Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 2001 2002 2003 2004 2005 1 2 3 4 5 6 Preopen Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 2001 2002 2003 2004 2005 1 2 3 4 5 Postclose Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 2001 2002 2003 2004 2005 2 4 6 8 10 12 Overnight Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 2001 2002 2003 2004 2005 0 .0 0 0 .0 8 0 .1 6
Figure 1. Time Series of the MSFT Volatilities for Different Time Periods
These are realized volatilities for regular hours, preopen, postclose, and the square root of overnight returns. The volatilities are in percentages.
33
Lag A C F 0 50 100 150 200 250 300 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Series : day.rv.ts Lag A C F 0 50 100 150 200 250 300 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Series : preopen.rv.ts Lag AC F 0 50 100 150 200 250 300 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Series : postclose.rv.ts Lag AC F 0 50 100 150 200 250 300 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 Series : overnight.return.ts^0.5Figure 2. Autocorrelations of the MSFT Volatilities for Different Time Periods
These are the autocorrelations of realized volatilities for regular hours, preopen, postclose and the square root of overnight returns.
Regular Hours
Preopen
Postclose
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Figure 3. MSFT 1-Step-Ahead Volatility Forecasting for Day GARCH (1,1) by Different Models
The thick line (blue) represents the ex post realized volatility series, and the thin line (red) represents the forecast conditional volatility. On July 9, 2004, there is an outlier, resulting from relatively large changes in a couple of recorded transaction prices. These changes in prices passed the data filtering algorithm we designed for cleaning the data. Although we cannot tell if these are recording errors, the statistics and forecasting results are qualitatively the same even after removing this observation. We therefore remove this outlier to make the rest of volatility and forecasting in Figure 3 more informative.
GARCH (1,1)
GARCH (1,1) with Close-to-Open Variance GARCH (1,1) with Preopen Variance
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Footnotes
1
We use regular-hours, day, and open-to-close interchangeably to represent the regular trading period between 9:30 am and 4:00 pm EST in this study. In a similar fashion, we use after-hours and close-to-open interchangeably to represent the period that is outside of the regular trading period.
2 The daily volume weighted price for MSFT is $39.44 for the sample period, and 25% of which would be about $10. 3 Note, however, that this trading at close activity only represents 15% of trades in postclose.
4
This rule of thumb is often close to the optimal sampling frequency advocated by Bandi and Russell (2008).
5 Including leverage effects (e.g., EGARCH, GJR GARCH, or TGARCH) might improve the volatility forecasting.
However, our main focus is on the after-hours information. Therefore, for the purpose of model simplicity, we do not consider this asymmetric effect.