Optimal Linear Filtering, Smoothing and Trend Extraction for m-period Differences of Processes with a Unit Root

In this paper I consider the problem of optimal linear filtering, smoothing and trend extraction for m-period differences of processes with a unit root. 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) and their form and properties are examined in detail. In particular, I find explicit representations for the asymptotic decomposition of the covariance matrix and show that the first two leading eigenvalues of the decomposition account for over 90% of the variability of the process. I examine the structure of the impulse and frequency response functions finding that the optimal filter has a “permanent” and a “transitory component” with the corresponding smoother being the sum of two such components. I also find explicit representations for the extrapolation coefficients that can be used in out-of-sample prediction. The methodology of the paper is illustrated with three short empirical applications using data on U.S. inflation and real GDP growth and data on the Euro/US dollar exchange rate. Finally, the paper contains a new technical result: I derive explicit representations for the filtering weights in the context of SSA for an arbitrary covariance matrix. This result allows one to examine specific effects of smoothing in any situation and has not appeared so far, to the best of my knowledge, in the related literature.