Heterogeneity and negative duration dependence
The effect of unobservable covariates can be measured by comparing models with and without heterogeneity. In this section we perform such a comparison using the exponential model. For simplicity we do not include observable covariates in the model. The conditional distribution of the duration variable given the heterogeneity factor p, is exponential with parameter X, = p, whereas the marginal distribution of the heterogeneity is n. Therefore, the conditional and marginal survivor functions are:
S( Уі 1 Pi> = ^pHPiy^
The corresponding hazard functions are:
X(Vi I Pi) = Pi,
X(V.) = – d log S(Vi) =______ — dS( Vi)
‘ dV S(Vi) dV
= Jo exp(-py,.-)pn(p)dp /0°exp(-pVi)n(p)dp ‘
The marginal hazard rate is an average of the individual hazard rates p, with respect to a modified probability distribution with pdf:
We also get:
X(Vi) = E%[X(Vi | Pi)] = EnK(Pi). (21.14)
This marginal hazard function features negative duration dependence. Indeed, by taking the first-order derivative we find:
dy^Vi) = – Jo p2 exp(-pVi)n(p)dp + [JQ°exp(-pVi)pn(p)dp]2
dV /0° exp(-pVi )n(p)dp [/0° exp(-pVi )n(p)dp]2
= – E% P2 + [En, i(Pi)]2 = – var%P; ^ o.
We note that the negative duration dependence at level Vi is related to the magnitude of heterogeneity with respect to a modified probability.
To illustrate previous results let us consider a sample of individuals belonging to two categories with respective exit rates p1 > p2. The individuals in the first category with a high exit rate are called movers, whereas we call stayers the individuals belonging to the second category. The structure of the whole population at date 0 is n1 = n, n2 = 1 – n. The marginal hazard rate derived in the previous section becomes:
X(V) = П 1S1(У)Ці + n2S2(V)p2 . (2115)
П 1S1( V) + n2S2( V)
Between 0 and v some individuals exit from the population. The proportions of those who leave differ in the two subpopulations; they are given by S1( y) = exp(-p1y) < S2( y) = exp(-p 2y), which implies a modified structure of remaining individuals at date V. This modified structure is:
Пі( y) = niSi( y)/[n1S1( y) + n 2S2( y)], n 2( y) = 1 – Пі( y). (21.16)
Since S1(y) < S2(y), the proportion of movers is lower at date y than at date 0, which implies X(y) < ^(0) = п1р1 + n2p2. Finally, we note that, for large y, n2(y) tends to one and the remaining population becomes homogenous including stayers only.