Kolmogorov–Smirnov simultaneous confidence bands for time series distribution function

Jie Li, Jiangyan Wang and Lijian Yang
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Jie Li: Tsinghua University
Jiangyan Wang: Nanjing Audit University

Computational Statistics, 2022, vol. 37, issue 3, No 2, 1015-1039

Abstract: Abstract Claims about distributions of time series are often unproven assertions instead of substantiated conclusions for lack of hypotheses testing tools. In this work, Kolmogorov–Smirnov type simultaneous confidence bands (SCBs) are constructed based on simple random samples (SRSs) drawn from realizations of time series, together with smooth SCBs using kernel distribution estimator (KDE) instead of empirical cumulative distribution function of the SRS. All SCBs are shown to enjoy the same limiting distribution as the standard Kolmogorov–Smirnov for i.i.d. sample, which is validated in simulation experiments on various time series. Computing these SCBs for the standardized S&P 500 daily returns data leads to some rather unexpected findings, i.e., student’s t-distributions with degrees of freedom no less than 3 and the normal distribution are all acceptable versions of the standardized daily returns series’ distribution, with proper rescaling. These findings present challenges to the long held belief that daily financial returns distribution is fat-tailed and leptokurtic.

Keywords: Bandwidth; Brownian bridge; Kernel; Kolmogorov distribution; Simple random sample; Stationarity (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s00180-021-01149-5

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