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We propose a novel method to extract global and local features of functional time series. The global features concerning the dominant modes of variation over the entire function domain, and local features of function variations over particular short intervals within function domain, are both important in functional data analysis. Functional principal component analysis (FPCA), though a key feature extraction tool, only focus on capturing the dominant global features, neglecting highly localized features. We introduce a FPCA-BTW method that initially extracts global features of functional data via FPCA, and then extracts local features by block thresholding of wavelet (BTW) coefficients. Using Monte Carlo simulations, along with an empirical application on near-infrared spectroscopy data of wood panels, we illustrate that the proposed method outperforms competing methods including FPCA and sparse FPCA in the estimation functional processes. Moreover, extracted local features inheriting serial dependence of the original functional time series contribute to more accurate forecasts. Finally, we develop asymptotic properties of FPCA-BTW estimators, discovering the interaction between convergence rates of global and local features.

Taylor's law is a widely observed empirical pattern that relates the variances to the means of population densities. We present four extensions of the classical Taylor's law (TL): (1) a cubic extension of the linear TL describes the mean¿variance relationship of human mortality at subnational levels well; (2) in a time series, long-run variance measures not only variance but also autocovariance, and it is a more suitable measure than variance alone to capture temporal/spatial correlation; (3) an extension of the classical equally weighted spatial variance takes account of synchrony and proximity; (4) robust linear regression estimators of TL parameters reduce vulnerability to outliers. Applying the proposed methods to age-specific Japanese subnational death rates from 1975 to 2018, we study temporal and spatial variations, compare different coefficient estimators, and interpret the implications. We apply a clustering algorithm to the estimated TL coefficients and find that cluster memberships are strongly related to prefectural gross domestic product. The time series of spatial TL coefficients has a decreasing trend that confirms the narrowing gap between rural and urban mortality in Japan.

In this paper, we propose an Adaptive Hyperbolic EGARCH (A-HYEGARCH) model to estimate the long memory of high frequency time series with potential structural breaks. Based on the original HYGARCH model, we use the logarithm transformation to ensure the positivity of conditional variance. The structural change is further allowed via a flexible time-dependent intercept in the conditional variance equation. To demonstrate its effectiveness, we perform a range of Monte Carlo studies considering various data generating processes with and without structural changes. Empirical testing of the A-HYEGARCH model is also conducted using high frequency returns of S&P 500, FTSE 100, ASX 200 and Nikkei 225. Our simulation and empirical evidence demonstrate that the proposed A-HYEGARCH model outperforms various competing specifications and can effectively control for structural breaks. Therefore, our model may provide more reliable estimates of long memory and could be a widely useful tool for modelling financial volatility in other contexts.