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This paper studies the impact of ambiguity and ambiguity aversion on equilibrium asset prices and portfolio holdings in competitive financial markets. It argues that attitudes toward ambiguity are heterogeneous across the population, just as attitudes toward risk are heterogeneous across the population, but that heterogeneity of attitudes toward ambiguity has different implications than heterogeneity of attitudes toward risk. In particular, when some state probabilities are not known, agents who are sufficiently ambiguity averse find open sets of prices for which they refuse to hold an ambiguous portfolio. This suggests a different cross-section of portfolio choices, a wider range of state price/probability ratios and different rankings of state price/probability ratios than would be predicted if state probabilities were known. Experiments confirm all of these suggestions. Our findings contradict the claim that investors who have cognitive biases do not affect prices because they are infra-marginal: ambiguity averse investors have an indirect effect on prices because they change the per-capita amount of risk that is to be shared among the marginal investors. Our experimental data also suggest a positive correlation between risk aversion and ambiguity aversion that might explain the "value effect" in historical data. JEL Classification: C91, D53, D81, G11, G12

Survival analysis (time-to-event analysis) is widely used in economics and finance, engineering, medicine and many other areas. A fundamental problem is to understand the relationship between the covariates and the (distribution of) survival times(times-to-event). Much of the previous work has approached the problem by viewing the survival time as the first hitting time of a stochastic process, assuming a specific form for the underlying stochastic process, using available data to learn the relationship between the covariates and the parameters of the model, and then deducing the relationship between covariates and the distribution of first hitting times (the risk). However, previous models rely on strong parametric assumptions that are often violated. This paper proposes a very different approach to survival analysis, DeepHit, that uses a deep neural network to learn the distribution of survival times directly.DeepHit makes no assumptions about the underlying stochastic process and allows for the possibility that the relationship between covariates and risk(s) changes over time. Most importantly, DeepHit smoothly handles competing risks; i.e. settings in which there is more than one possible event of interest.Comparisons with previous models on the basis of real and synthetic datasets demonstrate that DeepHit achieves large and statistically significant performance improvements over previous state-of-the-art methods.

Missing data is a ubiquitous problem. It is especially challenging in medical settings because many streams of measurements are collected at different -and often irregular -times. Accurate estimation of those missing measurements is critical for many reasons, including diagnosis, prognosis and treatment. Existing methods address this estimation problem by interpolating within data streams or imputing across data streams (both of which ignore important information) or ignoring the temporal aspect of the data and imposing strong assumptions about the nature of the data-generating process and/or the pattern of missing data (both of which are especially problematic for medical data). We propose a new approach, based on a novel deep learning architecture that we call a Multi-directional Recurrent Neural Network (M-RNN) that interpolates within data streams and imputes across data streams. We demonstrate the power of our approach by applying it to five real-world medical datasets. We show that it provides dramatically improved estimation of missing measurements in comparison to 11 state-of-the-art benchmarks (including Spline and Cubic Interpolations, MICE, MissForest, matrix completion and several RNN methods); typical improvements in Root Mean Square Error are between 35% -50%. Additional experiments based on the same five datasets demonstrate that the improvements provided by our method are extremely robust.