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

·  Working Papers on GMM

·  Published Work on GMM

Research on Robustness

 

Working Papers on Robustness [top]

Abstract: 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 over time can aggravate or attenuate long run risks. The arrival of signals induces the consumer to alter his posterior distribution over models and parameters. The consumer copes with doubts about probabilities by slanting them in directions that have pessimistic consequences for value functions. His twisted over model probabilities give rise to model uncertainty premia that contribute a time-varying component to what is ordinarily measured as the market price of risk.

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 likelihood ratios (entropies). Martingales reprent alternative models. A decision maker construct a sequence of decision rules by pretending that a sequence of minimizing players choose increments to a martingale and distortion 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 (1995) to problems that contains hidden states. The worst case martingale is overdetermined, expressing a temporal inconsistency of worst case belief about the hidden state, but not about observables.

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.

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.

Link to the volume

 

Published Papers on Robustness [top]

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.

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.

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.

 

 

Research on Risk and Valuation

 

Working Papers on Risk and Valuation [top]

Abstract: In this paper I propose to augment the toolkit for economic dyn valuation with methods that will reveal economic import of long-run ture. These tools enable informative decompositions of a model’s dyna for valuation. The methods I feature build in part on Perron-Frobeniu to valuation operators that explicitly incorporate stochastic growth operators are indexed by the gap of time between when a payo? is real is priced. Appropriately adapted Perron-Frobenius theory gives a ch the valuation behavior when this gap becomes large. Using such metho erational decompositions of value implications of economic models inc of parameter sensitivity and characterizations of long-run risk prices.

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 the semigroup property or the The Law of Iterated Values. The operator formulation we develop provides a way to link short term risk adjustments to what happens in the medium and long term. We apply this apparatus to give a precise notion of a long term risk-return tradeoff.

 

Link to the volume

Published Papers on Risk and Valuation [top]

Abstract: We characterize and measure a long-run risk return tradeoff for valuation of financial cash flows that are exposed to fluctuations in macroeconomic growth. This tradeoff features cash flow components that are realized far into the future but are still reflected in current asset values. We use the recursive utility model with empirical inputs from vector autoregressions to quantify this tradeoff; and we study the long-run risk differences in aggregate securities and in portfolios constructed based on the ration of book equity to market equity. We isolate features of the economic model needed for the long run valuation differences among this portfolios to be sizable. Finally, we show how the resulting measurements vary when we consider alternative statistical specifications of cash flow and consumption growth.

 

 

Research on Operator Methods

 

Working Papers on Operator Methods [top]

Abstract: Nonlinearities in the drift and diffusion coefficients influence temporal dependence in scalar diffusion models. We study this link using two notions of temporal dependence: ? ? mixing and ? ? mixing. We show that ? ? mixing and ? ? mixing with exponential decay are essentially equivalent concepts for scalar diffusions. For stationary diffusions that fail to be ??mixing, we show that they are still ??mixing except that the decay rates are slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. Some have spectral densities that diverge at frequency zero in a manner similar to that of stochastic processes with long memory. Finally we show how nonlinear, state-dependent, Poisson sampling alters the unconditional distribution as well as the temporal dependence.

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 the semigroup property or the The Law of Iterated Values. The operator formulation we develop provides a way to link short term risk adjustments to what happens in the medium and long term. We apply this apparatus to give a precise notion of a long term risk-return tradeoff.

 

Abstract: 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 multivariate densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these 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; this supports semiparametric identification 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 non-reversible diffusion processes.

Published Papers on Operator Methods [top]

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.

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.

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.

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]

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]

Description: GMM entry for the forthcoming Palgrave dictionary

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