Five number summary approach to functional ARCH for high-frequency intraday volatility

This paper proposes a five-number summary approach for intraday volatility, employing a moving average concept and five-number summary including quartiles based on recent observations. Functional analysis is discussed within the functional ARCH (fARCH) model to suit high-frequency time series for intraday volatility. Using both simulated and real high-frequency data, we show that the first and third quartiles from the five-number summary approach outperform the Hörmann et al. (2013) fARCH model under various comparison metrics. To validate the proposed method, we analyse one-minute high-frequency Korea Composite Stock Price Index (KOSPI) as well as five-minute high-frequency S&P 500 major stock data. Results demonstrate that the first and third quartiles from the five-number summary approach provide superior predictions of intraday volatility compared to the fARCH model in terms of MAE (mean absolute error), WAPE (weighted average percentage error) and Directional Accuracy (DA) measures. Furthermore, it is empirically found that B-spline basis functions outperform Fourier basis functions in functional ARCH across all tested stocks. This research highlights that the first and third quartiles can serve as a more efficient alternative for forecasting intraday volatility in high-frequency financial time series, with recommending the use of B-spline basis functions over Fourier basis functions.