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 9x t✻55OLFEmp.UI Tr.UI✲JRPWel. Tr.Welfare0τ 1τ 2Fig. 3. Labour market history of a hypothetical individual.τ e✲tThe sequence may be more compactly rewritten as:wherey l =r =(y 1 ,...,y n ),{(ul , x τl ), if 1 ≤ l ≤ n − 1,(u n ,0), if l = n.The initial state, x 0 , is the same for each individual in our sample and is exogenouslydetermined by school attendance laws. Consequently, there is no need to explicitly modelthe initial state in which individuals are observed.3.1 Likelihood functionEach individual contributes a sequence r = (y 1 ,...,y n ) to the likelihood function. Thecontribution can be written conditionally on a vector of exogenous variables, z, and anunobserved heterogeneity factor, ν.Let l v (θ) denote the conditional contribution of the sequence r. We have,nl v (θ) = ∏ f (y l | y 1 ,...,y l−1 ; z; ν; θ),l=1where f (y l | y 1 ,...,y l−1 ; z; ν; θ) is the conditional density of y l given y 1 , ..., y l−1 , z and ν,and θ ∈ Θ ⊂ IR p is a vector of parameters. Naturally, the destination state of the last spell isunknown since the duration is censored. Its contribution to the conditional likelihood functionis limited to the survivor function of the observed duration.The random variable ν is assumed to be independently and identically distributedacross individuals, and independent from the exogenous variables z. If the unobserved
56 Advances in Econometrics - Theory and Applications10 Will-be-set-by-IN-TECHheterogeneity can take only a finite number of values, ν 1 ,...,ν J , the contribution of a sequencer to the likelihood function isl(θ) =J n∑ ∏ f (y l | y 1 ,...,y l−1 ; z; ν j ; θ) π j , (1)j=1 l=1where π j is the probability that the unobserved heterogeneity term takes the value ν j (0 ≤π j ≤ 1, ∑ J j=1 π j = 1).If ν is a continuous random variable, then∫l(θ) =n∏ f (y l | y 1 ,...,y l−1 ; z; ν; θ) g(ν; γ) d ν, (2)Vl=1where g(ν; γ) is a density probability function and V is the support of ν.Furthermore, if we assume that Y l is independent of Y 1 ,...,Y l−2 , given Y l−1 = y l−1 , Z = zand the value of the unobserved term ν, thenf (y l | y 1 ,...,y l−1 ; z; ν; θ) = f (y l | y l−1 ; z; ν; θ).Given the history of the process, the joint distribution of the duration of spell l and thedestination state only depends on the current state on the labour market. This assumptionwill be relaxed by introducing other characteristics of the history of the process.3.2 Modeling individual spellsIn this section we focus on the conditional distribution of y l= (u l , x τl ), where u l is theduration of the l th spell in state x τl−1 . Define u ∗ l,k as the waiting time before leaving state x τ l−1for state x τl . At the end of the l th spell, the individual will enter into the state correspondingto the smallest latent duration u ∗ l,k ′ . We will assume that these K latent durations areindependently distributed. Thus the duration of spell l is given by 12u l = infk ′ u ∗ l,k ′.Let f j (u | y 1 ,...,y l−1 ; z; ν; θ) denote the probability density function (p.d.f.) of the latentduration u ∗ l,j , given the history of the process up to time τ l−1, ν and covariates z. Let S j (u |y 1 ,...,y l−1 ; z; ν; θ) be the corresponding survivor function:S j (u | y 1 ,...,y l−1 ; z; ν; θ) =∫ +∞f j (s | y 1 ,...,y l−1 ; z; ν; θ) ds.u12 If the transition from state i to state j cannot be observed, we assume that the corresponding latentdistribution is defective and set a probability mass equal to one on +∞.
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