A simple two-component model for the distribution of intraday returns

Abstract

We model the conditional distribution of high-frequency financial returns by means of a two-component quantile regression model. Using three years of 30 minute returns, we show that the conditional distribution depends on past returns and on the time of the day. Two practical applications illustrate the usefulness of the model. First, we provide quantile-based measures of conditional volatility, asymmetry and kurtosis that do not depend on the existence of moments. We find seasonal patterns and time dependencies beyond volatility. Second, we estimate and forecast intraday Value at Risk. The two-component model is able to provide good-risk assessments and to outperform GARCH-based Value at Risk evaluations.

Keywords:

• intraday returns
• quantile regression
• intraday VaR

JEL Classifications: :

• C14
• C22
• C53
• G10

Acknowledgements

We are grateful to the editor Ingmar Nolte and a referee for the insightful remarks that have improved the paper. We are also grateful to Luc Bauwens, Valentina Corradi, Catherine Dehon, Pierre Giot, Marc Hallin, Peter R. Hansen, Roger Koenker, Francesco Lisi, Simone Manganelli, Alexander McNeil, Maria Pacurar, Roberto Pascual, Sergio Pastorello, and the participants at the ECARES internal seminar, the Zeuthen workshop 2006, the ESEM 2007 and CEF 2007 conferences for the insightful remarks. Any remaining errors and inaccuracies are ours.

Notes

High-frequency trading has received an increasing attention among practitioners and the media press. For instance, Charles Duhigg, from The New York Times, published on 23 July 2009, the article ‘Stock Traders Find Speed Pays, in Milliseconds’. The second line reads ‘It is called high-frequency trading – and it is suddenly one of the most talked-about and mysterious forces in the markets’.

For a comparison of the two approaches, see Martens, Chang, and Taylor (2002).

A different frequency choice is possible. However, at higher frequencies illiquid stocks present many periods without any trade and thus zero returns. This implies spurious autocorrelations in the return series. Yet, all the estimation results presented in the paper are qualitatively similar at higher frequencies, but the forecast results for the medium liquid and illiquid stocks improve significantly at 30 min, or lower, frequency. An alternative is to choose the frequency as a function of the liquidity of the stock, e.g. 15 min for the most liquid stocks and 30 min for the medium liquid or illiquid stocks.

http://personalpages.manchester.ac.uk/staff/laura.coroneo or www.ecares.org/veredas.html

The Jarque–Bera and the Shapiro–Wilk normality tests also reject normality.

We tried higher orders of the Fourier series, but they turned out not to be statistically significant.

To account for possible leverage effects, we have also introduced r _t−1. It turned out not to be statistically significant. We also included more lags of absolute returns. Results did not improve substantially.

For ease of exposition, we omit the graphical results for the other two stocks. They are qualitatively the same as for TEF.

This is confirmed in Section 5.1, where we compute quantile-based shape measures. The intuition is as follows. The 50% conditional quantile is zero and the conditional quantiles to the left of the median (i.e. τ<0.5) are negative, while the ones to the right (i.e. τ>0.5) are positive. This means that the seasonal component at a τ smaller than 0.5 will be added to a negative conditional quantile, while the seasonal component at a τ bigger than 0.5 will be added to a positive conditional quantile. If at a τ>0.5 we have a positive seasonal component, the τ conditional quantile increases and hence the dispersion increases. If instead the seasonal component at a τ>0.5 is negative, the τ conditional quantile decreases and the dispersion decreases. The same reasoning holds for any τ<0.5 where a negative seasonal component decreases the τ conditional quantile (more dispersion), while a positive seasonal component increases the τ conditional quantile, decreasing the dispersion.

Other tests for independence against different alternatives are possible. The duration-based approach in Pelletier and Christoffersen (2004) allows for testing against more general forms of dependence. Candelon et al. (2011) have developed a more robust procedure which does not need a specific distributional assumption for the time intervals between violations under the alternative. Moreover, Hurlin and Tokpavi (2006) develop a test procedure to jointly test the absence of autocorrelation in the hit variables for various coverage rates.

In this article we adopt the terminology that volatility is defined as a measure of dispersion. Likewise, asymmetry is understood as a measure of the difference in the probability mass between both sides of the mode, and kurtosis is defined as a measure of the thickness of the tails of the distribution.

For more details about quantile-based skewness and kurtosis measures, see Groeneveld and Meeden (1984), Brys, Hubert, and Struyf (2006) and Dominicy and Veredas (2010).

A GARCH model for intraday data (explained at length in Section 5.2.1) is given by , where g _t is the conditional volatility and φ _d is the intraday seasonal component. The conditional quantile of returns is , where denotes the τ quantile of the distribution of . Substituting in (9) and (10), the term cancels out since it is the same for all the quantiles. Therefore, the theoretical conditional quantile-based asymmetry and kurtosis for a GARCH model are constant. To show that this may not be a tenable assumption, we compute (8)–(10) on the standardized residuals of a GARCH model.

We have tried with smaller and larger rolling windows sizes (namely 1000, 1200, 1500, 1600, 1800, 1900 and 2000 observations). Results do not change qualitatively and are available upon request.

Note that the confidence levels 2.5% and 1% were not in the grid of quantiles of the in-sample estimation results. We introduce them now since they are two common quantiles in the measurement of downside risk.