Long-Range Dependent Curve Time Series

Abstract

We introduce methods and theory for functional or curve time series with longrange dependence. The temporal sum of the curve process is shown to be asymptotically normally distributed, the conditions for this covering a functional version of fractionally integrated autoregressive moving averages. We also construct an estimate of the long-run covariance function, which we use, via functional principal component analysis, in estimating the orthonormal functions spanning the dominant sub-space of the curves. In a semiparametric context, we propose an estimate of the memory parameter and establish its consistency. A Monte-Carlo study of finitesample performance is included, along with two empirical applications. The first of these finds a degree of stability and persistence in intra-day stock returns. The second finds similarity in the extent of long memory in incremental age-specific fertility rates across some developed nations.

Granger causality of bivariate stationary curve time series

“…We refer to such data structure as sliced functional time series, examples of which include intraday stock price curves (Kokoszka, Rice, & Shang, 2017) and intraday particulate matter (Shang, 2017). On the other hand, when the continuum is not a time variable, functional time series can also arise when observations over a period are considered as finitedimensional realizations of an underlying continuous function (e.g., yearly age-specific mortality rates; see Li et al 2020).…”

“…that is, r j t ðu i Þ is the log-return for the jth company at the middle of time interval i at day t (e.g., see Li et al 2020;Shang et al 2019). In a given day, there are 96 5-minute price observations from 9:30 to 17:20 Eastern time, resulting in 95 values of the logreturns.…”

“…In these time series, the data frequency is high enough to model itself as a curve time series and gives rise to the analysis of functional time series (see, e.g., Horváth & Kokoszka, 2012;Kokoszka & Reimherr, 2017). Examples of functional time series include intraday stock price curves, with each functional observation defined as a pricing function of time points within a day (e.g., see Horváth, Kokoszka, & Rice, 2014;Li, Robinson, & Shang, 2020), and intraday volatility curves, with each functional observation defined as a volatility function of time points within a day (e.g., see Shang, Yang, & Kearney, 2019).…”

Local Whittle estimation of long‐range dependence for functional time series

“…The bulk of the literature studies functional data which are either independent or stationary SRD (e.g., Bosq, 2000;Ramsay and Silverman, 2005;Ferraty and Vieu, 2006;Hörmann and Kokoszka, 2010;Horváth and Kokoszka, 2012;Berkes et al, 2013;Hsing and Eubank, 2015). Li et al (2020) is among the first to extend the functional framework from SRD to LRD (see also Characiejus and Rackauskas, 2014;Düker, 2018). They not only establish the central limit theorem for a temporal sum of LRD functional observations, but also develop functional principal component analysis (FPCA) and estimate the memory parameter for the projected process via semiparametric R/S.…”

Forecasting intraday S&P 500 index returns: A functional time series approach

“…Klepsch, Klüppelberg, and Wei (2017) extended the FAR and FMA processes to a functional autoregressive moving average (FARMA). Recently, Li, Robinson, and Shang (2017) considered long-range dependent functional time series, and developed a functional autoregressive integrated moving average process. Adopting a nonparametric perspective, Besse, Cardot, and Stephenson (2000) suggested that functional kernel regression be used to measure the temporal dependence by a similarity measure characterized by notions of neighborhood distance and bandwidth.…”