Performance Sampling and Bimodal Duration Dependence

Abstract

Performance sampling models of duration dependence in employee turnover and firm exit predict that hazard rates will initially be low, gradually rise to a maximum, and then fall. Some empirical duration distributions have bimodal hazard rates, however. In this paper, we present a generalization of the performance sampling model that can account for such deviations from unimodality. While the standard model of performance sampling assumes that the mean and the standard deviation of performance are constant over time, we allow them to change in time, to reflect the fact that tasks may change over time. We derive the hazard rate implied by this more general model and show that it can be bimodal. Using data on turnover in law firms, we show that the hazard rate predicted by these models fit data better than existing models.

Keywords:

• duration dependence
• hazard rates
• learning
• performance sampling

Acknowledgments

This paper has benefited from comments by Bill Barnett, Glenn Carroll, Boyan Jovanovic, Dan Levinthal, Jim March, Huggy Rao, and Ezra Zuckerman.

Notes

¹Formally, we assume that S(t) = μt + σ(t)W(t), where σ(t) is a continuous function of time, and W(t) is the standard Brownian motion. Here σ(t) is the infinitesimal standard deviation of task performance, defined as the square root of [lim]²|S(0) = 0} (Karlin and Taylor, 1981).

² Formally, we assume that S _i (t) = μ _i (t)t + σW(t), where the instantaneous drift rate, u _i (t), is a continuous function of time, and W(t) is the standard Brownian motion.

³This possibility was suggested to us by Boyan Jovanovic.

*Number of observations is 1999, of which 786 are censored.

⁴The censored observations here consist mainly of those individuals who were promoted to partnership.

⁵For the numerical optimizations we used both Mathematica and the Solver add-in to Excel. We experimented with numerous possible starting values for the parameters, to ensure a global rather than a local maximum. We found identical solutions using the different packages.

⁶The log likelihood for the standard performance sampling model is then –3549.91.

⁷As formulated, the models are not identical. It is likely, however, that they would be identical if a different function was used for the change in the drift rates.

⁸To see this, note that the assumption that σ²(t) is the infinitesimal variance means that

For example, Karlin and Taylor (1981, p. 159). Since the variance of the increment, S(t + Δt) − S(t), during an interval Δt, is E{[S(t + Δt) − S ⁽ t)]2|S(0) = 0}, and since increments are assumed to be independent, the variance of S(t) at time t is equal to

which is approximately ∫ σ²(t)dt.