland.

O wen, A . (19 88). Empiricallikelihood, A nnals ofStatistics .

R ilstone, P. & V eall, M . (19 9 6). U sing bootstrapped con…dence intervals forimproved inferences with seeminglyunrelated regression equations, EconometricT heory12 (3): 57 0–581.

Shao, J. & T u, D . (19 9 5). T he Jackknife and B ootstrap, SpringerV erlag, B erlin.

van G iersbergen, N . & Kiviet, J. F. (19 9 4). H owto implementbootstrap tests in staticand dynamicregression models, D iscussion P aper7 -9 4- 130, T inbergenInstitute.

W ong, K.-F. (19 9 6). B ootstrapping hauman’s exogeneity test, Economic L etters53: 139 –143.