Click on the items listed below to see some recent research material.
· Working Papers on Robustness
· Published Papers on Robustness
· Working Papers on Risk and Valuation
· Published Papers on Risk and Valuation
· Working Papers on Operator Methods
· Published Papers on Operator Methods
Research on Robustness
Working Papers on Robustness [top]
- "Robustness, Estimation and Detection," with T. J. Sargent
(November 7
We propose adjustments for model ambiguity that survive in continuous time limits. Our formulations emerge from continuous-time versions of two discrete-time recursive models that Hansen and Sargent (2007) used for a decision maker who knows neither models nor distributions of unobserved states. We use expectations of likelihood ratios (relative entropies) to measure concerns about model misspecification and exploit the role of relative entropies in statistical methods for using historical data to discriminate between probability models. Statistical detection motivates our adjustment for model ambiguity and also helps us calibrate parameters that measure model ambiguity.
- "Doubts or Variability?" with F. Barillas and T. J. Sargent, forthcoming in Journal of Economic Theory. [LINK]
Abstract: Reinterpreting most of the market price of risk as a price of model uncertainty eradicates a link between asset prices and measures of the welfare costs of aggregate fluctuations that was proposed by Hansen et al., Tallarini, and Alvarez and Jermann [Lars Peter Hansen, Thomas Sargent, Thomas Tallarini, Robust permanent income and pricing, Rev. Econ. Stud. 66 (1999) 873–907; Thomas D. Tallarini, Risk-sensitive real business cycles, J. Monet. Econ. 45 (3) (2000) 507–532; Fernando Alvarez, Urban J. Jermann, Using asset prices to measure the cost of business cycles, J. Polit. Econ. 112 (6) (2004) 1223–1256]. Prices of model uncertainty contain information about the benefits of removing model uncertainty, not the consumption fluctuations that Lucas [Robert E. Lucas Jr., Models of Business Cycles, Basil Blackwell, Oxford and New York, 1987; Robert E. Lucas Jr., Macroeconomic priorities, American Economic Review, Papers and Proceedings 93 (2003) 1–14] studied. 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 instead of risk-aversion experiments to calibrate that parameter. Plausible values of detection error probabilities give prices of model uncertainty that approach the Hansen and Jagannathan [Lars Peter Hansen, Ravi Jagannathan, Implications of security market data for models of dynamic economies, J. Polit. Econ.99 (1991) 225–262] bounds. Fixed detection error probabilities give rise to virtually identical asset prices as well as virtually identical costs of model uncertainty for Tallarini's two models of consumption growth.
- "Fragile Beliefs and
the price of uncertainty," with T. J. Sargent
(March 5
Abstract: Concerns about misspecification and an enduring model selection problem in which one of the models has long run risks give rise to countercyclical risk premia. We use two risk-sensitivity operators to construct the stochastic discount factor for a representative consumer who evaluates consumption streams in light of model selection and parameter estimation problems that can aggravate or attenuate long run risks as time passes. The arrival of signals induces the consumer to alter his posterior distribution over models and parameters. The consumer copes with specification doubts by slanting probabilities pessimistically. These pessimistic model probabilities induce model uncertainty premia that contribute a time-varying component to what is ordinarily measured as the market price of risk.
- "Risk and Robustness
in Equilibrium," with E. W. Anderson and T. J. Sargent ,(March 8, 1998)
This early paper gives a continuous-time, stochastic formulation of robust control theory and a characterization of prices. While this paper was substantially revised and given a new title, the original manuscript remains interesting and is cited in our subsequent work. This paper provided the impetus for much of our subsequent work.
Published Papers on Robustness [top]
- “Beliefs, Doubts and Learning: Valuing Macroeconomic Risk,” Richard T. Ely Lecture, The American Economic Review Vol. 97, No. 2, May 2007. [LINK] [Ely Lecture Errata ]
Abstract: This essay examines the problem of inference
within a rational expectations model from two
perspectives: that of an econometrician and that
of the economic agents within the model. The
assumption of rational expectations has been
and remains an important component to quantitative
research. It endows economic decision
makers with knowledge of the probability law
implied by the economic model. As such, it is an
equilibrium concept. Imposing rational expectations
removed from consideration the need for
separately specifying beliefs or subjective components
of uncertainty. Thus, it simplified model
specification and implied an array of testable
implications that are different from those considered
previously. It reframed policy analysis
by questioning the effectiveness of policy levers
that induce outcomes that differ systematically
from individual beliefs.
Abstract: We study how a concern for robustness modifies a policy
marker’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 is misspecified.
We compute decision rules that are robust to misspecifications of each submodel and of the 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.
- "Robust Estimation
and Control without Commitment," with T. J. Sargent,
Journal of Economic Theory, 136
(2007) 1-27.
Abstract: In a Markov decision problem with hidden
state variables, a posterior distribution serves as a state variable and Bayes’ law under an 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 when measured by
their expected log likelihood ratios (entropies). Martingales represent
alternative models. A decision maker constructs a sequence of robust decision
rules by pretending that a sequence of minimizing players choose increments to
martingales and distortions to the prior over the hidden state. A risk
sensitivity operator induces robustness to perturbations of the approximating
model conditioned on the hidden state. Another risk sensitivity operator
induces robustness to the prior distribution over the hidden state. We use these
operators to extend the approach of Hansen and Sargent
[Discounted linear exponential quadratic Gaussian control, IEEE Trans. Automat.
Control 40(5) (1995) 968–971] to problems that contain hidden states.
- "Robust Control and
Model Misspecification," with T. J. Sargent,
G. Turmuhambetova, and N. Williams, Journal
of Economic Theory, 128 (2006) 45-90.
Abstract: A decision maker fears that data are generated by a staistical perturbation of an approximating model that is either a controlled difussion or a controlled measure over continuous functions of time. A perturbation is constrained by relative entropy. Several two-palyer zero-sum games yield robust decision rules and are related to one another and to max-min expected utility theory of Gilboa and Schmeidler (1989). Alternative sequential and non-sequential versions of robust control theory present identical robust decision rules taht are dynamically consistent in a useful sense.
- "Recursive Robust
Estimation and Control under Commitment," with T. J. Sargent Journal of Economic Theory, 124 (2005)
258-301.
Abstract: This paper studies robust decision problems with hidden state variables. It gives the recursive implementation of the commitment solution with discounting from robust control theory. The recursive implication shows formally how discounting and commitment are encoded in the robust decision rules. We suggest alternative recursive formulations of the decision problem that are attractive alternatives to the commitment solution.
- "Comment on Exotic
Preferences for Macroeconomics by D. K. Backus, B. R. Routledge
and S. E. Zin" , NBER Macroeconomics
Annual 2004. Edited by M. Gertler and K. Rogoff.
- "A Quartet of Semigroups for Model Specification, Robustness, Prices of Risk and Model Detection," E. W. Anderson, L. P. Hansen and T. J. Sargent, Journal of the European Economic Association, Volume 1, Issue 1, 2003. Link to the Issue
Abstract: 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, 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 of Forward Looking Models," L. P. Hansen and T. J. Sargent, Journal of Monetary Economics, Vol 50, Issue 3, 2003. Link to the Issue
Abstract: This
paper shows how to formulate and compute robust Ramsey (aka Stackelberg)
plans for linear models with forward-looking private agents. The leader and teh followers share a common approximating model and both
have preferences for robust decision rules because both doubt the model. Since
their preferences differ, the leader's and follower's decision rules are
fragile to different misspecifications of the approximating model. To compute a
Stackelberg equilibrium we formulate a Bellman
equation that is associated to an artificial single-agent robust control
problem. The artificial Bellman equation contains a description of the implementability constraints that include Euler equations
that describe the worst-case analysis of the followers. As an example, the
paper analyzes a model of a monopoly facing a competitive fringe.
- "Robust Permanent Income and Pricing," L. P. Hansen, T. J. Sargent and T. D. Tallarini, Jr., Review of Economic Studies, 66, 1999. JSTOR link to the Issue
Research on Risk and Valuation
Working Papers on Risk and Valuation [top]
- "Risk Price Dynamics," with José Scheinkman, Jaroslav Borovička, and Mark Hendricks (November 11, 2009).
We present a novel approach to depicting asset pricing dynamics by characterizing
shock exposures and prices for alternative investment horizons. We quantify the shock
exposures in terms of elasticities that measure the impact of a current shock on future
cash-flow growth. The elasticities are designed to accommodate nonlinearities in the
stochastic evolution modeled as a Markov process. Stochastic growth in the underlying
macroeconomy and stochastic discounting in the representation of asset values are
central ingredients in our investigation. We provide elasticity calculations in a series
of examples featuring consumption externalities, recursive utility, and jump risk.
- "Pricing Growth-Rate Risk," with José Scheinkman (November 11, 2009).
We characterize the compensation demanded by investors in equilibrium for in-
cremental exposure to growth-rate risk. Given an underlying Markov di usion that
governs the state variables in the economy, the economic model implies a stochas-
tic discount factor process S and a reference stochastic growth process G for the
macroeconomy. Both are modeled conveniently as multiplicative functionals of a multi-
dimensional Brownian motion. To study pricing we consider the pricing implications
of parameterized family of growth processes Gε, with Gº = G, as ε is made small. This
parameterization de nes a direction of growth-rate risk exposure that is priced using
the stochastic discount factor S. By changing the investment horizon we trace a term
structure of risk prices that shows how the valuation of risky cash
ows depends on the investment horizon. Using methods of Hansen and Scheinkman (2009), we characterize
the limiting behavior of the risk prices as the investment horizon is made arbitrarily
long.
- "Pricing Kernels and Stochastic Discount Factors," with Eric Renault, forthcoming in the Encyclopedia of Quantitative Finance. (May 22, 2009).
In this entry we characterize pricing kernels or stochastic discount factors that are used
to represent valuation operators in dynamic stochastic economies. A kernel is commonly used
mathematical term used to represent an operator. The term stochastic discount factor
extends concepts from economics and finance to include adjustments for risk. As we will
see, there is a tight connection between the two terms. The terms pricing kernel and
stochastic discount factor are often used interchangeably. After deriving convenient
representations for prices, we provide several examples of stochastic discount factors
and discuss econometric methods for estimation and testing of
asset pricing models that restrict the stochastic discount factors.
- "Modeling the Long
Run: Valuation in Dynamic Stochastic Economies" , (August 5,
2008).
I explore the equilibrium value implications of
economic models that incorporate reactions to a stochastic environment. I
propose a dynamic value decomposition (DVD) designed to distinguish components
of an underlying economic model that influence values over long horizons from
components that impact only the short run. To quantify the role of parameter
sensitivity and to impute long-term risk prices, I develop an associated
perturbation technique. Finally, I use DVD methods to study formally some
example economies and to speculate about others. A DVD is enabled by
constructing operators indexed by the elapsed time between the date of pricing
and the date of the future payoff (i.e. the future realization of a consumption
claim). Thus formulated, methods from applied mathematics permit me to
characterize valuation behavior as the time between price determination and
payoff realization becomes large. An outcome of this analysis is the construction
of a multiplicative martingale component of a process that is used to represent
valuation in a dynamic economy with stochastic growth. I contrast the
differences in the applicability between this multiplicative martingale method
and an additive martingale method familiar from time series analysis that is
used to identify shocks with long-run economic consequences.
- "Comment on Exotic
Preferences for Macroeconomics by D. K. Backus, B. R. Routledge
and S. E. Zin" , NBER Macroeconomics
Annual 2004. Edited by M. Gertler and K. Rogoff.
Published Papers on Risk and Valuation [top]
- "Long Term Risk: an
Operator Approach" , with J. Scheinkman,
Econometrica, Econometric Society, vol. 77(1), pages 177-234.
Abstract: We
build a familiy of valuation operators indexed by the
increment of time between the payoff date and the current period value. These
operators are necessarily related by what is know as
· "Consumption Strikes Back?: Measuring Long Run Risk", with J.C. Heaton and N. Li Journal of Political Economy, Vol 116, Issue 2, 2008, 260-302. Link to the Issue
Abstract: We study structural models of stochastic discount factors and explore alternative methods of estimating such models using data on macroeconomic risk and asset returns. Particular attention is devoted to recursive utility models in which risk aversion can be modified without altering intertemporal substitution. We characterize the impact of changing the intertemporal substitution and risk aversion parameters on equilibrium short-run and long-run risk prices and on equilibrium wealth.
· "Intertemporal Substitution and Risk Aversion", with John Heaton, Junghoon Lee, Nikolai Roussanov, Handbook of Econometrics, Vol 6, Part 1, 2007, 3967-4056. Link to the Issue
Abstract: We characterize
and measure a long-term risk-return trade-off for the valuation of cash flows
exposed to fluctuations in macroeconomic growth. This trade-off features risk
prices of cash flows that are realized far into the future but continue to be
reflected in asset values. We apply this analysis to claims on aggregate cash
flows and to cash flows from value and growth portfolios by imputing values to
the long-run dynamic responses of cash flows to macroeconomic shocks. We
explore the sensitivity of our results to features of the economic valuation
model and of the model cash flow dynamics.
- "Intangible
Risk?" , L.P. Hansen, J.C. Heaton and N. Li, published in Measuring Capital in the New Economy (NBER Books), Corrado, Haltiwanger and Sichel, eds., pages 111-152, 2005
- "Value in an
Uncertain Economy" L.P. Hansen, Address at the 474th Convocation
at the
Research on Operator Methods
Working Papers on Operator Methods [top]
- "Nonlinear Principal Components
and Long Run Implications of Mutivariate
Diffusions", with X. Chen and J. Scheinkman.
(Annals of Statistics, Vol.37, No 6B, pp. 4279-4312 (2009))
We investigate a method for extracting nonlinear principal components
(NPCs). These NPCs maximize variation subject to smoothness and orthogonality
constraints; but we allow for a general class of constraints and multivariate
probability densities, including densities without compact support and
even densities with algebraic tails. We provide primitive sufficient conditions
for the existence of these NPCs. By exploiting the theory of continuous-time,
reversible Markov diffusion processes, we give a different interpretation of
these NPCs and the smoothness constraints. When the diffusion matrix is
used to enforce smoothness, the NPCs maximize long-run variation relative
to the overall variation subject to orthogonality constraints. Moreover, the
NPCs behave as scalar autoregressions with heteroskedastic innovations; this
supports semiparametric identification and estimation of a multivariate reversible
diffusion process and tests of the overidentifying restrictions implied
by such a process from low-frequency data. We also explore implications for
stationary, possibly nonreversible diffusion processes. Finally, we suggest a
sieve method to estimate the NPCs from discretely-sampled data.
- "Operator Methods for
Continuous-Time Markov Processes", with Y. Ait-Sahalia
and J. Scheinkman (August 18, 2008). Forthcoming in the Handbook of Financial Econometrics.
Published Papers on Operator Methods [top]
- "Nonlinearity and
Temporal Dependence," with Xiaohong
Chen and Marine Carrasco, (October 4, 2009), (forthcoming in Journal of Econometrics.)
Nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models. We study this link using three measures of temporal dependence: ρ−mixing, β−mixing and α−mixing. Stationary diffusions that are ρ − mixing have mixing coefficients that decay exponentially to zero. When they fail to be ρ−mixing, they are still β−mixing and α−mixing; but coefficient decay is slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state-dependent, Poisson sampling alters the temporal dependence.
- "Long Term Risk: an
Operator Approach," with J. Scheinkman
(January 2009). Econometrica, Vol. 77, No. 1, pp. 177-234.
We create an analytical structure that reveals the long-run risk-return relationship for nonlinear continuous-time Markov environments. We do so by studying an eigenvalue problem associated with a positive eigenfunction for a conveniently chosen family of valuation operators. The members of this family are indexed by the elapsed time between payoff and valuation dates, and they are necessarily related via a mathematical structure called a semigroup.We represent the semigroup using a positive process with three components: an exponential term constructed from the eigenvalue, a martingale, and a transient eigenfunction term. The eigenvalue encodes the risk adjustment, the martingale alters the probability measure to capture long-run approximation, and the eigenfunction gives the long-run dependence on the Markov state.We discuss sufficient conditions for the existence and uniqueness of the relevant eigenvalue and eigenfunction. By showing how changes in the stochastic growth components of cash flows induce changes in the corresponding eigenvalues and eigenfunctions, we reveal a long-run riskreturn trade-off.
- "Spectral Methods for Identifying Scalar Diffusions" with Jose A. Scheinkman and Nizar Touzi, Journal of Econometrics 86, 1998, 1-32.
Abstract: This
paper shows how to identify nonparametrically scalar
stationary difussions from discrete-time data. The
local evolution of the diffusion is characterized by a drift and difussion coefficient along with the specification of
boundary behavior. We recover this local evolution from two objects that can be
inferred directly from discrete-time data: the stationary density and a
conveniently chosen eigenvalue-eigenfunction pair of
the conditional expectation operator over a unit interval of time. This
construction also lends itself to a spectral characterization of the
over-identifying restrictions implied by a scalar diffusion model of a
discrete-time Markov process.
- "Short-Term Interest Rates as Subordinated Diffusions", with T. G. Conley, E. G. J. Luttmer and J. A. Scheinkman, Review of Financial Studies Paper 10:3, Fall 1997, 525-577.
Abstract: In this
article we characterize and estimate the process for short-term interest rates
using federal funds interest rate data. We presume we are observing a
discrete-time sample of a stationary scalar diffusion. We concentrate in a
class of models in which the local volatility elasticity is constant and the
drift has a flexible specification. To accommodate missing observations and to
break the link between "economic time" and calendar time, we model
the sampling scheme as an increasing process that is not directly observed. We
propose and implement two methods for estimation. We find evidence for a
volatility elasticity between one and one-half and two. When interest rates are
high, local mean reversion is small and the mechanics for introducing stationarity is the increased volatility of the diffusion
process.
- "Bootstrapping the Long Run" with T. G. Conley and W. F. Liu, Macroeconomic Dynamics Volume 1, Issue 2, 1997 279-311.(Link to the Issue)
Abstract: We
develop and apply bootstrap methods for diffusion models when fitted to the
long run as characterized by the stationary distribution of the data. To obtain
bootstrap refinements to statistical inference, we simulate candidate diffusion
processes. We use these bootstrap methods to assess measurements of local mean
reversion or “pull” to the center of the distribution for short-term interest
rates. We also use them to evaluate the fit of the model to the empirical
density.
- "Back to the Future: Generating Moment Implications for Continuous Time Markov Processes" with Jose A. Scheinkman, Econometrica 63:4, 1995, 767-804.
Abstract: Continuos-time Markov processes can be characterized
conveniently by their infinitesimal generators. For such processes there exist
forward and reverse-time generators. We show how to use those generators to
construct moment conditions implied by stationary Markov processes. Generalized
methods of moments estimators and tests can be constructed using these moment
conditions. The resulting econometric mothods are to
be applied to discrete-time data obtained by sampling continuous-time Markov
processes.
Research on GMM
Working Papers on GMM [top]
- "Underidentification?" with Manuel Arellano and Enrique Sentana (July 31, 2009)
Abstract: We develop methods for testing the hypothesis that an econometric model is underidentified and inferring the nature of the failed identification. By adopting a generalized method-of moments perspective, we feature directly the structural relations and we allow for nonlinearity in the econometric specification. We establish the link between a test for overidenti cation and our proposed test for underidentification. If, after attempting to replicate the structural relation, we find substantial evidence against the overidentifying restrictions of an augmented model, this is evidence against underidentification of the original model.
Description: These are unpublished proofs for the Econometrica, 1982, paper : "Large Sample Properties of Generalized Method of Moments Estimators", Econometrica , Vol. 50, No. 4 (Jul., 1982), pp. 1029-1054. If you have access to JSTOR, click here to see the paper
Published Material on GMM [top]
- "Generalized Methods
of Moments Estimation " (June 17, 2007)
Description: GMM entry for the forthcoming Palgrave dictionary
- "Generalized Methods
of Moments: A Time Series Perspective " (2000)
Description: It gives a perspective on the time series formulation and application of Generalized Method of Moments estimation. This file corresponds to the original paper that appeared later in the Encycopledia, somewhat modified and under the new title of 'Method of Moments'. The full reference for the published version is: International Encyclopedia of the Social and Behavioral Sciences, N. J. Smelser and P. B. Bates (editors), Pergamon: Oxford, 2000
- "Interview with Lars Peter Hansen", E. Ghysels and A. Hall, Journal of Business and Economic Statistics, 4, 2002.