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

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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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