A robust-filtering method for noisy non-stationary multivariate time series with econometric applications

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<jats:title>Abstract</jats:title><jats:p>We investigate a new filtering method to estimate the hidden states of random variables for multiple non-stationary time series data. This helps in analyzing small sample non-stationary macro-economic time series in particular and it is based on the frequency domain application of the separating information maximum likelihood (SIML) method, developed by Kunitomo et al. (Separating Information Maximum Likelihood Estimation for High Frequency Financial Data. Springer, New York, 2018), and Kunitomo et al. (Japan J Statistics Data Sci 2:73–101, 2020), and Nishimura et al. (Asic-Pacific Financial Markets, 2019). We solve the filtering problem of hidden random variables of trend-cycle, seasonal and measurement-errors components, and propose a method to handle macro-economic time series. We develop the asymptotic theory based on the frequency domain analysis for non-stationary time series. We illustrate applications, including some properties of the method of Müller and Watson (Econometrica 86-3:775–804, 2018), and analyses of some macro-economic data in Japan.</jats:p>

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