The Limits of Mathematics and NP Estimation in ... - Chichilnisky

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The Impact of Government-SponsoredTraining Programs on the Labor Market Transitions of Disadvantaged MenThe Impact of Government-Sponsored Training Programs on the Labor Market Transitions of Disadvantaged Men 1157The conditional joint density of the duration of spell l and the destination state k is given bythe following expressionf (u, k|y 1 ,...,y l−1 ; z; ν; θ) = f k (u | y 1 ,...,y l−1 ; z; ν; θ)KS j (u | y 1 ,...,y l−1 ; z; ν; θ),∏j=1j̸=k= h k (u|y 1 ,...,y l−1 ; z; ν; θ) S(u|y 1 ,...,y l−1 ; z; ν; θ),where h k (u | y 1 ,...,y l−1 ; z; ν; θ) is the hazard function associated with the latent duration u ∗ l,kand S(u | y 1 ,...,y l−1 ; z; ν; θ) is the survivor function of the duration of the l th spell. Becausethe latent durations are assumed to be conditionally independent we haveS(u | y 1 ,...,y l−1 ; z; ν; θ) =K∏ S j (u | y 1 ,...,y l−1 ; z; ν; θ),j=1where u ≥ 0. The expression represents the conditional probability that the duration of spell lis at least equal to u or, equivalently, that all latent durations are at least equal to u. Therefore,the conditional contribution of a given sequence to the likelihood function is:n Kl v (θ) = ∏ ∏ h k (u | y 1 ,...,y l−1 ; z; ν; θ) δ l,kS k (u | y 1 ,...,y l−1 ; z; ν; θ),l=1 k=1where δ l,k is equal to 1 if the individual enters into state k at the end of spell l and to 0otherwise :l = 1,...,n.δ l,k ={ 1, if xτl = k,0, otherwise,3.3 Unobserved heterogeneitySo far the discussion surrounding the unobserved heterogeneity components has voluntarilybeen kept general. The use of maximum likelihood procedures requires that we specifydistribution functions for these components. Most applications rely on the work of Heckman& Singer (1984) and approximate arbitrary continuous distributions using a finite number ofmass points (see Gritz (1993), Ham & Rea (1987), Doiron & Gorgens (2008)). More recentpapers use richer specifications that allow the heterogeneity terms to be correlated acrossstates (see Bonnal et al. (1997), Ham & LaLonde (1996)). These specifications are sometimesreferred to as single or two-factor loading distributions and are also based on a finite set ofmass points. In our work, we wish to investigate the robustness of the parameter estimatesto various distributional assumptions. We will use two and three-factor loading distributionsas in the aforementioned papers. Additionally, we will investigate the consequences on theslope parameters of using various continuous distributions instead of the usual finite sets ofmass points.
58 Advances in Econometrics - Theory and Applications12 Will-be-set-by-IN-TECHTo fix ideas, let w =(w 1 ,...,w K ) be a vector of unobserved heterogeneity variables, with w k adestination-specific component (k = 1,...,K). Ideally, the joint distribution of the unobservedheterogeneity terms should not be independent.Consider first a two-factor loading model (see Van den Berg (1997)) such thatw k = exp(a k v 1 + b k v 2 ), (3)where v 1 ∈{−2, c 2 }, v 2 ∈{c 1 , c 2 }, b k ∈ IR, a k = 1I[k ≥ 2] and b 1 = 1. The random variablesv 1 and v 2 are assumed to be independent. The constraints imposed on the support of v 1 andv 2 are sufficient for identification and to allow the correlation between log(w k ) and log(w k ′ )to span the interval [−1; 1].Moreover, assume thatProb[(V 1 , V 2 )=(v 0 1 , v0 2 )] = ⎧⎪ ⎨⎪ ⎩p 2 , if v 0 1 = −2 and v0 2 = c 1,p ∗ (1 − p), if v 0 1 = −2 and v0 2 = c 2,(1 − p) ∗ p, if v 0 1 = c 2 and v 0 2 = c 1,(1 − p) 2 , if v 0 1 = c 2 and v 0 2 = c 2,where c 1 , c 2 ∈ IR and the probability p is defined as(4)p =exp(d)1 + exp(d) ,where d ∈ IR is a parameter.The correlation between log(w k ) and log(w k ′ ), denoted ρ k,k ′,isρ k,k ′ =a k a k ′ σ 2 v 1+ b k b k ′ σ 2 v 2√a 2 k σ2 v 1+ b 2 k σ2 v 2√a 2 k ′ σ 2 v 1+ b 2 k ′ σ 2 v 2, (5)where k, k ′ = 1, . . . , K and σv 2 jis the variance of v j , j=1,2. A positive correlation coefficientbetween w j and w k implies that those who are likely to have high transition rates between anygiven state and state j will also have high transition rates into state k.A two-factor loading model with two independent heterogeneity terms with commoncontinuous distribution can also be derived from this specification. As before, let w k denotethe heterogeneity term for destination k:w k = exp(a k v 1 + b k v 2 ),where a k and b k are parameters (a k = 1I[k ≥ 2] and b 1 = 1).Here v 1 and v 2 are assumed to be independently and identically distributed. Let q(v; γ) bethe p.d.f. of v 1 and v 2 . The correlations between log(w k ) and log(w k ′ ) are given by the sameexpression as in (5). In principle, q(v; γ) represents any well-behaved distribution function.The above specification can be further generalized to a three-factor loading model withcommon continuous distribution. In this case the unobserved components depend on thedestination state as well as the current state. Let w j,k be specific to the transition betweenorigin j and destination k.w j,k = w ′ j w k = exp(a ′ j v 3 + b ′ j v 2) × exp(a k v 1 + b k v 2 ), (6)where a ′ j , b′ j , a k and b k are parameters (a ′ j = a k = 1I[k ≥ 2], b 1 = 1).
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