Adaptive Optimal Scaling of Metropolis-Hastings Algorithms Using the Robbins-Monro Process

We present an adaptive method for the automatic scaling of Random-Walk Metropolis-Hastings algorithms, which quickly and robustly identifies the scaling factor that yields a specified overall sampler acceptance probability. Our method relies on the use of the Robbins-Monro search process, whose performance is determined by an unknown steplength constant. We give a very simple estimator of this constant for proposal distributions that are univariate or multivariate normal, together with a sampling algorithm for automating the method. The effectiveness of the algorithm is demonstrated with both simulated and real data examples. This approach could be implemented as a useful component in more complex adaptive Markov chain Monte Carlo algorithms, or as part of automated software packages

Extracting the Italian output gap: a Bayesian approach

During the last decades particular effort has been directed towards understanding and predicting the relevant state of the business cycle with the objective of decomposing permanent shocks from those having only a transitory impact on real output. This trend--cycle decomposition has a relevant impact on several economic and fiscal variables and constitutes by itself an important indicator for policy purposes. This paper deals with trend--cycle decomposition for the Italian economy having some interesting peculiarities which makes it attractive to analyse from both a statistic and an historical perspective. We propose an univariate model for the quarterly real GDP, subsequently extended to include the price dynamics through a Phillips curve. This study considers a series of the Italian quarterly real GDP recently released by OECD which includes both the 1960s and the recent global financial crisis of 2007--2008. Parameters estimate as well as the signal extraction are performed within the Bayesian paradigm which effectively handles complex models where the parameters enter the log--likelihood function in a strongly nonlinear way. A new Adaptive Independent Metropolis--within--Gibbs sampler is then developed to efficiently simulate the parameters of the unobserved cycle. Our results suggest that inflation influences the Output Gap estimate, making the extracted Italian OG an important indicator of inflation pressures on the real side of the economy, as stated by the Phillips theory. Moreover, our estimate of the sequence of peaks and troughs of the Output Gap is in line with the OECD official dating of the Italian business cycle