Predicting Stock Volatility Using After-Hours Information: Evidence. from the NASDAQ Actively Traded Stocks

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1 Predicting Stock Volatility Using After-Hours Information: Evidence

from the NASDAQ Actively Traded Stocks

Chun-Hung Chen

1

Office of the Comptroller of the Currency

Wei-Choun Yu

2

Winona State University

Eric Zivot

3

University of Washington

March 3, 2011

Abstract

We use realized volatilities based on after-hours high frequency stock returns to predict next day stock volatility. We extend the GARCH model to include additional information: the whole after hours, the preopen realized variance, the postclose realized variance, and the overnight squared return. For the thirty most active NASDAQ stocks, we find that most of the stocks exhibit positive and significant preopen coefficients and that the inclusion of the preopen variance can mostly improve the out-of-sample forecastability of the next day conditional day volatility. The inclusions of postclose variance and overnight squared return do provide some predictive power for the next day conditional volatility but to a lesser degree; their predictive abilities are inferior to preopen variance. Our findings support the results of prior studies: traders trade mostly for non-information reasons in the postclose period and trade mostly for information reasons in the preopen period.

Keywords: Financial Markets, GARCH Model, Evaluating Forecasts, High-Frequency Data, Realized Variance.

1_{Corresponding Author: Economics Risk Analysis Division, 250 E St. SE, Washington, DC 20219. Email:}

[email protected]. Tel: +1-202-874-4056. We thank seminar participants from the Department of Finance at National Central University and Office of Comptroller of Currency and Mike Wenz for their helpful comments.

2_{Department of Economics, Winona State University, Somsen 319E, Winona, MN 55987, USA. Email:}

[email protected]. Tel: +1-507-457-2982. Fax: +1-507-457-5697.

3_{Department of Economics, University of Washington, Box 353330, Condon 401, Seattle, WA 98195.}

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2 Predicting Stock Volatility Using After-Hours Information: Evidence

from the NASDAQ Actively Traded Stocks

1. Introduction

Volatility modeling has received much attention over the past two decades in finance literature not only because it relates directly to the profits of traders but also because it is important to the valuation of derivative instruments. The goal of modeling and forecasting volatility is to have better risk management, more accurate derivative prices, and more efficient portfolio allocations. Good financial decision making relies on an accurate prediction of the second moment of the underlying financial instrument.

Among various volatility modeling techniques, the most popular models are the generalized autoregressive conditional heteroskedasticity (GARCH) models developed by Engle (1982) and Bollerslev (1986). This family of models can explain well the stylized facts of financial return volatility: persistence, mean reversion, and the leverage effect. Moreover, as Andersen and Bollerslev (1998) and others have shown, when using appropriate ex post volatility measures, the GARCH model can provide good forecastability of the conditional volatility. Besides modeling techniques or methods, the forecastability also relies on a useful information set. Until recently, the most commonly used information set for modeling daily volatility was the historical daily closing prices. Recent research (e.g., Andersen and Bollerslev, 1998, Andersen, Bollerslev, and Lange, 1999, Fuertes, Izzeldin and Kalotychou, 2009, and Martens, Dijk, and Pooter, 2009) has shown that the use of intra-day high frequency data can substantially improve the measurement and forecastability of daily volatility. The majority of these studies used intra-day data observed during normal trading hours. We contribute to this literature by considering intra-night data

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observed during the whole after-hours period1 and each of the subperiods, which will be defined in later sections.

Although several studies have documented the importance of after-hours information for volatility modeling (e.g., Oldfield and Rogalski, 1980; Greene and Watts, 1996; Cao, Ghysels, and Hatheway, 2000; Masulis and Shivakumar, 2002; Taylor, 2007; and Tsiakas, 2008), only a few have actually employed high-frequency data in the analysis. One such paper is Taylor (2007), which uses high-frequency overnight S&P 500 futures volatility information to predict S&P 500 stocks volatility. In this paper, we study individual stocks instead of market indices. As

mentioned in Campbell et al. (2001), there are several motives for studying the volatilities of individual stocks. For instance, many investors have large holdings of individual stocks that have not been diversified and therefore are subject to idiosyncratic volatility.

According to the findings from Barclay and Hendershott (2008), the after-hours transactions become meaningful, i.e., generate price discovery, only if the stock has sufficient after-hours trading activity. Since most stocks are thinly traded during after hours, for our study we choose the thirty most actively traded NASDAQ stocks in 2004. We utilize the availability of after-hours trading opportunities to the public and the recording of high frequency after-hours transaction data of thirty NASDAQ stocks from 2001 to 2004, our sample period, to examine how this extended information set could be effectively used to improve the modeling and the forecasting of next day conditional volatility.

We employ the GARCH model, which is used for daily returns with after-hours realized variance as an additional exogenous variable included in the conditional variance equation. We also use realized volatility as the ex post measure for our forecasting evaluation. Our results show that with the inclusion of realized variance for the whole after-hours period in the information set, out of 30 stocks, 16 of them do provide better forecasting of next day volatility compared to the ones without the inclusion. Moreover, when breaking up the whole after-hours period into three subperiods, we find that preopen coefficients in the GARCH (1,1) model are positive and

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significant for 23 stocks in the in-sample analysis, and the inclusion of the preopen variance can improve the out-of-sample forecastability of the conditional day volatility for 18 stocks. The postclose variance and the overnight squared return exhibit predictive power to a lesser degree, and their out-of-sample forecasting abilities are mostly inferior to preopen variance. The evidence, by and large, supports the results of prior studies showing that traders trade mostly for

non-information reasons in the postclose period but trade mostly for non-information reasons in the preopen period and that the inclusion of preopen variance could enhance the predictability of next day conditional volatility.

Our study contributes to the existing literature in the following ways. First, we use high-frequency intra-night transaction data that has not yet been systematically exploited for the modeling and forecasting of daily conditional volatility. Second, we refine the after-hours information by segmenting the whole after-hours period into subperiods based on their different information densities. Third, past research has often focused on in-sample forecasting evaluation while we evaluate our models’ predictability for out-of-sample as well.

The rest of the paper proceeds as follows. Section 2 reviews the volatility literature using after-hours information. Section 3 explains our data and realized volatility construction. The results of modeling and forecasting conditional volatility based on the GARCH model are provided in Sections 4. Section 5 contains our concluding remarks.

2. Literature Review

Conditional return volatility models such as GARCH demonstrate that past return shocks and volatilities contain information about the evolution of future volatilities and therefore can be used for forecasting purposes. One explanation for this result is the Mixture of Distribution Hypothesis (MDH), suggested by Clark (1973), Tauchen and Pitts (1983), and Kalev et al. (2004). They attribute dependence in volatilities to the serial correlation of the news arrival rate. Lamoureux

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and Lastrapes (1990) use the MDH proposed by Clark (1973) to explain the persistent nature of daily conditional volatility in the GARCH model. They assume that a stochastic model can be derived by considering the daily return in day t, εt, as a sum of i.i.d. (0, 

2 ) intra-day price increments, δit, 1 t n t it i    



, (1)

where i denotes the ith intraday price movement, and the random variable nt is a mixing variable

that denotes the arrival rate of information in day t. Clark (1973) assumes that εt is drawn from a

mixture of distributions, of which the variances depend on nt,

) , 0 ( ~ | 2 t t t n N  n  . (2)

When the arrival rate of information is serially correlated, nt can be expressed as

1

( )

t t t

n  a b L n_ u , (3)

where a is a constant and ut is white noise. The conditional variance becomes

t t t t t t E n n a b L u 2 1 2 2 2 ) ( ] | [











  



  _ (4)

which demonstrates the persistence in the conditional variance captured in the GARCH model. To examine this hypothesis, Lamoureux and Lastrapes (1990), Sharma, Mougoue, and Kamath (1996), and Brooks (1998) use trading volume as a proxy for the information arrival rate and include it as an exogenous variable in the GARCH (1,1) specification for daily volatility. They show that the inclusion of volume greatly reduces the persistence parameter of the estimated GARCH model. Moreover, Brooks (1998) shows that including trading volume in a GARCH model does not improve volatility forecasts because it does not provide additional information which is not already captured by past conditional volatility. In order to check whether the

inclusion of the after-hours volatility can improve the forecastability of the future conditional day volatility, we need to check whether the information contained in the after-hours volatility provides additional explanatory power to the future conditional day volatility. If the after-hours

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volatility provides additional information rather than substituting for information already

incorporated in the past conditional volatility or volume, it could be used to improve forecasts of next day volatility.

It is well known in microstructure literature that information and announcements frequently occur after normal trading hours. If there are no trading opportunities during after hours, this occurrence and accumulation of information during the close-to-open period should contribute to the upcoming day (open-to-close) volatility. In other words, when after-hours trading is not available, the information will be realized at the opening hours. Therefore, the occurrence of after-hours information or news implies higher-than-usual volatility during the following regular trading hours.

Even when trading is available for all or part of the after-hours period, as is true for our sample period, we can still expect information in this period to have an impact on the following regular trading hours for two reasons. The first reason is the spillover effect. If the market is not fully efficient, it would take some time for the information to be incorporated into prices. This phenomenon could be more prominent due to the relatively illiquid nature of the after-hours trading environment. Since it takes trades to facilitate price discovery (Barclay and Hendershott, 2003), the information might not be fully incorporated into the price until the regular-trading hour when the trading volume is much higher. The second reason is the informed nature of trades in after hours. Barclay and Hendershott (2004) indicate that the traders in after hours are mainly professional and institutional. Many of them trade for short-lived private information. It is likely that they trade for private or scheduled news that has yet to be announced. Therefore, it is rational to expect that a highly volatile after-hours trading would lead to a highly volatile day trading in the next day.

Gallo and Pacini (1998) study the impact of close-to-open returns, measured as the percentage difference of the previous daily closing price and current daily opening price, on the following day (open-to-close) volatility for the six major market indices using a GARCH (1,1)

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model with the close-to-open returns as an exogenous variable. Examining three London International Financial Futures Exchange contracts, Brooks, Clare and Persand (2000) find that the inclusion of close-to-open returns substantially reduces the excessive volatility persistence in the GARCH model. Their study suggests that the overnight volatility might improve the

calculations of capital requirements for risk management of banks. Martens (2002) examines whether GARCH (1,1) models that include different functional forms of the after-hours volatility can improve the forecasts of the following day volatility for the S&P 500 index futures

transactions. Gallo and Pacini (1998) find that the inclusion of close-to-open returns improves forecastability of conditional volatility for some stock indices, while Martens (2002) finds that the inclusion cannot improve forecastability. This mixed evidence could come from the poor

exploitation of after-hours information. We propose a refinement of the after-hours information by segmenting the whole after-hours period into several subperiods according to information densities. Barclay and Hendershott (2003, 2004) have previously proposed this segmentation in their after-hours papers, but we are the first to apply it to forecasting volatility.

Since volatility is not directly observable, the methodology on which measuring ex post volatility is based is important for evaluating volatility forecasts. Past studies, such as Cumby, Figlewski and Hasbrouck (1993), Figlewski (1997), and Jorion (1995), show that standard volatility models such as GARCH perform poorly in out-of-sample forecasting evaluation when squared returns are used to proxy for ex post volatility measures. Andersen and Bollerslev (1998) point out that while the squared return is an unbiased estimate for unobserved volatility, it is a very noisy estimate. This, therefore, can potentially explain why those volatility models appear to produce poor forecasts. They further show that realized volatility, which is defined as the sum of squared returns sampled at high intra-daily frequencies, provides a much more reliable ex post volatility measure than the squared returns. The GARCH forecasts significantly improve when evaluated against the realized volatility measures. Furthermore, Andersen et al. (1999) show that

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the forecasting performance of standard volatility models can be greatly enhanced by utilizing high-frequency data.

3. Data and Volatility Measurement

3.1. Data

Our high frequency data is taken from the Trade and Quote (TAQ) database, which provides data on tick-level transaction prices and quotes from 8:00 am until 6:30 pm EST, when the NASDAQ Trade Dissemination Service (NTDS) is on. Since trading volume is relatively low for stocks in after hours, we have chosen thirty active stocks that have the highest trading activities during the after-hours period based on the aggregate trading volume in the whole year of 2004 in NASDAQ. The thirty stocks are reported in Table 1. Because Microsoft (MSFT) has the highest trading volume, we use it as the principal stock under investigation and present more detailed results pertaining to it. The sample period is from January 2001 to December 2004, during which time the after-hours trading information is recorded through NTDS and is available to the public. We choose the first three and a half years as the in-sample period for modeling volatility and the later half-year as the out-of-sample period to evaluate forecasting performance.

The TAQ data typically contain some recording errors. We therefore remove any recorded trades that have a change of positive or negative 25% from their immediately prior trades in a day2. We also remove dates in which either preopen, postclose, or day transaction data is missing or in which stock splits occur.

3.2. After-Hours Subperiods

Barclay and Hendershott (2003, 2004) break the after-hours period into three subperiods: the postclose period (4:00 to 6:00 pm EST), the overnight period (6:00 pm to 8:00 am EST), and the preopen period (8:00 to 9:30 am EST). They investigate the information structure of the

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postclose and preopen and find that the probability of an informed trade is much higher in the latter period than in the former period. They find that about 80% of all trading volume in the postclose occurs at the closing price or within the closing quotes at 4:00 pm EST3. This implies that traders tend to trade for liquidity demands right after the regular trading hour is closed. Furthermore, Barclay and Hendershott (2003) use the probability of informed trade measure developed by Easley, Kiefer, and O’Hara (1997) to show that trading is highly informed during the preopen, which implies that traders are more likely to trade for information reasons in this period. Even though traders can still trade through an electronic communication network (ECN) or a market maker during the overnight period, there is no formal analysis on the information structure for this period. The overnight data is usually not available from the reporting service provided by NTDS. Using their proprietary dataset, Barclay and Hendershott (2003) find that only 1% of total after-hours trades occur during that period.

The uneven information in each after-hours subperiod leads us to hypothesize that the volatility in each subperiod should have different effects on the following day volatility. We expect that the postclose volatility contains little to no information, while the volatility in the preopen contains new information about the following day volatility. This means that the inclusion of the preopen volatility in the information set may improve the forecastability of a volatility model. The impact of volatility in the overnight period on conditional day volatility, however, is less obvious. If the preopen trades have realized most or all of the information that occurred in the overnight period, or if the overnight squared return measure is very noisy, we would expect overnight volatility to have little or no predictability on the day volatility.

3.3. Volatility Measurement

Realized variance is a more accurate measure of conditional variance than the squared return. We use it to measure trading day variance and variance during the after-hours periods as well as to evaluate our volatility predictions. Following Bollerslev and Wright (2001), Andersen

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et al. (2001a), and Andersen et al. (2003), we construct realized variance by summing up

intra-period high frequency squared returns:

, , , ( 1) 1 , , 1 1/ 2 2 , , 1 i t n i t n i t n i t i t n n i t i t n n r p p r r r                    



(5)

where p denotes the logarithmic stock price; i is either the regular hour, the preopen, or the postclose period; r is the intra-day return; 1/Δ is the number of observations for each of the periods (Δ is 5 minutes in regular hours and is 15 minutes in after-hours); and σi,t

2

is the estimated realized variance for period i in day t. Realized volatility is computed as the square root of

realized variance. Since there is no data for trades in the overnight period, we measure the variance based on the first trade of preopen and the last trade of the previous day’s postclose:

2 2

Overnight,t (pFirst Trade of Preopen,t pLast Trade of Postclose,t1)

    (6)

Andersen et al. (2001b) shows that as sampling frequency increases, realized variance approximates to integrated variance, which is the actual realized return variation over a given horizon for a continuous time diffusion process and an unbiased estimate of conditional variance. Although it is theoretically demonstrated that the measurement error associated with the

estimation of the realized variance becomes very small as the sampling frequency increases, in reality the existence of market microstructure frictions (e.g., bid-ask bounce, price discreteness, and infrequent trading) can create large biases. To avoid this problem, Andersen et al. (2001a) propose sampling the intra-day observations at 5-minute intervals4. Since the trading environment in after-hours is known to have much larger microstructure frictions than during regular-hours, we sample observations at a 5-minute frequency for regular-hours and at a 15-minute frequency for after-hours. The resulting numbers of inter-period observations for the regular-hours, the preopen, and the postclose are 78, 6, and 8, respectively.

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Figure 1 shows the regular-hours, the preopen, and the postclose realized volatility, in addition to the overnight absolute return for MSFT from January 2001 to December 2004. Table 2 lists some descriptive statistics of realized volatility measures for MSFT. These measures represent the total amount of volatility per day in each period. Similar to the distribution of returns, the distributions of volatilities are all skewed to the right and have fat tails. The autocorrelation plots in the four periods are shown in Figure 2. The daily and preopen realized volatility series both exhibit the commonly known characteristic of long-memory, or persistence. In contrast, we do not observe this feature in the postclose realized volatility series and overnight absolute return.

Barclay and Hendershott (2003) find that price changes are larger in the preopen than in the postclose, which indicates more private information and less noise in the preopen period. Table 3 provides volatilities per hour and per trade for the preopen and the postclose periods. The average volatilities per hour for the preopen and postclose are 0.39% and 0.29%, respectively, and the average volatilities per trade are 0.0102% and 0.00592%, respectively. The numbers show that volatility is higher in the preopen than in the postclose, which is consistent with the result of Barclay and Hendershott (2003). Both the median volatilities per hour and per trade provide the same qualitative results.

4. GARCH Modeling and Forecasting

The GARCH framework is the most common approach to modeling and forecasting volatility. We use the GARCH (p,q) model5

2 2 2 1 1 t t t t t p q t i t i j t j i j r z h h h      _  _       







(7)

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where μ and ω are constants in the conditional mean equation and the conditional variance equation, respectively; εt is a serially uncorrelated residual term (news shock) with mean zero; zt

is an i.i.d. random variable with mean zero and unit variance; and ht 2

is the conditional variance at time t. While there are many variations of the GARCH (p,q) model, a GARCH (1,1) model is usually sufficient for most financial time series applications (Andersen and Bollerslev, 1998; Hansen and Lunde, 2004).

Table 4 shows the Akaike information criterion, Bayesian information criterion, and the log-likelihood for all GARCH (p,q) models with p≤2 and q≤2 for the daily MSFT return series in the in-sample period, which is from January 2001 to June 2004. The GARCH (1,1) with Student’s

t error distribution for zt appears to be the most appropriate model. The first column of Table 5

reports the coefficients for the daily GARCH (1,1) model in regular hours. The sum of the estimates of α and β is 0.997, which shows that the conditional volatility is quite persistent. This result is very similar to 0.9986 reported by Martens (2002) for S&P 500 futures. The ARCH and Ljung-Box tests on the squared residuals are employed to check for the adequacy of the fitted model. We find that the GARCH (1,1) specification fits the in-sample return series of the MSFT well. The rest of stocks’ daily GARCH (1,1) estimates are reported in the first three columns of Table 6.

4.1. GARCH Model for Day Returns with Night Variance

The GARCH model offers flexibility in that the additional exogenous variables that are thought to have impact on conditional volatility can be included in the conditional variance equation. The modified GARCH (1,1) is:

2 2 2 1 1 1 t t t t t t t t t r z h h h x                 (8)

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where xt represents the additional exogenous variable in the conditional variance equation. Both

Gallo and Pacini (1998) and Martens (2002) use this approach by including the close-to-open squared return as the additional exogenous variable in the conditional variance equation. Martens (2002) finds the coefficient on the additional variable to be statistically insignificant. On the other hand, Gallo and Pacini (1998) find the coefficient to be statistically significant for most of the major market indices, with the sign of the coefficient being positive for some indices and negative for others. If the impact of after-hours information on the volatility of regular hours is caused by the possibility of the informed traders trading private information before the news is publicly announced during the regular hours, a higher after-hours volatility should lead to a higher volatility the following day. Therefore, we expect the sign of the coefficient for the overnight variance variable to be statistically significant and positive.

To investigate the impact of after-hours information in the GARCH (1,1) model (8), we use the following four exogenous variables:

All three subperiods together (AN): 2_, 2_, 2_,

1 1 1.5 2 14 17.5 17.5 17.5 PO PC PO PC ON PO PC N N t t n t n t n n n x r r r   







 (9)

The preopen period only (PO): 2 , 1 PO PO PO N t t n n x r  



(10)

The postclose period only (PC): 2 , 1 PC PC PC N t t n n x r  



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The overnight period only (ON): 2_, ON

t t n

x r (12)

where PO, PC, and ON denote the preopen, the postclose, and the overnight period, respectively. The variable xt defined in (9) is a time-weighted average realized variance of the close-to-open

(the whole after-hours) period.

The second through fifth columns of Table 5 show the estimation results of the modified GARCH (1,1) model (8) using the exogenous variables defined in (9) – (12) for MSFT. First, we find that the estimated coefficient (standard error) for the whole after-hours period realized

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variance (9) is 0.065 (0.064), which is positive but statistically insignificant. This result agrees with Martens (2002) in that the close-to-open variance does not provide significant explanatory power for the next day conditional variance. Second, we find that the coefficient for the postclose variance (11) is negative and statistically insignificant. This result is consistent with the

hypothesis that traders primarily trade in the postclose for non-information reasons, and therefore there is no information to be carried over into the next day volatility. Another possibility is that if there were any information in the postclose period, it could be subsequently incorporated into the price through trading activities during the following overnight and the preopen periods. The only explanatory variable that we find to be statistically significant is the preopen realized variance (10), which has an estimated coefficient (standard error) of 0.221 (0.090). Hence, a 1% increase in the preopen realized variance would lead to a 0.221% increase in the following regular-hour conditional variance. This result is consistent with our hypothesis in that the coefficient should be positive and significant.

Finally, we find that the estimated persistence parameters are 0.997 and 0.971 for the GARCH (1,1) model (7) and the modified GARCH (1,1) model (8) with the preopen variance, respectively. As discussed above concerning the MDH and Gallo and Pacini (1998), this slight decrease in the persistence parameter indicates that the preopen variance appears to provide independent information from that contained in the past day returns. This result enhances our hypothesis that the addition of the preopen variance into the model would improve forecasts of the next day conditional day volatility.

Table 6 gives the estimation results of the modified GARCH (1,1) model (8) for all thirty stocks. As summarized in Column 1 of Table 9, out of thirty stocks, 13 stocks show significant and positive coefficients for the whole after-hours period. In Column 2 of Table 9, 23 stocks show significant and positive coefficients for the preopen period. In Column 3 of Table 9, 10 stocks show significant and positive coefficients for the postclose period. In Column 4 of Table 9,

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10 stocks show significant and positive coefficients for the overnight period. The in-sample

evidence shows that preopen trading volatility is most likely to impact the following day volatility.

4.2. Out-of-sample Forecast Evaluation

Figure 3 shows the out-of-sample ex post realized volatility series and the forecasted 1-step-ahead conditional volatility series of various fitted GARCH (1,1) models for MSFT. The

forecasts with the preopen variance as an exogenous variable in the conditional variance equation appear to track the realized volatility series the best.

We formally evaluate the forecasting performance of the different models using the following metrics: Mincer-Zarnowitz Regression:

 

_{t k}2_ 1 2 a₀a h₁

 

_{t k t}2_ _| 1 2 u_{t k}_ (13) RMSE:





1 2 2 2 2 | 1 1 T t k t k t t h T      _    



 (14) MAE: 2 2 _| 1 1 T t k t k t t h T    



(15) HRMSE:





1 2 2 2 2 | 1 1 1 T t k t k t t h T         



 (16) HMAE: 2 2 _| 1 1 1 T t k t k t t h T    



(17) LL:



2 2 _|



1 1 log T t k t k t t h T       



(18)

where



_t2__kdenotes realized variance in day t+k and where

h

_t2_k_|tdenotes the conditional variance

forecast for day t+k based on information available in day t. In the Mincer-Zarnowitz regression proposed by Mincer and Zarnowitz (1969), if the conditional volatility model is correctly specified, we should have a0 and a1 equal to zero and one, respectively, with high statistical

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suffer from a standard errors-in-variables problem, which would make interpretation difficult. Nonetheless, they argue that the R2 of the regression can be used to evaluate the variability of the ex post volatility that is explained by the forecasted conditional volatility. The root mean square error (RMSE) and the mean absolute error (MAE) are two commonly used criteria. To check whether the results are reliable in a nonlinear and heteroskedastic environment, we follow Andersen et al. (1999) and use the heteroskedasticity adjusted RMSE (HRMSE), MAE (HMAE), and the logarithmic loss function (LL).

As stressed by both Andersen and Bollerslev (1998) and Hansen and Lunde (2006), it is crucial to choose the right ex post volatility measure to serve as the benchmark for the forecast evaluation since volatility is not directly observed. Several past studies, such as Figlewski (1997) and Jorion (1995), have used daily squared returns as the proxy for the ex post volatility measure and conclude that standard volatility models explain little of the variability in the ex post

volatility. Andersen and Bollerslev (1998) and Hansen and Lunde (2006) use realized volatility and demonstrate that it provides a more reliable and accurate measure of the true volatility and that its use in forecast evaluation statistics leads to more accurate inferences regarding forecasting accuracy.

To perform the forecast evaluations, we first estimate the parameters of the GARCH (1,1) models (8) and (9) from the in-sample data and then compute 1-step-ahead predictions for conditional volatility over a rolling window. Table 7 lists the 1-step-ahead forecast evaluation results by the Mincer-Zarnowitz regression (13). For MSFT, we see that the GARCH (1,1) with the preopen realized variance provides the best forecasting performance in terms both of accuracy and of explaining the variability in the ex post measures. The estimated coefficient on a1 is 0.976

when forecasts are computed from a GARCH (1,1) model with preopen realized variance, compared to 0.899 for forecasts computed from the GARCH (1,1) model without the inclusion (day GARCH (1,1)). The GARCH (1,1) with the preopen realized variance has a substantially higher R2 = 0.157 than the R2 = 0.091 from the day GARCH (1,1).

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Table 9 summarizes the other stocks’ Mincer-Zarnowitz results for which the inclusion of after-hours information both increases R2 and makes a1 closer to one. Column 5 of Table 9 shows

that out of 30 stocks, 7 stocks improve their out-of-sample forecasting results for the whole after-hours period. Column 6 of Table 9 shows that out of 30 stocks, 14 stocks improve their out-of-sample forecasting results for the preopen period. Column 7 of Table 9 shows that out of 30 stocks, 6 stocks improve their out-of-sample forecasting results for the postclose period. Column 8 of Table 9 shows that out of 30 stocks, 6 stocks improve their out-of-sample forecasting results for the overnight period. The preopen period trading provides much more enhanced forecasting results compared to other after-hours subperiods based on the observed Mincer-Zarnowitz results.

Table 8 presents forecast evaluation statistics from equations (14)-(18). For MSFT, the forecasts provided by the GARCH (1,1) with the preopen realized variance (column 3) are superior to those from the day GARCH (1,1) only (column 1). Their forecast errors for MSE, MAE, HMSE, HMAE, and LL are the lowest. We also see that the forecasts from the GARCH (1,1) with the whole after-hours period (column 2), preopen period (column 4), and overnight period (column 5) perform relatively poorly compared to those from the day GARCH (1,1) only. Columns 9 to 12 of Table 9 summarize the out-of-sample performance for the inclusion of after-hours subperiods for the other stocks. The asterisk represents that including each subperiod will improve the forecasting performance compared to the forecasting without any after-hours information (day GARCH (1,1) only). Since we use five forecasting evaluation methods, if there are at least three methods that perform better, we count this subperiod as improving the

forecasting outcome. Column 9 of Table 9 shows that out of 30 stocks, 16 stocks improve their forecasting for the whole after-hours period; Column 10 shows that 18 stocks improve their forecasting for the preopen period; Column 11 shows that 9 stocks improve their forecasting for the postclose period; and Column 12 shows that 16 stocks improve their forecasting for the overnight period.

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To investigate the best after-hours subperiod that could enhance the forecasting performance, the double asterisks denote the subperiod that produces lowest forecasting errors. Among 16 stocks that improve next day volatility forecasting by including the whole after-hours period, only one stock stands out as superior to other subperiods. Among 18 stocks improving next day volatility forecasting by including the preopen period, 15 stocks are superior to other subperiods. Among nine stocks improving next day volatility forecasting by including the postclose period, only three stocks are superior to other subperiods. Among 16 stocks improving next day volatility forecasting by including the overnight period, only four stocks are superior to other subperiods.

In summary, based on the in-sample and out-of-sample results, it is apparent that the preopen period provides additional information and therefore improves the next day forecasting performance.

5. Conclusion

Most of the volatility forecast literature has focused on comparing the forecast performance of different volatility models. In this study, we concentrate on whether an expanded information set can increase the forecastability of a conditional day volatility model. While the typical information in previous literature is the daily return and/or variance measures, we add

information of after-hours variance using high-frequency observations. We present the GARCH model for daily volatility by four measures of after-hours information: the whole after-hours, the preopen variances, the postclose variances, and the overnight squared return. By examining the thirty most actively traded NASDAQ stocks, we find that preopen coefficients in the GARCH (1,1) model are positive and significant for 23 out of 30 stocks in the in-sample analysis, and the inclusion of the preopen variance can substantially improve the out-of-sample forecastability of the conditional day volatility for 18 stocks. The postclose variance and the overnight squared

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return exhibit predictive power for future conditional volatility to a lesser degree, and their out-of-sample forecasting abilities are mostly inferior to preopen variance. The evidence, by and large, supports the results of prior studies showing that traders trade mostly for non-information reasons in the postclose period but trade mostly for information reasons in the preopen period.

We propose two reasons for why the preopen variance can be used to improve the

predictability of the model. The first is the spillover effect, and the second is the possibility of the informed traders trading private information that is yet to be released during the following regular hours. One extension of our analysis is to examine how the preopen variance affects the

volatilities in different intra-day periods. If the predictive power of the preopen variance comes from the spillover information from the preopen period to the regular hours, we can expect the highest impact to occur in the opening hours. If the time of day affected appears to be random, it is more likely due to the second conjecture, assuming that the timing of the private information becoming public is also random throughout the day.

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Table 1. The Thirty Most Actively Traded NASDAQ Stocks

Ticker Company Name Aggregated Trading Volume (in 100s)

MSFT Microsoft 173,337,986

SIRI Sirius XM Radio 170,203,390

INTC Intel 166,982,231

CSCO Cisco 139,513,698

ORCL Oracle 114,794,458

SUNW Sun Microsystems 103,675,149

JDSU JDS Uniphase 82,977,355

AMAT Applied Materials 82,499,633

YHOO Yahoo! 40,999,796 DELL Dell 40,003,828 CIEN CIENA 32,400,220 NXTL Nextel Communications 32,299,303 JNPR Juniper Networks 31,213,448 CNXT Conexant Systems 30,375,528 QCOM QUALCOMM 27,718,095 BRCM Broadcom 25,715,821

ADCT ADC Telecommunications 25,086,690

EBAY eBay 22,397,597

AAPL Apple 21,945,886

SEBL Siebel Systems 21,940,453

AMZN Amazon.com 21,730,820

LVLT Level 3 Communications 21,416,579

BEAS BEA Systems 21,406,136

AMGN Amgen 21,384,813

VRTS Virtus Investment Partners 21,106,127

ATML Atmel 20,806,448

RFMD RF Micro Devices 20,465,717

BRCD Brocade Communications Systems 20,178,726 RIMM Research In Motion Limited 19,832,282

TLAB Tellabs 18,729,020

The above thirty active stocks have the highest trading activities based on the aggregate trading volume in 2004 in NASDAQ. We excluded three stocks, ICGE, TASR and CMCSA, due to their lack of after-hours trading data for at least three years in the sample period.

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Table 2. Summary Statistics of Daily Realized Return and Volatility for MSFT

Min. Mean Median Max. St. dev. Skew. Kurt.

Return -0.0771 0.0001 -0.0007 0.1058 0.0190 0.4731 5.3961 Volatility Reg. Hour 0.4183 1.7717 1.5852 6.4343 0.8706 1.4916 6.3789 Preopen 0.0367 0.5818 0.4700 5.7256 0.4269 3.2429 26.8472 Postclose 0.0330 0.5888 0.3473 12.5415 0.8427 5.9118 58.0698 Overnight 0.0000 0.0053 0.0008 0.2190 0.0155 7.4075 77.7413 The realized volatilities are all in percentage terms.

Table 3. Average Volume and Volatility for MSFT

Volume (daily) Volume (hourly) Volatility (hourly) Volatility (per trade) Reg. Hour 978959 150609 0.273 0.000181 Preopen 5701 3801 0.388 0.0102 Postclose 9925 4963 0.294 0.00592

All reported volumes are in terms of trades, and all reported volatilities are in percentage terms.

Table 4. GARCH Model Selection for MSFT

Normal Error Distribution

GARCH(1,1) GARCH(1,2) GARCH(2,1) GARCH(2,2)

AIC -4459 -4457 -4458 -4453

BIC -4440 -4433 -4434 -4424

Likelihood 2233 2234 2234 2232

Student’s t Error Distribution

GARCH(1,1) GARCH(1,2) GARCH(2,1) GARCH(2,2)

AIC -4463 -4461 -4461 -4442

BIC -4439 -4433 -4433 -4409

Likelihood 2237 2237 2237 2228

The GARCH selection is based on the Akaike information criterion (AIC), Bayesian information criterion (BIC), and the log-likelihood of the model.

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Table 5. Day GARCH (1,1) Parameter Estimates of MSFT

GARCH (1,1) GARCH (1,1) with Close-to-Open GARCH (1,1) with Preopen GARCH (1,1) with Postclose GARCH (1,1) with Overnight μ -6.37e-4 (5.16e-4) -5.85e-4 (5.18e-4) -4.52e-4 (5.29e-4) -6.24e-4 (5.21e-4) -6.02e-4 (5.15e-4) ω 1.32e-6 (1.25e-6) 8.85e-7 (1.56e-6) -1.06e-6 (2.02e-6) 1.47e-6 (1.32e-6) 1.05e-6 (1.48e-6) α 0.065*** (0.017) 0.066*** (0.018) 0.068*** (0.019) 0.066*** (0.017) 0.066*** (1.75e-2) β 0.932*** (0.017) 0.922*** (0.020) 0.903*** (0.022) 0.932*** (0.017) 0.923*** (0.020) ρ _(0.064)0.065 0.221*** _(0.090) _(0.015)-0.007 _(0.057)0.057 Degree of Freedom 19.23 28.85 28.84 21.28 19.50 Likelihood 2237 2245 2240 2239 2238 ARCH test (P-value) 0.121 0.228 0.154 0.143 0.222 Ljung-Box Test (P-value) 0.638 0.660 0.535 0.645 0.657

The reported coefficients are based on quasi-maximum likelihood estimations of a Student’s t GARCH (1,1) model estimated from the in-sample period:

2 2 2 1 1 1 t t t t t t t t t r z h h h x

 



 

_



_



_       

The standard error is in the parenthesis. ARCH test and Ljung-Box Test are performed to check for the ARCH effect and autocorrelation of the residuals. *** denotes significance at a 1% level, ** denotes significance at a 5% level, and * denotes significance at a 10% level.

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Table 6. GARCH Model for Day Returns with Night Variance In-Sample Results

Regular Day Hour Close-to-Open (AN) Preopen (PO) Postclose (PC) Overnight (ON)

Name mu alpha beta mu alpha beta rho mu alpha beta rho mu alpha beta rho mu alpha beta rho

MSFT Value -0.001 0.065*** 0.932*** -0.001 0.066*** 0.922*** 0.065 0.000 0.068*** 0.903*** 0.221*** -0.001 0.066*** 0.932*** -0.007 -0.001 0.066*** 0.923*** 0.057 SE 0.001 0.017 0.017 0.001 0.018 0.020 0.064 0.001 0.019 0.022 0.090 0.001 0.017 0.017 0.015 0.001 0.018 0.020 0.057 SIRI Value -0.009*** 0.174*** 0.867*** -0.008*** 0.150*** 0.839*** 0.200** -0.008*** 0.171*** 0.806*** 0.164** -0.009*** 0.162*** 0.863*** 0.049 -0.008*** 0.155*** 0.849*** 0.160** SE 0.001 0.050 0.025 0.001 0.047 0.033 0.100 0.001 0.051 0.037 0.075 0.001 0.049 0.027 0.054 0.001 0.046 0.030 0.082 INTC Value -0.001 0.043*** 0.954*** -0.001 0.029** 0.935*** 0.151** -0.001 0.040 0.942*** 0.089* -0.001 0.029** 0.942*** 0.064*** -0.001 0.036*** 0.940*** 0.104** SE 0.001 0.012 0.013 0.001 0.014 0.017 0.063 0.001 0.013 0.017 0.053 0.001 0.013 0.015 0.024 0.001 0.014 0.016 0.051 CSCO Value -0.001 0.038*** 0.958*** -0.001 0.024 0.934*** 0.202** -0.001 0.041*** 0.929*** 0.155** -0.001 0.037** 0.949*** 0.063 -0.001 0.030 0.937*** 0.144** SE 0.001 0.012 0.013 0.001 0.014 0.020 0.079 0.001 0.015 0.021 0.073 0.001 0.015 0.016 0.046 0.001 0.014 0.020 0.060 ORCL Value -0.001* 0.041*** 0.955*** -0.001* 0.041*** 0.953*** 0.012 -0.001* 0.041*** 0.957*** -0.001 -0.001** 0.027** 0.955*** 0.050** -0.001 0.043*** 0.950*** 0.032 SE 0.001 0.011 0.011 0.001 0.012 0.011 0.024 0.001 0.011 0.010 0.002 0.001 0.011 0.011 0.021 0.001 0.012 0.012 0.040 SUNW Value -0.003*** 0.024*** 0.976*** -0.003*** 0.022** 0.954*** 0.175*** -0.003*** 0.035*** 0.943*** 0.136*** -0.003*** 0.024*** 0.976*** -0.002 -0.003*** 0.021** 0.957*** 0.153*** SE 0.001 0.008 0.008 0.001 0.010 0.012 0.058 0.001 0.013 0.015 0.053 0.001 0.008 0.008 0.008 0.001 0.009 0.011 0.052 JDSU Value -0.004*** 0.059*** 0.941*** -0.004*** 0.059*** 0.938*** 0.018 -0.004*** 0.045*** 0.933*** 0.096** -0.004*** 0.059*** 0.944*** -0.017 -0.004*** 0.060*** 0.939*** 0.011 SE 0.001 0.015 0.013 0.001 0.016 0.014 0.029 0.001 0.015 0.015 0.043 0.001 0.014 0.013 0.014 0.001 0.016 0.014 0.022 AMAT Value -0.001 0.042*** 0.953*** -0.001 0.025 0.905*** 0.430*** -0.001 0.024* 0.947*** 0.153** -0.001 0.017 0.923*** 0.146*** -0.001 0.035*** 0.930*** 0.288*** SE 0.001 0.013 0.014 0.001 0.015 0.026 0.135 0.001 0.013 0.018 0.065 0.001 0.013 0.021 0.047 0.001 0.014 0.020 0.104 YHOO Value 0.002 0.048*** 0.948*** 0.002 0.049*** 0.946*** 0.000 0.001 0.046*** 0.920*** 0.163** 0.002 0.036*** 0.957*** 0.037 0.002 0.049*** 0.946*** 0.000 SE 0.001 0.013 0.013 0.001 0.013 0.014 0.001 0.001 0.016 0.022 0.083 0.001 0.011 0.011 0.026 0.001 0.013 0.014 0.001 DELL Value 0.001 0.065*** 0.932*** 0.001 0.057*** 0.885*** 0.254*** 0.001 0.061*** 0.901*** 0.133** 0.001 0.063*** 0.912*** 0.042** 0.001 0.059*** 0.907*** 0.186** SE 0.001 0.016 0.016 0.001 0.021 0.028 0.097 0.001 0.019 0.022 0.062 0.001 0.019 0.021 0.021 0.001 0.019 0.022 0.081 CIEN Value -0.006*** 0.070*** 0.935*** -0.006*** 0.067*** 0.935*** 0.018 -0.006*** 0.050*** 0.932*** 0.097** -0.006*** 0.067*** 0.936*** 0.016 -0.006*** 0.069*** 0.935*** 0.007 SE 0.001 0.019 0.015 0.001 0.019 0.015 0.057 0.001 0.018 0.016 0.044 0.001 0.020 0.014 0.029 0.001 0.019 0.015 0.055 NXTL Value -0.003*** 0.035*** 0.965*** -0.003*** 0.031*** 0.961*** 0.043** -0.003*** 0.027** 0.950*** 0.070*** -0.003*** 0.031*** 0.961*** 0.035 -0.003*** 0.033*** 0.963*** 0.030 SE 0.001 0.010 0.009 0.001 0.011 0.009 0.022 0.001 0.012 0.012 0.025 0.001 0.011 0.009 0.025 0.001 0.011 0.009 0.019 JNPR Value -0.002* 0.054*** 0.941*** -0.003** 0.036*** 0.935*** 0.134** -0.002* 0.050*** 0.938*** 0.024 -0.002** 0.036*** 0.944*** 0.073** -0.003** 0.042*** 0.935*** 0.106** SE 0.001 0.015 0.015 0.001 0.014 0.018 0.061 0.001 0.016 0.018 0.028 0.001 0.014 0.015 0.034 0.001 0.014 0.017 0.052 CNXT Value -0.006*** 0.057*** 0.928*** -0.006*** 0.045*** 0.936*** 0.028 -0.006*** 0.035*** 0.950*** 0.007 -0.006*** 0.044*** 0.930*** 0.035 -0.006*** 0.053*** 0.930*** 0.023 SE 0.001 0.016 0.018 0.001 0.015 0.017 0.020 0.001 0.013 0.014 0.005 0.001 0.017 0.018 0.028 0.001 0.016 0.018 0.022 QCOM Value 0.002** 0.032*** 0.965*** 0.002** 0.032*** 0.967*** -0.012 0.002** 0.031*** 0.970*** -0.014 0.002** 0.034*** 0.969*** -0.019 0.002** 0.032*** 0.965*** -0.005 SE 0.001 0.011 0.011 0.001 0.011 0.011 0.027 0.001 0.010 0.011 0.021 0.001 0.011 0.010 0.017 0.001 0.011 0.011 0.024 BRCM Value -0.001 0.032*** 0.966*** -0.001 0.038*** 0.948*** 0.105 -0.001 0.025** 0.941*** 0.202** -0.001 0.033*** 0.966*** -0.003 -0.001 0.033*** 0.955*** 0.113 SE 0.001 0.010 0.009 0.001 0.012 0.014 0.067 0.001 0.011 0.016 0.084 0.001 0.010 0.009 0.007 0.001 0.011 0.013 0.070

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We use GARCH (1,1) model to compute the stock volatility from January 2001 to June 2004. The regular day hour is based on the GARCH model 2 2 2

1 1 p q t i t i j t j i j h   _ h_    







.

The GARCH (1,1) day returns with night variance (AN, PO,PC, or ON) are based on h_t2   _t2_₁h_t2_₁x_t_₁. *** denotes significance at a 1% level, ** denotes significance at a 5% level, and * denotes significance at a 10% level.

ADCT Value -0.004*** 0.044*** 0.951*** -0.004*** 0.046*** 0.927*** 0.114* -0.004*** 0.029* 0.934*** 0.112** -0.004*** 0.044*** 0.945*** 0.024 -0.004*** 0.048*** 0.940*** 0.047 SE 0.001 0.014 0.014 0.001 0.018 0.021 0.062 0.001 0.016 0.019 0.051 0.001 0.015 0.016 0.021 0.001 0.016 0.017 0.053 EBAY Value 0.003*** 0.040*** 0.954*** 0.002*** 0.039*** 0.955*** 0.000 0.002*** 0.044*** 0.924*** 0.141** 0.003*** 0.038*** 0.951*** 0.022 0.002*** 0.039*** 0.955*** 0.000 SE 0.001 0.011 0.011 0.001 0.011 0.011 0.001 0.001 0.015 0.020 0.071 0.001 0.012 0.012 0.015 0.001 0.011 0.011 0.000 AAPL Value 0.001 0.021*** 0.978*** 0.001 0.035*** 0.942*** 0.067** 0.001 0.049*** 0.906*** 0.092** 0.001 0.062*** 0.918*** 0.000 0.001 0.048*** 0.926*** 0.074* SE 0.001 0.008 0.009 0.001 0.014 0.020 0.035 0.001 0.018 0.029 0.046 0.001 0.020 0.027 0.016 0.001 0.017 0.024 0.039 SEBL Value -0.004*** 0.036*** 0.961*** -0.004*** 0.034*** 0.951*** 0.110** -0.004*** 0.032*** 0.951*** 0.076** -0.004*** 0.026*** 0.967*** 0.036 -0.004*** 0.037*** 0.950*** 0.090* SE 0.001 0.011 0.011 0.001 0.012 0.013 0.054 0.001 0.012 0.014 0.039 0.001 0.010 0.010 0.022 0.001 0.013 0.014 0.048 AMZN Value 0.003*** 0.029*** 0.968*** 0.003*** 0.026*** 0.966*** 0.018 0.003*** 0.025*** 0.974*** -0.003 0.003*** 0.016* 0.967*** 0.052** 0.003*** 0.025*** 0.968*** 0.018 SE 0.001 0.010 0.010 0.001 0.009 0.011 0.027 0.001 0.009 0.009 0.003 0.001 0.010 0.010 0.024 0.001 0.009 0.010 0.024 LVLT Value -0.005*** 0.123*** 0.882*** -0.005*** 0.097*** 0.873*** 0.186** -0.005*** 0.113*** 0.859*** 0.131** -0.005*** 0.119*** 0.871*** 0.054 -0.005*** 0.096*** 0.882*** 0.151** SE 0.001 0.033 0.024 0.001 0.029 0.027 0.084 0.001 0.033 0.029 0.065 0.001 0.033 0.027 0.044 0.001 0.028 0.024 0.073 BEAS Value -0.003*** 0.045*** 0.955*** -0.003*** 0.047*** 0.948*** 0.042 -0.003*** 0.049*** 0.924*** 0.144** -0.003*** 0.037*** 0.951*** 0.044** -0.003*** 0.045*** 0.954*** 0.003 SE 0.001 0.012 0.011 0.001 0.013 0.012 0.045 0.001 0.016 0.018 0.074 0.001 0.012 0.011 0.022 0.001 0.012 0.011 0.038 AMGN Value -0.001 0.062*** 0.929*** -0.001 0.062*** 0.921*** 0.048 0.000 0.061*** 0.907*** 0.140** -0.001 0.065*** 0.929*** -0.014 -0.001 0.062*** 0.920*** 0.050 SE 0.001 0.016 0.017 0.001 0.017 0.020 0.046 0.001 0.019 0.023 0.081 0.001 0.016 0.017 0.015 0.001 0.017 0.019 0.042 VRTS Value 0.000 0.044*** 0.953*** 0.000 0.043*** 0.946*** 0.071 0.000 0.035*** 0.944*** 0.078** 0.000 0.044*** 0.956*** -0.011 0.000 0.047*** 0.944*** 0.074 SE 0.001 0.013 0.013 0.001 0.014 0.015 0.059 0.001 0.013 0.015 0.040 0.001 0.013 0.013 0.024 0.001 0.014 0.015 0.069 ATML Value -0.003** 0.082*** 0.901*** -0.003** 0.081*** 0.897*** 0.024 -0.003** 0.082*** 0.895*** 0.019 -0.003** 0.029** 0.934*** 0.093*** -0.003** 0.079*** 0.908*** -0.025 SE 0.001 0.023 0.025 0.001 0.023 0.026 0.057 0.001 0.023 0.027 0.041 0.001 0.014 0.018 0.032 0.001 0.022 0.024 0.029 RFMD Value -0.005*** 0.068*** 0.927*** -0.005*** 0.056*** 0.921*** 0.159** -0.005*** 0.036*** 0.930*** 0.120*** -0.005*** 0.066*** 0.928*** 0.008 -0.005*** 0.069*** 0.921*** 0.047 SE 0.001 0.016 0.016 0.001 0.016 0.018 0.073 0.001 0.014 0.017 0.040 0.001 0.015 0.016 0.013 0.001 0.017 0.019 0.081 BRCD Value -0.004*** 0.062*** 0.934*** -0.004*** 0.061*** 0.928*** 0.065 -0.004*** 0.067*** 0.920*** 0.059 -0.004*** 0.061*** 0.929*** 0.035 -0.004*** 0.062*** 0.931*** 0.035 SE 0.001 0.016 0.016 0.001 0.018 0.016 0.052 0.001 0.019 0.019 0.063 0.001 0.017 0.016 0.027 0.001 0.017 0.016 0.041 RIMM Value 0.001 0.028*** 0.968*** 0.001 0.030*** 0.964*** 0.010 0.001 0.031** 0.942*** 0.093* 0.000 0.030*** 0.956*** 0.043* 0.001 0.030*** 0.965*** 0.006 SE 0.001 0.010 0.011 0.001 0.011 0.013 0.019 0.001 0.013 0.020 0.050 0.001 0.012 0.014 0.024 0.001 0.010 0.012 0.016 TLAB Value -0.002 0.036*** 0.966*** -0.002* 0.078*** 0.890*** 0.075 -0.002** 0.048*** 0.916*** 0.118** -0.002** 0.031** 0.952*** 0.045** -0.002* 0.036*** 0.965*** 0.007 SE 0.001 0.011 0.011 0.001 0.025 0.032 0.058 0.001 0.019 0.026 0.057 0.001 0.013 0.015 0.022 0.001 0.011 0.012 0.030

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Table 7. Mincer-Zarnowitz Regression Results

MSFT Value Std Err t-value Adj R2

Day a0 0.097 0.122 0.798 0.091 Day a1 0.899 0.130 6.904 AN a0 0.410 0.089 4.584 0.083 AN a1 0.573 0.106 5.387 PO a0 0.033 0.141 0.236 0.157 PO a1 0.977 0.175 5.567 PC a0 0.437 0.086 5.113 0.087 PC a1 0.501 0.077 6.537 ON a0 0.536 0.072 7.487 0.083 ON a1 0.293 0.046 6.373

SIRI Value Std Err t-value Adj R2

INTC Value Std Err t-value Adj R2

Day a0 0.473 0.322 1.469 0.072 Day a1 0.655 0.204 3.206 AN a0 1.185 0.392 3.026 -0.004 AN a1 0.189 0.245 0.773 PO a0 0.385 0.338 1.138 0.074 PO a1 0.742 0.222 3.346 PC a0 1.321 0.167 7.891 0.000 PC a1 0.093 0.094 0.989 ON a0 0.937 0.390 2.404 0.007 ON a1 0.356 0.248 1.434

CSCO Value Std Err t-value Adj R2

Day a0 -0.403 0.796 -0.506 0.039 Day a1 1.149 0.335 3.430 AN a0 0.704 0.424 1.658 0.013 AN a1 0.518 0.236 2.192 PO a0 -0.499 0.461 -1.082 0.054 PO a1 1.217 0.245 4.967 PC a0 0.858 0.353 2.433 0.006 PC a1 0.399 0.196 2.038 ON a0 0.402 0.440 0.912 0.023 ON a1 0.696 0.244 2.857

ORCL Value Std Err t-value Adj R2

SUNW Value Std Err t-value Adj R2

JDSU Value Std Err t-value Adj R2

AMAT Value Std Err t-value Adj R2

Day a0 1.982 0.768 2.580 -0.008 Day a1 -0.082 0.398 -0.205 AN a0 1.790 0.376 4.761 -0.008 AN a1 0.019 0.184 0.102 PO a0 1.692 0.768 2.204 -0.008 PO a1 0.076 0.429 0.178 PC a0 2.175 0.212 10.276 0.009 PC a1 -0.162 0.094 -1.727 ON a0 1.495 0.377 3.963 -0.001 ON a1 0.174 0.191 0.910

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YHOO Value Std Err t-value Adj R2

Day a0 0.148 0.107 1.383 0.209 Day a1 0.782 0.020 39.100 AN a0 0.205 0.329 0.623 0.198 AN a1 0.761 0.142 5.343 PO a0 -0.219 0.069 -3.174 0.373 PO a1 0.940 0.013 72.308 PC a0 0.437 0.211 2.076 0.335 PC a1 0.624 0.087 7.161 ON a0 0.204 0.329 0.620 0.199 ON a1 0.761 0.142 5.349

DELL Value Std Err t-value Adj R2

CIEN Value Std Err t-value Adj R2

Day a0 3.514 0.677 5.188 -0.006 Day a1 0.097 0.202 0.482 AN a0 3.534 0.691 5.116 -0.006 AN a1 0.090 0.204 0.442 PO a0 3.558 0.688 5.173 -0.007 PO a1 0.084 0.207 0.407 PC a0 3.395 0.673 5.040 -0.004 PC a1 0.131 0.195 0.670 ON a0 3.530 0.685 5.154 -0.006 ON a1 0.092 0.203 0.453

NXTL Value Std Err t-value Adj R2

Day a0 1.478 0.604 2.448 -0.006 Day a1 0.167 0.367 0.455 AN a0 1.442 0.375 3.846 -0.002 AN a1 0.190 0.230 0.826 PO a0 1.289 0.344 3.747 0.004 PO a1 0.281 0.208 1.350 PC a0 2.485 0.425 5.851 0.015 PC a1 -0.394 0.239 -1.652 ON a0 1.317 0.379 3.473 0.002 ON a1 0.266 0.236 1.129

JNPR Value Std Err t-value Adj R2

CNXT Value Std Err t-value Adj R2

QCOM Value Std Err t-value Adj R2

Day a0 -0.507 0.512 -0.989 0.128 Day a1 1.300 0.306 4.254 AN a0 1.669 0.040 41.315 -0.007 AN a1 -0.015 0.039 -0.396 PO a0 0.167 0.355 0.471 0.087 PO a1 0.924 0.222 4.157 PC a0 0.007 0.359 0.020 0.144 PC a1 1.009 0.220 4.594 ON a0 1.711 0.052 33.045 0.009 ON a1 -0.068 0.046 -1.484

BREM Value Std Err t-value Adj R2

Day a0 -0.349 0.711 -0.490 0.108 Day a1 1.044 0.283 3.691 AN a0 -0.011 0.668 -0.016 0.105 AN a1 0.884 0.258 3.423 PO a0 -0.491 0.687 -0.716 0.129 PO a1 1.128 0.280 4.027 PC a0 -0.396 0.709 -0.558 0.111 PC a1 1.062 0.282 3.767 ON a0 -0.635 0.773 -0.821 0.120 ON a1 1.136 0.303 3.750

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ADCT Value Std Err t-value Adj R2

Day a0 3.847 1.103 3.489 -0.008 Day a1 -0.144 0.384 -0.375 AN a0 4.425 0.437 10.135 0.025 AN a1 -0.329 0.142 -2.318 PO a0 4.670 0.686 6.813 0.009 PO a1 -0.440 0.241 -1.827 PC a0 4.868 0.500 9.733 0.058 PC a1 -0.498 0.168 -2.974 ON a0 4.126 0.666 6.192 -0.002 ON a1 -0.235 0.224 -1.049

EBAY Value Std Err t-value Adj R2

AAPL Value Std Err t-value Adj R2

Day a0 2.945 1.556 1.894 -0.003 Day a1 -0.572 0.802 -0.712 AN a0 1.644 0.845 1.946 -0.008 AN a1 0.098 0.404 0.242 PO a0 1.755 0.842 2.084 -0.009 PO a1 0.044 0.408 0.107 PC a0 2.214 0.689 3.215 -0.006 PC a1 -0.175 0.320 -0.549 ON a0 1.612 0.735 2.194 -0.008 ON a1 0.110 0.340 0.324

SEBL Value Std Err t-value Adj R2

Day a0 1.227 1.020 1.203 0.003 Day a1 0.507 0.476 1.065 AN a0 1.387 0.340 4.084 0.051 AN a1 0.417 0.151 2.771 PO a0 1.766 0.960 1.839 -0.006 PO a1 0.244 0.417 0.585 PC a0 1.536 0.199 7.704 0.122 PC a1 0.306 0.073 4.209 ON a0 2.271 0.629 3.610 -0.008 ON a1 0.022 0.300 0.074

AMZN Value Std Err t-value Adj R2

Day a0 -0.987 0.568 -1.738 0.228 Day a1 1.326 0.252 5.264 AN a0 -1.125 0.596 -1.886 0.233 AN a1 1.388 0.265 5.243 PO a0 -1.104 0.585 -1.886 0.225 PO a1 1.390 0.262 5.310 PC a0 0.048 0.616 0.079 0.109 PC a1 0.821 0.255 3.213 ON a0 -1.164 0.597 -1.948 0.230 ON a1 1.411 0.266 5.299

LVLT Value Std Err t-value Adj R2

BEAS Value Std Err t-value Adj R2

AMGN Value Std Err t-value Adj R2

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VRTS Value Std Err t-value Adj R2

Day a0 3.071 0.809 3.796 0.000 Day a1 -0.427 0.353 -1.209 AN a0 2.471 0.411 6.018 0.006 AN a1 -0.153 0.178 -0.858 PO a0 1.547 0.559 2.769 -0.002 PO a1 0.249 0.241 1.034 PC a0 3.165 0.951 3.329 0.000 PC a1 -0.468 0.413 -1.133 ON a0 2.440 0.366 6.660 0.007 ON a1 -0.137 0.156 -0.876

ATML Value Std Err t-value Adj R2

RFMD Value Std Err t-value Adj R2

Day a0 3.237 0.908 3.565 -0.007 Day a1 -0.115 0.318 -0.363 AN a0 1.989 0.981 2.028 -0.002 AN a1 0.330 0.341 0.967 PO a0 0.428 0.657 0.652 0.067 PO a1 0.926 0.246 3.761 PC a0 3.230 0.986 3.276 -0.008 PC a1 -0.113 0.346 -0.326 ON a0 2.957 0.888 3.331 -0.008 ON a1 -0.014 0.308 -0.047

BRCD Value Std Err t-value Adj R2

RIMM Value Std Err t-value Adj R2

Day a0 -0.131 0.815 -0.161 0.026 Day a1 0.939 0.330 2.847 AN a0 -0.033 0.738 -0.045 0.026 AN a1 0.905 0.299 3.031 PO a0 0.004 0.698 0.005 0.027 PO a1 0.903 0.261 3.463 PC a0 0.916 0.686 1.335 0.006 PC a1 0.528 0.250 2.107 ON a0 -0.057 0.747 -0.076 0.027 ON a1 0.916 0.304 3.015

TLAB Value Std Err t-value Adj R2

Day a0 0.727 0.495 1.468 0.097 Day a1 0.828 0.279 2.970 AN a0 1.151 0.408 2.823 0.081 AN a1 0.421 0.162 2.598 PO a0 0.920 0.505 1.823 0.076 PO a1 0.604 0.238 2.538 PC a0 0.837 0.637 1.314 0.051 PC a1 0.732 0.339 2.155 ON a0 0.904 0.409 2.209 0.104 ON a1 0.723 0.229 3.159 Mincer-Zarnowitz Regression:

 

₂ 1 2

 

₂ 1 2 0 1 | t k a a ht k t ut k _   _  _

where _{t k}2_ is the realized day volatility and h_{t k t}2_ _| is the forecast day volatility.

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Table 8. Out-of-Sample Forecast Performance

MSFT Day AN PO PC ON MSE 0.440 0.454 0.424 0.474 0.765 MAE 0.206 0.239 0.202 0.256 0.596 HMSE 0.442 0.426 0.391 0.420 0.443 HMAE 0.208 0.236 0.204 0.227 0.370 LL -0.049 -0.046 -0.029 -0.128 -0.448 SIRI Day AN PO PC ON MSE 1.566 1.532 1.535 1.530 1.548 MAE 1.082 1.034 1.038 1.032 1.056 HMSE 0.378 0.406 0.375 0.373 0.408 HMAE 0.282 0.289 0.276 0.273 0.293 LL -0.122 -0.082 -0.105 -0.109 -0.090 INTC Day AN PO PC ON MSE 0.299 0.327 0.290 0.483 0.310 MAE 0.235 0.252 0.230 0.372 0.244 HMSE 0.197 0.207 0.201 0.251 0.206 HMAE 0.153 0.159 0.157 0.201 0.160 LL -0.059 -0.085 -0.018 -0.181 -0.054 CSCO Day AN PO PC ON MSE 0.673 0.681 0.665 0.714 0.674 MAE 0.376 0.369 0.409 0.437 0.361 HMSE 0.391 0.416 0.338 0.391 0.412 HMAE 0.219 0.219 0.231 0.239 0.215 LL -0.137 -0.111 -0.126 -0.178 -0.114 ORCL Day AN PO PC ON MSE 0.345 0.346 0.346 0.349 0.352 MAE 0.265 0.265 0.267 0.268 0.271 HMSE 0.220 0.221 0.224 0.226 0.226 HMAE 0.169 0.170 0.172 0.171 0.174 LL -0.006 -0.005 0.006 -0.022 -0.002 SUNW Day AN PO PC ON MSE 0.449 0.513 0.511 0.450 0.562 MAE 0.351 0.394 0.391 0.353 0.460 HMSE 0.189 0.220 0.248 0.188 0.220 HMAE 0.148 0.167 0.181 0.148 0.180 LL -0.024 -0.042 0.026 -0.029 -0.107 JDSU Day AN PO PC ON MSE 0.904 0.900 0.902 0.883 0.899 MAE 0.661 0.659 0.667 0.639 0.657 HMSE 0.422 0.418 0.448 0.394 0.416 HMAE 0.307 0.305 0.325 0.286 0.304 LL 0.164 0.164 0.186 0.149 0.162 AMAT Day AN PO PC ON MSE 0.487 0.528 0.478 0.708 0.505 MAE 0.384 0.421 0.368 0.538 0.385 HMSE 0.258 0.277 0.276 0.299 0.270 HMAE 0.203 0.219 0.210 0.239 0.204 LL -0.075 -0.090 0.004 -0.181 -0.066 YHOO Day AN PO PC ON MSE 0.597 0.595 0.552 0.685 0.595 MAE 0.487 0.485 0.450 0.559 0.485 HMSE 0.251 0.251 0.234 0.263 0.251 HMAE 0.207 0.207 0.193 0.221 0.207 LL -0.196 -0.192 -0.193 -0.242 -0.192 DELL Day AN PO PC ON MSE 0.310 0.305 0.312 0.320 0.299 MAE 0.232 0.229 0.236 0.239 0.223 HMSE 0.319 0.306 0.323 0.332 0.296 HMAE 0.236 0.225 0.241 0.244 0.217 LL 0.069 0.038 0.079 0.074 0.029 CIEN Day AN PO PC ON MSE 1.348 1.337 1.352 1.307 1.343 MAE 0.903 0.891 0.911 0.860 0.896 HMSE 0.470 0.462 0.468 0.442 0.466 HMAE 0.296 0.289 0.299 0.273 0.292 LL 0.152 0.143 0.158 0.125 0.147 NXTL Day AN PO PC ON MSE 0.669 0.682 0.675 0.734 0.676 MAE 0.445 0.462 0.467 0.484 0.459 HMSE 0.375 0.397 0.402 0.372 0.399 HMAE 0.253 0.268 0.275 0.252 0.270 LL -0.030 -0.017 -0.009 -0.089 -0.004 JNPR Day AN PO PC ON MSE 0.593 0.659 0.587 0.796 0.641 MAE 0.462 0.530 0.456 0.675 0.512 HMSE 0.262 0.280 0.261 0.305 0.278 HMAE 0.204 0.225 0.203 0.260 0.222 LL -0.137 -0.176 -0.131 -0.255 -0.156 CNXT Day AN PO PC ON MSE 1.704 1.685 1.734 1.724 1.677 MAE 1.080 1.078 1.099 1.086 1.075 HMSE 0.407 0.393 0.414 0.413 0.392 HMAE 0.258 0.252 0.263 0.257 0.251 LL 0.136 0.124 0.150 0.133 0.119 QCOM Day AN PO PC ON MSE 0.343 0.343 0.352 0.339 0.343 MAE 0.257 0.260 0.264 0.261 0.259 HMSE 0.207 0.205 0.217 0.208 0.206 HMAE 0.155 0.155 0.163 0.160 0.155 LL -0.027 -0.036 0.003 -0.008 -0.033 BRCM Day AN PO PC ON MSE 0.631 0.664 0.604 0.629 0.645 MAE 0.522 0.560 0.492 0.522 0.543 HMSE 0.248 0.252 0.244 0.248 0.249 HMAE 0.205 0.213 0.199 0.205 0.210 LL -0.128 -0.156 -0.103 -0.127 -0.145