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

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1 Predicing Sock Volailiy Using Afer-Hours Informaion: Evidence from he NASDAQ Acively Traded Socks Chun-Hung Chen 1 Office of he Comproller of he Currency Wei-Choun Yu 2 Winona Sae Universiy Eric Zivo 3 Universiy of Washingon March 3, 2011 Absrac We use realized volailiies based on afer-hours high frequency sock reurns o predic nex day sock volailiy. We exend he GARCH model o include addiional informaion: he whole afer hours, he preopen realized variance, he posclose realized variance, and he overnigh squared reurn. For he hiry mos acive NASDAQ socks, we find ha mos of he socks exhibi posiive and significan preopen coefficiens and ha he inclusion of he preopen variance can mosly improve he ou-of-sample forecasabiliy of he nex day condiional day volailiy. The inclusions of posclose variance and overnigh squared reurn do provide some predicive power for he nex day condiional volailiy bu o a lesser degree; heir predicive abiliies are inferior o preopen variance. Our findings suppor he resuls of prior sudies: raders rade mosly for non-informaion reasons in he posclose period and rade mosly for informaion reasons in he preopen period. Keywords: Financial Markes, GARCH Model, Evaluaing Forecass, High-Frequency Daa, Realized Variance. 1 Corresponding Auhor: Economics Risk Analysis Division, 250 E S. SE, Washingon, DC chunhung.chen@occ.reas.gov. Tel: We hank seminar paricipans from he Deparmen of Finance a Naional Cenral Universiy and Office of Comproller of Currency and Mike Wenz for heir helpful commens. 2 Deparmen of Economics, Winona Sae Universiy, Somsen 319E, Winona, MN 55987, USA. wyu@winona.edu. Tel: Fax: Deparmen of Economics, Universiy of Washingon, Box , Condon 401, Seale, WA ezivo@u.washingon.edu. Tel: Fax:

2 Predicing Sock Volailiy Using Afer-Hours Informaion: Evidence from he NASDAQ Acively Traded Socks 1. Inroducion Volailiy modeling has received much aenion over he pas wo decades in finance lieraure no only because i relaes direcly o he profis of raders bu also because i is imporan o he valuaion of derivaive insrumens. The goal of modeling and forecasing volailiy is o have beer risk managemen, more accurae derivaive prices, and more efficien porfolio allocaions. Good financial decision making relies on an accurae predicion of he second momen of he underlying financial insrumen. Among various volailiy modeling echniques, he mos popular models are he generalized auoregressive condiional heeroskedasiciy (GARCH) models developed by Engle (1982) and Bollerslev (1986). This family of models can explain well he sylized facs of financial reurn volailiy: persisence, mean reversion, and he leverage effec. Moreover, as Andersen and Bollerslev (1998) and ohers have shown, when using appropriae ex pos volailiy measures, he GARCH model can provide good forecasabiliy of he condiional volailiy. Besides modeling echniques or mehods, he forecasabiliy also relies on a useful informaion se. Unil recenly, he mos commonly used informaion se for modeling daily volailiy was he hisorical daily closing prices. Recen research (e.g., Andersen and Bollerslev, 1998, Andersen, Bollerslev, and Lange, 1999, Fueres, Izzeldin and Kaloychou, 2009, and Marens, Dijk, and Pooer, 2009) has shown ha he use of inra-day high frequency daa can subsanially improve he measuremen and forecasabiliy of daily volailiy. The majoriy of hese sudies used inra-day daa observed during normal rading hours. We conribue o his lieraure by considering inra-nigh daa 2

3 observed during he whole afer-hours period 1 and each of he subperiods, which will be defined in laer secions. Alhough several sudies have documened he imporance of afer-hours informaion for volailiy modeling (e.g., Oldfield and Rogalski, 1980; Greene and Was, 1996; Cao, Ghysels, and Haheway, 2000; Masulis and Shivakumar, 2002; Taylor, 2007; and Tsiakas, 2008), only a few have acually employed high-frequency daa in he analysis. One such paper is Taylor (2007), which uses high-frequency overnigh S&P 500 fuures volailiy informaion o predic S&P 500 socks volailiy. In his paper, we sudy individual socks insead of marke indices. As menioned in Campbell e al. (2001), here are several moives for sudying he volailiies of individual socks. For insance, many invesors have large holdings of individual socks ha have no been diversified and herefore are subjec o idiosyncraic volailiy. According o he findings from Barclay and Hendersho (2008), he afer-hours ransacions become meaningful, i.e., generae price discovery, only if he sock has sufficien afer-hours rading aciviy. Since mos socks are hinly raded during afer hours, for our sudy we choose he hiry mos acively raded NASDAQ socks in We uilize he availabiliy of afer-hours rading opporuniies o he public and he recording of high frequency afer-hours ransacion daa of hiry NASDAQ socks from 2001 o 2004, our sample period, o examine how his exended informaion se could be effecively used o improve he modeling and he forecasing of nex day condiional volailiy. We employ he GARCH model, which is used for daily reurns wih afer-hours realized variance as an addiional exogenous variable included in he condiional variance equaion. We also use realized volailiy as he ex pos measure for our forecasing evaluaion. Our resuls show ha wih he inclusion of realized variance for he whole afer-hours period in he informaion se, ou of 30 socks, 16 of hem do provide beer forecasing of nex day volailiy compared o he ones wihou he inclusion. Moreover, when breaking up he whole afer-hours period ino hree subperiods, we find ha preopen coefficiens in he GARCH (1,1) model are posiive and 3

4 significan for 23 socks in he in-sample analysis, and he inclusion of he preopen variance can improve he ou-of-sample forecasabiliy of he condiional day volailiy for 18 socks. The posclose variance and he overnigh squared reurn exhibi predicive power o a lesser degree, and heir ou-of-sample forecasing abiliies are mosly inferior o preopen variance. The evidence, by and large, suppors he resuls of prior sudies showing ha raders rade mosly for noninformaion reasons in he posclose period bu rade mosly for informaion reasons in he preopen period and ha he inclusion of preopen variance could enhance he predicabiliy of nex day condiional volailiy. Our sudy conribues o he exising lieraure in he following ways. Firs, we use highfrequency inra-nigh ransacion daa ha has no ye been sysemaically exploied for he modeling and forecasing of daily condiional volailiy. Second, we refine he afer-hours informaion by segmening he whole afer-hours period ino subperiods based on heir differen informaion densiies. Third, pas research has ofen focused on in-sample forecasing evaluaion while we evaluae our models predicabiliy for ou-of-sample as well. The res of he paper proceeds as follows. Secion 2 reviews he volailiy lieraure using afer-hours informaion. Secion 3 explains our daa and realized volailiy consrucion. The resuls of modeling and forecasing condiional volailiy based on he GARCH model are provided in Secions 4. Secion 5 conains our concluding remarks. 2. Lieraure Review Condiional reurn volailiy models such as GARCH demonsrae ha pas reurn shocks and volailiies conain informaion abou he evoluion of fuure volailiies and herefore can be used for forecasing purposes. One explanaion for his resul is he Mixure of Disribuion Hypohesis (MDH), suggesed by Clark (1973), Tauchen and Pis (1983), and Kalev e al. (2004). They aribue dependence in volailiies o he serial correlaion of he news arrival rae. Lamoureux 4

5 and Lasrapes (1990) use he MDH proposed by Clark (1973) o explain he persisen naure of daily condiional volailiy in he GARCH model. They assume ha a sochasic model can be derived by considering he daily reurn in day, ε, as a sum of i.i.d. (0, 2 ) inra-day price incremens, δ i, n, (1) i1 i where i denoes he i h inraday price movemen, and he random variable n is a mixing variable ha denoes he arrival rae of informaion in day. Clark (1973) assumes ha ε is drawn from a mixure of disribuions, of which he variances depend on n, 2 n ~ N(0, n ). (2) When he arrival rae of informaion is serially correlaed, n can be expressed as n a b( L) n u, (3) 1 where a is a consan and u is whie noise. The condiional variance becomes E[ n ] n a b( L) u (4) 2 1 which demonsraes he persisence in he condiional variance capured in he GARCH model. To examine his hypohesis, Lamoureux and Lasrapes (1990), Sharma, Mougoue, and Kamah (1996), and Brooks (1998) use rading volume as a proxy for he informaion arrival rae and include i as an exogenous variable in he GARCH (1,1) specificaion for daily volailiy. They show ha he inclusion of volume grealy reduces he persisence parameer of he esimaed GARCH model. Moreover, Brooks (1998) shows ha including rading volume in a GARCH model does no improve volailiy forecass because i does no provide addiional informaion which is no already capured by pas condiional volailiy. In order o check wheher he inclusion of he afer-hours volailiy can improve he forecasabiliy of he fuure condiional day volailiy, we need o check wheher he informaion conained in he afer-hours volailiy provides addiional explanaory power o he fuure condiional day volailiy. If he afer-hours 5

6 volailiy provides addiional informaion raher han subsiuing for informaion already incorporaed in he pas condiional volailiy or volume, i could be used o improve forecass of nex day volailiy. I is well known in microsrucure lieraure ha informaion and announcemens frequenly occur afer normal rading hours. If here are no rading opporuniies during afer hours, his occurrence and accumulaion of informaion during he close-o-open period should conribue o he upcoming day (open-o-close) volailiy. In oher words, when afer-hours rading is no available, he informaion will be realized a he opening hours. Therefore, he occurrence of afer-hours informaion or news implies higher-han-usual volailiy during he following regular rading hours. Even when rading is available for all or par of he afer-hours period, as is rue for our sample period, we can sill expec informaion in his period o have an impac on he following regular rading hours for wo reasons. The firs reason is he spillover effec. If he marke is no fully efficien, i would ake some ime for he informaion o be incorporaed ino prices. This phenomenon could be more prominen due o he relaively illiquid naure of he afer-hours rading environmen. Since i akes rades o faciliae price discovery (Barclay and Hendersho, 2003), he informaion migh no be fully incorporaed ino he price unil he regular-rading hour when he rading volume is much higher. The second reason is he informed naure of rades in afer hours. Barclay and Hendersho (2004) indicae ha he raders in afer hours are mainly professional and insiuional. Many of hem rade for shor-lived privae informaion. I is likely ha hey rade for privae or scheduled news ha has ye o be announced. Therefore, i is raional o expec ha a highly volaile afer-hours rading would lead o a highly volaile day rading in he nex day. Gallo and Pacini (1998) sudy he impac of close-o-open reurns, measured as he percenage difference of he previous daily closing price and curren daily opening price, on he following day (open-o-close) volailiy for he six major marke indices using a GARCH (1,1) 6

7 model wih he close-o-open reurns as an exogenous variable. Examining hree London Inernaional Financial Fuures Exchange conracs, Brooks, Clare and Persand (2000) find ha he inclusion of close-o-open reurns subsanially reduces he excessive volailiy persisence in he GARCH model. Their sudy suggess ha he overnigh volailiy migh improve he calculaions of capial requiremens for risk managemen of banks. Marens (2002) examines wheher GARCH (1,1) models ha include differen funcional forms of he afer-hours volailiy can improve he forecass of he following day volailiy for he S&P 500 index fuures ransacions. Gallo and Pacini (1998) find ha he inclusion of close-o-open reurns improves forecasabiliy of condiional volailiy for some sock indices, while Marens (2002) finds ha he inclusion canno improve forecasabiliy. This mixed evidence could come from he poor exploiaion of afer-hours informaion. We propose a refinemen of he afer-hours informaion by segmening he whole afer-hours period ino several subperiods according o informaion densiies. Barclay and Hendersho (2003, 2004) have previously proposed his segmenaion in heir afer-hours papers, bu we are he firs o apply i o forecasing volailiy. Since volailiy is no direcly observable, he mehodology on which measuring ex pos volailiy is based is imporan for evaluaing volailiy forecass. Pas sudies, such as Cumby, Figlewski and Hasbrouck (1993), Figlewski (1997), and Jorion (1995), show ha sandard volailiy models such as GARCH perform poorly in ou-of-sample forecasing evaluaion when squared reurns are used o proxy for ex pos volailiy measures. Andersen and Bollerslev (1998) poin ou ha while he squared reurn is an unbiased esimae for unobserved volailiy, i is a very noisy esimae. This, herefore, can poenially explain why hose volailiy models appear o produce poor forecass. They furher show ha realized volailiy, which is defined as he sum of squared reurns sampled a high inra-daily frequencies, provides a much more reliable ex pos volailiy measure han he squared reurns. The GARCH forecass significanly improve when evaluaed agains he realized volailiy measures. Furhermore, Andersen e al. (1999) show ha 7

8 he forecasing performance of sandard volailiy models can be grealy enhanced by uilizing high-frequency daa. 3. Daa and Volailiy Measuremen 3.1. Daa Our high frequency daa is aken from he Trade and Quoe (TAQ) daabase, which provides daa on ick-level ransacion prices and quoes from 8:00 am unil 6:30 pm EST, when he NASDAQ Trade Disseminaion Service (NTDS) is on. Since rading volume is relaively low for socks in afer hours, we have chosen hiry acive socks ha have he highes rading aciviies during he afer-hours period based on he aggregae rading volume in he whole year of 2004 in NASDAQ. The hiry socks are repored in Table 1. Because Microsof (MSFT) has he highes rading volume, we use i as he principal sock under invesigaion and presen more deailed resuls peraining o i. The sample period is from January 2001 o December 2004, during which ime he afer-hours rading informaion is recorded hrough NTDS and is available o he public. We choose he firs hree and a half years as he in-sample period for modeling volailiy and he laer half-year as he ou-of-sample period o evaluae forecasing performance. The TAQ daa ypically conain some recording errors. We herefore remove any recorded rades ha have a change of posiive or negaive 25% from heir immediaely prior rades in a day 2. We also remove daes in which eiher preopen, posclose, or day ransacion daa is missing or in which sock splis occur Afer-Hours Subperiods Barclay and Hendersho (2003, 2004) break he afer-hours period ino hree subperiods: he posclose period (4:00 o 6:00 pm EST), he overnigh period (6:00 pm o 8:00 am EST), and he preopen period (8:00 o 9:30 am EST). They invesigae he informaion srucure of he 8

9 posclose and preopen and find ha he probabiliy of an informed rade is much higher in he laer period han in he former period. They find ha abou 80% of all rading volume in he posclose occurs a he closing price or wihin he closing quoes a 4:00 pm EST 3. This implies ha raders end o rade for liquidiy demands righ afer he regular rading hour is closed. Furhermore, Barclay and Hendersho (2003) use he probabiliy of informed rade measure developed by Easley, Kiefer, and O Hara (1997) o show ha rading is highly informed during he preopen, which implies ha raders are more likely o rade for informaion reasons in his period. Even hough raders can sill rade hrough an elecronic communicaion nework (ECN) or a marke maker during he overnigh period, here is no formal analysis on he informaion srucure for his period. The overnigh daa is usually no available from he reporing service provided by NTDS. Using heir proprieary daase, Barclay and Hendersho (2003) find ha only 1% of oal afer-hours rades occur during ha period. The uneven informaion in each afer-hours subperiod leads us o hypohesize ha he volailiy in each subperiod should have differen effecs on he following day volailiy. We expec ha he posclose volailiy conains lile o no informaion, while he volailiy in he preopen conains new informaion abou he following day volailiy. This means ha he inclusion of he preopen volailiy in he informaion se may improve he forecasabiliy of a volailiy model. The impac of volailiy in he overnigh period on condiional day volailiy, however, is less obvious. If he preopen rades have realized mos or all of he informaion ha occurred in he overnigh period, or if he overnigh squared reurn measure is very noisy, we would expec overnigh volailiy o have lile or no predicabiliy on he day volailiy Volailiy Measuremen Realized variance is a more accurae measure of condiional variance han he squared reurn. We use i o measure rading day variance and variance during he afer-hours periods as well as o evaluae our volailiy predicions. Following Bollerslev and Wrigh (2001), Andersen 9

10 e al. (2001a), and Andersen e al. (2003), we consruc realized variance by summing up inraperiod high frequency squared reurns: r p p i, n i, n i, ( n1) r 1 r i, i, n n1 1/ 2 2 i, ri, n n1 (5) where p denoes he logarihmic sock price; i is eiher he regular hour, he preopen, or he posclose period; r is he inra-day reurn; 1/Δ is he number of observaions for each of he periods (Δ is 5 minues in regular hours and is 15 minues in afer-hours); and σ 2 i, is he esimaed realized variance for period i in day. Realized volailiy is compued as he square roo of realized variance. Since here is no daa for rades in he overnigh period, we measure he variance based on he firs rade of preopen and he las rade of he previous day s posclose: ( p p ) (6) 2 2 Overnigh, Firs Trade of Preopen, Las Trade of Posclose, 1 Andersen e al. (2001b) shows ha as sampling frequency increases, realized variance approximaes o inegraed variance, which is he acual realized reurn variaion over a given horizon for a coninuous ime diffusion process and an unbiased esimae of condiional variance. Alhough i is heoreically demonsraed ha he measuremen error associaed wih he esimaion of he realized variance becomes very small as he sampling frequency increases, in realiy he exisence of marke microsrucure fricions (e.g., bid-ask bounce, price discreeness, and infrequen rading) can creae large biases. To avoid his problem, Andersen e al. (2001a) propose sampling he inra-day observaions a 5-minue inervals 4. Since he rading environmen in afer-hours is known o have much larger microsrucure fricions han during regular-hours, we sample observaions a a 5-minue frequency for regular-hours and a a 15-minue frequency for afer-hours. The resuling numbers of iner-period observaions for he regular-hours, he preopen, and he posclose are 78, 6, and 8, respecively. 10

11 Figure 1 shows he regular-hours, he preopen, and he posclose realized volailiy, in addiion o he overnigh absolue reurn for MSFT from January 2001 o December Table 2 liss some descripive saisics of realized volailiy measures for MSFT. These measures represen he oal amoun of volailiy per day in each period. Similar o he disribuion of reurns, he disribuions of volailiies are all skewed o he righ and have fa ails. The auocorrelaion plos in he four periods are shown in Figure 2. The daily and preopen realized volailiy series boh exhibi he commonly known characerisic of long-memory, or persisence. In conras, we do no observe his feaure in he posclose realized volailiy series and overnigh absolue reurn. Barclay and Hendersho (2003) find ha price changes are larger in he preopen han in he posclose, which indicaes more privae informaion and less noise in he preopen period. Table 3 provides volailiies per hour and per rade for he preopen and he posclose periods. The average volailiies per hour for he preopen and posclose are 0.39% and 0.29%, respecively, and he average volailiies per rade are % and %, respecively. The numbers show ha volailiy is higher in he preopen han in he posclose, which is consisen wih he resul of Barclay and Hendersho (2003). Boh he median volailiies per hour and per rade provide he same qualiaive resuls. 4. GARCH Modeling and Forecasing The GARCH framework is he mos common approach o modeling and forecasing volailiy. We use he GARCH (p,q) model 5 r zh h p q ii jh j i1 j1 (7) 11

12 where μ and ω are consans in he condiional mean equaion and he condiional variance equaion, respecively; ε is a serially uncorrelaed residual erm (news shock) wih mean zero; z is an i.i.d. random variable wih mean zero and uni variance; and h 2 is he condiional variance a ime. While here are many variaions of he GARCH (p,q) model, a GARCH (1,1) model is usually sufficien for mos financial ime series applicaions (Andersen and Bollerslev, 1998; Hansen and Lunde, 2004). Table 4 shows he Akaike informaion crierion, Bayesian informaion crierion, and he log-likelihood for all GARCH (p,q) models wih p 2 and q 2 for he daily MSFT reurn series in he in-sample period, which is from January 2001 o June The GARCH (1,1) wih Suden s error disribuion for z appears o be he mos appropriae model. The firs column of Table 5 repors he coefficiens for he daily GARCH (1,1) model in regular hours. The sum of he esimaes of α and β is 0.997, which shows ha he condiional volailiy is quie persisen. This resul is very similar o repored by Marens (2002) for S&P 500 fuures. The ARCH and Ljung-Box ess on he squared residuals are employed o check for he adequacy of he fied model. We find ha he GARCH (1,1) specificaion fis he in-sample reurn series of he MSFT well. The res of socks daily GARCH (1,1) esimaes are repored in he firs hree columns of Table GARCH Model for Day Reurns wih Nigh Variance The GARCH model offers flexibiliy in ha he addiional exogenous variables ha are hough o have impac on condiional volailiy can be included in he condiional variance equaion. The modified GARCH (1,1) is: r zh h h x (8) 12

13 where x represens he addiional exogenous variable in he condiional variance equaion. Boh Gallo and Pacini (1998) and Marens (2002) use his approach by including he close-o-open squared reurn as he addiional exogenous variable in he condiional variance equaion. Marens (2002) finds he coefficien on he addiional variable o be saisically insignifican. On he oher hand, Gallo and Pacini (1998) find he coefficien o be saisically significan for mos of he major marke indices, wih he sign of he coefficien being posiive for some indices and negaive for ohers. If he impac of afer-hours informaion on he volailiy of regular hours is caused by he possibiliy of he informed raders rading privae informaion before he news is publicly announced during he regular hours, a higher afer-hours volailiy should lead o a higher volailiy he following day. Therefore, we expec he sign of he coefficien for he overnigh variance variable o be saisically significan and posiive. To invesigae he impac of afer-hours informaion in he GARCH (1,1) model (8), we use he following four exogenous variables: All hree subperiods ogeher (AN): x r r r NPO NPC 2 2 2, n PO, n PC, non 17.5 n PO npc (9) The preopen period only (PO): The posclose period only (PC): x x NPO 2 r, npo npo 1 (10) NPC 2 r, npc npc 1 (11) The overnigh period only (ON): x r (12) 2, n ON where PO, PC, and ON denoe he preopen, he posclose, and he overnigh period, respecively. The variable x defined in (9) is a ime-weighed average realized variance of he close-o-open (he whole afer-hours) period. The second hrough fifh columns of Table 5 show he esimaion resuls of he modified GARCH (1,1) model (8) using he exogenous variables defined in (9) (12) for MSFT. Firs, we find ha he esimaed coefficien (sandard error) for he whole afer-hours period realized 13

14 variance (9) is (0.064), which is posiive bu saisically insignifican. This resul agrees wih Marens (2002) in ha he close-o-open variance does no provide significan explanaory power for he nex day condiional variance. Second, we find ha he coefficien for he posclose variance (11) is negaive and saisically insignifican. This resul is consisen wih he hypohesis ha raders primarily rade in he posclose for non-informaion reasons, and herefore here is no informaion o be carried over ino he nex day volailiy. Anoher possibiliy is ha if here were any informaion in he posclose period, i could be subsequenly incorporaed ino he price hrough rading aciviies during he following overnigh and he preopen periods. The only explanaory variable ha we find o be saisically significan is he preopen realized variance (10), which has an esimaed coefficien (sandard error) of (0.090). Hence, a 1% increase in he preopen realized variance would lead o a 0.221% increase in he following regular-hour condiional variance. This resul is consisen wih our hypohesis in ha he coefficien should be posiive and significan. Finally, we find ha he esimaed persisence parameers are and for he GARCH (1,1) model (7) and he modified GARCH (1,1) model (8) wih he preopen variance, respecively. As discussed above concerning he MDH and Gallo and Pacini (1998), his sligh decrease in he persisence parameer indicaes ha he preopen variance appears o provide independen informaion from ha conained in he pas day reurns. This resul enhances our hypohesis ha he addiion of he preopen variance ino he model would improve forecass of he nex day condiional day volailiy. Table 6 gives he esimaion resuls of he modified GARCH (1,1) model (8) for all hiry socks. As summarized in Column 1 of Table 9, ou of hiry socks, 13 socks show significan and posiive coefficiens for he whole afer-hours period. In Column 2 of Table 9, 23 socks show significan and posiive coefficiens for he preopen period. In Column 3 of Table 9, 10 socks show significan and posiive coefficiens for he posclose period. In Column 4 of Table 9, 14

15 10 socks show significan and posiive coefficiens for he overnigh period. The in-sample evidence shows ha preopen rading volailiy is mos likely o impac he following day volailiy Ou-of-sample Forecas Evaluaion Figure 3 shows he ou-of-sample ex pos realized volailiy series and he forecased 1-sepahead condiional volailiy series of various fied GARCH (1,1) models for MSFT. The forecass wih he preopen variance as an exogenous variable in he condiional variance equaion appear o rack he realized volailiy series he bes. We formally evaluae he forecasing performance of he differen models using he following merics: k a a h k u k 2 2 Mincer-Zarnowiz Regression: RMSE: h (13) 12 1 T 2 k k T (14) 1 MAE: 1 T 2 2 k h k T 1 (15) 1 T 12 2 (16) 1 T 2 2 HRMSE: 1 k h k HMAE: 1 T T k h k (17) 1 1 T (18) T 2 2 LL: log k h k 1 where 2 k denoes realized variance in day +k and where 2 h k denoes he condiional variance forecas for day +k based on informaion available in day. In he Mincer-Zarnowiz regression proposed by Mincer and Zarnowiz (1969), if he condiional volailiy model is correcly specified, we should have a 0 and a 1 equal o zero and one, respecively, wih high saisical significance. However, Andersen and Bollerslev (1998) poin ou ha he coefficiens could 15

16 suffer from a sandard errors-in-variables problem, which would make inerpreaion difficul. Noneheless, hey argue ha he R 2 of he regression can be used o evaluae he variabiliy of he ex pos volailiy ha is explained by he forecased condiional volailiy. The roo mean square error (RMSE) and he mean absolue error (MAE) are wo commonly used crieria. To check wheher he resuls are reliable in a nonlinear and heeroskedasic environmen, we follow Andersen e al. (1999) and use he heeroskedasiciy adjused RMSE (HRMSE), MAE (HMAE), and he logarihmic loss funcion (LL). As sressed by boh Andersen and Bollerslev (1998) and Hansen and Lunde (2006), i is crucial o choose he righ ex pos volailiy measure o serve as he benchmark for he forecas evaluaion since volailiy is no direcly observed. Several pas sudies, such as Figlewski (1997) and Jorion (1995), have used daily squared reurns as he proxy for he ex pos volailiy measure and conclude ha sandard volailiy models explain lile of he variabiliy in he ex pos volailiy. Andersen and Bollerslev (1998) and Hansen and Lunde (2006) use realized volailiy and demonsrae ha i provides a more reliable and accurae measure of he rue volailiy and ha is use in forecas evaluaion saisics leads o more accurae inferences regarding forecasing accuracy. To perform he forecas evaluaions, we firs esimae he parameers of he GARCH (1,1) models (8) and (9) from he in-sample daa and hen compue 1-sep-ahead predicions for condiional volailiy over a rolling window. Table 7 liss he 1-sep-ahead forecas evaluaion resuls by he Mincer-Zarnowiz regression (13). For MSFT, we see ha he GARCH (1,1) wih he preopen realized variance provides he bes forecasing performance in erms boh of accuracy and of explaining he variabiliy in he ex pos measures. The esimaed coefficien on a 1 is when forecass are compued from a GARCH (1,1) model wih preopen realized variance, compared o for forecass compued from he GARCH (1,1) model wihou he inclusion (day GARCH (1,1)). The GARCH (1,1) wih he preopen realized variance has a subsanially higher R 2 = han he R 2 = from he day GARCH (1,1). 16

17 Table 9 summarizes he oher socks Mincer-Zarnowiz resuls for which he inclusion of afer-hours informaion boh increases R 2 and makes a 1 closer o one. Column 5 of Table 9 shows ha ou of 30 socks, 7 socks improve heir ou-of-sample forecasing resuls for he whole aferhours period. Column 6 of Table 9 shows ha ou of 30 socks, 14 socks improve heir ou-ofsample forecasing resuls for he preopen period. Column 7 of Table 9 shows ha ou of 30 socks, 6 socks improve heir ou-of-sample forecasing resuls for he posclose period. Column 8 of Table 9 shows ha ou of 30 socks, 6 socks improve heir ou-of-sample forecasing resuls for he overnigh period. The preopen period rading provides much more enhanced forecasing resuls compared o oher afer-hours subperiods based on he observed Mincer-Zarnowiz resuls. Table 8 presens forecas evaluaion saisics from equaions (14)-(18). For MSFT, he forecass provided by he GARCH (1,1) wih he preopen realized variance (column 3) are superior o hose from he day GARCH (1,1) only (column 1). Their forecas errors for MSE, MAE, HMSE, HMAE, and LL are he lowes. We also see ha he forecass from he GARCH (1,1) wih he whole afer-hours period (column 2), preopen period (column 4), and overnigh period (column 5) perform relaively poorly compared o hose from he day GARCH (1,1) only. Columns 9 o 12 of Table 9 summarize he ou-of-sample performance for he inclusion of aferhours subperiods for he oher socks. The aserisk represens ha including each subperiod will improve he forecasing performance compared o he forecasing wihou any afer-hours informaion (day GARCH (1,1) only). Since we use five forecasing evaluaion mehods, if here are a leas hree mehods ha perform beer, we coun his subperiod as improving he forecasing oucome. Column 9 of Table 9 shows ha ou of 30 socks, 16 socks improve heir forecasing for he whole afer-hours period; Column 10 shows ha 18 socks improve heir forecasing for he preopen period; Column 11 shows ha 9 socks improve heir forecasing for he posclose period; and Column 12 shows ha 16 socks improve heir forecasing for he overnigh period. 17

18 To invesigae he bes afer-hours subperiod ha could enhance he forecasing performance, he double aserisks denoe he subperiod ha produces lowes forecasing errors. Among 16 socks ha improve nex day volailiy forecasing by including he whole afer-hours period, only one sock sands ou as superior o oher subperiods. Among 18 socks improving nex day volailiy forecasing by including he preopen period, 15 socks are superior o oher subperiods. Among nine socks improving nex day volailiy forecasing by including he posclose period, only hree socks are superior o oher subperiods. Among 16 socks improving nex day volailiy forecasing by including he overnigh period, only four socks are superior o oher subperiods. In summary, based on he in-sample and ou-of-sample resuls, i is apparen ha he preopen period provides addiional informaion and herefore improves he nex day forecasing performance. 5. Conclusion Mos of he volailiy forecas lieraure has focused on comparing he forecas performance of differen volailiy models. In his sudy, we concenrae on wheher an expanded informaion se can increase he forecasabiliy of a condiional day volailiy model. While he ypical informaion in previous lieraure is he daily reurn and/or variance measures, we add informaion of afer-hours variance using high-frequency observaions. We presen he GARCH model for daily volailiy by four measures of afer-hours informaion: he whole afer-hours, he preopen variances, he posclose variances, and he overnigh squared reurn. By examining he hiry mos acively raded NASDAQ socks, we find ha preopen coefficiens in he GARCH (1,1) model are posiive and significan for 23 ou of 30 socks in he in-sample analysis, and he inclusion of he preopen variance can subsanially improve he ou-of-sample forecasabiliy of he condiional day volailiy for 18 socks. The posclose variance and he overnigh squared 18

19 reurn exhibi predicive power for fuure condiional volailiy o a lesser degree, and heir ouof-sample forecasing abiliies are mosly inferior o preopen variance. The evidence, by and large, suppors he resuls of prior sudies showing ha raders rade mosly for non-informaion reasons in he posclose period bu rade mosly for informaion reasons in he preopen period. We propose wo reasons for why he preopen variance can be used o improve he predicabiliy of he model. The firs is he spillover effec, and he second is he possibiliy of he informed raders rading privae informaion ha is ye o be released during he following regular hours. One exension of our analysis is o examine how he preopen variance affecs he volailiies in differen inra-day periods. If he predicive power of he preopen variance comes from he spillover informaion from he preopen period o he regular hours, we can expec he highes impac o occur in he opening hours. If he ime of day affeced appears o be random, i is more likely due o he second conjecure, assuming ha he iming of he privae informaion becoming public is also random hroughou he day. 19

20 Table 1. The Thiry Mos Acively Traded NASDAQ Socks Ticker Company Name Aggregaed Trading Volume (in 100s) MSFT Microsof 173,337,986 SIRI Sirius XM Radio 170,203,390 INTC Inel 166,982,231 CSCO Cisco 139,513,698 ORCL Oracle 114,794,458 SUNW Sun Microsysems 103,675,149 JDSU JDS Uniphase 82,977,355 AMAT Applied Maerials 82,499,633 YHOO Yahoo! 40,999,796 DELL Dell 40,003,828 CIEN CIENA 32,400,220 NXTL Nexel Communicaions 32,299,303 JNPR Juniper Neworks 31,213,448 CNXT Conexan Sysems 30,375,528 QCOM QUALCOMM 27,718,095 BRCM Broadcom 25,715,821 ADCT ADC Telecommunicaions 25,086,690 EBAY ebay 22,397,597 AAPL Apple 21,945,886 SEBL Siebel Sysems 21,940,453 AMZN Amazon.com 21,730,820 LVLT Level 3 Communicaions 21,416,579 BEAS BEA Sysems 21,406,136 AMGN Amgen 21,384,813 VRTS Virus Invesmen Parners 21,106,127 ATML Amel 20,806,448 RFMD RF Micro Devices 20,465,717 BRCD Brocade Communicaions Sysems 20,178,726 RIMM Research In Moion Limied 19,832,282 TLAB Tellabs 18,729,020 The above hiry acive socks have he highes rading aciviies based on he aggregae rading volume in 2004 in NASDAQ. We excluded hree socks, ICGE, TASR and CMCSA, due o heir lack of afer-hours rading daa for a leas hree years in he sample period. 20

21 Table 2. Summary Saisics of Daily Realized Reurn and Volailiy for MSFT Min. Mean Median Max. S. dev. Skew. Kur. Reurn Volailiy Reg. Hour Preopen Posclose Overnigh The realized volailiies are all in percenage erms. Table 3. Average Volume and Volailiy for MSFT Volume (daily) Volume (hourly) Volailiy (hourly) Volailiy (per rade) Reg. Hour Preopen Posclose All repored volumes are in erms of rades, and all repored volailiies are in percenage erms. Table 4. GARCH Model Selecion for MSFT Normal Error Disribuion GARCH(1,1) GARCH(1,2) GARCH(2,1) GARCH(2,2) AIC BIC Likelihood Suden s Error Disribuion GARCH(1,1) GARCH(1,2) GARCH(2,1) GARCH(2,2) AIC BIC Likelihood The GARCH selecion is based on he Akaike informaion crierion (AIC), Bayesian informaion crierion (BIC), and he log-likelihood of he model. 21

22 Table 5. Day GARCH (1,1) Parameer Esimaes of MSFT GARCH (1,1) GARCH (1,1) GARCH (1,1) wih wih Preopen Close-o-Open μ -6.37e e e-4 (5.16e-4) (5.18e-4) (5.29e-4) ω 1.32e e e-6 (1.25e-6) (1.56e-6) (2.02e-6) α 0.065*** 0.066*** 0.068*** GARCH (1,1) wih Posclose GARCH (1,1) wih Overnigh -6.24e-4 (5.21e-4) -6.02e-4 (5.15e-4) 1.47e e-6 (1.32e-6) (1.48e-6) 0.066*** 0.066*** (0.017) (0.018) (0.019) (0.017) (1.75e-2) β 0.932*** 0.922*** 0.903*** 0.932*** 0.923*** (0.017) (0.020) (0.022) (0.017) (0.020) ρ *** (0.064) (0.090) (0.015) (0.057) Degree of Freedom Likelihood ARCH es (P-value) Ljung-Box Tes (P-value) The repored coefficiens are based on quasi-maximum likelihood esimaions of a Suden s GARCH (1,1) model esimaed from he in-sample period: r zh h h x The sandard error is in he parenhesis. ARCH es and Ljung-Box Tes are performed o check for he ARCH effec and auocorrelaion of he residuals. *** denoes significance a a 1% level, ** denoes significance a a 5% level, and * denoes significance a a 10% level. 22

23 Table 6. GARCH Model for Day Reurns wih Nigh Variance In-Sample Resuls Regular Day Hour Close-o-Open (AN) Preopen (PO) Posclose (PC) Overnigh (ON) Name mu alpha bea mu alpha bea rho mu alpha bea rho mu alpha bea rho mu alpha bea rho MSFT Value *** 0.932*** *** 0.922*** *** 0.903*** 0.221*** *** 0.932*** *** 0.923*** SE SIRI Value *** 0.174*** 0.867*** *** 0.150*** 0.839*** 0.200** *** 0.171*** 0.806*** 0.164** *** 0.162*** 0.863*** *** 0.155*** 0.849*** 0.160** SE INTC Value *** 0.954*** ** 0.935*** 0.151** *** 0.089* ** 0.942*** 0.064*** *** 0.940*** 0.104** SE CSCO Value *** 0.958*** *** 0.202** *** 0.929*** 0.155** ** 0.949*** *** 0.144** SE ORCL Value * 0.041*** 0.955*** * 0.041*** 0.953*** * 0.041*** 0.957*** ** 0.027** 0.955*** 0.050** *** 0.950*** SE SUNW Value *** 0.024*** 0.976*** *** 0.022** 0.954*** 0.175*** *** 0.035*** 0.943*** 0.136*** *** 0.024*** 0.976*** *** 0.021** 0.957*** 0.153*** SE JDSU Value *** 0.059*** 0.941*** *** 0.059*** 0.938*** *** 0.045*** 0.933*** 0.096** *** 0.059*** 0.944*** *** 0.060*** 0.939*** SE AMAT Value *** 0.953*** *** 0.430*** * 0.947*** 0.153** *** 0.146*** *** 0.930*** 0.288*** SE YHOO Value *** 0.948*** *** 0.946*** *** 0.920*** 0.163** *** 0.957*** *** 0.946*** SE DELL Value *** 0.932*** *** 0.885*** 0.254*** *** 0.901*** 0.133** *** 0.912*** 0.042** *** 0.907*** 0.186** SE CIEN Value *** 0.070*** 0.935*** *** 0.067*** 0.935*** *** 0.050*** 0.932*** 0.097** *** 0.067*** 0.936*** *** 0.069*** 0.935*** SE NXTL Value *** 0.035*** 0.965*** *** 0.031*** 0.961*** 0.043** *** 0.027** 0.950*** 0.070*** *** 0.031*** 0.961*** *** 0.033*** 0.963*** SE JNPR Value * 0.054*** 0.941*** ** 0.036*** 0.935*** 0.134** * 0.050*** 0.938*** ** 0.036*** 0.944*** 0.073** ** 0.042*** 0.935*** 0.106** SE CNXT Value *** 0.057*** 0.928*** *** 0.045*** 0.936*** *** 0.035*** 0.950*** *** 0.044*** 0.930*** *** 0.053*** 0.930*** SE QCOM Value 0.002** 0.032*** 0.965*** 0.002** 0.032*** 0.967*** ** 0.031*** 0.970*** ** 0.034*** 0.969*** ** 0.032*** 0.965*** SE BRCM Value *** 0.966*** *** 0.948*** ** 0.941*** 0.202** *** 0.966*** *** 0.955*** SE

24 ADCT Value *** 0.044*** 0.951*** *** 0.046*** 0.927*** 0.114* *** 0.029* 0.934*** 0.112** *** 0.044*** 0.945*** *** 0.048*** 0.940*** SE EBAY Value 0.003*** 0.040*** 0.954*** 0.002*** 0.039*** 0.955*** *** 0.044*** 0.924*** 0.141** 0.003*** 0.038*** 0.951*** *** 0.039*** 0.955*** SE AAPL Value *** 0.978*** *** 0.942*** 0.067** *** 0.906*** 0.092** *** 0.918*** *** 0.926*** 0.074* SE SEBL Value *** 0.036*** 0.961*** *** 0.034*** 0.951*** 0.110** *** 0.032*** 0.951*** 0.076** *** 0.026*** 0.967*** *** 0.037*** 0.950*** 0.090* SE AMZN Value 0.003*** 0.029*** 0.968*** 0.003*** 0.026*** 0.966*** *** 0.025*** 0.974*** *** 0.016* 0.967*** 0.052** 0.003*** 0.025*** 0.968*** SE LVLT Value *** 0.123*** 0.882*** *** 0.097*** 0.873*** 0.186** *** 0.113*** 0.859*** 0.131** *** 0.119*** 0.871*** *** 0.096*** 0.882*** 0.151** SE BEAS Value *** 0.045*** 0.955*** *** 0.047*** 0.948*** *** 0.049*** 0.924*** 0.144** *** 0.037*** 0.951*** 0.044** *** 0.045*** 0.954*** SE AMGN Value *** 0.929*** *** 0.921*** *** 0.907*** 0.140** *** 0.929*** *** 0.920*** SE VRTS Value *** 0.953*** *** 0.946*** *** 0.944*** 0.078** *** 0.956*** *** 0.944*** SE ATML Value ** 0.082*** 0.901*** ** 0.081*** 0.897*** ** 0.082*** 0.895*** ** 0.029** 0.934*** 0.093*** ** 0.079*** 0.908*** SE RFMD Value *** 0.068*** 0.927*** *** 0.056*** 0.921*** 0.159** *** 0.036*** 0.930*** 0.120*** *** 0.066*** 0.928*** *** 0.069*** 0.921*** SE BRCD Value *** 0.062*** 0.934*** *** 0.061*** 0.928*** *** 0.067*** 0.920*** *** 0.061*** 0.929*** *** 0.062*** 0.931*** SE RIMM Value *** 0.968*** *** 0.964*** ** 0.942*** 0.093* *** 0.956*** 0.043* *** 0.965*** SE TLAB Value *** 0.966*** * 0.078*** 0.890*** ** 0.048*** 0.916*** 0.118** ** 0.031** 0.952*** 0.045** * 0.036*** 0.965*** SE We use GARCH (1,1) model o compue he sock volailiy from January 2001 o June The regular day hour is based on he GARCH model h p q h i i j j i1 j The GARCH (1,1) day reurns wih nigh variance (AN, PO,PC, or ON) are based on h h x. *** denoes significance a a 1% level, ** denoes significance a a 5% level, and * denoes significance a a 10% level

25 Table 7. Mincer-Zarnowiz Regression Resuls MSFT Value Sd Err -value Adj R2 Day a Day a AN a AN a PO a PO a PC a PC a ON a ON a SIRI Value Sd Err -value Adj R2 Day a Day a AN a AN a PO a PO a PC a PC a ON a ON a INTC Value Sd Err -value Adj R2 Day a Day a AN a AN a PO a PO a PC a PC a ON a ON a CSCO Value Sd Err -value Adj R2 Day a Day a AN a AN a PO a PO a PC a PC a ON a ON a ORCL Value Sd Err -value Adj R2 Day a Day a AN a AN a PO a PO a PC a PC a ON a ON a SUNW Value Sd Err -value Adj R2 Day a Day a AN a AN a PO a PO a PC a PC a ON a ON a JDSU Value Sd Err -value Adj R2 Day a Day a AN a AN a PO a PO a PC a PC a ON a ON a AMAT Value Sd Err -value Adj R2 Day a Day a AN a AN a PO a PO a PC a PC a ON a ON a