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

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).