Abstract
What are “deep uncertainties,” and how should their presence influence prudent decisions? To address these questions, we bring ideas from robust control theory into statistical decision theory. Decision theory has its origins in axiomatic formulations by von Neumann and Morgenstern, Wald, and Savage. After Savage, decision theorists constructed axioms that formalize a notion of ambiguity aversion. Meanwhile, control theorists constructed decision rules that are robust to some model misspecifications. We reinterpret axiomatic foundations of decision theories to express ambiguity about a prior over a family of models along with concerns about misspecifications of the corresponding likelihood functions.
Keywords— deep uncertainty, ambiguity, misspecification, variational preferences, statistical divergence, relative entropy, prior, likelihood
JEL Codes— C10, C14, C18
Abstract
We study asset pricing implications of a revealing and tractable formulation of smooth ambiguity investor preferences in a continuous-time environment. Investors do not observe a hidden Markov state and instead make inferences about this state using past data. We show that ambiguity about this hidden state distribution alters investor decisions and equi-librium asset prices. Our continuous-time formulation allows us to apply recursive filtering and Hamilton-Jacobi-Bellman methods to solve the modified decision problem. Using such methods, we show how characterizations of portfolio allocations and local uncertainty-return trade-offs change when investors are ambiguity-averse.
Keywords— Risk, ambiguity, robustness, asset pricing, portfolio allocation, continuous time
Related: Read Research Reflection by Hansen – “Navigating Uncertainty” March 11, 2022
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Geophysicists examine and document the repercussions for the earth’s climate induced by alternative emission scenarios and model specifications. Using simplified approximations, they produce tractable characterizations of the associated uncertainty. Meanwhile, economists write simplified damage functions to assess uncertain feedbacks from climate change back to the economic opportunities for the macroeconomy. How can we assess both climate and emissions impacts, as well as uncertainty in the broadest sense, in social decision-making? We provide a framework for answering this question by embracing recent decision theory and tools from asset pricing, and we apply this structure with its interacting components in a revealing quantitative illustration.
Online Appendix
Associated Results and Python Scripts – GitHub
A decision maker is averse to not knowing a prior over a set of restricted structured models (ambiguity) and suspects that each structured model is misspecified. The decision maker evaluates intertemporal plans under all of the structured models and, to recognize possible misspecifications, under unstructured alternatives that are statistically close to them. Likelihood ratio processes are used to represent unstructured alternative models, while relative entropy restricts a set of unstructured models. A set of structured models might be finite or indexed by a finite-dimensional vector of unknown parameters that could vary in unknown ways over time. We model such a decision maker with a dynamic version of variational preferences and revisit topics including dynamic consistency and admissibility.
A decision maker constructs a convex set of nonnegative martingales to use as likeli-hood ratios that represent alternatives that are statistically close to a decision maker’s baseline model. The set is twisted to include some specific models of interest. 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. Three quantitative illustrations start with baseline models having exogenous long-run risks in technology shocks. These put endogenous long-run risks into con-sumption dynamics that differ in details that depend on how shocks affect returns to capital stocks. We describe sets of alternatives to a baseline model that generate countercyclical prices of uncertainty.
Keywords— Risk, uncertainty, relative entropy, robustness, asset prices, exponential quadratic stochastic discount factor
JEL Classification— C52, C58, D81, D84, G12
Paper
Associated Paper Results and Code
In many dynamic economic settings, a decision maker finds it challenging to quantify the uncertainty or to assess the potential for mistakes in models. We explore alternative ways of acknowledging these challenges by drawing on insights from decision theory as conceptualized and implemented in statistics, engineering, and economics. Building on prior research, we suggest tractable and revealing ways to incorporate behavioral responses to uncertainty, broadly conceived. Our analysis adopts recursive intertemporal preferences for decision makers that allow them to be ambiguity averse and concerned about the potential misspecification of subjective uncertainty. By design, these representations have revealing implications for continuous-time environments with Brownian information structures. Problems where uncertainty’s structure is obscure such as macroeconomics, finance and climate change are promising areas for application of these tools.
Supplemental Index
A planner and agent in a permanent-income economy cannot observe part of the state, regard their model as an approximation, and value decision rules that are robust across a set of models. They use robust decision theory to choose allocations. Equilibrium prices reflect the preference for robustness and so embody a “market price of Knightian uncertainty.” We compute market prices of risk and compare them with a model that assumes that the state is fully observed. We use detection error probabilities to constrain a single parameter that governs the taste for robustness.
We study how decision-makers’ concerns about robustness affect prices and quantities in a stochastic growth model. In the model economy, growth rates in technology are altered by infrequent large shocks and continuous small shocks. An investor observes movements in the technology level but cannot perfectly distinguish their sources. Instead the investor solves a signal extraction problem. We depart from most of the macroeconomics and finance literature by presuming that the investor treats the specification of technology evolution as an approximation. To promote a decision rule that is robust to model misspecification, an investor acts as if a malevolent player threatens to perturb the actual data-generating process relative to his approximating model. We study how a concern about robustness alters asset prices. We show that the dynamic evolution of the risk-return trade-off is dominated by movements in the growth-state probabilities and that the evolution of the dividend-price ratio is driven primarily by the capital-technology ratio.
