P. Deb, PK Trivedi/Journal of Health Economics 21(2002)601-625 one that distinguishes non-users and users(TPM) while minimizing all other sources of variation. From the nB density shown in Eq(2. 1 )one can derive the probability of being a non-user as Pr1(y=0x,61) where the subscript 1 denotes parameters associated with the first part of the TPM, A1.i exp(x'P1) and (1/ a1)ii. The probability of being a user is calculated as(1- PrI(i=OLxi, 01)). The first part involves only binary information so the parameters(B1) of the mean function and the parameter aI are not separately identifiable. We set a1=1 without loss of generality In the second part of the TPM, the distribution of utilization conditional on some us is assumed to follow a truncated NB distribution. After some algebraic manipulation, one gets r(y+v)「/x2;+v2吻 f2(ylx,y>0,62)= T(2 i/(i+1) 入2,+v2,i as the conditional density of use. Note that, al though the first and second parts are derived from the nB density, the parameters are allowed to be different The first and second parts of the TPM enter multiplicatively in the likelihood functio Therefore, the likelihood function associated with the binary choice can be maximized separately from the second part, which is estimated using the truncated subsample of positive observations of yi. The mean of the count variable in this TPM is given by E(w|x)=Pr1y>0x,61)2 Pr and the variance by Pr1(y>0|x,61) 2-k Pr1(y>0|x;,61) Pr2(y7>0|x;,62) Pr2(y>0|x;,62) Both the mean and the variance in the TPM are, in general, different from their standard NB counterparts. The TPM can accommodate over and underdispersed data relative to the NBM I Although we have chosen to derive both parts of the hurdle model from parent NB distributions, we rece that users may sometimes choose to estimate the binary choice part using more familiar logit or probit models This choice is typically not significant, because, as is commonly known, the exact choice of distribution in binary choice models makes very little difference to the estimated probabilities In our case, we have also estimated logit models with almost identical resultsP. Deb, P.K. Trivedi / Journal of Health Economics 21 (2002) 601–625 605 one that distinguishes non-users and users (TPM) while minimizing all other sources of variation. From the NB density shown in Eq. (2.1)one can derive the probability of being a non-user as Pr1(yi = 0|xi, θ 1) =
ψ1,i λ1,i + ψ1,i ψ1,i , (2.4) where the subscript 1 denotes parameters associated with the first part of the TPM, λ1,i = exp(x iβ1) and ψ1,i = (1/α1)λk 1,i. The probability of being a user is calculated as (1 − Pr1(yi = 0|xi, θ 1)). The first part involves only binary information so the parameters (β1) of the mean function and the parameter α1 are not separately identifiable. We set α1 = 1 without loss of generality. In the second part of the TPM, the distribution of utilization conditional on some use is assumed to follow a truncated NB distribution. After some algebraic manipulation, one gets f2(yi|xi, yi > 0, θ 2) = Γ (yi + ψ2,i) Γ (ψ2,i)Γ (yi + 1)
λ2,i + ψ2,i ψ2,i ψ2,i − 1 −1 ×
λ2,i λ2,i + ψ2,i yi (2.5) as the conditional density of use.1 Note that, although the first and second parts are derived from the NB density, the parameters are allowed to be different. The first and second parts of the TPM enter multiplicatively in the likelihood function. Therefore, the likelihood function associated with the binary choice can be maximized separately from the second part, which is estimated using the truncated subsample of positive observations of yi. The mean of the count variable in this TPM is given by E(yi|xi) = Pr1(yi > 0|xi, θ 1) Pr2(yi > 0|xi, θ 2) λ2,i (2.6) and the variance by V(yi|xi)= Pr1(yi > 0|xi, θ 1) Pr2(yi > 0|xi, θ 2)  λ2,i+α2λ2−k 2,i +
1− Pr1(yi > 0|xi, θ 1) Pr2(yi > 0|xi, θ 2) λ2 2,i . (2.7) Both the mean and the variance in the TPM are, in general, different from their standard NB counterparts. The TPM can accommodate over and underdispersed data relative to the NBM. 1 Although we have chosen to derive both parts of the hurdle model from parent NB distributions, we recognize that users may sometimes choose to estimate the binary choice part using more familiar logit or probit models. This choice is typically not significant, because, as is commonly known, the exact choice of distribution in binary choice models makes very little difference to the estimated probabilities. In our case, we have also estimated logit models with almost identical results.