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.

We examine asset prices and returns in the context of a pure exchange 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 complicated economies. Our analysis suggests that capital gains play a crucial role in generating 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 bound for a given asset menu is the average continuously compounded return on its growth-optimal portfolio. We use this bound to evaluate the plausibility of various parametric asset pricing models to characterize financial market puzzles such as the equity premium puzzle and the risk-free rate puzzle. We show that the insights offered by the growth-optimal bounds differ substantially from those obtained by other nonparametric bounds.

I develop an algorithm for solving dynamic models in which individual decision rules and the cross-sectional distribution of agents' characteristics influence each other. To illustrate the algorithm, I solve an endowment economy with incomplete markets, a continuum of heterogeneous agents, and aggregate shocks. The key innovation of the algorithm is to parameterize the (cross-sectional) density with a flexible functional form, which makes it possible to avoid simulation techniques. The paper shows how to check for accuracy and establishes links between the properties of the incomplete-markets economy and the aspects involved in obtaining a numerical solution.

We derive asset-pricing and portfolio-choice implications of a dynamic incomplete-markets model in which consumers are heterogeneous in several respects: labor income, asset wealth, and preferences. In contrast to earlier papers, we insist on at least roughly matching the model's implications for heterogeneity — notably, the equilibrium distributions of income and wealth — with those in U.S. data. This approach seems natural: Models that rely critically on heterogeneity for explaining asset prices are not convincing unless the heterogeneity is quantitatively reasonable. We find that the class of models we consider here is very far from success in explaining the equity premium when parameters are restricted to produce reasonable equilibrium heterogeneity. We express the equity premium as a product of two factors: the standard deviation of the excess return and the market price 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 the constraints on borrowing. If substantial borrowing is allowed, the market 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 observed value.

In answer to the question “Will borrowing constraints necessarily intensify aggregate fluctuations and aggregate cyclical variability?” it has been found that complete markets equilibrium displays aggregate fluctuations that may be dampened when borrowing constraints are introduced. Like others, I find that variability in the distribution of labor productivity shocks amplifies aggregate fluctuations. I also find that allowing agents to have different permanent incomes amplifies aggregate fluctuations, enriching the asset-pricing implications of the complete contingent claims model when demand aggregation is not possible. Although agents are able to equalize their intertemporal marginal rates of substitution (IMRS) of consumption state-by-state, the IMRS of labor is not equalized across agents, creating gains from specialization. To determine how frictions affect aggregate variability, two types of borrowing constraints are studied. In the first model, dividend payments are restricted and, in the second, nonhuman wealth is restricted to be positive. Either type of borrowing constraint can dampen aggregate fluctuations.

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

Barnett originated the Divisia monetary aggregates, which in continuous time exactly track any monetary aggregator function under perfect certainty. With user costs measuring the prices of the services of components, Barnett's aggregates are based on Francois Divisia's derivation of the Divisia line integral from the first-order conditions for optimizing behavior by economic agents under perfect certainty. We derive an extended Divisia index from the first-order conditions (Euler equations) that apply under risk. Our extended Divisia index is the first extension of index number theory into the domain of decision making under risk and thereby produces a route for the extension of all index number theory to permit non-risk-neutrality. We generate simulated data from a modeled rational consumer and investigate the tracking accuracy of the extended Divisia index to the consumer's exact aggregator function.

The Divisia monetary index weights each monetary asset by its expenditure 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 asset return data, and so, the index is model-free. In the presence of uncertainty, however, the true user cost of a monetary asset depends on the covariance of the asset's return with the marginal utility of wealth. Intuitively, an asset's value depends both on its expected payoff and on the asset's ability to hedge wealth fluctuations. As a result, the true user cost cannot be computed without an explicit model of preferences. It follows that the exact Divisia monetary index in a stochastic economy is not model-free, but depends on the underlying preference assumptions.

Economic theory indicates the need for nonlinear structural models to study medium-term and long-run dynamic behavior of an economy. This paper argues that economic systems can be better specified and estimated using differential-equation rather than difference-equation systems and briefly reviews the estimators of continuous models. This approach of specifying structural models on the basis of economic theory and institutional structure explicitly and then testing the underlying hypothesis to verify the structural form is contrasted with a general-to-specific approach of successively more restricted VARMAX processes. Previous analyses of stability about the steady state or fixed point in phase space are extended to more general attractors to allow an investigation of complexity in economic systems. The critical dependence of some attractors, and particularly strange attractors, on parameter values emphasizes the need for consistent, efficient estimation. A structural approach provides a rigorous alternative to using single time series to determine whether economic systems exhibit aperiodic or chaotic dynamical behavior.