Robust Functional Data Analysis of Dollar Exchange Rates Using Functional Median Polish

Abstract* We employ the 34-year monthly time series of the US dollar exchange rate to explore the existence of outliers and nonlinearity which confine traditional decomposition methods for large datasets. The data was modelled as a smoothed functional over a Fourier basis of seven functions. Based on the model presented, one-way and two-way Functional Median Polish (FMP) was applied to disentangle the series into a central function, annual and monthly effects, and residuals. Functional ANOVA tests were performed to determine the significance of each of these components. The central function was 0.64–0.65 on the functional scale, and it is observed that the annual and monthly effects were not significant after functional smoothing. With short-term shocks, the residuals had a centralized size around zero and relatively heavy tails, unlike the normally present seasonal events. Such outcomes verify that FMP is a strong and interpretable working framework that avoids the adverse effects of outliers significantly and that it is appropriate in the analysis of economic time series.