We develop a linear GMM estimator of the discrete-time mixed proportional hazard (MPH) model of duration with an arbitrary distribution of unobserved heterogeneity. We allow for competing risks, observable characteristics, and censoring. We prove our estimator is consistent and apply it to the duration of price spells. We find substantial unobserved heterogeneity with economically meaningful implications for the response of output to a monetary policy shock in a model with time-dependent pricing rules and for the degree of state dependence in a model of price plans.

For an incompatible patient–donor pair, kidney exchanges often forbid receipt-before-donation (the patient receives a kidney before the donor donates) and donation-before-receipt, causing a double-coincidence-of-wants problem. We study an algorithm, the Unpaired kidney exchange algorithm, which eliminates this problem. In a dynamic matching model, we show that the waiting time of patients under Unpaired is close to optimal and substantially shorter than under widely used algorithms. Using a rich administrative dataset from France, we show that Unpaired achieves a match rate of 63% and an average waiting time of 176 days for transplanted patients. The (infeasible) optimal algorithm is only slightly better (64% and 144 days); widely used algorithms deliver less than 40% match rate and at least 232 days waiting times. We discuss a range of solutions that can address the potential practical incentive challenges of Unpaired. In particular, we extend our analysis to an environment where a deceased donor waitlist can be integrated to improve the performance of algorithms. We show that our theoretical and empirical comparisons continue to hold. Finally, based on these analyses, we propose a practical version of the Unpaired algorithm.