On Financial Time Series Decompositions with Applications to Volatility

We consider decompositions of financial time series that identify important modes of variation in the series. The first term in the decomposition measures long-term trends and focuses on large-scale features of variability. The second term measures short-term trends and local features of variability remaining after the long-term trend has been removed. The third term measures the irregularity left in the series after the long- and short-term trends have been subtracted out. This term is further broken down by regressing it on its own lagged values. One goal of this decomposition is to transform a "raw" time series into three interpretable terms plus a term that is approximately noise. In this paper, the methodology is applied to the exchange rates of Japanese Yen (JY), German Marks (GM), Swiss Francs (SF), and British Pounds (BP) in the unit of U.S. dollars. Similarities and differences in the trends between these currencies as well as their volatilities are discussed.