Partner Choice and the Marital College Premium

Transcription

1 Partner Choe and the Martal College Premum Perre-André Chappor Bernard Salané Yoram Wess September 6, 2012 Abstrat We onstrut a strutural model of household deson-makng and mathng and estmate the returns to shoolng wthn marrage. We onsder agents wth dosynrat preferenes for marrage that may be orrelated wth eduaton and allow the eduaton levels of spouses to nterat n produng surplus. Usng US data on marrages of ndvduals born between 1943 and 1972, we show that hanges n preferenes towards assortatve mathng are not neessary to explan hanges n mathng patterns. A onstant supermodularty of the surplus funton wth varable addtve shfts fts the data very losely. In lne wth theoretal predtons, we fnd that the martal ollege premum has nreased for women but not for men. We thank Gary Beker, Steve Levtt, Marelo Morera, Kevn Murphy and semnar audenes n Amsterdam, Buffalo, Chago, Delaware, CEMFI Madrd, LSE, Mlan (Boon), Pars, Penn State, Rand Santa Mona, San Dego, Toronto and Vanderblt for omments. Fnanal support from the NSF (Grant SES ) s gratefully aknowledged. Columba Unversty. Columba Unversty, orrespondng author. Emal: bsalane@olumba.edu. Tel Avv Unversty. 1

2 1 Introduton 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 nreased for both genders, although at a faster pae for men. Aordng to Clauda Goldn and Larry Katz (2008), male and female ollege attendane rates were about 10% for the generaton born n 1900, and reahed respetvely 55% and 50% for men and women born n Ths ommon trend, however, broke 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 have expeted ther ollege attendane rate to keep nreasng, possbly at a faster pae. Ths predton s satsfed for women: 70% of the generaton born n 75 attended ollege. On the ontrary, the male ollege attendane rate nreased at a muh slower rate, f at all. As a result, n reent ohorts women are more eduated than men, by an nreasng margn. To explan these strkngly asymmetr responses to seemngly dental nentves, Perre-André Chappor, Murat Iygun and Yoram Wess (2009, from now on CIW) 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 (see CIW for more detals). Seondly, eduaton has an mpat on a person s stuaton on the marrage market; t affets the probablty of gettng marred, the haratersts of the future spouse, and the sze and dstrbuton of the surplus generated wthn marrage. CIW advane the hypothess that, unlke the market ollege premum, ths martal ollege premum may have evolved n a hghly asymmetr way between genders. In ther paper, agents have heterogeneous atttudes towards marrage and heterogeneous osts of human aptal aquston; ther nvestment n eduaton s based on ther (ratonal) expetatons about the total (standard and martal) returns, whh n turns are determned at equlbrum by the dstrbuton 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 of eduaton by gender. Varous tehnologal hanges redung household hores, (as n Jeremy Greenwood et al., 2005) as well as progress n brth ontrol tehnologes, medal tehnques and nfant feedng methods (stressed by Robert Mhael, 2000, Goldn and Katz, 2002, and Stefana Albanes et al., 2009) an trgger a hange n equlbrum, leadng to martal returns to eduaton that are hgher for women than for men. Ths types of asymmetry an then generate dsrepanes n the demand for hgher eduaton of women and men. Whle ths argument s theoretally onsstent, establshng emprally ts relevane s a hallengng task. In ontrast to the returns to shoolng n the labor market, whh an be reovered from observed wages data, the returns to shoolng wthn marrage are not dretly observed and an only be estmated ndretly from the marrage patterns of ndvduals wth dfferent levels of shoolng. Our paper provdes the frst suh estmates. Spefally, we onsder a frtonless mathng framework wth Transferable Utlty (TU). The analyss of the marrage market as a mathng proess, whh dates bak to Gary Beker s semnal ontrbutons (see Beker 1973, 1974, 1991) has reently attrated renewed attenton 2. Jeremy Fox (2010a,b) provdes a nonparametr approah that does not expltly model the stohast struture of the jont surplus; nstead, t reles on a rank order property, whh postulates that assgnments that generate more surplus n a determnst model are more lkely to be observed when stohast aspets are ntrodued. In ontrast, our approah s expltly strutural, n the lne of the semnal ontrbuton by Eugene Choo and Aloysus Sow (2006). They use a spefaton of the stohast elements n the jont surplus that yelds a very smple nverson formula, from observed mathng patterns to the underlyng jont surplus funton. They appled ths approah to study the response of the US marrage market to the legalzaton of aborton; see Marstella Bottn and Sow (2008) and Sow (2009) for other applatons. Alfred Galhon and Bernard Salané (2012) generalze the Choo and Sow framework to arbtrary separable stohast dstrbutons; they also provde a theoretal and eonometr analyss of multrteron mathng under the same separablty assumpton. We extend the empral mathng lterature n three dretons. Frst, we larfy the 2 The reader s referred to Graham (2011) for a general presentaton. 3

4 underlyng theoretal struture needed for these approahes. We onsder a strutural model of mathng on the marrage market that s lose, n sprt, to that adopted by CIW. The model provdes an explt representaton of household behavor based on a olletve framework, wth ndvdual preferenes belongng to Theodore Bergstrom and Rhard Cornes s Generalzed Quas Lnear (GQL) famly (2003). Suh preferenes are neessary and suffent for a TU framework;.e., they admt a ardnal representaton n whh the Pareto fronter s a straght lne wth slope -1, and whose nterept s an nreasng onvex funton of the household s total nome. Agents math after hoosng ther eduaton level, but before ther permanent nome s revealed; they therefore onsder ther expeted surplus ondtonal on ther eduatonal level and that of potental partners. In addton, stll followng CIW, we assume that eah ndvdual has dosynrat preferenes for marrage, whh s known before nvestment n human aptal s deded. We work out the mplatons of ths framework for the key endogenous varables, namely ndvdual utltes at the stable math. The theoretal struture that we use reles on separablty between systemat omplementary trats and addtve random elements n the gans from marrage. Ths property that we all separablty s suffent to fully haraterze the stohast dstrbuton of the endogenous varables; moreover, the mathng equlbrum ondtons translate nto a smple dsrete hoe struture. Seond, we onsder a general eonometr spefaton that allows lass-spef dstrbutons of the random omponents - n ontrast to Choo and Sow (2006), who assumed that the dstrbutons of unobservable preferenes for marrage are dental aross eduaton lasses. Class-spef dstrbutons are learly requred by our theoretal bakground; ndeed, an mmedate onsequene of our model (and atually of CIW) s that seleton nto eduaton annot be ndependent from (unobservable) preferenes for marrage. We show that, n ths extended ontext, the martal ollege premum has a dret nterpretaton n terms of dfferenes n ex ante expeted utlty ondtonal on eduaton; moreover, our strutural approah stll leads to a losed form haraterzaton of these dfferenes. Our thrd ontrbuton s to extend the approah to a mult-market framework 3. We onsder several ohorts of men and women, whh ntrodues varaton n the proportons 3 See Fox (2010a, 2010b) and Fox and Yang (2012) for dfferent approahes to poolng data from many markets. 4

5 of men and women at all eduaton levels. We mpose a smple (and emprally testable) restrton, whh posts that preferenes for assortatve mathng reman onstant aross ohorts (although the surplus generated may vary, and possbly n an eduaton-spef manner). We show that the model s then vastly overdentfed, even wthout ndependene between preferenes for marrage and eduatonal hoes. In fat, one an dentfy a more general struture, n whh the systemat omponent of the surplus nvolves lass-spef temporal drfts; ths generalzed model stll generates strong overdentfaton restrtons. 4 In summary, our framework allows us to study the evoluton of mathng patterns throughout tme, allowng the gans from marrage and the ntra-household alloaton of these gans to evolve over tme n a lass-spef way. From ths nformaton, we an extrat the tme patterns of the martal eduaton premums of men and women. We apply our model to the US populaton, for the ohorts born between 1940 and We show that the martal ollege premum evolved non monotonally over the perod. Spefally, we fnd that n the begnnng of that perod (for ohorts born before the md-1950s), the premum 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. These fndngs are not based on a model of ndvdual demand for eduaton: our martal ollege premum s estmated exlusvely from the observed marrage patterns. Yet they losely ft the argument of CIW (seton II.C), whh s based on a theoretal analyss of nvestment n hgher eduaton. Ths buttresses ther lam that the nrease n the martal omponent of the eduaton premum for women ould explan the spetaular nrease n female demand for hgher eduaton. Our strutural approah allows us to break down the hanges n the martal ollege premum nto ther omponents. In partular, we dentfy the group spef pres (.e., dual varables of the mathng proess) 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 nrease n the number of eduated women relatve to 4 As s well known, the model of Choo and Sow s just dentfed wth ross-setonal data; allowng for eduaton-spef dstrbutons would therefore lead to serous (under) dentfaton problems n suh ontext. 5

6 eduated men. Ths happened beause the margnal ontrbuton of eduated women to the surplus wth eduated men has rsen over tme. Ths n turn s manly due to the varable omponent: eduated women beame more produtve (of surplus) relatve to less eduated women n all marrages, rrespetve of the type of the husband. In the end, eduated women have ganed relatve to uneduated women n three ways: by marryng at hgher rates and by reevng a hgher share of a larger martal surplus. Our fndngs are fully onsstent wth explanatons based on the lberatng effets of new tehnologes; n addton, we fnd that these effets vary onsderably wth the wfe s eduaton, but are more or less ndependent of the shoolng of the husband. We also fnd that the gans generated by marrages wth equally eduated partners have delned for all eduaton levels, refletng the general reduton n marrage over tme. However, the smallest delne s n mathes n whh 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.) * * * Our results also shed lght on the evoluton of marrage patterns between the 1960s and the 2000s. They have hanged n omplex ways. Overall, the perentage of ouples n whh both spouses have a ollege degree has sgnfantly nreased over the perod; however, as women wth ollege degree beame more abundant, the proporton of eduated women who marry eduated men has delned, as some eduated women had to marry downwards (wth less eduated men.) Many observers have nevertheless onluded that assortatve mathng on eduaton s stronger now than four deades ago 5 ; Burtless (1999) for nstane argues that ths evoluton omplements the nrease n the labor market ollege premum n explanng nreased nterhousehold nome nequalty. Part of the nrease n the proporton of ouples where both partners have smlar eduatons reflets the shfts n the eduaton of women; but some of t may also derve from hanges n preferenes towards assortatve mathng. An mportant advantage of our strutural approah s that t allows to formally dsentangle these two effets, notably by 5 One of the dffultes n measurng hanges n assortatve mathng s that none of the ndexes that are used n the lterature have a very onvnng foundaton. 6

7 provdng a strutural nterpretaton of the noton of preferenes for assortatveness as ndated by the supermodularty of the surplus funton that stems from the ndvdual preferenes under onsderaton. Our dentfyng assumpton s that preferenes for assortatveness dd not hange over ohorts; therefore, the qualty of the empral ft of our model s a dret test of the relevane of ths assumpton. Based on our fndngs, the onluson s lear-ut; the model fts the data remarkably well, ndatng that one an explan the hangng marrage patterns of the last deades wthout appealng to hanges n preferenes towards assortatve mathng. Ths fndng seems to ontradt results n the soologal lterature that argue 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, log-lnear 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 that s ommon n the soologal lterature. The results spurously suggest that preferenes for homogamy have hanged, even though our model rules out suh hanges. These fndngs demonstrate the mportane of a strutural approah to gude the nterpretaton of the empral results. * * * 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. Our empral fndngs are presented n Seton 7. 2 The Data We begn by desrbng our data and some stylzed fats about the evoluton of mathng by eduaton over the last deades n the US. 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 7

8 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) ths last ategory nludes anyone who attended ollege, whether they graduated or not 6. When studyng mathng patterns, we have to dede whh math to 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 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. 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 6 A fner lassfaton would be desrable, but ell szes shrnk fast. 8

9 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 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 are more lkely than men to attend 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 whle husbands used to marry down, husbands born after 1955 are more lkely to be marred to a wfe wth a hgher level of eduaton than thers 7. 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 ollege-eduated 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 marry at all. On the other hand, the marrage patterns of women are remarkably stable, as evdened n Fgure 4. 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 hgh-shool dropouts on the other hand have faed a very steep delne n marrage rates. 7 Note that these results are exatly n lne wth the exstng lterature (see for nstane Goldn and Katz (2008, p. 252), suggestng that the seleton nto our sample does not affet the man patterns under onsderaton. 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 Proporton Husband more eduated Same eduaton Husband less eduated Year of brth of husband Fgure 2: Relatve eduaton of partners 10

11 HSD HSG SC Born Born Man: SC Man: HSG Man: HSD Proporton Fgure 3: Marrage patterns of men who marry HSD HSG SC Born Born Woman: SC Woman: HSG Woman: HSD Proporton Fgure 4: Marrage patterns of women who marry 11

12 Proporton Hgh Shool Dropout Hgh Shool Graduate Some College Year of brth Fgure 5: Proporton of men who never marred Proporton Hgh Shool Dropout Hgh Shool Graduate Some College Year of brth Fgure 6: Proporton of women who never marred 12

13 3 Theoretal framework Our model derves from CIW. Consder an eonomy wth two perods and large numbers of men and women. In perod one, agents draw a ost of nvestment n human aptal and (a vetor of) martal preferenes from some random dstrbutons; then they nvest n eduaton by hoosng from a fnte set of possble eduatonal levels (and payng the orrespondng, person-spef ost). In perod 2, agents math on a frtonless marrage market wth transferable utlty; they eah reeve an nome, the realzaton of whh depends on the agent s eduaton; and they onsume, aordng to an alloaton of resoures that was part of the mathng agreement. When nvestng n human aptal, agents must antpate the outome of ther nvestment. Ths outome has two dstnt omponents. One s a larger future nome. In our framework, ths effet s taken to be exogenous, and t benefts the agents rrespetve of ther martal stuaton. Seond, a hgher eduatonal level has an mpat on martal prospets; t affets the probablty of gettng marred, the expeted nome of the future spouse, the total utlty generated wthn the household, and the ntra-ouple alloaton of ths utlty. These martal gans, however, depend on the equlbrum reahed on the marrage market; ths n turn depends on the dstrbuton of eduaton n the two populatons, and ultmately of the nvestment desons made n the frst perod. As usual, the model an be solved bakwards usng a ratonal expetatons assumpton; equlbrum s reahed when the martal gans resultng from gven dstrbutons of eduaton for men and women trgger frst perod nvestment desons that exatly generate these dstrbutons. Note that even f martal preferenes and nvestment ost were ndependent ex ante, eduaton desons made durng the frst perod must be orrelated wth preferenes for marrage ex post: agents wth stronger preferenes for marrage are more lkely to reeve the martal gan than agents who prefer to stay sngle, therefore have stronger nentves to nvest n eduaton. In the present paper, we am at estmatng and testng the seond perod behavor desrbed by ths model. Ths hoe s mostly dtated by avalable data: whle prvate osts of human aptal nvestment are not observable, the resultng dstrbuton of eduaton by gender s. In addton, onentratng on the seond perod allows to ntrodue a slghtly more general framework whle addressng the empral ontent of the key theoretal on- 13

14 ept: the noton of a martal ollege premum. We therefore onsder the stuaton at the begnnng of the seond perod. Agents are eah haraterzed by ther level of eduaton, whh belongs to some fnte set and s observable by all, and by ther preferenes for marrage, whh s observed by ther potental mates but not by the eonometran. One an then observe mathng patterns; the goal s to dentfy the underlyng struture, and n partular the martal gans assoated wth eah eduatonal level. 3.1 The model Preferenes The eonomy onssts of a male populaton M, endowed wth some ontnuous, atomless measure dµ M, and a female populaton F, endowed wth some ontnuous, atomless measure dµ F. Eah populaton s parttoned nto a fnte number of lasses, I = 1,..., I for men and J = 1,..., J for women, orrespondng to the varous eduaton levels avalable. The eonomy has (n + N) ommodtes, of whh n are prvately onsumed by eah ndvdual and N may be publly onsumed by a ouple. The preferenes over ommodtes of eah ndvdual are of the GQL form (Bergstrom and Cornes 1983): ( u (q, Q) = a q 1, Q ) + q 1 B (Q) where q = ( q 1,..., ) qn s the vetor of prvate onsumptons by agent, q 1 = ( q 2,..., ) qn, Q = ( Q 1,..., Q N) s the vetor of household s publ onsumpton, and a, B are nreasng funtons. Let us normalze all pres to 1 for smplty. If the ndvdual s sngle and has an nome x, she would hoose (q, Q) to solve max q,q a ( q 1, Q ) + q 1 B (Q) suh that n q l + l l=1 Q l = x. Denote V (x ) the value of ths program. In a ouple (, j), t s well-known that wth GQL preferenes, any Pareto effent onsumpton suh that q 1 q1 j S j (x + x j ) = max q,q j,q u (q, Q) + u j (q j, Q) s.t. > 0 must maxmze the sum of utltes. We therefore defne: 14 k ( ) q k + qj k + l Q l = x + x j.

15 If ommodtes are normal, S s nreasng and strtly onvex. 8 In partular, the seond ross dervatve of S j n x and x j s always postve: nomes are omplementary n the produton of the jont surplus S j (x + x j ) V (x ) V j (x j ). In addton to preferenes over ommodtes, eah ndvdual has martal preferenes whh we model by random vetors. For nstane, a woman j belongng to lass J has a vetor of martal preferenes b J j = ( b 1J j,..., b IJ ) j where b nj j denotes the extra utlty j derves from marryng a spouse wth an eduaton n, as opposed to stayng sngle. For notatonal onvenene, we defne b 0J j = 0 for all J. Smlarly, man s dosynrat martal preferenes are desrbed by the vetor a I = ( a I1,..., a IJ ) where I denotes s eduaton; as above, we defne a I0 = 0 for all I. Note that the dstrbuton of the a I and bj j vetors typally depend on eah person s eduaton (I and J respetvely), as ndated by the supersrpt; ths dependene reflets, among other thngs, the fat that the deson to nvest n eduaton was partly drven by martal preferenes. To reflet ths, we defne A IJ = E ( a IJ I ) and B IJ = E ( b IJ j j J ) ; (1) A IJ, for nstane, represents the average preferene for women of eduaton J of men of eduaton I, takng nto aount that these men onsdered ther marrage prospets when hoosng ther level of eduaton. Fnally, we let α IJ = a IJ A IJ and β IJ j = b IJ B IJ denote the wthn-eduaton varaton n martal preferenes Surplus funton The gan generated by the math of man, belongng to lass I, and woman j, belongng to lass J, therefore s the sum of two omponents. One s the expeted eonom gan 8 See for nstane Brownng, Chappor and Wess (2012, h.7.) 15

16 generated by jont onsumpton; the other onssts of (the sum of) the spouses dosynrat preferenes for marrage wth a spouse belongng to that partular lass. Sne ndvduals math after hoosng ther human aptal nvestment but before nome s realzed, the frst omponent s the expeted value of the surplus S j (x + y j ), ondtonal on the spouses levels of eduaton: The total gan s therefore: S IJ = E [S j (x + y j ) I, j J] gj IJ = S IJ + E ( a IJ + b IJ j I, j J ) = S IJ + A IJ + α IJ = G IJ + α IJ + β IJ j + B IJ + β IJ j where G IJ = S IJ + A IJ + B IJ and E ( α IJ ) ( ) = E β IJ j = 0. Alternatvely, and j may hoose to reman sngle; eah gan then s g I0 = E [V (x ) I] = G I0 and g 0J j = E [V j (x j ) j J] = G 0J As always, mathng patterns are drven by the surplus z j generated by the mathng of man and woman j, defned as the dfferene between the agents total gan when marred and the sum of ther ndvdual gans as sngles: z j = g IJ j = Z IJ + α IJ G I0 G 0J + β IJ j where Z IJ = G IJ G I0 G 0J The surplus of a sngle agent s normalzed to zero, by defnton: z 0 = z 0,j = 0 for all, j and Z I0 = Z 0J = 0 for all I, J The matrx Z = ( Z IJ) wll play a rual role n what follows. As we shall see, the equlbrum mathng wll depend on preferenes through the matrx Z and the dstrbuton 16

17 of the α s and β s. From the defntons above, Z IJ = E [S j (x + y j ) I, j J]+E ( a IJ V (x ) a I0 I ) +E ( b IJ j V j (x j ) b 0J j j J ) reflets the dstrbuton of nome and preferenes over ommodtes of spouses who hose eduaton levels I and J (and eah other), as well as the dstrbuton of ther martal preferenes. It s therefore a omplex objet; but t s the rual onstrut that determnes martal patterns n our ontext. Our goal s to hek under whh ondtons t s dentfable from mathng patterns. 3.2 Mathng A mathng onssts of () a measure dµ on the set M F, suh that the margnal of dµ 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 = z j for any (, j) Supp (dµ) 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 z j generated by ther math. The numbers u and v j are the expeted utltes man and woman j get on the marrage market, on top of ther utltes when they reman sngle; for any par that marres wth postve probablty, they must add up to the total surplus generated by the unon Stable 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 0, v j 0 and (2) u + v j z j for any (, j) M F. (3) 17

18 The two ondtons n (2) mples that marred agents would not prefer remanng sngle; the thrd (ondton (3)) translates the fat that for any possble math (, j), the realzed surplus z j annot exeed the sum of utltes respetvely reahed by and j n ther urrent stuaton (.e., a volaton of ths ondton would mply that and j ould both strtly nrease ther utlty by mathng together). As s well known, a stable mathng of ths type s equvalent to a maxmzaton problem; spefally, a math s stable f and only f t maxmzes total surplus, zdµ, over the set of measures whose margnal over M (resp. F) s dµ M (resp. dµ F ). A frst onsequene s that exstene s guaranteed under mld assumptons. Moreover, the dual of ths maxmzaton problem generates, for eah man (resp. woman j), a dual varable or shadow pre u (resp. v j ), and the dual onstrants these varables must satsfy are exatly (2): 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 (2). However, when the populatons beome large, the ntervals wthn whh u and v j may vary typally shrnk; n the lmt of ontnuous and atomless 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 A bas lemma From an eonom perspetve, our man nterest les n the dual varables u and v. Indeed, v j s the addtonal utlty provded to woman j by her equlbrum marrage outome. Whle ths value s ndvdual-spef (t depends on Mrs. j s preferenes for marrage), ts expeted value ondtonal of j havng reahed a gven level of eduaton J s dretly related to the martal premum assoated wth eduaton J (more on ths below). In our ontext, there exsts a smple and powerful haraterzaton of these dual varables; t s gven by the followng Lemma: Lemma 1 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 (4) 18

19 satsfyng the followng property: for any mathed ouple (, j) suh that I and j J, u = U IJ + α IJ and v j = V IJ + β IJ j (L) Proof. Assume that and both belong to I, and ther partners j and j both belong 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 + β IJ j (1) + β IJ j (2) + βij j (3) + βij j (4) β IJ j βij j v j v j β IJ j βij j hene v j v j = β IJ j βij j It follows that the dfferene v j β IJ j does not depend on j,.e.: The proof for u s dental. v j β IJ j = V IJ for all I, j J In words, Lemma 1 states that the dual utlty v j of woman j, belongng to lass J and marred wth a husband n eduaton lass I, s the sum of two terms. One s woman j s dosynrat preferene for a spouse wth eduaton I, β IJ j ; the seond term, V IJ, only depends on the spouses lasses, not on who they are. In terms of surplus dvson, therefore, the U IJ and V IJ denote how the determnst omponent of the surplus, Z IJ, 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 Lemma 1 s also vald for sngles f we set U I0 = Z I0 = 0 and V 0J = Z 0J = 0. 19

20 3.2.3 Stable mathng: a haraterzaton An mmedate onsequene of Lemma 1 s that the stable mathng has a smple haraterzaton n terms of ndvdual hoes: Proposton 2 A set of neessary and suffent ondtons for stablty s that 1. for any mathed ouple ( I, j J) one has and α IJ β IJ j α IK U IK U IJ for all K (5) α IJ U IJ (6) β KJ j V KJ V IJ for all K (7) β IJ j V IJ (8) 2. for any sngle man I one has 3. for any sngle woman j J one has α IJ U IJ for all J (9) β IJ j V IJ for all J (10) Proof. The proof s n several steps. Let ( I, j J) be a mathed ouple. Then: 1. Frst, man must better off than beng sngle, whh gves: U IJ + α IJ 0 hene α IJ U IJ and the same must hold wth woman j. Ths shows that (6), (8), (9) and (10) are neessary. 2. Take some woman 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 20 + β IJ j ( V IJ + β IJ j )

21 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 woman k n K J, urrently marred to some n I. Then s better off mathed wth j than k gves: U IJ + α IJ z k v k = z IK + α IK + β IK k ( V IK + β IK ) k whh s equvalent to α IJ α IK U IK U IJ and we have proved that the ondtons (5) are neessary. The proof s dental for (7). 4. We now show that these ondtons are suffent. Assume, ndeed, that they are satsfed. We want to show two propertes. Frst, take some woman 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 ) s a dret onsequene of (7) appled to l. Fnally, take some woman 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 (7) appled to k we have that: + β IK j ( V LK + β LK ) k β LK k β IK k V IK V LK and from (5) appled to : α IJ α IK U IK U IJ and the requred nequalty s just the sum of the prevous two. 21

22 Stablty thus readly translates nto a set of nequaltes n our framework; and eah of these nequaltes 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. Ths separaton s possble beause the endogenous fators U IJ and V IJ adjust to make the separate ndvdual hoes onsstent wth eah other. 3.3 Interpretaton The noton of surplus s rual n analyzng mathng patterns. It also has an mportant eonom nterpretaton that goes bak to the theoretal bakground provded by CIW. Labor eonoms defnes the ollege premum as the perentage nrease n expeted wage warranted by a ollege eduaton (as opposed to, say, a hgh shool dploma). Ths wage premum an readly be measured usng avalable data (and ontrollng for seleton nto ollege); exstng empral work suggests that, at the frst order, t s smlar for sngles and marred persons and for men and women (although, learly, the number of hours, and therefore the resultng gan n labor nome, may markedly dffer aross these populatons). The pont made by CIW s that, n addton to the standard wage premum, there exsts a martal ollege premum, whereby a ollege eduaton enhanes an ndvdual s martal prospets, nludng not only the probablty of beng marred and the expeted eduaton (or nome) of the spouse, but also the sze of the surplus generated and ts dvson wthn the ouple. In other words, t s well-understood that ollege eduaton provdes benefts to ndvduals n terms of hgher wages, better areer prospets, et. What we explore here s whether ollege-eduated ndvduals reeve addtonal benefts from ther eduaton on the marrage market); and whether ths an ontrbute to explan the observed asymmetry between men and women n terms of demand for eduaton. An obvous problem s that the martal ollege premum s qute dffult to estmate emprally, beause ntrahousehold alloaton annot be dretly observed. The man purpose of the present paper s to provde a methodology for suh estmaton. The rual remark, at ths pont, s that the notons prevously defned allow a lear defnton of the ollege premum. Indeed, the surplus s omputed as the dfferene between the total utlty generated wthn the ouple and the sum of ndvdual utltes of the spouses f sngle, thus 22

23 apturng exatly the addtonal gans from eduaton that only beneft marred people. Regardng ndvdual well-beng, 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 therefore of hs preferene vetor wthn the lass). Note, however, that ths value does not onde wth the average utlty of men n lass I who end up beng marred to women J at a stable mathng; the latter value s larger than U IJ, refletng the fat that an agent s hoe of hs spouse s lass s endogenous. Formally, the expeted surplus of an agent wth eduaton I s n fat: ū I = E max J=0,1,...,J ( U IJ + α IJ) where the expetaton s taken upon the dstrbuton of the preferene shok α. In partular, ths expeted surplus depends on the dstrbuton of the preferene shoks; t wll be omputed below under a spef assumpton on ths dstrbuton 9. Fnally, the dfferene ū I ū J denotes the dfferene n expeted surplus obtaned by reahng the eduaton level I nstead of J. It therefore represents exatly the martal premum generated by that hange n eduaton level that s, the gan that arues to marred people, n addton to the benefts reeved by sngles. 4 Empral mplementaton 4.1 Probabltes Havng transformed the problem nto a standard dsrete hoe problem, t s natural to make the followng assumpton 10 : 9 Wth a ontnuum of agents, whle the α s and β s are random, the U IJ and V IJ are not. 10 Ths dstrbuton s also referred to as type-i extreme value dstrbuton. It was frst ntrodued nto eonoms by Danel MFadden (1973) to deal wth statstal nferene on a model of ndvdual hoe behavor from data obtaned by samplng from a populaton and where the ndvdual hoe depends on unobserved haratersts. spefaton n marrage market analyss. Dagsvk (200) and Choo and Sow (2006) were the frst to apply ths 23

24 Assumpton 1 HG (Heteroskedast Gumbel): The random terms α and β are suh that where the α IJ and the Euler onstant. α IJ = σ I. α IJ β IJ = µ J. β IJ β IJ j follow ndependent Gumbel dstrbutons G ( k, 1), wth k β IJ j In partular, the α IJ and β IJ j have mean zero and varane π2 6, therefore the αij ( σ I ) 2 ( and π 2 6 µ J ) 2. have mean zero and respetve varane π2 6 A dret onsequene of Proposton 2 s that, for any I and any I: and and γ IJ Pr ( mathed wth a woman n J) exp ( U IJ /σ I) = K exp (U IK /σ I ) + 1 γ I0 Pr ( sngle) = 1 K exp (U IK /σ I ) + 1 where we normalze U I0 to 0. Smlarly, for any J and any woman j J: δ IJ P (j mathed wth a man n I) (11) exp ( V IJ /µ J) = K exp (V KJ /µ J ) + exp (V 0J /µ J and (12) ) δ 0J P (j sngle) = 1 K exp (V KJ /µ J ) + 1 where V 0J = 0. These formulas an be nverted to gve: exp ( U IJ /σ I) = γ IJ 1 K γik (13) and exp ( V IJ /µ J) = δ IJ 1 δ KJ (14) 24

25 therefore: ( U IJ = σ I γ IJ ln 1 K γik ( V IJ = µ J ln δ IJ 1 δ KJ ) ) In what follows, we assume that there are sngles n eah lass: γ I0 > 0 and δ 0J > 0 for eah I, J, mplyng that K γik < 1 and K δkj < 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, we an readly ompute the lass-spef expeted utltes desrbed above: [ ū I ( = E max U IJ + σ I α IJ ) ] J 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: [ 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 ( γ I0) (15) J and smlarly ( v J = µ J ln exp ( ) V IJ /µ J) + 1 = µ J ln ( δ 0J) (16) I 4.2 Heteroskedastty: a short dsusson An mportant property of the model just presented s heteroskedastty: the varane of the unobserved heterogenety parameters s lass-spef. Ths s a key dfferene wth the framework adopted by Choo and Sow (2006), n whh shoks are assumed homoskedast. In our ontext, heteroskedastty s a dret onsequene of the theoretal bakground desrbed above. An agent s nvestment n human aptal depends, among other thngs, of ther martal preferenes; even f the dstrbuton of preferenes ex ante (.e. before the agent hooses an eduaton level) s d, the ondtonal dstrbuton gven the hosen 25

26 level of eduaton wll depend on ths level, beause of the seleton operated on these preferenes. Heteroskedastty, n turn, has several mplatons. Frst, the expeted utlty of an arbtrary agent n lass I, as gven by (15), 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 maxmum 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 ( γ I0) term). Ths remark, n turn, has mportant onsequenes for measurng the martal ollege premum. To see that, 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 (15) then mples that ( ) γ 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 and remans sngle. The hgher the mean, the smaller s the proporton (see Fgure 7). For nstane, f one sees that ollege graduates are more lkely to reman sngle than hgh shool graduates (γ I0 > γ K0, mplyng that ln ( γ K0 /γ I0) < 0), we would then 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. γ I0 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 t may also be that the varane s larger (even wth a hgher mean), as llustrated n Fgure 8. The one-to-one relatonshp no longer holds and a hgher perentage does not neessarly 26

27 Fgure 7: Homoskedast random gans mply a smaller mean. One has to ompute the respetve varanes whh, n turn, may affet the omputaton of the martal ollege premum. Spefally, we now have that: ū I ū K = σ K ln ( γ K0) σ I ln ( γ I0) (17) If γ I0 > γ 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. Fgure 8: Heteroskedast random gans Generally, eduaton nfluenes martal prospets through four dfferent hannels: t nreases marrage probabltes; t hanges the potental qualty (here eduaton) of the future spouses; and t affets the sze and the dstrbuton of surplus wthn the household. In the speal homoskedast verson of the model, these three hannels are ntrnsally mxed, and the expeted utlty of eah spouse s fully determned by the perentage of 27

28 persons n the same eduaton lass that remans sngle. The heteroskedast verson s muh rher; welfare mpats go beyond the sole probablty of marrage, and nvolve other onsderatons. Consequently, the onlusons drawn from the model 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. 11 There are of ourse many ways to go beyond the orgnal model of Choo and Sow (2006). Heteroskedastty s qute natural n our framework: eduaton s partly determned by martal preferenes, so that the dstrbuton of martal preferenes depends on eduaton n a omplex way. Ther ondtonal expetaton s subsumed n the matrx Z; but even f all α s and β s were ex ante..d, no reasonable assumpton would make ther ondtonal varanes onstant aross eduaton levels. A more omplex statstal model would also allow for orrelaton aross the ondtonal dstrbutons of martal preferene shoks; the framework of Galhon and Salané (2012) shows how suh models an be analyzed and estmated, provded that the data s rh enough and/or enough dentfyng assumptons are mposed to reover the prmtves of the model. 4.3 Extenson: Covarates The bas framework just desrbed an be extended to the presene of ovarates;.e., we may spefy the α s and β s as funtons of observed ndvdual haratersts (other than ther lass). 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, we dsregard heteroskedastty): α IJ = X.ζ IJ m + α IJ 11 Note however from equatons (15) and (16) that f the varanes are assumed to be 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. 28

29 where ζ IJ m, ζ IJ f ζ 0J f = 0, and where as above the α IJ dstrbutons G ( k, 1). I: 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 are vetor parameters, wth the normalzaton U I0 = ζ I0 m = 0 and V 0J = IJ (resp. β j ) follow ndependent, type 1 extreme values Then the omputatons are as above, and we an estmate for γ IJ exp ( U IJ + X.ζ IJ ) m = Pr ( mathed wth a woman n J) = K exp ( U IK + X.ζ IK ) ( m + exp U I0 + X.ζ I0 ) m γ I0 exp ( U I0 + X.ζ I0 ) m = Pr ( sngle) = K exp ( U IK + X.ζ IK ) ( m + exp U I0 + X.ζ I0 ) m and the onlusons follow. In partular, these models an be estmated runnng standard (multnomal) logts. 5 Identfaton We now onsder the dentfaton problem. In prate, we observe realzed mathng.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 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. 29

30 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 where the α IJ and β IJ j z j = Z IJ + σ I α IJ + µ J βij j 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 3 Assume that a model M = ( Z IJ, σ I, µ J) generates some mathng probabltes ( γ IJ, δ IJ), and let U IJ, V IJ denote the orrespondng dual varables. Then U IJ = σ I log γ IJ 1 K γik (18) and therefore Z IJ = σ I log V IJ = µ J log γ IJ 1 K γik + µj log δ IJ 1 K δkj (19) δ IJ 1 K δkj Moreover, for any ( σ I, µ J) R +, the model N = ( ZIJ, σ I, µ J) where Z IJ = σi σ I U IJ + µj µ J V IJ (20) generates the same mathng probabltes, and the orrespondng, dual varables are Ū IJ = σi σ I U IJ (21) V IJ = µj µ J V IJ (22) 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 (20), (21) and (22) must hold. Proof. From the prevous alulatons, there s a one-to-one relatonshp between the γ IJ and the υ IJ ; the result follows. 30

31 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 men and women 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 marrage market outomes 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 onsder a stable heteroskedast strutural model M = ( Z IJ, σ I, µ J) that holds for dfferent ohorts of agents, = 1,..., T, wth varyng lass ompostons. The bas struture beomes: z j, = Z IJ + σ I α IJ, + µ J β IJ j, Also, assume for the tme beng that eah man marres a woman wthn hs ohort. 12 The model then defnes, for eah ohort, a mathng problem assoated wth shadow pres that are ohort spef, leadng to the defnton of U IJ γ IJ = Pr ( I mathed wth a woman n J n ohort ) = and V IJ. Then exp ( U IJ ) /σ I 1 + K exp (U KJ /σ K ) 12 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 ( + 1) a fat that wll be taken nto aount n the empral applaton, but an be gnored for the tme beng. 31

32 γ I0 = Pr ( I sngle) = therefore and smlarly: K exp (U KJ /σ I ) exp ( U IJ ) γ IJ /σ I = 1 K γik δ IJ = Pr (j J mathed wth a woman n I n ohort ) = δ I0 = Pr (j J sngle) = mplyng that Moreover, we have K exp (V IK /µ K ) exp ( V IJ ) δ IJ /µ J = 1 K δik (23) exp ( V IJ ) /µ J 1 + K exp (V IK /µ K ) (24) U IJ + V IJ = Z IJ (25) Now, let p IJ (24), the p IJ = U IJ /σ I and q IJ and q IJ = V IJ /µ J. The rual remark s that from (23) and are dretly observable from the data. It follows that (25) has a dret, testable mplaton. Indeed, defne the vetors: p IJ = ( p IJ 1,..., p IJ ) T q IJ = ( q1 IJ,..., qt IJ ) 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 (26) If T 3, ths generates a frst testable restrton namely that for eah (I, J), any 3 3 determnant extrated from the matrx M IJ = ( p IJ, q IJ, 1 ) must be zero. 32