Abstract
Spurred by the initial research of Andersen and Bollerslev (1998) and Barndorff-Nielsen and Shephard (2001) high frequency intraday returns are increasingly considered for the purpose of approximating realised volatility. The notion that daily ex-post volatility is better approximated when based on cumulative squared intraday return data is supported by the theory that the measurement noise contained in daily squared returns prevents the observation of the actual volatility process but is reduced as the sampling frequency of the return series from which volatility is calculated is increased1. As such, it therefore theoretically justifies and extends the earlier work of French, Schwert and Stambaugh (1987), amongst others. Andersen and Bollerslev (1998) also showed that daily Generalised Autoregressive Conditional Heteroskedasticity (GARCH) volatility forecasts of exchange rates, when evaluated against intraday volatility measures, are far more accurate than had been previously assumed. These findings were subsequently confirmed with regard to stock index data by Blair, Poon and Taylor (2001) who examined the predictive accuracy of out-of-sample volatility forecasts based on GARCH models; we reached the same conclusion with regard to Stochastic Volatility (SV) models in Chapter 6.
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References
See, for example, Andersen, Bollerslev, Diebold and Labys (2001a), Andersen, Bollerslev, Diebold and Labys (2001b), Barndorff-Nielsen and Shephard (2001) and Barndorff-Nielsen and Shephard (2002).
Due to strong periodic intraday patterns the GARCH models are not actually estimated at the higher intraday level, see e.g. Andersen and Bollerslev (1997). Instead the parameter estimates of the GARCH models at the higher frequencies are inferred from the temporal aggregation results of Drost and Nijman (1993).
Fractionally integrated models are also advocated in the context of financial market data by, for example, Andersen, Bollerslev, Diebold and Labys (1999) and Andersen et al. (2001a).
Provided that the asset is sufficiently liquid, the 5-minute frequency is acknowledged as the highest frequency at which the effects of market microstructure biases, such as bid-ask bounces and discrete price observations, are not too distorting. Also see: Andersen et al. (2001a), Andersen et al. (1999), Ebens (1999) and Andersen and Bollerslev (1997).
Three trading days are missing from our data sample: 11 February 1998, 28 August 1998 and 14 January 1999.
After accounting for early market closures, 559 price notations are ”missing” of which 92 are 9.30 a.m. prices.
Theoretically, the volatility estimates become free of measurement noise as the sampling frequency interval becomes infinitesimally small; see the references in note 1. is the sample autocorrelation coefficient at lag t with asymptotic standard error and Q(t) is the Box-Ljung portmanteau statistic based on t squared autocorrelations.
Andersen, Bollerslev, Diebold and Ebens (2001) and Andersen and Bollerslev (1997) use this definition of realised volatility in their stock market studies.
Also see Ebens (1999) and Andersen et al. (2001) for similar autocorrelation coefficients and comparable Q statistic values.
See e.g. Martens (2001) and Blair et al. (2001).
More specifically, the parameters in the GARCH model were estimated using the G@RCH 1.1 package of Laurent and Peters (2002).
The data series and the programs used for the estimation of the UC-RV, SV and SVX models can be downloaded from the Internte at also see Appendix C.
Ebens (1999), for example, never estimates the autoregressive parameter in his ARFI¬MAX(p,d,q) model and Andersen et al. (2001b) fix the value of the d parameter prior to the estimation of the other parameters in the ARFIMA model.
Also see Oomen (2001) who encounters the same problem with regard to a ten-year sample of FTSE-100 index returns.
See Harvey (1993), pp.30–32. The fitted autocorrelation function for τ 2 is given by
Unlike Martens (2002), we find a slightly better in-sample fit when the overnight returns are excluded from the realised volatility measure.
Type models still benefit from inclusion of both daily and 5-minute returns but that for GARCH 20These GARCH model findings confirm the empirical results of Blair et al. (2001) who examine Standard and Poor’s 100 stock index returns over the earlier 1987 to 1992 period and find values for γ similar to ours. In contrast, Martens (2002) reports on much smaller and statistically insignificant γ estimates for returns on Standard and Poor’s 500 futures. Also see: Taylor and Xu (1997).
The SVX and GX models are based on the intraday volatility measure of equation (7.5). Adding the squared overnight returns to the intraday volatility measure produces very similar volatility forecasts although they are slightly less accurate.
Also see: Andersen et al. (2001b) who make the observation that daily GARCH models are not very good at assessing the current volatility level.
We also calculated the RMSE and MAE error statistics which react symmetrically to under-and overestimations; in terms of model ranking little changed but relative differences between error statistics decreased considerably.
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© 2003 Springer Science+Business Media Dordrecht
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Hol, E.M.J.H. (2003). Stock Index Volatility Forecasting with High Frequency Data. In: Empirical Studies on Volatility in International Stock Markets. Dynamic Modeling and Econometrics in Economics and Finance, vol 6. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-5129-1_7
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DOI: https://doi.org/10.1007/978-1-4757-5129-1_7
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