RichaRd T. Ely LecTure Beliefs, Doubts and Learning: Valuing Macroeconomic Risk By Lars Peter Hansen* This essay examines the problem of inference sophisticated investment decisions. Should we within a rational expectations model from two put econometricians and economic agents on perspectives: that of an econometrician and that comparable footing, or should we endow eco- of the economic agents within the model. The nomic agents with much more refined statistical assumption of rational expectations has been knowledge? and remains an important component to quan- From an econometric standpoint, the out- titative research. It endows economic decision come of the rational expectations approach is makers with knowledge of the probability law the availability of extra information about the implied by the economic model. As such, it is an underlying economic model. This information equilibrium concept. Imposing rational expecta- is reflected in an extensive set of cross-equation tions removed from consideration the need for restrictions. These restrictions allow an econo- separately specifying beliefs or subjective com- metrician to extract more precise information ponents of uncertainty. Thus, it simplified model about parameters or to refine the specification specification and implied an array of testable of exogenous processes for the model builder. implications that are different from those con- To understand the nature of these restrictions, sidered previously. It reframed policy analysis consider a dynamic model in which economic by questioning the effectiveness of policy levers agents must make investment decisions in physi- that induce outcomes that differ systematically cal, human, or financial capital. The decision to from individual beliefs. invest is forward-looking because an investment I consider two related problems. The first is made today has ramifications for the future the problem of an econometrician who follows capital stock. The forward-looking nature of John F. Muth (1961), Robert E. Lucas, Jr., and investment induces decision makers to make Edward C. Prescott (1971), Lucas (1972a), predictions or forecasts as part of their current Thomas J. Sargent (1973), and an extensive body period choice of investment. The forward-look- of research by adopting an assumption of rational ing perspective affects equilibrium outcomes expectations on the part of economic agents. In including market valuations of capital assets. implementing this approach, researchers abstract Rational expectations econometrics presumes from hard statistical questions that pertain to that agents know the probabilities determin- model specification and estimation. The second ing exogenous shocks as they formulate their problem is that of economic decision makers or choices. This translates to an extensive set of investors who must forecast the future to make cross-equation restrictions that can be exploited to aid identification and inference. * Department of Economics, University of Chicago, 1126 The cross-equation restrictions broadly con- E. 59th St., Chicago, IL 60637 (e-mail: l-hansen@uchicago. ceived are a powerful tool, but to what extent edu). The topics I cover in this paper have been influenced by should we as applied researchers rely on it? As an extensive collaboration with Thomas Sargent. I greatly applied time series econometricians, we routinely appreciate conversations with John Heaton, Ravi Jagannathan, confront challenging problems in model specifi- Monika Piazzesi, and Martin Schneider. I owe a special acknowledgement to Thomas Sargent and Grace Tsiang cation. How do we model stochastic dynamics in who provided many valuable comments on preliminary the short and long run? What variables are best drafts of this paper. Also I want to thank participants at forecasters? How do we select among competing workshops at New York University and the Federal Reserve models? Bank of Chicago. Junghoon Lee and Ricardo Mayer pro- vided expert research assistance. This material is based A heuristic defense for rational expectations upon work supported by the National Science Foundation appeals to a Law of Large Numbers and gives under award SES0519372. agents a wealth of data. This allows, at least as 2 AEA PAPERS AND PROCEEDINGS MAY 2007 an approximation, for us the model builders to • How is learning altered when decision mak- presume investor knowledge of a probability ers admit that the models are misspecified or model and its parameters. But statistical infer- simplified? ence, estimation, and learning can be difficult in practice. In actual decision making, we may By answering these questions, we will see be required to learn about moving targets, to how statistical ambiguity alters the predicted make parametric inferences, to compare model risk-return relation, and we will see when performance, or to gauge the importance of learning induces model uncertainty premia long-run components of uncertainty. As the that are large when macroeconomic growth is statistical problem that agents confront in our sluggish. model is made complex, rational expectations’ presumed confidence in their knowledge of the I. Rational Expectations and Econometrics probability specification becomes more tenu- ous. This leads me to ask: (a) how can we bur- The cross-equation restrictions are the novel den the investors with some of the specification component to rational expectations economet- problems that challenge the econometrician, rics. They are derived by assuming investor and (b) when would doing so have important knowledge of parameters and solving for equi- quantitative implications? I confront these ques- librium decision rules and prices. I consider tions formally by exploring tools that quantify two examples of such restrictions from the asset when learning problems are hard, by examining pricing literature, and review some estimation the Bayesian solution to such problems and by methods designed for estimating models subject speculating on alternative approaches. to such restrictions. One example is the equilib- In this essay, I use the literature that links mac- rium wealth-consumption ratio and the other is roeconomics and asset pricing as a laboratory for a depiction of risk prices. examining the role of expectations and learning. The linkage of macroeconomics and finance is a A. Cross-Equation Restrictions natural choice for study. Even with a rich array of security markets, the macroeconomic risks Consider an environment in which equilib- cannot be diversified away (averaged out across rium consumption evolves as: investors), and hence are reflected in equilib- rium asset prices. Exposure to such risks must be (1) ct11 2 ct 5 mc 1 a · t 1 scut11 rewarded by the marketplace. By studying asset pricing, we, as model builders, specify the for- zt11 5 Azt 1 szut11, ward-looking beliefs of investors and how they cope with risk and uncertainty. Prior to develop- where ct is the logarithm of consumption, {ut} ing asset pricing applications, we consider some is an i.i.d. sequence of normally distributed stylized statistical decision and inferential prob- random vectors with mean zero and covariance lems that turn out to be informative. matrix I, and { t} is the process used to forecast I ask five questions that are pertinent to mod- consumption growth rates. I take equation (1) as eling the linkages between asset pricing and the equilibrium law of motion for consumption. macroeconomics: Following David M. Kreps and Evan L. Porteus (1978) and Larry G. Epstein and Stanley • When is estimation difficult? E. Zin (1989), I use a model of investor prefer- ences in which the intertemporal composition • What are the consequences for the econo- of risk matters. I will have more to say about metrician? such preferences subsequently. As emphasized by Epstein and Zin (1989), such preferences give • What are the consequence for economic a convenient way to separate risk and intertem- agents and for equilibrium outcomes? poral substitution. John Y. Campbell (1996) and others have used log linear models with such • What are the real time consequences of investor preferences to study cross-sectional learning? returns. VOL. 97 NO. 2 RicHard T. Ely LectUre Wealth-Consumption Ratio.—Let r be the Shadow Risk Prices.—Assume a unitary inverse of the intertemporal elasticity of substi- elasticity of substitution and a recursive utility tution and b be the subjective discount factor. risk parameter g and a discount factor b and Approximate (around r 5 1): the same consumption dynamics. Consider the price of the one-period exposure to the shock vector . Following the convention in (2) w 2 c < 2log11 2 b2 ut11 t t finance, let the price be quoted in terms of the 2 mean reward for being exposed to uncertainty. 1 2 r ba 2 b 1 1 m , 11 2 3 r1I A2 zt v 4 For Kreps and Porteus (1978) preferences, the intertemporal composition of risk matters, and where wt is log wealth. The constant term mv as a consequence the consumption dynamics are includes a risk adjustment. A key part of this reflected in the equilibrium prices, including the relation is the solution to a prediction problem: one-period risk prices. This linkage has been a focal point of work by Ravi Bansal and Amir ` Yaron (2004) and others. Specifically, the one- E bj 1 2 2 m 2 0 period price vector is c a ct1j ct1j21 c zt d j51 21 p 5 sc 1 3b1g 2 12ar1I 2 bA2 sz 4. 21 5 ba9(I 2 bA) zt. Later, I will add more detail about the construc- tion of such prices. For now, I simply observe Formula (2) uses the fact that preferences I con- that while this price vector is independent of the sider are represented recursively with an aggre- state vector zt, it depends on the vectors sc and sz gator that is homogeneous of degree one. As a along with the A matrix. Again, we have cross- consequence, Euler’s theorem gives a simple equation restrictions, but now the coefficients relation between the shadow value of the con- that govern variability also come into play. sumption process and the continuation value Pricing a claim to the next period shock is for that process.