Technical Note on "Assessing Bayesian Model Comparison in Small Samples"

Abstract

This technical note is developed as a companion to the paper ‘Assessing Bayesian Model Comparison in Small Samples’ (Globalization and Monetary Policy Institute working paper no. 189). Taking the workhorse open-economy model of Martinez-Garcia and Wynne (2010) with nominal rigidities under monopolistic competition as our Data-Generating Process, we investigate with simulated data how Bayesian model comparison based on posterior odds performs when the model becomes arbitrarily close to a closed-economy and/or an economy with flexible prices and perfect competition. This technical note elaborates on three key technical points relevant for Martinez-Garcia and Wynne (2014). First, we explain the building blocks of the open-economy model of Martinez-Garcia and Wynne (2010). We also derive the equilibrium conditions (and the steady state) under producer-currency pricing. Second, we discuss the log-linearization of the equilibrium conditions around the deterministic steady state and our benchmark parameterization. The linear rational expectations model that results from the log-linearization is used to simulate the data under our benchmark parameterization. These simulated data is used in Martinez-Garcia and Wynne (2014) to conduct their Bayesian model comparison exercises. Third, we describe the Bayesian estimation and model comparison techniques with special emphasis on the questions of: (a) how we elicit priors on the models themselves and the parameters of a given model, and (b) how we compute posterior model probabilities. Simultaneously, commentary is provided whenever appropriate to clarify the economic significance of the assumptions embedded in our workhorse open-economy model. JEL codes: C11, C13, F41 * Enrique Martinez-Garcia, Research Department, Federal Reserve Bank of Dallas, 2200 N. Pearl Street, Dallas, TX 75201. 214-922-5262. enrique.martinez-garcia@dal.frb.org. Mark A. Wynne, Research Department, Federal Reserve Bank of Dallas, 2200 N. Pearl Street, Dallas, TX 75201. 214-922-5159. Mark.A.Wynne@dal.frgb.org. We would like to thank Nathan Balke, Maria Teresa Martinez-Garcia and Valentin Martinez Mira for helpful suggestions. Diego Vilan was a co-author in a related project and contributed to the early stages of development of this paper. We gratefully acknowledge the outstanding research assistance provided by Valerie Grossman, the help of Kuhu Parasrampuria, and the Federal Reserve Bank of Dallas support. The views in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System. 1 New Open Economy Macro (NOEM) Model The NOEM model that we use in Martinez-Garcia and Wynne (2014) is a variant of the workhorse model of Clarida et al. (2002) introduced in Martinez-Garcia and Wynne (2010). This is a symmetric two-country model with a continuum of unit mass of households and consumption varieties, equally divided between the Home country and the Foreign country. We employ this framework because it integrates an open-economy New Keynesian Phillips curve that ‡eshes out the content of the global slack hypothesis into a stylized dynamic stochastic general equilibrium model, but also because it nests model speci cations without nominal rigidities (monetary neutrality) and/or under autarky (closed-economies) as limiting cases. We abstract from a number of relevant modelling features like capital and investment (see, e.g., Chari et al. (2002), and Martinez-Garcia and Sondergaard (2008)), durable goods (see, e.g., Engel and Wang (2011)), and monopolistically competitive suppliers of labor (see, e.g., Clarida et al. (2002)) in order to assess Bayesian model comparison solely in the presence of e¤ects from increased trade openness and monetary non-neutrality induced by nominal rigidities. Households. The lifetime utility for the representative household in the Home country is additively separable in consumption, Ct, and labor, Lt, i.e., X+1 =0 Et ln (Ct+ ) 1 1 + ' (Lt+ ) 1+' ; (1) and similarly the lifetime utility for the representative household in the Foreign country is additively separable in consumption, C t , and labor, L t , i.e., X+1 =0 Et ln C t+ 1 1 + ' L t+ 1+' ; (2) where 0 0 is the inverse of the Frisch elasticity of labor supply. The inverse of the intertemporal elasticity of substitution is equal to 1 under the assumption of log-utility on consumption. The Home household maximizes its lifetime utility subject to the sequence of budget constraints,