This paper studies the time-series behavior of asset returns and aggregate consumption. Using a representative consumer model and imposing restrictions on preferences and the joint distribution of consumption and returns, we deduce a restricted log-linear time-series representation. Preference parameters for the representative agent are estimated and the implied restrictions are tested using postwar data.
Research Topic: Econometrics
Misspecified Recovery
Asset prices contain information about the probability distribution of future states and the stochastic discounting of those states as used by investors. To better understand the challenge in distinguishing investors’ beliefs from risk‐adjusted discounting, we use Perron–Frobenius Theory to isolate a positive martingale component of the stochastic discount factor process. This component recovers a probability measure that absorbs long‐term risk adjustments. When the martingale is not degenerate, surmising that this recovered probability captures investors’ beliefs distorts inference about risk‐return tradeoffs. Stochastic discount factors in many structural models of asset prices have empirically relevant martingale components.
Uncertainty Outside and Inside Economic Models (Nobel Lecture)
“We must infer what the future situation would be without our interference, and what changes will be wrought by our actions. Fortunately, or unfortunately, none of these processes is infallible, or indeed ever accurate and complete.” Knight (1921)
Recursive Models of Dynamic Linear Economies
A common set of mathematical tools underlies dynamic optimization, dynamic estimation, and filtering. In Recursive Models of Dynamic Linear Economies, Lars Peter Hansen and Thomas Sargent use these tools to create a class of econometrically tractable models of prices and quantities.
They present examples from microeconomics, macroeconomics, and asset pricing. The models are cast in terms of a representative consumer. While Hansen and Sargent demonstrate the analytical benefits acquired when an analysis with a representative consumer is possible, they also characterize the restrictiveness of assumptions under which a representative household justifies a purely aggregative analysis.
Based on the 2012 Gorman lectures, the authors unite economic theory with a workable econometrics while going beyond and beneath demand and supply curves for dynamic economies. They construct and apply competitive equilibria for a class of linear-quadratic-Gaussian dynamic economies with complete markets. Their book stresses heterogeneity, aggregation, and how a common structure unites what superficially appear to be diverse applications. An appendix describes MATLAB® programs that apply to the book’s calculations.
Proofs for Large Sample Properties of Generalized Method of Moments Estimators
I present proofs for the consistency of generalized method of moments (GMM) estimators presented in Hansen (1982). Some basic approximation results provide the groundwork for the analysis of a class of such estimators. Using these results, I establish the large sample convergence of GMM estimators under alternative restrictions on the estimation problem.
Underidentification?
We develop methods for testing that an econometric model is underidentified and for estimating the nature of the failed identification. We adopt a generalized-method-of moments perspective in a possibly non-linear econometric specification. If, after attempting to replicate the structural relation, we find substantial evidence against the overidentifying restrictions of an augmented model, this is evidence against underidentification of the original model. To diagnose how identification might fail, we study the estimation of a one-dimensional curve that gives the parameter configurations that provide the greatest challenge to identification, and we illustrate this calculation in an empirical example.
Nonlinearity and Temporal Dependence
Nonlinearities in the drift and diffusion coefficients influence temporal dependence in diffusion models. We study this link using three measures of temporal dependence: ?-mixing, ?-mixing and ?-mixing. Stationary diffusions that are ?-mixing have mixing coefficients that decay exponentially to zero. When they fail to be ??-mixing, they are still ?-mixing and ?-mixing; but coefficient decay is slower than exponential. For such processes we find transformations of the Markov states that have finite variances but infinite spectral densities at frequency zero. The resulting spectral densities behave like those of stochastic processes with long memory. Finally we show how state dependent, Poisson sampling alters the temporal dependence.
Operator Methods for Continuous-Time Markov Processes
This chapter surveys relevant tools, based on operator methods, to describe the evolution in time of continuous-time stochastic process, over different time horizons. Applications include modeling the long-run stationary distribution of the process, modeling the short or intermediate run transition dynamics of the process, estimating parametric models via maximum-likelihood, implications of the spectral decomposition of the generator, and various observable implications and tests of the characteristics of the process.
Nonlinear Principal Components and Long-Run Implications of Multivariate Diffusions
We investigate a method for extracting nonlinear principal components. These principal components maximize variation subject to smoothness and orthogonality constraints; but we allow for a general class of constraints and densities, including densities without compact support and even densities with algebraic tails. We provide primitive sufficient conditions for the existence of these principal components. We also characterize the limiting behavior of the associated eigenvalues, the objects used to quantify the incremental importance of the principal components. By exploiting the theory of continuous-time, reversible Markov processes, we give a different interpretation of the principal components and the smoothness constraints. When the diffusion matrix is used to enforce smoothness, the principal components maximize long-run variation relative to the overall variation subject to orthogonality constraints. Moreover, the principal components behave as scalar autoregressions with heteroskedastic innovations. Finally, we explore implications for a more general class of stationary, multivariate diffusion processes.
Generalized Method of Moments Estimation
Generalized Method of Moments (GMM) refers to a class of estimators which are constructed from exploiting the sample moment counterparts of population moment conditions (sometimes known as orthogonality conditions) of the data generating model. GMM estimators have become widely used, for the following reasons:
- GMM estimators have large sample properties that are easy to characterize in ways that facilitate comparison. A family of such estimators can be studied a priori in ways that make asymptotic efficiency comparisons easy. The method also provides a natural way to construct tests which take account of both sampling and estimation error.
- In practice, researchers find it useful that GMM estimators can be constructed without specifying the full data generating process (which would be required to write down the maximum likelihood estimator.) This characteristic has been exploited in analyzing partially specified economic models, in studying potentially misspecified dynamic mod- els designed to match target moments, and in constructing stochastic discount factor models that link asset pricing to sources of macroeconomic risk.Books with good discussions of GMM estimation with a wide array of applications in- clude: Cochrane (2001), Arellano (2003), Hall (2005), and Singleton (2006). For a theoretical treatment of this method see Hansen (1982) along with the self contained discussions in the books. See also Ogaki (1993) for a general discussion of GMM estimation and applications, and see Hansen (2001) for a complementary entry that, among other things, links GMM estimation to related literatures in statistics. For a collection of recent methodological ad- vances related to GMM estimation see Ghysels and Hall (2002). While some of these other references explore the range of substantive applications, in what follows we focus more on the methodology.