Smoothing Non-Stationary Time Series Using the Discrete Cosine Transform

Abstract

This paper considers the problem of smoothing a non-stationary time series (having either deterministic and/or stochastic trends) using the discrete cosine transform (DCT). The DCT is a powerful tool which has found fruitful applications in filtering and smoothing as it can closely approximate the optimal Karhunen-Loeve transform (KLT). In fact, it is known that it almost corresponds to the KLT for first-order autoregressive processes with a root close to unity: This is the case with most economic and financial time series. A number of new results are derived in the paper: (a) The explicit form of the linear smoother based on the DCT, which is found to have time-varying weights and that uses all observations; (b) the extrapolation of the DCT-smoothed series; (c) the form of the average frequency response function, which is shown to approximate the frequency response of the ideal low pass filter; (d) the asymptotic distribution of the DCT coefficients under the assumptions of deterministic or stochastic trends; (e) two news method for selecting an appropriate degree of smoothing, in general and under the assumptions in (d). These findings are applied and illustrated using several real world economic and financial time series. The results indicate that the DCT-based smoother that is proposed can find many useful applications in economic and financial time series.