A Method to Estimate the Human Capital from Sample Surveys on ...

A <strong>Method</strong> for The Estimation of The Distribution of<strong>Human</strong> <strong>Capital</strong> <strong>from</strong> <strong>Sample</strong> <strong>Surveys</strong> on Income andWealthGiorgio Vittadini, University of Bicocca Milan, Italy Camilo Dagum, University of Bologna, Italy and University ofOttawa, Canada, Pietro Giorgio Lovaglio, University of Bicocca-Milan, Italy, Michele Costa, University of Bologna,Italy.Giorgio Vittadini, Department of Statistics, University of Bicocca-Milan,Via Bicocca degli Arcimboldi 8, 20126MILAN, ITALY.KEY WORDS: <strong>Human</strong> <strong>Capital</strong>,Latent Variable, Pls, Lisrel,Optimal ScalingIntroductionThis paper proposes a method for estimating <strong>the</strong> 1983U.S. household distribution of <strong>Human</strong> <strong>Capital</strong>. From<strong>the</strong> statistical point of view, <strong>the</strong> HC is defined as aLatent Variable measured by a set of observed mixedindica<strong>to</strong>rs in a Path Analysis Model. The HC estimatesconsider <strong>the</strong> definitions advanced for a Latent Variablein a Path Analysis with respect <strong>to</strong> formative andreflective indica<strong>to</strong>rs.The set of indica<strong>to</strong>rs and <strong>the</strong>ir links with HCThe concept of <strong>Human</strong> <strong>Capital</strong> (HC), <strong>the</strong>oretically andsystematically developed over <strong>the</strong> last 50 years (Mincer1958, 1970; Becker 1962, 1964 and Schultz 1959,1961) has been estimated in literature by ei<strong>the</strong>r <strong>the</strong>retrospective (Kendrick 1976; Eisner, 1985) orprospective methods (Jorgenson and Fraumeni 1989).The first, dealing with <strong>the</strong> cost of production, isinsufficient for various reasons, because it does nottake in<strong>to</strong> account <strong>the</strong> social costs, such as publicinvestment in education, <strong>the</strong> variables concerning homeconditions and community environments, and <strong>the</strong>genetic contribution <strong>to</strong> HC, including health conditions(Dagum and Vittadini 1996). Moreover, <strong>the</strong> actualeffects of <strong>the</strong> investment in HC on <strong>the</strong> income andwealth of <strong>the</strong> households are not considered.In <strong>the</strong> prospective method <strong>the</strong> HC can be defined as <strong>the</strong>present actuarial value of an individual’s expectedincome related <strong>to</strong> his skill, acquired abilities, andeducation (Dagum and Slottje 2000). However, <strong>the</strong>prospective method reduces <strong>the</strong> HC investment <strong>to</strong> itsmonetary value in terms of an assumed flow of income,and it ignores <strong>the</strong> amount of investment in education,job training and o<strong>the</strong>r investments. It is also difficult <strong>to</strong>predict future income.We now present a new methodology <strong>to</strong> estimate <strong>the</strong>distribution of HC in families, giving greater emphasis<strong>to</strong> economic issues because <strong>the</strong> definition of HCinvolves both its investment amounts on families andits effect on income. In this case, instead of quantitativefinancial indica<strong>to</strong>rs, we have a composite set ofqualitative and quantitative indica<strong>to</strong>rs (Table 1) with<strong>the</strong> Path Analysis diagram showing <strong>the</strong>ir causal links.The ”indirect” set of indica<strong>to</strong>rs Γ =(x 2, x 3, x 6, y 8, y 9, y 10,y 11, y 12, y 13 ) involved in <strong>the</strong> Path Analysis is composedof a set of causal links between <strong>the</strong>mselves and a set ofindica<strong>to</strong>rs [Ψ=(x 1, x 4, x 5, x 7, y 1, y 2, y 3, y 4, y 5, y 6, y 7, y 15 )]and y 14 (<strong>to</strong>tal wealth), which measure <strong>the</strong> investment ineducation and are directly connected with HC in (2).As proposed in statistical literature (Tenenhaus, 1995),<strong>the</strong>se indica<strong>to</strong>rs can be defined as formative indica<strong>to</strong>rsbecause <strong>the</strong>y “form” or “cause” <strong>the</strong> multidimensionalconstruct HC in equation (2). HC is a “consequence”of <strong>the</strong> investment in education. It is measured by a se<strong>to</strong>f formative indica<strong>to</strong>rs F = (y 14 ,Ψ)The most important formative indica<strong>to</strong>rs are years ofeducation, years of full time and part time employment,and wealth. Marital status, gender, region and age areinvolved as well, because <strong>the</strong> actual value ofinvestment in HC is influenced by <strong>the</strong>se personal andenvironmental conditions.The effects on income of <strong>the</strong> households HC andwealth are presented in (3). Therefore y 17 can beclassified as reflective indica<strong>to</strong>r, because it reflects HC,in <strong>the</strong> sense that it is a consequence of <strong>the</strong> amount andtype of <strong>the</strong> investment in education. Wealth (y 14 ) isboth a formative indica<strong>to</strong>r and an independent cause ofincome.The econometric specification and analysis was madeby Dagum (1994) and Dagum et al. (2003).

Indirect Indica<strong>to</strong>rs Γx 2 =H Gender; x 3 = H Race; x 6 = S Age; y 5 = H Yearsof Not Full-Time Work; y 7 = S Years of Not Full-Time Work; y 8 = H Job Status; y 9 = H Occupation;y 10 = H Industry; y 11 = S Job Status; y 12 = S Occupationy 13 = S IndustryFormative indica<strong>to</strong>rs F = (Ψ, y 14 )Ψ: x 1 = H Age; x 4 = Region ;x 5 = H Marital Status;x 7 = S Gender; y 1 = H Years of Schooling;y 2 = S Years of Schooling; y 3 = Number of Children;y 4 = H Years of Full-Time Work;y 6 = S Years of Full-Time Work; y 14 = HouseholdTotal Wealth; y 15 = Household Total Debts.Reflective indica<strong>to</strong>ry 17 = Household IncomeH: Household Head;S: SpouseTable 1 Observed indica<strong>to</strong>rsThe statistical definition of <strong>the</strong> LV HCWe have already stated (Dagum and Vittadini 1996)that, <strong>from</strong> <strong>the</strong> statistical point of view, HC can beexpressed as an LV. But <strong>the</strong>re are different ways an LVcan be defined. Traditionally, a variable can be definedas an LV if <strong>the</strong> equations cannot be manipulated in<strong>to</strong>expressing <strong>the</strong> variable as a function of manifestvariables (Bentler 1982). In o<strong>the</strong>r words, in thisdefinition, an LV is a fac<strong>to</strong>r that underlies and causesreflective indica<strong>to</strong>rs and accounts for <strong>the</strong>ir observedvariance in a measurement model (typically <strong>the</strong> fac<strong>to</strong>rmodel) given <strong>the</strong> effects of o<strong>the</strong>r explicative indica<strong>to</strong>rs(in this case <strong>the</strong> reflective indica<strong>to</strong>r Income, given <strong>the</strong>effect of <strong>the</strong> explicative indica<strong>to</strong>r wealth in equation(3)). O<strong>the</strong>rwise we can define HC as a latent variablecaused and measured (with errors) by a linearcombination of <strong>the</strong> formative indica<strong>to</strong>rs F in equation(2). Finally we can propose a third, more complete,definition of an LV, as in this case where it isconnected with both formative and reflectiveindica<strong>to</strong>rs in a Path Diagram. Hence <strong>the</strong> latent variableHC can be defined as a linear combination offormative indica<strong>to</strong>rs F that best fits <strong>the</strong> reflectiveindica<strong>to</strong>r earning income, as in equations (2)-(3).The proposed methodologyThis approach completes <strong>the</strong> methodology proposed byDagum and Slottje (2000) where <strong>the</strong>y combine azerodimensional latent variable approach (part A) andan actuarial ma<strong>the</strong>matical approach (part B).The Latent Variable approach proposes a newmethodology able <strong>to</strong> obtain <strong>the</strong> zerodimensional HClatent variable, <strong>the</strong>n transforms <strong>the</strong> estimated latentvariable in<strong>to</strong> an accounting monetary value, and finallyestimates <strong>the</strong> mean value of HC. The Path Analysisand <strong>the</strong> Latent Variable Approach are shown inFigure1.The Actuarial Ma<strong>the</strong>matical approach starts with <strong>the</strong>actuarial estimation, in monetary values, of <strong>the</strong> averagehuman capital by age of economic units and finallyestimates <strong>the</strong> average of <strong>the</strong> population in monetaryunits. The syn<strong>the</strong>sis gives <strong>the</strong> final HC estimation anddistribution of American Household.INDIRECTINDICATORS ΓINDICATORS OFHOUSEHOLD INVESTMENT INEDUCATION:FORMATIVE INDICATORS ΨH S YEARS OF SCHOOLING;H S YEARS OF TOTAL TIME WORK; HAGE;REGION; H MARITAL STATUS;S GENDER; NUMBER OF CHILDRENHCWEALTH y 14LATENT VARIABLE HUMAN CAPITALFigureINDICATOR OF EFFECTS OFHC: REFLECTIVEINDICATOR INCOME y 17Figure 1: Path Analysis and Latent Variablesapproachy 1 = g 1 (x 1 , x 3 , x 4 , x 5 ) + u 1y 2 = g 2 (x 1 , x 2 , x 3 , x 4 , x 5 , y 1 ) + u 2y 3 = g 3 (x 1 , x 3 , x 4 , x 5 , y 1 ) + u 3y 4 = g 4 (x 1 , x 2 , x 3 , x 4 , x 5 , y 2 , y 3 ) + u 4y 5 = g 5 (x 1 , x 2 , x 3 , x 4 , x 5 , y 1 , y 4 ) + u 5y 6 = g 6 (x 2 , x 3 , x 4 , x 5 , x 6 , y 2 , y 3 , y 4 ) + u 6y 7 = g 7 (x 2 , x 4 , x 5 , x 6 , y 2 , y 5 , y 6 )+ u 7y 8 = g 8 (x 1 , x 2 , x 3 , x 4 , x 5 , y 1 , y 3 , y 4 ) + u 8 (1)y 9 = g 9 (x 1 , x 2 , x 3 , y 8 ) + u 9(1)y 10 = g 10 (x 2 , x 3 , x 4 , y 1 , y 4 , y 5 , y 9 ) + u 10y 11 = g 11 (x 1 , x 2 , x 4 , x 5 , x 6 , y 2 , y 3 , y 6 , y 9 ) + u 11y 12 = g 12 (x 1 , x 2 , x 4 , x 5 , x 6 , y 2 , y 3 , y 9 , y 11 ) + u 12y 13 = g 13 (x 3 , x 4 , y 6 , y 12 ) + u 13y 14 = g 14 (x 4 , y 1 , y 2 , y 4 , y 7 , y 8 , y 9 , y 10 , y 11 , y 12 , y 13 )+ u 14y 15 = g 15 (x 1, x 3 , x 4 , y 1 , y 2 , y 3 , y 4 , y 9 , y 10 , y 12 , y 14 ) + u 15HC = Fg = [y 14 ,Ψ] g + u 16 (2)

y 17 = y 14 k 1 + HC k 2 + u 17 (3)The Latent Variable Approach: previous proposalThe traditional proposal in statistical literature is <strong>to</strong>obtain <strong>the</strong> latent variable HC as a latent cause whichunderlies observed indica<strong>to</strong>rs by means of Fac<strong>to</strong>rAnalysis . In this case starting <strong>from</strong> (3), we obtain:Q y14y 17 = HC k 2 # + u 17 (4)where Q y14=I– Py 14is <strong>the</strong> orthogonal complement of<strong>the</strong> column spaces of y14 with Py 14= y14 (y 14 ′ y 14 ) -1 y 14 ′.By means of Fac<strong>to</strong>r Analysis we obtain HC as <strong>the</strong>latent cause of <strong>the</strong> reflective indica<strong>to</strong>r earning income.First of all, in this way we define <strong>the</strong> HC withouttaking in<strong>to</strong> account <strong>the</strong> amount of investment ineducation measured by <strong>the</strong> formative indica<strong>to</strong>rs F.Secondly, under general conditions, given earningincome Wealth Q y14y17, <strong>the</strong> parameter k # 2 is notidentified and <strong>the</strong> scores of <strong>the</strong> latent variable HC arenot unique. In a Fac<strong>to</strong>rial or in a Structural model when<strong>the</strong> expected values of latent variables are null, <strong>the</strong>identification problem is essentially whe<strong>the</strong>r or notvec<strong>to</strong>r ϑ of parameters and of variances andcovariances of latent variables and errors is uniquelydetermined by <strong>the</strong> covariance matrix Σ of indica<strong>to</strong>rswhose elements are σ ij . In o<strong>the</strong>r words if a vec<strong>to</strong>r ϑ canbe uniquely determined <strong>from</strong> Σ ( and <strong>the</strong>refore if Σ isgenerated by one and only one vec<strong>to</strong>r ϑ) <strong>the</strong>n solving<strong>the</strong> equations σ ij = σ ij (ϑ), i ≤ j (with p manifestvariables, <strong>the</strong>re are ½ p(p + 1) equations in n(θ)unknown parameters), or a subset of <strong>the</strong>m, this vec<strong>to</strong>rof parameter is identified and <strong>the</strong> whole model is said<strong>to</strong> be identified; o<strong>the</strong>rwise it is not. Anderson andRubin showed that a necessary condition foridentification is that <strong>the</strong> number of equations σ ij =σ ij (ϑ), i ≤ j must be greater than <strong>the</strong> order of <strong>the</strong> vec<strong>to</strong>rϑ: p ≥ 2t n(θ)+1. However, since <strong>the</strong> equations aboveare often non-linear, <strong>the</strong> solution is often complicatedand tedious, and explicit solutions for all ϑ’s seldomexist. “No general and practically useful necessary andsufficient conditions for identification are available”(Everitt 1984).If a model is not completely identified, appropriaterestrictions may be imposed on ϑ <strong>to</strong> make itidentifiable. The choice of restrictions may affect <strong>the</strong>interpretation of <strong>the</strong> results of an estimated model.Under general conditions for <strong>the</strong> Fac<strong>to</strong>r Model, if wedo not consider a few very restricted cases in whichconditions for identifiability are studied analytically,e.g. where <strong>the</strong> endogenous variables are measuredwithout error (Geraci 1976), <strong>the</strong> problem cannot beresolved. In practice, it is suggested (Jöreskog, 1981b)that “The identification problem can be studied on acase by case basis by examining <strong>the</strong> equations”,choosing <strong>the</strong> restriction, not only in number but also inposition, in order <strong>to</strong> obtain unique solutions. This isalso true in <strong>the</strong> case of local identifiability of <strong>the</strong>parameters (Wegge 1965, 1991 Fisher 1976,Ro<strong>the</strong>nberg 1971, Geraci 1976, Bekker and Pollock1986, Shapiro 1985, Bekker 1989, 1991, Wegge andFeldman, 1983).In our case, we have one equation σ ij = σ ij (ϑ):σ = (k 2 # ) 2 + σ u17 (5)y Q y1417With two unknown values, <strong>the</strong> square of <strong>the</strong>#parameter k 2 (k # 2 ) 2 and <strong>the</strong> variance of <strong>the</strong> error u 17(σ u17 ). Therefore, under general conditions, when <strong>the</strong>Reliability Ratio between σ and (k2 # ) 2 isQ y14 y17unknown or <strong>the</strong> variance of <strong>the</strong> error σ u17 orInstrumental Variables are not available, <strong>the</strong> model (4)is not identifiable (Fuller, 1987).Regarding <strong>the</strong> problem of indeterminacy we can verifythat, under general conditions, <strong>the</strong> matrix of observedindica<strong>to</strong>rs is less than <strong>the</strong> matrix of latent scores anderrors. Therefore, it can be demonstrated that even if<strong>the</strong> model is identified <strong>the</strong> latent scores areindeterminate. There are infinite sets of latent scoresfor <strong>the</strong> same identified model. It can be proved thatsome of <strong>the</strong>m can be ei<strong>the</strong>r negatively correlated <strong>to</strong>each o<strong>the</strong>r (Reiersol 1950; Guttmann 1955; Andersonand Rubin 1956; Lawley and Maxwell 1963; Joreskog1967; Schonemann and Wang 1972; Schonemann andSteiger 1978; Steiger 1979; Schonemann and Haagen1987). In this case, given y17 and k # 2 , we canQ y14obtain infinite set of scores of HC; moreover some of<strong>the</strong>m can be negatively correlated.An alternative proposal is given by <strong>the</strong> Partial LeastSquares <strong>Method</strong> (<strong>from</strong> here on referred <strong>to</strong> as PLS):PLS provides estimates of parameters g in (2) definingand estimating an LV “by deliberate approximation asa linear aggregate of its observed indica<strong>to</strong>rs” (Wold1982). In this definition <strong>the</strong> HC appearing in (2) is nota fac<strong>to</strong>r of <strong>the</strong> observed reflective indica<strong>to</strong>rs (3) but anunobserved <strong>the</strong>oretical construct, approximated by alinear combination of observed formative indica<strong>to</strong>rs,e.g. following equation (2):H Ĉ = F ĝ(6)where HĈ is <strong>the</strong> proxy obtained by reducing <strong>the</strong> loss ofinformation with respect <strong>to</strong> <strong>the</strong> unobservable HC.There are two alternatives for obtaining <strong>the</strong> solutions ofHĈ in (6) by means of <strong>the</strong> PLS. The PLS mode A isbased on iterative multivariate regressions of <strong>the</strong> LV’son <strong>the</strong> observed indica<strong>to</strong>rs; <strong>the</strong>refore, if <strong>the</strong>re is a singleLV, it cannot be used, because it causes “circularsolutions” without improvements in <strong>the</strong> iterations. ThePLS mode B is based on simple iterative regressions on<strong>the</strong> observed indica<strong>to</strong>rs F=(y 14 ; Ψ). It can be provedthat <strong>the</strong> estimate of HĈ is equivalent <strong>to</strong> <strong>the</strong> firstprincipal component of F (Wold 1982). Therefore wehave in (6):HĈ = Fv 1 = y 14 v 11 + Ψ v 12 (7)

where F =(y 14 , Ψ) and v 1 ′ =(v 11 , v 12 )′ is <strong>the</strong> firsteigenvec<strong>to</strong>r of F′F, HĈ is <strong>the</strong> first principal componen<strong>to</strong>f F after its standardization <strong>to</strong> unit variance(Var(HĈ)=1), v 11 contains <strong>the</strong> element of <strong>the</strong> firsteigenvec<strong>to</strong>r connected with y 14 , v 12 is <strong>the</strong> sub-vec<strong>to</strong>r ofv 1 connected with Ψ.First of all, <strong>the</strong> estimate HĈ of HC does not take in<strong>to</strong>account <strong>the</strong> actual effects of <strong>the</strong> investment in HC on<strong>the</strong> income and wealth of <strong>the</strong> households.Secondly, also <strong>from</strong> <strong>the</strong> statistical point of view, <strong>the</strong>reare some general critique about <strong>the</strong> solutions obtainedby means of PLS (Garthwaite 1994).In this case, in particular, every solution that can beobtained in (3) starting <strong>from</strong> (7) is logicallyinconsistent. In effect, substituting HC^ obtained by(7) in (4) we obtain:and <strong>from</strong> (7) we have:Q y14y 17 = HĈ k 2 # + u 17 (8)k # 2 = HC^′ Qy y17 =(y 14 v 11 + Ψ v 12 )′ 14Q y14y17=v 12 ′ Ψ ′ Qy y 17 (9)14#<strong>from</strong> which k 2 cannot consider <strong>the</strong> whole HCcontribution <strong>to</strong> earned income y17 because, byQ y14definition, <strong>the</strong> indirect contribution of Wealth onIncome y 17 by means of HĈ is null.However, if we consider equation (3) where <strong>the</strong>dependent variable is Income y 17 we have, substitutingHĈ obtained by (7):y 17 = [y 14 , y 14 , Ψ] ⎜ ⎜⎛10⎜⎝00 ⎞⎟ ⎛ k1⎞v11⎟⎜⎟v⎟ ⎝k2 ⎠12 ⎠+ u 17 (10)In (10) we observe <strong>the</strong> presence of collinearity betweenregressors, and if we join <strong>the</strong> parameters, we cannotdivide <strong>the</strong> direct contribution of Invested Wealth onIncome and <strong>the</strong> indirect contribution of Wealth bymeans of HĈ. In effect,y 17 = y 14 [ k 1 + v 11 k 2 ] + Ψv 12 k 2 + u 17 (11)The Latent Variable Approach: a new proposalIt has been shown that <strong>the</strong> solutions obtained by meansof <strong>the</strong> Fac<strong>to</strong>r Model are not unique and that <strong>the</strong>solutions obtained by <strong>the</strong> PLS <strong>Method</strong> are not logicallyconsistent (Lovaglio 2003). In order <strong>to</strong> overcome thisproblem, a solution can be found in <strong>the</strong> use of all <strong>the</strong>information embedded in <strong>the</strong> Path Analysis model (2)(3). In this way, <strong>the</strong> HC is not previously obtained inequation (3) but, respecting <strong>the</strong> economic relationshipsis simultaneously obtained <strong>from</strong> reflective andformative indica<strong>to</strong>rs. In this perspective, observing <strong>the</strong>Path Analysis model (2) and (3), HC can be defined asa multidimensional construct approximated by <strong>the</strong>linear combination of its formative indica<strong>to</strong>rs (y 14 ,Ψ)that better fits <strong>the</strong> only reflective indica<strong>to</strong>rthat we can define as <strong>the</strong> earned income effect.Therefore we have <strong>from</strong> (2) :Premultiplying <strong>the</strong> equation (13) by F and taking in<strong>to</strong>account (2) we obtain:yQ y14 17,Q y14y 17 = Fgk 2 +u 17 =F k 3 +u 17 where k 3 = gk 2 (12)In (12) we obtain k * 3by means of an ordinary LeastSquares Regression of Q y y17 on F. The k * 143vec<strong>to</strong>rcontains <strong>the</strong> effects of <strong>the</strong> formative indica<strong>to</strong>rs F onearned income yQ y14 17:k * -13= g k 2 = S F F′ Qy y17 where S F = F′F (13)14F k * 3= F g k 2 = HC k 2 (14)Remembering that Var(HC) = S HC =1 we reach:* *2k3′SF k3= k2 S HC k 2 = k 2(15)From (15) we obtain k 2 * , <strong>the</strong> effect of HC on incomenet of wealth yQ y14 17:k 2 * = [(y 17 ′ Q y14F SF -1 F′ Q y14y17] 1/2 == [y 17 ′ Qy PF y14Q y14 17] 1/2 (16)where P F =F(F′ F) -1 F′.Therefore, <strong>from</strong> (13) and (16), we obtain g*, <strong>the</strong> effec<strong>to</strong>f <strong>the</strong> formative indica<strong>to</strong>rs F on HC:g*= k * / k =3*2= [y 17 ′ Q y14PF Q y14y 17 ] -1/2 S -1 F F′ Q y14y17 (17)At this point <strong>from</strong> (2) and (17) we obtain <strong>the</strong> estimationof HC scores (HC*) :HC* = F g* (18)The Latent Variable Approach: mixed indica<strong>to</strong>rsIn our case, some of <strong>the</strong> formative indica<strong>to</strong>rs arecategorical.Therefore we partition <strong>the</strong> vec<strong>to</strong>r of formativeindica<strong>to</strong>rs in<strong>to</strong> quantitative (contained in <strong>the</strong> column ofmatrix F q ) and categorical indica<strong>to</strong>rs F c in order <strong>to</strong>obtain consistent solutions with <strong>the</strong> quantitative case.We express <strong>the</strong> equation (2) in <strong>the</strong> following way:HC = F c g c + F q g q + u 16 , (F c = x 3 , x 4 , x 5 , x 7 ) (19)where F = (F c ,F q ), g = (g c ,g q )

We avoid <strong>the</strong> approach of <strong>the</strong> Item Fac<strong>to</strong>r Model(Chris<strong>to</strong>ffersson 1975; Olsson 1979; Mu<strong>the</strong>n andChris<strong>to</strong>ffersson 1981) consisting of a Fac<strong>to</strong>r Modelwith qualitative and categorical indica<strong>to</strong>rs. In effect,this model increases <strong>the</strong> difficulties concerning <strong>the</strong>original fac<strong>to</strong>r model. The non realistic hypo<strong>the</strong>sis of<strong>the</strong> normality of qualitative indica<strong>to</strong>rs, and somerestrictive assumptions, determine an underestimationof <strong>the</strong> true correlation between different indica<strong>to</strong>rs, <strong>the</strong>asymp<strong>to</strong>tical dis<strong>to</strong>rtions of standard errors and <strong>the</strong> nonchi-square distribution of goodness of fit statistics(Quiroga 1991; Vittadini 1999).We choose <strong>the</strong> multidimensional scaling methodALSOS, which alternatevely estimates, in separatesteps <strong>the</strong> parameter vec<strong>to</strong>r g and quantifies <strong>the</strong>categorical indica<strong>to</strong>rs F c by means of a uniquealgorithm., inside a specified model, <strong>the</strong> MultipleRegression Analysis (De Leeuw, Young and Takane1976; De Leeuw and Young 1978; De Leeuw and VanRijckevorsel 1980; Young 1981; Gifi 1981; Keller andWaansbeek 1983).We adapt this methodology in order <strong>to</strong> obtain <strong>the</strong> HC*,with <strong>the</strong> same methodology proposed in <strong>the</strong>quantitative case. If at <strong>the</strong> first step we arbitrarilychoose <strong>the</strong> parameters g c (0)* , we obtain <strong>the</strong> firstquantification of F c (0)* F cF c (0)* = F c g c (0)* with F (0)* = (F c (0* ,F q ) (20)At this point, we introduce F (0)* in (11) usingequations (13)-(17), we obtain <strong>the</strong> first estimates of <strong>the</strong>(1)* (1)*parameters k 2 , k 3 , g (1)* = (g (1)* c , g * q ) and bymeans of (18) <strong>the</strong> first estimates of HC, HC (1)* . Using(1)(1)g c in (19) we obtain a new quantification F c ofindica<strong>to</strong>rs F c . The iterative process continues until we* *have no more changes in k 2 , k 3 , g * , HC*, F*Therefore, in this way, <strong>the</strong> case of mixed indica<strong>to</strong>rs istreated similarly <strong>to</strong> <strong>the</strong> case of quantitative indica<strong>to</strong>rs.<strong>Human</strong> <strong>Capital</strong> in monetary units.As a quantitative multidimensional construct, <strong>the</strong>proposed methodology estimates HC* by a linearcombination of mixed formative indica<strong>to</strong>rs (y 14 ,Ψ) thatbest fits <strong>the</strong> reflective indica<strong>to</strong>rs Q y14y17. Itsestimation is consistent with well established economic<strong>the</strong>ory.Using <strong>the</strong> 1983 Federal Reserve Survey of 4,103households as a representative stratified sample of83,422,111 American households (Avery andElliehausen 1985) we obtain <strong>the</strong> HC* scores anddistribution which represent <strong>the</strong> estimated HCstandardized scores and distribution of <strong>the</strong> Americanhouseholds (Figure 2).6005004003002001000-3,5 - -3,2-2,8 - -2,5-2,1 - -1,8-1,4 - -1,1-,7 - -,4-,0 - ,3,7 - 1,01,4 - 1,72,1 - 2,42,8 - 3,1Figure 2: Standardized distribution of HCAt this point, as shown in Dagum and Slottje (2000),we transform <strong>the</strong> estimated standardized latent variableHC* in<strong>to</strong> an accounting monetary value applying <strong>the</strong>following transformationThen we estimate its mean value :HC°(i) = exp [HC*(i) ] (21)∑ n 1 i=∑ n 1 i=µ(HC°) = HC°(i) f(i) / f(i) (22)where f(i) is <strong>the</strong> number of households in <strong>the</strong> entirepopulation of American households that <strong>the</strong> i-thsampled household represents, HC°(i) is <strong>the</strong>accounting monetary value of HC*(i) and n is <strong>the</strong>sample size.By an actuarial approach <strong>the</strong> authors estimate <strong>the</strong> realHC upon <strong>the</strong> idea that an individual’s expected meanincome at age x+t of a person of age x should be equal<strong>to</strong> <strong>the</strong> mean earned income of individuals being at <strong>the</strong>present x+t years old; <strong>the</strong>refore <strong>the</strong> average humancapital h(x) of households head of age x is equal <strong>to</strong> <strong>the</strong>average expected earned income by age of <strong>the</strong>households head actualised at a given discount rate andweighted by <strong>the</strong> survival probability. Hence, <strong>the</strong>average human capital h(x) of <strong>the</strong> households head ofage x (assumed <strong>to</strong> stay in <strong>the</strong> labour market until age70) is:h(x) = Σ t y x+t p x, x+t (1+ i) -t t =0,..70-x (23)where y x+t is <strong>the</strong> mean income (real) of <strong>the</strong> householdshead of age x + t, p x,x+t is <strong>the</strong> probability of survival atage x+t of a person of age x and, i is <strong>the</strong> discount rate(estimated <strong>to</strong> be 0.08). Therefore <strong>the</strong> estimation of <strong>the</strong>average HC of <strong>the</strong> population of American families inmonetary units was obtained by Dagum and Slottje(2000) as <strong>the</strong> weighted mean of h(x):∑ 70 20 x =∑ 70 20 x =µ(h) = h(x)f(x) / f(x) (24)

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