Forecasting Asymmetries in Aggregate Stock Market Returns: Evidence from Conditional Skewness

Transcription

1 Forecasing Asymmeries in Aggregae Sock Marke Reurns: Evidence from Condiional Skewness C. James Hueng Wesern Michigan Universiy James B. McDonald Brigham Young Universiy November, 4 Corresponding auhor: C. James Hueng Deparmen of Economics 54 Friedmann Hall Wesern Michigan Universiy Kalamazoo, MI Phone: (69) Fax: (69) James.Hueng@wmich.edu i

2 Forecasing Asymmeries in Sock Reurns: Evidence from Condiional Densiies Absrac This paper provides a ime-series es for he Differences-of-Opinion heory proposed by Hong and Sein (3) in he aggregae marke, hus exending Chen, Hong, and Sein s () crosssecional es for his heory across individual socks. An auoregressive condiional densiy model wih a skewed- disribuion is used o esimae he effecs of pas rading volume on reurn asymmery. Using NYSE and AMEX daa from 96 o, we find ha he predicion of he Hong-Sein model ha negaive skewness will be mos pronounced under high rading volume condiions is no suppored in our ime-series analysis wih marke daa. JEL Classificaion: C5, G. Keywords: differences of opinion; asymmery; skewed- disribuion; auoregressive condiional densiy models. ii

3 Inroducion An exensive lieraure documens he asymmerical disribuion of sock reurns. Several economic heories have been proposed o explain he mechanism generaing his asymmery, including leverage effecs, volailiy feedback mechanism, and sochasic bubbles models. Since he heories all focus on mechanisms in he aggregae, hey could be formulaed as a represenaive-agen model. Hong and Sein (3), however, propose an alernaive heory based on invesor heerogeneiy. Their model assumes ha differences of opinion exis among invesors and ha some invesors face shor-sale consrains. When disagreemen is high, i is more likely ha bearish invesors do no iniially paricipae in he marke, and heir informaion is no fully incorporaed ino prices, because of heir shor-sale consrains. If he marke receives posiive news, bullish invesors informaion is sill revealed in prices, while bearish invesors informaion remains hidden. On he oher hand, if he marke receives negaive news and he previously bullish invesors have a change of hear and bail ou of he marke, hose previously bearish invesors may become he marginal "suppor buyers" and hence reveal more of heir informaion. Thus, accumulaed hidden informaion ends o come ou when he marke is falling. Tha is, condiional on high invesor heerogeneiy, volailiy is higher when he reurn is low, which explains why reurns are negaively skewed. A disincive feaure of he Hong-Sein model no shared by represenaive-agen models is ha i provides esable empirical implicaions. Their model predics ha negaive skewness will be mos pronounced under high rading volume condiions. Specifically, he Hong-Sein model shows ha rading volume rises wih he exen of heerogeneiy among invesors, which is consisen wih lieraure saing ha differences of opinion drive rading volumes [Harris and Raviv (993), Kandel and Pearson (995), and Odean (998)]. Therefore, when disagreemen among invesors is high, rading volume is high. Based on his informaion, we would expec he reurn o be from a more negaively skewed disribuion. See Chen, Hong, and Sein () for a summary of hose mechanisms. 3

4 Moivaed by he Hong-Sein model, Chen, Hong, and Sein () develop a series of cross-secional regressions in an aemp o forecas asymmery in he daily reurns for individual socks. Their baseline measure of asymmery is he sample skewness of marke-adjused reurns over a period of six monhs. The key forecasing variables are he pas urnover (or relaive volume he raio of rading volume o he shares ousanding) and reurns. The esimaed coefficiens from he regressions of negaive skewness on pas urnover and reurns are always posiive and srongly significan. They conclude ha negaive skewness is mos pronounced in socks ha have experienced eiher an increase in urnover over he prior six monhs or high pas reurns over he prior hiry-six monhs. Chen, Hong, and Sein focus mainly on he cross-secional regressions for individual socks and only briefly experimen wih a ime-series regression for he sock marke as a whole, which hey repor would have been a more ineresing pracice. They argue ha he saisical power issue resrics he ime-series analysis. A he individual firm level, here is enough crosssecional daa, so hey compue he sample skewness in daily reurns measured a nonoverlapping six-monh inervals. However, in he ime-series regressions wih aggregae marke daa, he limied sample size would yield a loss in saisical power. This paper uses an alernaive measure of asymmery so ha Chen, Hong, and Sein s empirical work can be exended o a rigorous ime-series analysis. Specifically, insead of using he sample skewness in six-monh inervals, we propose a parameric model in which a parameer ha measures asymmery is changing everyday and depends on he daily available informaion on forecasing variables. Chen, Hong, and Sein assume ha he populaion skewness is fixed wihin a given sixmonh period, bu is ime-varying across six-monh periods. However, he Hong-Sein model does no imply any specific ime horizon during which he populaion skewness is fixed. The choice of six-monh inervals is based on concerns of measuremen errors; i.e., he horizon canno be oo shor because he calculaion of a higher-order sample momen would be srongly 4

5 influenced by ouliers in he sample. In our approach, he populaion skewness changes everyday. We apply an auoregressive condiional densiy (ARCD) model suggesed by Hansen (994) o esimae a condiional parameer ha measures he daily condiional skewness. By specifying he disribuional parameer as a funcion of he forecasing variables, we have sufficien saisical power o es he significance of he effecs of he forecasing variables on asymmery. Our framework builds on a GARCH model wih a flexible parameric disribuion for residuals, where he parameers in he condiional densiy are ime-varying and depend on he forecasing variables. This modeling sraegy permis parameric specificaions for condiional dependence beyond he firs wo momens. To model he variaion in he condiional disribuion beyond he mean and variance, we use Hansen s skewed suden (ST) disribuion o describe he condiional densiy. The ST disribuion is a parsimonious wo-parameer disribuion, bu also a flexible one, able o model no only lepokurosis bu also asymmery. 3 Using NYSE and AMEX daily daa from July 96 o December, we find ha, in conras o he predicion of he Hong-Sein model, higher prior urnover eiher predics a more posiively skewed disribuion for fuure marke reurns, or does no have a saisically significan Numerous mehods exis for esimaing unknown probabiliy disribuions. These include kernel esimaors, semi-nonparameric (SNP) mehods, and flexible parameric forms o capure he daa characerisics such as skewness, ail-hickness, and peakedness. While kernel esimaors and SNP have greaer flexibiliy in modeling an unknown fixed disribuion, hey do no easily lend hemselves o modeling he behavior of higher order momens. Ofen he flexible parameric represenaions have sufficien flexibiliy o represen daa characerisics and faciliae modeling dynamic behavior of he momens hrough heir funcional relaionship wih he disribuional parameers. 3 An alernaive model o deal wih skewness in he disribuion of reurns is he SPARCH model, which uses a mixure of wo disribuions. The SPARCH model, however, does no have significan advanages over our GARCH-ST model. For example, Bekaer and Harvey (997) use a mixure of wo normal disribuions o model he emerging marke reurns. Their SPARCH model has a hree-parameer disribuion. Our ST disribuion only has wo parameers o be esimaed. Besides, heir SPARCH model does no offer a parameer ha direcly measures he asymmery of he disribuion like our model does. In oher SPARCH models, he choice of a weighing scheme beween he wo disribuions is somewha arbirary. 5

6 predicing power on marke skewness, depending on he differen measures of urnover ha we use o proxy for differences of opinion. Therefore, he insignifican resul shown in Chen, Hong, and Sein s ime-series analysis on marke daa is no due o he lack of saisical power because our model provides enough saisical power. We argue ha eiher he predicion of he Hong- Sein model is no suppored in he marke level, or rading volume is simply a bad proxy for opinion divergence. On he oher hand, as a byproduc of our analysis, we also find significanly negaive effecs of prior 36-monh reurns on fuure skewness, consisen wih Chen, Hong, and Sein s empirical sudy. Tha is, when pas reurns have been high, we predic ha he fuure reurn disribuion is more negaively skewed. This resul is suggesed by models of sochasic bubbles [see, for example, Blanchard and Wason (98)]: afer a long period of ime ha he bubble has been building up, a large drop when i pops is expeced. The remainder of he paper is organized as follows. Secion I inroduces he ARCD-ST model. We presen and discuss he empirical resuls in Secion II. Secion III concludes he paper. I. Model The excess reurn is modeled as a GARCH-in-Mean process including lagged reurns in he condiional mean, in an aemp o esimae he zero-mean, serially uncorrelaed residuals: () y = µ θi y i ζ h ε, m i= where y is he daily reurn in excess of he risk-free rae a ime, and h ( = E ε ) is he condiional variance of he error erm ε based on he informaion available a ime -. The lag lengh m is chosen by he Ljung-Box Q ess as he minimum lag ha renders serially uncorrelaed residuals (a he 5% significance level) up o 3 lags from an OLS auoregression of 6

7 y. To keep our model parsimonious, we also remove he lags wih insignifican esimaed coefficiens based on he OLS auoregression. Noe ha he OLS auoregression is used only o deermine he number of lagged reurns included in he condiional mean. The coefficiens in he condiional mean equaion will be joinly esimaed wih he condiional variance equaion using full informaion maximum-likelihood (FIML) esimaion. The condiional variance h follows a GARCH process. I is well documened in he finance lieraure ha sock reurns have asymmeric effecs on predicable volailiies; see among ohers, Glosen, Jaganahan, and Runkle (993) and Bekaer and Wu (). Therefore, we use an asymmeric GARCH model proposed by Glosen, Jaganahan, and Runkle (he GJR model), which is claimed o be he bes parameric model among a wide range of predicable volailiy models experimened by Engle and Ng (993). For a robusness check, we also experimen wih anoher popular asymmeric GARCH model, he EGARCH model, proposed by Nelson (99). In addiion o he asymmeric variance specificaion, we also include a Monday dummy in he GARCH process, as suggesed by Harvey and Siddique (999), who argue ha Mondays are characerized by subsanially greaer volailiy han he oher days of he week. Finally, o conrol he effec of urnover on variance, we also include pas urnover in he GARCH process. The role of urnover in forecasing volailiies is suggesed by he mixure hypohesis in he finance lieraure [see Lamoureux and Lasrapes (99) and Laux and Ng (993)], which saes ha he GARCH effec in daily sock reurns reflecs ime dependence in he process generaing informaion flow o he marke, and urnover proxies for informaion arrival ime. Our asymmeric GARCH model wih he GJR specificaion is as follows: = () h κ α h β ε γ I ε δ TO φ MON, where I = if ε > and I = if ε <, TO is a measure of pas urnover (o be defined laer), and MON is he Monday dummy. Similarly, our EGARCH model is specified as 7

8 8 ( ). ) log( ) log( MON TO h I h h h = φ δ ε γ ε β α κ The relaionship beween he reurn residuals and is condiional variance can be specified as, v h = ε where v is a zero-mean and uni-variance random variable. The specificaion of he disribuion of he sochasic process {v } deermines he disribuion of y. In response o he levels of kurosis found in sock reurn daa, Bollerslev (987) combined a -disribuion and a GARCH (,) model. However, he -GARCH model is sill symmeric and unable o model he skewness in sock reurns. Hansen (994) proposes a skewed suden s (ST) disribuion for v. 4 The densiy funcion of he ST disribuion is wrien as (3) < =, for, for ), ( σ µ λ µ σ σ σ µ λ µ σ σ λ ST v v c v v c v g where < <, < < λ, 4 = λ µ c, 3 µ λ σ =, and Γ Γ = ) ( π c. For deails on he ST densiy, see he appendix in Hansen (994). The ST disribuion is able o model no only lepokurosis bu also asymmery. The parameer conrols he ails and he peak of he densiy and λ conrols he rae of descen of he densiy around v =. Specifically, when λ >, he mode of he densiy is o he lef of zero and he disribuion skews o he righ, and vice-versa when λ <. When λ =, he disribuion is symmeric. 4 Oher candidaes would include he exponenial generalized bea of he second kind (EGB) and ransformaions of normally disribued variables discussed by Johnson (949). Wang e al. () apply he EGB o GARCH models. Hansen, McDonald, and Theodossiou () compare he performance of several asymmeric disribuions wih GARCH models and find similar performance. Thus he ST appears o be an excellen choice for aemping o model dynamic behavior of higher-order momens.

9 In he GARCH lieraure, he condiional disribuion of he sochasic process {v } is simply assumed o be independen of he condiioning informaion a -. The only feaures of he condiional disribuion of y ha depend on he informaion a - are he mean and he variance. Hansen (994) suggess ha condiional dependence should be allowed beyond he firs wo momens. His "auoregressive condiional densiy (ARCD) modeling sraegy is o model he parameers in he condiional densiy funcion as funcions of he elemens of he informaion se so ha he higher momens also depend on he condiioning informaion. We build our model based on Hansen s seup, bu our model differs from Hansen s in he specificaion of he laws of moion for he ime-varying parameers. 5 Hansen conjecures ha since GARCH models make he condiional second momen a funcion of he lagged errors, i is reasonable o believe ha his sraegy could also work well for he oher momens. Therefore, he models he disribuion parameers as quadraic funcions of he lagged error erms. This paper uilizes he heoreical guidance from he Hong-Sein model and he empirical evidence from Chen, Hong, and Sein () o specify he laws of moion for he ime-varying parameers. Tha is, we assume ha ν = ε / h ) follows he ST disribuion (3) wih ime-varying parameers: ( 3. (4) =., ω, = a b TO c RET d h e ω,, exp( ω ),.99 (.99) (5) λ =. 99, ω = a b TO c RET d h e ω,, exp( ω ), 5 Harvey and Siddique (999) use an alernaive mehod o esimae condiional skewness. Insead of modeling he condiional skewness hrough he disribuional parameer as Hansen (994) and we do, hey specify a condiional skewness process ha is similar o he condiional variance process; i.e., he condiional skewness is a funcion of is own pas value and he cube of he pas residuals. Their mehodology, however, is more compuaionally difficul han ours in ha hey firs obain he esimaed condiional momens and hen use hem o compue he disribuional parameers o be used in he likelihood funcion. By using he ST disribuion, hey need o solve wo equaions for wo unknowns in each ieraion of he MLE. Therefore, our approach has a compuaional advanage over he Harvey-Siddique model. 9

10 where he logisic ransformaion ( U L) θ = L is used o se consrains on he exp( ω) parameers. Wih his ransformaion, even if ω is allowed o vary over he enire real line, θ will be consrained o lie in he region [L, U]. The key variable in he specificaions of he ime-varying parameers (4) and (5) is he pas urnover TO. According o he Hong-Sein model, when differences of opinion are large, hose bearish invesors who are subjec o he shor-sales consrain will si ou of he marke and heir informaion is no revealed. This accumulaed hidden informaion ends o come ou during marke declines. Therefore, he Hong-Sein model predics ha negaive skewness in reurns will be mos pronounced afer periods of heavy rading, because rading volume (and herefore urnover) proxies for he differences of opinion. However, he lengh of his informaion accumulaion period is a subjec of debae. A shor informaion accumulaion period may make TO a noisy proxy for differences of opinion. Chen, Hong, and Sein () sugges he use of he average over he pas six-monh period. Therefore, we define TO as he average of urnover raios from dae -5 o dae -. The oher variables in (4) and (5) are included as conrol variables in an aemp o isolae he effecs of urnover. In paricular, Chen, Hong, and Sein () find ha when pas reurns have been high, skewness is forecased o be more negaive. They claim ha reurns as far back as 36 monhs sill have some abiliy o predic negaive skewness. Therefore, we define RET as he average of he reurns from dae -75 o dae -. The selecion of he variables ha affec he disribuion parameers is based on Chen, Hong, and Sein (). This is, however, by no means he complee lis of facors ha influence he asymmery. Our goal is o see wheher pas urnover raio has a significanly negaive effec on asymmery, as suggesed by he Hong-Sein model, in he ime-series seup. Some omied variables ha have impacs on asymmery and are correlaed wih urnover may exaggerae he impacs of urnover. However, due o he lack of heoreical jusificaions in he lieraure on

11 possible impacs of oher variables on asymmery, and o keep our model parsimonious, we only include variables similar o hose used in Chen, Hong, and Sein (). Recall ha λ governs he asymmery of he disribuion. When λ >, he disribuion is posiively skewed. When λ <, he disribuion is negaively skewed. Therefore, he parameer of paricular ineres is b, he effec of he average of prior urnover raios on λ. The Hong- Sein model predics ha b is negaive if he prior urnover proxies he differences of opinion, negaive skewness is more pronounced if prior urnover is higher. On he oher hand, Chen, Hong, and Sein s () empirical evidence also emphasizes ha c is negaive: when pas reurns have been high, skewness is forecased o become more negaive. II. Esimaion Resuls A. Daa and Specificaion Tes We collec daily daa on sock prices, reurns, rading volumes, and shares ousanding for all NYSE and AMEX firms from he CRSP daabank for he period from July 96 o December. The marke reurns and urnover raios are value-weighed based on he marke values of individual socks. The risk-free rae is defined as he hree-monh T-bill raes a daily raes aken from he FRED daabase published by he Federal Reserve Bank of S. Louis. The hisorical marke excess reurns and urnover raios are ploed in Figures and. As expeced, all he values around Ocober 9, 987 are very volaile. As is common pracice in he lieraure, o address he concern ha he enormous daily movemens during Ocober 987 may dominae our inferences, we also show he regression resuls omiing Ocober 987. Since he specificaion of he condiional densiy plays an imporan role in our paper, before esimaing he model we conduc an ou-of-sample condiional densiy forecass evaluaion proposed by Diebold, Gunher, and Tay (998). Their mehod evaluaes he probabiliy inegral ransform series. This series is obained as follows. Firs, he sample is spli ino in-sample and

12 ou-of-sample periods for model esimaion and densiy forecas evaluaion. The daa up o 99 are used as he in-sample observaions and he daa from he las en years of our sample (99- ) are used for ou-of-sample densiy forecass. We use he in-sample observaions o esimae he model and hen freeze he esimaed model. Based on he esimaed model and he available observaions of he variables, he forecas of he nex day s condiional densiy is formed. Afer he nex day s reurn is realized, we calculae he implied CDF value of his reurn in he forecased disribuion. The ime-series of CDFs generaed by his process in he ou-ofsample period is he probabiliy inegral ransform series. This series should follow an i.i.d. U(,) disribuion. Denoe he inegral ransform series as z. In addiion o he sandard ess such as he Kolmogorov-Smirnov es of i.i.d. U(,), Diebold, Gunher, and Tay (998) sugges he use of hisogram of z and he correlograms of z z ), ( z z ), ( 3 ( z z ), and 4 z z ) o ( evaluae he condiional densiy specificaion. Figures 3a-d plos he hisograms of z wih bins for each of he four models we consider: he GJR specificaion and he EGARCH specificaion wih and wihou Ocober 987 observaions. Figure 4a-d shows he correlograms. The dashed lines superimposed on he chars are approximae 95% confidence inervals under he null ha z is i.i.d. U(,). 6 In general, he hisograms are very close o uniform and no buerfly paern is revealed, which shows he abiliy of he ST disribuion in modeling fa ails and asymmery of he disribuion. On he oher hand, he correlograms also do no reveal srong serial correlaions. Mos of he auocorrelaions are wihin he 95% confidence inervals. For hose saisically significan observaions, he auocorrelaions are all below.. Finally, he Kolmogorov-Smirnov es of i.i.d. U(,) fails o rejec he null a he 5% level for wo of he four models. The es saisics from he oher wo models are very close o he 5% criical value. 6 The 95% confidence inervals for he bin heighs in he hisograms are obained from, Mone Carlo simulaions. The 95% confidence inervals for he correlograms are obained by using Barle s formula.

13 Overall, boh he GJR and EGARCH specificaions are able o pick up he dynamics of he momens, and wheher or no o include he Ocober 987 observaions does no change he resuls of he specificaion ess. These findings jusify our specificaion of he condiional densiy. B. Esimaion Resuls Table shows he esimaion resuls for he four models menioned above using he whole sample. Firs of all, he condiional mean and variance equaions fi very well for all four models. All he coefficiens have he expeced signs and are almos all saisically significan a he 5% level. In he condiional mean equaion, he condiional variance has a posiive effec on reurns, which is consisen wih he finance heory ha an asse wih a higher perceived risk would pay a higher reurn on average. In he condiional variance equaion, he signs on he esimaed coefficiens of he GARCH process and Monday dummy are consisen wih he findings by Harvey and Siddique (999), who also use a GJR model and an EGARCH model (wihou he urnover variable) for value-weighed daily reurns on he S&P5 index from 969 o 997. Firs, he GARCH process is highly persisen. Second, he reurn shocks have asymmeric effecs on predicable variance. Negaive shocks end o cause more volailiies han posiive shocks do. Finally, he condiional variance ends o be higher on Mondays and when prior urnover is high. Now urn o he law of moions of he disribuion parameers. The esimaed coefficiens on he lagged parameers are saisically significan and close o one, indicaing ha he dynamic processes of he parameers, and herefore he higher momens, are very persisen. The condiional variance has a posiive and significan effec on λ in he GJR model including Ocober 987 observaions, and on in he EGARCH model excluding Ocober 987 observaions. 3

14 Our focus is on he law of moion for λ. Especially, he effecs of prior urnover raios on λ (b ) are all posiive, a resul ha conradics he predicion of he Hong-Sein model. The posiive effecs are only marginally significan hough, wih P-values equal o.9,.,.5, and.97, respecively in hese four models. On he oher hand, consisen wih he findings of Chen, Hong, and Sein (), prior 36-monh reurns predic a more negaively skewed reurn: he esimaed c are all negaive and saisically significan. This resul can be explained by models of sochasic bubbles. Tha is, higher pas reurns imply ha he bubble has been building up for a while and a larger drop is expeced when i pops. C. Esimaion Resuls from Alernaive Measures of Trading Volume As menioned in he previous secion, urnover raios are a proxy for he differences of opinion, he variable ha acually predics negaive skewness in he Hong-Sein model. The average of prior six-monh urnover raio we use o obain he above resuls may be a noisy and crude proxy for opinion divergence and herefore, we may no provide a fair es for he Hong- Sein model. To address his concern, we experimen wih several measures of urnover raios o check he robusness of our resuls. Firs, he Hong-Sein model predics ha negaive skewness in reurns will be mos pronounced afer periods of heavy rading. However, he lengh of he periods is a subjec of debae. The choice of he six-monh period used by us and Chen, Hong, and Sein () is arbirary. Therefore, in Table, we ry differen informaion accumulaion periods for TO and rerun he esimaions using he GJR specificaion for he whole sample. In paricular, we experimen wih prior one-day (daily), five-day (weekly), -day (monhly), 5-day (annual), and 75-day (36-monh) average urnover raios. As can be seen from Table, he resuls are very consisen across differen definiions of TO. An imporan difference is ha he effecs of prior urnover raios on asymmery are no only posiive, bu also significanly differen from zero a he 5% level when he prior daily, weekly, monhly, and 36-monh average urnover raios are 4

15 used. This finding srenghens our argumen ha Hong and Sein s predicion is no suppored in he aggregae marke. The second es for robusness we perform is suggesed by he empirical rading volume lieraure [e.g., Campbell, Grossman, and Wang (993) and Lee and Swaminahan ()]. Turnover is used as he volume measure in mos previous sudies because i reduces he lowfrequency variaion in volume. However, as can be seen from Figure, i does no eliminae i compleely. Turnover sill has an upward rend in he lae 96 s, in he period beween he eliminaion of fixed commissions in 975 and he crash of 987, and during he marke boom in he 99 s. These rends in he level of urnover may be due in par by echnology innovaion or increased sock marke paricipaion. Lee and Swaminahan () sugges ha he informaion conen of urnover raios be due o ineremporal variaions in normal rading aciviy. Therefore, hey compue he change in urnover raios as a measure of he abnormal rading aciviy. Following heir mehodology, we define he change in urnover raios as he average daily urnover raio over he pas six monhs minus he average daily urnover raio from dae -5 o dae - (approximaely a one-year period from four years ago). The use of he four-year horizon is driven by he empirical fac ha he level of urnover is a very slowly mean revering process. And because of his reason, for comparison, Lee and Swaminahan also compue resuls by using he lagged level of urnover raios from four years ago. Panel (A) of Table 3 compares he esimaion resuls from he GJR model wih hree differen measures of rading volume. The full sample is used. The firs column repeas he resuls of he firs column of Table and he fourh column of Table, which use he average urnover raios from he pas six monh. The second column replaces he level of urnover raios wih he changes in urnover raios over he pas four years. The hird column uses he lagged urnover raios over a six-monh period from four years ago. To conrol he reurn effec, in he hird model we also lag he pas 36-monh average reurn by four years. The resuls are very consisen across hree alernaive measures of urnover, wih only hree noiceable differences 5

16 from he firs model o he second and he hird models. Firs of all, he new definiions of rading volumes do no have a significan effec on he condiional variance anymore. Secondly, he esimaed effecs of pas reurns on marke skewness (c ) are now saisically insignifican. The sign even changes in he lagged-urnover model. Finally, and mos imporanly, he esimaes of b, he effecs of prior urnover raios on marke skewness, are negaive for models wih changes in urnover raios and lagged urnover raios. However, he esimaes are no saisically significan. This observaion is consisen wih he findings of Chen, Hong, and Sein s () ime-series es: using aggregae marke daa, pas urnover raios have a negaive effec on marke skewness, bu is saisically insignifican. Finally, we experimen wih models excluding pas reurns from he dynamics of he disribuional parameers. This experimen is moivaed by he fac ha we find a negaive relaion beween pas reurns and skewness. As suggesed by models of sochasic bubbles, pas reurns pick up periods of overpricing. Therefore, he parameer b can be inerpreed as elling us wheher pas rading volume predics a marke crash afer conrolling for he level of overpricing. Dropping pas reurns allows us o answer he quesion of wheher high pas rading volume picks up when he aggregae marke is overpriced due o increased opinion divergence and is hence more likely o crash. The resuls are shown in Panel (B) of Table3. All he esimaes he effecs of prior urnover raios on skewness are negaive, bu are saisically insignifican. Therefore, even if we do no conrol he effecs of pas reurns on skewness, we sill canno find saisically significan predicing power of pas urnover raios on fuure marke crash. In sum, we find ha higher pas urnover raios eiher predic more posiively skewed marke reurns, or have an insignifican negaive effec on skewness of marke reurns. Tha is, he predicion of he Hong-Sein model ha negaive skewness will be mos pronounced under high rading volume condiions is no suppored in our ime-series analysis wih 6

17 marke daa. The insignificanly negaive effecs of pas urnover on skewness, hough, are consisen wih Chen, Hong, and Sein s ime-series resuls using marke daa. Therefore, heir claim ha heir insignifican resul is due o he lack of saisical power is no legiimae because our model provides enough saisical power. D. Esimaion Resuls from a porfolio Excluding he S&P 5 Socks The Hong-Sein model predics ha, due o shor-sale consrains, negaive skewness will be mos pronounced under high rading volume condiions. However, he lieraure on opions rading argues ha inroducing opions rading can poenially reduce or even eliminae he informaional effec of shor-sale consrains. For example, Figlewski and Webb (993) presen empirical evidence ha rading in opions allows invesors who face shor-sale consrains o ake equivalen opion posiions. Therefore, one possible explanaion for he lack of a negaive relaion beween pas urnover and marke skewness is ha he shor-sale consrain may no be binding a he marke level, given liquid fuure and opion markes. Even hough our focus is on esing he Hong-Sein implicaion a he marke level over ime, i will be ineresing o deviae from he marke analysis and es for his conjecure by analyzing a sub-marke porfolio where shor-sale consrains are more likely o bind. Previous empirical sudies sugges ha small socks wih low insiuional ownership are more expensive o shor han large socks wih high insiuional ownership. Tha is, our resuls using he marke daa may be dominaed by economically imporan socks in he marke porfolio ha are easy and cheap o shor. Therefore, we form a porfolio by excluding he S&P 5 socks from he marke porfolio in an aemp o mimic a porfolio consising of socks ha are hard for invesors o shor. 7 We use his new daa se o repea he ess parallel o hose in Tables 3 and 4 and repor he resuls in Tables 4 and 5. I is shown ha, even if we exclude he S&P 5 socks from 7 On each dae of he sample period, we ook ou he socks ha were in he S&P 5 on ha dae from he marke porfolio we formed in Secion II.A. 7

18 he marke porfolio, our conclusion is no changed pas urnover raios sill have no saisically significan predicing power on porfolio skewness in he ime-series analysis. III. Discussion and Conclusion Chen, Hong, and Sein s () ime-series es on Hong-Sein s (3) Differences-of- Opinion model is no able o generae saisically significan resuls, and hey claim ha i is because of he lack of saisical power. This paper proposes a model ha can be used o conduc his ime-series es wih sufficien saisical power. We build a parameric model in which a disribuional parameer conrols he asymmery of he disribuion. This parameer is modeled as ime-varying and depends on he daily observaion of he forecasing variables. Therefore, daily observaions of skewness are available and we are able o esimae he effecs of he forecasing variables on asymmery. Differen from he cross-secional resuls in Chen, Hong, and Sein, in which he effec of prior urnover on skewness is found o be negaive and herefore suppors he Hong-Sein model, his paper provides ime-series evidence for he aggregae marke ha conradics he predicion of he Hong-Sein model. Higher pas urnover eiher predics more posiively skewed marke reurns or does no have a saisically significan predicing power on marke skewness, depending on he differen measures of pas urnover being used. The resuls of his research, however, do no dispue he cross-secional resuls obained by Chen, Hong, and Sein using individual socks. Insead, we argue ha he insignifican resul shown in heir ime-series analysis on aggregae marke is no due o he lack of saisical power because our model provides enough saisical power. Sharing he same concerns wih hose indicaed in Chen, Hong, and Sein, our ime-series analysis may no be a igh es of he Hong-Sein model. Specifically, i is sill subjec o debae wheher our urnover variables are a good proxy for he inensiy of disagreemen among invesors. The urnover variables may as well capure oher facors ha are no conrolled in our 8

19 parsimonious model, such as changes in rading coss. Noneheless, we believe ha we provide a heoreical framework which yields a ime-series analysis comparable o he cross-secional analysis conduced by Chen, Hong, and Sein. This paper idenifies some issues associaed wih aggregae daa which need o be addressed in he heory of reurn asymmery. The role of he effec of fuure and opion markes on shor-sale consrains in he Hong- Sein model is a opic of ineres for fuure sudies. Since our focus is on he ime-series analysis of he aggregae marke daa, we only briefly analyze a sub-marke porfolio where shor-sale consrains are more likely o bind. Even hough we show ha his experimen does no change our conclusion, i is possible ha our approximaion o he porfolio facing shor-sale consrains is no a good one. On he oher hand, i is also possible ha he effec of he shor-sale consrains disappears in some way when a porfolio is formed. Furher insigh can be gained by analyzing comparaive porfolios ha conain socks wih and wihou fuure and opion rading in boh cross-secional and ime-series analyses. 9

20 Reference: Bekaer, G. and C. R. Harvey, 997, Emerging Equiy Marke Volailiy. Journal of Financial Economics, 43, Bekaer, G. and G. Wu,, Asymmeric Volailiy and Risk in Equiy Markes. Review of Financial Sudies, 3, -4. Blanchard, O.J. and M. Wason, 98, Bubbles, Raional Expecaions, and Financial Marke. In Wachel, P (Ed.), Crises in Economic and Financial Srucure. Lexingon Books, Lexingon, MA, Bollerslev, T., 987, A Condiionally Heeroskedasic Time Series Model for Speculaive Prices and Raes of Reurn. Review of Economics and Saisics, 69, Campbell, J., S. Grossman, and J. Wang, 993, Trading Volume and Serial Correlaion in Sock Reurns. The Quarerly Journal of Economics, 8, Chen, J., H. Hong, and J. C. Sein,, Forecasing crashes: rading volume, pas reurns and condiional skewness in sock prices. Journal of Financial Economics, 6, Diebold, F. X., T. A. Gunher, and A. S. Tay, 998, Evaluaing Densiy Forecass wih Applicaions o Financial Risk Managemen. Inernaional Economic Review, 39, Engle, R. F. and V. K. Ng, 993, Measuring and Tesing he Impac of News on Volailiy. Journal of Finance, 48, Figlewski S. and G. P. Webb, 993, Opions, Shor Sales, and Marke Compleeness. Journal of Finance, 48, Glosen, L. R., R. Jagannahan, and D. E. Runkle, 993, On he Relaion beween he Expeced Value and he Volailiy of he Normal Excess Reurn on Socks. Journal of Finance, 48, Greene, W. H., 3, Economeric Analysis, fifh ediion, New Jersey: Prenice Hall. Hansen, B. E., 994, Auoregressive condiional densiy esimaion. Inernaional Economic Review, 35, Hansen, C. B., J. B. McDonald, and P. Theodossiou,, Flexible Parameric Disribuions for Financial Daa. Brigham Young Universiy. Harris, M., and A. Raviv, 993, Differences of opinion make a horse race. Review of Financial Sudies, 6, Harvey C. R., and A. Siddique, 999, Auoregressive Condiional Skewness. Journal of Financial and Quaniaive Analysis, 34, Hong, H., and J. C. Sein, 3, Differences of Opinion, Shor-Sales Consrains and Marke Crashes. Review of Financial Sudies, 6,

21 Johnson, N. L., 949, Sysems of Frequency Curves Generaed by Mehods of Translaion. Biomerika, 36, Kandel, E., and N. D. Pearson, 995, Differenial inerpreaion of public signals and rade in speculaive markes. Journal of Poliical Economy, 3, Lamoureux, C. G., and W. D. Lasrapes, 99, Heeroskedasiciy in Sock Reurn Daa: Volume versus GARCH Effecs. Journal of Finance, 45, -9. Laux, P. A., and L. K. Ng, 993, The source of GARCH: empirical evidence from an inraday reurns model incorporaing sysemaic and unique risks. Journal of Inernaional Money and Finance,, Lee, C., and B. Swaminahan,, Price Momenum and Trading Volume. Journal of Finance, 55, Nelson, D. B., 99, Condiional Heeroskedasiciy in Asse Reurn: A New Approach. Economerica, 59, Odean, T., 998, Volume, volailiy, price and profi when all raders are above average. Journal of Finance, 53, Wang, K. L., C. Fawson, C. Barre, and J. B. McDonald,, A Flexible Paramerric GARCH Model wih an Applicaion o Exchange Raes. Journal of Applied Economerics, 6,

22 Table : Esimaion Resuls The parameers are shown in equaions ()--(5): () = = m i i i h y y ε ζ θ µ,, v h = ε () GJR MON TO I h h = φ δ ε γ ε β α κ, ( ) EGARCH MON TO h I h h h = φ δ ε γ ε β α κ ) log( ) log(, (3) < =, for, for ), ( σ µ λ µ σ σ σ µ λ µ σ σ λ ST v v c v v c v g (4),, ) exp(. 3.,,, = = e h d RET c TO b a ω ω ω (5),,, ) exp(.99) ( = = e h d RET c TO b a ω ω ω λ. Model GJR EGARCH GJR Excluding /987 EGARCH Excluding / µ (consan) (.9) (.) (.6) (.49) θ (AR) (.) (.) (.) (.) θ (AR) (.9) (.5) (.9) (.) θ 6 (AR6) (.54) (.79) Mean Equaion ζ (GARCH-in- Mean) (.) (.) (.) (.37) The numbers in parenheses are P-values. A value of. indicaes ha he rue value is smaller han.5. All compuaions were performed using GAUSS MAXLIK module wih he BHHH (Bernd, Hall, Hall, and Hausman s) algorihm using he defaul convergence crierion ( -5 ) on he enries of he gradien vecor. The esimaed sandard errors were calculaed robus sandard errors corresponding o resuls summarized in Greene (3, P.5).

23 Table : Esimaion Resuls (coninued) Variance Equaion: Disribuion Parameers: ( ) ( λ ) Model κ (consan) α (GARCH) β (ARCH) γ (asymmery) δ (urnover) φ (Monday) a (consan) b (urnover) c (reurn) d (GARCH) e (lag) a (consan) b (urnover) c (reurn) d (GARCH) e (lag) GJR EGARCH GJR Excluding /987 EGARCH Excluding / E E (.4) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.) (.9) (.) (.8) (.) (.453) (.887) (.8) (.65) (.) (.7) (.69) (.63) (.35) (.46) (.948) (.74) (.54) (.89) (.58) (.8) (.) (.) (.) (.) (.69) (.) (.86) (.95) (.9) (.) (.5) (.97) (.4) (.) (.) (.47) (.3) (.) (.5) (.43) (.) (.) (.) (.) The numbers in parenheses are P-values. All compuaions were performed using GAUSS MAXLIK module wih he BHHH (Bernd, Hall, Hall, and Hausman s) algorihm using he defaul convergence crierion ( -5 ) on he enries of he gradien vecor. The esimaed sandard errors were calculaed robus sandard errors corresponding o resuls summarized in Greene (3, P.5). 3

24 Table : Esimaion Resuls wih Differen Horizons of Pas Turnover GJR model wih full sample Mean Equaion: Variance Equaion: Disribuion Parameers: ( ) µ (consan) θ (AR) θ (AR) Prior Daily Turnover Prior Weekly Turnover Prior Monhly Turnover Prior 6-Monh Turnover Prior Annual Turnover Prior 36-Monh Turnover (.59) (.49) (.54) (.9) (.9) (.8) (.) (.) (.) (.9) (.9) (.9) ζ (GARCHin-Mean) (.9) (.3) (.8) (.) (.) (.) κ (consan) α (GARCH) β (ARCH) γ (asymmery) δ (urnover) φ (Monday) a (consan) 3.7E-7 3.7E-7 3.7E-7 3.7E-7 3.7E-7 3.7E-7 (.6) (.53) (.99) (.4) (.6) (.) (.4) (.3) (.3) (.9) (.6) (.5) (.838) (.8) (.9) (.453) (.46) (.8) b (urnover) (.9) (.7) (.97) (.) (.8) (.) c (reurn) (.95) (.99) (.957) (.35) (.37) (.84) d (GARCH) (.3) (.5) (.37) (.54) (.56) (.4) e (lag) ( λ ) a (consan) (.75) (.75) (.86) (.69) (.68) (.53) See he noes on Table b (urnover) (.37) (.37) (.44) (.9) (.66) (.38) c (reurn) (.4) (.4) (.) (.4) (.3) (.) d (GARCH) (.3) (.3) (.4) (.3) (.5) (.8) e (lag) 4

25 Table 3: Esimaion Resuls wih Differen Definiions of Pas Turnover GJR model wih full sample Mean Equaion: Variance Equaion: Disribuion Parameers: ( ) ( λ ) (A) Including reurns in he dynamics of disribuional parameers (B) Excluding reurns in he dynamics of disribuional parameers wih pas wih wih lagged wih pas wih wih lagged 6-monh changes in Turnover and 6-monh changes in Turnover a Turnover b Reurn c Turnover a Turnover b Turnover µ (consan) (.9) (.4) (.39) (.) (.3) (.8) θ (AR) θ (AR) (.9) (.) (.) (.) (.5) (.5) ζ (GARCHin-Mean) (.) (.4) (.3) (.) (.3) (.) κ (consan) 3.7E-7 3.7E-7 3.6E-7 3.7E-7 3.7E-7 3.6E-7 (.4) (.7) (.) (.987) α (GARCH) β (ARCH) γ (asymmery) δ (urnover) (.) (.6) (.3) (.) (.55) (.346) φ (Monday) (.9) (.) (.) (.) (.) (.3) a (consan) (.453) (.7) (.385) (.696) (.3) (.6) b (urnover) (.) (.5) (.36) (.75) (.76) (.64) c (reurn) (.35) (.37) (.67) d (GARCH) (.54) (.6) (.44) (.6) (.7) (.58) e (lag) a (consan) (.69) (.) (.43) (.3) (.) (.57) b (urnover) (.9) (.59) (.86) (.3) (.66) (.7) c (reurn) (.4) (.73) (.9) d (GARCH) (.3) (.) (.47) (.) (.8) (.63) e (lag) See he noes on Table. a. This is he same model as ha in he firs column of Table and he fourh column of Table. b. This model replaces he level of urnover wih he change in urnover defined as he average daily urnover from dae o dae 5 (pas six monhs) minus he average daily urnover from dae o dae 5 (four years ago). c. This model replaces he level of urnover wih he lagged urnover defined as he average daily urnover from dae o dae 5 (four-year-lagged six-monh horizon). The reurn variable in he disribuional parameer funcions (RET ) is changed o he average daily reurn from dae o dae 75 (four-year-lagged 36-monh horizon) 5

26 Table 4: Esimaion Resuls wih Differen Horizons of Pas Turnover from a Porfolio Excluding S&P 5 Socks GJR model wih full sample Mean Equaion: Variance Equaion: Disribuion Parameers: ( ) ( λ ) µ (consan) θ (AR) θ (AR) θ 3 (AR3) Prior Daily Turnover Prior Weekly Turnover Prior Monhly Turnover Prior 6-Monh Turnover Prior Annual Turnover Prior 36-Monh Turnover (.) (.) (.) (.) (.) (.) ζ (GARCHin-Mean) (.48) (.54) (.66) (.65) (.73) (.4) κ (consan) α (GARCH) β (ARCH) γ (asymmery) δ (urnover) φ (Monday) a (consan) b (urnover) c (reurn) d (GARCH) e (lag) a (consan) b (urnover) c (reurn) d (GARCH) e (lag) See he noes on Table. 3.7E-7 3.7E-7 3.7E-7 3.7E-7 3.7E-7 3.7E (.5) (.9) (.5) (.4) (.4) (.3) (.7) (.59) (.45) (.6) (.8) (.5) (.35) (.47) (.6) (.93) (.97) (.8) (.49) (.47) (.6) (.7) (.75) (.) (.7) (.3) (.3) (.35) (.38) (.) (.389) (.388) (.39) (.46) (.48) (.9) (.58) (.63) (.68) (.575) (.49) (.96) (.9) (.59) (.) (.8) (.8) (.8) (.6) (.33) (.97) (.5) (.8) (.8)

27 Table 5: Esimaion Resuls wih Differen Definiions of Pas Turnover from a Porfolio Excluding S&P 5 Socks GJR model wih full sample Variance Equaion: Disribuion Parameers: ( ) ( λ ) See he noes on Table 3. (A) Including reurns in he dynamics of disribuional parameers wih wih pas wih lagged 6-monh changes in Turnover a Turnover b Turnover & Reurn c (B) Excluding reurns in he dynamics of disribuional parameers wih pas 6-monh Turnover a wih changes in Turnover b wih lagged Turnover µ (consan) θ (AR) θ (AR) θ 3 (AR3) (.) (.3) (.9) (.) (.) (.8) ζ (GARCHin-Mean) (.65) (.6) (.) (.7) (.) (.) κ (consan) 3.7E-7 3.7E-7 3.6E-7 3.7E-7 3.7E-7 3.7E α (GARCH) β (ARCH) γ (asymmery) δ (urnover) (.4) (.66) (.3) (.9) (.76) (.79) φ (Monday) a (consan) (.6) (.645) (.7) (.6) (.) (.78) b (urnover) (.93) (.7) (.4) (.8) (.77) (.) c (reurn) (.7) (.48) (.) d (GARCH) (.35) (.44) (.8) (.) (.) (.5) e (lag) (.) (.) (.) (.) (.) (.) a (consan) (.46) (.5) (.3) (.9) (.34) (.49) b (urnover) (.575) (.696) (.76) (.35) (.69) (.45) c (reurn) (.8) (.38) (.) d (GARCH) (.5) (.54) (.) (.56) (.9) (.) e (lag) 7

28 Figure : Excess reurns (%) [Mean:.54 Sandard Deviaion:.855] (//987) (/9/987) -8 7/3/6 /5/65 8//67 3/4/7 /6/7 5//75 /3/77 6/7/8 //83 8/4/85 /7/88 9//9 4/5/93 /8/95 6//98 /5/ Figure : Turnover raios (%) [Mean:.85 Sandard Deviaion:.3].7. (/9/987) /5/964 3//968 4/8/97 5/4/976 6/3/98 8/5/984 9//988 /7/99 /3/996 /9/ 8

29 Figure 3: Esimae of he Densiy of z (a) GJR model including Ocober 987 (b) EGARCH model including Ocober 987 (Kolmogorov-Smirnov es saisic =.*) (Kolmogorov-Smirnov es saisic =.3*) (c) GJR model excluding Ocober 987 (d) EGARCH model excluding Ocober 987 (Kolmogorov-Smirnov es saisic =.9*) (Kolmogorov-Smirnov es saisic =.6*) See ex for deails of he definiion of he probabiliy inegral ransform series z. * The %, 5%, and % criical values for he Kolmogorov-Smirnov es saisics are.4,.7, and.3, respecively, for (a) and (b), and.4,.7, and.33, respecively, for (c) and (d). 9

30 z z ) ( (a) The GJR model including Ocober 987 observaions Figure 4: Esimaes of he auocorrelaion funcions of powers of z 3 z z ) z z ) ( ( 4 ( z z ) (b) The EGARCH model including Ocober 987 observaions (c) The GJR model excluding Ocober 987 observaions (d) The EGARCH model excluding Ocober 987 observaions