
Journal of Finance (2016) 71 (6): 24932544.
Asset prices contain information about the probability distribution of future states and the stochastic discounting of those states as used by investors. To better understand the challenge in distinguishing investors' beliefs from riskadjusted discounting, we use PerronFrobenius Theory to isolate a positive martingale component of the stochastic discount factor process. This component recovers a probability measure that absorbs longterm risk adjustments. When the martingale is not degenerate, surmising that this recovered probability captures investors' beliefs distorts inference about riskreturn tradeoffs. Stochastic discount factors in many structural models of asset prices have empirically relevant martingale components.

Handbook of Macroeconomics: Volume 2B (2016) Chapter 20, Elsevier B.V., 16411696.
Dynamic economic models make predictions about impulse responses that characterize how macroeconomic processes respond to alternative shocks over different horizons. From the perspective of asset pricing, impulse responses quantify the exposure of macroeconomic processes and other cash flows to macroeconomic shocks. Financial markets provide compensations to investors who are exposed to these shocks. Adopting an asset pricing vantage point, we describe and apply methods for computing exposures to macroeconomic shocks and the implied compensations represented as elasticities over alternative payoff horizons. The outcome is a term structure of macroeconomic uncertainty.

Mathematics and Financial Economics (2014) 8 (4): 333354.
We construct shock elasticities that are pricing counterparts to impulse response functions. Recall that impulse response functions measure the importance of nextperiod shocks for future values of a time series. Shock elasticities measure the contributions to the price and to the expected future cash flow from changes in the exposure to a shock in the next period. They are elasticities because their measurements compute proportionate changes. We show a particularly close link between these objects in environments with Brownian information structures.

Journal of Econometrics (2014) 183 (1): 6790.
We develop new methods for representing the assetpricing implications of stochastic general equilibrium models. We provide assetpricing counterparts to impulse response functions and the resulting dynamic value decompositions (DVDs). These methods quantify the exposures of macroeconomic cash flows to shocks over alternative investment horizons and the corresponding prices or investors' compensations. We extend the continuoustime methods developed in Hansen and Scheinkman (2012) and Borovička, Hansen, Hendricks, and Scheinkman (2011) by constructing discretetime, statedependent, shockexposure and shockprice elasticities as functions of the investment horizon.
Our methods are applicable to economic models that are nonlinear, including models with stochastic volatility.

Journal of Financial Econometrics (2011) 9 (1): 365.
We present a novel approach to depicting asset pricing dynamics by characterizing shock exposures and prices for alternative investment horizons. We quantify the shock exposures in terms of elasticities that measure the impact of a current shock on future
cashflow growth. The elasticities are designed to accommodate nonlinearities in the stochastic evolution modeled as a Markov process. Stochastic growth in the underlying macroeconomy and stochastic discounting in the representation of asset values are central ingredients in our
investigation. We provide elasticity calculations in a series of examples featuring consumption externalities, recursive utility, and jump risk.