Today we start a three series lecture on education. The general outline is as follows:

Transcription

1 Lecture Educaton Today we start a three seres lecture on educaton. The general outlne s as follows: The Human Captal Model Heterogenety n returns to schoolng Instrumental Varables Approaches Lqudty Constrants Sub-optmal Indvdual choce Evdence of Sgnallng (Acemoglu) It s hard to know where to start wth coverng educaton. There has been so much wrtten about t, and not just n economcs. It s a hot topc because t s an easly dentfable publc polcy ssue, wth tons of data avalable for lookng at t. The correlaton between wages and schoolng s one of the most strkng relatonshps n labor economcs. We spend 5 years n school. It s well worth understandng ndvdual s reasons for schoolng, ts mpact on ndvduals, the wage dstrbuton, and externaltes. The standard human captal earnngs functon, or Mnceran model takes on the followng form: log y a + bs + cx + dx where y s some earnngs measure: sometmes hourly, weekly, annual. S s ndvdual years of schoolng. The U.S. census and BLS stopped askng number of years of school and swtched to hghest degree completed n 99. Often ths new defnton s converted to years of schoolng by guessng tme to completon for each degree. X s experence, often measured as the number of predcted years n the labor force based on school attanment. Ths s potental experence, usually age years of school 6. Ths may not be an accurate measure as, more recently, ndvduals move Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture

2 n and out of the labor market. The model assumes the same experence profle for all educaton levels. Recent research uses a quartc term for experence: 3 X and 4 X The Mncer equaton assumes schoolng only affects the experence profle n levels, and that the log earnngs ncreases wth schoolng at a lnear rate. Ths does not have to be the case, but emprcally, these two functonal form specfcatons seems to hold up reasonably well (e.g. see Card s Handbook Chapter). Several ssues arse n relaton to ths equaton and the nterpretaton of b. What s b? ) an ncrease n productvty? ) a sgnal of productvty?: employers use schoolng to derve productvty expectatons 3) omtted varables bas, measurement error? Does b dffer by: 4) SES background? 5) school qualty? 6) level? (s the process really lnear?) 7) How do people choose S? 8) Are there constrants n choosng S? 9) Are there externaltes from choosng S? The U.S. government spends about 4.5 percent of GDP on elementary and secondary educaton and percent of GDP on post-secondary educaton. The U.S. spends about 6.5 percent of GDP on educaton annually. That s a lot of money. An mportant role n research of educaton s to answer the questons above to help determne how publc funds mght affect short and long run outcomes of nterest. Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture

3 Grllches, 977 Econometrca Card s Model: (Econometrca, Handbook Chapter, and Earnngs, Schoolng, and Ablty Revsted, Research n Labor Economcs, Vol. 4, pp 3-48, 995) Grllches s classc paper covers many of the central ssues underlyng the human captal earnngs model: ablty bas (both postve and negatve), measurement error, and nterpretaton of the Mnceran returns to schoolng. Card bulds on ths, and focuses on what measures of the effects of schoolng are we nterested n, what does OLS measures, IV, measure, and other emprcal approaches. Assume that ndvduals have an nfnte plannng horzon that starts at t. Lfetme utlty s: V ρt ( S, c( t)) u( c( t)) e dt subject to the budget constrant: Rt c ( t) e dt S e Rt dt e R RS The model s been wrtten up wth earnngs not a functon of age. Wth schoolng addtvely separable n age we can gnore earnngs growth that does not depend on schoolng. Ths s just a smplfyng assumpton. As I ve wrtten t, s does not appear n the utlty functon. Therefore, the educaton decson s smply to choose how much to maxmze the budget set: choose educaton to maxmze ncome. The frst order condtons are: Rs RS e e W '( R + y'( R R Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture 3

4 Or: y' ( R Whch states that ndvduals nvest n schoolng untl the margnal return s equal to the nterest rate. Ths expresson s where the term returns to schoolng comes from. The left hand sde of ths expresson s the margnal benefts from an nvestment n schoolng (per dollar of foregone earnngs) and the rght hand sde as the margnal cost (the opportunty cost of the dollar nvestment, n ths case) Suppose that y a + b S ( e descrbes the human captal producton functon. Then from F.O.C., log α + b S α + RS y'( b R Wth ths functonal form, ether b R, and an ndvdual s ndfferent to how much schoolng they get, or b > R and the ndvdual obtans an nfnte amount of schoolng, or b < R and the ndvdual obtans no schoolng. Ths pecularty, wth no soluton to optmal schoolng, arses because costs of schoolng (R) are ndependent of the benefts. We need to ntroduce some curvature n ether the margnal benefts or margnal costs of school to avod corner solutons. <draw fgure> Suppose nstead that: a + b S ks, e So that the margnal ncrease n log earnngs from schoolng falls wth addtonal schoolng. Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture 4

5 y'( b k S R Now we have a closed form soluton for optmal schoolng: b R S* k <draw fgure> Heterogenety n the return to schoolng, b, leads to dfferent optmal amounts. Heterogenety n a does not affect schoolng, but ths s just because of the functonal form. If nstead, b S k S b S k S y'( e y ( S ) e + A, ( b ks ) R b S ks e + A <draw fgure> Heterogenety n the levels of ntal earnngs wll tend to lower optmal schoolng. Thnk Bll Gates, Mck Jagger, pro-athletes. Ths functonal form s admttedly messy, but t does get across one of Grllches man ponts, whch s that there are two forms of ablty bas dfferences n ablty endowments (n the levels) whch tends to lower optmal educaton attanment, and dfferences n the nteracton between ablty and schoolng (n the slopes) whch tends to rase optmal educaton. There s no apror reason for belevng one domnates over the other, but most researchers are more concerned wth ablty bas from dfferences n b. As Card does, I wll gnore the mportance of ablty dfferences n the levels. The relatonshp between log earnngs and schoolng does not appear, at frst blush, to be lnear, as n the Mncer equaton. Sub n for S*, however, and notce that the ablty bas n b leads to convexty n logy and S, whle the quadratc expresson leads to a concave relatonshp. The two may offset each other to generate a lnear cross-secton relatonshp wth a heterogeneous populaton. See Card (95) for another paragraph and fgure about ths. Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture 5

6 We can also ntroduce heterogenety n costs. Let s modfy the lfetme utlty functon to nclude dstaste for school: V ( S, c( t)) S ( u( c( t)) ρt ϕ ( t))) e dt + u( c( t)) S e ρt dt, where ϕ (t) s convex and reflects the dsutlty from school (we mght run nto trouble f ndvduals lke school and ϕ (t) was concave). Now, the schoolng decson affects utlty drectly, and we have to set up the Lagrangan when calculatng the F.O.C. Let u ( c( t)) log c( t). The frst order condtons are: y'( R + ρe ϕ ( r + k S ρ S I ve also ntroduced heterogenety n the nterest rate. A hgh R s often used to proxy for lqudty constrants. Indvduals that need to borrow to attend school but cannot easly do so can be modelled as ndvduals facng hgh values of R. Indvdual dfferences n optmal schoolng can arse from dfferences n the economc benefts from schoolng, represented by: y'( b ks and dfferences n the margnal costs, represented by: r + k S Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture 6

7 <draw fgure> b r S*, k k + k k Loosely speakng, varaton n b corresponds to varaton n ablty, whereas varaton n r corresponds to varaton n access to funds or tastes for schoolng. Note agan that we are not lookng at nnate ablty dfferences, ndependent of schoolng (e.g. athletc ablty), rather we are focussng on ablty dfferences n the slope of the returns to schoolng. What are the man reasons for schoolng dfferences? Note specal case: when r r and k, we have equalty of opportunty: schoolng dfferences only arse from ablty dfferences, but costs assocated wth schoolng the same (although, not clear what generates dfferences n b ). Heckman suggests we have equalty of opportunty, and b determned by young age. What s the causal effect of educaton? The margnal effect of addtonal school on earnngs, b k S *, dffers across ndvduals. The k k average margnal return s β E( b k S * ) b + r. Ths s the expected ncrease n average log k k earnngs f a random sample of the populaton acqured an addtonal unt of educaton. Ths s the Average Treatment Effect (ATE) of schoolng. The ATE may not be all that relevant for wantng to evaluate a partcular schoolng nterventon that affects a specfc sub-populaton. We mght be nterested n the average and local average mpact from addtonal educaton, but also why educaton attanment dffers n the frst place: that s, how does b and r dffer across the populaton and why? Let s use the defnton of β as a benchmark for comparng OLS and IV coeffcents on the returns to educaton. Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture 7

8 To see the mplcatons of ths when tryng to estmate the average returns to schoolng, consder ths model mples a regresson equaton: log y a where α + b S + b S k S k S a ) + + a + ( b b ) S λ ( S S u, and b b ψ ( S S ) + v λ s the tradtonal ablty bas that s postve n ths model when cov( r, ) < (and/or cov( b, a ) > ). Note, λ could be negatve n the Grllches case of Bll Gates and pro-athletes examples. a (check: cov( a, S ) cov( a,( b r ) / k) λ var( S ) var( b r ) / k) cov( a, b ) cov( a, r ) k var( b ) + var( r ) + cov( b, r ) ) ψ s the comparatve advantage bas that arses from dfferences n the slope of the earnngsschoolng functon. Ordnary Least Squares: Our coeffcent estmate for the relatonshp s: cov(log, S ) var( S ) b k S + λ +ψ S, assumng that b and r are jonly symmetrc, whch means: E[( b 3 3 b ) ] E[( r r) ] E[( b b )( r r) ]. Use laws of expectatons to verfy ths for yourselves (t s a bt of messy algebra). b k S s the ATE. Our estmate s based (relatve to ATE) by λ + S ψ, A cross-sectonal regresson of earnngs on schoolng therefore also yelds an upward based estmate of the average Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture 8

9 margnal return to schoolng, even gnorng ablty bas. Even wthout varaton n the ntercept of the earnngs functon, we over estmate the ATE wth OLS (and Card was the frst to pont ths out). Instrumental Varables Consder an exogenous change n college expenses, Z, as an nstrument that affects r, but more for people wth hgh values of r. Let k. The log earnngs, schoolng equaton s: log y α + b S + e Note, from a change n schoolng: log y b S Wth a vald nstrument, Z, p lmb v E[log y Z ] E[log y Z ] E[ S Z ] E[ S Z ] E[log y Z ] E[log y Z ] + E[ log y ] E[ S Z ] E[ S Z ] + E[ S ] E[ b S ] E[ S ] Suppose the drop n college expenses leads to a fall n r and an ncrease n S, but mostly among persons wth hgh B. If B s hgher for people who tend to have hgh values of r, such an nstrument wll dentfy hgher than average B (LATE). Lang (93), and Card refer to ths as dscount rate bas. <draw fgure> Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture 9

10 Card suggests ths as a reason why IV estmates from supply-sde varaton tend to be hgher than OLS estmates. (Measurement error mght also be a possblty, but generally IV>OLS more than me would suggest). Example: Card: Usng Geographc Varaton n College Proxmty to Estmate the Return to Schoolng Card takes data from the U.S. Natonal Longtudnal Survey of Young Men (NLSYM), whch began wth 555 men aged 4-4 n 966 and occasonally ntervewed them later. 83% of these ndvduals were ntervewed n 976. The NLSYM contans some geographc nformaton, ncludng an ndcator varable for whether an accredted 4-year college (unversty) s located close by. 7% of ndvduals dd lve close by to such a college. Card s frst stage equaton s: S γ X + v where X s a dummy varable for the presence of a nearby college. For ths nstrument to be vald, ths varable must be correlated wth educaton attanment, but uncorrelated wth v, other factors that could affect both schoolng and Y. Card s dea for ths nstrument s that students who grow up n an area wthout a college face a hgher cost of college educaton, snce the opton of lvng at home s precluded. Once would expect ths hgher cost to reduce nvestments n hgher educaton, at least among chldren from relatvely low-ncome famles. We use the frst stage to predct ndvdual schoolng, Sˆ γˆ X, and then use these predctons to estmate the second stage: Y β + Sˆ + e β Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture

11 I want to emphasze here that we are estmatng the average effect of schoolng on earnngs only off of the varaton n schoolng from lvng close to or far away from a 4-year college. Therefore, we are only estmatng the average effect off of those affected by the nstrument, and not the average effect for the entre sample. Who are the people affected by ths nstrument? Lkely those from low-ncome parents who can t afford, or don t want to, send ther chldren to a school out of town. Thus, for ths group, the margnal cost to attend school may be hgher than the average margnal cost f lqudty constrants play a role. The margnal benefts from educaton may also dffer compared to the average n the sample, or for ndvduals under other crcumstances. Instrumental varable approaches estmate causal effects only off of a partcular group affected by the nstrument. We therefore can only draw conclusons from our results for ths group, and not for the rest of the sample. What does Card fnd? Tables 3 and 4. The potental problem wth an nstrumental varables strategy you should always frst consder s the valdty of the nstrument. Is t really true that college proxmty s unrelated to unobservable factors that correlate wth Y? College proxmty s obvously related to geography n partcular urban versus rural locaton. If nnate ablty or demographc characterstcs that mpact educaton attanment choces (but not through resdence costs) are related to beng n an urban or rural area, than ths nstrument s nvald we stll face omtted varable problems, and cannot draw defnte conclusons from the results. Card realzes ths, and argues that f ths nstrument explans schoolng dfferences only through the effect of lowerng costs to some n attendng school, then we should expect to see a frst stage effect among ndvduals wth low-ncome famly background versus hgh-ncome famly background. See Fgure : Ths s what he shows: The presence of a nearby college has ts strongest effect on men wth lowest propenstes to contnue ther educaton (men from sngle headed famles wth low parental educaton n rural southern areas). Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture

12 Is ths convncng proof that the nstrument s vald? If average famly background characterstcs for those lvng n rural or urban areas are the same regardless of ncome-class, then we would expect the same omtted varables bas when examnng the effect of schoolng from ndvduals of low or hgh ncome groups wth ths nstrument. The fndng that we do not suggests the omtted varables bas could be small, and we are, ndeed, estmatng the effect of schoolng off of lvng away from a 4- year college. However, f dfferences n urban/rural background characterstcs vary themselves by ncome group, then we may stll face omtted varable bas n these estmates. If low ncome famles lvng n urban areas are more lkely to place a strong emphass on educaton than those lvng n rural areas, but ths s not true for hgh ncome famles n urban and rural areas, Card s robustness check does not work. He tres to add addtonal controls, such as AFQT ablty measures, and broader regon controls. You decde. Card fnds that the IV estmates for the return to schoolng are larger than the OLS results. Possbltes: ) heterogeneous returns to educaton, and margnal returns hgher to those that obtan more school from beng close to college ) Measurement Error n OLS 3) OLS ablty bas downwards (Grllches argument) More snster reasons: 4) specfcaton bas : researchers workng wth IV and hgher standard errors end up preferrng specfcatons that generate sgnfcant t-statstcs, whch tends to preferrng hgher estmates (see Ashenfelter) 5) Instruments are nvald Phlp Oreopoulos Labor Economcs Notes for 4.66 Fall 4-5 Lecture