
with Lars Peter Hansen and Balint Szoke and Lloyd S. Han
December 2018
A decision maker constructs a convex set of nonnegative martingales to use as likelihood 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. Maxmin expected utility
over that set gives rise to equilibrium prices of model uncertainty expressed as worstcase distortions to drifts in a representative
investor's baseline model. Three quantitative illustrations start with baseline models having exogenous longrun risks in technology shocks.
These put endogenous longrun risks into consumption 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.

with Lars Peter Hansen
December 2018
A representative investor confronts two levels of model uncertainty. The investor has a set of well defined parametric ``structured models'' but does
not know which of them is best. The investor also suspects that all of the structured models are misspecified. These uncertainties about
probability distributions of risks give rise to components of equilibrium prices that differ from the well understood risk prices widely used in
asset pricing theory. A quantitative example highlights a representative investor's uncertainties about the size and persistence of
macroeconomic growth rates. Our model puts nonlinearities into marginal valuations that induce time variations in market prices of uncertainty.
These arise because the representative investor especially fears high persistence of low growth rate states and low persistence of high growth
rate states.

with Lars Peter Hansen
November 2018
An ambiguity averse decision maker evaluates plans under a restricted family of what we call structured models and unstructured
alternatives that are statistically close to them. The structured models can include parametric models in which parameter values vary
over time in ways that the decision maker cannot describe probabilistically. Because he suspects that all parametric models are
misspecified, the decision maker also evaluates plans under alternative probability distributions with much less structure.

with Lars Peter Hansen
May 2014
This paper studies alternative ways of representing uncertainty about a law of motion in a version of a classic macroeconomic targeting problem of Milton Friedman (1953). We study both "unstructured uncertainty"  ignorance of the conditional distribution of the target next period as a function of states and controls  and more "structured uncertainty"  ignorance of the probability distribution of a response coefficient in an otherwise fully trusted specification of the conditional distribution of next period's target. We study whether and how different uncertainties affect Friedman's advice to be cautious in using a quantitative model to fine tune macroeconomic outcomes.

with Martin Ellison
July 2012
The welfare cost of random consumption fluctuations is known from De Santis (2007) to be increasing in the level of individual consumption risk in the economy. It is also known from Barillas et al. (2009) to increase if agents in the economy care about robustness to model misspecification. In this paper, we combine these two effects and calculate the cost of business cycles in an economy with consumers who face individual consumption risk and who fear model misspecification. We find that individual risk has a greater impact on the cost of business cycles if agents already have a preference for robustness. Correspondingly, we find that endowing agents with concerns about a preference for robustness is more costly if there is already individual risk in the economy. The combined effect exceeds the sum of the individual effects.

with Lars Peter Hansen
July 2012
For each of three types of ambiguity, we compute a robust Ramsey plan and an associated worstcase probability model. Ex post, ambiguity of type I implies endogenously distorted homogeneous beliefs, while ambiguities of types II and III imply distorted heterogeneous beliefs. Martingales characterize alternative probability specifications and clarify distinctions among the three types of ambiguity. We use recursive formulations of Ramsey problems to impose local predictability of commitment multipliers directly. To reduce the dimension of the state in a recursive formulation, we transform the commitment multiplier to accommodate the heterogeneous beliefs that arise with ambiguity of types II and III. Our formulations facilitate comparisons of the consequences of these alternative types of ambiguity.

with Lars Peter Hansen
January 2011
We formulate two continuoustime hidden Markov models in which a decision maker distrusts both his model of state dynamics and a prior distribution of unobserved states. We use relative entropy's role in statistical model discrimination % using historical data, we use measures of statistical model detection to modify Bellman equations in light of model ambiguity and to calibrate parameters that measure ambiguity. We construct two continuous time models that are counterparts of two discretetime recursive models of \cite{hansensargent07}. In one, hidden states appear in continuation value functions, while in the other, they do not. The formulation in which continuation values do not depend on hidden states shares features of the smooth ambiguity model of Klibanoff, Marinacci, and Mukerji. For this model, we use our statistical detection calculations to guide how to adjust contributions to entropy coming from hidden states as we take a continuous time limit.

June 28, 2010
Keynote address at ICORES10, Prague, June 28, 2010 given by Professor Stephen Stigler of the University of Chicago.

May 2010
This paper with Lars Hansen corrects typos that appeared in the version that was published in 2007 in the Journal of Economic Theory. The corrections appear in blue.

May 2010
This paper with Lars Hansen corrects typos that appeared in the version that was published in 2005 in the Journal of Economic Theory. The corrections appear in blue.

with Martin Ellison
July 2010
In this thoroughly revised version, we defend the forecasting performance of the FOMC from the recent criticism of Christina and David Romer. One argument is just to graph the data and note that the discrepancies spotted by Romer and Romer are small, expecially after Greenspan took over from Volcker. We spend most of our time on another more sophisticated argument. This argument is that the FOMC forecasts a worstcase scenario that it uses to design decisions that will work well enough (are robust) despite possible misspecification of its model. Because these FOMC forecasts are not predictions of what the FOMC expects to occur under its model, it is inappropriate to compare their performance in a horse race against other forecasts. Our interpretation of the FOMC as a robust policymaker can explain all the findings of the Romers and rationalises differences between FOMC forecasts and forecasts published in the Greenbook by the staff of the Federal Reserve System.

with Lars Peter Hansen
May 2010
This is a survey paper about exponential twisting as a model of model distrust. We feature examples from macroeconomics and finance. The paper is for a handbook of Monetary Economics edited by Benjamin Friedman and Michael Woodford.

by Anastasios G. Karantounias (with Lars Peter Hansen and Thomas J. Sargent)
October 2009
This paper studies an optimal fiscal policy problem of Lucas and Stokey (1983) but in a situation in which the representative agent's distrust of the probability model for government expenditures puts model uncertainty premia into historycontingent prices. This gives rise to a motive for expectation management that is absent within rational expectations and a novel incentive for the planner to smooth the shadow value of the agent's subjective beliefs in order to manipulate the equilibrium price of government debt. Unlike the Lucas and Stokey (1983) model, the optimal allocation, tax rate, and debt all become history dependent despite complete markets and Markov government expenditures.

with Lars Peter Hansen
January 2009
This paper is a comprehensive overhaul of our earlier paper ``Fragile Beliefs and the Price of Model Uncertainty’. A representative consumer uses Bayes' law to learn about parameters and to construct probabilities with which to perform ongoing model averaging. The arrival of signals induces the consumer to alter his posterior distribution over parameters and models. The consumer copes with specification doubts by slanting probabilities pessimistically. One of his models puts longrun risks in consumption growth. The pessimistic probabilities slant toward this model and contribute a countercyclical and signalhistorydependent component to prices of risk We use detection error probabilities to discipline risksensitivity parameters.

with Lars Peter Hansen
December 2008
We use two risksensitivity operators to construct the stochastic discount factor for a representative consumer who evaluates consumption streams in light of parameter estimation and model selection problems that present long run risks. The arrival of signals induces the consumer to alter his posterior distribution over models and parameters. The consumer expresses his doubts about model specifications and priors by slanting them in directions that are pessimistic in terms of value functions. His twistings over model probabilities give rise to timevarying model uncertainty premia that contribute a volatile timevarying component to the marketprice of model uncertainty.

with Lars Peter Hansen and Ricardo Mayer
October 30, 2008
For linear quadratic Gaussian problems, this paper uses two risksensitivity operators defined by Hansen and Sargent to construct decision rules that are robust to misspecifications of (1) transition dynamics for possibly hidden state variables, and (2) a probability density over hidden states induced by Bayes' law. Duality of risksensitivity to the `multiplier preferences’ minmax expected utility theory of Hansen and Sargent allows us to compute risksensitivity operators by solving twoplayer zerosum games. That the approximating model is a Gaussian joint probability density over sequences of signals and states gives important computational simplifications. We exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. In Games I, II, and III, the minimizing players' worstcase densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of \citet{hs2005a} that builds in time consistency. We describe how detection error probabilities can be used to calibrate the risksensitivity parameters that govern fear of model misspecification in hidden Markov models.

with Timothy Cogley, Riccardo Colacito, and Lars Peter Hansen
January 11, 2008
We study how a concern for robustness modifies a policy maker's incentive to experiment. A policy maker has a prior over two submodels of inflationunemployment dynamics. One submodel implies an exploitable tradeoff, the other does not. Bayes' law gives the policy maker an incentive to experiment. The policy maker fears that both submodels and his prior probability distribution over them are misspecified. We compute decision rules that are robust to misspecifications of each submodel and of a prior distribution over submodels. We compare robust rules to ones that Cogley, Colacito, and Sargent (2007) computed assuming that the models and the prior distribution are correctly specified. We explain how the policy maker's desires to protect against misspecifications of the submodels, on the one hand, and misspecifications of the prior over them, on the other, have different effects on the decision rule.

with Lars Peter Hansen
November 22, 2006
Responding to criticisms of Larry Epstein and his coauthors, this paper describes senses in which various representations of preferences from robust control are or are not time consistent. We argue that the senses in which preferences are not time consistent do not hinder applications.

with Francisco Barillas and Lars Peter Hansen
July 2008
Reinterpreting most of the market price of risk as a market price of model uncertainty eradicates the link between asset prices and measures of the welfare costs of aggregate fluctuations that were proposed by Hansen, Sargent, and Tallarini (1999), Tallarini (2000), and Alvarez and Jermann (2004). Market prices of model uncertainty contain informationabout compensation for removing model uncertainty, not the consumption fluctuations that Lucas (1987, 2003) studied. By using the preference specification of Kreps and Porteus with intertemporal elasticity of one put the mean and standard deviation of the stochastic discount factor close to the bounds of Hansen and Jagannathan (1991), but only for very high values of a risk aversion parameter, and he needed a substantially higher risk aversion parameter for a trendstationary model of consumption than for a random walk model. A maxmin expected utility theory lets us reinterpret Tallarini's riskaversion parameter as measuring a representative consumer's doubts about the model specification. We use model detection error probabilities instead of riskaversion experiments to calibrate that parameter. Values of detection error probabilities that imply a somewhat but not overly cautious representative consumer give market prices of model uncertainty that approach the HansenJagannathan bounds. Fixed detection error probabilities give rise to virtually identical asset prices for Tallarini's two models of consumption growth. We calculate the welfare costs of removing model uncertainty and find that they are large.

with Lars Peter Hansen
June 2005
In a Markov decision problem with hidden state variables, a decision maker expresses fears that his model is misspecified by surrounding it with a set of alternatives that are nearby as measured by their expected log likelihood ratios (entropies).Sets of martingales represent alternative models. Within a twoplayer zerosum game under commitment, a minimizing player chooses a martingale at time $0$.Probability distributions that solve distorted filtering problems serve as state variables, much like the posterior in problems without concerns about misspecification. We state conditions under which an equilibrium of the zerosum game with commitment has a recursive representation that can be cast in terms of two risksensitivity operators. We apply our results to a linear quadratic example that makes contact with the analysis of Basar and Bernhard (1995) and Whittle (1990).

with Lars Peter Hansen
May 2006
In a Markov decision problem with hidden state variables, a posterior distribution serves as a state variable and Bayes' law under the approximating model gives its law of motion. A decision maker expresses fear that his model is misspecified by surrounding it with a set of alternatives that are nearby as measured by their expected log likelihood ratios (entropies). Sets of martingales represent alternative models. A decision maker constructs a sequence of robust decision rules by pretending that there is a sequence of minimizing players who choose increments to a martingale from within this set. One risk sensitivity operator induces robustness to perturbations of the approximating model conditioned on the hidden state. Another risk sensitivity operator induces robustness with respect to a prior distribution over the hidden state. We thereby extend the approach of Hansen and Sargent (IEEE Transactions on Automatic Control, 1995) to problems that contain hidden states. We study linear quadratic examples.

with Lars Peter Hansen, Gauhar Turmuhambetova, and Noah Williams
September 2005
This paper integrates a variety of results in robust control theory in the context of an approximating model that is a diffusion. The paper is partly a response to some criticisms of Anderson, Hansen, and Sargent (see below) by Chen and Epstein. It formulates two robust control problems  a multiplier problem from the literature on robust control and a constraint formulation that looks like GilboaSchmeidler's minmax expected utility theory. The paper studies the connection between the two problems, states an observational equivalence result for them, links both problems to `risk sensitive' optimal control, and discusses time consistency of the preference orderings associated with the two robust control problems.

with Lars Hansen
2004
Prepared for a Fed conference in honor of Dale Henderson, Richard Porter, and Peter Tinsley
The paper reviews how the structure of the SimonTheil certainty equivalence result extends to models that incorporate a preference for robustness to model uncertainty. A model of precautionary savings is used an example.

with Evan Anderson and Lars Hansen
April 2003
This paper supersedes `Risk and Robustness in Equilibrium’, also on this web page. A representative agent fears that his model, a continuous time Markov process with jump and diffusion components,is misspecified and therefore uses robust control theory to make decisions. Under the decision maker's approximating model, that cautious behavior puts adjustments for model misspecification into market prices for risk factors. We use a statistical theory of detection to quantify how much model misspecification the decision maker should fear, given his historical data record. A semigroup is a collection of objects connected by something like the law of iterated expectations. The law of iterated expectations defines the semigroup for a Markov process, while similar laws define other semigroups. Related semigroups describe (1) an approximating model; (2) a model misspecification adjustment to the continuation value in the decision maker's Bellman equation;(3) asset prices; and (4) the behavior of the model detection statistics that we use to calibrate how much robustness the decision maker prefers. Semigroups 2, 3, and 4 establish a tight link between the market price of uncertainty and a bound on the error in statistically discriminating between an approximating and a worst case model.

with Lars Hansen
November 19, 2002
This is a comprehensive revision of an earlier paper with the same title. We describe an equilibrium concept for models with multiple agents who, as under rational expectations share a common model, but all of whom doubt their model, unlike rational expectations. Agents all fear model misspecification and perform their own worstcase analyses to construct robust decision rules. Although the agents share the approximating models, their differing preferences cause their worstcase models to diverge. We show how to compute Stackelberg (or Ramsey) plans where both leaders and followers fear model misspecification.

with Lars Peter Hansen
January 22, 2001
Paper prepared for presentation at the meetings of the American Economic Association in New Orleans , Jan 5, 2001 . This paper is a summary of results presented in more detail in Hansen, Sargent, Turmuhambetova, and Williams (2001)  see below. That paper formulates two robust control problems  a multiplier problem from the literature on robust control and a constraint formulation that looks like GilboaSchmeidler's minmax expected utility theory.

with Marco Cagetti, Lars Peter Hansen, and Noah Williams
January 2001
A continuous time asset pricing model with robust nonlinear filtering of a hidden Markov state.

with Lars Peter Hansen
December 2000
The text of Sargent's Frisch lecture at the 2000 World Congress of the Econometric Society; also the basis for Sargent's plenary lecture at the Society for Economic Dynamics in Costa Rica, June 2000.

with Lars Peter Hansen and Neng Wang
August 25, 2000
This paper reformulate Hansen, Sargent, and Tallarini's 1999 (RESTud) model by concealing elements of the state from the planner and the agents, forcing them to filter. The paper describes how jointly to do robust filtering and control, then computes the appropriate `market prices of Knightian uncertainty.' Detection error probabilities are used to discipline the one free parameter that robust decision making adds to the standard rational expectations paradigm.

Risk and Robustness in General Equilibrium
with Evan Anderson and Lars Hansen
March 1998
This paper describes a preference for robust decision rules in discrete time and continuous time models. The paper extends earlier work of Hansen, Sargent, and Tallarini in several ways. It permits nonlinearquadratic Gaussian set ups. It develops links between asset prices and preferences for robustness. It links premia in asset prices from Knightian uncertainty to detection error statistics for discriminating between models.