An AFD-based adaptive impulse response model of time series

Abstract

In this paper, an adaptive impulse response model of time series is proposed based on adaptive Fourier decomposition (AFD), which characterizes the data-driven impulse response model with “energy-resolution” and is applied to the effect analysis of shocks on time series. First, according to the observed time series, the frequency points of impulse response function are estimated, and the approximate expression of impulse response function in frequency domain is given. Second, AFD is carried out on the approximate impulse response function level by level, and the poles and coefficients of basis function of each level are obtained adaptively by using the criterion of energy maximization. Third, when the goodness of fit is greater than a certain threshold, all decomposition levels are determined and the adaptive impulse response model is established. Each decomposition level depicts the energy-based resolution characteristics of the impulse response model. Based on gold price return and US dollar index return in daily frequency from 2017 to 2022, this paper carried out AFD-based effect analysis with energy-resolution year by year. Results show that the proposed AFD-based adaptive impulse response model in this paper can describe the persistence effect of shocks of gold price on the US dollar index, and further measure the plainness of linkage relationship between these two time series. Decomposing impulse response to each decomposition level reveals consistence, convergency, and sufficiency of the impulse response model. The AFD-based adaptive data-driven model provides a new way for the analysis of time series in time-frequency domain.