Businesses often face the problem of providing incentives for agents to work effectively together on projects that develop over time. The agents' costly and unobservable effort jointly affects the survival of the project and thus the expected value of its cash now. A key feature of many contracting problems with multiple agents is that the agents exert effort at different times: some at the outset and some over time. The optimal timing of compensation reflects the timing of effort with payment for up-front effort preceding compensation for continuous effort. Deferring payment for agents exerting effort over time improves their incentives without impairing incentives for the up-front effort because this effort is sunk once the project is set up. The exact pattern of compensation between the agents with continuous effort depends on the relative severity of their moral hazard problems. In a special case where moral hazards are equally severe, the agents equally split the cash flow once it becomes available. This study suggests an approach to understanding a broad set of contracting problems in economics and finance. It rationalizes business conventions such as deferred compensation for top executives, the 50:50 split between law firm partners, and profit shares of influential directors (or lead actors) and residual claims of producers in the movie industry. Furthermore, the model predicts business failures such as the crisis in the mortgage industry due to the lack of characteristics suggested in the optimal contract.

We develop a nonparametric specification test for continuous-time models using the transition density. Using a data transform and correcting for boundary bias of kernel estimators, our test is robust to serial dependence in data and provides excellent finite sample performance. Besides univariate diffusion models, our test is applicable to a wide variety of continuous-time and discretetime dynamic models, including time-inhomogeneous diffusion, GARCH, stochastic volatility, regimeswitching,jump-diffusion, and multivariate diffusion models. A class of separate inference procedures is also proposed to help gauge possible sources of model misspecification. We strongly reject a variety of univariate diffusion models for daily Eurodollar spot rates and some popular multivariate affine term structure models for monthly U.S. Treasury yields.

We propose two nonparametric transition density-based speciÞcation tests for continuous-time models. Unlike the marginal density used in the literature, the transition density can capture the full dynamics of a continuous-time process. To address the concerns of the Þnite sample perfor- mance of nonparametric methods in the literature, we introduce an appropriate data transfor- mation and correct the boundary bias of kernel estimators. As a result, our tests are robust to persistent dependence in data and provide reliable inferences for sample sizes often encountered in empirical Þnance. Simulation studies show that even for data with highly persistent depen- dence, our tests have reasonable size and good power against a variety of alternatives in Þnite samples. Besides one-factor diffusion models, our tests can be applied to a broad class of dynamic models, including discrete-time dynamic models, time-inhomogeneous diffusion models, stochas- tic volatility models, jump-diffusion models, and multi-factor diffusion models. When applied to Eurodollar interest rates, our tests overwhelmingly reject a variety of popular one-factor diffusion models. We Þnd that introducing nonlinear drift does not signiÞcantly improve the goodness of Þt, and the main reason for the rejection of one-factor diffusion models is the violation of the Markov assumption. Some popular non-Markovian models with GARCH, regime switching and jumps perform signiÞcantly better than one-factor diffusion models, but they are still far from being adequate to fully capture the interest rate dynamics. Our study shows that, contrary to the general perception in the literature, nonparametric methods are a reliable and powerful tool for analyzing Þnancial data.