Optimal Linear Filtering, Smoothing and Trend Extraction for the m-th Differences of a Unit Root Process: A Singular Spectrum Analysis Approach

The problem of optimal linear filtering, smoothing and trend extraction for m-period differences of processes with a unit root is studied. Such processes arise naturally in economics and finance, in the form of rates of change (price inflation, economic growth, financial returns) and finding an appropriate smoother is thus of immediate practical interest. The filter and resulting smoother are based on the methodology of Singular Spectrum Analysis (SSA). An explicit representation for the asymptotic decomposition of the covariance matrix is obtained. The structure of the impulse and frequency response functions indicates that the optimal filter has a “permanent” and a “transitory component”, with the corresponding smoother being the sum of two such components. Moreover, a particular form for the extrapolation coefficients that can be used in out-of-sample prediction is proposed. In addition, an explicit representation for the filtering weights in the context of SSA for an arbitrary covariance matrix is derived. This result allows one to examine the specific effects of smoothing in any situation. The theoretical results are llustrated using different data sets, namely U.S. inflation and real GDP growth.