October 2010 | Article

Robust Hidden Markov LQG Problems

Lars Peter Hansen, Ricardo Mayer, Thomas Sargent

For linear quadratic Gaussian problems, this paper uses two risk-sensitivity operators defined by Hansen and Sargent (2007b) to construct decision rules that are robust to misspecifications of (1) transition dynamics for state variables and (2) a probability density over hidden states induced by Bayes’ law. Duality of risk sensitivity to the multiplier version of min–max expected utility theory of Hansen and Sargent (2001) allows us to compute risk-sensitivity operators by solving two-player zero-sum games. Because the approximating model is a Gaussian probability density over sequences of signals and states, we can exploit a modified certainty equivalence principle to solve four games that differ in continuation value functions and discounting of time t increments to entropy. The different games express different dimensions of concerns about robustness. All four games give rise to time consistent worst-case distributions for observed signals. But in Games I–III, the minimizing players’ worst-case densities over hidden states are time inconsistent, while Game IV is an LQG version of a game of Hansen and Sargent (2005) that builds in time consistency. We show how detection error probabilities can be used to calibrate the risk-sensitivity parameters that govern fear of model misspecification in hidden Markov models.

Journal: Journal of Economic Dynamics & Control|Volume: 34|Issue Number: 10|Pages: 1951-1966|Tags: Risk, Robustness and Ambiguity|Export BibTeX >
  title={Robust Hidden Markov LQG Problems},
  author={Hansen, Lars Peter and Mayer, Ricardo and Sargent, Thomas},
  journal={Journal of Economic Dynamics and Control},
October 2010 | Working Paper

Modeling and Measuring Systemic Risk

Markus Brunnermeier, Lars Peter Hansen, Anil Kachyap, Arvind Krishnamurthy, Andrew W. Lo

An important challenge worthy of NSF support is to quantify systemic financial risk. There are at least three major components to this challenge: modeling, measurement, and data accessibility. Progress on this challenge will require extending existing research in many directions and will require collaboration between economists, statisticians, decision theorists, sociologists, psychologists, and neuroscientists.

Tags: Financial Market Linkages to the Macroeconomy|Export BibTeX >
  title={Modeling and Measuring Systemic Risk},
  author={Brunnermeier, Markus K. and Hansen, Lars Peter and Kashyap, Anil K. and Krishnamurthy, Arvind and Lo, Andrew W},
July 2010 | Article

Fragile Beliefs and the Price of Model Uncertainty

Lars Peter Hansen, Thomas J. Sargent

A representative consumer uses Bayes’ law to learn about parameters of several models and to construct probabilities with which to perform ongoing model averaging. The arrival of signals induces the consumer to alter his posterior distribution over models and parameters. The consumer’s specification doubts induce him to slant probabilities pessimistically. The pessimistic probabilities tilt toward a model that puts long-run risks into consumption growth. That contributes a countercyclical history-dependent component to prices of risk.

Journal: Quantitative Economics|Volume: 1|Issue Number: 1|Pages: 129-162|Tags: Financial Market Linkages to the Macroeconomy, Risk, Robustness and Ambiguity|Export BibTeX >
  title={Fragile Beliefs and the Price of Uncertainty},
  author={Hansen, Lars Peter and Sargent, Thomas J.},
  journal={Quantitative Economics},
  publisher={Wiley Online Library}
May 2010 | Chapter

Pricing Kernels and Stochastic Discount Factors

Lars Peter Hansen, Eric Renault
Title of book: Encyclopedia of Quantitative Finance|Editor(s): Rama Cont|Place of Publication: Hoboken, NJ|Publisher: Wiley|Tags: Uncertainty and Valuation|Export BibTeX >
  title={Pricing Kernels and Stochastic Discount Factors},
  author={Hansen, Lars P and Renault, Eric},
  journal={Encyclopedia of Quantitative Finance},
  publisher={Wiley Hoboken, NJ}
April 2010 | Article

Nonlinearity and Temporal Dependence

Xiaohong Chen, Lars Peter Hansen, Marine Carrasco

Nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models. We study this link using three measures of temporal dependence: ?-mixing, ?-mixing and ?-mixing. Stationary diffusions that are ?-mixing have mixing coefficients that decay exponentially to zero. When they fail to be ??-mixing, they are still ?-mixing and ?-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.

Journal: Journal of Econometrics|Volume: 155|Issue Number: 2|Pages: 155-169|Tags: Econometrics|Export BibTeX >
  title={Nonlinearity and Temporal Dependence},
  author={Chen, Xiaohong and Hansen, Lars Peter and Carrasco, Marine},
  journal={Journal of Econometrics},
January 2010

Operator Methods for Continuous-Time Markov Processes

Yacine Aït-Sahalia, Lars Peter Hansen, José A. Scheinkman

This chapter surveys relevant tools, based on operator methods, to describe the evolution in time of continuous-time stochastic process, over different time horizons. Applications include modeling the long-run stationary distribution of the process, modeling the short or intermediate run transition dynamics of the process, estimating parametric models via maximum-likelihood, implications of the spectral decomposition of the generator, and various observable implications and tests of the characteristics of the process.

Pages: 1-66|Title of book: Handbook of Financial Econometrics, Vol. 1: Tools and Tehcniques|Editor(s): Yacine Aït-Sahalia and Lars Peter Hansen|Place of Publication: Amsterdam|Publisher: North-Holland|Tags: Econometrics|Export BibTeX >
  title={Operator Methods for Continuous-Time Markov Processes},
  author={A{"i}t-Sahalia, Yacine and Hansen, Lars P and Scheinkman, Jos{'e} A},
  journal={Handbook of financial econometrics},
December 2009 | Article

Nonlinear Principal Components and Long Run Implications of Multivariate Diffusions

Xiaohong Chen, Lars Peter Hansen, José A. Scheinkman

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.

Journal: Annals of Statistics|Volume: 37|Issue Number: 6B|Pages: 4279-4312|Tags: Uncertainty and Valuation|Export BibTeX >
  title={Nonlinear Principal Components and Long-Run Implications of Multivariate Diffusions},
  author={Chen, Xiaohong and Hansen, Lars Peter and Scheinkman, Jos{'e}},
  journal={The Annals of Statistics},
November 2009 | Article

Doubts or Variability?

Francisco Barillas, Lars Peter Hansen, Thomas J. Sargent

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, Sargent, and Tallarini [17], Tallarini [30], Alvarez and Jermann [1]. Prices of model uncertainty contain information about the benefits of removing model uncertainty, not the consumption fluctuations that Lucas [22] and [23] 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 [11] 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.

Journal: Journal of Economic Theory|Volume: 144|Issue Number: 6|Pages: 2388-2418|Tags: Risk, Robustness and Ambiguity|Export BibTeX >
  title={Doubts or Variability?},
  author={Barillas, Francisco and Hansen, Lars Peter and Sargent, Thomas J},
  journal={journal of economic theory},
January 2009 | Article

Long Term Risk: an Operator Approach

Lars Peter Hansen, José A. Scheinkman

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 risk-return trade-off.

Journal: Econometrica|Volume: 77|Issue Number: 1|Pages: 177-234|Tags: Uncertainty and Valuation|Export BibTeX >
  title={Long-Term Risk: An Operator Approach},
  author={Hansen, Lars Peter and Scheinkman, Jos{'e} A},
  publisher={Wiley Online Library}
December 2008 | Article

Robustness and U.S. Monetary Experimentation

Timothy Cogley, Riccardo Colacito, Lars Peter Hansen, Thomas J. Sargent

We study how a concern for robustness modifies a policymaker’s incentive to experiment. A policymaker 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 policymaker an incentive to experiment. The policymaker 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 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 policymaker’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.

Journal: Journal of Money Credit and Banking|Volume: 40|Issue Number: 8|Pages: 1559-1623|Tags: Risk, Robustness and Ambiguity|Export BibTeX >
  title={Robustness and US Monetary Policy Experimentation},
  author={Cogley, Timothy and Colacito, Riccardo and Hansen, Lars Peter and Sargent, Thomas J},
  journal={Journal of Money, Credit and Banking},
  publisher={Wiley Online Library}