Noah Williams

Robust control and model misspecification

Abstract 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.

Escaping Nash Inflation

Mean dynamics govern convergence to rational expectations equilibria of self-referential systems under least squares learning. We highlight escape dynamics that propel away from a rational expectations equilibrium under fixed-gain recursive learning schemes. These learning schemes discount past observations. In a model with a unique self-confirming equilibrium, we show that the destination of the escape dynamics is an outcome associated with government discovery of too strong a version of the natural rate hypothesis. That destination is not sustainable as a self-confirming equilibrium but is visited recurrently. The escape route dynamics cause recurrent outcomes close to the Ramsey (commitment) inflation rate in a model with an adaptive government. JEL Classification: E3, E52, E58

Impacts of Priors on Convergence and Escapes from Nash Inflation

Recent papers have analyzed how adaptive agents may converge to and escape from self-confirming equilibria. All of these papers have imputed to agents a particular prior about drifting coefficients. In the context of a model of monetary policy, this paper analyzes dynamics that govern both convergence and escape under a more general class of priors for the government. The authors characterize how the shape of the prior influences the dynamics in important ways. There are priors for which the E-stability condition is not enough to assure local convergence to a self-confirming equilibrium. Their analysis also tracks down the source of differences in the sustainability of Ramsey inflation encountered in the analyses of Sims (1988) and Chung (1990), on the one hand, and Cho, Williams, and Sargent (2002), on the other.

The Conquest of South American Inflation

We infer determinants of Latin American hyperinflations and stabilizations by using the method of maximum likelihood to estimate a hidden Markov model that assigns roles both to fundamentals in the form of government deficits that are financed by money creation and to destabilizing expectations dynamics that can occasionally divorce inflation from fundamentals. Levels and conditional volatilities of monetized deficits drove most hyperinflations and stabilizations, with a notable exception in Peru, where a cosmetic reform of the type emphasized by Marcet and Nicolini occurred.

A solvable continuous time dynamic principal-agent model

I study the provision of incentives in a continuous time dynamic moral hazard model with hidden actions and hidden states. I consider a principal–agent model with linear production and exponential utility, whose explicit solution allows me to show how allocations are distorted for incentive reasons, and how access to hidden savings further alters allocations. I solve the model by applying a stochastic maximum principle, where the co-state variables from the agents optimization problem become state variables for the principals problem of choosing an optimal contract. I show that the main effect of moral hazard is a distortion on the effort margin, with a smaller effect on the intertemporal consumption allocation. Access to hidden savings shuts down the intertemporal distortions and increases the effort distortion. I also show how the optimal contracts can be implemented via a constant equity share, a constant flow payment, and a constant tax on savings.