Partner Choice and the Marital College Premium

Transcription

1 Partner Choe and the Martal College Premum Perre-André Chappor Bernard Salané Yoram Wess January 30, 2011 Abstrat Several theoretal ontrbutons have argued that the returns to shoolng wthn marrage play a rual role for human aptal nvestments. Our paper empral nvestgates the evoluton of these returns over the last deades. We onsder a frtonless mathng framework à la Beker-Shapley-Shubk, n whh the gan generated by a math between two ndvduals s the sum of a systemat effet that only depends on the spouses eduaton lasses and a math-spef term that we treat as random; followng Choo and Sow (2006), we assume the latter omponent has an addtvely separable struture. We derve a omplete, theoretal haraterzaton of the model. We show that f the supermodularty of the surplus funton s nvarant over tme and errors have extreme value dstrbutons, the model s overdentfed even f the surplus funton vares over tme. We apply our method to US data on ndvduals born between 1943 and Our model fts the data very losely; moreover, we fnd that the determnst part of the surplus s ndeed supermodular and that, n lne wth theoretal predtons, the martal ollege premum has nreased more for women than for men over the perod. Columba Unversty Columba Unversty. Tel Avv Unversty. 1

2 1 Introduton Martal ollege premum and the demand for hgher eduaton The jont evoluton of US male and female demand for ollege eduaton over the reent deades rases an nterestng puzzle. Durng the frst half of the entury, ollege attendane nreases for both genders, although at a faster pae for men. Aordng to Goldn and Katz (2008), male and female ollege attendane rates are about 10% for the generaton born n 1900, and reah respetvely 55% and 50% for men and women born n Ths ommon trend, however, breaks down for the ohorts born n the 50s and later. These ndvduals faed a market rate of return to shoolng (the ollege premum ) that was substantally hgher than ther predeessors; therefore one would expet ther attendane rate to keep nreasng, possbly at a faster pae. Ths predton s satsfed for women, whose rate reahes 70% for the generaton born n 75. On the ontrary, the male rate s all but flat. As a result, n the reent ohorts women are more eduated than men. To explan ths strongly asymmetr responses to dental nentves, Chappor et al (2009) stress the role of gender dfferenes n the returns to shoolng wthn marrage 1. They argue that the return to eduaton has two dstnt omponents. One s the standard market ollege premum, whereby a ollege degree sgnfantly nrease wages; ths omponent has evolved n a largely smlar way for men and women. Seondly, eduaton has an mpat on a person s stuaton on the marrage market; t affets ther probablty of gettng marred, but also the haratersts of the future spouse and the ntrahousehold dstrbuton of the martal surplus wthn marrage. Chappor et al (2009) suggest that ths martal ollege premum may have evolved n a totally asymmetr way between genders, a lam supported by the substantal mprovements n household and brth ontrol tehnology, as well as by the hangng roles of women wthn the household, and that ths asymmetry may explan the dsrepanes 1 Another, largely omplementary explanaton proposed by Beker, Hubbard and Murphy (2009) reles on the dfferenes between male and female dstrbutons of unobserved ablty. Stll, these authors also emphasze that eduated women must have reeved some addtonal, ntrahousehold return to ther eduaton. It s presely that addtonal term that our approah allows to evaluate. 2

3 n demand for hgher eduaton. Whle the martal ollege premum argument s theoretally onsstent, provng t emprally s a hallengng task (see Greenwood et al, 2004). In ontrast to the returns to shoolng n the market that an be estmated from observed wages data, the returns to shoolng wthn marrage are not dretly observed and an only be nferred ndretly from the marrage patterns of ndvduals wth dfferent levels of shoolng. In ths paper, we provde suh estmates. Our empral approah s based on a strutural model of mathng on the marrage market that s lose, n sprt, to that adopted by Chappor et al. (2009). Spefally, we onsder a frtonless mathng framework a la Beker-Shapley-Shubk, n whh the gan generated by the math of male and female j s the sum of a systemat effet, that only depends on the spouses eduaton lasses, and a math-spef term that we treat as random. Our rual dentfyng assumpton, smlar to that n Choo and Sow (2006), s that the latter term s addtvely separable nto a male-spef and a female-spef omponents. A natural nterpretaton s that the omplementarty propertes of the model, whh drve the assortatveness of the stable mathng, operate only between lasses, and are not affeted by the unobservable varables. Whle undoubtedly restrtve, ths assumpton allows us to fous on our man top of nterest, namely mathng between eduaton lasses; n that sense, our model s essentally motvated by a onern for parsmony. Moreover, our separablty assumpton generates testable restrtons that are onsstent wth the data 2. Under ths separablty assumpton, we derve a set of neessary and suffent ondtons for stablty, and show that these ondtons an be nterpreted n terms of a standard, dsrete hoe framework. We then dsuss the dentfablty of our theoretal setup. In a ross-setonal ontext, the smplest verson, whh reles on a strong homoskedastty assumptons, s exatly dentfed; so that we annot dentfy any 2 Whle the frtonless nature of our model would be a strong assumpton n many ontexts, we beleve that t s probably more aeptable n our framework, presely beause of the separablty assumpton. In our separable world, the absene of frtons only means that any agent an meet at least one potental mate from eah of the eduaton lasses under onsderaton at (almost) no ost. 3

4 pattern of heteroskedastty. If, however, the same struture (as summarzed by the matrx of eonom gans by spouses eduaton lasses) s observed for subpopulatons wth dfferent ompostons, then the bas model s (vastly) overdentfed. In fat, one an dentfy a more general struture, n whh the systemat omponent of the surplus nvolves lass-spef temporal drfts; moreover, ths generalzed model stll generates strong overdentfaton restrtons. 3 We apply our model to the US populaton, for ohorts born between 1935 and 1975 and marred between ages 18 and 35. We show that the martal returns to shoolng evolved non monotonally over the perod. Spefally, we fnd that n the begnnng of that perod (for ohorts born before the md 50s), the return dereases for both men and women. For the followng ohorts, however, the evoluton s gender-spef; we fnd that the martal premum has nreased sharply for women over the perod, whle they have not hanged muh for men. Eduated women have ganed relatve to uneduated women n two ways: by marryng at hgher rates and by reevng a hgher share of the martal surplus. Interestngly, although these fndngs are not based on a model of ndvdual demand for eduaton (the premum s estmated exlusvely from the observed mathng patterns), they exatly ft the predton of the theoretal analyss n Chappor et al (2009). Fnally, we also fnd that the gans generated by marrages wth equally eduated partners have delned for all types of marrage, refletng the general reduton n marrage over tme. However, the smallest delne s n mathes when one or both partners have ollege eduaton. Ths fndng an be related to empral work showng that suh marrages are also less lkely to break (see Wess and Wlls 1997 and Bruze, Svarer and Wess 2010). The evoluton of assortatve mathng A related ssue s what happened to assortatve mathng. The observed patterns are qute omplex. Overall, the perentage of ouples n whh both spouses have a ollege degree has sgnfantly nreased 3 We show that an even more general struture, n whh the sale of ndvdual heterogenety may vary by eduaton lass, ould also be dentfed. In prate, however, the assumpton of dental homogenety s not rejeted by the data. 4

5 over the perod; however, as women wth ollege degree beame more abundant, the proporton of eduated women who marry eduated men has delned (beause some eduated women had to marry downwards wth less eduated men), whle men wth hgh shool degree shfted upwards from marryng women wth hgh shool degree to marryng more often women wth ollege degree. All n all, many observers have nevertheless onluded that assortatve mathng was stronger now than four deades ago, and that ths evoluton had a deep mpat on ntrahousehold nequalty (see for nstane Burtless 1999). An nterestng queston s whether ths phenomenon s entrely due to the mehanal effet of the nrease n female eduaton, or whether t also reflets a shft n preferenes towards assortatve mathng (as would be the ase, for nstane, f the share of publ goods n households rses wth tme - or nome - and smlar eduaton faltates agreements on the omposton and level of these publ goods). An mportant advantage of our strutural approah s that t allows to formally dsentangle the two aspets. In ths respet, our onlusons are lear-ut. We do not fnd any evdene for a hange n preferenes for assortatve mathng. In fat, we do not rejet the null that the nteraton n martal gans by level of shoolng (as summarzed by the supermodularty of the matrx of systemat gans) has remaned stable over tme. To the extent that we fnd an nreasng proporton of ouples n whh both partners are eduated, ths s not beause the gans from havng a ollege degree for both partners (ompared wth only one partner havng a ollege degree) have rsen over tme. We fnd strong evdene that for eduated women, the addtve gans from marrage have shfted over tme. One possble nterpretaton s that t beame less ostly for eduated women to marry manly beause household hores have been redued, so that marred women an partpate more n the labor market (see Greenwood, Seshadr and Yorukoglu 2005), and also beause brth ontrol tehnologes have drastally mproved over the perod, allowng for better plannng of famly fertlty (see Mhael, 2000, and Goldn and Katz 2002). However, our fndngs suggest that these lberatng effets are more or less ndependent of the shoolng of the husband. As a by-produt of our nvestgaton, we an dentfy the matrx of systemat gans; we fnd that t s, ndeed, sgnfantly supermodular. 5

6 Ths fndng seems to ontradt results n the soologal lterature that have shown, usng log lnear models, that even after aountng for hanges n the relatve number of men and women n eah skll group, homogamy has nreased n the US and several other ountres (see for nstane Shwartz and Mare, 2005). However, these onlusons were drawn from redued-form models wth no dret eonom nterpretaton and an therefore be qute msleadng. To hek ths, we used our model to generate marrage data and we ran t through the type of log-lnear regresson adopted n the soologal lterature. The results (spurously) suggest that preferenes for homogamy have hanged, even though our model rules out suh hanges. These fndngs further outlne the mportane of a strutural approah to gude the nterpretaton of the empral results. Fnally, another outome of our strutural approah s the dentfaton of the group spef pres that determne the dvson of the gans from marrage between husbands and wves of dfferent types. We fnd that n ouples n whh both spouses have a ollege degree, the share of the wfe n the gans from marrage has nreased over tme, despte the rse n the number of eduated women relatve to eduated men. Ths happened beause the margnal ontrbuton of eduated women to the surplus wth eduated men has rsen over tme. We fnd that the nrease s manly due to the varable omponent: eduated women beame more produtve relatve to less eduated women n all marrages, rrespetve of the type of the husband. Ths fndng onfrms the analyss of Chappor et al. (2009), aordng to whh the nrease n the martal omponent of the eduaton premum for women ould explan the spetaular nrease n female demand for hgher eduaton. Related lterature The analyss of the marrage market as a mathng proess, whh dates bak to Beker s semnal ontrbutons (see Beker 1981), has reently attrated renewed attenton. Probably the most mportant empral work s due to Choo and Sow (2006), who propose one of the frst mplementatons of a Beker- Shapley-Shubk model based on a dsrete hoe model. Our paper extends ther ontrbuton n three dretons. Frst, we larfy the underlyng theoretal struture, n partular by workng out the assumptons needed on the fundamentals of the model 6

7 (.e., the matrx of systemat gan) and ther mplatons for the endogenous varables (ndvdual utltes at the stable math). Seondly, we onsder a model that allows for nterlass heteroskedastty of the random omponents. Thrdly, we study the evoluton of mathng patterns throughout tme, n a framework that also allows the gans for marrage to evolve n a lass-spef way. Our ultmate goal s to study mathng on eduaton, and more spefally to provde a dynam perspetve on the evoluton of the orrespondng patterns over several deades. These varous extensons are neessary for our purpose. Evaluatng how the martal ollege premum evolves over tme obvously neesstates a dynam analyss. It also requres omparng (expeted) utltes between lasses, a task for whh the homoskedastty assumpton of the standard verson s potentally napproprate. In Choo and Sow s approah, the bas, homoskedast verson s exatly dentfed, mplyng that any generalzaton wll fae severe dentfablty problems. We show that these problems an however be overome n a more dynam ontext, provded that the strutures drvng assortatve mathng remans onstant. In partular, our dentfaton strategy s orgnal. Another related approah s due to Galhon and Salané (2010), who provde a theoretal and eonometr analyss of multrteron mathng under the same separablty assumpton. Ther fous s dfferent: whle our paper onsders a small number of lasses, they seek to provde a general method to estmate and test flexble parametr spefatons of the gans from marrage when many ovarates are avalable 4. Seton 2 presents some stylzed fats. Then we ntrodue our theoretal framework n Seton 3, and seton 4 desrbes the bas prnples underlyng ts empral mplementaton. In Seton 5, we dsuss dentfaton ssues and present our man theoretal results on that top. Seton 6 desrbes the mathng patterns n the data, and our empral fndngs are presented n Seton 7. 4 In another paper, Galhon and Salané (2011) generalze the Choo and Sow framework to arbtrary dstrbutons; they also derve the soal surplus funton, and they endogenze the hoe of partner gender. 7

8 2 Some stylzed fats We frst brefly desrbe some raw fats about the evoluton of mathng by eduaton over the last deades. To do ths, we use the Ameran Communty Survey, a representatve extrat of the Census, whh we downloaded from IPUMS (see Ruggles et al (2008).) Unlke earler waves of the survey, the 2008 survey has nformaton on urrent marrage status, number of marrages, and year of urrent marrage. Of the 3,000,057 observatons n our orgnal sample, we only keep whte adults (aged 18 to 70) who are out of shool; the resultng sample at ths stage has 1,307,465 observatons and s 49.5% male. We used the detaled eduaton varable of the ACS to defne three subategores: 1. Hgh Shool Dropouts (HSD) 2. Hgh Shool Graduates (HSG) 3. Some College (SC). Our ategory some ollege aggregates all ndvduals who at least started ollege. The drawbak s that our hghest eduaton ategory nludes 2-year and 4-year ollege graduates, along wth ollege students who dd not graduate. One may want to separate 4-year ollege graduates nstead, and aggregate the rest of our thrd ategory wth hghshool graduates; the results are qualtatvely smlar. A fner lassfaton would be desrable, but ell szes shrnk fast. When studyng mathng patterns, we have to dede whh math we onsder: the urrent math of a ouple, or earler unons n whh the urrent partners entered? also, do we defne a sngle as someone who never marred, or as someone who s urrently not marred? It s notorously hard to model dvore and remarrage n an emprally redble manner. Sne ths s not the objet of ths paper, we hose nstead to only keep frst mathes, and never-marred sngles. Gven ths sample seleton, n eah ohort we mss: those ndvduals who ded before the 2008 Survey; those who are sngle n 2008 but were marred before: there are 8

9 36,094 ndvduals who are separated from ther spouse 218,839 who are dvored 143,963 who are wdowed. those who are marred n 2008, but not n a frst marrage more presely, n Table 1, we only kept the top left ell. Number of marrages Total 1 384,291 42,147 5, , ,773 56,210 14, , ,250 15,334 9,069 31,653 Total 438, ,691 29, ,165 Table 1: Men n rows, women n olumns Outomes are trunated n our data, sne young men and women who are sngle n 2008 may stll marry; n our fgures (and later n our estmates) we rumvent ths dffulty by stoppng at the ohort born n 1972 the frst unon ours before age 35 for most men and women. To examne marrage patterns, we dropped the small number of ouples where one partner marred before age 16 or after age 35 (reall that these are frst unons.) Ths leaves us wth 179,353 ouples, 44,344 sngle men, and 32,985 sngle women. The nreasng level of eduaton of women s shown on Fgure 1: n ohorts born after 1955 women graduate more from ollege. Not ondentally, the proporton of marrages n whh the husband s more eduated than the wfe has fallen qute dramatally. Fgure 2 shows that sne the early 1980s, there are now more marrages n whh the wfe has a hgher level of eduaton (ths fgure uses 4 levels of eduaton.) Fgures 3 and 4 desrbe hanges n the level of eduaton of the partners of marred men (resp. women) between the earler ohorts (born n the early 40s) and the most reent ohorts n our sample (born n the early 70s.) Fgure 3 shows that ollegeeduated men now fnd a ollege-eduated wfe muh more easly; and n fat even less-eduated men are now more lkely to marry a ollege-eduated woman f they 9

10 Proporton Hgh Shool Dropout Men Hgh Shool Dropout Women Hgh Shool Graduate Men Hgh Shool Graduate Women Some College Men Some College Women Year of brth Fgure 1: Eduaton levels of men and women 10

11 Proporton Husband more eduated Same eduaton Husband less eduated Year of marrage Fgure 2: Relatve eduaton of partners 11

12 marry at all. On the other hand, the marrage patterns of women are remarkably stable, as evdened n Fgure 4. HSD HSG SC Born Born Man: SC Man: HSG Man: HSD Proporton Fgure 3: Marrage patterns of men who marry We llustrate the delne n marrages by plottng the perentage of ndvduals of a gven ohort who never marred n Fgures 5 and 6. They show that a hgher eduaton has tempered the delne n marrage, espeally for women; and that hghshool dropouts on the other hand have faed a very steep delne n marrage rates. 12

13 HSD HSG SC Born Born Woman: SC Woman: HSG Woman: HSD Proporton Fgure 4: Marrage patterns of women who marry 13

14 Proporton Hgh Shool Dropout Hgh Shool Graduate Some College Year of brth Fgure 5: Proporton of men who never marred 14

15 Proporton Hgh Shool Dropout Hgh Shool Graduate Some College Year of brth Fgure 6: Proporton of women who never marred 15

16 3 Theoretal framework The bas struture We onsder a frtonless, Beker-Shapley-Shubk mathng game between a male populaton M, endowed wth some measure dµ M, and a female populaton F, endowed wth some measure dµ F. Eah populaton s parttoned nto a fnte number of lasses, I = 1,..., N for men and J = 1,..., M for women. The gan generated by the math of Mr., belongng to lass I, and Mrs. j, belongng to lass J, s the sum of two omponents, one ommon to all ndvduals n the same lass, the other math spef: g j = Z IJ + ε IJ j wth the notaton I = 0, J = 0 for sngles; here, Z IJ denotes the ommon omponent and ε IJ j s a random shok wth mean zero. A mathng onssts of () a measure dµ on the set M F, suh that the margnal of δµ over M (resp. F ) s dµ M (dµ F ), and () a set of payoffs (or mputatons) {u, M} and {v j, j F } suh that u + v j = g j for any (, j) Supp (µ) In words, a mathng ndates who marres whom (note that the alloaton may be random, hene the measure), and how any marred ouple shares the gan generated by ther math. A mathng s stable f one an fnd nether a man who s urrently marred but would rather be sngle, nor a woman j who s urrently marred but would rather be sngle, nor a woman j and a man who are not urrently marred together but would both rather be marred together than reman n ther urrent stuaton. Formally, we must have that: u + v j g j for any (, j) M F (1) whh translate the fat that for any possble math (, j), the realzed gan g j annot exeed the sum of utltes respetvely reahed by and j n ther urrent stuaton. As s well known, a mathng model of ths type s equvalent to a maxmzaton problem; spefally, a math s stable f and only f t maxmzes total gan, gdµ, over the set of measures whose margnal over M (resp. F ) s dµ M (dµ F ). A frst 16

17 onsequene s that exstene s guaranteed under mld assumptons. Moreover, the dual of ths maxmzaton problem generates, for eah male (resp. female j), a shadow pre u (resp. v j ), and the dual onstrants these varables must satsfy are exatly (1); n other words, the dual varables exatly onde wth payoffs assoated to the mathng problem. Fnally, s the stable mathng unque? Wth fnte populatons, the answer s no; n general, the payoffs u and v j an be margnally altered wthout volatng the (fnte) set of nequaltes (1). However, when the populatons beome large, the ntervals wthn whh u and v j may vary typally shrnk; n the lmt of ontnuous populatons, (the dstrbutons of) ndvdual payoffs are exatly determned. On all these ssues, the reader s referred to Chappor, MCann and Neshem (2009) for prese statements. The man empral assumpton that wll be rual n what follows: Assumpton S (separablty): the dosynrat omponent ε j s addtvely separable: We now ntrodue a smplfyng assumpton where E [ α IJ ] ] = E [β IJ j = 0. ε IJ j = α IJ + β IJ j (S) In words, the math spef term s the sum of two ontrbutons. The male ontrbuton s ndvdual spef and may depend on both hs and hs spouse s lass - but t does not depend on the prese dentty of s spouse; and the same property holds for the female ontrbuton. Note that ths assumpton s equvalent to the followng property: for any, í I and any j, j J, g j + g j = g j + g j Ths property mples that wthn eah par of lasses, (I, J), any mathng would be stable. In prate, ths means that we exlusvely onentrate on the martal patterns between lasses (although ths an be relaxed by the ntroduton of ovarates, see below). 17

18 Eah male s thus fully haraterzed by the realzaton of the vetor α = ( ) α 11,..., αmn. For notatonal onssteny, we defne α I0 = ε I0 0 and β 0J j = ε 0J 0j (and smlarly for women). Then we have the followng Lemma: Lemma 1 Assume the ε satsfy the separablty property (S). For any stable mathng, there exst numbers U IJ and V IJ, I = 1,..., M, J = 1,..., N, wth U IJ + V IJ = Z IJ (2) satsfyng the followng property: for any mathed ouple (, j) suh that I and j J, u = U IJ + α IJ and (L) Proof. v j = V IJ + β IJ j Assume that and both belong to I, and are both mathed wth a spouse (resp. j and j ) belongng to J. Stablty requres that: u + v j = Z IJ + α IJ u + v j Z IJ + α IJ u + v j = Z IJ + α IJ u + v j Z IJ + α IJ Subtratng (1) from (2) and (4) from (3) gves hene β IJ j βij j It follows that the dfferene v j β IJ j The proof for u s dental. v j v j β IJ j v j v j = β IJ j βij j + β IJ j (1) + β IJ j (2) + βij j (3) + βij j (4) βij j does not depend on j,.e.: v j β IJ j = V IJ for all I, j J 18

19 In words: the dfferenes u α IJ and v j β IJ j only depend on the spouses lasses, not on who they are. The U IJ and V IJ denote how the ommon omponent of the gan s dvded between spouses; then a spouse s utlty s the sum of ther share of the ommon omponent and ther own, dosynrat ontrbuton. Note, ndentally, that (L) s also vald for sngles f we set U I0 = Z I0 and V 0J = Z 0J. An ntutve nterpretaton of U IJ (or equvalently of V IJ ) would be the followng. Assume that a man randomly pked n lass I s fored to marry a woman belongng to lass J (assumng that the populatons are large, so that ths small devaton from stablty does not affet the equlbrum payoffs). Then hs expeted utlty s exatly U IJ (the expetaton beng taken over the random hoe of the ndvdual wthn the lass). Note, however, that ths value does not onde wth the average utlty of men n lass I marred to women J at a stable mathng. The latter value s larger than U IJ (refletng the fat that an agent hooses hs wfe s lass), and wll be omputed below. Stable mathngs: a haraterzaton Under ths separablty assumpton, the empral haraterzaton of the stable math beomes muh easer. We frst provde a smple translaton of the stablty propertes: Lemma 2 A set of neessary and suffent ondtons for stablty s that 1. for any mathed ouple ( I, j J) one has α IJ α IJ α IK U IK U IJ for all K (3) α I0 U I0 U IJ (4) and β IJ j β IJ j β KJ j V KJ V IJ for all K (5) β 0J j V 0J V IJ (6) 2. for any sngle male I one has α IJ α I0 U I0 U IJ for all J (7) 19

20 3. for any sngle female j J one has β IJ j β 0J j V 0J V IJ for all J (8) Proof. The proof s n several steps. Let ( I, j J) be a mathed ouple. Then: 1. Frst, male must better off than beng sngle, whh gves: U IJ + α IJ U I0 + α I0 α IJ hene α I0 U I0 U IJ and the same must hold wth female j. Ths shows that 4, 6, 7 and 8 are neessary. 2. Take some female j n J, urrently marred to some n I. Then must be better off mathed wth j than j, whh gves: ( ) U IJ + α IJ z j v j = z IJ + α IJ + β IJ j V IJ + β IJ j and one an readly hek that ths nequalty s always satsfed as an equalty, refletng the fat that s ndfferent between j and j, and symmetrally j s ndfferent between and. 3. Take some female k n K J, urrently marred to some n I. Then s better off mathed wth j than k gves: U IJ + α IJ whh s equvalent to z k v k = z IK + α IK α IJ α IK + β IK k U IK U IJ ( V IK + β IK k and we have proved that the ondtons 3 are neessary. The proof s dental for We now show that these ondtons are suffent. Assume, ndeed, that they are satsfed. We want to show two propertes. Frst, take some female j n J, urrently marred to some l n L I. Then s better off mathed wth j than j. Indeed, ( ) U IJ + α IJ z j v j = z IJ + α IJ + β IJ j V LJ + β LJ j ) 20

21 s a dret onsequene of 5 appled to l. Fnally, take some female k n K J, urrently marred to some l n L I. Then s better off mathed wth j than j. Indeed, t s suffent to show that U IJ + α IJ z k v k = z IK + α IK But from 5 appled to k we have that: and from 3 appled to : β LK k α IJ β IK k α IK + β IK j V IK V LK U IK U IJ ( V LK + β LK k and the requred nequalty s just the sum of the prevous two. ) In summary, under our separablty assumpton, stablty an readly be translated nto a set of nequaltes, eah of whh relates to one agent only. Ths property s rual, beause t mples that the model an be estmated usng standard statstal proedures appled at the ndvdual level, wthout onsderng ondtons on ouples. We now see how these nsghts an be mplemented n prate. 4 Empral mplementaton 4.1 Probabltes Assume, frst, that the lasses are large, so that whle the α and β are random the U IJ and V IJ are not. Gven the omputatons above, t s natural to make the followng assumpton 5 : Assumpton HG (Heteroskedast Gumbel): The random terms α and β are suh that α IJ = σ I. α IJ β IJ = µ J. β IJ 5 Gumbel dstrbutons are better known to eonomsts under the lumser name of type-i extreme value dstrbutons. 21

22 where the α IJ and β IJ In partular, the α IJ and β IJ j j follow ndependent Gumbel dstrbutons G ( k, 1). and β IJ j have mean zero and respetve varane π2 6 Lemma then mples: have mean zero and varane π2 6, therefore the αij ( ) σ I 2 ( ) and π 2 µ J 2. The prevous Lemma 3 A set of neessary and suffent ondtons for stablty s that 1. for all mathed ouple ( I, j J) one has and α IJ α IJ β IJ j β IJ j 2. for all sngle male I one has α IK U IK U IJ σ I for all K (9) α I0 U I0 U IJ σ I (10) β KJ j V KJ V IJ µ J for all K (11) α IJ β 0J j V 0J V IJ µ J (12) α I0 3. for all sngle female j J one has β IJ j β 0J j Therefore, for any I and any I: U I0 U IJ σ I for all J (13) V 0J V IJ µ J for all J (14) 6 and a IJ = Pr ( mathed wth a female n J) exp ( U IJ /σ I) = K exp (U IK /σ I ) + 1 a I0 = Pr ( sngle) = 1 K exp (U IK /σ I )

23 where U I0 has been normalzed to 0. Smlarly, for any J and any female j J: where V 0J = 0. and b IJ = P (j mathed wth a male n I) (15) exp ( V IJ /µ J) = K exp (V KJ /µ J ) + exp (V 0J /µ J and (16) ) b 0J exp ( V 0J /µ J) = P (j sngle) = K exp (V KJ /µ J ) + exp (V 0J /µ J ) These formulas an be nverted to gve: therefore: exp ( U IJ /σ I) = exp ( V IJ /µ J) = ( U IJ = σ I a IJ ln 1 K aik ( V IJ = µ J ln a IJ 1 K aik (17) b IJ 1 b KJ (18) b IJ 1 b KJ In what follows, we assume that there are sngles n eah lass: a I0 > 0 and b 0J > 0 for eah I, J, mplyng that K aik < 1 and K bkj < 1 for all I, J. Note that a dret onsequene of these results s that, knowng the Z IJ and the populaton szes, we an algebraally ompute U IJ /σ I and V IJ /µ J for all (I, J). Fnally, defne: [ ū I = E max J ) ) ( U IJ + σ I α IJ ) ] In words, ū I s the expeted utlty of an agent n lass I, gven that ths agent wll hose a spouse n hs preferred lass. From the propertes of Gumbel dstrbutons, we have that: and smlarly [ ū I = σ I ( E max U IJ /σ I + α IJ ) ] J ( = σ I ln exp ( ) U IJ /σ I) + 1 = σ I ln ( a I0) (19) J ( v J = µ J ln exp ( ) V IJ /µ J) + 1 = µ J ln ( b 0J) (20) I 23

24 4.2 Why does heteroskedastty matter? An mportant property of the model just presented s heteroskedastty: the varane of the unobserved heterogenety parameters s lass-spef. Ths property may n prnple matter for varous reasons. For one thng, the expeted utlty of an arbtrary agent n lass I, as gven by (19), s dretly proportonal to the standard devaton of the random shok. Indeed, remember that the agent hooses the lass of hs spouses so as to maxmze hs utlty; and the expetaton of the max of..d varables nreases wth the varane. It follows that the utlty generated by the aess to the marrage market annot be exlusvely measured by the probablty of remanng sngle (refleted n the ln ( a I0) term). Ths remark, n turn, has mportant onsequenes for measurng the martal ollege premum. To see how, start from a model n whh the random omponent of the martal gan s homoskedastally dstrbuted (.e., the varane s the same aross ategores: σ I = µ J = 1 for all I, J). The martal ollege premum s measured by the dfferene ū I ū K, where I s the ollege eduaton lass whereas K s the hgh shool graduate one. Condton (19) then mples that ( ) a ū I ū K K0 = ln In words, the gan an dretly be measured by the (log) rato of snglehood probabltes n the two lasses. The ntuton s that people marry f and only f ther (dosynrat) gan s larger than some threshold. If these random gans are homoskedastally dstrbuted, then there s a one-to-one orrespondene between the mean of the dstrbuton for a partular lass and the perentage of that lass that s below the threshold,.e. that remans sngle: the hgher the mean, the smaller the proporton (see Fgure 7). For nstane, f one sees that ollege graduate are more lkely to reman sngle than hgh shool graduates (a I0 > a K0, mplyng that ln ( a K0 /a I0) < 0), we an onlude that the expeted martal gan s smaller for ollege graduates (ū I < ū K ), therefore that the martal ollege premum s negatve. Consder, now, the heteroskedast verson. Thngs are dfferent here, beause the perentage of sngle depends on both the mean and the varane. If eduated women are more lkely to reman sngle, t may be beause the gan s on average smaller, but a I0 24

25 Fgure 7: Homoskedast random gans t may also be that the varane s larger (even wth a hgher mean), as llustrated n Fgure 8. The one-to-one relatonshp needs not hold, and a hgher perentage does not neessarly mply a smaller mean. One has to ompute the respetve varanes - whh, n turn, may affet the omputaton of the martal ollege premum. Tehnally, we now have that: ū I ū K = σ K ln ( a K0) σ I ln ( a I0) (21) If a I0 > a K0 and σ I σ K, one an onlude that ū I ū K < 0; but whenever σ I > σ K the onluson s not granted, and depends on the prese estmates. In other words, eduaton operates on martal prospets through three dfferent hannels: t nreases marrage probabltes; t hanges the potental qualty (here eduaton) of the future spouses; and t affets the dstrbuton of surplus wthn the household. In the bas, homoskedast verson of the model, due to the assumptons made on the dstrbutons of the random terms, these three hannels are ntrnsally mxed, and the expeted utlty of eah spouse s fully determned by the perentage of persons n the same eduaton lass that remans sngle. The heteroskedast ver- 25

26 Fgure 8: Heteroskedast random gans son s muh rher; welfare mpats go beyond the sole probablty of marrage, and nvolve other onsderatons. Clearly, the onlusons drawn from the model may sgnfantly depend on the assumptons made regardng ts homoskedastty propertes. It s therefore mportant that these assumptons be testable rather than ad ho -.e., that homoskedastty be mposed by the data (or at least ompatble wth them) rather than assumed a pror. In that sense, the estmaton of the varanes s a rual part of the dentfaton proess Extenson: Covarates The bas framework just desrbed an be extended to the presene of ovarates;.e., we may spefy the ε k (hene the α and β) as a funton of ndvdual haratersts (other than the mathng ones). Let X be a vetor of suh haratersts of man, and Y j of woman j. We may use the followng stohast struture (where, for smplty, 6 Note, however, that f the varanes are assumed onstant aross tme, then the varatons n snglehood probablty must stll reflet smlar hanges n the expeted gans from marrage. In other words, f we fnd that the perentage of, say, unsklled women remanng sngle has nreased between two ohorts and, we an unambguously onlude that the gans from marrage have dmnshed for these women over the perod. 26

27 we dsregard heteroskedastty): α IJ = X.ζ IJ m + α IJ α I0 = X.ζ I0 m + α I0 β IJ j = Y j.ζ IJ IJ f + β j β 0J j = Y j.ζ 0J 0J f + β j where ζ IJ m, ζ IJ f are vetor parameters, wth the normalzaton U I0 = ζ I0 m V 0J = ζ 0J f = 0, and where as above the α IJ = 0 and (resp. βij j ) follow ndependent, type 1 extreme values dstrbutons G ( k, 1). Then the omputatons are as above. In other words, we an estmate for I: ( ) exp U IJ + X.ζ IJ a IJ m = Pr ( mathed wth a female n J) = ) ( K (U exp IK + X.ζ IK + exp a I0 = Pr ( sngle) = and the onlusons follow. standard (multnomal) logts. ( exp K exp (U IK + X.ζ IK m ) U I0 + X.ζ I0 m ) ( + exp m U I0 + X.ζ I0 m In partular, these models an be estmated runnng ) U I0 + X.ζ I0 m ) 5 Identfaton We now onsder the dentfaton problem. In prate, we observe realzed mathngs -.e., populatons n eah lasses and the orrespondng martal patterns. To what extend an one reover the fundamentals -.e., the surplus matrx Z and the heteroskedastty parameters σ and µ - rually depends on the type of data avalable. We frst onsder a stat ontext, n whh populaton szes are fxed. We show that n that ase, the model s exatly dentfed f we assume omplete homoskedastty, and not dentfed otherwse. Muh more nterestng s the stuaton n whh populaton szes vary over tme whle (some of) the strutural parameters reman onstant. Then one an dentfy both the surplus matrx Z and the heteroskedastty parameters σ 27

28 and µ, provded that they reman onstant over tme; atually, one an even ntrodue ether tme varyng heteroskedastty or a drft n the surplus matrx wthout losng dentfablty; and fnally, the model generates strong overdentfyng restrtons. We onsder the two ases suessvely. 5.1 The stat framework We start wth a purely stat framework. Defne a model M as a set ( Z IJ, σ I, µ J) suh that wth and where the α IJ and β IJ j ε IJ j g j = Z IJ + ε IJ j = σ I α IJ + µ J β IJ j (S) follow ndependent Gumbel dstrbutons G ( k, 1). Note that the model s learly nvarant when the ( Z IJ, σ I, µ J) are all multpled by a ommon, postve onstant; for that reason, n what follows we normalze σ 1 to be 1. The followng result s vald for stat (ross-setonal) data: Proposton 4 Assume that a model M = ( Z IJ, σ I, µ J) generates some mathng probabltes ( a IJ, b IJ), and let U IJ, V IJ denote the orrespondng dual varables. Then and therefore Z IJ = σ I log U IJ = σ I log V IJ = µ J log a IJ 1 K aik (22) a IJ 1 K aik + µj log b IJ 1 K bkj (23) b IJ 1 K bkj Moreover, for any ( σ I, µ J) R +, the model N = ( ZIJ, σ I, µ J) where σ I σ I U IJ + µj µ J V IJ = Z IJ (24) generates the same mathng probabltes, and the orrespondng, dual varables are Ū IJ = σi σ I U IJ (25) V IJ = µj µ J V IJ (26) 28

29 Conversely, f two models M = ( Z IJ, σ I, µ J) and N = ( ZIJ, σ I, µ J) generate the same mathng probabltes, then the ondtons (24), (25) and (26) must hold. Proof. From the prevous alulatons, there s a one-to-one relatonshp between the a IJ and the υ IJ ; the result follows. The prevous result s essentally negatve; t states that n a stat ontext, the heteroskedast verson of the model s not dentfed. The heteroskedastty parameters ( σ I, µ J) an be hosen arbtrarly; for any value of these parameters, one an fnd values { Z IJ, I = 1,..., N, J = 1,..., M } that exatly ratonalze the data. An nterpretaton of the non dentfablty result s n terms of utlty sales. The unt n whh the Us and V s are measured s not determned unless we make assumptons on the varanes of the αs and βs. Ths negatve result s mportant, n partular, for welfare omparsons. In a ross-setonal settng, omparng welfare between males and females or between ndvduals belongng to dfferent lasses s hghly problemat, sne t an only rely on arbtrary hoes of the unts. 5.2 Changes n populaton szes Muh more promsng s a stuaton n whh one an observe the market over dfferent perods (or for dfferent ohorts), when the varous populatons hange n respetve szes over the perods. Then a rher model an atually be estmated. We start wth the benhmark ase, then onsder the generalzed verson that wll be taken to data later The benhmark verson Let us now assume that the prevous, heteroskedast strutural model M = ( Z IJ, σ I, µ J) holds for dfferent ohorts of agents, = 1,..., T, wth varyng lass ompostons. The bas struture beomes: g j, = Z IJ + ε IJ j, wth ε IJ j, = σ I α IJ, + µ J β IJ j, (S) 29

30 Also, assume for the tme beng that eah man marres a woman wthn hs ohort. 7 As before, the mathng model defnes, for eah ohort, a mathng problem assoated to shadow pres; the latter are now ohort spef. Under the same assumptons as above, the prevous onstrut apples for eah ohort, leadng to the defnton of U IJ and V IJ. Then a IJ = Pr ( I mathed wth a female n J n ohort ) = a I0 = Pr ( I sngle) = therefore and smlarly: K exp (U KJ /σ I ) exp ( U IJ ) a IJ /σ I = 1 K aik b IJ = Pr (j J mathed wth a female n I n ohort ) = b I0 = Pr (j J sngle) = mplyng that Moreover, we have K exp (V IK /µ K ) exp ( V IJ ) b IJ /µ J = 1 K bik exp ( ) U IJ /σ I 1 + K exp (U KJ /σ K ) (27) exp ( ) V IJ /µ J 1 + K exp (V IK /µ K ) (28) Now, let p IJ and (28), the p IJ U IJ + V IJ = Z IJ (29) = U IJ /σ I and q IJ = V IJ /µ J. The rual remark s that from (27) and q IJ are dretly observable from the data. It follows that (29) has a dret, testable mplatons. Indeed, defne the vetors: p IJ = ( p IJ 1,..., p IJ ) T q IJ = ( q1 IJ,..., qt IJ ) 7 Emprally, ths s not exatly rght; women tend to marry slghtly older men, so that n the applaton the wfe of a man n ohort typally belongs to ohort ( + 2) - a fat that wll be taken nto aount n the empral applaton, but an be gnored for the tme beng. 30

31 and 1 = (1,..., 1) Then for eah par (I, J), the vetors p IJ, q IJ and 1 must be olnear: σ I p IJ + µ J q IJ Z IJ 1 = 0 (30) whh generates a frst testable restrton - namely that for eah (I, J), the determnant D IJ = p IJ, q IJ, 1 must be zero. If that restrton s satsfed, assume that ether p IJ or q IJ s not onstant over the ohorts. Then the vetors p IJ and 1 (or q IJ and 1) are lnearly ndependent, so that the lnear ombnaton n (30) s unque up to a ommon multplatve onstant. Sne, n our ase, the onstant s pnned down by the normalzaton σ 1 = 1, we onlude that for eah par (I, J), the regresson exatly dentfes σ I, µ J and Z IJ. Fnally, sne eah σ I but σ 1 (resp. eah µ J ) s dentfed from N (M) dfferent regressons, the model generates a seond set of overdentfyng restrtons. Fnally, a more parsmonous verson of the model obtans by mposng that the σs and the µs are dental aross lasses (.e., σ I = σ for all I and µ J = µ for all J), although these values may be dfferent between gender (.e., we do not mpose that σ = µ). Condton (30) s then strengthened: f we defne the vetors p, q and 1 IJ n R N M T by: p = ( p 11,..., p NM ), q = ( q 11,..., q NM ) and 1 IJ = (0,...0, 1,..., 1, 0,...0) then (keepng the normalzaton σ = 1): p = µ q + Z IJ 1 IJ (31) I,J Ths requres that (2 + NM) vetors be olnear n a spae of dmenson NMT, a strong restrton as soon as T 2; moreover, f ths property s satsfed, then µ and the Z IJ are dentfed. We onlude that whenever the populatons are not onstant aross ohorts, both the homoskedast and the heteroskedast versons of the benhmark strutural model are (vastly) overdentfed. 31

32 5.2.2 Extenson: ategory-spef drfts The prevous, overdentfaton result suggest that a more general verson of the model may atually be dentfable. We now proeed to show that ths s ndeed the ase. Spefally, we now relax the assumpton that the Z IJ are onstant aross ohorts; we therefore ntrodue ategory-spef drfts, whereby the Z IJ s vary aordng to: Z IJ = ζ I + ξ J + Z IJ (32) Ths s equvalent to assumng that, for all (I, J) and (K, L), the seond dfferene: Z IJ Z IL Z KJ + Z KL = Z IJ Z IL Z KJ + Z KL s ndependent of. Clearly, what we are assumng s therefore that the supermodularty propertes of the martal gans are onstant over tme. It s mportant to stress what ths extenson allows and what t rules out. Under (32), the benefts of marrage may evolve over tme (although the varanes do not); and these evolutons may be both gender- and eduaton- spef. In words, we allow, for nstane, the gans generated by marrage to derease less for an eduated women than for an unsklled man. However, the omponents refletng omplementarty (or supermodularty) between eduaton lasses - the seond dfferenes ( Z IJ Z IL) ( Z KJ Z KL) - are left nvarant. In partular, the fores drvng the assortatveness of the math are supposed to be onstant for the varous ohorts. Our hallenge s presely to test whether ths hypothess s ompatble wth the evolutons n martal patterns observed over the last deades. Normalzatons The form (32) requres addtonal normalzatons. We normalze ζ I 1 = ξ J 1 = 0 so that Z IJ = Z IJ 1. Also, note that for any > 1, the ζ I and ξ J are only defned up to a (ommon) addtve onstant;.e. for any gven salar k, one an replae ( ) ( ) ζ I, ξ J wth ζ I + k, ξ J k for all (I, J) wthout hangng (32). We an therefore normalze ξ 1 to be zero for all. Testng the framework Under (32), equaton (29) beomes: σ I p IJ + µ J q IJ = ζ I + ξ J + Z IJ I, J, (33) 32

33 Ths mples that for all I and all J 2, we have: σ I ( p IJ p I1 ) + µ J q IJ µ 1 q I1 = ξ J + Z IJ Z I1 (34) Computng ths expresson for I = 1 and dfferenng: σ I ( p IJ p I1 ) ( σ1 p 1J p 11 ) +µ J ( q IJ q 1J ) µ 1 ( q I1 q 11 ) = Z IJ Z I1 Z 1J +Z 11. Ths requres a normalzaton sne all terms an be multpled by the same fator. We ould hoose for nstane σ 1 = 1, so that p 1J p 11 = σ I ( p IJ p I1 ) +µ J ( q IJ (35) q 1J ) µ 1 ( q I1 q 11 ) ( Z IJ Z I1 Z 1J + Z 11) From ths, we derve a frst testable restrton. To smplfy notaton, denote D 2 Z IJ = Z IJ Z I1 Z 1J + Z 11 the seond dfferene of the mean surplus; and defne the vetors: Then for eah par (I > 1, J > 1): and R IJ P IJ = ( p IJ 1 p I1 1,..., p IJ T Q IJ = ( q1 IJ q1 1J,..., qt IJ p I1 ) T ) q 1J T R IJ = ( p 1J 1 p 11 1,..., p 1J T p 11 T and 1 = (1,..., 1) R IJ = σ I P IJ + µ J Q IJ µ 1 Q I1 D 2 Z IJ 1 (36) belongs to the subspae generated by { P IJ, Q IJ, Q I1, 1 }, a frst testable restrton for eah (I > 1, J > 1). A seond set of testable restrtons omes from the fat that when we deompose R IJ over the bass { P IJ, Q IJ, Q I1, 1 }, the oeffent of P IJ (resp. Q IJ,resp. Q I1 ) does not depend on J (resp. I, resp. s onstant). In prate, we frst estmate the probabltes of the varous martal outomes dretly from the data, and we use them to onstrut estmates of the vetors P, Q and R; then we hoose the heterogenety parameters ((σ I ), (µ J )) and the seond dfferenes ) 33

34 (D 2 Z IJ ) so as to mnmze the devatons from the ondtons n (36). Ths mnmum dstane estmaton tehnque also allows us to test the model by evaluatng the dstane funton at ts mnmum. In our applaton there are 116 ondtons n (36), and only 9 free parameters; ths s qute a strngent test sne the probabltes of the varous mathes are estmated from a large sample and thus very presely. One we have estmated the heterogenety parameters σ I and µ J we an also reonstrut the left-hand sde of equaton (33):  IJ = ˆσ I p IJ + ˆµ J q IJ. Our theory states that n an ANOVA regresson of ths ÂIJ, only 1- way and 2-way effets should appear. To put ths n terms more famlar to appled eonometrans: a regresson of ÂIJ on fxed effets for I, for J, and for (the 1-way effets) and on fxed effets for the nteratons (I, J), (I, ) and (J, ) (the 2-way effets) should have an R 2 of one. Ths s an alternatve way of evaluatng departures from the theory, based more on eonom sgnfane than on statstal sgnfane. Identfaton: the man result Fnally, should we fal to rejet, the model s dentfed. To see why, note that the deomposton of R IJ over { P IJ, Q IJ, Q I1, 1 } s generally unque; the σ I and µ J are therefore (over) dentfed as the respetve oeffents of the frst two vetors n the deomposton, and µ 1 as mnus the oeffent of the thrd. Rewrtng (33) for = 1 gves σ I p IJ 1 + µ J q IJ 1 = Z IJ whh shows that the Z IJ are dentfed. Last, applyng (33) dentfes ζ I for all I sne we set ξ 1 0; and (34) then dentfes ξ J for all J 2. A more parsmonous verson Comng bak to the parsmonous verson ntrodued above (σ I = σ for all I and µ J = µ for all J), ondton (35) beomes (wth the same notatons as above): σ (( p IJ p I1 ) ( p 1J p 11 )) ( ( +µ q IJ q 1J ) ( q I1 q 11 )) = Z IJ Z I1 Z 1J +Z 11 34

35 In ths ase, the omputaton of µ has a smple and ntutve nterpretaton. For any (I 1, J 1), let 2 a IJ denote the seond dfferene of the log probablty a IJ that a man n I marres a woman n J, takng for nstane the frst ategory as a benhmark for both genders: 2 a IJ = ln a IJ ln a I1 ln a 1J + ln a 11 Clearly, the use of suh seond dfferenes refers to the supermodularty propertes of the (log) probabltes. In partular, f ln a IJ then 2 a IJ Now, let 3 a IJ = 0 for all (I, J, ). We an smlarly defne 2 b IJ ln a IJ = s I + t J s addtvely separable: denote the varaton of ths seond dfferene over ohorts: 3 a IJ and 3 b IJ In other words, the rato 3 a IJ / 3 b IJ = 2 a IJ +1 2 a IJ 3 a IJ 3 b IJ for women. Then our model mples that: = µ σ should not depend on the lasses I and J nor on the ohort - and the rato µ/σ has then a natural nterpretaton n terms of mnus ths rato (remember that some normalzaton, say σ = 1, s stll needed). For nstane, the rato s lose to zero f the seond dfferene 2 vares muh less for men than for women. 8 Atually, more omplex models an n prnple be tested and estmated n ths framework. For nstane, one may assume a unform drft n the Zs but allow for ohort-spef varanes; the model would then beome: g j, = Z IJ + ζ + σ I α IJ, + µ J β IJ j, 8 Ths property ould n prnple be used to onstrut both a spefaton test and a non parametr estmator of the rato. In our data, however, the power of the test s qute weak, due to nsuffent varatons n the seond dfferene aross ohorts. 35

36 Agan, one an show that ths model () generates testable restrtons and () s dentfed up to smple normalzatons (a formal proof s avalable from the authors). 6 Results We estmate the Pr(J I, ) and Pr(I J, ) probabltes by the obvous nonparametr tehnque of ountng numbers of marrages n ells, assumng that a man of ohort marres a woman of ohort ( + 1) (the one-year gap s both the mode and the medan age dfferene at marrage.) We ran the analyss for ohorts of men born between 1943 to Then we reonsttute the p and q terms and we run the mnmum dstane proedure, takng I and J = 3 rather than 1 as referene, sne ategory 1 (hgh-shool dropouts) beomes less numerous over tme. We also found t more onvenent to normalze estmates usng the restrton Z 33 + Z 11 Z 13 Z 31 = 1, whh sales the onstant part of the jont surplus by makng the largest ross-dfferene term equal to one. Ths allows us to mantan the symmetry between men and women. Mnmum dstane estmaton amounts to hoosng the heterogenety parameters and the seond dfferene so as to mnmze the length of the resduals n (36). As usual, the optmal hoe of a norm s the nverse of the varane-ovarane matrx of the resduals. Sne we use 29 ohorts and we have three ategores, the vetor of resduals has dmenson 29 (3 1) (3 1) = 116, and ts varane-ovarane matrx s rather unweldy. To avod relyng too muh on mprese estmates of some off-dagonal elements of the varane matrx, we only used ts dagonal elements 9. Usng the full matrx does not materally alter our results. 9 Our estmator of these dagonal elements reles on a frst-step mnmum dstane estmator based on weghtng the resduals by the observed number of marrages. In omputng t, we neglet the orrelaton between the estmated P and Q. 36

37 6.1 Tests The hypotheses mpled by our model s very roundly rejeted. Whle ths sounds lke a dsappontng outome, the ANOVA proedure desrbed n seton gves muh more postve results. When we reonstruted the ÂIJ fator, we found that n the 2-way ANOVA regresson 10 the man effets were the 1-way effets on I and J (for a total of 46.2% of the varane), the 1-way ohort effet (for 13.8%), and the 2-way (I, J) effet (for 37.4%). As t turns out, the resdual, whh measures the devaton from our theory, aounts for only % of the varane of ÂIJ. Ths s a remarkably small number, sne the 3-way nteraton terms omprse 104 degrees of freedom, for = 261 observatons. As an llustraton, we generated randomly 1,000 samples of 261 d N(0, 1) varables; the 3-way nteraton aounts for 43% of the varane on average, wth a dsperson of 3%. These apparently dvergent results are a strkng llustraton of the dfferene between statstal sgnfane and eonom sgnfane. samples of men and women, the odds ratos p IJ and q IJ Sne we use rather large are very presely estmated, and any small devaton from the theory (the 0.5% of the varane above) results n a very large value of the test statst, and thus a spetaular statstal rejeton. Thus the statstal rejeton of our theory s a mnor dstraton, and we pursue our analyss of the 99.5% of the varane n marrage patterns that we manage to explan. 6.2 Estmated Heterogenetes Table 2 gves our estmates of the σ I and µ J terms. The model n Choo-Sow (2006) mposes that they all be equal; on the ontrary, we fnd lear and sgnfant varatons aross our estmates. In partular, eah estmated µ s larger than the orrespondng σ; and the hypothess that eah σ equals the orrespondng µ s strongly rejeted. There also appears to be muh less heterogenety among hgh-shool graduates than for the other two ategores; gven the dsusson of seton 4.2, ths wll play an mportant role n what follows. 10 We weghted eah (I, J, ) observaton by the orrespondng number of marrages n the data. 37