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A decision maker constructs a convex set of nonnegative martingales to use as likelihood ratios that represent parametric alternatives to a baseline model and also non-parametric models statistically close to both the baseline model and the parametric alternatives. Max-min expected utility over that set gives rise to equilibrium prices of model uncertainty expressed as worst-case distortions to drifts in a representative investor's baseline model. We offer quantitative illustrations for baseline models of consumption dynamics that display long-run risk. We describe a set of parametric alternatives that generates countercyclical prices of uncertainty.
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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.
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Asset prices contain information about the probability distribution of future states and the stochastic discounting of these states. Without additional assumptions, probabilities and stochastic discounting cannot be separately identified. Ross (2013) introduced a set of assumptions that restrict the dynamics of the stochastic discount factor in a way that allows for the recovery of the underlying probabilities. We use decomposition results for stochastic discount factors from Hansen and Scheinkman (2009) to explain when this procedure leads to misspecified recovery. We also argue that the empirical evidence on asset prices indicates that the recovered measure would differ substantially from the actual probability distribution and that interpreting this measure as the true probability distribution may severely bias our inference about risk premia, investors' aversion to risk, and the welfare cost of economic fluctuations.
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The design and conduct of climate change policy necessarily confronts uncertainty along multiple fronts. We explore the consequences of ambiguity over various sources and configurations of models that impact how economic opportunities could be damaged in the future. We appeal to decision theory under risk, model ambiguity and misspecification concerns to provide an economically motivated approach to uncertainty quantification. We show how this approach reduces the many facets of uncertainty into a low dimensional characterization that depends on the uncertainty aversion of a decision-maker or fictitious social planner. In our computations, we take inventory of three alternative channels of uncertainty and provide a novel way to assess them. These include i) carbon dynamics that capture how carbon emissions impact atmospheric carbon in future time periods; ii) temperature dynamics that depict how atmospheric carbon alters temperature in future time periods; iii) damage functions that quantify how temperature changes diminish economic opportunities. We appeal to geoscientific modeling to quantify the first two channels. We show how these uncertainty sources interact for a social planner looking to design a prudent approach to the social pricing of carbon emissions.
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Asset prices contain information about the probability distribution of future states and the stochastic discounting of these states. Without additional assumptions, probabilities and stochastic discounting cannot be separately identified. Ross (2013) introduced a set of assumptions that restrict the dynamics of the stochastic discount factor in a way that allows for the recovery of the underlying probabilities. We use decomposition results for stochastic discount factors from Hansen and Scheinkman (2009) to explain when this procedure leads to misspecified recovery. We also argue that the empirical evidence on asset prices indicates that the recovered measure would differ substantially from the actual probability distribution and that interpreting this measure as the true probability distribution may severely bias our inference about risk premia, investors' aversion to risk, and the welfare cost of economic fluctuations.
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A decision maker constructs a convex set of nonnegative martingales to use as likelihood ratios that represent parametric alternatives to a baseline model and also non-parametric models statistically close to both the baseline model and the parametric alternatives. Max-min expected utility over that set gives rise to equilibrium prices of model uncertainty expressed as worst-case distortions to drifts in a representative investor's baseline model. We offer quantitative illustrations for baseline models of consumption dynamics that display long-run risk. We describe a set of parametric alternatives that generates countercyclical prices of uncertainty.
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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.
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This paper develops a new method informed by data and models to recover information about investor beliefs. Our approach uses information embedded in forward-looking asset prices in conjunction with asset pricing models. We step back from presuming rational expectations and entertain potential belief distortions bounded by a statistical measure of discrepancy. Additionally, our method allows for the direct use of sparse survey evidence to make these bounds more informative. Within our framework, market-implied beliefs may differ from those implied by rational expectations due to behavioral/psychological biases of investors, ambiguity aversion, or omitted permanent components to valuation. Formally, we represent evidence about investor beliefs using a novel nonlinear expectation function deduced using model-implied moment conditions and bounds on statistical divergence. We illustrate our method with a prototypical example from macro-finance using asset market data to infer belief restrictions for macroeconomic growth rates.
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