Shift-Share Designs: Theory and Inference *

Transcription

1 Shft-Share Degn: Theory and Inference * Rodrgo Adão Mchal Koleár Eduardo Morale Augut 13, 2018 Abtract We tudy nference n hft-hare regreon degn, uch a when a regonal outcome regreed on a weghted average of oberved ectoral hock, ung regonal ector hare a weght. We conduct a placebo exerce n whch we etmate the effect of a hft-hare regreor contructed wth randomly generated ectoral hock on actual labor market outcome acro U.S. Commutng Zone. Tet baed on commonly ued tandard error wth 5% nomnal gnfcance level reject the null of no effect n up to 55% of the placebo ample. We ue a tylzed economc model to how that th overrejecton problem are becaue regreon redual are correlated acro regon wth mlar ectoral hare, ndependently of ther geographc locaton. We derve novel nference method that are vald under arbtrary cro-regonal correlaton n the regreon redual. We how that our method yeld ubtantally wder confdence nterval n popular applcaton of hft-hare regreon degn. *We thank Krll Boruyak, Peter Egger, Gordon Hanon, Bo Honoré, and emnar partcpant at Carleton Unverty, Prnceton Unverty, Yale Unverty, the Globalzaton & Inequalty BFI conference, IDB, GTDW, Unl, EESP-FGV, PUC- Ro, and the Prnceton-IES conference for very ueful comment. We thank Juan Manuel Catro Vncenz for excellent reearch atance. We thank Davd Autor, Davd Dorn and Gordon Hanon for harng ther code and data. All error are our own. Unverty of Chcago Booth School of Bune. Emal: radao@uchcago.edu Prnceton Unverty. Emal: mkolear@prnceton.edu Prnceton Unverty. Emal: ecmorale@prnceton.edu

2 1 Introducton We tudy nference n hft-hare degn: regreon pecfcaton n whch one tude the mpact of a et of hock, or hfter, on unt dfferentally expoed to them, and whoe dfferental expoure depend on a et of weght, or hare. Specfcally, hft-hare regreon have the form Y = βx + Z δ + ɛ, where X S w X, =1 and S w = 1. 1) =1 For example, n an nvetgaton of the mpact of ectoral demand hfter on regonal employment change, Y correpond to the change n employment n regon, the hfter X a meaure of the change n demand for the good produced by ector, and the hare w may be meaured a the ntal hare of regon employment n ector. Other oberved charactertc of regon are captured by the vector Z, whch nclude the ntercept, and ɛ the regreon redual. 1 Shft-hare pecfcaton can be very appealng n many context: they are mple to apply and have the potental to both crcumvent complcated endogenety ue and provde etmate of treatment effect that are robut to dfferent mcrofoundaton. A a reult, uch pecfcaton have been appled n numerou nfluental tude, ncludng Bartk 1991), Blanchard and Katz 1992), Card 2001) and Autor, Dorn and Hanon 2013). At the ame tme, two type of concern have been raed: frt, the degn may not be approprate n the preence of cro-regonal general equlbrum effect, and econd, the etmand polcy relevance unclear when the effect of the hfter X are heterogeneou acro ector and regon. In th paper, we put thee concern ade and focu on a dfferent queton: how do we perform nference n hft-hare regreon? We fnd that uual tandard error formula may ubtantally undertate the true varablty of OLS etmator of β n eq. 1). To llutrate the emprcal mportance of th problem, we conduct a placebo exerce. A outcome, we ue change n employment rate and average wage for 722 Commutng Zone n the Unted State. We buld a hft-hare regreor by combnng actual ectoral employment hare n 1990 wth randomly drawn ector-level hfter for dgt SIC manufacturng ector. We contruct n th way many placebo ample that dffer excluvely n the randomly drawn ectoral hfter. For each ample, we compute the OLS etmate of β n eq. 1) and tet f t true value zero. Snce the hfter are randomly generated, ther true effect ndeed zero. Vald 5% level gnfcance tet hould therefore reject the null of no effect n at mot 5% of the placebo ample. We fnd however that uual tandard error cluterng on tate a well a heterocedatcty-robut unclutered error are much maller than the true tandard devaton of the OLS etmator and, a a reult, lead to evere overrejecton. Dependng on the labor market outcome ued a the Y varable n eq. 1), the rejecton rate for 5% level tet can be a hgh a 55% f heterocedatcty-robut tandard error are ued and 45% for tandard error clutered on tate, and t never below 17%. In other word, uppoe that 100 reearcher receved data on our randomly generated hock, but were told ntead that thee are actual ectoral hock of nteret, uch a 1 For mplcty of expoton, we refer to the unt of obervaton at whch the outcome varable meaured a a regon, and the unt of obervaton at whch the hfter meaured a a ector. However, our reult apply to any regreon admttng the repreentaton n eq. 1). 1

3 change n trade flow, tarff, or mmgrant employment. Ideally, at mot 5 of them would report tattcally gnfcant, fale-potve reult. However, f thee reearcher were to ue tandard nference procedure, up to 55 of them would fnd a tattcally gnfcant effect of the randomly generated hock on labor market outcome acro U.S. Commutng Zone. The overrejecton even more evere when 2- and 3-dgt SIC code are ued to defne the ector, o that the total number of ector maller. To explan the ource of th overrejecton problem, we ntroduce a tylzed economc model. Our model feature multple regon, each of whch produce output n multple ector. The key ngredent of our tylzed model are a ector-regon labor demand and a regonal aggregate labor upply. We aume that labor demand n each ector-regon par ha an elatcty wth repect to wage that ector-pecfc and an ntercept that, crucally, aggregate everal ector-pecfc component e.g. ectoral productvte and demand hfter for the correpondng ectoral good). Aggregate laborupply n each regon upward-lopng and depend on a regon-pecfc ntercept. 2 We ue a potental outcome framework to repreent the mpact of a partcular ector-pecfc labor demand hock on change n regonal employment predcted by the model. Lettng Y x 1,..., x S ) denote change n aggregate employment n regon f the hock of nteret exogenouly et to x 1,..., x S ), our model mple that Y x 1,..., x S ) = Y 0) + S =1 w x β, 2) where Y 0) = Y 0,..., 0) regon employment change f the hock of nteret equal zero for all ector, and Y = Y X 1,..., X S ) the employment change for the realzed hock X 1,..., X S ). A key nght of our model that the potental outcome Y 0) nclude a hft-hare component that, ung the ame hare w, meaure the mpact on regon of all ector-level hock other than the hock of nteret X. The regreon redual ɛ n eq. 1) wll generally nhert the tructure of the potental outcome Y 0), and wll thu account for hft-hare component that aggregate all unoberved ector-level hock ung the ame hare w that enter the contructon of the regreor X. Conequently, whenever two regon have mlar hare, they wll not only have mlar expoure to the hfter X, but wll alo tend to have mlar value of the redual ɛ. Whle tradtonal nference method allow for ome form of dependence between the redual, uch a patal dependence wthn a tate, they do not drectly addre the poble dependence between redual generated by mlarty n the hare. Th why, n our placebo exerce, tradtonal nference method underetmate the varance of the OLS etmator of β, creatng the overrejecton problem. Motvated by the fndng of our placebo exerce, we tudy the properte of the OLS etmator of β n eq. 1) under repeated amplng of the ector-level hock X, condtonng on the realzed 2 In Appendx A, we how that a pecal cae of the model n Adão, Arkolak and Epoto 2018) mcrofound the labor upply and labor demand functon that we aume. In th mcrofoundaton, every regon produce a dfferentated varety of each ectoral good, varete are freely traded acro regon, labor the only factor of producton, and worker are both mmoble acro regon and equally productve n all ector wthn a regon. In Onlne Appendx C, we provde alternatve mcrofoundaton that feature a) ector-pecfc captal, a n Jone 1971) and Kovak 2013), and b) worker wth doyncratc ectoral productvte, a n Galle, Rodríguez-Clare and Y 2017), Lee 2017) and Burten, Morale and Vogel 2018a). We alo dcu n th Onlne Appendx the mplcaton of allowng for labor moblty acro regon. 2

4 hare w, control Z, and redual ɛ. Th amplng approach natural gven our nteret n the caual effect of the hfter X : we are ntereted n what would have happened f the ector-level hock of nteret had taken dfferent value, holdng everythng ele contant. The key aumpton we mpoe that, condtonal on the control Z and the hare w, the hfter X are a good a randomly agned and ndependent acro ector. Gven th aumpton, we how that the regreon etmand β n eq. 1) correpond to a weghted average of the heterogeneou parameter β n eq. 2), and derve novel confdence nterval that are vald n ample wth a large number of regon and ector under any correlaton tructure of the regreon redual acro regon. 3,4 Our tandard error formula eentally form ectoral cluter whoe varance depend on the varance of a weghted um of the regreon redual ɛ, wth weght that correpond to the hare w. To gan ntuton on th formula, t ueful to conder the pecal cae n whch each regon fully pecalzed n one ector.e. for every, w = 1 for ome ector ); n th cae, our procedure dentcal to ung the uual clutered tandard error formula, but wth cluter defned a group of regon pecalzed n the ame ector. Th n lne wth the rule of thumb that one hould cluter at the level of varaton of the regreor of nteret. 5 We llutrate the fnte-ample properte of our novel nference procedure by mplementng t on the ame placebo ample that we ue to llutrate the ba of uual tandard error formula. Our new formula delver etmate that are cloe to the true tandard devaton of the OLS etmator acro the placebo ample; conequently, when appled to perform gnfcance tet, they yeld rejecton rate that are cloe to the nomnal gnfcance level. A predcted by the theory, our tandard error formula reman accurate n the preence of a tate-level term n the regreon redual, and no matter whether the hfter X are homokedatc or heterokedatc. When the number of ector mall or a ector gnfcantly larger that the other one, our method overreject relatve to the nomnal gnfcance level, but t tll attenuate the overrejecton problem n comparon to uual tandard error formula. In the fnal part of the paper, we llutrate the mplcaton of our new nference procedure for three popular applcaton of hft-hare regreon. Frt, we the tudy of the effect of change n ector-level Chnee mport competton on labor market outcome acro U.S. Commutng Zone, a n Autor, Dorn and Hanon 2013). Second, we ue change n ector-level natonal employment to etmate the elatcty of regonal employment to regonal average wage, a n Bartk 1991). Latly, we ue change n the tock of mmgrant from varou orgn countre to nvetgate the mpact of mmgraton on employment and wage acro occupaton and Commutng Zone n the Unted State, a n the lterature poneered by Altonj and Card 1991) and Card 2001). In thee applcaton, our propoed confdence nterval are ubtantally wder than thoe mpled 3 Software mplementng our confdence nterval avalable at 4 Th reult mlar to that n Barro et al. 2012), who conder cro-ecton regreon etmated at an ndvdual level when the varable of nteret vare only acro group of ndvdual. They how that, a long a the hfter of nteret a good a randomly agned and ndependent acro thee ndvdual group, tandard error clutered on group are vald under any correlaton tructure of the redual. 5 In an extenon, we alo provde confdence nterval that are vald when the hfter X are ndependent only acro cluter of ector, allowng thu for any correlaton of thee hfter acro ector belongng to the ame cluter. We alo extend our methodology to ettng n whch the hft-hare regreor not the treatment of nteret but an ntrument n an ntrumental varable etmator. 3

5 by tate-clutered or heterocedatcty-robut tandard error. In partcular, the 95% confdence nterval for the etmated effect of Chnee competton on local labor market ncreae by 20% 70%, although thee effect reman tattcally gnfcant. We obtan mlar ncreae n the length of the 95% confdence nterval for the etmated mpact of mmgraton hock, whch are 20% 120% wder than thoe mpled by tradtonal method. In contrat, our confdence nterval for the labor upply elatcty etmated ung the procedure n Bartk 1991) are almot dentcal to thoe contructed ung tandard approache; ntutvely, the ectoral hfter ued n th applcaton the change n natonal employment by ector oak up mot ectoral hock affectng the outcome varable and, conequently, no hft-hare tructure left n the regreon redual. 6 Shft-hare degn have been appled to etmate the effect of a wde range of hock. Snce the applcaton are too numerou to comprehenvely enumerate, let u lt a few electve example. In emnal paper, Bartk 1991) and Blanchard and Katz 1992) ue hft-hare tratege to analyze the mpact on local labor market of hfter meaured a change n natonal ectoral employment. More recently, hft-hare tratege have been appled to nvetgate the local labor market conequence of varou obervable hock, ncludng nternatonal trade competton Topalova, 2007, 2010; Kovak, 2013; Autor, Dorn and Hanon, 2013; Dx-Carnero and Kovak, 2017; Perce and Schott, 2017), credt upply Greentone, Ma and Nguyen, 2015), technologcal change Acemoglu and Retrepo, 2017, 2018), and ndutry reallocaton Chodorow-Rech and Weland, 2018). Shft-hare regreor have alo been ued to tudy the mpact of the ame hock on other outcome, uch a poltcal preference Autor et al., 2017a; Che et al., 2017; Colantone and Stang, 2018), marrage pattern Autor, Dorn and Hanon, 2018), crme level Dx-Carnero, Soare and Ulyea, 2017), and nnovaton Acemoglu and Lnn, 2004; Autor et al., 2017b). Shft-hare regreor have been extenvely ued a well to etmate the mpact of mmgraton on labor market, a n Card 2001) and many other paper followng h approach; ee revew of th lterature n Lew and Per 2015) and Dutmann, Schönberg and Stuhler 2016). Furthermore, recent paper have explored veron of hft-hare tratege to etmate the effect on frm of hock to outourcng cot and foregn demand Hummel et al., 2014; Aghon et al., 2018). In addton to ung hft-hare degn to etmate the overall mpact of a hfter of nteret, other work ha ued thee degn a part of a more general tructural etmaton approach; ee Damond 2016), Adão 2016), Galle, Rodríguez-Clare and Y 2017), Burten et al. 2018b), Bartelme 2018). Baum-Snow and Ferrera 2015) revew addtonal applcaton of hft-hare ntrumental varable n the context of urban economc. 7 Independently of the am of the reearcher when etmatng a hft-hare regreon, and of the nterpretaton of the etmand β n eq. 1), uual tandard error formula wll generally be baed and, a long a the retrcton we mpoe on the data generatng proce hold, our novel nference procedure wll be aymptotcally vald. Our paper related to three other paper tudyng the tattcal properte of hft-hare pecf- 6 To llutrate th pont, we etmate the ame nvere labor upply elatcty ung ntead the hft-hare ntrument n Autor, Dorn and Hanon 2013). The ector hfter n th cae change n trade flow from Chna to developed countre other than the U.S. leave n the regreon redual other ectoral hock affectng U.S. labor market; conequently, our confdence nterval are n th cae 20% 250% wder than thoe mpled by tradtonal nference procedure. 7 Several paper ue a hft-hare approach that treat the hfter a unoberved, and for th reaon ue the hare drectly a regreor. Th approach ha been appled to nvetgate the mpact of technologcal hfter Autor and Dorn, 2013), credt upply hfter Huber, 2018), and mmgraton hfter Card and Dnardo, 2000; Monra, 2015). We treat the ectoral hare X a oberved and leave the extenon to the unoberved cae to future work. 4

6 caton. Frt, Goldmth-Pnkham, Sorkn and Swft 2018) focu on the cae n whch the hft-hare regreor ued a an ntrumental varable. Wthn th ettng, thee author tudy the uage of the full vector of hare w 1,..., w S ) a an ntrument for the endogenou treatment, and they conclude that th approach requre that th vector of hare be a good a randomly agned condtonal on the hfter, and ndependent acro regon or cluter of regon. Gven our nteret n explorng the mpact of a pecfc et of hfter, rather than the mpact of a et of hare, th approach not attractve n our ettng. That ad, there may be other ettng n whch th approach more appealng. Second, Boruyak, Hull and Jaravel 2018), alo focung on the ue of a hft-hare regreor a an ntrumental varable, how that t a vald ntrument f the et of hfter a good a randomly agned condtonal on the hare, and dcu contency of the ntrumental varable etmator n th context. Our approach to nference follow ther dentfcaton nght; th way of thnkng about the hft-hare degn alo natural gven our economc model. Thrd, Jaeger, Rut and Stuhler 2018) tudy complcaton wth the hft-hare ntrument when t correlated over tme and there a luggh adjutment of the outcome varable to change n t. The ret of th paper organzed a follow. Secton 2 preent the reult of a placebo exerce llutratng the properte of nference procedure prevouly ued n the lterature on hft-hare degn. Secton 3 ntroduce our tylzed economc model and map t mplcaton nto a potental outcome framework. Secton 4 etablhe the aymptotc properte of the OLS etmator of β n eq. 1), and provde a content etmator of t tandard error. Secton 5 preent the reult of a placebo exerce n whch we llutrate the performance of our novel nference procedure. Secton 6 revt the concluon from everal pror applcaton of hft-hare regreon analy, and Secton 7 conclude. Appendx A nclude a mcrofoundaton for the tylzed economc model ntroduced n Secton 3, and Appendx B contan proof for all propoton n Secton 4. Addtonal reult are collected n Onlne Appendce C, D and E. 2 Overrejecton of uual tandard error: placebo evdence In th ecton, we mplement a placebo exerce to evaluate the fnte-ample performance of the two nference method mot commonly appled n hft-hare regreon degn: a) Ecker-Hubert- Whte or heterokedatcty-robut tandard error, and b) tandard error clutered on group of regon geographcally cloe to each other. In our placebo, we regre oberved change n U.S. regonal labor market outcome on a hft-hare regreor that contructed by combnng actual data on ntal ectoral employment hare for each regon wth randomly generated ector-level hock. We decrbe the etup n Secton 2.1 and ummarze the reult n Secton Setup and Data We generate 30, 000 placebo ample ndexed by m. Each of them contan N = 722 regon and S = 397 ector. We dentfy each regon wth a U.S. Commutng Zone CZ), and each ector wth ether a 4-dgt SIC manufacturng ndutry or an aggregated non-manufacturng ector. We ndex manufacturng ndutre by = 1,..., S 1 and the non-manufacturng ector by = S. 5

7 Ung the notaton ntroduced n eq. 1), each placebo ample m ha dentcal value of the hare {w } N,S =1,=1, the outcome {Y } =1 N, and the non-manufacturng hfter X S; the placebo ample dffer excluvely n the vector of hfter for the manufacturng ector X1 m,..., Xm S 1 ). Specfcally, the hare correpond to employment hare n 1990, the outcome correpond to change n employment rate and average wage for dfferent ubet of the populaton between 2000 and 2007, and the hfter for the non-manufacturng ector alway et to zero, X S = 0. The vector of hfter for the manufacturng ector X1 m,..., Xm S 1 ) drawn..d. from a normal dtrbuton wth zero mean and varance varx m ) = 5 n each placebo ample m. Becaue the hfter are ndependent of both the outcome and the hare, the parameter β zero n every placebo ample m note t doen t matter what the dependence tructure between the outcome and hare themelve ). For each placebo ample m, gven the oberved outcome Y, the generated hft-hare regreor X m and a vector of control Z ncludng only an ntercept, we compute the OLS etmate of β, the heterokedatcty-robut tandard error whch we label a Robut), and the tandard error that cluter CZ n the ame tate wth label St-cluter). Our man ource of data on employment hare the County Bune Pattern, and our meaure of change n employment rate and average wage are baed on data from the Cenu Integrated Publc Ue Mcro Sample n 2000 and the Amercan Communty Survey for 2006 through Gven thee data ource, we contruct our varable followng the procedure decrbed n the Onlne Appendx of Autor, Dorn and Hanon 2013) Reult Table 1 preent the medan and tandard devaton of the emprcal dtrbuton of the OLS etmate of β acro the 30,000 placebo ample, along wth the medan length of the dfferent tandard error etmate, and rejecton rate for 5% gnfcance level tet of the null hypothe H 0 : β = 0. The hfter have no effect on the outcome and column 1) of Table 1 how that, up to mulaton error, the average of the etmated coeffcent ndeed zero for all outcome. Column 2) report the tandard devaton of the etmated coeffcent. Th dperon the target of the etmator of the tandard error of the OLS etmator. 9 Column 3) and 4) report the medan tandard error for Robut and St-cluter procedure, repectvely, and how that both tandard error etmator are downward baed relatve to the tandard devaton of the OLS etmator. On average acro all outcome, the medan magntude of the heterokedatcty-robut and tate-clutered tandard error are, repectvely, 41% and 30% lower than the true tandard devaton. The downward ba n the Robut and St-cluter tandard error tranlate nto a evere overrejecton of the null hypothe H 0 : β = 0. Snce the true value of β equal 0 by contructon, a correctly behaved tet tattc hould generate a rejecton rate of 5%. Column 5) and 6) n Table 1 how that tradtonal tandard error etmator yeld much hgher rejecton rate. For example, when the outcome varable the CZ employment rate, the rejecton rate for a 5% gnfcance level for the null hypothe H 0 : β = % and 38.3% when Robut and St-cluter tandard error are ued, 8 We are very grateful to the author for harng ther code and dataet wth u. 9 Fgure D.1 n Onlne Appendx D.2 report the emprcal dtrbuton of the OLS etmate when the dependent varable the change n each CZ employment rate. It dtrbuton reemble a normal dtrbuton centered around β = 0. 6

8 Table 1: Standard error and rejecton rate of the hypothe H 0 : β = 0 at 5% gnfcance level. Etmate Medan td. error Rejecton rate Mean Std. dev Robut St-cluter Robut St-cluter 1) 2) 3) 4) 5) 6) Panel A: Change n the hare of workng-age populaton Employed % 38.3% Employed n manufacturng % 44.4% Employed n non-manufacturng % 17.4% Panel B: Change n average log weekly wage Employed % 34.1% Employed n manufacturng % 16.8% Employed n non-manufacturng % 33.5% Note: For the outcome varable ndcated n the frt column, th table ndcate the medan and tandard devaton of the OLS etmate acro the placebo ample column 1) and 2)), the medan tandard error etmate column 3) and 4)), and the percentage of dataet for whch we reject the null hypothe H 0 : β = 0 ung a 5% gnfcance level tet column 5) and 6)). Robut the Ecker-Huber-Whte tandard error, and St-cluter the tandard error that cluter CZ n the ame tate. Reult are baed on 30,000 mulaton draw. repectvely. Thee rejecton rate are very mlar when the dependent varable ntead the change n the average log weekly wage. Thee reult are quanttatvely mportant. To ee th, conder the followng thought-experment. Suppoe we were to provde our 30, 000 mulated ample to 30, 000 reearcher wthout dclong to them the orgn of the data. Intead, we would tell them that the hfter correpond to change n a ectoral hock of nteret for ntance, trade flow, tarff, natonal employment or the number of foregn worker employed n an ndutry. If thee reearcher et out to evaluate the mpact of thee hock on U.S. CZ ung tandard nference procedure wth a 5% gnfcance level tet, then over a thrd of them would conclude that our computer generated hock had a tattcally gnfcant effect on the evoluton of employment rate between 2000 and The followng remark ummarze the reult of our placebo exerce. 10 Remark 1. In hft-hare regreon, tradtonal nference method uffer from a evere overrejecton problem and yeld confdence nterval that are too hort. To develop ome ntuton on the ource of th overrejecton problem, note that the tandard error etmator commonly appled n hft-hare regreon degn aume that the regreon redual are ether ndependent acro all regon for Robut), or between geographcally defned regonal group for St-cluter). Gven that hft-hare regreor are correlated acro regon wth mlar ectoral employment hare {w } S =1, thee method generally lead to a downward ba n the tandard error etmate whenever regon wth mlar ectoral employment hare {w } S =1 alo tend to have mlar regreon redual. In the next ecton, we conder the mplcaton of a tylzed economc model, and how that uch correlaton between the regreon redual are lkely to are becaue 10 In Secton 5, we extend our analy to a number of modfcaton of th baelne etup, ncludng alternatve defnton of ector and regon, allowng for a non-zero hock to the non-manufacturng ector, and allowng for correlaton between the hock to dfferent ector. The overrejecton problem alway at leat a evere a n th baelne etup. 7

9 regon are generally expoed to unoberved ector-level hock, n addton to the oberved hock X. Conequently, whenever a reearcher runnng a hft-hare regreon, both heterokedatctyrobut and tate-clutered tandard error wll generally be baed downward. 3 Stylzed economc model Th ecton preent a tylzed economc model mappng ector-level hock to labor market outcome for a et of regonal econome. The am of the model twofold. Frt, we how that the mpact of ectoral hfter on regonal labor market outcome have a hft-hare tructure, wth heterogeneou effect acro regon and ector. Second, we how that unoberved ectoral hfter ntroduce correlaton n the regreon redual acro regon wth mlar oberved hare. We decrbe the model fundamental n Secton 3.1, dcu t man mplcaton for the mpact of ectoral hock n Secton 3.2, and map thee mplcaton to a potental outcome framework n Secton Envronment We conder an economy wth multple ector = 1,..., S and multple regon = 1,..., J. We aume that the labor demand n ector and regon, L, gven by log L = σ log ω + log D, σ > 0, 3) where ω the wage rate n regon, σ the ector-pecfc labor demand elatcty, and D are regon- and ector-pecfc labor demand hfter. The hfter D may account for multple ectoral component. Snce our analy focue on the mpact of one partcular ectoral component, we decompoe D nto an oberved hfter of nteret, χ, other potentally unoberved) hfter that vary at a ectoral level and are grouped nto µ, and a redual regon- and ector-pecfc hfter η. That, wthout lo of generalty, we wrte log D = ρ log χ + log µ + log η. 4) We aume that the labor upply n regon gven by log L = φ log ω + log v, φ > 0, 5) where φ the labor upply elatcty, and v a regon-pecfc labor upply hfter. Worker are aumed to be mmoble acro regon, but freely moble acro ector. Thu, we defne the equlbrum a the wage {ω } J =1 that atfy the followng market clearng condton: L = S L, = 1,..., J. 6) =1 There are multple mcrofoundaton that are content wth the labor demand n eq. 3) and the labor upply n eq. 5). For our purpoe, the dfferent labor demand mcrofoundaton are mportant 8

10 only to the extent that they affect the nterpretaton of the ector- and regon-pecfc labor demand hfter D. For example, one could aume that labor the only factor of producton and that every regon a cloed economy and, n th cae, D may account both for demand hfter for ectorpecfc good and for ector-pecfc productvty hfter. Smlarly, a we how n Appendx A, we may alo allow good to be freely traded acro regon and aume that a ubet of the J regon are mall open econome; n th cae, the hfter D for thee mall open econome wll account for the world prce of ector, whch wll telf capture the mpact of foregn demand and productvty hock. We alo how n Appendx A that the labor upply n eq. 5) may be derved a the outcome of the utlty maxmzaton problem of ndvdual who, condtonal on beng employed, are ndfferent about the ector of employment, but have heterogeneou dutlte of beng employed at all. 3.2 Labor market mpact of ectoral hock We aume that, n any perod, our model characterze the labor market equlbrum n every regon = 1,..., J and that, acro perod, change n the labor market outcome {ω, L } J =1 are due to change n the ectoral hfter of nteret, {χ } S =1, other potental ectoral hfter {µ } S =1, ector- and regon-pecfc hfter {η } J,S =1,=1, and labor upply hfter, {v } J =1. Specfcally, n every perod, the value of thee hfter correpond to draw from an unknown jont dtrbuton F ): {χ, µ } S =1, {η } J,S =1,=1, {v } J =1 ) F ). 7) We ue ẑ = logz t /z 0 ) to denote log-change n a varable z between ome ntal perod t = 0 and any other perod t. Up to a frt-order approxmaton around the ntal equlbrum, eq. 3) to 6) mply that the change n employment n regon ˆL = S l 0 [β ˆχ + λ ˆµ + λ ˆη ] + 1 λ ) ˆv, 8) =1 where l 0 the ntal employment hare of ector n regon, λ φ [ φ + l 0 σ ] 1, and β ρ λ. Accordng to eq. 8), the mpact of ectoral hfter on equlbrum employment n regon depend both on the ntal ectoral employment hare {l 0 }S =1, and the regon- and hfter-pecfc elatcte {β, λ } S =1. Conequently, the employment change n eq. 8) nclude everal component wth a hft-hare tructure: the hare term alway the ntal employment hare n a ector l 0, and the hft term ether the ectoral hock of nteret, ˆχ, or alternatve labor demand hock, ˆµ. Th tructure wth multple hft-hare term, ome of them oberved and other potentally unoberved, central to undertandng the reult preented n Secton 2. Notce alo that, even condtonal on the ntal employment hare l 0, the mpact of a ector hfter on regon- employment may be heterogeneou acro ector and regon: β may vary acro and. 11 Whle tandard dataet wll uually contan nformaton on the ntal employment hare for every ector and regon {l 0 }J,S 1,=1, each parameter β not generally known or drectly 11 In our model, β doe not vary acro regon or ector f and only f all ector have the ame labor demand elatcty, σ = σ, and hock pa-through, ρ = ρ. 9

11 oberved, and thu, the mpact of the ectoral hfter need to be etmated. We ummarze th dcuon n the followng remark: Remark 2. The change n regonal employment wll generally combne multple hft-hare term, and the hfter effect depend on parameter that are heterogeneou acro ector and regon. The property that the mpact of a hfter n ector on employment n regon may be wrtten a l 0 β that underle Remark 2 doe not depend on the partcular mcrofoundaton of the labor demand and labor upply expreon n eq. 3) and 5). The only dfference acro thee mcrofoundaton how β depend on the tructural parameter of each mcrofounded model. Bede the llutratve example of a poble mcrofoundaton decrbed n Appendx A, we provde alternatve mcrofoundaton n Onlne Appendce C.2 and C.3. Specfcally, we how n Onlne Appendx C.2 that eq. 8) content wth a Jone 1971) model featurng ector-pecfc nput of producton. In Onlne Appendx C.3, we how that eq. 8) alo are n a Roy 1951) model n whch worker have heterogeneou preference for beng employed n the dfferent ector. We alo extend our model n Onlne Appendx C.4 to allow for mgraton acro regon. In th cae, the change n regonal employment ˆL n any gven regon = 1,..., J depend not only on the regon own hft-hare term ncluded n eq. 8), but alo on an endogenou component, common to all regon, that combne the hft-hare term correpondng to all regon = 1,..., J. Thu, n the preence of mgraton, l 0 β the partal effect of the hfter ˆχ on local employment that gnore cro-regonal pllover; conequently, t wll only capture the dfferental effect of the ector-pecfc hock ˆχ on regon relatve to all other regon. However, once we condton on fxed effect that aborb thee cro-regonal pllover, Remark 2 reman vald for the model wth mgraton. 3.3 From theory to nference We buld on the nght of Secton 3.2 to propoe a general framework to etmate the effect of hfter on an outcome of nteret that vare at a dfferent level than thee hfter. For concretene, we refer to the level at whch the hfter vary a ector, and the level at whch the outcome vare a regon, but our reult do not depend on thee partcular label. To make prece what we mean by the effect of hfter on an outcome, we ue the potental outcome notaton, wrtng Y x 1,..., x S ) to denote the potental counterfactual) outcome that would occur n regon f the hock to the S ector were exogenouly et to {x } S =1. Contently wth eq. 8), we aume that the potental outcome are lnear n the hock, Y x 1,..., x S ) = Y 0) + S =1 w x β, where S w = 1, 9) =1 and Y 0) Y 0,..., 0) denote the potental outcome n regon when all hock {x } S =1 are et to zero. Accordng to eq. 9), ncreang x by one unt whle holdng the hock to the other ector contant, lead to an ncreae n regon outcome of w β unt. Th the treatment effect of x on Y x 1,..., x S ). The actual oberved) outcome gven by Y = Y X 1,..., X S ), whch depend on 10

12 the realzaton of the hfter X 1,..., X S. To map eq. 8) nto eq. 9), defne Y = ˆL, w = l 0, x = ˆχ, Y 0) = S l 0 λ ˆµ + ˆη ) + 1 λ ) ˆv. 10) =1 Oberve that Y 0) aggregate all hfter other than the ectoral hfter of nteret ˆχ. In the ret of the paper, we aume that we oberve data for N regon and S ector on the ectoral hfter X, the regonal outcome Y, and the regon-ector hare w. 12,13 We are ntereted n the properte of the OLS etmator ˆβ of the coeffcent on the hft-hare regreor X = S =1 w X n a regreon of Y onto X. To help u focu on the key conceptual ue, we abtract away from any addtonal covarate or control for now, and aume that X and Y have been demeaned, o that we can omt the ntercept n a regreon of Y on X ee Secton 4.2 for the cae wth control). The OLS etmator of the coeffcent on X n th mplfed ettng gven by and we can wrte the regreon equaton a ˆβ = N =1 X Y =1 N, 11) X2 Y = βx + ɛ, where X S w X, =1 S w = 1, 12) =1 where β denote the populaton analog of ˆβ. The defnton of the etmand β and the properte of the etmator ˆβ wll depend on: a) what the populaton of nteret ; and b) how we thnk about repeated amplng. For a), we defne the populaton of nteret to be the oberved et of N regon, a oppoed to focung on a large uperpopulaton of regon from whch the N oberved regon are drawn. Conequently, we are ntereted n the parameter {β } N,S =1,=1 and the treatment effect {w β } N,S =1,=1 themelve, rather than the dtrbuton from whch they are drawn, whch would be the cae f we were ntereted n a uperpopulaton of regon. 14 For b), gven our nteret on etmatng the ceter parbu mpact of a pecfc et of hock X 1,..., X S, we conder repeated amplng of thee hock, whle holdng fxed the hare w, the parameter β, and the potental outcome Y 0). Gven our aumpton on the populaton of nteret and on the type of repeated amplng, the etmand β defned a the populaton analog of eq. 11) under repeated amplng of the hock X : β = N =1 E[X Y F 0 ] N =1 E[X2 F 0 ], wth F 0 = {Y 0), β, w } N,S =1,=1, 13) 12 We can thnk of the N oberved regon a a ubet of the J regon extng worldwde and whoe labor market equlbrum decrbed n Secton 3.1 and For mplcty, we aume that we have data on the hfter X drectly, rather than pobly noy etmate of them. 14 Th defnton of the populaton of nteret common n applcaton of the hft-hare approach. For example, the abtract of Autor, Dorn and Hanon 2013) read: We analyze the effect of rng Chnee mport competton between 1990 and 2007 on U.S. local labor market. Smlarly, the abtract of Dx-Carnero and Kovak 2017) read: We tudy the evoluton of trade lberalzaton effect on Brazlan local labor market emphae added). 11

13 and, gven eq. 9) and 12), the regreon error ɛ then defned a the redual where β defned a n eq. 13). ɛ = Y X β = Y 0) + S =1 w X β β), 14) Thu, the tattcal properte of the regreon redual ɛ depend on the properte of the potental outcome Y 0), the hfter {X } S =1, the hare {w } N,S =1,=1, and the dfference between the parameter {β } N,S =1,=1 and the etmand β. Importantly, the potental outcome Y 0) wll generally ncorporate term that have a hft-hare tructure analogou to that of the regreor of nteret, X. Specfcally, a llutrated n eq. 10), the model ntroduced n Secton 3.1 mple that Y 0) nclude a weghted average of unoberved ectoral labor-demand hock, S =1 l0 λ ˆµ. Hence, f two regon and have mlar hare {l 0 }S =1 and {l0 }S =1, they wll tend to have mlar regreor X and X and mlar potental outcome Y 0) and Y 0). It then follow from eq. 14) that the redual ɛ and ɛ wll be correlated. 15 We ummarze th dcuon n the followng remark. Remark 3. Correctly performng nference for the coeffcent on a hft-hare regreor β requre takng nto account that the regreon redual wll generally nhert the ame hft-hare tructure. Remark 3 ha mportant mplcaton for etmatng the varablty of ˆβ acro ample. In partcular, tradtonal nference procedure do not account for correlaton n ɛ among regon wth mlar hare and, therefore, tend to underetmate the varablty of ˆβ. A we dcu n all remanng ecton of the paper, th the man reaon for the overrejecton problem decrbed n Secton 2. 4 Aymptotc properte of hft-hare regreon In th ecton, we formulate the tattcal aumpton that we mpoe on the data generatng proce DGP), preent aymptotc reult that we derve ung thee aumpton, and ue the model ntroduced n Secton 3.1 to provde an economc nterpretaton for thee aumpton. We frt conder n Secton 4.1 the mple cae n whch there a ngle regreor wth a hft-hare tructure and no control, a n Secton 3.3. We ntroduce control n Secton 4.2. Secton 4.3 conder further extenon. All proof and techncal detal are n Appendx B. Followng the notaton ntroduced n Secton 3.3, we wrte ector-level varable uch a the hock X ) n crpt font tyle and regon-level aggregate uch a X ) n normal tyle. To compactly tate our aumpton and reult, we ue tandard matrx and vector notaton. In partcular, for a column) L-vector A that vare at the regonal level, A denote the N L matrx wth the th row gven by A. For an L-vector A that vare at the ectoral level, A denote the S L matrx wth the th row gven by A. If L = 1, then A and A are an N-vector and an S-vector, repectvely. Let W denote the N S matrx of hare, o that t, ) element gven by w, and let B denote the N S matrx wth, ) element gven by β. 15 A we dcu n Secton 4.2, when control are ncluded, th concluon wll tll hold unle the control account for all ectoral hock other than {X } S =1 that affect the outcome. 12

14 4.1 No control We tudy here the tattcal properte of the OLS etmator n eq. 11). We aume that, condtonally on the matrx of hare W, the hock are a good a randomly agned n that they are ndependent of the potental outcome Y x 1,..., x S ). Formally, gven the defnton of the potental outcome n eq. 9), we aume Y0), B) X W. 15) In the next ubecton, we weaken th aumpton by aumng that the hock are a good a randomly agned condtonally on ome control. A dcued n Secton 3.3, we conder the tattcal properte of ˆβ under repeated amplng of the hock X, and condton on the realzed value of the hare and on the potental outcome. Th approach analogou to the randomzaton-tyle nference n the lterature on nference n randomzed controlled tral ee Imben and Rubn, 2015, for a revew); t leverage the random agnment aumpton n eq. 15), and enure that the tandard error that we derve wll reman vald under any dependence tructure between the hare w acro ector and regon, and under any correlaton tructure of the potental outcome Y 0), or equvalently, of the regreon error ɛ, acro regon. In partcular, th approach allow but doe not requre) the redual to have a hft-hare tructure. We conder aymptotc wth the number of ector gong to nfnty, S, and aume that N a S. Formally, the number of regon N thu depend on S, but we keep th condtonng mplct. We do not retrct the rato N/S, o that the number of regon may grow at a fater rate than the number of ector. The aumpton needed for the propoton below are collected n Appendx B.1. The key aumpton underlyng our approach to nference that the hock X 1,..., X S ) are ndependent acro condtonal on the hare W ee Aumpton 1) n Appendx B.1). In contrat, Y 0) and the hare w can be correlated n an arbtrary manner acro. We alo do not requre X, or any other varable, to be dentcally dtrbuted the ector and regon may be heterogeneou. The man regularty condton that we need that each ector aymptotcally neglgble n the ene that max n /N 0, where n = =1 N w the aggregate ze of ector n the populaton of nteret ee Aumpton 2) n Appendx B.1). It generalze the tandard contency condton n the cluterng lterature that the larget cluter be aymptotcally neglgble. To ee the connecton, conder the pecal cae wth concentrated ector, n whch each regon pecalze n one ector ). Then w = 1 f = ) and w = 0 otherwe, and n the number of regon that pecalze n ector. In th cae, X = X ), o that, f eq. 15) hold, ˆβ equvalent to an OLS etmator n a randomzed controlled tral n whch the treatment vare at a cluter level; here the th cluter cont of regon that pecalze n ector. The condton max n /N 0 then reduce to the aumpton that the larget cluter be aymptotcally neglgble. Propoton 1. Suppoe Aumpton 1 and 2 n Appendx B.1 hold. Then β = N =1 S =1 π β =1 N S =1 π, and ˆβ = β + o p 1), 16) 13

15 where π = w 2 varx W). Th propoton gve two reult. Frt, t how that the etmand β n eq. 13) can be expreed a a weghted average of the regon- and ector-pecfc parameter {β } N,S =1,=1, wth weght that are ncreang n the hare and varance of the hock. Second, t how that the OLS etmator ˆβ converge to th etmand a S. The pecal cae wth concentrated ector agan ueful to undertand Propoton 1. Fully concentrated ector mply that S =1 π β = varx ) W)β ) and, therefore, the frt reult n Propoton 1 reduce to the tandard reult from the randomzed controlled tral lterature wth cluter-level randomzaton wth each cluter defned a all regon pecalzed n the ame ector) that the weght are proportonal to the varance of the hock. The etmand β doe not n general equal a weghted average of the heterogeneou treatment effect. A dcued earler, the effect on the outcome n regon of ncreang the value of the ector hock n one unt equal to w β ; the total effect of ncreang the hfter multaneouly n every ector by one unt S =1 w β. Conequently, for a et of regon- and ector-pecfc weght {ξ } N,S =1,=1, the correpondng weghted average treatment effect τ ξ N =1 S =1 ξ w β =1 N S =1 ξ, and a weghted total average treatment effect τζ T = =1 N ζ S =1 w β, where {ζ } =1 N are regonal weght that um to one. If β contant acro and, β = β, then β = τζ T, and τ ξ can be contently etmated a ˆτ ξ = ˆβ N =1 S =1 ξ w / N =1 S =1 ξ. On the other hand, f β vare acro regon and ector, then t not clear n general how to explot knowledge of the etmand β defned n eq. 16) to learn omethng about τ ξ or τζ T. A pecal cae n whch t poble to contently etmate τ ξ are when X homocedatc and ξ = w ; n th cae, a content etmate gven by ˆτ ξ = ˆβ N =1 S =1 w2 ˆσ2 / N =1 S =1 w ˆσ 2, where ˆσ 2 a content etmate of varx ). 16 Under lght trengthenng of the regularty condton ee Aumpton 3 n Appendx B.1), we obtan the followng dtrbutonal reult: Propoton 2. Suppoe Aumpton 1, 2 and 3 hold, and uppoe that V N = 1 S =1 n2 var N X ɛ Y0), B, W =1 converge n probablty to a non-random lmt, where n = N =1 w. Then N ˆβ β) = N 0, ) 2 + o p 1). S =1 n2 1 N N =1 X2 16 In general, one could contently etmate τ ξ or τ T ζ by mpong a mappng between β and tructural parameter and obtanng content etmate of thee tructural parameter. However, nce th mappng wll vary acro model, the contency of uch etmator wll not be robut to alternatve modelng aumpton, even f all thee modelng aumpton predct an equlbrum relatonhp lke that n eq. 8); e.g. ee expreon for β n Appendx A and n Onlne Appendce C.2 and C.3. V N ) 14

16 Th propoton how that ˆβ aymptotcally normal, wth a rate of convergence equal to N S =1 n2 ) 1/2. If the ector ze n are all equal to N/S, the rate of convergence equal to S. However, f the ze are unequal, the rate may be lower. Accordng to Propoton 2, the aymptotc varance formula ha the uual andwch form. Snce X oberved, to contruct a content tandard error etmate, t uffce to contruct a content etmate of V N, the mddle part of the andwch. Suppoe that β common acro regon and ector, β = β, then t follow from eq. 15) and the aumpton that X 1,..., X ) are ndependent acro that 17 V N = S =1 varx W)R 2 N S, R = w ɛ. 17) =1 n2 =1 Replacng varx W) by X 2, and ɛ by the regreon redual ˆɛ = Y X ˆβ, we obtan the tandard error etmate ŝe ˆβ) = S =1 X2 ˆR 2 N, ˆR = w ˆɛ. 18) =1 N =1 X2 To gan ntuton for the expreon n eq. 18), conder the cae wth concentrated ector uch that the formula become S =1 X2 ˆR 2 = S =1 N =1 I{) = }X ˆɛ ) 2. In th pecal cae, the tandard error formula n eq. 18) reduce to the uual cluter-robut tandard error, allowng for arbtrary correlaton acro regon pecalzed n the ame ector. 18 When regon are not fully pecalzed n a ector, the tandard error n eq. 18) account for the fact that regon wth mlar ectoral compoton wll generally have mlar error; only n the pecal cae n whch the regreon error ɛ = Y 0) ha no ectoral component o there are no unoberved ector-level hock), t wll be the cae that covx ɛ, X j ɛ j ) = 0 for = j. In contrat, the uual heterocedatcty-robut tandard error fal to account for th correlaton. Standard error clutered by group of regon defned by ther geographcal proxmty wll alo generally fal to account for th correlaton. In fact, they wll only capture t f and only f all regon are fully pecalzed n a ngle ector and the ector of pecalzaton the ame for regon belongng to the ame geographcally defned cluter. Remark 4. In the expreon for V N n eq. 17), the only expectaton taken over X we do not take any expectaton over the hare w or the redual ɛ. Th becaue our nference condtonal on the realzed value of the hare and on the potental outcome. In term of the regreon n eq. 12), th mean that we conder properte of ˆβ under repeated amplng of X = w X condtonal on the hare w and on the redual ɛ a oppoed to, ay, conderng properte of ˆβ under repeated amplng of the redual ɛ condtonal on X ). A a reult, our tandard error allow for arbtrary dependence between the redual ɛ. 17 The tandard error formula that we provde reman vald f β heterogeneou acro regon and ector, a long a ome mld retrcton on the form of heterogenety apply; ee Appendx B.6 for a dcuon. 18 Thu, n the cae wth concentrated ndutre, the uual approach to nference that conder repeated amplng of ɛ, holdng the regreor contant, would delver the ame tandard error formula f one aumed that ɛ were ndependent acro locaton pecalzng n dfferent ndutre. 15

17 4.1.1 Dcuon of aumpton In general, n order to dentfy a relatonhp a caual, one need a random agnment aumpton. In order to do nference and apply a central lmt theorem, one need an ndependence-type aumpton. 19 In our cae, the key dentfyng aumpton that the hfter {X } S =1 are a good a randomly agned condtonal on the hare {w } S,N =1,=1 ee eq. 15)). Th dentfcaton aumpton ha been prevouly uggeted by Boruyak, Hull and Jaravel 2018). For nference, we alo requre that the hock are ndependent acro ector. A llutrated through the economc model decrbed n Appendx A and Onlne Appendx C, thee aumpton generally mply retrcton on the tochatc proce of economc fundamental. How trong thee retrcton are wll depend on the pecfc context. For example, n a world n whch all N regon of nteret are cloed econome, the only ectoral hock are ether productvty or preference hock, and the hfter of nteret are the former, thee aumpton requre that, condtonal on the hare, the productvty hock are: a) ndependent of preference hock; and b) ndependent acro ector. In Secton 4.2, we llutrate how to relax aumpton a) by ncorporatng control nto the regreon pecfcaton and, n Secton 4.3.2, we how how to relax t by ung ntrumental varable. Addtonally, we how n Secton how to relax aumpton b) by allowng for a non-zero correlaton n the ectoral hock of nteret wthn cluter of ector. Goldmth-Pnkham, Sorkn and Swft 2018) nvetgate a dfferent approach to dentfcaton baed on the aumpton that the hare w 1,..., w S ) are a good a randomly agned condtonal on the hfter X. For nference, th approach requre that the hare w 1,..., w S ) be ndependent acro regon or cluter of regon. However, a llutrated through the tylzed economc model preented n Secton 3, thee hare are generally equlbrum object and, conequently, they are unlkely to be a good a randomly agned. For ntance, n the cae of the envronment decrbed n Secton 3.1, under the aumpton that σ = σ for all ector, t hold that l 0 = D0 /S k=1 D0 k ), where D 0 the labor demand hfter of ector n regon n the ntal equlbrum. However, a hown n eq. 4), 10) and 14), the regreon redual ɛ account for change n certan varable that alo affect the demand hfter D 0 and, conequently, l0 wll generally be correlated wth ɛ unle change n thoe varable are ndependent of ther pat ntal level. 20 Furthermore, a the demand hfter D 0 are lkely to depend on term that vary by ector ee eq. 4)), the labor hare l0 wll generally be correlated acro all regon = 1,..., N for a gven ector, complcatng the tak of dervng vald nference procedure n th ettng. The reult n Propoton 1 and 2 alo requre the aumpton that max n /N 0. In term of the economc model ntroduced n Secton 3, th aumpton mpoe that no one ector domnate the other n term of ntal employment at the natonal level;.e. =1 N l0 not too large for any one ector. A we llutrate n Secton 5.2, th condton atfed for the U.S. when only manufacturng ector are taken nto account; t would not hold f the non-manufacturng ector ncluded a one of 19 For example, for nference on average treatment effect, whch commonly the goal when runnng a regreon, one aume that the treatment a good a randomly agned condtonal on control, and typcally alo that the data on ndvdual..d., whch mple that the treatment ndependent acro ndvdual condtonal on the control. 20 Importantly, the correlaton between the hare {w } S =1 and the regreon redual ɛ doe not affect the contency of the OLS etmator of β f the hfter X are a good a randomly agned condtonal on the hare w. 16

18 the S ector ncorporated nto the analy unle the dtrbuton of X for the non-manufacturng ector degenerate at zero). 21 Fnally, Propoton 1 and 2 alo requre the number of ector and the number of regon to go to nfnty. Shft-hare degn are however ometme ued n ettng n whch the number of regon or the number of ector mall. Through placebo exerce, we llutrate n Secton 5 the fnteample properte of the tandard error etmator ntroduced n eq. 18): our etmate are very cloe to the true tandard devaton of the etmator ˆβ for ample ze employed n typcal applcaton. 4.2 General cae wth control In many applcaton of hft-hare regreon degn, a K-vector of regonal control Z ncluded n the regreon pecfcaton. We now tudy the properte of the OLS etmator of the coeffcent on X n a regreon of Y onto X and Z. To th end, let Z denote the N K matrx wth -th row gven by Z, and let Ẍ = X ZZ Z) 1 Z X denote an N-vector whoe -th element equal to the regreor X wth the control Z partalled out.e. the -th redual from regreng X onto Z). Then, by the Frch Waugh Lovell theorem, ˆβ equvalent to ˆβ = N =1 ẌY N =1 Ẍ2 = Ẍ Y Ẍ Ẍ, 19) and the OLS etmator of the coeffcent on Z equvalent to ˆδ = Z Z) 1 Z Y X ˆβ). The control Z may play two role. Frt, control may be ncluded to ncreae the precon of ˆβ. Second, and more mportantly, they may be ncluded to proxy for latent ector-level hock {Z } S =1 that have an ndependent effect on the outcome Y and are correlated wth the hfter {X } S =1. In the preence of uch hock, the hfter are only a good a randomly agned condtonal on them, and t neceary to control for them n order to prevent omtted varable ba. To account for the two poble role that control may play, we aume that the control Z admt the decompoton Z = S w Z + U. 20) =1 If the kth component Z k of Z ncluded for precon, then Z k = 0 for all = 1,..., S, and Z k ncluded becaue Y 0) and U k are correlated. Th the cae, for ntance, f Y 0) and U k contan regonal hock that are ndependent of the ectoral hfter of nteret X. If, on the other hand, Z k ncluded to proxy for a latent hock Z, then U k repreent the meaurement error n Z when controllng for Z and Z k a perfect only f U k = When analyzng the mpact of nternatonal trade on regonal labor market outcome, t tandard to ether et the hock of the non-manufacturng ector to zero Topalova, 2007, 2010; Autor, Dorn and Hanon, 2013; Hakobyan and McLaren, 2016) or to remove the non-manufacturng ector from the analy and recale the hare of all manufacturng ector o that they add up to one Kovak, 2013). Ether of thee approache wll atfy the retrcton that max n /N 0. 17