IMF-揭示溫度的局部影響 Shedding Light on the Local Impact of Temperature 2024

SheddingLightontheLocalImpactof

Temperature

DaHoang,DuongLe,HaNguyen,andNikolaSpatafora

WP/24/178

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progressbytheauthor(s)andarepublishedtoelicitcommentsandtoencouragedebate.

TheviewsexpressedinIMFWorkingPapersare

thoseoftheauthor(s)anddonotnecessarily

representtheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement.

NAT

2024

AUG

?2024InternationalMonetaryFundWP/24/178

IMFWorkingPaper

AfricanDepartmentandInstituteforCapacityDevelopment

SheddingLightontheLocalImpactofTemperature

PreparedbyDaHoang,DuongTrungLe,HaNguyen,andNikolaSpatafora

AuthorizedfordistributionbyLucEyraudandMercedesGarcia-EscribanoAugust2024

IMFWorkingPapersdescriberesearchinprogressbytheauthor(s)andarepublishedtoelicit

commentsandtoencouragedebate.TheviewsexpressedinIMFWorkingPapersarethoseofthe

author(s)anddonotnecessarilyrepresenttheviewsoftheIMF,itsExecutiveBoard,orIMFmanagement.

ABSTRACT:Weuseanewdatasettoestimatetheimpactoftemperatureoneconomicactivityatamore

geographicallyandtemporallydisaggregatedlevelthantheexistingliterature.Analyzing30-kilometergridcellsatamonthlyfrequency,temperaturehasanegative,highlystatisticallysignificant,andquantitativelylarge

effectonoutput:a1°Cincreaseinmonthlytemperatureisassociatedwitha0.77percentreductionin

nighttimelights,aproxyforlocaleconomicactivity.Theeffectsofevenatemporaryincreaseintemperaturepersistforalmostoneyearaftertheshock.Increasesintemperaturehaveanespeciallylarge,negativeimpactongrowthinpoorercountries,indicatingthattheyaremorevulnerabletotheimpactofclimatechange.

RECOMMENDEDCITATION:Hoang,Da,DuongTrungLe,HaNguyen,andNikolaSpatafora.2024.

“SheddingLightontheLocalImpactofTemperature”.IMFWorkingPaperNo.24/178,InternationalMonetaryFund,Washington,DC.

JELClassificationNumbers:

Q54;O47

Keywords:

Climatechange;temperature;economicgrowth;nighttimelights

Author’sE-MailAddress:

dhoang@,

dle6@,

hnguyen7@,

nspatafora@

IMFWORKINGPAPERSHigh-FrequencyImpactofTemperatureonEconomicActivity

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WORKINGPAPERS

SheddingLightontheLocalImpactofTemperature

PreparedbyDaHoang,DuongTrungLe,HaNguyen,andNikolaSpatafora

1

1Theauthor(s)wouldliketothankGuyAmouzouAgbe,RobertBeyer,LucEyraud,VladimirKlyuev,MercedesGarcia-Escribano,YaoJiaxiong,ThibaultLemaire,EmanueleMassetti,LucaRicci,GregorSchwerhoff,CanSever,FilipposTagklis,andseminar

participantsattheIMFandWorldBankforvaluablecommentsandfeedback.

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Contents

I.Introduction 5

II.DataandEmpiricalSpecification 7

II.1Data 7

II.2TheoreticalMotivation 9

II.3EmpiricalSpecification 10

ContemporaneousEffectsofTemperature 10

DynamicEffectsofTemperature 11

III.Findings 12

III.1ContemporaneousEffects 12

III.2DynamicEffects 15

IV.RobustnessChecks 16

V.Conclusions 19

References 21

Appendix 23

A.SupplementaryResults 23

B.DataDescription:theGoogleEarthEngine 29

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I.Introduction

Climatechangeisakeyglobalchallenge.Temperaturesarerising.Droughts,floods,wildfires,andmassive

stormsareoccurringmorefrequentlywithdevastatingeffects.Understandingtheimpactofrisingtemperature,themostbasicmanifestationofclimatechange,oneconomicactivityisfundamentaltoadaptationand

mitigationefforts.

Theeconomicliteraturehasgenerallyfoundthathighertemperaturehurtseconomicactivity,particularlyforhotandpoorcountries.Theearlyliteratureexaminedtherelationshipbetweenaveragetemperatureandaggregateeconomicvariables(e.g.,SachsandWarner,1997;Gallup,Sachs,andMellinger,1999).Itfoundthathotter

countriestendtobepoorer.However,thisrelationshipmightbedrivenbyomittedvariables,suchascountryinstitutions.Themorerecentliteratureusesfluctuationsintemperaturewithinacountrytocontrolforslow-

movingcountrycharacteristics.

2

Itfindsthathighertemperaturereduceseconomicgrowthinpoorcountries(Delletal.,2012;Acevedoetal.,2020)andtheUnitedStates(Colacitoetal.,2019).Thenegativeeffectsrunthroughreducedtotalfactorproductivitygrowth(LettaandTol,2019),reducedinvestmentandlabor

productivity(Acevedoetal.,2020;KalkuhlandWenz,2020)andreducedsectoralproductivity(LeporeandFernando,2023).Burkeetal.(2015)documentthenon-lineareffectoftemperature:economicgrowthriseswithaverageannualtemperatureuntilapproximately13°C,atwhichpointtherelationshipreverses.

Theliteraturetypicallyexaminestheimpactofaverageannualtemperatureonannualeconomicoutcomes.

3

However,temperatureatagivenlocationmayvarygreatlyacrossseasons.Forinstance,temperaturein

Washington,D.C.,variesgreatlywithintheyear

(AppendixFigure1)

;theannualaverageofcloseto21°Ccan

bemisleadinglymoderatebecauseWashingtonD.C.hasacoldwinterandahotsummer.Averaging

temperatureovertheentireyearmaythereforemissimportantfluctuations,suggestingtheimportanceofcarryingouttheanalysisatafrequencythatishigherthanannual.

Similarly,averagingtemperatureoveralargegeographicarea,suchasacountryorevenaprovince,maymissimportantdetails.Bothtemperatureanditsfluctuations,andthestructureofeconomicactivityanditssensitivitytotemperature,mayvarygreatlyacrossspace.Hence,examiningtheimpactoftemperaturerequiresanalysisatamoregranularspatiallevel.

Inthispaper,weexplicitlyacknowledgethatbothtemperatureanditseffectsarehighlyheterogeneousand

localized.Tothatend,weanalyzethelinkbetweentemperatureandeconomicactivityusingdatafinely

disaggregatedacrossbothtimeandspace.Weexaminetheimpactofmonthlytemperatureoneconomic

activitywithin30-kilometerby30-kilometergridcells.Goingtothegridandmonthlylevelsallowsustoestimatetheeffectsoftemperaturemorepreciselyacrossdifferentclimatezones.Thisanalysisoffersnewinsightsto

2See,forinstance,Kahnetal.,2021;Acevedoetal.,2020;Colacitoetal.,2019;LettaandTol,2019;Cashinetal.,2017;andDelletal.,2012.SeealsothesurveysbyChangetal.(2023),Auffhammer(2018),andDelletal.(2014).

3See,forinstance,Bergetal.,2023;Newelletal.,2021;Acevedoetal.,2020;KalkuhlandWenz,2020;Burkeetal.,2015;Delletal.,2012;andDeschênesandGreenstone,2007.Akyapietal.(2022)identify,amongalargenumberofannualclimateindicators,thosemostpredictiveofeconomicdamage.

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complementtheexistingliterature,whichfocusesonannualaveragetemperatureandeconomicoutcomesatthecountrylevel.

4

Granularanalysescomewiththeirchallenges.Thefirstandmostobviousissueisthelackofeconomicdataatthelocallevel.

5

Inthispaper,localeconomicactivityisproxiedbytheintensityofnighttimelights.Alarge

literaturepointstothevalueofnighttimelightsasanindicatorforeconomicactivity(ChenandNordhaus,2011;HuandYao,2022;Martinez,2022;Asheretal.,2021).Relatedtoourpaper,Felbermayretal.(2022)examinetheimpactofstorms,excessiveprecipitation,droughts,andcoldspellsoneconomicactivityproxiedby

nighttimelights.

Overall,wefindanegativeandhighlystatisticallysignificanteffectoftemperatureonthegrowthofnighttimelights.Relatedly,ourheterogeneityanalysisindicatesthatthissignificanteffectoftemperatureisfoundalsoinlocationswithrelativelyloweraveragetemperatures.Thissignificanteffectisobservedincountriesacrosstheincomedistribution,althoughitisstatisticallylargerinpoorerareas.

Underourfullyspecifiedmodel,a1°Cincreaseinmonthly-averagetemperaturereducesthe

contemporaneousyear-over-yeargrowthofmonthlynighttimelightsby0.77percent.CommonestimatesoftheelasticityofnighttimelightswithrespecttoGDPinturnsuggestthiswouldbeassociatedwithareductionin

GDPgrowth,relativetobaseline,ofbetween0.5and0.77percentagepoints.Thismagnitudeislargerthanthefindingsoftenreportedintheliteraturethatusesmoreaggregateddata.

6

Ourestimatesaremoreinlinewith

Acevedoetal.(2020),whichreportthata1°Cincreaseinannualtemperaturereducesoutputgrowthby0.6–1.2percentagepointsforlow-andmiddle-incomecountries,althoughtheeffectislargelynotstatistically

significantforadvancedeconomies.

Akeyquestionishowtheseeffectschangeacrossdifferentclimatezones.Sofar,thereislittleevidence

becauseofthelackofgranulardata.Theliteraturetendstofindstrongereffectsinpoorercountriesandin

warmerareas(Delletal.,2012;Burkeetal.,2015,Acevedoetal.,2020).Inanearlyandinfluentialpaper,

Burkeetal.(2015),usingcountry-averageandannualaveragetemperaturedata,findunevenimpactsof

temperatureonGDPgrowth:positiveforcountrieswithaveragetemperaturebelow13°C,andnegativefor

countrieswithaveragetemperaturesabovethat.However,recentresearchsuggeststhatthisfindingofuneveneffectsisnotrobust.Forinstance,Nathetal.(2023)showthattheuneveneffectsdisappearwhencontrollingforlaggedtemperature.Thefragilefindingswithannual-andcountry-averagedataonceagainindicatethe

needformoregranularanalyses.

Animportantandrelatedissueisthattheliteratureusingaggregatedataisoftenincapableofdisentanglingtheimpactofrisingtemperatureinhotlocationsfromthatinpoorerregions;thatis,itisuncleariftheimpact

reflectsahotclimateorloweconomicdevelopment.Werevisitthisdebateinthispaper.Usingverygranulardata,wecancontrolforbothfactors.Weassignanygivengridcellinanygivenmonthtoaspecific

4Nguyen(2024)examinestheimpactofUS-countylevelseasonaltemperatureoncountry-leveljobgrowth.Thepaperfindsthatahottersummerhurtsjobgrowth,butawarmerwinterhelps.

5Someeffortshavebeenmadetocomputecell-levelcrosscellproduct(seeGeigeretal.,2017,butonlyfor10-yearincrements).

6Forexample,usingcountryaverageandannualaveragetemperaturedata,Delletal.(2012)findthatonaverage,acrossall

countries,a1°Cincreaseintemperaturereducesacountry’sannualGDPgrowthby0.3percentagepoints,andthedeclineisnotstatisticallysignificant.

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temperaturebin,dependingonitsaveragemonthlytemperatureinthatmonth.

7

Wethenexaminetheimpactoftemperatureacrossdifferenttemperaturebins.Wefindthattheimpactoftemperatureongrowthinnighttimelightsisnegativeandstatisticallysignificant,exceptincolderareas.Inotherwords,ourfindingssupportthe

commonhypothesisthattemperatureaffectseconomicactivitymorenegativelyinrelativelywarmareas.

Atthesametime,wefindthat,whencontrollingfortemperature,incomelevelsalsomatter.Withineach

temperaturebin,wefindthattemperaturehasmorenegativeeffectsingridcellsthatarepartofalow-ormiddle-incomecountry.Ourresultthereforesuggeststhattemperaturedoeshavearelativelymoresevereeffectoneconomicactivityinpoorerareas.

Anotherimportantpointrelatestothelong-termeffectsoftemperatureoneconomicactivity.Couldatemporarytemperatureshockinducelong-lastinggrowtheffects?Toanswerthisquestion,werelyonlocalprojection

modelstoanalyzetheimpulseresponseofeconomicactivity,proxiedbynighttimelights,followinganincreaseintemperature.Wefindthatevenatemporaryshocktomonthlytemperatureshasarelativelypersistenteffectoneconomicactivity,whichremainsstatisticallysignificantforalmostayear,althoughtheeffectwanesasthetimehorizonexpands.

II.DataandEmpiricalSpecification

II.1Data

Dataforthemainanalysisvariables—temperatureandnighttimelights—arecollectedfromtwomainsources.ThetemperatureandprecipitationdataarecollectedfromtheERA5MonthlyAggregatesdataset(C3S,2017;Gorelicketal.,2017).ERA5isthefifth-generationglobalclimatereanalysisproducedbyEuropeanCentreforMedium-RangeWeatherForecasts(ECMWF).ERA5MonthlyprovidesaggregatedvaluesforeachmonthforsevenERA5climatereanalysisparameters:2mairtemperature(inKelvin,convertedtoCelsius),2mdewpointtemperature,totalprecipitation(inmeters),meansealevelpressure,surfacepressure,10mu-componentofwindand10mv-componentofwind.ThedatasetisavailablefromJanuary1979toJune2020andcoverstheentireglobewithagridcellresolutionofabout30kilometers(thatis,withdataon30-kilometerwidesquare

areas).Wecollectmonthlymaximumairtemperatureat2mandmonthlytotalprecipitation.

WecollectnighttimelightsdataasmonthlyaverageradiancecompositeimagesfromtheVisibleInfrared

ImagingRadiometerSuite(VIIRS)Day/NightBand(DNB)(Elvidgeetal.,2013).ThisversionisanalternativeconfigurationoftheVIIRSDNB,whichcorrectsforstraylight(dataneartheedgesoftheswathareexcluded).Dataimpactedbycloudcoverisalsoexcluded.And,sincetheusefulnessofnighttimelightstoproxyeconomicactivitycanbeaffectedbysnow,weexcludeobservationswheresnowcover(collectedfromERA5dataset)

exceeds50percentinamonth.

8

ThedatasetisavailablefromJanuary2014toJune2022withagridcell

7Specifically,eachgridcellinanygivencalendarmonthisassignedtooneofsixpossibletemperaturebins(lessthan0°C,0°C–10°C,10°C–20°C,20°C–30°C,30°C–35°C,andgreaterthan35°C),dependingonitsaveragetemperatureinthatmonth.Forexample,thegridcellscoveringWashingtonD.C.haveanaverageJanuarytemperaturebetween0°Cand10°C,andanaverageJunetemperaturebetween20°Cand30°C.

8Snowcoverinthepresenceofmoonlightcanmakeanareaappearverybright(Zhangetal,2023).Theresultsarealsorobusttoexcludingobservationswheresnowcoverexceeds40percentor60percent.

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resolutionofabout500meters.Tomergetemperatureandprecipitationdatawithnighttimelightdata,we

collectthemonthlydatafortheoverlappingperiodfromJanuary2014toJune2020(78months)usinggridcellsof30kilometersby30kilometers,whichareavailableforbothdatasets.Finally,wecollectgrid-cellleveldataonpopulation(CIESINandSEDAC,2018).AlldataarecollectedviaGoogleEarthEngine.

Table1

presentssummarystatisticsofnighttimelightsandtemperature.Acrossallcell-monthobservations,

themedianyear-on-year(YoY)growthrateofnighttimelightisapproximately4percent,withsignificant

variation(94percentagepoints).Theaveragetemperatureis28.5°C.Temperaturehasincreasedby0.10°C(or0.46percent)annuallyforthetypical(median)cell.

Figure1

illustratesthespatialandseasonalvariationsintemperatureacrosstheglobe;bluetonesdenotelowertemperatures,redtoneshighertemperatures.

Table1:SummaryStatistics.

YoYGrowthinNighttimeLights

Temperature(°C)

YoYChangeinTemperature(°C)

Totalmonthly

precipitation(meter)

Observations(cell

Xmonth)

8,438,699

12,517,650

10,585,842

12,517,650

Numberofcells

197,166

200,648

200,648

200,648

Mean

0.1167

28.5310

0.1293

0.0758

Std

0.9360

9.3457

3.0510

0.1017

Min

-24.2884

-9.9429

-22.2279

0.0000

P1

-2.5226

4.8381

-8.1817

0.0000

P5

-1.1159

10.6698

-4.7809

0.0000

P25

-0.2597

22.7557

-1.4121

0.0089

Median

0.0405

30.3510

0.0980

0.0434

P75

0.4174

34.8485

1.6333

0.1001

P95

1.6501

42.0870

5.2489

0.2760

P99

3.1880

45.4611

8.6147

0.4515

Max

14.4289

53.7555

20.9840

4.0605

Note:Summarystatisticsofnighttimelight(YoYgrowth)andtemperature(inlevelandYoYchange).Unitsarepercentagepoints,exceptfortherows“Observations(cellXmonth)”and“Numberofcells”.Eachgridcellisa30-kilometerby30-kilometersquare.MonthlyobservationsfromJanuary2014toJune2020(78months).

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Figure1:Averagetemperaturebyselectedcalendarmonth,acrosstheworld.JanuaryApril

JulyOctober

Source:C3S,2017;Gorelicketal.,2017;Elvidgeetal.,2013;andIMFstaffcalculations.

Note:Themapsexcludecellswithzeroormissingnighttimelights.

II.2TheoreticalMotivation

Tostudytherelationshipbetweentemperatureandeconomicgrowth,wefirstpresentatheoreticalmotivation,basedonDelletal.(2012).Thiswillinformoursubsequentempiricalspecification.Leteconomicactivity

changeaccordingto

yt=eβTtA(1)

whereYdenoteseconomicactivity,Tdenotestemperature,Aisaproductivityterm,andtdenotesthemonth.Here,temperaturehasadirectleveleffectoneconomicactivity,representedbyβ.Inaddition,productivity

affectseconomicactivity,asrepresentedbyα.Andtemperaturemayalsoaffectannualproductivitygrowth:

log(At)?log(At?12)=g+λTt+σTt?12(2)

whereλdenotestheshort-runeffectoftemperatureonproductivitygrowth,and(λ+σ)thelong-runeffect.Thespecificationallowsforeitherpositiveornegativeeffectsoftemperatureonproductivitygrowth.Takinglogs

andyear-over-yearfirstdifferencesof

(1)

,andcombiningwith

(2)

,yields

log(yt)?log(yt?12)=β(Tt?T12)+α(g+λTt+σTt?12)

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?log(yt)?log(yt?12)=αg+(β+αλ)Tt+(ασ?β)Tt?12(3)

So,temperatureaffectsthegrowthrateofoutput;theshort-rungrowtheffectisgivenby(β+αλ),andthelong-runeffectby[α(λ+σ)].Equation

(3)

formsthebasisfortheempiricalspecification.

II.3EmpiricalSpecification

ContemporaneousEffectsofTemperature

Inlinewiththetheoreticalmotivation,themainempiricalregressiontakestheform

log(Lc,t)?log(Lc,t?12)=β0+β1Tc,t+β2Tc,t?12+σ1pc,t+FEt+FEc,m+Ec,t(4)

whereLdenotesnighttimelights,Tdenotestheaveragetemperatureofthemonth,cindexesthe(30-kilometerby30-kilometer)gridcells,andtdenotesthemonth.Changesinnighttimelightsproxyforchangesineconomicactivity.FEtaremonth-specificfixedeffectsthatcaptureglobalfactorsaffectingthegrowthofnighttimelightsacrossallcellsinaparticularmonth.FEc,marecell–by–calendar-monthfixedeffects,whichcontrolsforthecalendar-month-specificgrowthofnighttimelightsforeachgridcell.

9

Theyaddressdifferentseasonalityfordifferentcells.Forinstance,theyhandlethefactthatthenorthernandsouthernhemisphereshaveoppositeseasons,orthatnighttimelightsmaygrowespeciallyfastduringsummermonthsforcellslocatedincoldclimatezones.Ourpreferredspecificationfurthercontrolsforprecipitationasacontemporaneouscovariateclimatefactor,wherepc,tdenotesavectorcomprisingtwodummiesindicatingwhetheragridcellexperiencedabnormallyhigh(onestandarddeviationabovethelong-termaverageprecipitationinaparticularcalendarmonth)orlow(onestandarddeviationbelow)inmontht.

10

Weuseequation

(4)

toanalyzethecontemporaneouseffectoftemperatureontheyear-over-yeargrowthofnighttimelights.Ttisthemainexplanatoryvariable,capturingmontht’stemperature.Thekeycoefficientofinterestisβ1.

11

Weviewtheestimatedeffectoflocaltemperatureonlocaleconomicactivityascausal,sincelocaltemperatureisbestviewedasexogenouswithrespecttolocaleconomicactivity.Inparticular,thereislittleriskofreversecausalityfromlocaleconomicactivitytolocaltemperature.

12

ThecontrolvariableTt?12capturesthe

9Essentially,FEc,mincludes12fixedeffectspergridcell,eachcorrespondingtoacalendarmonthinayear.

10Morespecifically,theprecipitationdummiesaredefinedasfollow:

High_Pc,t=1ifpc,t>average(pc,t)+std(pc,t),and=0otherwise;andLow_Pc,t=1ifpc,t<average(pc,t)?std(pc,t),and=0otherwise;

wherepc,tdenotesthetotalprecipitationlevelofthemonth,andbothaveragesandstandarddeviationsarecell–calendar-monthspecific.

11Whichcorrespondstotheparameters(β+αλ)inequation(3).

12Inprinciple,someomitted,climate-relatedvariablemightbothbesystematicallycorrelatedwithlocaltemperature,andaffectlocaleconomicactivity.Themostobvioussuchvariableisprecipitation;asdiscussed,wethereforecontrolforthis.

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impactoftemperatureinthesamemonthofthepreviousyear.Standarderrorsareclusteredattheprovince(specifically,first-tieradministrative)levelineachcountrywheredataareavailable.

Wethenextendtheanalysis,investigatingseparatelytheimpactofpositiveandnegativechangesintemperature:

log(Lc,t)?log(Lc,t?12)=β0+β1Tc,t+β2Tc,t?12+y1psitivec,t+y2Tc,t×psitivec,t+σ1pc,t+FEt+FEc,m+Ec,t(5)

wheretheindicatorPositivec,t=1if(Tc,t?Tc,t?12)≥0,and=0otherwise.Here,thecontemporaneouseffectoftemperatureontheyear-over-yeargrowthofnighttimelightsisgivenbyβ1fornegativetemperatureshocks,and(β1+y2)forpositivetemperatureshocks.

Inananalogousmanner,wealsoinvestigateseparatelytheimpactoflargepositiveandlargenegativechangesintemperature,definedasthosechangesthataremorethanonestandarddeviationawayfromtheaveragetemperaturechange:

log(Lc,t)?log(Lc,t?12)=β0+β1Tc,t+β2Tc,t?12+δ1Tc,t×Hig?psitivec,t+δ2Tc,t×Hig?Negativec,t+μ1Hig?psitivec,t+μ2Hig?Negativec,t+σ1pc,t+FEt+FEc,m+Ec,t(6)

wheretheindicators

HighPositivec,t=1if(Tc,t?Tc,t?12)>average(Tc,t?Tc,t?12)+std(Tc,t?Tc,t?12),and=0otherwise;and

HighNegativec,t=1if(Tc,t?Tc,t?12)<average(Tc,t?Tc,t?12)?std(Tc,t?Tc,t?12),and=0otherwise;andbothaveragesandstandarddeviationsarecell-specific.

DynamicEffectsofTemperature

Wealsowanttoexaminethedynamicimpactoftemperatureonnighttimelights,includingthelong-runeffect.Tothisend,weemployaflexibleempiricalspecification,adoptingthelocallinearprojectionsmethodology

(Jordà,2005).Impulseresponseanalysisusinglocalprojectionshasbecomeapopularalternativetothe

traditionalstructuralvectorautoregressive(SVAR)models.Inparticular,localprojectionsofferthefollowing

benefitsoverthetraditionalapproach:(i)theyaremorerobusttomisspecification,sincetheydonotconstraintheshapeofimpulseresponsefunctions;(ii)theycanaccommodatenonlinearandmoreflexiblespecifications;

and(iii)theyareeasiertoimplement,becausetheyonlyrequirestandardlinearregressions(Jordà,2005).Ourspecificationtakestheform:

log(Lc,t+?)?log(Lc,t?1)=αc,?+β?Tc,t+Xc,t+σ1pc,t+FEt+FEc,m+Ec,t(7)

wherehdenotestheprojectionhorizon,andlog(Lc,t+?)?log(Lc,t?1)capturesthegrowthrateofnigh