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
Applied researchers using structural models under rational expectations (RE) often confront empirical evidence of misspecification. In this paper we consider a generic dynamic model that is posed as a vector of unconditional moment restrictions. We suppose that the model is globally misspecified under RE, and thus empirically flawed in a way that is not econometrically subtle. We relax the RE restriction by allowing subjective beliefs to differ from the data-generating probability (DGP) model while still maintaining that the moment conditions are satisfied under the subjective beliefs of economic agents. We use statistical measures of divergence relative to RE to bound the set of subjective probabilities. This form of misspecification alters econometric identification and inferences in a substantial way, leading us to construct robust confidence sets for various set identified functionals.
JEL Classification: C14, C15, C31, C33, G40
Keywords: Subjective beliefs, bounded rationality, misspecification sets, nonlinear expectation, divergence, Lagrange multipliers, stochastic dual programming, confidence sets
Ninety years ago, Slutsky (1927) and Yule (1927) opened the door to the use of probability models in the analysis of economic time series. Their vision was to view economic time series as linear responses to current and past independent and identically distributed impulses or shocks. In distinct contributions, they showed how to generate approximate cycles with such models. Each had a unique background and perspective. Yule was an eminent statistician who, in the words of Stigler (1986), among his many contributions, managed effectively to invent modern time series analysis.” Yule constructed and estimated what we call a second-order model and applied it to study the time series behavior of sunspots. Slutsky wrote his paper in Russia in the 1920s motivated by the study of business cycles. Much later, his paper was published in Econometrica, but it was already on the radar screen of economists, such as Frisch. Indeed Frisch was keenly aware of both Slutsky (1927) and Yule (1927) and acknowledged both in his seminal paper Frisch (1933) on the impulse and propagation problem. Building on insights from Slutsky and Yule, Frisch pioneered the use of impulse response functions in economic dynamics. His ambition was to provide explicit economic interpretations for how current period shocks alter economic time series in current and future time periods. The Journal of Political Economy (JPE) provided an important platform for research that confronts Frisch’s ambition in substantively interesting ways. Read full paper here.
Investors face uncertainty over models when they do not know which member of a set of well-defined “structured models” is best. They face uncertainty about mod-els when they suspect that all of the structured models might be misspecified. We refer to worries about the first type of ignorance as ambiguity concerns and worries about the second type as misspecification concerns. These two types of ignorance about probability distributions of risks add what we call uncertainty components to equilibrium prices of those risks. A quantitative example highlights a representa-tive investor’s uncertainties about the size and persistence of macroeconomic growth rates. Our model of preferences under concerns about model ambiguity and misspec-ification puts nonlinearities into marginal valuations that induce time variations in market prices of uncertainty. These reflect the representative investor’s fears of high persistence of low growth rate states and low persistence of high growth rate states.
For the Non-Expert:
Vox EU: Acknowledging and pricing macroeconomic uncertainties by Lars Peter Hansen and Thomas J. Sargent
This paper studies estimators that make sample analogues of population orthogonality conditions close to zero. Strong consistency and asymptotic normality of such estimators is established under the assumption that the observable variables are stationary and ergodic. Since many linear and nonlinear econometric estimators reside within the class of estimators studied in this paper, a convenient summary of the large sample properties of these estimators, including some whose large sample properties have not heretofore been discussed, is provided.
This paper studies estimators that make sample analogues of population orthogonality conditions close to zero. Strong consistency and asymptotic normality of such estimators is established under the assumption that the observable variables are stationary and ergodic. Since many linear and nonlinear econometric estimators reside within the class of estimators studied in this paper, a convenient summary of the large sample properties of these estimators, including some whose large sample properties have not heretofore been discussed, is provided.
This paper proves two propositions about identification in a continuous time version of a linear stochastic rational expectations model. The model is a continuous time version of Lucas and Prescott (1971), in which’the equilibrium can be interpreted.~ the solution of a ‘stochastic control problem, either of a collection of private agents or of a fictitious “socialplanner. Estimation is directed toward isolating the parameters of the agent’s objective function and of the stochastic processes of the forcing functions that the agent faces. This approach has been advocated by Lucas (1967, 1976), Lucas and Prescott (1971), and Lucas and Sargent (1981) as offering the potential to analyze an interesting class of policy interventions promised by structural models, while meeting the criticisms of most econometric policy evaluation methods that were made by Lucas (1976). At the same time, inspired by the work of Sims (1971), Geweke (1978), and P.C.B. Phillips (1972, 1973, 1974), we want to estimate models in which optimizing economic agents make decisions at finer time intervals than the interval of time between the observations used by the econometrician. We adopt a continuous time theoretical framework both because it is an interesting limiting case because it has received extensive attention in the theoretical and the econometric literatures.
