## Speculation And Wealth When Investors Have Diverse Beliefs And Financial Markets Are Incomplete

with Timothy Cogley and Viktor Tsyrennikov

July 2012

In our heterogenous-beliefs incomplete-markets models, precautionary and speculative motives coexist. Missing
markets for Arrow securities affect the size and avenues for precautionary savings.
Survival dynamics suggested by Friedman (1953) and studied by Blume and Easley (2006) depend on whether agents can
trade a disaster-state security. When the market for a disaster-state security is closed, precautionary savings
flow into risk-free bonds, prompting less-informed investors to accumulate wealth. Because speculation motives
are strongest for the disaster-state Arrow security, opening this market brings outcomes close to those for a complete-markets
benchmark where instead it is well-informed investors who accumulate wealth. Speculation is more limited in other cases,
and outcomes for wealth dynamics are closer to those in an economy in which only a risk-free bond can be traded.

## Bayesian Model Averaging, Learning and Model Selection

with George Evans, Seppo Honkapohja, and Noah Williams

January 2012

Agents have two forecasting models, one consistent with the unique
rational expectations equilibrium, another that assumes a time-varying
parameter structure. When agents use Bayesian updating to choose
between models in a self-referential system, we find that learning dynamics
lead to selection of one of the two models. However, there
are parameter regions for which the non-rational forecasting model is
selected in the long-run. A key structural parameter governing outcomes
measures the degree of expectations feedback in Muth’s model
of price determination.

## Market Prices of Risk with Diverse Beliefs, Learning, and Catastrophes

with Timothy Cogley and Viktor Tsyrennikov

December 2011

This paper studies market prices of risk in an economy with two types of agents with diverse beliefs. The paper studies both a complete markets
economy and a risk-free bonds only (Bewley) economy.

## Wealth Dynamics in a Bond Economy with Heterogeneous Beliefs

with Timothy Cogley and Viktor Tsyrennikov

November 2011

We study an economy in which two types of agents have diverse
beliefs about the law of motion for an exogenous endowment. One type knows the
true law of motion, and the other learns about it via Bayes’s
theorem. Financial markets are incomplete, the only traded asset being a risk-free
bond. Borrowing limits are imposed to ensure the existence of an
equilibrium. We analyze how financial-market structure affects the distribution of
financial wealth and survival of the two agents. When markets are complete, the
learning agent loses wealth during the learning transition and eventually exits
the economy Blume and Easley 2006). In contrast, in a bond-only economy,
the learning agent accumulates wealth, and both agents survive
asymptotically, with the knowledgeable agent being driven to his debt limit. The absence
of markets for certain Arrow securities is central to reversing the direction
in which wealth is transferred.

## Inflation-Gap Persistence in the U.S.

with Timothy Cogley and Giorgio E. Primiceri

December 2007

We use Bayesian Markov Chain Monte Carlo methods to estimate two models of post WWII U.S. inflation rates with drifting stochastic volatility and drifting coefficients. One model is univariate, the other a multivariate autoregression. We define the inflation gap as the deviation of inflation from a pure random walk component of inflation and use both of our models to study changes over time in the persistence of the inflation gap measured in terms of short- to medium-term predicability. We present evidence that our measure of the persistence of the inflation gap increased until Volcker brought mean inflation down in the early 1980s and that it then fell during the chairmanships of Volcker and Greenspan. Stronger evidence for movements in inflation gap persistence emerges from the VAR than from the univariate model. We interpret these changes in terms of a simple dynamic new Keynesian model that allows us to distinguish altered monetary policy rules and altered private sector parameters.

## Diverse Beliefs, Survival, and the Market Price of Risk

with Timothy Cogley

July 2008

We study prices and allocations in a complete-markets, pure endowment economy in which agents have heterogenous beliefs. Aggregate consumption growth evolves exogenously according to a two-state Markov process. The economy is populated by two types of agents, one that learns about transition probabilities and another that knows them. We examine how the presence of the better-informed agent influences allocations, the market price of risk, and the rate at which asset prices converge to values that would be computed under the typical assumption that all agents know the transition probabilities.

## The Conquest of South American Inflation

with Noah Williams and Tao Zha

October 2008

This is an extensively revised and shortened version of our 2006 paper. We infer determinants of Latin American hyperinflations and stabilizations in 5 countries 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 in 2003 seems to have been at work.

## Self-Confirming Equilibrium

with In-Koo Cho

December 2006

Prepared for the New Palgrave Dictionary of Economics.

This is an extensively revised and shortened version of our 2006 paper. We infer determinants of Latin American hyperinflations and stabilizations in 5 countries 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 in 2003 seems to have been at work.

This is an extensively revised and shortened version of our 2006 paper. We infer determinants of Latin American hyperinflations and stabilizations in 5 countries 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 in 2003 seems to have been at work.

## The Benefits from U.S. Monetary Policy Experimentation in the Days of Samuelson and Solow and Lucas

with Timothy Cogley and Riccardo Colacito

September, 2005

A policy maker knows two models of inflation-unemployment dynamics. One implies an exploitable trade-off, the other does not. The policy maker's prior probability over the two models is part of his state vector. Bayes' law converts the prior into a posterior at each date and gives the policy maker an incentive to experiment. For a model calibrated to U.S. data through the early 1960s, we isolate the component of government policy that is due to experimentation by comparing the outcomes from two Bellman equations, the first of which `experiments and learns', the second of which `learns but doesn't experiment'. We interpret the second as an `anticipated utility' model and study how well its outcomes approximate those from the `experiment and learn' Bellman equation. The approximation is good. We use the model to study rates at which false models would be disposed of starting from initial conditions designed to emulate (a) a situation in which the Samuelson-Solow model is true, but some prior probability is attached to Lucas’s model, and (b) a situation in which the Lucas model is true but some prior probability attaches to the Lucas model. The rates differ in interesting ways.

## The Market Price of Risk and the Equity Premium: A Legacy of the Great Depression?

with Timothy Cogley

August, 2007

A representative consumer is endowed with a prior that is pessimistic relative to a true data generating mechanism. True consumption growth follows a two state Markov chain with probabilities calibrated by Cechetti, Lam, and Mark. We obtain a pessimistic prior by using a calculation from robust control. Then we endow agents with Bayes’ Law and let time and chance remove their pessimism. From the view point of a rational expectations econometrician, the stochastic discount factor has a multiplicative adjustment, the Radon-Nikodym derivative of the decision maker’s model relative to the true one. We use this framework to study how market prices of risk behave as Bayes’ Law causes the `legacy of the Great Depression’ gradually to wear off. We obtain a high equity premium that gradually falls during the post WWII period.

## Anticipated Utility and Rational Expectations as Approximations of Bayesian Decision Making

with Timothy Cogley

March, 2005

We regard as an `anticipated utility’ model like ones advocated by David Kreps as an approximation to a model with a Bayesian agent and study the quality of approximation in several contexts of interest to a macroeconomist. We display some examples in which the approximation is quite good.

## Shocks and Government Beliefs: The Rise and Fall of American Inflation

with Noah Williams and Tao Zha

April 2005

We use a Bayesian Markov Chain Monte Carlo algorithm jointly to estimate the parameters of a `true' data generating mechanism and those of a sequence of approximating models that a monetary authority uses to guide its decisions. Gaps between a true expectational Phillips curve and the monetary authority's approximating non-expectational Phillips curve models unleash inflation that a monetary authority that knows the true model would avoid. A sequence of dynamic programming problems implies that the monetary authority's inflation target evolves as its estimated Phillips curve moves. Our estimates attribute the rise and fall of post WWII inflation in the US to an intricate interaction between the monetary authority's beliefs and economic shocks. Shocks in the 1970s made the monetary authority perceive a tradeoff between inflation and unemployment that ignited big inflation. The monetary authority's beliefs about the Phillips curve changed in ways that account for Volcker's conquest of US inflation.

## The Conquest of U.S. Inflation: Learning, Model Uncertainty, and Robustness

with Timothy Cogley

November, 2003

An adaptive Fed’s model is a mixture of three models. The Fed uses Bayesian methods to update estimates of three models of the Phillips curve: a Samuelson-Solow model, a Solow-Tobin model, and a Lucas model. Each period, the central bank also updates the probability that it assigns to each of these three models, then determines its first-period decision by solving a `Bayesian linear regulator problem’. Although by the mid 1970s the U.S. data induce the Fed to assign very high probability to the Lucas model, the government refrains from adopting its low-inflation policy recommendation because that policy has very bad consequences under one of the other low (but not zero) probability models. The statistical model is thus able to explain the puzzling delay in the Fed’s decision to deflate after learning the natural rate hypothesis.

## Knowing the Forecasts of Others

with Joseph G. Pearlman

April 10, 2004

This paper uses recursive methods to compute an equilibrium of the notorious model in section 8 of Townsend’s 1983 JPE paper `Forecasting the Forecasts of Others’. The equilibrium is of finite (and even low) dimension. Market prices fully reveal traders’ private information, making the equilibrium equivalent with one in which traders pool their information before trading. This means that the problem of forecasting the forecasts of others disappears in equilibrium. There is no need to keep track of higher order beliefs.

## Impacts of Priors on Convergence and Escape from Nash Inflation

with Noah Williams

July 2004

This paper generalizes the learning model that Cho, Williams, and Sargent (2002) and Sargent (1999) attributed to the government, then calculates rates of convergence to and escape from self-confirming equilibria.

## Drifts and Volatilities: Monetary Policies and Outcomes in the Post WWII U.S

with Timothy Cogley

August 26, 2002

This paper answers criticisms of our 2001 NBER Macro Annual paper made by Sims and Stock. We enrich our specification of a drifting coefficient VAR to allow stochastic volatility and study whether our earlier evidence for drifting coefficients survives this generalization. It does. We investigate the power of various tests that have been used to test time invariance of the autoregressive coefficients of VARs against alternatives like ours. All except one have low power. These results about power help reconcile our results with those of Sims and Bernanke and Mihov. We also study evidence that monetary policy rules have drifted.

## Bayesian Fan Charts for U.K. Inflation: Forecasting and Sources of Uncertainty in an Evolving Monetary System

with Timothy Cogley and Sergei Morozov

September, 2003

We use Bayesian methods to estimate a VAR with drifting coefficients and volatilities then construct fan charts that we compare with those of the Bank of England’s MPC. Our fan charts incorporate several sources of uncertainty, including a form of model uncertainty that is represented by drifting coefficients and volatilities.

## Evolving Post-World War II U.S. Inflation Dynamics

with Timothy Cogley

June 2001

This paper uses Bayesian methods and post World War II data on inflation, unemployment, and an interest rate to estimate a `drifting coefficients' model. The model is used to construct characterizations of the data that make contact with various points in Lucas's Critique and Sargent's The Conquest of American Inflation published by Princeton University Press. The paper constructs various measures of posterior means and variances of inflation and unemployment and studies their relationships. This paper will appear in the 2001 Macroeconomic Annual.

## Escaping Nash Inflation

with In-Koo Cho and Noah Williams

May 31, 2001

This paper analytically computes the `escape route' for a special case of the model in my Marshall lectures The Conquest of American Inflation published by Princeton University Press. We show that theoretical computations of the mean dynamics and escape dynamics do a good job of explaining simulations like those in The Conquest of American Inflation. The paper uses the mysterious insight of Michael Harrison: `If an unlikely event occurs, it is very likely to occur in the most likely way' .

## Laboratory Experiments with an Expectational Phillips Curve

with Jasmina Arifovic

August 22, 2001

Experiments with human subjects in a Kydland-Prescott Phillips curve economy.

## Learning to be Credible

with In-Koo Cho

## Informationally Decentralized Learning Algorithms for Finite Player, Finite Action Games of Incomplete Information

with William Brock, Ramon Marimon, and John Rust

April 1988