Extracting a robust U.S. business cycle using a time-varying multivariate model-based bandpass filter

Concurrent research documents sizeable changes in the volatility of U.S. macroeconomic time series; e.g., see Kim and Nelson (1999), McConnell and Pérez-Quirós (2000), Stock and Watson (2002), and Sensier and van Dijk (2004). Most of the evidence from this literature suggests a sizeable reduction in volatility for many series; many of them used to construct business cycle indicators. With the exception of the beginning and the end of the series, the gain function of a lowpass filter such as the Hodrick-Prescott filter or a bandpass filter remains constant through time. Consequently, the estimates provided by these filters do not account for the great moderation. The methodological contribution of this paper is the construction of a business cycle indicator that has bandpass filter properties, accounts for time varying volatility and utilizes data sampled at different frequencies. Our indicator is constructed from the multivariate unobserved components time series model of Valle e Azevedo et al. (2006), where we extend the model to include stochastic volatility in both the common cycle and irregular components of the model. This enables us to account for the heteroskedasticity present in the data. The trend components are flexible stochastic functions of random walk processes. The common cycle is a higher-order stochastic cycle formulated by Harvey and Trimbur (2003) which ensures that the extracted business cycle has bandpass filter properties. We further adjust the stochastic cycle for phase shift and amplitude between series using the device of Rünstler (2004). Finally, we introduce mixture distributions for the innovations of the trend component and for the stochastic volatility processes. Although the coefficients of the mixture distributions are given known values, the specification remains sufficiently flexible and appears to be robust to other values for the coefficients and for aberrant observations. We use Markov chain Monte Carlo (MCMC) methods for the estimation of all parameters including the trend, cycle, irregular, and stochastic volatility components.