State-Observation Sampling and the Econometrics of Learning Models

In nonlinear state-space models, sequential learning about the hidden state can proceed by particle filtering when the density of the observation conditional on the state is available analytically (e.g. Gordon et al., 1993). This condition need not hold in complex environments, such as the incomplete-information equilibrium models considered in financial economics. In this paper, we make two contributions to the learning literature. First, we introduce a new filtering method, the state-observation sampling (SOS) filter, for general state-space models with intractable observation densities. Second, we develop an indirect inference-based estimator for a large class of incomplete-information economies. We demonstrate the good performance of these techniques on an asset pricing model with investor learning applied to over 80 years of daily equity returns

Regime-Switching and the Estimation of Multifractal Processes

We propose a discrete-time stochastic volatility model in which regimeswitching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Regime-Switching and the Estimation of Multifractal Processes

We propose a discrete-time stochastic volatility model in which regime switching serves three purposes. First, changes in regimes capture low frequency variations, which is their traditional role. Second, they specify intermediate frequency dynamics that are usually assigned to smooth autoregressive processes. Finally, high frequency switches generate substantial outliers. Thus, a single mechanism captures three important features of the data that are typically addressed as distinct phenomena in the literature. Maximum likelihood estimation is developed and shown to perform well in finite sample. We estimate on exchange rate data a version of the process with four parameters and more than a thousand states. The estimated model compares favorably to earlier specifications both in- and out-of-sample. Multifractal forecasts slightly improve on GARCH(1,1) at daily and weekly intervals, and provide considerable gains in accuracy at horizons of 10 to 50 days.

Idiosyncratic Production Risk, Growth, and the Business Cycle

We introduce a neoclassical growth economy with idiosyncratic production risk and incomplete markets. Each agent is an entrepreneur operating her own neoclassical technology with her own capital stock. The general equilibrium is characterized in closed form. Idiosyncratic production shocks introduce a risk premium on private equity and reduce the demand for investment. The steady state is characterized by a lower capital stock due to entrepreneurial risk and a lower interest rate due to precautionary savings as compared to complete markets. The private equity premium is endogenously countercyclical: the anticipation of low savings and high interest rates in the future feed back to high risk premia and low investment in the present. Countercyclicality in risk taking slows down convergence to the steady state and amplifies the magnitude and persistence of the business cycle. These results, which contrast sharply with those obtained in Bewley models, highlight the macroeconomic significance of missing markets in production and investment risk.

Large Deviations and the Distribution of Price Changes

The Multifractal Model of Asset Returns ("MMAR," see Mandelbrot, Fisher, and Calvet, 1997) proposes a class of multifractal processes for the modelling of financial returns. In that paper, multifractal processes are defined by a scaling law for moments of the processes' increments over finite time intervals. In the present paper, we discuss the local behavior of multifractal processes. We employ local Holder exponents, a fundamental concept in real analysis that describes the local scaling properties of a realized path at any point in time. In contrast with the standard models of continuous time finance, multifractal processes contain a multiplicity of local Holder exponents within any finite time interval. We characterize the distribution of Holder exponents by the multifractal spectrum of the process. For a broad class of multifractal processes, this distribution can be obtained by an application of Cramer's Large Deviation Theory. In an alternative interpretation, the multifractal spectrum describes the fractal dimension of the set of points having a given local Holder exponent. Finally, we show how to obtain processes with varied spectra. This allows the applied researcher to relate an empirical estimate of the multifractal spectrum back to a particular construction of the Stochastic process.Multifractal model of asset returns, multifractal spectrum, compound stochastic process, subordinated stochastic process, time deformation, scaling laws, self-similarity, self-affinity

Idiosyncratic Production Risk, Growth and the Business Cycle

We introduce a neoclassical growth economy with idiosyncratic production risk and incomplete markets. The general equilibrium is characterized in closed form. Uninsurable production shocks introduce a risk premium on private equity and typically result in a lower steady-state level of capital than under complete markets. In the presence of such risks, the anticipation of low investment and high interest rates in the future discourages risk-taking and feeds back into low investment in the present. An endogenous macroeconomic complementarity thus arises, which slows down convergence and amplifies the magnitude and persistence of the business cycle. These results — contrasting sharply with those of Aiyagari (1994) and Krusell and Smith (1998) — highlight that idiosyncratic production or capital-income risk can have significant adverse effects on capital accumulation and aggregate volatility. Keywords: Capital income, Entrepreneurial risk, Fluctuations, Growth, Investment, Precautionary savings.

Incomplete Market Dynamics in a Neoclassical Production Economy

We investigate a neoclassical economy with heterogeneous agents, convex technologies and idiosyncratic production risk. Combined with precautionary savings, investment risk generates rich effects that do not arise in the presence of pure endowment risk. Under a finite horizon, multiple growth paths and endogenous fluctuations can exist even when agents are very patient. In infinite-horizon economies, multiple steady states may arise from the endogeneity of risktaking and interest rates instead of the usual wealth effects. Depending on the economy's parameters, the local dynamics around a steady state are locally unique, totally unstable or locally undetermined, and the equilibrium path can be attracted to a limit cycle. The model generates closed-form expressions for the equilibrium dynamics and easily extends to a variety of environments, including heterogeneous capital types and multiple sectors.

Incomplete Markets, Growth, and the Business Cycle

We introduce a Ramsey growth model with incomplete markets, decentralized production, and idiosyncratic technological risk. The combination of uninsurable shocks with the precautionary motive can slow down capital accumulation or give rise to persistent fluctuations even when agents are very patient and technology is strictly convex. The model generates closed-form expressions for the equilibrium dynamics under a finite or infinite horizon. Multiple steady states and poverty traps can arise from the endogeneity of the interest rate instead of the usual wealth effect. Depending on the economy's parameters, the local dynamics around a steady state are locally unique, totally unstable or locally undetermined, and the equilibrium path can be attracted to a limit cycle. In calibrated examples, financial incompleteness substantially slows down convergence to the steady state and thus increases the persistence of aggregate shocks.

Incomplete Market Dynamics in a Neoclassical Production Economy

We investigate a neoclassical economy with heterogeneous agents, convex technologies and idiosyncratic production risk. Combined with precautionary savings, investment risk generates rich effects that do not arise in the presence of pure endowment risk. Under a finite horizon, multiple growth paths and endogenous fluctuations can exist even when agents are very patient. In infinite-horizon economies, multiple steady states may arise from the endogeneity of risk-taking and interest rates instead of the usual wealth effects. Depending on the economy’s parameters, the local dynamics around a steady state are locally unique, totally unstable or locally undetermined, and the equilibrium path can be attracted to a limit cycle. The model generates closed-form expressions for the equilibrium dynamics and easily extends to a variety of environments, including heterogeneous capital types and multiple sectors.

Multifrequency Jump-Diffusions: An Equilibrium Approach

This paper proposes that equilibrium valuation is a powerful method to generate endogenous jumps in asset prices, which provides a structural alternative to traditional reduced-form specifications with exogenous discontinuities. We specify an economy with continuous consumption and dividend paths, in which endogenous price jumps originate from the market impact of regime-switches in the drifts and volatilities of fundamentals. We parsimoniously incorporate shocks of heterogeneous durations in consumption and dividends while keeping constant the number of parameters. Equilibrium valuation creates an endogenous relation between a shock's persistence and the magnitude of the induced price jump. As the number of frequencies driving fundamentals goes to infinity, the price process converges to a novel stochastic process, which we call a multifractal jump-diffusion.