# Working papers

- Necessary and Sufficient Conditions for Existence and Uniqueness of Recursive Utilities (with John Stachurski)We study existence, uniqueness and stability of solutions for a class of discrete time recursive utilities models. By combining two streams of the recent literature on recursive preferences—one that analyzes principal eigenvalues of valuation operators and another that exploits the theory of monotone concave operators—we obtain conditions that are both necessary and sufficient for existence and uniqueness. We also show that the natural iterative algorithm is convergent if and only if a solution exists. Consumption processes are allowed to be nonstationary.
- Survey Data and Subjective Beliefs in Business Cycle Models (with Anmol Bhandari and Paul Ho)Previously circulated as NBER WP No. 22225 "
*Identifying ambiguity shocks in business cycle models using survey data*"Survey data on household forecasts for unemployment and inflation rates reveal large upward biases that are positively correlated and countercyclical. We develop a framework to analyze general equilibrium settings where agents' subjective beliefs are endogenous and shaped by time-varying concerns for model misspecification. Applying our framework to a New-Keynesian model with frictional labor markets, we find that, consistent with the survey evidence, an increase in concerns for model uncertainty generates large belief distortions, which reduce aggregate demand and propagate through frictional goods and labor market to cause a contraction. As part of our analysis we also develop solution techniques that preserve the effects of time-varying concerns for model misspecification in the class of linear solutions. -
*Revise and resubmit, Journal of Political Economy*[PDF, 16 March 2018] [bibtex] Circulated under the title*Heterogeneous Beliefs under Recursive Preferences*.Online appendix with additional discussion and resultsI study the long-run behavior of an economy with two types of agents who differ in their beliefs and are endowed with homothetic recursive preferences of the Duffie-Epstein-Zin 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 long-run outcomes where both agents survive, or more incorrect agents dominate. I derive analytical conditions for the existence of nondegenerate long-run 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 consumption-saving choice of agents with heterogeneous beliefs, and the role of equilibrium prices in shaping long-run outcomes.

# Work in progress

**Discount Rates and Employment Fluctuations**(with Katarína Borovičková)**Optimal Policies with Robust Concerns**(with Anmol Bhandari and Paul Ho)**Robust Preference Expansions**(with Lars Peter Hansen, available on request)**Pricing Rare Events**(with Lars Peter Hansen and José Scheinkman)

# Publications

- Misspecified Recovery (with Lars Peter Hansen and José Scheinkman)
**Journal of Finance (2016) 71 (6): 2493-2544.***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 risk-adjusted discounting, we use Perron-Frobenius Theory to isolate a positive martingale component of the stochastic discount factor process. This component recovers a probability measure that absorbs long-term risk adjustments. When the martingale is not degenerate, surmising that this recovered probability captures investors' beliefs distorts inference about risk-return tradeoffs. Stochastic discount factors in many structural models of asset prices have empirically relevant martingale components. - Term Structure of Uncertainty in the Macroeconomy (with Lars Peter Hansen)
**Handbook of Macroeconomics: Volume 2B (2016) Chapter 20, Elsevier B.V., 1641-1696.**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. - Shock Elasticities and Impulse Responses (with Lars Peter Hansen and José Scheinkman)
**Mathematics and Financial Economics (2014) 8 (4): 333-354.**We construct shock elasticities that are pricing counterparts to impulse response functions. Recall that impulse response functions measure the importance of next-period 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. - Examining Macroeconomic Models through the Lens of Asset Pricing (with Lars Peter Hansen)
**Journal of Econometrics (2014) 183 (1): 67-90.**Shock elasticity toolbox for the computation of shock elasticities in models solved by Dynare/Dynare++ can be downloaded from the software page.We develop new methods for representing the asset-pricing implications of stochastic general equilibrium models. We provide asset-pricing 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 continuous-time methods developed in*Hansen and Scheinkman (2012)*and*Borovička, Hansen, Hendricks, and Scheinkman (2011)*by constructing discrete-time, state-dependent, shock-exposure and shock-price elasticities as functions of the investment horizon. Our methods are applicable to economic models that are nonlinear, including models with stochastic volatility.Dynare model code [ZIP file] for the example in Section 7, based on the paper by Ai, Croce and Li (2010). Use in conjunction with the shock elasticity toolbox to produce the shock elasticities. - Risk-Price Dynamics (with Lars Peter Hansen, Mark Hendricks, and José Scheinkman)
**Journal of Financial Econometrics (2011) 9 (1): 3-65.**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 cash-flow 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.