Effects of intervaling on high-frequency realized higher-order moments

Abstract

In high-frequency finance, the statistical terms ‘realized skewness’ and ‘realized kurtosis’ refer to the realized third- and fourth-order moments of high-frequency returns data normalized (or divided) by ‘realized variance’. In particular, before any computations of these two normalized realized moments are carried out, one often predetermines the holding-interval and sampling-interval and thus implicitly influencing the actual magnitudes of the computed values of the normalized realized higher-order moments i.e. they have been found to be interval-variant. To-date, little theoretical or empirical studies have been undertaken in the high-frequency finance literature to properly investigate and understand the effects of these two types of intervalings on the behaviour of the ensuring measures of realized skewness and realized kurtosis. This paper fills this gap by theoretically and empirically analyzing as to why and how these two normalized realized higher-order moments of market returns are influenced by the selected holding-interval and sampling-interval. Using simulated and price index data from the G7 countries, we then proceed to illustrate via count-based signature plots, the theoretical and empirical relationships between the realized higher-order moments and the sampling-intervals and holding-intervals.

Keywords:

• High-frequency data analysis
• Intervaling effect
• Holding-interval
• Sampling-interval
• Realized skewness
• Realized kurtosis

JEL Classification:

• C10
• G10

Acknowledgments

The authors are grateful to the participants at the Quantitative Methods in Finance (QMF) Conference, Sydney, Australia (12th–15th December, 2017) for their comments on an associated paper titled: ‘Optimal Sampling Frequencies for Realized Variance, Realized Skewness and Realized Kurtosis’. In particular, we thank the Session Chair, Marcello Rambaldi, for raising the seminal question that provided the motivation for this paper: ‘Is there an optimal number of (infill) observations, N, for computing realized skewness and realized kurtosis?.’ Our answer: ‘As the RS and RK values are interval-variant and insensitive to market microstructure; there is no optimal number of infill observations per se. However, as RV is sensitive to market microstructure noise at high sampling-frequencies, it would be prudent to select the infill number of observations based on the optimal sampling-interval selected for computing RV, τ, and from the relationship $N\equiv h/\tau$.’ We would also like to thank the editor and two anonymous referees who kindly reviewed the earlier versions of this manuscript and provided invaluable comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The holding-intervals and the sampling-intervals in high-frequency finance are generally not equivalent. Holding-intervals are commonly daily, weekly or monthly intervals; whereas sampling-intervals are the intraday-intervals at which the high-frequency data is being sampled; for example at 1-minute, 5-minute or 30-minute intervals and so on. The holding-interval effect in low-frequency finance has also been called the interval effect, the investment interval problem, the holding period problem, etc.

2 The G7-countries consist of Canada (GSPTSE index), France (CAC 40 index), Germany (GDAXIP index), Italy (FTMIB index), Japan (N500 index), United Kingdom (FTSE 100 Index), and United States (S&P 500 Index). Shown in parenthesis are Reuters instrument codes for the indexes.

3 The central limit theorem states that the skewness goes to zero and the kurtosis approaches three, as the number of observation approaches infinity. We observe that this only holds for stock prices generated from a pure diffusive process with no jumps.

4 In the case of realized variance, the greater the holding-interval the higher the variations and hence the realized variance increases. However, the estimates of realized variance become more efficient (and converges) as the sampling-interval decreases i.e at higher frequencies.

5 See the errata in the online Technical Appendices to Chapter 3 of Meucci (2007), where the author has explicitly stated that ‘the multiplicative relation does not hold for all raw moments and all central moments, but only holds for the projection of expected values and covariances’. The online Technical Appendices is available at https://www.arpm.co/symmys-articles/AMeucciRiskAndAssetAllocationTechnicalAppendices.pdf.

6 The sample period for US, UK, Japan, and Canada starts from 1st May, 2002 to 15th November, 2017. For France, Germany, and Italy due to the fact that the data with the same length as those mentioned earlier was not available from the Thompson Reuters Tick History, the data sample for France starts 2nd May, 2002 to 15th November, 2017, Germany from 6th May, 2002 to 15th November, 2017 and Italy from 1st June, 2009 to 15th November, 2017.

7 Using Monte Carlo techniques and 1-second data, Amaya et al. (2015) verified that estimates of the realized higher moments are reliable in finite samples and that at 1-minute return series are robust to the presence of market microstructure noise.

8 Ideally, one should construct realized higher-order moments from high-frequency intra-day return data of the true price process, $p^{\ast }\left(t\right)$. However, $p^{\ast }\left(t\right)$ is unobservable and hence the realized higher-order moment are computed from observed prices $p\left(t\right)$ which are contaminated with market microstructure noise (Hansen and Lunde 2004). The observed price is defined as $p\left(t\right)\equiv p^{\ast }\left(t\right)+u\left(t\right)$, where $u\left(t\right)$ is the noise component that may arise from the bid-ask bounce, price discreteness, rounding errors and price reporting error (see Bai 2000, Andreou and Ghysels 2002, Oomen 2004). In this paper, we assume there are no market microstructure noise effects and consequently, for our empirical work, we download raw data at the 1-minute sampling frequency and not at the higher 1-second frequency. As such our findings are confined to ‘high’ frequency data and not to ‘very high’ frequency data; this being left for a subsequent study and paper.

9 In order to obtain weekly observations, Amaya et al. (2015) first construct daily realized moments and then take the average in a 5-days window to obtain their weekly realized moments. These so-called weekly observations are just an average of daily realized moments. Our weekly and monthly data are computed using the relevant full holding-interval.