The `Pile-up Problem' in Trend-Cycle Decomposition of Real GDP: Classical and Bayesian Perspectives

In the case of a flat prior, a conventional wisdom is that Bayesian inference may not be very different from classical inference, as the likelihood dominates the posterior density. This paper shows that there are cases in which this conventional wisdom does not apply. An ARMA model of real GDP growth estimated by Perron and Wada (2009) is an example. While their maximum likelihood estimation of the model implies that real GDP may be a trend stationary process, Bayesian estimation of the same model implies that most of the variations in real GDP can be explained by the stochastic trend component, as in Nelson and Plosser (1982) and Morley et al. (2003). We show such dramatically different results stem from the differences in how the nuisance parameters are handled between the two approaches, especially when the parameter estimate of interest is dependent upon the estimates of the nuisance parameters for small samples. For the maximum likelihood approach, as the number of the nuisance parameters increases, we have higher probability that the moving-average root may be estimated to be one even when its true value is less than one, spuriously indicating that the data is `over-differenced.' However, the Bayesian approach is relatively free from this pile-up problem, as the posterior distribution is not dependent upon the nuisance parameters.