The analysis of Section 18.2 shows the effects of unobserved heterogeneity on the unconditional or averaged hazard function. The result that neglected heterogeneity leads to under-estimation of the slope of the average hazard function is emphasized. Let the conditional hazard function be ${\lambda }_c(t\mid v)=\nu {\lambda }_0(t)$, where ${\lambda }_0$ denotes the baseline or unconditional hazard function. Show that (i) the unconditional hazard ${\lambda }_U(t)<{\lambda }_0(t)$ and (ii) $\mathrm{\partial }{\lambda }_U(t)/\mathrm{\partial }t<0$ in each of the following examples.

(a) $v\sim$ Uniform $[0,1]$ and ${\lambda }_0(t)=1\mathrm{\forall }t$.

(b) $v$ follows a unit exponential distribution with pdf $g(v)=e^{-v}$ and ${\lambda }_0(t)=$ $\rho \exp (\gamma t),\rho >0,\gamma <0$.