Evaluating Real Business Cycle Models Using Likelihood Methods

One of the biggest issues in using likelihood methods to evaluate and compare Real Business Cycle models is the lack of stochastic components, or lack of dimensionality, in the models. There have been a number of approaches to remedy this problem. One method is to add stochastic elements such as measurement errors to the model so that there are as many stochastic terms in the model as variables. Another approach is to compute the likelihood using only subsets of variables at a time.This paper describes a method that directly computes a likelihood function for an RBC model, via its state-space representation, using all of the information available without having to add arbitrary stochastic elements. The state-space representation for the model is obtained using a linear-quadratic approximation. In the state-space representation all of the variables are represented as linear functions of the state variables. The state variables include the stochastic elements of the model. Independent indices of the variables are then used to compute the likelihood function thus allowing for information from all of the variables in the modle to be use in computing the likelihood function. The independent indices are computed using a canonical decomposition procedure. It is shown how this method can be extended to the problem where the relationship between the state variables and the observed variables are non-linear. A standard RBC model is evaluated and the linear and non-linear case is compared.Once a full dimensional likelihood function is calculated it is possible to use Bayesian non-nested model comparison techniques to compare RBC and non-RBC models directly. It is also possible to directly compare non-nested models across sub-samples of the data as well as across the whole sample. This paper shows how these procedures can be implemented using standard Markov chain Monte Carlo techniques.