The Kinetics of the Stock Markets

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1 Asia Pacific Managemen Review (00) 7(1), 1-4 The Kineics of he Sock Markes Hsinan Hsu * and Bin-Juin Lin ** (received July 001; revision received Ocober 001;acceped November 001) This paper applies he kinemaic and kineic heories of physics o derive wo imporan price behavior equaions for he sock markes: he equaion of moion and he work-energy equaion. Daily daa of he Taiwan Sock Exchange index, rading values, and oal unis of buy and sale orders are used o es he heories. Empirical resuls indicae ha hese wo equaions provide a powerful and good descripion of he behavior of sock prices. The conribuion of his paper is o formally heorize some phenomena ha are ofen heard from he praciioners of sock markes. Keywords: he equaion of moion of he sock prices; he work-energy equaion of he sock prices; kineic energy of sock markes; excess demand; rading values. 1. Inroducion In he field of modern finance, heoriss have borrowed some imporan ideas from physics, such as random walk, chaos, Brownian moion, ec., o describe he behavior of sock prices. Some phenomenal connecions beween he physical world and he sock markes seem o be ineresing and prevailing. In his paper, again, we will borrow he principles of dynamics in physics o explain he behavior of sock prices. The empirical resuls show ha hese connecions are wonderful and asonishing. Dynamics is concerned wih bodies in moion under he acion of forces. Dynamics may subdivide ino wo branches of sudy: kinemaics and kineics. Kinemaics deals wih only he geomery of moion. Specifically, i deals wih he mahemaical descripion of moion in erms of posiion, velociy, and acceleraion; whereas, kineics considers he effecs of forces on he moion of bodies. The fundamenal properies of force and he relaionship beween force and acceleraion are governed by he Newon s Laws, 1 * Corresponding auhor. Deparmen of Business Adminisraion, Naional Cheng Kung Universiy, Tainan, Taiwan, R.O.C. Tel: Ex , Fax: ** Deparmen of Business Adminisraion, Naional Cheng Kung Universiy, Tainan, Taiwan, R.O.C. 1 The Newon s firs law, law of ineria, describes he naural sae of moion of a free body on which no exernal force is acing. Whereas he Newon s hird law, law of acion and reacion, says ha forces always come in equal and opposiely direced pairs. Forces always occur in pairs; each of hem can no exis wihou he oher. 1

2 especially he Newon s Second Law, which relaes he exernal forces acing on a body o is mass and acceleraion. This relaionship is called he equaion of moion. However, when he forces acing on he body can be expressed as funcions of space coordinaes, he equaion of moion is inegraed wih respec o he displacemen of he body. The resuling equaion represens he principle of work and kineic energy. Like he prices of oher commodiies, changes in sock prices are deermined by he changes in demand and supply of he sock markes. If here exiss an excess demand for a sock, he sock price will go up; conversely, if here exiss an excess supply, he sock price will go down. Thus, he relaionship beween excess demand and price change for socks is jus as he relaionship beween exernal force and acceleraion for bodies. The firs purpose of his paper is o esablish he relaionship beween forces acing on socks and he resuling changes in is sock price by he applicaion of he Newon s Second Law of moion. This relaionship is called he equaion of moion for sock markes. Furhermore, he erminology of kineic energy of he sock markes is ofen heard in he invesmen pracice. However, i seems o be never invesigaed, o our knowledge, in he academic lieraure. The second purpose of his paper is o formally heorize he work-kineic energy relaionship for he sock markes. In addiion o he developmen of hese wo imporan equaions for he sock prices, we also conduc an empirical examinaion o see how well he heories can explain he acual daa. The res of his paper is organized as follows: The second secion is o apply dynamic principle o derive he equaion of moion and he equaion of work and kineic energy for he sock prices. The hird secion describes he empirical mehodology. In secion four, resuls are presened and analyzed. Finally, secion five is conclusions.. Models for he Dynamics of he Sock Markes The models derived in his paper explicily direc aenion o cerain resemblance beween he heoreical eniies of he dynamics of he sock marke and he real physical subjec of he dynamics of a body. We will concepualize he siuaion as involving he use of analogy. Analogies can lead o he formulaion of heories. They are someimes an uerly essenial When he forces acing on he sock are expressed as a funcion of ime, he equaion of moion is inegraed wih respec o ime and yields he impulse-momenum equaion.

3 par of heories. A beer example is perhaps provided by he familiar dynamic meaphor of paricles in financial analysis, wih he behavior of sock prices as random walk, or wih he dynamics of sock prices as he geomeric Brownian moion, and so on. No heory is o be condemned as merely an analogy jus because i makes use of one. The poin o be considered is wheher or no here is somehing else o be learned from he analogy if we do choose o draw i. The derivaion of he models in his paper are based on he following assumpions: A1: The dynamic of sock prices for a sock is similar o he dynamic of a paricle subjec o exernal forces. The following analogies are used, wih shares of a sock as he mass of a paricle, he sock price as he disance of he movemen of a paricle, he change in excess demand for a sock as he exernal force exering on a paricle, he rading values of a sock as he kineic energy. A: The excess demand for a sock is measurable (or wih cerainy) raher han sochasic. In he case of sochasic excess demand for a sock, he behavior of sock prices is also sochasic. A3: No price limis are imposed in he sock marke. A4: There is a fricion force for a sock, so ha he disappearance of he exernal force (excess demand) exering on a sock makes he sock suck and he sock price unchanged..1 Kinemaics of Sock Markes Kinemaics is he sudy of he geomery of moion. I deals wih he mahemaical descripion of moion in erms of posiion, velociy, and acceleraion. Kinemaics serves as a prelude o dynamics. In his subsecion, we will concern only wih ranslaional moion of sock price, which is defined as changes in sock price as a funcion of ime. I is convenien o view a sock as an ideal paricle. 3 In his case, prices as a funcion of ime gives a complee descripion of he kinemaics of socks. Le S be he sock price a ime. The insananeous velociy, V, of sock price a a given ime is he ime rae of change of price a ime. The 3 An ideal paricle is a body wih no size and no inernal srucure. Viewing a sock as an ideal paricle is implicily defined in financial heories, such as random walk, Brownian moion, ec. o describe he behavior of sock prices. Alhough an ideal paricle has no size and no inernal srucure, i has mass. Similarly, a sock has is mass, which is he shares ousanding. When an exernal force acs on he sock, he moion of a sock (i.e., price change) will occur. 3

4 insananeous acceleraion, a, of sock price is he ime rae of change of velociy of sock price a a given ime. The mahemaical expressions for velociy and acceleraion of sock price are and ds V = d dv a = = d d S d When he velociy V of he sock price is consan, hen he moion is called he uniform moion. Rearranging equaion (1), we have ds = Vd Inegraing boh sides, and seing he iniial sock price o be S 0, we obain (1) () S d ζ = S0 0 Vd τ (3) From his, we have S = S V (4) When he acceleraion, a, of he sock price is consan, hen he moion is referred o as he uniform acceleraed moion. Rearranging equaion (), we have dv = ad Inegraing boh sides, and seing he iniial velociy of sock price o be V o, we obain V d ν = V0 0 ad τ (5) or V = V 0 + a (6) In he case of consan acceleraion of sock price, he posiion of sock price can be deermined by subsiuing (6) ino (3). Tha is, S a S τ ) dζ = Vdτ = V + ( dτ 4

5 or 1 S = S 0 + V 0 + a (7) The relaionship beween he velociy V and he posiion (price) S, wihou direc reference o ime, can be obained as follows: Rearranging equaion (), we wrie dv dv ds dv a = = = V d ds d ds or ads = VdV Inegraing boh sides, we have S S adζ = V V 0 0 νdν or a( S S V 1 ) = ( V 0 V0 0 ) = V + a( S S ) (8) 0. Kineics of Sock Markes: Newon s Second Law of Moion In his subsecion, we will consider he effecs of forces on he moion of sock price. We will esablish he relaionship beween he forces acing on he sock and he resuling change in he moion of sock price by direcly applying he Newon s Second Law of moion. This relaionship is called he equaion of moion. Like oher commodiies, sock price is deermined by boh he demand and supply of he sock marke. Equilibrium price is reached when demand equals supply. However, if here exiss an excess demand, hen he sock price will go up. Conversely, if here exiss an excess supply (or negaive excess demand), hen he sock price will go down. The poin is ha i is he excess demand causing he sock price o move. 4 This relaionship is like he 4 Of course, here mus be some facors (e.g., macroeconomic, indusrial, and company-specific facors) affecing a change in excess demand for a sock. However, hese inrinsic facors are no our ineres in his paper. A his momen, his conclusion is based on he basic economic principles. Laer on we will poin ou ha he excess demand a he ime of sock marke 5

6 exernal forces causing a body o move. The fundamenal properies of force and he relaionship beween force and acceleraion are governed by he Newon s hree laws of moion, especially by he Newon s Second Law. The Newon s Second Law esablishes he relaion beween he force, F, acing on a body and acceleraion, a, caused by his force. Mahemaical expression of he Newon s Second Law is as follows: F a = (9) m where m is he mass of he body. Newon s Second Law is usually called he equaion of moion. In sock markes, he excess demand acs as he exernal force ha causes he sock price o change. In his paper, he excess demand, ED, is defined as he oal unis of buy orders minus he oal unis of sale orders for a sock. Since some buy and sale orders are ineffecive orders, 5 he excess demand a he ime of sock marke opening in each rading day usually provides lile explanaion power. This means ha a negaive excess demand a he opening of sock marke does no necessarily imply ha he sock price will go down. However, changes in excess demand in he subsequen rading ime do provide explanaory power. 6 An increasing excess demand will follow an up of he sock price, and vice versa. From his discussion, change in excess demand, ΔED, will be used hereafer as exernal force acing on he sock price. Thus, according o he Newon's Second Law of moion, we obain he following equaion of moion for he sock markes. a = 1 ED (10) m where a is he acceleraion of sock price, and m is he mass of he sock. Equaion (10) saes ha he acceleraion of sock price is proporional o he change in excess demand and inversely proporional o he mass of opening in each rading day does no necessarily imply ha sock price will change in he same direcion; however, changes in excess demand do provide explanaory power. Therefore, laer on change in excess demand will be used as he exernal force acing on he sock price. 5 A buy order is no an effecive order if he price specified by he order is lower han he prevailing marke price. Similarly, a sale order is no an effecive order if he price specified by he order is higher han he prevailing marke price. 6 Again, par of changes in excess demand are ineffecive and par of changes in excess demand are effecive. We assume ha he raio of effecive changes in excess demand o ineffecive changes in excess demand keeps consan. 6

7 he sock. Thus, oher hing being equal, we expec ha he more he change in excess demand, he larger he acceleraion of sock price; he smaller he size of he sock, he larger he acceleraion of sock price..3 Kineics of Sock Markes: Work and Energy When he force acing on he sock can be expressed as funcions of space coordinaes, he equaion of moion for he sock marke is inegraed wih respec o he displacemen of he sock price. The resuling equaion represens he principle of work and kineic energy for he sock markes. The work done by a force on he sock price is defined as he force imes he displacemen of he sock price in he direcion of he force. The expression for he infiniesimal work, dw, done by he force, F, on he sock during is infiniesimal displacemen, ds, is defined as dw = FdS (11) The oal work W done by he force F on he sock price, as i moves beween poins A and B is defined as W = B FdS A (1) Rewrie he Newon s Second Law of moion as dv F = ma = m d (13) Subsiuing equaion (13) ino equaion (1), yields dv = B B W m Vd = mv dv A d A or 1 1 W = mv B mv A (14) Equaion (14) is he equaion of work and kineic energy for he sock 1 markes. Noe ha mv is he kineic energy (KE). The equaion (14) says ha he work done on he sock price is equal o he change in kineic energy, or B KE = FdS (15) A 7

8 If F is consan, hen equaion (15) becomes KE = F( S S ) = F S (16) B A Subsiuing ΔED for F in equaion (16), yields KE= ED S Usually, ED is proporional o S, and has he same direcion as S. Therefore, an increase of he change in excess demand will follow an increase of he kineic energy of he sock markes; and vise versa..4 Kineics of Sock Markes: Impulse and Momenum When he forces acing on he sock are expressed as a funcion of ime, he equaion of moion for he sock marke is inegraed wih respec o ime and yields he impulse-momenum equaion for he sock markes. Assuming ha he mass of a sock is consan, equaion (13) can be arranged o read d F = (mv ) (17) d where mv is called he momenum of he sock markes. This equaion saes ha he resulan force acing on a sock is equal o he ime rae of change of he momenum. Muliplying boh sides of equaion (17) by d and inegraing beween ime 1 and, yields 1 Fdτ = V V 1 d( mν ) = mv mv 1 (18) The inegral Fd is called he impulse acing on he sock. Equaion (18) 1 saes ha he change of momenum of he sock marke equals he impulse of he exernal forces. 3. Empirical Mehodology 3.1 The Daa Because he daa for buy and sale orders of individual socks are unavailable, we use he Taiwan Sock Exchange (TSE) index o implemen an empirical sudy. Variables in his sudy include daily rading values of he TSE, oal unis of buy orders, oal unis of sale orders (1 uni = 1,000 8

9 shares), and he indexes. The daily daa for rading values and he TSE indexes are colleced from he EPS/AREMOS daa base. The daily daa for oal unis of buy orders and oal unis of sale orders for all socks lising in TSE are obained from he Taiwan Sock Exchange. The daa covers a sevenyear period, December 8, March 31, A oal of,350 rading days are included. 3. Empirical Mehodology For he pracical ineres, we will es wo principles: he equaion of moion for he sock markes and equaion of work and kineic energy for he sock markes. A. Tes of he equaion of moion for he sock markes In he previous secion, we menion ha he relaion of change in excess demand o he acceleraion of sock prices is jus as he relaion of exernal forces o he acceleraion of a body. The excess demand for socks a ime τ is defined as he difference beween he oal unis of buy orders a τ and he oal unis of sale orders a τ. i.e., ED(T,τ) = TB(T,τ) - TS(T,τ) (19) where ED(T,τ) represens he excess demand a ime τ on dae T, TB(T,τ) represens he oal unis of buy orders a ime τ on dae T, and TS(T,τ) represens he oal unis of sale orders a ime τ on dae T. Nex, he change in excess demand on dae T is defined as he difference in he excess demand beween he opening ime of he sock marke and he close ime of he sock marke on dae T. Tha is ED(T,9:00am) = TB(T,9:00am) - TS(T,9:00am) ED(T,1:00noon) = TB(T,1:00noon) - TS(T,1:00noon) and ΔED(T) = ED(T,1:00noon) - ED(T,9:00noon) (0) Like oher commodiies, sock price is deermined by he change in demand and supply of he sock marke. If here exiss a change in excess demand, hen he sock price will go up. Conversely, if here exiss a change in excess supply, hen he sock price will go down. Le ΔS(T) represen he change in sock price beween dae T and dae T-1, i.e., ΔS(T) = S(T) - S(T-1) (1) From equaion (7), and noing ha V 0 = 0 and a = 0 a he opening ime of a 9

10 sock marke in each rading day, 7 we obain he price change in sock a dae as follows: or where S S 1 = a 0 τdτ 1 S = a 1 () a is he acceleraion of sock price a dae. Subsiuing he Newon s Second Law (equaion (9)) ino (), yields 1 S( T ) = ED( T ) (3) m To es he equaion of moion for he sock markes, equaion (3), he following regression equaion is used 8 S = α + β ED + u (4) If he heory is correc, hen α= 0, and β 0. Because he daa for excess demand of individual socks are unavailable, we use he aggregae of oal excess demand daa for all socks in TSE o es his principle and predicion. B. Tes of he Equaion of Work and Kineic Energy for he Sock Markes The principle of work and kineic energy for he sock markes saes ha he work done by exernal force (i.e., change in excess demand) on he sock marke is equal o he change in kineic energy. Firsly, he work done by he exernal force a dae T is W(T) =ΔED(T) ΔS(T) (5) Nex, he oal rading value of TSE a dae T is used as a surrogae of he kineic energy of sock marke. The daily rading value is also he change in kineic energy a dae T because he rading value a he open of sock marke is zero. Therefore, he equaion of work and kineic energy for he 7 Equaion (7) represens he case of coninuous moion. However, in he case of sock marke, sock price changes sar from he opening of a sock marke in each rading day. Therefore, a he opening of he sock marke, V(, 9:00am) = V 0 = 0 and a = 0. 8 From equaion (7), i seems ha here are wo facors, ΔCD(T) and m, affecing he sock price change, ΔS(T). However, in he empirical es, we only se one explanaory variable, ΔCD, in regression (4) for he following reasons: one is he fac ha he oal mass of he whole sock marke (i.e., he oal shares ousanding of socks lised in he TSE) in a shor period keeps almos unchanged. The oher is ha he magniude of will reflec he magniude of m. is inversely relaed o m. 10

11 sock marke is ΔKE(T)=ΔED(T) ΔS(T) (6) To es he adequacy of he equaion of work and kineic energy, he following regression equaion is applied ln( KE ) = α + β ln ED + γ ln S + ε (7) The absolue values of ΔED and ΔS is required and correc because: (a) he definiion of logarihmic funcions requires nonnegaive values as is domain, and (b) ΔED and ΔS are in he same direcions (i.e., same signs). If he rading value a dae T can ruly represen he change in kineic energy, hen, in heory, α = 0, β = 1, and γ= 1. However, if he rading value a dae T is only a proxy, hen β and γ are no necessarily equal o 1. C. Economeric consideraions Because regression equaions (4) and (7) are used for esing he adequacy of he wo heories. The ime series of daa may violae he assumpions of regression equaions. Thus, he following seps are implemened for beer esimaion and esing. 1. Tes of he sabiliy of regression parameers Because he daa covers a long period of ime, he parameers of regression equaions may change. To consider his possibiliy, he dummy variable echnique is used for esimaion in differen marke condiions (bull and bear). For his purpose, he whole sample period is divided ino hree subperiods by wo ime poins, Ocober 1, 1990 and July 31, 1997, on which a bull marke and a bear marke are roughly divided. The firs and hird periods belong o bear markes and he second period is a bull marke.. Tes of homoscedasiciy Because rading values and sock volailiies in differen marke condiions are likely o be differen, a es of homoscedasiciy of regression residuals is needed. The Glejser mehod 9 is used for his purpose. 3. Tes of auocorrelaion To deec he auocorrelaion of regression equaion, he Durbin- 9 See Glejser [4] (pp ) for he mehod. 11

12 Wason es is used. If presence of auocorrelaion is found, hen he Cochrane-Orcu procedure is used for esimaion. 4. Uni-Roo Tes To es wheher he ime series daa is saionary for regression, he augmened Dickey-Fuller (ADF) es is used. If he daa exiss an uni roo, hen he daa is no saionary and he difference-saionary procedure is used. 5. OLS vs. GARCH Esimaion Finally, if he regression residuals show an exisence of auoregressional condiional heeroscedasiciy (ARCH), hen he GARCH mehod of Bollerslve (1986) is used for he esimaion; oherwise, OLS mehod is used for esimaion. 4. Resuls and Analysis 4.1 Tes of he Equaion of Moion for he Sock Markes For convenience of reading, we repea he equaion of moion for he sock markes as S = α + β ED + u (4) Table 1 shows he regression resuls for he esimaion of he coefficiens of he equaion of moion. In general, all β s are highly saisically significan a.005 level (or beer) for he whole period and each subperiods. Adjused R-square shows ha he variabiliy of ΔED (changes in excess demand) could explain abou 40% variabiliy of ΔS (sock price change). This resul is consisen wih Hsu [5]. Values of β s range from o wih average value of abou This means ha an uni increase in excess demand will raise he TSE index abou poins. Since he values of β in differen marke condiions (bull and bear) and in differen subperiods are differen, he effecs of one uni change in excess demand on he change in TSE index in differen periods are differen. I is ineresing o menion ha he effec of one uni change in excess demand in period 1990/01/ /10/01 is wice more han hose of oher periods. This resul can be explained as follows: Firs, changes in excess demand for socks in differen periods are likely o differ because of differen business cycles for individual indusries and individual socks. Therefore, 1

13 some socks are likely o be acively raded in one period and oher socks are likely o be acively raded in anoher periods. Second, he "masses" of individual socks are differen, and he daa for excess demand are aggregae daa. If he disribuion of percenages of change in excess demand among individual socks varies over ime, hen β should be changed over ime. There is sill a hird explanaion ha he fricion in differen marke rends is differen. For example, in a rend of sharply rising in sock price, bid prices in buy orders are likely o be much higher han he prevailing marke price. On he oher hand, in a rend of sharply sliding in sock price, offer prices in sale orders are likely o be much lower han he prevailing marke price. Thus, he sensiiviy of sock price change o he change in excess demand in differen marke rends is differen. 10 Table 1 Summary for he Regressional Resuls for he Equaion of Moion for Sock Marke (Unresriced) Regression Coefficiens Periods R α β The Whole Period 90/01/04-98/03/31 Subperiods 90/01/04-90/1/7 91/01/03-91/1/8 9/01/04-9/1/9 93/01/05-93/1/31 94/01/05-94/1/31 95/01/05-95/1/30 96/01/04-96/1/31 97/01/04-98/03/31 Bear-marke Period 90/01/04-90/10/01 Bull-marke Period 90/10/0-97/07/31 Bear-marke Period 97/08/01-98/03/ ( ) ( )*** ( ) ( ) (.99779)*** ( )*** ( )** (.45863)* ( ) ( )* ( ) ( ) ( )*** ( )*** ( )*** ( )*** ( )*** ( )*** ( )*** ( )*** ( )*** ( )*** (3.8808)*** ( )*** Noe: 1. Figures in parenheses ( ) are -values. * significan a 0.05, ** significan a 0.01, *** significan a or beer. 10 We hank an anonymous referee for poining ou his explanaion. 13

14 In heorem, regression equaion (4) should have no inercep (by he Newon's Second Law). From able 1, for he whole period, α is no saisically significan. For he hree bull/bear periods, only he firs period has significan α. However, for some years, αs are saisically significan. Table shows he resuls for he regression wih resricion on α (=0). From ables 1 and, i is seen ha he corresponding βs are almos he same regardless he regression is resriced or no. Table Summary for he Regressional Resuls for he Equaion of Moion for Sock Markes (Resriced) Periods β -value R The Whole Period 90/01/04-98/03/ *** Subperiods 90/01/04-90/1/ *** /01/03-91/1/ *** /01/04-9/1/ *** /01/05-93/1/ *** /01/05-94/1/ *** /01/05-95/1/ *** /01/04-96/1/ *** /01/04-98/03/ *** Bear-marke Period 90/01/04-90/10/ *** Bull-marke Period 90/10/0-97/07/ *** Bear-marke Period 97/08/01-98/03/ *** Noe: * significan a 0.05, ** significan a 0.01, *** significan a or beer. A. Model Modificaion 1. Tes of he sabiliy of regression parameers Dummy variable echnique is used o es he sabiliy of regression coefficiens as follows: S ED D D D ED D ED + U = α + β + γ 1 + δ + ζ 1 + η D 1 = D = 1 for 90/10/0 97/07/31 0 oherwise 1 for 97/08/01 98/03/31 0 oherwise 14

15 The esimaed regression equaion is S = ED D D (-4.76) *** (9.08) *** (4.833) *** (.619) ** D 1 ED D ED R =0.48 ( ) *** ( ) *** From above equaion, i is seen ha a srucural change exiss. Therefore, he esimaed regression equaions for he hree bull/bear markes should be differen. The resuls are as follows: For period 1990/01/ /10/31 S1 = ED For period 1990/10/0-1997/07/31 S = For period 1997/08/ /03/ ED S3 = Tes of heeroscedasiciy ED3 Using Glejser mehod for esing heeroscedasiciy of he above regression residuals, U. Tha is, U = α + β + µ ED H 0 :β = 0 The esimaed resuls are 1 U 1 = ED heeroscedasiciy 1 (16.63) *** (-.09) * U = ED homoscedasiciy (47.305) *** (-0.603) U 3 = ED homoscedasiciy 3 (87.037) *** (1.35) 15

16 1. Figures in parenheses ( ) are -values. *: significan a 0.05, **: significan a 0.01, ***: significan a or beer From hese resuls, we conclude ha heeroscedasiciy is exised in he firs period 3. Tes of auocorrelaion Firs, Durbin-Wason es is applied o he above regression. The D-W values for he hree subperiods are 1.595, 1.8, and 1.98, respecively. Then, he Cochrane-Orcu procedure is used o correc auocorrelaion. The final D-W values are.01,.004, and 1.981, respecively. This shows ha auocorrelaion for he hree regression no longer exis. Finally, he Glejser mehod is applied again, he heeroscedasiciy also disappears. 4. Uni-roo es Since all ADF- values are greaer han heir Mackinnon criical values, he null hypohesis ha he ime series daa has uni roo is rejeced. Therefore, he daa for regression is saionary. B. Price Predicion Using he Equaion of Moion Firs, four ime-lengh periods (1-year, -year, 3-year, and 4-year periods) of daa are used for esimaing he parameers of he equaion of moion for he sock markes (Eq.(4)) o obain four esimaed regression equaions. Then, each esimaed regression is used o predic he daily sock indexes of four subsequen periods (1-monh, -monh, 3-monh, and 4- monh lenghs), respecively. For each esimaion period and is subsequen predicion period, he process is rolling over. For example, for one-year esimaion period and one-monh predicion period, period January December 1990 daa are firs used for esimaion of regression funcion. Then daily sock indexes in January 1991 are prediced. Afer his, period February 1990-January 1991 daa are used for esimaion and hen daily sock indexes in February 1991 are prediced... and so on. The prediced values are compared wih he acual sock prices, and hen percenage of predicion errors (=(acual index prediced index)/ prediced index) are calculaed. Table 3 repors he resuls of predicion errors (in percenage) by using OLS. Panels A hrough D separae he resuls for he four esimaion periods. Columns hrough 5 give he resuls for he four predicion periods. From Table 3, we find ha maximum predicion errors range from.48% (for he 3- year esimaion period, 6-monh predicion period) o 1.3% (for he 3-year esimaion period, 1-monh predicion period). Average predicion errors 16

17 range from % (for he 4-year esimaion period, 1-monh predicion period) o.0501% (for he 3-year esimaion period, -monh predicion period). All predicion errors are no saisically significanly differen from zero. Table 3 Predicion Errors of he Equaion of Moion for Sock Markes (OLS Mehod) Predicion Period One Monh Two Monh Three Monh Six Monh Panel A: One-Year Esimaion Period Maximum 1.194% 1.053% 1.040% 0.766% Minimum 0.001% 0.008% % 0.05% Mean % % % % Variance value p-value Panel B: Two-Year Esimaion Period Maximum 1.068% 0.807% 0.618% -0.54% Minimum % % 0.08% % Mean 0.030% % % 0.030% Variance value p-value Panel C: Three-Year Esimaion Period Maximum 1.34% 0.950% 0.787% 0.477% Minimum % 0.09% 0.001% % Mean % % % % Variance value p-value Panel D: Four-Year Esimaion Period Maximum 0.948% 0.767% 0.688% 0.667% Minimum 0.014% % 0.001% % Mean % % % % Variance value p-value In general, he longer he esimaion period and he shorer he predicion period, he more accurae for he predicion of sock prices. Using he daa of 4 years o esimae he equaion of moion for he sock markes gives he bes predicion (predicion errors range from % o %). 17

18 Table 4 gives he resuls of predicion errors by using GARCH. In general, using GARCH model o esimae gives more accurae resuls han by using OLS mehod. However, for he GARCH mehod, he -year esimaion period gives he bes resuls. In his case, he longer he predicion period, he more accurae for he predicion. Table 4 Predicion Errors of he Equaion of Moion for Sock Markes (GARCH Mehod) Predicion Period One-Monh Two-Monh Three-Monh Six-Monh Panel A: One-Year Esimaion Period Maximum 1.169% 1.048% 1.035% 0.761% Minimum % 0.006% 0.007% -0.0% Mean 0.03% 0.08% 0.030% 0.03% Variance value p-value Panel B: Two-Year Esimaion Period Maximum 1.054% 0.695% 0.586% 0.440% Minimum 0.016% % 0.001% % Mean % 0.001% % % Variance value p-value Panel C: Three-Year Esimaion Period Maximum 1.143% 0.779% 0.593% % Minimum 0.013% 0.016% % % Mean % 0.033% % 0.007% Variance value p-value Panel D: Four-Year Esimaion Period Maximum 0.776% % 0.49% 0.461% Minimum 0.013% 0.016% 0.000% % Mean % % % % Variance value p-value

19 4. Tes of Equaion of Work and Kineic Energy For convenience of reading, we repea he equaion of work and energy as follows ln( KE ) = a + b ln ED + c ln S + ε (7) Table 5 shows he preliminary resuls of he above regression, which seem no o give a saisfacory explanaion because of low R s (ranging from.034 o.4). A furher model modificaion is as follows: Table 5 Regression of Equaion of Work and Kineic Energy (Eq.(7)) Periods Regression Coefficiens a b c R The Whole Period 90/01/04-98/03/31 Subperiods 90/01/04-90/1/7 91/01/03-91/1/8 9/01/04-9/1/9 93/01/05-93/1/31 94/01/05-94/1/31 95/01/05-95/1/30 96/01/04-96/1/31 97/01/04-98/03/31 Bear-marke Period 90/01/04-90/10/01 Bull-marke Period 90/10/0-97/07/31 Bear-marke Period 97/08/01-98/03/ (53.400)*** (1.1596)*** ( )*** (.3367)*** ( )*** (37.587)*** ( )*** ( )*** ( )*** ( )*** ( )*** (44.137)*** (14.13)*** ( )*** ( )*** ( ) ( ) ( )*** ( )*** ( )*** ( )*** ( )*** ( )*** ( )* ( ) (.13803)* (.1888)* ( )*** ( )*** ( ) ( )*** ( )*** ( )** ( ) ( )*** (-0.364) Noe : 1. Figures in parenheses ( ) are -values. * significan a 0.05, ** significan a 0.01, *** significan a or beer 19

20 A. Model Modificaion 1. Tes of he sabiliy of regression parameers Dummy variable echnique is used o es he sabiliy of regression coefficiens as follows: ln( KE ) = a + b ln ED + c ln S + dd1 + ed + fd1 + gd S id ln S ln ED + hd1 + + θ ln ED D 1 = D = 1 for 90/10/0 97/07/31 0 oherwise 1 for 97/08/01 98/03/31 0 oherwise The esimaed regression equaion is ln( KE ) = ln ED ln S D 1 (18.645) *** (3.855) *** (-0.841) (-.550) * + ln D D1 ln ED D ED (.613)** (0.511) (-1.788) + ln D1 ln S D S (3.41) *** (0.50) Noe : 1. Figures in parenheses ( ) are -values. * significan a 0.05, ** significan a 0.01, *** significan a or beer From above equaion, i is seen ha a srucural change exiss. Therefore, he esimaed regression equaions for he hree bull/bear markes should be differen. The resuls are as follows: For period 1990/01/ /10/31, ln( KE 1) = ln ED ln For period 1990/10/0-1997/07/31, ln( KE S ) = ln ED ln 1 S 0

21 For period 1997/08/ /03/31, ln( KE 3) = ln ED ln. Tes of heeroscedasiciy Using Glejser mehod for esing heeroscedasiciy of he above regression residuals, θ. Tha is, θ = a + b ln ED + c ln S + ε S 3 H 0 :a = b = 0 The esimaed resuls are i.e., homoscedasiciy θ 1 = ln ln ED S1 (3.566) *** (-1.96) (0.616) homoscedasiciy θ θ = ln ED ln 51 S (10.084) *** (-3.65) *** (-.803) ** 3 = ln ED ln 87 S 3 heeroscedasiciy homoscedasiciy (3.681) *** (-1.537) (0.338) Noe : 1. Figures in parenheses ( ) are -values. * significan a 0.05, ** significan a 0.01, *** significan a or beer From hese resuls, we conclude ha heeroscedasiciy is exised in he second period. 3. Tes of auocorrelaion Firs, Durbin-Wason es is applied o he above regression. The D-W values for he hree subperiods are.354,.36, and.47, respecively. Then, he Cochrane-Orcu procedure is used o correc auocorrelaion. The final D-W values are.164,.41, and.43, respecively. This shows ha auocorrelaion for he hree regression no longer exis. Finally, he Glejser mehod is applied again, he heeroscedasiciy also disappears. 4. Uni-roo es Since all ADF- values are smaller han heir Mackinnon criical values, he null hypohesis ha he ime series daa has uni roo is acceped. Therefore, he daa for regression is none saionary. This implies he firs 1

22 differences for he daa should be aken. B. Resuls and Analysis Afer he modificaion and adjusion of he original regression equaion, hough here sill exiss an indecisive auocorrelaion in he second period regression, he basic assumpions regarding all he regression equaions are, in general, accepable. Furhermore, afer aking he firs-order difference of he regression variables, he daa for regression is saionary and he explanaion powers for all regressions are highly improved (wih adjused R =.90 for he whole period). Table 6 shows he regression resuls for he equaion of work and kineic energy afer aking he firs-order difference of regression variables. A high explanaion power probably implies ha he correc work-energy principle for he sock markes is ha he variables should be expressed as percenages. Tha is, The change in kineic energy (i.e., percenage change in rading values) is equal he work done (i.e., he absolue value of percenage change in excess demand imes he absolue value of percenage change in sock prices). Mahemaically, ln( KE ) ln( KE 1) = a + b(ln ED ln ED 1 ) + c(ln S ln S 1 ) or ln( KE / KE 1) = a + b ln( ED / Ed 1 ) + c ln( S / S 1 ) (8) Table 6 Summary of Regression for he Equaion of Work and Kineic Energy (Afer aking he firs-order differences for variables in Eq.(8)) Coefficien â bˆ ĉ F Adj-R The Whole Period 90/01/04-98/03/ (98.17)*** (6.0165)*** (10.338)*** Subperiods 90/01/04-90/10/ (33.706)*** (-1.81) (1.417) /10/0-97/07/ (97.81)*** (6.954)*** (10.404)*** /08/01-98/03/ (79.903)*** (1.607) (1.043) Noe : 1. Figures in parenheses ( ) are -values. * significan a 0.05, ** significan a 0.01, *** significan a or beer

23 3. Before aking he firs-order difference, R are , 0.09, , and , respecively, for he four periods. 5. Conclusions This paper applies he kinemaic and kineic heories of physics o develop wo imporan behavior equaions for he sock prices: he equaion of moion and he equaion of work and kineic energy. Empirical evidence shows ha hese wo equaions provide a powerful and good descripion of he behavior of sock prices. The conribuions of his paper are o formally heorize some language (e.g., energy of sock marke, price-volume relaion, ec.) ha is ofen heard from he praciioners of sock markes, and o furher undersand he behavior of sock prices. Empirical evidence regarding he equaion of moion shows ha an uni change in excess demand (unis of buy orders minus unis of sale orders) will cause he TSE index o change abou poins wih he same direcion of he change in excess demand. If he equaion is used for predicion, we find ha he longer he esimaion period, he more accurae for he predicion. Using OLS mehod, he 4-year esimaion period gives he bes predicion resuls; however, using GARCH mehod, he -year esimaion period gives he bes predicion resuls. Ineresingly, all average predicion errors are, in general, small and no saisically significan. Empirical evidence regarding he equaion of work and kineic energy shows ha change in kineic energy (i.e., percenage change in rading values) is equal he work done (i.e., he absolue value of percenage change in excess demand imes he absolue value of percenage change in sock prices). This principle provides a raher high explanaory power for he Taiwan sock marke. Alhough hese wo principles for he sock markes provide a very good descripion of sock price behavior for he hisorical daa of he Taiwan sock marke, his does no imply ha he developmen of he wo equaions violaes he hypohesis of marke efficiency since we assume ha changes in excess demand are deerminisic. However, small predicion errors for he equaion of moion imply ha sock prices are predicable for some persons who own he informaion of buy and sale orders. To avoid his opporuniy and o provide a fair marke, he ime inerval beween wo successive maches of ransacions should be shorened. The shorer, he beer. Also, he developmen of hese wo principles does no rejec he applicaion of he Brownian moion o he dynamic of sock prices. Indeed, if changes in excess demand are random (because he arrival of informaion is random), hen he 3

24 dynamic of sock prices mus also be sochasic. Ex pos daa provides good fis of he heories, his does no mean sock prices are predicable. Fuure research can focus on he applicaions of he equaion of work and kineic energy and on he empirical evidence of he principle of impulse and momenum for sock markes. If daa for individual socks are available, empirical sudies can focus on he implicaions of he equaions. For example, oher hings being equal, he larger he mass of a sock, he smaller he change in sock price. Finally, fuure applicaions of hese models can be explored in sock markes and fuures markes o enhance profis for invesors. Acknowledgemens The auhors would like o hank wo anonymous referees for heir commens and valuable suggesions. All errors are, of course, ours. References [1] Bollerslve T (1986) Generalized Auoregressive Condiional Heeroscedasiciy. Journal of Economerics, 31, [] Bollerslve T, Chou RY, and Kroner KF (199) ARCH Modeling in Finance. Journal of Economerics, 5, [3] Das B, Kassimali A, and Sami S (1994) Engineering Mechanics-- Dynamics, Irwin. [4] Glejser, H (1969) A New Tes for Homoscedasiciy. Journal of American Saisical Associaion, [5] Hsu H (1993) The Acceleraion of Sock Prices in he Taiwan Sock Markes. Proceedings on he Second Conference on he Theories and Pracice of Securiies and Financial Markes, [6] Maddala GS (199) Inroducion o Economerics, nd ed., Prenice Hall, New York. [7] McGill DJ and Wilon WK (1995) An Inroducion o Dynamics, 3rd ed., PWS Publishing Company, Boson. [8] Ohanian, HC (1989) Physics, W.W. Noron & Company, New York. 4

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