Forward and backward smoothing for noisy nonstationary time series with an application of detecting recent change points

We propose a novel smoothing approach for time series analysis to estimate the hidden states of random variables and handle noisy, nonstationary time series data. The method is applicable even when the sample size is small, as is often the case with major macroeconomic time series data. Our approach is based on the frequency decomposition of nonstationary time series, and we address the smoothing challenges specific to such data. In particular, we introduce forward SIML smoothing and backward SIML smoothing, designed to resolve the initial value problem in nonstationary time series analysis. The proposed smoothing methods offer interpretations in both the time and frequency domains. To demonstrate the effectiveness of our approach, we apply our method to the problem of detecting recent breaks in macroeconomic consumption trends.