The paper concerns the design of nonparametric low-pass filters that have the property of reproducing a polynomial of a given degree. Two approaches are considered. The first is locally weighted polynomial regression (LWPR), which leads to linear filters depending on three parameters: the...

The chapter deals with parametric models for the measurement of the business cycle in economic time series. It presents univariate methods based on parametric trend{cycle decom- positions and multivariate models featuring a Phillips type relationship between the output gap and inflation and the...

This note is concerned with the spectral properties of matrices associated with linear smoothers. We derive analytical results on the eigenvalues and eigenvectors of smoothing matrices by interpreting the latter as perturbations of matrices belonging to algebras with known spectral properties,...

We apply a recently proposed Bayesian model selection technique, known as stochastic model specification search, for characterising the nature of the trend in macroeconomic time series. We illustrate that the methodology can be quite successfully applied to discriminate between stochastic and...

Trend estimation deals with the characterization of the underlying, or long–run, evolution of a time series. Despite being a very pervasive theme in time series analysis since its inception, it still raises a lot of controversies. The difficulties, or better, the challenges, lie in the...

The Beveridge-Nelson decomposition defines the trend component in terms of the eventual forecast function, as the value the series would take if it were on its long-run path. The paper introduces the multistep Beveridge-Nelson decomposition, which arises when the forecast function is obtained by...

A univariate first order stochastic cycle can be represented as an element of a bivariate first order vector autoregressive process, or VAR(1), where the transition matrix is associated with a Givens rotation. From the geometrical viewpoint, the kernel of the cyclical dynamics is described by a...

The paper establishes the conditions under which the generalised least squares estimator of the regression parameters is equivalent to the weighted least squares estimator. The equivalence conditions have interesting applications in local polynomial regression and kernel smoothing. Specifically,...

The local level model with stochastic volatility, recently proposed for U.S. by Stock and Watson (Why Has U.S. Inflation Become Harder to Forecast?, Journal of Money, Credit and Banking, Supplement to Vol. 39, No. 1, February 2007), provides a simple yet sufficently rich framework for...

The series on average hours worked in the manufacturing sector is a key leading indicator of the U.S. business cycle. The paper deals with robust estimation of the cyclical component for the seasonally adjusted time series. This is achieved by an unobserved components model featuring an...