## Robust Control and Model Misspecification

A decision maker fears that data are generated by a statistical perturbation of an approximating model that is either a controlled diffusion or a controlled measure over continuous functions of time. A perturbation is constrained in terms of its relative entropy. Several different two-player zero-sum games that yield robust decision rules are related to one another, to the max–min expected utility theory of Gilboa and Schmeidler [Maxmin expected utility with non-unique prior, J. Math. Econ. 18 (1989) 141–153], and to the recursive risk-sensitivity criterion described in discrete time by Hansen and Sargent [Discounted linear exponential quadratic Gaussian control, IEEE Trans. Automat. Control 40 (5) (1995) 968–971]. To represent perturbed models, we use martingales on the probability space associated with the approximating model. Alternative sequential and nonsequential versions of robust control theory imply identical robust decision rules that are dynamically consistent in a useful sense.

**Journal:**Journal of Economic Theory|

**Volume:**128|

**Issue Number:**1|

**Pages:**45-90|

**Tags:**Risk, Robustness and Ambiguity|Export BibTeX >

@article{hstw:2006,

Author = {Lars Peter Hansen and Thomas J. Sargent and Guahar A. Turmuhambetova and Noah Williams},

Journal = {Journal of Economic Theory},

Pages = {45-90},

Title = {Robust Control and Model Misspecification},

Volume = {128},

Year = {2006}}

## Introduction to Model Uncertainty and Robustness

This article introduces the symposium on model uncertainty and robustness.

**Journal:**Journal of Economic Theory|

**Volume:**128|

**Issue Number:**1|

**Pages:**1-3|

**Tags:**Risk, Robustness and Ambiguity, Uncertainty and Valuation|Export BibTeX >

@article{hmrss:2006, title={Introduction to Model Uncertainty and Robustness}, author={Hansen, Lars Peter and Maenhout, Pascal and Rustichini, Aldo and Sargent, Thomas J and Siniscalchi, Marciano M}, journal={Journal of Economic Theory}, volume={128}, number={1}, pages={1--3}, year={2006}, publisher={Elsevier} }✕

## Nonlinear Principal Components and Long-Run Implications of Multivariate Diffusions

We investigate a method for extracting nonlinear principal components. These principal components maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these principal components. We also characterize the limiting behavior of the associated eigenvalues, the objects used to quantify the incremental importance of the principal components. By exploiting the theory of continuous-time, reversible Markov processes, we give a different interpretation of the principal components and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the principal components maximize long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the principal components behave as scalar autoregressions with heteroskedastic innovations. Finally, we explore implications for a more general class of stationary, multivariate diffusion processes.

**Journal:**Annals of Statistics|

**Tags:**Econometrics|Export BibTeX >

@article{hansen2000principal, title={Principal Components and the Long Run}, author={Xiaohong Chen, Lars Peter Hansen, and José A. Scheinkman}, year={2000}, publisher={Citeseer} }✕

## Robust Estimation and Control Under Commitment

In a Markov decision problem with hidden state variables, 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. 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 findings of T. Ba?ar and P. Bernhard [H?-Optimal Control and Related Minimax Design Problems, second ed., Birkhauser, Basel, 1995] and P. Whittle [Risk-sensitive Optimal Control, Wiley, New York, 1990].

**Journal:**Journal of Economic Theory|

**Volume:**124|

**Issue Number:**2|

**Pages:**258-301|

**Tags:**Risk, Robustness and Ambiguity|Export BibTeX >

@article{hansen2005robust, title={Robust Estimation and Control Under Commitment}, author={Hansen, Lars Peter and Sargent, Thomas J}, journal={Journal of economic Theory}, volume={124}, number={2}, pages={258--301}, year={2005}, publisher={Elsevier} }✕

## Intangible Risk?

**Pages:**111-152|

**Title of book:**Measuring Capital in the New Economy|

**Editor(s):**Carol Corrado, John C Haltiwanger, and Daniel E Sichel|

**Place of Publication:**Chicago|

**Publisher:**University of Chicago Press|

**Series Name:**NBER Studies in Income and Wealth|

**Tags:**Uncertainty and Valuation|Export BibTeX >

@incollection{hansen2005intangible, title={Intangible Risk}, author={Hansen, Lars Peter and Heaton, John C and Li, Nan}, booktitle={Measuring Capital in the New Economy}, pages={111--152}, year={2005}, publisher={University of Chicago Press} }✕

## Robust Control of Forward Looking Models

This paper shows how to formulate and compute robust Ramsey (aka Stackelberg) plans for linear models with forward-looking private agents. The leaders and the 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 followers’ define a Stackelberg equilibrium with robust decision makers in which the leader and follower have different worst-case models despite sharing a common approximating model. To compute a Stackelberg equilibrium we formulate a Bellman equation that is associated with an artificial single-agent robust control problem. The artificial Bellman equation contains a description of 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.

**Journal:**Monetary Economics|

**Volume:**50|

**Issue Number:**3|

**Pages:**581-604|Export BibTeX >

@article{hansen:2003robust, title={Robust Control of Forward-Looking Models}, author={Hansen, Lars Peter and Sargent, Thomas J.}, journal={Journal of Monetary Economics}, volume={50}, number={3}, pages={581--604}, year={2003}, publisher={Elsevier} }✕

## A Quartet of Semigroups for Model Specification, Robustness, Prices of Risk and Model Detection

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.

**Journal:**Journal of the European Economic Association|

**Volume:**1|

**Issue Number:**1|

**Pages:**68-123|

**Tags:**Risk, Robustness and Ambiguity, Uncertainty and Valuation|Export BibTeX >

@article{anderson2003quartet, title={A quartet of Semigroups for Model Specification, Robustness, Prices of Risk, and Model Detection}, author={Anderson, Evan W and Hansen, Lars Peter and Sargent, Thomas J}, journal={Journal of the European Economic Association}, volume={1}, number={1}, pages={68--123}, year={2003}, publisher={Wiley Online Library} }✕

## Robust Permanent Income and Pricing with Filtering

A planner and agent in a permanent-income economy cannot observe part of the state, regard their model as an approximation, and value decision rules that are robust across a set of models. They use robust decision theory to choose allocations. Equilibrium prices reflect the preference for robustness and so embody a “market price of Knightian uncertainty.” We compute market prices of risk and compare them with a model that assumes that the state is fully observed. We use detection error probabilities to constrain a single parameter that governs the taste for robustness.

**Journal:**Macroeconomic Dynamics|

**Volume:**6|

**Pages:**40-84|

**Tags:**Risk, Robustness and Ambiguity|Export BibTeX >

@article{hsw:2002,

title={Robust Permanent Income and Pricing with Filtering},

author={Hansen, Lars Peter and Sargent, Thomas J. and Wang, Neng E},

journal={Macroeconomic Dynamics},

volume={6},

number={01},

pages={40–84},

year={2002},

publisher={Cambridge Univ Press}

}

✕## Robustness and Pricing with Uncertain Growth

We study how decision-makers’ concerns about robustness affect prices and quantities in a stochastic growth model. In the model economy, growth rates in technology are altered by infrequent large shocks and continuous small shocks. An investor observes movements in the technology level but cannot perfectly distinguish their sources. Instead the investor solves a signal extraction problem. We depart from most of the macroeconomics and finance literature by presuming that the investor treats the specification of technology evolution as an approximation. To promote a decision rule that is robust to model misspecification, an investor acts as if a malevolent player threatens to perturb the actual data-generating process relative to his approximating model. We study how a concern about robustness alters asset prices. We show that the dynamic evolution of the risk-return trade-off is dominated by movements in the growth-state probabilities and that the evolution of the dividend-price ratio is driven primarily by the capital-technology ratio.

**Journal:**Review of Financial Studies|

**Volume:**15|

**Issue Number:**2|

**Pages:**363-404|

**Tags:**Risk, Robustness and Ambiguity|Export BibTeX >

@article{chsw:2002, title={Robustness and Pricing With Uncertain Growth}, author={Cagetti, Marco and Hansen, Lars Peter and Sargent, Thomas and Williams, Noah}, journal={Review of Financial Studies}, volume={15}, number={2}, pages={363--404}, year={2002}, publisher={Soc Financial Studies} }✕

## Generalized Method of Moments Estimation: A Time Series Perspective (published title “Method of Moments”)

This entry describes empirical methods for estimating dynamic economic systems using time-series data. By design, the methods target specific feature of the dynamic system and do not require a complete specification of the time-series evolution. The resulting generalized-method-of-moments estimation and inference methods use estimating equations implied by some components of a dynamic economic system. This entry describes the statistical methods and some applications of these methods.

**Title of book:**International Encyclopedia of the Social and Behavioral Sciences|

**Editor(s):**N. J. Smelser and P. B. Bates|

**Place of Publication:**Amsterdam|

**Publisher:**Elsevier|

**Tags:**Econometrics|Export BibTeX >

@article{hansen2001generalized, title={Generalized Method of Moments Estimation: A Time Series Perspective}, author={Hansen, Lars Peter}, journal={International Encyclopedia of Social and Behavioral Sciences}, year={2001} }✕