## Sets of Models and Prices of Uncertainty

with Lars Peter Hansen

March 2015

A representative consumer expresses distrust of a baseline probability model by using
a convex set of martingales as likelihood ratios to represent other probability models.
The consumer constructs the set to include martingales that represent particular
parametric alternatives to the baseline model as well as others representing only
vaguely specified models statistically close to the baseline model. The representative
consumer’s max-min expected utility over that set gives rise to equilibrium prices
of model uncertainty expressed as worst-case distortions to the drifts in his baseline
model. We calibrate a quantitative example to aggregate US consumption data.

## Four Types of Ignorance

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
targetting 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 coeffcient in an otherwise fully
trusted specication 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.

## Welfare Cost of Business Cycles in Economies with Individual Consumption Risk

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.

## Three Types of Ambiguity

with Lars Peter Hansen

July 2012

For each of three types of ambiguity, we compute a robust Ramsey plan and an associated
worst-case 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.

## Robustness and ambiguity in continuous time

with Lars Peter Hansen

January 2011

We formulate two continuous-time 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 discrete-time 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.

## The Changing History of Robustness, by Stephen Stigler

June 28, 2010

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

## Recursive Robust Estimation and Control without Commitment

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.

## Robust Estimation and Control under Commitment

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.

## A defence of the FOMC

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
worst-case 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.

## Wanting robustness in macroeconomics

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.

## Managing expectations and fiscal policy

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 history-contingent
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.

## Fragile beliefs and the price of uncertainty

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 long-run risks in consumption growth. The pessimistic probabilities slant toward this model and contribute a counter-cyclical and signal-history-dependent component to prices of risk We use detection error probabilities to discipline risk-sensitivity parameters.

## Fragile beliefs and the price of model uncertainty

with Lars Peter Hansen

December 2008

We use two risk-sensitivity 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 time-varying model uncertainty premia that contribute a volatile time-varying component to the marketprice of model uncertainty.

## Robust Hidden Markov LQG Problems

with Lars Peter Hansen and Ricardo Mayer

October 30, 2008

For linear quadratic Gaussian problems, this paper uses two risk-sensitivity 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 risk-sensitivity to the `multiplier preferences’ min-max expected utility theory of Hansen and Sargent allows us to compute risk-sensitivity operators by solving two-player zero-sum 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' worst-case 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 risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

## Robustness and U.S. Monetary Policy Experimentation

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 inflation-unemployment dynamics. One submodel implies an exploitable trade-off, 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.

## Time Inconsistency of Robust Control?

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.

## Doubts or Variability? (Formerly titled Reinterpreting a graph of Tallarini)

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 trend-stationary model of consumption than for a random walk model. A max-min expected utility theory lets us reinterpret Tallarini's risk-aversion parameter as measuring a representative consumer's doubts about the model specification. We use model detection error probabilities instead of risk-aversion 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 Hansen-Jagannathan 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.

## Robust Estimation and Control under Commitment

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 two-player zero-sum 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 zero-sum game with commitment has a recursive representation that can be cast in terms of two risk-sensitivity operators. We apply our results to a linear quadratic example that makes contact with the analysis of Basar and Bernhard (1995) and Whittle (1990).

## Recursive Robust Estimation and Control Without Commitment

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.

## Robust Control and Misspecification

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 Gilboa-Schmeidler's min-max 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.

## `Certainty equivalence’ and `model uncertainty’

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 Simon-Theil certainty equivalence result extends to models that incorporate a preference for robustness to model uncertainty. A model of precautionary savings is used an example.

The paper reviews how the structure of the Simon-Theil certainty equivalence result extends to models that incorporate a preference for robustness to model uncertainty. A model of precautionary savings is used an example.

## A Quartet of Semi-Groups for Model Specification, Robustness, Prices of Risk, and Model Detection

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.

## Robust control and filtering of forward-looking models

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 worst-case analyses to construct robust decision rules. Although the agents share the approximating models, their differing preferences cause their worst-case models to diverge. We show how to compute Stackelberg (or Ramsey) plans where both leaders and followers fear model misspecification.

## Robust Control and Model Uncertainty

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 Gilboa-Schmeidler's min-max expected utility theory.

## Robust Pricing with Uncertain Growth

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.

## Acknowledging Misspecification in Macroeconomic Theory

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.

## Robust Permanent Income and Pricing with Filtering

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 non-linear-quadratic 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.

## Discounted Linear Exponential Quadratic Gaussian Control

with Lars Peter Hansen