Combined Minimum Relative Entropy and Maximum Likelihood estimation of dynamic models

This paper proposes an approach to estimating dynamic economic models that is based on a combination of relative entropy minimization (MRE) and log-likelihood maximization (MLE). It is assumed that there is not enough data to estimate it in the conventional manner, but that it is nevertheless required to fit the available data, given a priori knowledge regarding the stochastic distribution of the parameters. The proposed MRE estimation procedure is derived from the Kullback–Leibler divergence, assuming a particular data generation process. Compared to the generalized maximum entropy (GME) approach of Golan, Judge and Karp (1996), MRE does not require any reformulation of the parameter space and has a more natural interpretation. The stochastic distribution of the parameters is assumed to have a continuous density function, rather than the discrete distribution implied by GME. In the class of continuous densities, the triangular probability density function is chosen for several reasons: it is a simple and intuitive way to formulate a priori beliefs, it simplifies the optimization problem to be solved, and yields an objective function for estimation that resembles the log-likelihood function. This last feature allows it to be combined with maximization of the log-likelihood of the observed data, yielding an objective function for estimation that combines MRE and MLE with triangular densities. It fits the available data and uses the a priori information on the parameters, both simultaneously and in an optimal way. Finally, the usefulness and power of the proposed procedure is exemplified by estimating the parameters that characterize capital formation in the Brazilian economy from 1991 to 2004.