Hyperparameter estimation in forecast models
A large number of non-linear time series models can be more easily analyzed using traditional linear methods by considering explicitly the difference between parameters of interest, or just parameters, and hyperparameters. One example is the class of conditionally Gaussian dynamic linear models. Bayesian vector autoregressive models and non-linear transfer function models are also important examples in the literature. Until recently, a two-step procedure was broadly used to estimate such models. In the first step maximum likelihood estimation was used to find the best value of the hyperparameter, which turned to be used in the second step where a conditionally linear model was fitted. The main drawback of such an algorithm is that it does not take into account any kind of uncertainty that might have been brought, and usually was, to the modeling at the first step. In other words and more practically speaking, the variances of the parameters are underestimated. Another problem, more philosophical, is the violation of the likelihood principle by using the sample information twice. In this paper we apply sampling importance resampling (SIR) techniques ( Rubin, 1988) to obtain a numerical approximation to the full posterior distribution of the hyperparameters. Then, instead of conditioning in a particular value of that distribution we integrate the hyperparameters out in order to obtain the marginal posterior distributions of the parameters. We used SIR to model a set of Brazilian macroeconomic time-series in three different, but important, contexts. We also compare the forecast performance of our approach with traditional ones.
hyperparameter sampling importance resampling (sir) posterior distribution littermans prior dynamic modeling bayes factor