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

60 Advances in Econometrics - Theory and Applications14 Will-be-set-by-IN-TECHIf α j,k > 1 then the hazard function is increasing then decreasing with respect of u. Ifα j,k ≤ 1 then the hazard function is decreasing.2. Piecewise-Constant Hazard ModelThe expression of the baseline hazard function ish 0 j,k (u; θ) =α j,k1I[u < u 0 1 ]+β j,k1I[u 0 1 ≤ u < u0 2 ]+γ j,k1I[u 0 2 ≤ u],where α j,k , β j,k , γ j,k ∈ IR + . u 0 1 and u0 2 are fixed.The baseline hazard function can be increasing then decreasing,increasing, strictly increasing or strictly decreasing.3. Weibull DistributionThe baseline hazard function isdecreasing thenh 0 j,k (u; θ) =α j,k β j,k u α j,k−1 ,α j,k , β j,k ∈ IR + .If α j,k > 1 then the hazard function is increasing with respect of u. If α j,k < 1 then thehazard function is decreasing with respect of u and if α j,k = 1 this conditional hazardfunction is constant.3.5 EstimationWe consider three alternative specifications for the unobserved heterogeneity distribution.1. Two-Factor Loading and Discrete DistributionThe log likelihood isNlog(L(θ)) = ∑ log(l i (θ)), (11)i=1where l i (θ) is obtained by substituting the sequence r i =(y 1,i ,...,y ni ,i) and the observedvector of covariates z i in (1). N is the size of the sample.In equation (1) π j is set equal to 13⎧⎨ p 2 , if j = 1,π j = p ∗ (1 − p), if j = 2, 3,⎩(1 − p) 2 , if j = 4,where p ∈ [0; 1] is a parameter. The log-likelihood is then maximized with respect of θ(θ ∈ Θ).2. Two-Factor Loading and Continuous DistributionThe model includes two unobserved heterogeneity terms v 1 and v 2 (v j > 0, j = 1, 2). Weassume these terms to be independently and identically distributed. Let q(v; γ) be thep.d.f. of v j , j = 1, 2.The contribution of a given realization to the likelihood function is given by equation (2),where ν =(v 1 , v 2 ) ′ , V = IR + × IR + and g(ν; γ) =q(v 1 ; γ) q(v 2 ; γ). The log-likelihood is13 See section 4.