
Prepared for the Palgrave volume on University of Chicago economics.

(with Anmol Bhandari and Paul Ho)
Revise and resubmit, Review of Economic Studies
Previously circulated as NBER WP No. 22225 "Identifying ambiguity shocks in business cycle models using survey data"
This paper develops a theory of subjective beliefs that departs from rational expectations, and shows that biases in household beliefs have quantitatively large effects on macroeconomic aggregates. The departures are formalized using modelconsistent notions of pessimism and optimism and are disciplined by data on household forecasts. The role of subjective beliefs is quantified in a business cycle model with goods and labor market frictions. Consistent with the survey evidence, an increase in pessimism generates upward biases in unemployment and inflation forecasts and lowers economic activity. The underlying belief distortions reduce aggregate demand and propagate through frictional goods and labor markets. As a byproduct of the analysis, solution techniques that preserve the effects of timevarying belief distortions in the class of linear solutions are developed.

(with Katarína Borovičková)
We study the role of fluctuations in discount rates for the joint dynamics of expected returns in the stock market and employment dynamics. We construct a nonparametric bound on the predictability and timevariation in conditional volatility of the firm's profit flow that must be met to rationalize the observed businesscycle fluctuations in vacancyfilling rates. A stochastic discount factor consistent with conditional moments of the riskfree rate and expected returns on risky assets only partly alleviates the need for an excessively volatile model of the expected profit flow.

(with Lars Peter Hansen)
We propose an approximation method for solving dynamic stochastic general equilibrium models in which agents are concerned about model misspecification. The method relies on a perturbation that treats this robust concern as a firstorder concept that is preserved as the volatility of the shocks vanishes. The approximation has a clear economic interpretation and generates solutions with consequences of robust preferences that standard perturbation methods only capture using higherorder terms. In particular, our method generates risk premia in the linear solution and time variation in these risk premia and stochastic volatility effects in the secondorder approximation.

(with John Stachurski)
Journal of Economic Theory (2021) 193: 105227.
We obtain an exact necessary and sufficient condition for the existence and uniqueness of equilibrium asset prices in infinite horizon, discretetime, arbitrage free environments. Using local spectral radius methods, we connect the condition, and hence the problem of existence and uniqueness of asset prices, with the recent literature on stochastic discount factor decompositions. Our results include a globally convergent method for computing prices whenever they exist. Convergence of this iterative method itself implies both existence and uniqueness of equilibrium asset prices.

(with John Stachurski)
Journal of Finance (2020) 75 (3): 14571493.
We obtain exact necessary and sufficient conditions for existence and uniqueness of solutions of a class of homothetic recursive utility models postulated by Epstein and Zin (1989). The conditions center on a single test value with a natural economic interpretation. The test sheds light on the relationship between valuation of cash flows, impatience, risk adjustment, and intertemporal substitution of consumption. We propose two methods to compute the test value when an analytical solution is not available. We further provide several applications.

Journal of Political Economy (2020) 128 (1): 206251.
Circulated under the title Heterogeneous Beliefs under Recursive Preferences.
I study the longrun behavior of an economy with two types of agents who differ in their beliefs and are endowed with homothetic recursive preferences of the DuffieEpsteinZin type. Contrary to models with separable preferences in which the wealth of agents with incorrect beliefs vanishes in the long run, recursive preference specifications lead to longrun outcomes where both agents survive, or more incorrect agents dominate. I derive analytical conditions for the existence of nondegenerate longrun equilibria in which agents with differently accurate beliefs coexist in the long run, and show that these equilibria exist for broad ranges of plausible parameterizations when risk aversion is larger than the inverse of the intertemporal elasticity of substitution. The results highlight a crucial interaction between risk sharing, speculative behavior and consumptionsaving choice of agents with heterogeneous beliefs, and the role of equilibrium prices in shaping longrun outcomes.

(with Lars Peter Hansen and José Scheinkman)
Journal of Finance (2016) 71 (6): 24932544.
2017 Amundi Smith Breeden Prize Distinguished Paper
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.

(with Lars Peter Hansen)
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.

(with Lars Peter Hansen and José Scheinkman)
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.

(with Lars Peter Hansen)
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.

(with Lars Peter Hansen, Mark Hendricks, and José Scheinkman)
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.