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

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 >

@article{chs:2009, 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}, pages={4279--4312}, year={2009}, publisher={JSTOR} }✕

## Doubts or Variability?

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 >

@article{bhs:2009, title={Doubts or Variability?}, author={Barillas, Francisco and Hansen, Lars Peter and Sargent, Thomas J}, journal={journal of economic theory}, volume={144}, number={6}, pages={2388--2418}, year={2009}, publisher={Elsevier} }✕

## Long Term Risk: an Operator Approach

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 >

@article{hansenscheinkman:2009, title={Long-Term Risk: An Operator Approach}, author={Hansen, Lars Peter and Scheinkman, Jos{'e} A}, journal={Econometrica}, volume={77}, number={1}, pages={177--234}, year={2009}, publisher={Wiley Online Library} }✕

## Robustness and U.S. Monetary Experimentation

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 >

@article{cchs:2008, 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}, volume={40}, number={8}, pages={1599--1623}, year={2008}, publisher={Wiley Online Library} }✕

## Consumption Strikes Back? Measuring Long Run Risk

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.

**Journal:**Journal of Political Economy|

**Volume:**116|

**Issue Number:**2|

**Pages:**260-302|

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

@article{hhl:2008, title={Consumption Strikes Back? Measuring Long-Run Risk}, author={Hansen, Lars Peter and Heaton, John C and Li, Nan}, journal={Journal of Political economy}, volume={116}, number={2}, pages={260--302}, year={2008}, publisher={The University of Chicago Press} }✕

## Recursive Robust Estimation and Control without Commitment

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.

**Journal:**Journal of Economic Theory|

**Volume:**136|

**Issue Number:**1|

**Pages:**1/27/2016|

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

@article{hansensargent:2007},

Author = {Lars Peter Hansen and Thomas J. Sargent},

Date-Added = {2016-03-28 00:37:21 +0000},

Date-Modified = {2016-03-28 00:40:39 +0000},

Journal = {Journal of Economic Theory},

Pages = {1-27},

Title = {Recursive Robust Estimation and Control without Commitment},

Volume = {136},

Year = {2007}}

## Generalized Method of Moments Estimation

Generalized Method of Moments (GMM) refers to a class of estimators which are constructed from exploiting the sample moment counterparts of population moment conditions (sometimes known as orthogonality conditions) of the data generating model. GMM estimators have become widely used, for the following reasons:

- GMM estimators have large sample properties that are easy to characterize in ways that facilitate comparison. A family of such estimators can be studied a priori in ways that make asymptotic efficiency comparisons easy. The method also provides a natural way to construct tests which take account of both sampling and estimation error.
- In practice, researchers find it useful that GMM estimators can be constructed without specifying the full data generating process (which would be required to write down the maximum likelihood estimator.) This characteristic has been exploited in analyzing partially specified economic models, in studying potentially misspecified dynamic mod- els designed to match target moments, and in constructing stochastic discount factor models that link asset pricing to sources of macroeconomic risk.Books with good discussions of GMM estimation with a wide array of applications in- clude: Cochrane (2001), Arellano (2003), Hall (2005), and Singleton (2006). For a theoretical treatment of this method see Hansen (1982) along with the self contained discussions in the books. See also Ogaki (1993) for a general discussion of GMM estimation and applications, and see Hansen (2001) for a complementary entry that, among other things, links GMM estimation to related literatures in statistics. For a collection of recent methodological ad- vances related to GMM estimation see Ghysels and Hall (2002). While some of these other references explore the range of substantive applications, in what follows we focus more on the methodology.

**Title of book:**The New Palgrave Dictionary of Economics|

**Editor(s):**Steven N Durlauf and Lawrence Blume|

**Place of Publication:**Basingstoke, Hampshire|

**Publisher:**Palgrave Macmillan|

**Tags:**Econometrics|Export BibTeX >

@incollection{hansen2010generalized, title={Generalized method of moments estimation}, author={Hansen, Lars Peter}, booktitle={Macroeconometrics and Time Series Analysis}, pages={105--118}, year={2010}, publisher={Springer} }✕

## Beliefs, Doubts and Learning: Valuing Macroeconomic Risk; Richard T. Ely Lecture

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.

**Journal:**American Economic Review|

**Volume:**97|

**Issue Number:**2|

**Pages:**1-30|

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

@article{hansen2007beliefs, title={Beliefs, Doubts and Learning: Valuing Macroeconomic Risk}, author={Hansen, Lars Peter}, journal={The American Economic Review}, volume={97}, number={2}, pages={1--30}, year={2007}, publisher={JSTOR} }✕

## Intertemporal Substitution and Risk Aversion

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.

**Pages:**3967-4056|

**Title of book:**Handbook of Econometrics, Volume 6A|

**Editor(s):**James Heckman and Edward Leamer|

**Place of Publication:**Amsterdam|

**Publisher:**Elsevier|

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

@article{hansen2007intertemporal, title={Intertemporal Substitution and Risk Aversion}, author={Hansen, Lars Peter and Heaton, John and Lee, Junghoon and Roussanov, Nikolai}, journal={Handbook of econometrics}, volume={6}, pages={3967--4056}, year={2007}, publisher={Elsevier} }✕

## 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}}