Robert Shimer, University of Chicago
Abstract: We consider a discrete time version of the mixed proportional hazard (MPH) model of duration. Building on results in Honoré (1993) for multi-spell data, we show that the baseline hazard and the frailty distribution of unobserved heterogeneity are identified. We then express these results as moment conditions and develop a GMM estimator of the baseline hazard and moments of the frailty distribution. Our setup easily incorporates right-censored spells in panel data. The GMM specification is linear in the baseline hazard rate parameters, and then sequentially linear in the moments of the frailty distribution, which makes estimation and inference straightforward. We also develop tests of whether the MPH model is the data generating process. Testing and estimation requires specification of neither the frailty distribution nor the shape of the baseline hazard function.
We apply our estimation procedure to the duration of price spells in weekly scanner price data from Nielsen-IRI. We find substantial unobserved heterogeneity, accounting for a large fraction of the decrease in the aggregate raw hazard rate. This contrasts with the estimates that we obtain using the methods most commonly used in the existing literature, which find a small role for heterogeneity. We argue that this difference comes from the fact that the commonly-used estimation method does not account for rounding in measured duration.