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with Lars Peter Hansen
September 2023
What are ``deep uncertainties'' and how should their presence alter
prudent courses of action? To help answer these questions, we bring ideas from robust
control theory into statistical decision theory. Decision theory in economics has its
origins in axiomatic formulations by von Neumann and Morgenstern as well as the
statisticians Wald and Savage. Since Savage's fundamental work, economists have provided
alternative axioms that formalize a notion of ambiguity aversion. Meanwhile, control
theorists created another way to construct decision rules that are robust to potential model
misspecifications. We reinterpret axiomatic foundations of some modern decision theories to
include ambiguity about a prior to put on a family of models simultaneously with concerns
about misspecifications of the corresponding likelihood functions. By building on ideas from
dynamic programming, our representations have recursive structures that preserve dynamic
consistency.
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with Lars Peter Hansen
November 2020
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.
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with Lars Peter Hansen
March 2020
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 models 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 representative 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.
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with Lars Peter Hansen and Balint Szoke and Lloyd S. Han
February 2020
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. 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 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.
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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.
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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.
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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.
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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.
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June 28, 2010
Keynote address at ICORES10, Prague, June 28, 2010 given by Professor Stephen Stigler of the
University of Chicago.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.