The exponential model with heterogeneity

The exponential regression model can easily be extended by introducing unobservable variables. We express the individual hazard rate as:

Хі = р,- exp(x!-0), (21.11)

where р is a latent variable representing the heterogeneity of individuals in the sample, called the heterogeneity factor. We assume that the heterogeneity factor is gamma distributed у (a, a), with two identical parameters to ensure Ep; = 1. The conditional duration distribution given the observable covariates is found by integrating out the unobservable heterogeneity.

Подпись:Подпись:f (yі xі, р; 0) п (р; a)dр


р exp(x і0) exp[—Уір exp(x0)] a Ц rexp(-ap)d^ 0 r(a)

aa exp(x!0) Г(а + 1)

[a + y і exp(x!0)]a+1 Г(а) [a + y і exp(x!0)]a+1

We find that the conditional duration distribution is Pareto translated. The associated conditional survivor function is:

Подпись:a~’exP<x8> du

[a + ц exp(x;0>]a+’


[a + y і exp(x;0>]a л whereas the hazard function is:

Подпись: X(y І | %i) 0, a)a exp(xi0>
a + y і exp(xi0>

The hazard function of the Pareto distribution with drift is a decreasing func­tion of y, and features negative duration dependence at the level of a representa­tive individual. Hence, by aggregating exponentially distributed durations with constant hazards across infinitely many different individuals with a gamma dis­tributed heterogeneity, we obtain a decreasing aggregate hazard function. The heterogeneity parameter a provides a natural measure of the negative duration dependence: the smaller a, the stronger the negative duration dependence. In the limiting case a = +«>, we get p, = ‘, and X(yi|xi; 0, a) = exp(xi0>; there is no duration dependence and the Pareto regression model reduces to the exponential regression model.

For the Pareto regression model, the loglikelihood function is:


logl(y1 x; % a> = Y log f(Уі 1 xi; ^ a>

i =’


= Y {(a + ‘)log a + x,0 – (a + ‘)log [a + y, exp(xi0>]}.

i =’

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