Estimating Nonlinear Dynamic Equilibrium Economies: A Likelihood Approach

Abstract

Abstract: This paper presents a framework to undertake likelihood-based inference in nonlinear dynamic equilibrium economies. The authors develop a sequential Monte Carlo algorithm that delivers an estimate of the likelihood function of the model using simulation methods. This likelihood can be used for parameter estimation and for model comparison. The algorithm can deal both with nonlinearities of the economy and with the presence of non-normal shocks. The authors show consistency of the estimate and its good performance in finite simulations. This new algorithm is important because the existing empirical literature that wanted to follow a likelihood approach was limited to the estimation of linear models with Gaussian innovations. The authors apply their procedure to estimate the structural parameters of the neoclassical growth model. JEL classification: C63, C68, E37 Key words: dynamic equilibrium economies, likelihood function, nonlinear solution methods 1. Introduction This paper presents a method to undertake likelihood based inference in nonlinear dynamic equilibrium models. We show how we can use sequential monte carlo methods to estimate the structural parameters of the model, those describing preferences and technology, and to compare different economies. Both tasks can be implemented from either a Bayesian or a classical perspective. Economist now routinely use dynamic general equilibrium economies to answer quantitative questions. However they employ much less often formal econometrics to take these models to the data. Part of the reason might have been the shortcomings of existing tools. To estimates these economies, the empirical literature has been forced to use either limited-information moment methods or likelihood techniques on linearized versions of the model. This situation is unsatisfactory. Moment procedures may suffer from strong small samples biases and may not use efficiently all the existing information. Linearization techniques depend crucially on the shape of the true policy function being accurately approximated by a linear relation and on the presence of gaussian shocks. The main obstacle for a more standard likelihood-based inference is the difficulty in evaluating the likelihood function implied by a nonlinear dynamic equilibrium economy. Beyond a few particular cases,(1) it is not possible to evaluate this function. Moment methods avoid the problem by moving away from full information approaches to inference. Linearization renounces evaluating the true likelihood function of the model and concentrates instead on the likelihood of an associated, more tractable, linear approximation to the economy. We propose a Sequential Monte Carlo method to solve this problem. We describe how this technique can be applied to evaluate the likelihood function implied by the nonlinear solution of a dynamic equilibrium economy even if the driving shocks of the model are non-normal (although the algorithm is general enough that it can also deal with linear models with or without normal shocks). To do so we borrow from a growing literature on nonlinear filtering(see the seminal paper by Gordon, Salmond and Smith, 1993 and the review of the literature in Doucet, de Freitas and Gordon, 2001). We adapt this know-how to deal with the likelihood functions of dynamic equilibrium models and we show how we get accurate and stable evaluations of the likelihood function. With these evaluations available, the door for likelihood-based inference is open, either by searching for a maximum of the function(Quasi-Maximum Likelihood estimation) or by simulating the posterior distribution of the parameters using a Markov Chain Monte Carlo algorithm(Bayesian estimation). The general idea of the procedure is as follows. First, for a given set of parameter values, we compute the equilibrium policy functions of the model. Since we want to conduct inference in the nonlinear model and not in a linear approximation, we rely on a nonlinear solution method to find the policy functions. …