Journal of Econometrics (2004), 123, pp 153-187.
Joint with Jesus Fernandez-Villaverde (University of Pennsylvania).
This paper studies the properties of the bayesian approach to estimation and comparison of dynamic equilibrium economies. Both tasks can be performed even if the models are nonnested, misspecified and nonlinear. First we show that bayesian methods have a classical interpretation: asymptotically the parameter point estimates converge to their pseudotrue values and the best model under the Kullback-Leibler will have the highest posterior probability. Second, we illustrate the strong small sample behavior of the approach using a well-known application: the U.S. cattle cycle. Bayesian estimates outperform Maximum Likelihood results and the proposed model is easily compared with a set of BVARs.
Review of Economic Studies (2007), 74, pp 1059-1087.
Joint with Jesus Fernandez-Villaverde (University of Pennsylvania).
This paper presents a framework to undertake likelihood-based inference in nonlinear and/or non-normal dynamic macroeconomic models. We apply a particle filter to estimate the likelihood function of the model. This likelihood can be used for parameter estimation and model comparison. We show consistency of the estimate of the likelihood function and its good performance in simulations. The algorithm is important because the literature can only evaluate the likelihood of linear macroeconomic models with Gaussian innovations. We apply our procedure to the neoclassical growth model.
Journal of Applied Econometrics (2005), 20, pp 891-910.
Joint with Jesus Fernandez-Villaverde (University of Pennsylvania).
This paper compares two methods for undertaking likelihood-based inference in dynamic equilibrium economies: a sequential Monte Carlo filter and the Kalman filter. The sequential Monte Carlo filter exploits the nonlinear structure of the economy and evaluates the likelihood function of the model by simulation methods. The Kalman filter estimates a linearization of the economy around the steady state. We report two main results. First, both for simulated and for real data, the sequential Monte Carlo filter delivers a substantially better fit of the model to the data as measured by the marginal likelihood. This is true even for a nearly linear case. Second, the differences in terms of point estimates, although relatively small in absolute values, have important effects on the moments of the model. We conclude that the nonlinear filter is a superior procedure for taking models to the data.
Journal of Monetary Economics (2005), 52, pp 1151-1166.
Joint with Pau Rabanal (IMF).
The baseline New Keynesian model cannot replicate the observed persistence in inflation, output, and real wages for sensible parameter values. As a result, several extensions have been suggestedto improve its fit to the data. We use a Bayesian approach to estimate and compare the baseline sticky price model of Calvo.s [1983. Staggeredpri ces in a utility maximizing framework. Journal of Monetary Economics 12, 383.398.] andt hree extensions. Our empirical results are as follows. First, we find that adding price indexation improves the fit of Calvo.s [1983. Staggered prices in a utility maximizing framework. Journal of Monetary Economics 12, 383.398.] model. Second, models with both staggered price and wage setting dominate models with only price rigidities. Third, introducing wage indexation does not significantly improve the fit. Fourth, all model estimates suggest a high degree of price stickiness. Fifth, the estimates of labor supply elasticity are higher in models with both staggeredpr ice andw age contracts. Finally, the estimatedin flation parameters of the Taylor rule are stable across models.
Spanish Economic Review (2008), 10, pp 23-40.
Joint with Pau Rabanal (IMF).
This paper estimates and compares four versions of the sticky price New Keynesian model for the Euro area using a Bayesian approach. The main results are: First, we find that the average duration of price contracts is between five and eight quarters and that price indexation is important. Second, average duration of wage contracts is estimated to be between two and three quarters, while wage indexation is unimportant. Third, the marginal likelihood indicates that sticky wages are important in the Euro area. Finally, using Smets and Wouters (2003) more informative priors, we present results that may indicate that data is not informative and, therefore, priors have a big influence on posteriors estimates.
Journal of Economic Dynamics and Control (2006), 30, pp. 2447-2508.
Joint with S. Boragan Aruoba (University of Maryland) and Jesus Fernandez-Villaverde (University of Pennsylvania).
This paper compares solution methods for dynamic equilibrium economies. We compute and simulate the stochastic neoclassical growth model with leisure choice using Undetermined Coefficients in levels and in logs, Finite Elements, Chebyshev Polynomials, Second and Fifth Order Perturbations and Value Function iteration for several calibrations. We document the performance of the methods in terms of computing time, implementation complexity and accuracy and we present some conclusions about our preferred approaches based on the reported evidence.
Click on this link to go to the companion web page where you can find the codes used in this paper.
Journal of Economic Dynamics and Control (2006), 30, pp. 2509-2531.
Joint with Jesus Fernandez-Villaverde (University of Pennsylvania).
This paper explores the application of the changes of variables technique to solve the stochastic neoclassical growth model. We use the method of Judd (2003) to change variables in the computed policy functions that characterize the behavior of the economy. We report how the optimal change of variables reduces the average absolute Euler equation errors of the solution of the model by a factor of three. We also demonstrate how changes of variables correct for variations in the volatility of the economy even if we work with first order policy functions and how we can keep a linear representation of the laws of motion of the model if we use a nearly optimal transformation. We discuss how to apply our results to estimate dynamic equilibrium economies.
Econometrica (2006), 74, pp 93-119.
Joint with Jesus Fernandez-Villaverde (University of Pennsylvania) and Manuel Santos (Arizona State University).
This paper studies the econometrics of computed dynamic models. Since these models generally lack a closed-form solution, their policy functions are approximated by numerical methods. Hence, the researcher can only evaluate an approximated likelihood associated with the approximated policy function rather than the exact likelihood implied by the exact policy function. What are the consequences for inference of the use of approximated likelihoods? First, we find conditions under which, as the approximated policy function converges to the exact policy, the approximated likelihood also converges to the exact likelihood. Second, we show that second order approximation errors in the policy function, which almost always are ignored by researchers, have first order effects on the likelihood function. Third, we discuss convergence of Bayesian and classical estimates. Finally, we propose to use a likelihood ratio test as a diagnostic device for problems derived from the use of approximated likelihoods.
Click on this link to go to the companion web page where you can find a working paper version, a technical appendix, and the codes used in this paper.
Review of Economic Studies conditional acceptance.
Joint with Daniel F. Waggoner (Federal Reserve Bank of Atlanta) and Tao Zha (Federal Reserve Bank of Atlanta).
SVARs are widely used for policy analysis and to provide stylized facts for dynamic general equilibrium models. Yet there have been no workable rank conditions to ascertain whether an SVAR is globally identified. When identifying restrictions, such as long-run restrictions, are imposed on impulse responses, there have been no efficient algorithms for small-sample estimation and inference. To fill these important gaps in the literature, this paper makes four contributions. First, we establish general rank conditions for global identification of both overidentified and exactly identified models. Second, we show that these conditions can be checked as a simple matrix-filling exercise and that they apply to a wide class of identifying restrictions, including linear and certain nonlinear restrictions. Third, we establish a very simple rank condition for exactly identified models that amounts to a straightforward counting exercise. Fourth, we develop a number of efficient algorithms for small-sample estimation and inference.
American Economic Review (2007), 97, pp 1021-1026.
Joint with Jesus Fernandez-Villaverde (University of Pennsylvania), Thomas Sargent (New York University) and Mark Watson (Princeton University).
An approximation to the equilibrium of a complete dynamic stochastic economic model can be expressed in terms of matrices (A,B,C,D) that define a state space system. An associated state space system (A,K,C,I) determines a vector autoregression for fixed observables available to an econometrician. We review circumstances under which the impulse response of the VAR resembles the impulse response associated with the economic model. We give four examples that illustrate a simple special condition for checking whether the mapping from VAR shocks to economic shocks is invertible.
Applied Economics Letters (2007), 14, pp 549-554.
Joint with Jesus Fernandez-Villaverde (University of Pennsylvania).
Recent work by Greenwood, Hercowitz, and Krusell (1997 and 2000) and Fisher (2003) has emphasized the importance of investment-specific technological change as a main driving force behind long-run growth and the business cycle. This paper shows how the growth model with investment-specific technological change has a closed-form solution if capital fully depreciates. This solution furthers our understanding of the model and it constitutes a useful benchmark to check the accuracy of numerical procedures to solve dynamic macroeconomic models in cases with several state variables.
Work in progress.
Joint with Jules H. van Binsbergen (Stanford University), Jesus Fernandez-Villaverde (University of Pennsylvania), and Ralph S.J. Koijen (University of Chicago).
This paper illustrates how to perform likelihood-based inference in dynamic stochastic general equilibrium (DSGE) models with Epstein-Zin preferences. This class of preferences has recently become a popular device to account for asset pricing observations and other phenomena that are challenging to address within the traditional state-separable utility framework. However, there has been little econometric work in the area, particularly from a likelihood perspective, because of the difficulty in computing an equilibrium solution to the model and in deriving the likelihood function. To fill this gap, we build a real business cycle model with Epstein-Zin preferences and long-run growth, solve it with perturbation techniques, and evaluate its likelihood with the particle .lter. We estimate the model using U.S. macro and yield curve data. We discuss the ability of the model to explain the business cycle, asset prices, the comovements between these two, and the implications of our point estimates for the welfare cost of the business cycle .
Work in progress.
Joint with Pau Rabanal (IMF) and Vicente Tuesta.
A central puzzle in international macroeconomics is that observed real exchange rates are highly volatile. Standard International Real Business Cycle (IRBC) models cannot reproduce this fact when calibrated using conventional parameterizations, and can only generate one fourth of the real exchange rate volatility observed in the data. Typically, IRBC models are solved assuming that total factor productivity (TFP) processes are stationary. In this paper, we .rst show that TFP processes for the U.S. and the .rest of the world. have a unit root, are cointegrated, and can be jointly characterized with a Vector Error Correction Model (VECM). Then, we explore the implications of extending an otherwise standard international real business cycle model that allows for cointegrated technology shocks. We show that the model can account for the high real exchange rate volatility observed in the data without having to rely on any particular nominal or real friction. Also, we show that the increase of relative volatility of the real exchange rate with respect to output in the last 20 years can be explained by changes in the parameter estimates of the VECM.
Work in progress.
Joint with Jesus Fernandez-Villaverde (University of Pennsylvania), Pablo Guerron-Quintana (North Carolina State University), and Martin Uribe (Columbia University),
This paper shows how changes in the volatility of the real interest rate at which small open emerging economies borrow have a quantitatively important e¤ect on real variables like output, consumption, investment, and hours worked. To motivate our investigation, we document the strong evidence of time-varying volatility in the real interest rates faced by a sample of four emerging small open economies: Argentina, Ecuador, Venezuela, and Brazil. We postulate a stochastic volatility process for real interest rates using T-bill rates and country spreads and estimate it with the help of the Particle .lter and Bayesian methods. Then, we feed the estimated stochastic volatility process for real interest rates in an otherwise standard small open economy business cycle model. We calibrate eight versions of our model to match basic aggregate observations, two versions for each of the four countries in our sample. We .nd that an increase in real interest rate volatility triggers a fall in output, consumption, investment, and hours worked, and a notable change in the current account of the economy.