Seasonal adjustment of time series observed at mixed frequencies using singular value decomposition with wavelet thresholding

In this paper, we propose a novel seasonal adjustment method that accommodates time series observed at mixed frequencies and possessing possibly multiple abrupt changes in seasonality. We assume that the observed series is a known linear transformation of an underlying high frequency series whose seasonal component can be represented as a matrix with a low rank Singular Value Decomposition (SVD) structure, and the nonseasonal component is difference stationary. The right and left singular vectors of the SVD correspond respectively to the seasonal patterns and their time-varying amplitudes. We propose a penalized optimization framework to estimate the seasonality where a penalty is defined to shrink towards zero the wavelet coefficients of the discrete wavelet transformation of the left singular vectors. A novel Alternating Direction Method of Multiplier (ADMM) algorithm that can handle the non-smooth penalty and the manifold structure of the parameter space is developed for efficient computation. Using both simulated and real data, we find that (i) when the seasonality is moderate or strong our proposed method performs well and correctly detects the underlying seasonality structure; and (ii) for single frequency time series, the performance of our proposed method compares well with those of the traditional X-12-ARIMA and SEATS methods, especially in the case when the seasonality is strong.