Data Lletin Analysis and MultifiatialStudies

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5 tmuu ReV wrking paper department f enmis EMPIRICAL STRATEGIES IN LABOR ECONOMICS Jshua D. Angrist Alan B. Krueger Otber 1998 massahusetts institute f tehnlgy 50 memrial drive Cambridge, mass

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7 WORKING PAPER DEPARTMENT OF ECONOMICS EMPIRICAL STRATEGIES IN LABOR ECONOMICS Jshua D. Angrist Alan B. Krueger N Rev. Otber 1998 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 50 MEMORIAL DRIVE CAMBRIDGE, MASS

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9 C:\... \prjets\handbk\transfer\hapterl098.wpd Otber 27, 1998 Empirial Strategies in Labr Enmis Jshua D. Angrist MIT and NBER and Alan B. Krueger Prinetn University and NBER Otber 1998 *We thank Eri Bettinger, Luia Breierva, Kristen Harknett, Aarn Siskind, Diane Whitmre, Eri Wang, and Steve Wu fr researh assistane. Fr helpful mments and disussins we thank Albert Abadie, Darn Aemglu, Jere Behrman, David Card, Angus Deatn, Jeff Kling, Guid Imbens, Chns Mazing, Steve Pishke, and Ceilia Ruse. Of urse, errrs and missins are slely the wrk f the authrs. This paper was prepared fr the Handbk f Labr Enmis.

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11 EMPIRICAL STRATEGIES IN LABOR ECONOMICS JOSHUA D. ANGR1ST AND ALAN B. KRUEGER Massahusetts Institute f Tehnlgy and Prinetn University Cntents 1. Intrdutin 2. Identifiatin strategies fr ausal relatinships 2.1 The range f ausal questins 2.2 Identifiatin in regressin mdels Cntrl fr nfunding variables Fixed-effets and differenes-in-differenes Instrumental variables Regressin-disntinuity designs 2.3 Cnsequenes f hetergeneity and nnlinearity 2.4 Refutability 3. Data lletin strategies Regressin and the nditinal expetatin funtin Mathing instead f regressin Mathing using the prpensity sre Interpreting instrumental variables estimates 3.1 Sendary sures 3.2 Primary data lletin and survey methds 3.3 Administrative data and rerd linkage 3.4 Cmbining samples 4. Measurement issues 4.1 Measurement errr mdels 4.2 The extent f measurement errr in labr data 4.3 Weighting and allated values 5. Summary Appendix Referenes

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13 ABSTRACT Empirial Strategies in Labr Enmis This hapter prvides an verview f the methdlgial and pratial issues that arise when estimating ausal relatinships that are f interest t labr enmists. The subjet matter inludes identifiatin, data lletin, and measurement prblems. Fur identifiatin strategies are disussed, and five empirial examples - the effets f shling, unins, immigratin, military servie and lass size - illustrate the methdlgial pints. In disussing eah example, we adpt an experimentalist perspetive that draws a lear distintin between variables that have ausal effets, ntrl variables, and utme variables. The hapter als disusses sendary data sets, primary data lletin strategies, and administrative data. The setin n measurement issues fuses n reent empirial examples, presents a summary f empirial findings n the reliability f key labr market data, and briefly reviews the rle f survey sampling weights and the allatin f missing values in empirial researh. JEL Numbers: J00, J31, CIO, C81

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15 1. Intrdutin Empirial analysis is mre mmn and relies n mre diverse sures f data in labr enmis than in enmis mre generally. Table 1, whih updates Staffrd's (1986, Table 7.2) survey f researh in labr enmis, bears ut this laim. Indeed, almst 80% f reent artiles published in labr enmis ntain sme empirial wrk, and a striking tw-thirds analyzed mir data. In the 1970s, mir data beame mre mmn in studies f the labr market than time-series data, and by the mid-90s the use f mir data utnumbered time-series data by a fatr f ver ten t ne. The use f mir and time-series data is mre evenly split in ther fields f enmis. In additin t using mir data mre ften, labr enmists have me t rely n a wider range f data sets than ther enmists. The fratin f published papers using data ther than what is in standard publi-use files reahed 38% perent in the perid frm 1994 t The files in the "all ther mir data sets" ategry in Table 1 inlude primary data sets lleted by individual researhers, ustmized publi use files, administrative rerds, and administrative-survey links. This is ntewrthy beause abut ten years ag, in his Handbk f Enmetris survey f enmi data issues, Grilihes (1986, p. 1466) bserved: "... sine it is the 'badness' f the data that prvides us with ur living, perhaps it is nt at all surprising that we have shwn little interest in imprving it, in getting invlved in the grubby task f designing and lleting riginal data sets f ur wn." The grwing list f papers invlving sme srt f riginal data lletin suggests this situatin may be hanging; examples inlude Freeman and Hall (1986), Ashenfelter and Krueger (1994), Andersn and Meyer (1994), Card and Krueger (1994, 1998), Dminitz and Manski (1997), Imbens, Rubin and Saerdte (1997), and Angrist( 1998). Labr enmis has als me t be distinguished by the use f utting edge enmetri and statistial methds. This laim is supprted by the bservatin that utside f time-series enmetris, many and perhaps mst innvatins in enmetri tehnique and style sine the 1970s were largely mtivated by researh n labr-related tpis. These innvatins inlude sample seletin mdels, nnparametri methds fr ensred data and survival analysis, quantile regressin, and the renewed interest in statistial and

16 2 identifiatin prblems related t instrumental variables estimatrs and quasi -experimental methds. What d labr enmists d with all the data they analyze? A brad distintin an be made between tw types f empirial researh in labr enmis: desriptive analysis and ausal inferene. Desriptive analysis an establish fats abut the labr market that need t be explained by theretial reasning and yield new insights int enmi trends. The imprtane f stensibly mundane desriptive analysis an be aptured by Sherlk Hlmes's admnitin that: "It is a apital ffense t therize befre all the fats are in." A great deal f imprtant researh falls under the desriptive heading, inluding wrk n trends in pverty rates, labr fre partiipatin, and wage levels. A gd example f desriptive researh f majr imprtane is the wrk dumenting the inrease in wage dispersin in the 1980s (see e.g.. Levy, 1987, Murphy and Welh, 1992; Katz and Murphy, 1992; Juhn, Murphy, and Piere, 1993). This researh has inspired a vigrus searh fr the auses f hanges in the wage distributin. In ntrast with desriptive analysis, ausal inferene researh seeks t determine the effets f partiular interventins r pliies, r t estimate features f the behaviral relatinships suggested by enmi thery. Causal inferene and desriptive analysis are nt mpeting methds; indeed, they are ften mplementary. In the example mentined abve, mpelling evidene that wage dispersin inreased in the 1980s inspired a searh fr auses f these hanges. Causal inferene is ften mre diffiult than desriptive analysis, and nsequently mre ntrversial. Mst labr enmists seem t share a mmn view f the imprtane f desriptive researh, but there are differenes in views regarding the rle enmi thery an r shuld play in ausal mdeling. This divisin is illustrated by the debate ver sial experimentatin (Burtless, 1995; Hekman and Smith, 1995), in ntrasting apprahes t studying the impat f immigratin n the earnings f natives (Card, 1990; Brjas, Freeman and Katz, 1997), and in reent sympsia illustrating alternative researh styles (Angrist, 1995a; Keane and Wlpin, 1 997). Researh in a struturalist style relies heavily n enmi thery t guide empirial wrk r t make preditins. Keane and Wlpin (1 997, p. Ill) desribe strutural wrk as trying t d ne f tw

17 3 things: (a) rever the primitives f enmi thery (parameters determining preferenes and tehnlgy); (b) estimate deisin rules derived frm enmi mdels. Given suess in either f these endeavrs, it is usually lear hw t make ausal statements and t generalize frm the speifi relatinships and ppulatins studied in any partiular appliatin. An alternative t strutural mdeling, ften alled the quasi-experimental r simply the "experimentalist" apprah, als uses enmi thery t frame ausal questins. But this apprah puts frnt and enter the prblem f identifying the ausal effets frm speifi events r situatins. The prblem f generalizatin f findings is ften left t be takled later, perhaps with the aid f enmi thery r infrmal reasning. Often this press invlves the analysis f additinal quasi-experiments, as in reent wrk n the returns t shling (see, e.g., the papers surveyed by Card in this vlume). In his methdlgial survey, Meyer (1995) desribes quasi-experimental researh as "an utburst f wrk in enmis that adpts the language and neptual framewrk f randmized experiments." Here, the ideal researh design is expliitly taken t be a randmized trial and the bservatinal study is ffered as an attempt t apprximate the fre f evidene generated by an atual experiment. In either a strutural r quasi-experimental framewrk, the researher's task is t estimate features f the ausa! relatinships f interest. This hapter fuses n the empirial strategies mmnly used t estimate features f the ausal relatinships that are f interest t labr enmists. The hapter prvides an verview f the methdlgial and pratial issues that arise in implementing an empirial strategy. We use the term empirial strategy bradly, beginning with the statement f a ausal questin, and extending t identifiatin strategies and enmetri methds, seletin f data sures, measurement issues, and sensitivity tests. The hie f tpis was guided by ur wn experienes as empirial researhers and ur researh interests. As far as enmetri methds g, hwever, ur verview is espeially seletive; fr the

18 4 mst part we ignre strutural mdeling sine that tpi is well vered elsewhere. 1 Of urse, there is nsiderable verlap between strutural and quasi-experimental apprahes t ausal mdeling, espeially when it mes t data and measurement issues. The differene is primarily ne f emphasis, beause strutural mdeling generally relies n assumptins abut exgenus variability in ertain variables and quasiexperimental analyses require sme theretial assumptins. The attentin we devte t quasi-experimental methds is als mtivated by skeptiism abut the redibility f empirial researh in enmis. Fr example, in a ritique f the pratie f mdem enmetris, Lester Thurw (1983, pp ) argued: "Enmi thery almst never speifies what sendary variables (ther than the primary nes under investigatin) shuld be held nstant in rder t islate the primary effets.... When we lk at the impat f eduatin n individual earnings, what else shuld be held nstant: IQ, wrk effrt, upatinal hie, family bakgrund? Enmi thery des nt say. Yet the effiients f the primary variables almst always depend n preisely what ther variables are entered in the equatin t "hld everything else nstant." This view f applied researh strikes us as being verly pessimisti, but we agree with the fus n mitted variables. In labr enmis, at least, the urrent ppularity f quasi-experiments stems preisely frm this nern: beause it is typially impssible t ntrl adequately fr all relevant variables, it is ften desirable t seek situatins where ne has a reasnable presumptin that the mitted variables are unrrelated with the variables f interest. Suh situatins may arise if the researher an use randm assignment, r if the fres f nature r human institutins prvide smething lse t randm assignment. The next setin reviews fur identifiatin strategies that are mmnly used t answer ausal questins in ntemprary labr enmis. Five empirial examples -- the effets f shling, unins, immigratin, military servie, and lass size - illustrate the methdlgial pints thrughut the hapter. In keeping with ur experimentalist perspetive, we attempt t draw lear distintins between variables that have 'See, fr example, Hekman and MaCurdy's (1986) Handbk f Enmetris hapter, whih "utlines the enmetri framewrk develped by labr enmists wh have built theretially mtivated mdels t explain the new data." (p. 1918). We als have little t say abut desriptive analysis beause desriptive statistis are mmnly disussed in statistis urses and bks (see, e.g., Tufte, 1992, r Tukey, 1977).

19 5 ausal effets, ntrl variables, and utme variables in eah example. In Setin 3 we turn t a disussin f sendary data sets and primary data lletin strategies. The fus here is n data fr the United States. 2 Setin 3 als ffers a brief review f issues that arise when nduting an riginal survey and suggestins fr assembling administrative data sets. Beause existing publi-use data sets have already been extensively analyzed, primary data lletin is likely t be a grwth industry fr labr enmists in the future. Fllwing the disussin f data sets, Setin 4 disusses measurement issues, inluding a brief review f lassial mdels fr measurement errr and sme extensins. Sine mst f this theretial material is vered elsewhere, inluding the Grilihes (1986) hapter mentined previusly, ur fus is n reent empirial examples. This setin als presents a summary f empirial findings n the reliability f labr market data, and reviews the rle f survey sampling weights and the allatin f missing values in empirial researh. 2. Identifiatin strategies fr ausal relatinships The bjet f siene is the disvery f relatins... f whih the mplex may be dedued frm the simple. Jhn Pringle Nihl, 1840 (quted in Lrd Kelvin's lass ntes). 2.1 The range f ausal questins The mst hallenging empirial questins in enmis invlve "what if statements abut unterfatual utmes. Classi examples f "what if questins in labr market researh nern the effets f areer deisins like llege attendane, unin membership, and military servie. Interest in these questins is mtivated by immediate pliy nerns, theretial nsideratins, and prblems faing individual deisin makers. Fr example, pliy makers wuld like t knw whether military utbaks will redue the earnings 'Overviews f data sures fr develping untries appear in Deatn's (1995) hapter in The Handbk f Develpment Enmis, Grsh and Glewwe (1996, 1998), and Kremer (1997). We are nt aware f a mprehensive survey f mir data sets fr labr market researh in Eurpe, thugh a few sures and studies are referened in Westergard-Nielsn (1989).

20 6 f minrity men wh have traditinally seen military servie as a majr areer pprtunity. Additinally, many new high shl graduates wuld like t knw what the nsequenes f serving in the military are likely t be fr them. Finally, the thery f n-the-jb training generates preditins abut the relatinship between time spent serving in the military and ivilian earnings. Regardless f the mtivatin fr studying the effets f areer deisins, the ausal relatinships at the heart f these questins invlve mparisns f unterfatual states f the wrld. Smene - the gvernment, an individual deisin maker, r an aademi enmist - wuld like t knw what utmes wuld have been bserved if a variable were manipulated r hanged in sme way. Lewis's (1986) study f the effets f unin wage effets gives a nise desriptin f this type f inferene prblem (p. 2): "At any given date and set f wrking nditins, there is fr eah wrker a pair f wage figures, ne fr uninized status and the ther fr nnunin status". Differenes in these tw ptential utmes define the ausal effets f interest in Lewis's wrk, whih uses regressin t estimate the average gap between them. 3 At first glane, the idea f unbserved ptential utmes seems straightfrward, but in pratie it is nt always lear exatly hw t define a unterfatual wrld. In the ase f unin status, fr example, the unterfatual is likely t be ambiguus. Is the effet defined relative t a wrld where uninizatin rates are what they are nw, a wrld where everyne is uninized, a wrld where everyne in the wrker's firm r industry is uninized, r a wrld where n ne is uninized? Simple mir-enmi analysis suggests that the answers t these questins differ. This pint is at the heart f Lewis's (1986) distintin between unin wage gaps, whih refers t ausal effets n individuals, and wage gains, whih refers t mparisns f equilibria in a wrld with and withut unins. In pratie, hwever, the prblem f ambiguus unterfatuals is typially reslved by fusing n the nsequenes f hypthetial manipulatins in the wrld as is, i.e., assuming there 3 See als Rubin (1974, 1977) and Hlland (1986) fr frmal disussins f unterfatual utmes in ausal researh.

21 7 are n general equilibrium effets. 4 Even if ambiguities in the definitin f unterfatual states an be reslved, it is still diffiult t leam abut differenes in unterfatual utmes beause the utme f ne senari is all that is ever bserved fr any ne unit f bservatin (e.g., a persn, State, r firm). Given this basi diffiulty, hw d researhers learn abut unterfatual states f the wrld in pratie? In many fields, and espeially in medial researh, the prevailing view is that the best evidene abut unterfatuals is generated by randmized trials beause randmizatin ensures that utmes in the ntrl grup really d apture the unterfatual fr a treatment grup. Thus, Federal guidelines fr a new drug appliatin require that effiay and safety be assessed by randmly assigning the drug being studied r a plaeb t treatment and ntrl grups (Center fr Drug Evaluatin and Researh, 1988). Learner (1982) suggested that the absene f randmizatin is the main reasn why enmetri researh ften appears less nvining than researh in ther mre experimental sienes. Randmized trials are ertainly rarer in enmis than in medial researh, but labr enmists are inreasingly likely t use randmizatin t study the effets f labr market interventins (Passell, 1992). In fat, a reent survey f enmists by Fuhs, Krueger, and Pterba (1998) finds that mst labr enmists plae mre redene in studies f the effet f gvernment training prgrams n partiipants' inme if the researh design entails randm assignment than if the researh design is based n strutural mdeling. Unfrtunately, enmists rarely have the pprtunity t randmize variables like eduatinal attainment, immigratin, r minimum wages. Empirial researhers must therefre rely n bservatinal studies that typially fail t generate the same fre f evidene as a randmized experiment. But the bjet f an bservatinal study, like an experimental study, an still be t make mparisns that prvide evidene abut ausal effets. Observatinal studies attempt t amplish this by ntrlling fr bservable differenes between mparisn grups using regressin r mathing tehniques, using pre-pst mparisns n the same 'Lewis's (1963) earlier bk disussed ausal effets in terms f industries and setrs, and made a distintin between "diret" and "indiret" effets f unins similar t the distintin between wage gaps and wage gains. Hekman, Lhner, and Taber (1998) disuss general equilibrium effets that arise in the evaluatin f llege tuitin subsidies.

22 8 units f bservatin t redue bias frm unbserved differenes, and by using instrumental variables as a sure f quasi-experimental variatin. Randmized trials frm a neptual benhmark fr assessing the suess r failure f bservatinal study designs that make use f these ideas, even when it is lear that it may be impssible r at least impratial t study sme questins using randm assignment. In almst every bservatinal study, it makes sense t ask whether the researh design is a gd "natural experiment." 5 A sampling f ausal questins that enmists have studied withut benefit f a randmized experiment appears in Table 2, whih haraterizes a few bservatinal studies gruped arding t the sure f variatin used t make ausal inferenes abut a single "ausing variable." The distintin between ausing variables and ntrl variables in Table 2 is ne differene between the disussin in this hapter and traditinal enmetri texts, whih tend t treat all variables symmetrially. The mbinatin f a learly labeled sure f identifying variatin in a ausal variable and the use f a partiular enmetri tehnique t explit this infrmatin is what we all an identifiatin strategy. Studies were seleted fr Table 2 primarily beause the sure r type f variatin that is being used t make ausal statements is learly labeled. The fur apprahes t identifiatin desribed in the table are: Cntrl fr Cnfunding Variables, Fixedeffets and Differenes-in-differenes, Instrumental Variables, and Regressin Disntinuity methds. This taxnmy prvides an utline fr the next setin Identifiatin in regressin mdels Cntrlfr nfunding variables Labr enmists have lng been nerned with the questin f whether the bserved psitive assiatin between shling and earnings is a ausal relatinship. This questin riginates partly in the 5 This pint is als made by Freeman (1989). The ntin that experimentatin is an ideal researh design fr Enmis ges bak at least t the Cwles Cmmissin. See, fr example, Girshik and Haavelm (1947), wh wrte (p. 79): "In enmi thery... the ttal demand fr the mmdity may be nsidered a funtin f all pries and f ttal dispsable inme f all nsumers. The ideal methd f verifying this hypthesis and btaining a piture f the demand funtin invlved wuld be t ndut a large-sale experiment, impsing alternative pries and levels f inme n the nsumers and studying their reatins."

23 9 bservatin that peple with mre shling appear t have ther harateristis, suh as wealthier parents, that are als assiated with higher earnings. Als, the thery f human apital identifies unbserved earnings ptential r "ability" as ne f the prinipal determinants f eduatinal attainment (see, e.g, Willis and Rsen, 1979). The mst mmn identifiatin strategy in researh n shling (and in enmis in general) attempts t redue bias in naive mparisns by using regressin t ntrl fr variables that are nfunded with (i.e., related t) shling. The typial estimating equatin in this ntext is, (1) Y,= 'X I 'Pr +p I S l + «i. where Y ( is persn i's lg wage r earnings, Xjis a kxl vetr f ntrl variables, inluding measures f ability and family bakgrund, Sj is years f eduatinal attainment, and e, is the regressin errr. The vetr f ppulatin parameters is [0/ p r]'. The "r" subsript n the parameters signifies that these are regressin effiients. The questin f ausality nerns the interpretatin f these effiients. Fr example, they an always be viewed as prviding the best (i.e., minimum-mean-squared-errr) linear preditr f Y (. 6 The best linear preditr need nt have ausal r behaviral signifiane; the resulting residual is unrreted with the regressrs simply beause the first-rder nditins fr the preditin prblem are Ej>;Xj]=0 and EfS^O. Regressin estimates frm five early studies f the relatinship between shling, ability, and earnings are summarized in Table 3. The first rw reprts estimates withut ability ntrls while the send rw reprts estimates that inlude sme kind f test sre in the X-vetr as a ntrl fr ability. Infrmatin abut the X-variables is given in the rws labeled "ability variable" and "ther ntrls". The first tw studies, Ashenfelter and Mney (1968) and Hansen, Weisbrd, and Sanln (1970) use data n individuals at the extremes f the ability distributin (graduate students and military rejets), while the thers use mre representative samples. Results frm the last tw studies, Grilihes and Masn (1972) and Chamberlain (1978), are reprted fr mdels with and withut family bakgrund ntrls. The shling effiients in Table 3 are smaller than the effiient estimates we are used t seeing The best linear preditr is the slutin t Min E[(Y, -X/b -S 2 b. ( ) ]. See, e.g., White (1980), r Gldberger (1991).

24 10 in studies using mre reent data (see, e.g., Card's survey in this vlume). This is partly beause the assiatin between earnings and shling has inreased, partly beause the samples used in the papers summarized in the table inlude nly yung men, and partly beause the mdels used fr estimatin ntrl fr age and nt ptential experiene (age-eduatin-6). The latter parameterizatin leads t larger effiient estimates sine, in a linear mdel, the shling effiient ntrlling fr age is equal t the shling effiient ntrlling fr experiene minus the experiene effiient. The nly speifiatin in Table 2 that ntrls fr ptential experiene is frm Grilihes (1977), whih als generates the highest estimate in the table (.065). The rrespnding estimate ntrlling fr age is.022. The table als shws that ntrlling fr ability and family bakgrund generally redues the magnitude f shling effiients, implying that at least sme f the assiatin between earnings and shling in these studies an be attributed t variables ther than shling. What nditins must be met fr regressin estimates like thse in Table 3 t have a ausal interpretatin? In this ase, ausality an be based n an underlying funtinal relatinship that desribes what a given individual wuld earn if he r she btained different levels f eduatin. This relatinship may be persn-speifi, s we write (2) Y a - US) t dente the ptential (r latent) earnings that persn i wuld reeive after btaining S years f eduatin. Nte that the funtin fs(5) has an "i" subsript n it while S des nt. This highlights the fat that althugh S is a variable, it is nt a randm variable. The funtin f^s) tells us what i wuld earn fr any value f shling, 5, and nt just fr the realized value, Sj. In ther wrds, f { (S) answers "what if questins. In the ntext f theretial mdels f the relatinship between human apital and earnings, the frm f f ; (5) may be determined by aspets f individual behavir and/r market fres. With r withut an expliit enmi mdel fr f((5), hwever, we an think f this funtin as desribing the earnings level f individual i if that persn were assigned shling level S (e.g., in an experiment).

25 11 One the ausal relatinship f interest, f^s), has been defined, it an be linked t the bserved assiatin between shling and earnings. A nvenient way t d this is with a linear mdel: (3) fi(s)=ptp5 + Hi- Iii additin t being linear, this equatin says that the funtinal relatinship f interest is the same fr all individuals. Again, S is written withut a subsript, beause equatin (3) tells us what persn i wuld earn fr any value f S and nt just the realized value, S ;. The nly individual-speifi and randm part f f ;(S) is a mean-zer errr mpnent, r\ it whih aptures unbserved fatrs that determine earnings. In pratie, regressin estimates have a ausal interpretatin under weaker funtinal-frm assumptins than this but we pstpne a detailed disussin f this pint until Setin 2.3. Nte that the earnings f smene with n shling at all is just p + r^ in this mdel. Substituting the bserved value S, fr S in equatin (3), we have (4) Y^P + ps. + r,, This lks like equatin (1) withut variates, exept that equatin (4) expliitly assiates the regressin effiients with a ausal relatinship. The OLS estimate f p in equatin (4) has prbability limit (5) C(Y S, SJ/VCSj) = p + C(Si, rij/vcsi). The term C(S;, nja^sj) is the effiient frm a regressin f rjj n Sj, and reflets any rrelatin between the realized and unbserved individual earnings ptential, whih in this ase is the same as rrelatin with Ss Tjj. If eduatinal attainment were randmly assigned, as in an experiment, then we wuld have C(S (, r i)=0 in the linear mdel. In pratie, hwever, shling is a nsequene f individual deisins and institutinal fres that are likely t generate rrelatin between r)j and shling. Cnsequently, it is nt autmati that OLS prvides a nsistent estimate f the parameter f interest. 7 Regressin strategies attempt t verme this rrelatin in a very simple way: in additin t the 'Enmetri textbks (e.g., Pindyk and Rubinfeld, 1991) smetimes refer t regressin mdels fr ausal relatinships as "true mdels," but this seems like ptentially misleading terminlgy sine nn-behaviral desriptive regressins uld als be desribed as being "true".

26 12 funtinal frm assumptin fr ptential utmes embdied in (3), the randm part f individual earnings ptential, r^, is dempsed int a linear funtin f the k bservable harateristis, X and an errr term,, ( ;, (6a) t li = Xi'P + i, where P is a vetr f ppulatin regressin effiients. This means that e, and Xj are unrreted by nstrutin. The key identifying assumptin is that the bservable harateristis, Xj, are the nly reasn why r)i and Sj (equivalently, f,(5) and S;) are rrelated, s (6b) E[S,ei]=0. This is the "seletin n bservables" assumptin disussed by Barnw, Cain, and Gldberger (1981), where the regressr f interest is assumed t be determined independently f ptential utmes after aunting fr a set f bservable harateristis. Cntinuing t maintain the seletin-n-bservables assumptin, a nsequene f (6a) and (6b) is that (7) C(Yi,S i )/V(S i ) = p + 4> Sx'P, where 4>sx is a kxl vetr effiients frm a regressin f eah element f X( n Sj. Equatin (7) is the wellknwn "mitted variables bias" frmula, whih relates a bivariate regressin effiient t the effiient n S, in a regressin that inludes additinal variates. If the mitted variables are psitively related t earnings (P>0) and psitively rrelated with shling (<J>SX>0), then C(Y jf S^/VCS,) is larger than the ausal effet f shling, p. A send nsequene f (6a) and (6b) is that the OLS estimate f p r in equatin (1) is in fat nsistent fr the ausal parameter, p. Nte, hwever, that the way we have develped the prblem f ausal inferene, E[Si i]=0 is an assumptin abut e- t and S whereas ErXjeJsO is a statement abut variates that is true by definitin. This suggests that it is imprtant t distinguish errr terms that represent the randm parts f mdels fr ptential utmes frm mehanial dempsitins where the relatinship between errrs and regressrs has n behaviral ntent. A key questin in any regressin study is whether the seletin-n-bservables assumptin is plausible.

27 13 The assumptin learly makes sense when there is atual randm assignment nditinal n X. Even ; withut randm assignment, hwever, seletin-n-bservables might make sense if we knw a lt abut the press generating the regressr f interest. We might knw, fr example, that appliants t a partiular llege r university are sreened using ertain harateristis, but nditinal n these harateristis all appliants are aeptable and hsen n a first-me/first-serve basis. This leads t a situatin like the ne desribed by Bamw, Cain, and Gldberger (1980, p. 47), where "Unbiasedness is attainable when the variables that determined the assignment are knwn, quantified, and inluded in the equatin." Similarly, Angrist (1998) argued that beause the military is knwn t sreen appliants n the basis f bserved harateristis, mparisns f veteran and nnveteran appliants that adjust fr these harateristis have a ausal interpretatin. The ase fr seletin-n-bservables in a generi shling equatin is less lear ut, whih is why s muh attentin has fused n the questin f mitted-variables bias in OLS estimates f shling effiients. Regressin pitfalls Shling is nt randmly assigned and, as in many ther prblems, we d nt have detailed institutinal knwledge abut the press that atually determines assignment. The hie f variates is therefre ruial. Obvius andidates inlude any variables that are rrelated with bth shling and earnings. Test sres are gd andidates beause many eduatinal institutins use tests t determine admissins and finanial aid. On the ther hand, it is dubtful that any partiular test sre is a perfet ntrl fr all the differenes in earnings ptential between mre and less eduated individuals. We see this in the fat that adding family bakgrund variables like parental inme further redues the size f shling effiients. A natural questin abut any regressin ntrl strategy is whether the estimates are highly sensitive t the inlusin f additinal ntrl variables. While ne shuld always be wary f drawing ausal inferenes frm a regressin with bservatinal data, sensitivity f the regressin results t hanges in the set f ntrl

28 14 variables is an extra reasn t wnder whether there might be unbserved variates that wuld hange the estimates even further. The previus disussin suggests that Table 3 an be interpreted as shwing that there is signifiant ability bias in OLS estimates f the ausal effet f shling n earnings. On the ther hand, a number f nerns less bvius than mitted-variables bias suggest this nlusin may be premature. A theme f the Grilihes and Chamberlain papers ited in the table is that the negative impat f ability measures n shling effiients is eliminated and even reversed ne ne aunts fr tw fatrs: measurement errr in the regressr f interest, and the use f endgenus test sre ntrls that are themselves affeted by shling. A standard result in the analysis f measurement errr is that if variables are measured with an additive errr that is unrrelated with rretly-measured values, this imparts an attenuatin bias that shrinks OLS estimates twards zer (see, e.g., Grilihes, 1986, Fuller, 1987, and Setin 4, belw). The prprtinate redutin is ne minus the rati f the variane f rretly-measured values t the variane f measured values. Furthermre, the inlusin f ntrl variables that are rrelated with atual values and unrrelated with the measurement errr tends t aggravate this attenuatin bias. The intuitin fr this result is that the residual variane f true values is redued by the inlusin f additinal ntrl variables while the residual variane f the measurement errr is left unhanged. Althugh studies f measurement errr in eduatin data suggest that nly 10 perent f the variane in measured eduatin is attributable t measurement errr, it turns ut that the dwnward bias in regressin mdels with ability and ther ntrls an still be substantial. 8 A send mpliatin raised in the early literature n regressin estimates f the returns t shling is that variables used t ntrl fr ability may be endgenus (see, e.g., Grilihes and Masn, 1972, r Chamberlain, 1977). If wages and test sres are bth utmes that are affeted by shling, then test sres annt play the rle f an exgenus, pre-determined ntrl variable in a wage equatin. T see this, 8 Fr a detailed elabratin f this pint, see Welh, 1975, r Grilihes, 1977, wh ntes (p. 13): "Clearly, the mre variables we put int the equatin whih are related t the systemati mpnents f shling, and the better we 'prtet' urselves against varius pssible biases, the wrse we make the errrs f measurement prblem." We present sme new evidene n attenuatin and variates in Setin 4, belw.

29 t and 15 nsider a simple example where the ausal relatinship f interest is (4), and C(S, r,)=0 s that a bivariate C regressin wuld in fat generate a nsistent estimate f the ausal effet. Suppse that shling affets test sres as well as earnings, and that the effet n test sres an be expressed using the mdel (8) A, = Y + Y.Si + T,i- This relatinship an be interpreted as refleting the fat that mre frmal shling tends t imprve test sres (s Yi>0). We als assume that CCS;, r,i)=0, s that OLS estimates f (8) wuld be nsistent fr y,. The questin is what happens if we add the utme variable, A,, t the shling equatin in a mistaken (in this ase) attempt t ntrl fr ability bias. Endgeneity f A, in this ntext means that r\ \] h are rrelated. Sine peple wh d well n standardized tests prbably earn mre fr reasns ther than the fat that they have mre shling, it seems reasnable t assume that C(r);, %)>(). In this ase, the effiient n S in a regressin f Yj n S and Aj ; s leads t an innsistent estimate f the effet f shling. Evaluatin f prbability limits shws that the OLS estimate f the shling effiient in a mdel that inludes A, nverges t (9) C(Y,S. i Ai )A^(S. = Aj ) p-y I (j) 01, where SM is the residual frm a regressin f Sj n A and ( 4>, is the effiient frm a regressin f ri; n r) H (see the Appendix fr details). Sine Yi>0 and 4>i>0. ntrlling fr the endgenus test sre variable tends t make the estimate f the returns t shling smaller, but this is nt beause f any mitted- variables bias in the equatin f interest. Rather it is a nsequene f the bias indued by nditining n an utme variable. 9 The prblems f measurement errr and endgenus regressrs generate identifiatin hallenges that lead researhers t use methds beynd the simple regressin-ntrl framewrk. The mst mmnly emplyed strategies fr dealing with these prblems invlve instrumental variables (TV), tw-stage least 9 A similar prblem may affet estimates f shling effiients in equatins that ntrl fr upatin. Like test sres and ther ability measures, upatin is itself a nsequene f shling that is prbably rrelated with unbserved earnings ptential.

30 16 squares (2SLS), and latent-variable mdels. We briefly mentin sme 2SLS and latent-variable estimates, but defer a detailed disussin f 2SLS and related IV strategies until Setin The majr pratial prblem in mdels f this type is t find valid instruments fr shling and ability. Panel B reprts Grilihes (1977) 2SLS estimates f equatin (1) treating bth shling and IQ sres as endgenus. The instruments are family bakgrund measures and a send ability prxy. Chamberlain (1978) develps an alternate apprah that uses panel data t identify the effets f endgenus shling in a latent-variable mdel fr unbserved ability. Bth the Chamberlain (1978) and Grilihes (1977) estimates are nsiderably larger than the rrespnding OLS estimates, a finding whih led these authrs t nlude that the empirial ase fr a negative ability bias in shling effiients is muh weaker than the OLS estimates suggest Fixed effets and differenes-in-dijferenes The main idea behind fixed-effets identifiatin strategies is t use repeated bservatins n individuals (r families) t ntrl fr unbserved and unhanging harateristis that are related t bth utmes and ausing variables. A lassi field f appliatin fr fixed-effets mdels is the attempt t estimate the effet f unin status. Suppse, fr example, that we wuld like t knw the effet f wrkers' unin status n their wages. That is, fr eah wrker, we imagine that there are tw ptential utmes, Y^ denting what the wrker wuld earn if nt a unin member, and Y,j denting what the wrker wuld earn as a unin member. This is just like Y 5 in the shling, example, exept that here "S " is the dihtmus variable, unin status. The effet f unin status n an individual wrker is Y^-Y^ but this is never bserved diretly sine nly ne ptential utme is ever bserved fr eah individual at any ne time." Mst analyses f the unin prblem begin with a nstant-effiients regressin mdel fr ptential '"Anther strand f the literature n ausal effets f shling uses sibling data t ntrl fr family effets that are shared by siblings (early studies are by Grseline, 1932 and Taubman, 1976; see als Grilihes's (1979) survey). Here the prblem f measurement errr is paramunt (see Setin and 4.1). "This ntatin fr unterfatual utmes was used by Rubin (1974, 1977). Siegfried and Sweeney (1980) and Chamberlain (1980) use a similar ntatin t disuss the effet f a lassrm interventin n test sres.

31 17 utmes, where (10) Y 0i = Xi 'P + 6 i, Y.^Ya + 5. As in the shling prblem, Yq, has been dempsed int a linear funtin f bserved variates, X/P, and a residual, e^ that is unnelated with X by ; nstrutin. Using U t indiate unin members, ( this leads t the regressin equatin, (11) Y.-X/p + U.fi + e,, whih desribes the ausal relatinship f interest. Many researhers wrking in this framewrk have argued that unin status is likely t be related t ptential nnunin wages, Y a, even after nditining n variates, X (see, e.g Abwd and Farber, 1982; ( r hapters 4 and 5 in Lewis, 1986). This means that Uj is rrelated with e, s OLS des nt estimate the { ausal effet, 6. An alternative t OLS uses panel data sets suh as mathed CPS rtatin grups, the Panel Study f Inme Dynamis, r the Natinal Lngitudinal Surveys and explits repeated bservatins n individuals t ntrl fr unbserved individual harateristis that are time-invariant. A well-knwn study in this genre is Freeman (1984). The fllwing mdel, similar t many in the literature n unin status, illustrates the fixed-effets apprah. Mdifying the previus ntatin t inrprate t=l,...,t bservatins n individuals, the fixedeffets slutin fr this prblem begins by writing (12) Y l = X it'p I + Xa i + $ ii where a, is an unbserved variable fr persn i, that we uld, in priniple, inlude as a ntrl if it were bserved. Equatin (12) is a regressin dempsitin with variates X and a it it s Jj u is unrreted with X and Oj by nstrutin it (X, an inlude harateristis frm different perids). The ausal/regressin mdel fr panel data is nw (13) Y it =X il'p l + UiA + ^i + 5 i..

32 18 where we have allwed the ausal effet f interest t be time-varying. The identifying assumptins are that the effiient X des nt vary arss perids and that (14) E[U it y=ofrs=l,...,t In ther wrds, whatever the sure f rrelatin is between Uj, and unbserved earnings ptential, it an be desribed by an additive time-invariant variate a it that has the same effiient eah perid. Sine differening eliminates Xa it OLS estimates f the differened equatin (15) Y it - Y it. k = X it X 'P, - it. k 'P,.k + U it6, - U,,A k + (5-5m) are nsistent fr the parameters f interest. Any transfrmatin f the data that eliminates the unbserved a { an be used t estimate the parameters f interest in this mdel. One f the mst ppular estimatrs in this ase is the deviatins-frmmeans r the Analysis f Cvariane (ANCOVA) estimatr, whih is mst ften used fr mdels where P, and 8, are assumed t be fixed. The analysis f variane estimatr is OLS applied t (16) Y jt - y, = PXXirXi) + 6(U U - Ui) + (5a - 1, where verbars dente persn-averages. Analysis f variane is preferable t differening n effiieny grunds in sme ases; fr mdels with nrmally distributed hmsedasti errrs, ANCOVA is the maximum likelihd estimatr. An alternative enmetri strategy fr the estimatin f mdels with individual effets uses repeated bservatins n hrt averages instead f repeated data n individuals. Fr details and examples see Ashenfelter (1984) r Deatn (1985). Finally, nte that while standard fixed-effets estimatrs an nly be used t estimate the effets f time-varying regressrs, Hausman and Taylr (1981) have develped a hybrid panel/tv predure fr mdels with time-invariant regressrs (like shling). It is als wrth nting that even if the ausing variable f interest is time-invariant, we an use standard fixed-effets estimatrs t estimate hanges in the effet f a time invariant variable. Fr example, the estimating equatin fr a mdel with fixed U, is (17) Y Y - i( it. k = X Xit'P, - it. k 'P,.k + UAA*) + (5* - 5m).

33 19 s (6,-6,. k) is identified. Angrist (1995b) used this methd t estimates hanges in shling effiients in the West Bank and Gaza Strip even thugh shling is apprximately time-invariant. Fixed-effets pitfalls The use f panel data t eliminate bias frm unbserved individual effets raises a number f enmetri and statistial issues. Sine this material is vered in Chamberlain's (1984) hapter in The Handbk f Enmetris, we limit ur disussin t an verview f prblems that have been f partiular nern t labr enmists. First, analysis f variane and differening estimatrs are nt nsistent when the press determining Uit invlves lagged dependent variables. This issue mes up in the analysis f training prgrams beause partiipants ften experiene a pre-prgram deline in earnings, a fat first nted by Ashenfelter (1978). If past earnings are bserved, the simplest strategy in this ase is simply t ntrl fr past earnings either by inluding lagged earnings as a regressr r in mathed treatment-ntrl mparisns (see, e.g., Dehejia and Wahba, 1995; Hekman, Ihimura, and Tdd, 1997). In fat, the questin f whether trainees and a andidate mparisn grup have similar lagged utmes is smetimes seen as a litmus test fr the legitimay f the mparisn grup in the evaluatin f training prgrams (see, e.g., Hekman and Htz, 1989). A prblem arises in this ntext, hwever, when the press determining U it invlves past utmes and an unbserved variate, a v Ashenfelter and Card (1985) disuss an example invlving the effet f training n the Sial Seurity-taxable earnings f trainees under the Cmprehensive Emplyment and Training At (CETA). They prpse a mdel f training status where individuals wh enter CETA training in year t d s beause they have lw a k and their earnings were unusually lw in year t-1. Suppse initially we ignre the fat that training status invlves past earnings, and estimate an equatin like (15). Ignring ther variates, this amunts t mparing the earnings grwth f trainees and ntrls. But whatever the true prgram effet is, the grwth in the earnings f CETA trainees frm year x-1 t year t+1 will tend t be larger

34 20 than the earnings grwth in a andidate ntrl grup simply beause f regressin-t-the-mean. This generates a spurius psitive training effet and the nventinal differening methd breaks dwn. 12 A natural strategy fr dealing with this prblem might seem t be t add Y it., t the list f ntrl variables, and then differene away the fixed effet in a mdel with Y jt., as regressr. The prblem is that nw any transfrmatin that eliminates the fixed effet will leave at least ne regressr - the lagged dependent variable - rrelated with the errrs in the transfrmed equatin. Althugh the lagged dependent variable is nt the regressr f interest, the fat that it is rrelated with the errr term in the transfrmed equatin means that the estimate f the effiient n U ii+, is biased as well. A detailed desriptin f this prblem, and the slutins that have been prpsed fr it, raises tehnial issues beynd the spe f this hapter. A useful referene is Nikell, 1981, espeially pages See als Card and Sullivan's (1988) study f the effet f CETA training n the emplyment rates f trainees, whih reprts bth fixed-effets estimates and mathing estimates that ntrl fr lagged utmes. A send ptential prblem with fixed-effets estimatrs is that bias frm measurement errr is usually aggravated by transfrmatins that eliminate the individual effets (see, e.g., Freeman, 1984; Grilihes and Hausman, 1986). This fat prvides an alternative explanatin fr why fixed-effets estimates ften turn ut t be smaller than estimates in levels. Finally, perhaps the mst imprtant prblem with this apprah is that the assumptin that mitted variables an be aptured by an additive, time-invariant individual effet is arbitrary in the sense that it usually des nt me frm enmi thery r frm infrmatin abut the relevant institutins.' 3 On the ther hand, the fixed-effets apprah has a superfiial plausibility ("whatever makes us speial is timeless") and an identifiatin payff that is hard t beat. Als, fixed-effets mdels lend themselves t a variety f speifiatin tests. See, fr example, Ashenfelter and Card (1985), Chamberlain (1984), Grilihes and Hausman (1986), Angrist and Newey (1991), and Jakubsn (1991). Many f these 12 Deviatins-frm-means estimatrs are als biased in this ase. l3 An exeptin is the literature n life-yle labr supply (e.g., MaCurdy, 1981; Altnji, 1986).

35 21 studies als fus n the unin example. The Differenes-in-Differenes (DD) mdel Differenes-in-differenes strategies are simple panel-data methds applied t sets f grup means in ases when ertain grups are expsed t the ausing variable f interest and thers are nt. This apprah, whih is transparent and ften at least superfiially plausible, is well-suited t estimating the effet f sharp hanges in the enmi envirnment r hanges in gvernment pliy. The DD methd has been used in hundreds f studies in enmis, espeially in the last tw deades, but the basi idea has a lng histry. An early example in labr enmis is Lester (1946), wh used the differenes-in-differenes tehnique t study emplyment effets f minimum wages. 14 The DD apprah is explained here using Card's (1990) study f the effet f immigratin n the emplyment f natives as an example. Sme bservers have argued that immigratin is undesirable beause lw-skilled immigrants may displae lw-skilled r less-eduated US itizens in the labr market. Anedtal evidene fr this laim inludes newspaper aunts f hstility between immigrants and natives in sme ities, but the empirial evidene is innlusive. See Friedberg and Hunt (1995) fr a survey f researh n this questin. As in ur earlier examples, the bjet f researh n immigratin is t find sme srt f mparisn that prvides a mpelling answer t 'what if questins abut the nsequenes f immigratin. Card's study used a sudden large-sale migratin frm Cuba t Miami knwn as the Mariel Batlift t make mparisns and answer unterfatual questins abut the nsequenes f immigratin. In partiular, Card asks whether the Mariel immigratin, whih inreased the Miami labr fre by abut 7 perent between May and September f 1980, redued the emplyment r wages f nn-immigrant grups. An imprtant mpnent f this identifiatin strategy is the seletin f mparisn ities that an be used "The DD methd ges by different names in different fields. Psyhlgist Campbell (1969) alls it the "nnequivalent ntrl-grup pretest-psttest design."

36 22 t estimate what wuld have happened in the Miami labr market absent the Mariel immigratin. The mparisn ities Card used in the Mariel Batlift study were Atlanta, Ls Angeles, Hustn, and Tampa-St. Petersburg. These ities were hsen beause, like Miami, they have large Blak and Hispani ppulatins and beause disussins f the impat f immigrants ften fuses n the nsequenes fr minrities. Mst imprtantly, these ities appear t have emplyment trends similar t thse in Miami at least sine This is dumented in Figure 1, whih is similar t a figure in Card's (1989) wrking paper that did nt appear in the published versin f his study. The figure plts mnthly bservatins n the lg f emplyment in Miami and the fur mparisn ities frm 1970 thrugh The tw series, whih are frm BLS establishment data, have been nrmalized by subtrating the 1970 value. Table 4 illustrates DD estimatin f the effet f Batlift immigrants n unemplyment rates, separately fr whites and blaks. The first lumn reprts unemplyment rates in 1979, the send lumn reprts unemplyment rates in 1981, and the third lumn reprts the differene. The rws give numbers fr Miami, the mparisn ities, and the differene between them. Fr example, between 1981 and 1979, the unemplyment rate fr Blaks in Miami rse by abut 1.3 perent, thugh this hange is nt signifiant. Unemplyment rates in the mparisns ities rse even mre, by 2.3 perent. The differene in these tw hanges, -1.0 perent, is a DD estimate f the effet f the Mariel immigrants n the unemplyment rate f Blaks in Miami. In this ase, the estimated effet n the unemplyment rate is atually negative, thugh nt signifiantly different frm zer. The ratinale fr this duble-differening strategy an be explained in terms f restritins n the nditinal mean funtin fr ptential utmes in the absene f immigratin. As in the unin example, let Yqj be i's emplyment status in the absene f immigratin and let Y n be i's emplyment status if the Mariel immigrants me t i's ity. The unemplyment rate in ity in year t is ErYJ, t], with n immigratin wave, and E[YJ, t] if there is an immigratin wave. In pratie, we knw that the Mariel immigratin happened in Miami in 1980, s that the nly values f E[Y,jl, t] we get t see are fr =Miami and r>1980.