Estimating Macroeconomic Models: A Likelihood Approach

Abstract

This paper shows how particle filtering facilitates likelihood-based inference in dynamic macroeconomic models. The economies can be non-linear and/or non-normal. We describe how to use the output from the particle filter to estimate the structural parameters of the model, those characterizing preferences and technology, and to compare different economies. Both tasks can be implemented from either a classical or a Bayesian perspective. We illustrate the technique by estimating a business cycle model with investment-specific technological change, preference shocks, and stochastic volatility. This paper shows how particle filtering facilitates likelihood-based inference in dynamic equilibrium models. The economies can be non-linear and/or non-normal. We describe how to use the particle filter to estimate the structural parameters of the model, those characterizing preferences and technology, and to compare different economies. Both tasks can be implemented from either a classical or a Bayesian perspective. We illustrate the technique by estimating a business cycle model with investment-specific technological change, preference shocks, and stochastic volatility. We highlight three results. First, there is strong evidence of stochastic volatility on U.S. aggregate data. Second, two periods of low and falling aggregate volatility, from the late 1950’s to the late 1960’s and from the mid 1980’s to today, were interrupted by a period of high and rising aggregate volatility from the late 1960’s to the early 1980’s. Third, variations in the volatility of preferences and investment-specific technological shocks account for most of the variation in the volatility of output growth over the last 50 years. Likelihood-based inference is a useful tool to take dynamic equilibrium models to the data (An and Schorfheide, 2007). However, most dynamic equilibrium models do not imply a likelihood function that can be evaluated analytically or numerically. To circumvent this problem, the literature has used the approximated likelihood derived from a linearized version of the model, instead of the exact likelihood. But linearization depends on the accurate approximation of the solution of the model by a linear relation and on the shocks to the economy being distributed normally. Both assumptions are problematic. First, the impact of linearization is grimmer than it appears. Fernandez-Villaverde, RubioRamirez and Santos (2006) prove that second-order approximation errors in the solution of the model have first-order effects on the likelihood function. Moreover, the error in the approximated likelihood gets compounded with the size of the sample. Period by period, small errors in the policy function accumulate at the same rate at which the sample size grows. Therefore,