We develop and apply bootstrap methods for diffusion models whenfitted to the long run as characterized by the stationarydistribution of the data. To obtain bootstrap refinements tostatistical inference, we simulate candidate diffusion processes. Weuse these bootstrap methods to assess measurements of local meanreversion or “pull” to the center of the distribution for short-terminterest rates. We also use them to evaluate the fit of the model to the empirical density.

We examine asset prices and returns in the context of a pureexchange economy. Our purpose is to identify the key channels by which changes in preferences affect the equity premium and the risk-free rate and to develop intuition that is useful for understanding asset pricing in more complicatedeconomies. Our analysis suggests that capital gains play a crucial role ingenerating empirically plausible mean equity premia.

We show that absence of arbitrage in frictionless markets implies a lower bound on the average of the logarithm of the reciprocal of the stochastic discount factor implicit in asset pricing models. The greatest lower boundfor a given asset menu is the average continuously compounded return on itsgrowth-optimal portfolio. We use this bound to evaluate the plausibility ofvarious parametric asset pricing models to characterize financial marketpuzzles such as the equity premium puzzle and the risk-free ratepuzzle. We show that the insights offered by the growth-optimal boundsdiffer substantially from those obtained by other nonparametric bounds.

I develop an algorithm for solving dynamic models in which individualdecision rules and the cross-sectional distribution of agents'characteristics influence each other. To illustrate the algorithm, Isolve an endowment economy with incomplete markets, a continuum ofheterogeneous agents, and aggregate shocks. The key innovation of thealgorithm is to parameterize the (cross-sectional) density with aflexible functional form, which makes it possible to avoid simulationtechniques. The paper shows how to check for accuracy and establisheslinks between the properties of the incomplete-markets economy andthe aspects involved in obtaining a numerical solution.

We derive asset-pricing and portfolio-choice implicationsof a dynamic incomplete-markets model in which consumers areheterogeneous in several respects: labor income, asset wealth, andpreferences. In contrast to earlier papers, we insist on at leastroughly matching the model's implications forheterogeneity — notably, the equilibrium distributions of income andwealth — with those in U.S. data. This approach seems natural:Models that rely critically on heterogeneity for explaining assetprices are not convincing unless the heterogeneity is quantitativelyreasonable. We find that the class of models we consider here isvery far from success in explaining the equity premium whenparameters are restricted to produce reasonable equilibriumheterogeneity. We express the equity premium as a product of twofactors: the standard deviation of the excess return and the marketprice of risk. The first factor, as expected, is much too low in the model. The size of the market price of risk depends crucially on theconstraints on borrowing. If substantial borrowing is allowed, themarket price of risk is about one one-hundredth of what it is in the data(and about 15% higher than in the representative-agent model).However, under the most severe borrowing constraints that we consider, the market price of risk is quite close to the observedvalue.

In answer to the question “Will borrowing constraints necessarilyintensify aggregate fluctuations and aggregate cyclical variability?” it has been found that complete markets equilibrium displaysaggregate fluctuations that may be dampened when borrowingconstraints are introduced. Like others, I find that variability inthe distribution of labor productivity shocks amplifies aggregatefluctuations. I also find that allowing agents to have differentpermanent incomes amplifies aggregate fluctuations, enriching theasset-pricing implications of the complete contingent claims modelwhen demand aggregation is not possible. Although agents are able toequalize their intertemporal marginal rates of substitution (IMRS) ofconsumption state-by-state, the IMRS of labor is not equalized acrossagents, creating gains from specialization. To determine howfrictions affect aggregate variability, two types of borrowingconstraints are studied. In the first model, dividend payments arerestricted and, in the second, nonhuman wealth is restricted to bepositive. Either type of borrowing constraint can dampen aggregatefluctuations.

The failure of consumption-based asset pricing models to match thestochastic properties of the equity premium and the risk-free ratehas been attributed by some authors to frictions, transaction costs,or durability. However, such frictions primarily would affect thehigher-frequency data components: Consumption-based pricing modelsthat concentrate on long-horizon returns should be more successful.We consider two consumption-based models: time-separable utility, andthe habit model of Constantinides. We estimate a vector ARCH modelthat includes the pricing kernel and the equity return, and use thefitted model to assess the model's implications for the equitypremium and for the risk-free rate. Neither model performs well at aquarterly horizon, but at longer horizons the Constantinides modelcan match the mean and the variance of the observed equity premium,captures time variation of the equity premium, and can better matchthe observed risk-free rate. We conclude that the equity-premium andrisk-free-rate puzzles are primarily problems for shorter-horizonreturns.

Barnett originated the Divisia monetary aggregates, which incontinuous time exactly track any monetary aggregator function underperfect certainty. With user costs measuring the prices of theservices of components, Barnett's aggregates are based on FrancoisDivisia's derivation of the Divisia line integral from thefirst-order conditions for optimizing behavior by economic agentsunder perfect certainty. We derive an extended Divisia index fromthe first-order conditions (Euler equations) that apply under risk.Our extended Divisia index is the first extension of index numbertheory into the domain of decision making under risk and therebyproduces a route for the extension of all index number theory topermit non-risk-neutrality. We generate simulated data from amodeled rational consumer and investigate the tracking accuracy ofthe extended Divisia index to the consumer's exact aggregatorfunction.

The Divisia monetary index weights each monetary asset by itsexpenditure share, evaluated at its user cost.See Barnett et al. [1997, equations (5) and (6)].In a world of perfect certainty, the user costs can be computed directly from assetreturn data, and so, the index is model-free. In the presence ofuncertainty, however, the true user cost of a monetary asset dependson the covariance of the asset's return with the marginal utility ofwealth. Intuitively, an asset's value depends both on its expectedpayoff and on the asset's ability to hedge wealth fluctuations. As aresult, the true user cost cannot be computed without an explicitmodel of preferences. It follows that the exact Divisia monetaryindex in a stochastic economy is not model-free, but depends on theunderlying preference assumptions.

Economic theory indicates the need for nonlinear structuralmodels to study medium-term and long-run dynamic behavior of aneconomy. This paper argues that economic systems can be betterspecified and estimated using differential-equation rather thandifference-equation systems and briefly reviews the estimators ofcontinuous models. This approach of specifying structural models onthe basis of economic theory and institutional structure explicitlyand then testing the underlying hypothesis to verify the structuralform is contrasted with a general-to-specific approach ofsuccessively more restricted VARMAX processes.Previous analyses ofstability about the steady state or fixed pointin phasespace are extended to more general attractors to allowan investigation of complexityin economicsystems. The critical dependence of some attractors, andparticularly strange attractors, on parameter valuesemphasizes the need for consistent, efficientestimation. A structural approachprovides a rigorous alternative to using single time series todetermine whether economic systems exhibit aperiodic or chaoticdynamical behavior.