A New Test for Randomness with Application to Stock Market Index Data

Transcription

1 Business and Economic Saisics Secion JSM 01 A New Tes for Randomness wih Applicaion o Sock Marke Index Daa Alicia G. Srandberg 1, Boris Iglewicz 1, Deparmen of Saisics, The Fox School of Business, Temple Universiy, Philadelphia, PA 191 Absrac Srandberg and Iglewicz (01) propose a es ha deecs deviaions from randomness, wihou an a priori disribuional assumpion. This nonparameric es is designed o deec deviaions of neighboring observaions from randomness, especially when he daase consiss of ime series observaions. The proposed es is especially effecive for larger daases. In our simulaion sudy, his es is compared o a number of variance raio and radiional saisical ess. The proposed es is shown o be a compeiive alernaive for a diverse choice of disribuions and daa models. In addiion, his es is able o successfully deec changes in variance, which can be informaive in shor erm invesing and opion rading. In our empirical applicaion, we review and compare several ransformaions while evaluaing common US sock marke indices. We consider wo commonly used ransformaions and a proposed new ransformaion from Srandberg and Iglewicz (01). This new ransformaion performs surprising well for sock marke index daa and is he only ransformaion o show consisence resuls among he considered ess. Key Words: Nonparameric, Randomness, Sock Marke Indices, Time Series, Variance Raio 1. Inroducion A new es, proposed by Srandberg and Iglewicz (01), is considered for sudying deparures from randomness of a series of independenly and idenically disribued (i.i.d.) random variables. There are many exising ess used o deermine if a ime series consiss of a random sample. Ofen hese ess have resricive disribuional assumpions, size disorions, or low power. The ineres here lies in developing an alernaive es wih minimal assumpions and a simple es saisic o deermine wheher a series consiss of a random sample. A es wih minimal assumpions is an imporan consideraion since daa disribuions are ofen unknown. Therefore, developing a es ha does no require a normaliy assumpion or a priori knowledge of he disribuion generaing hese daa is highly desirable. The proposed es is shown o work well for varied symmeric and skewed disribuions wih reliance on an upper and lower percenile and sraighforward condiional binomial probabiliy, wihou an a priori disribuional assumpion. The es is hen empirically compared wih wo popular exising ess and shown o be nicely compeiive. In our comparison varied sample sizes and disribuional models are considered. The proposed es is also evaluaed in an applied applicaion using sock marke index daa. These daa are no normally disribued; raher hey are nonsaionary wih ypically an increasing long erm rend. These daa are aypical because such daa 907

2 Business and Economic Saisics Secion JSM 01 ofen consiss of srucural breakdowns where a relaively large number of adjacen ime series observaions follow a regular paern followed by highly irregular periods wih changes in mean and variance (Chu, Sinchcombe and Whie 1996, Bandyopadhyay, Biswas, and Mukherjee 008). Even afer making a pracical ransformaion he disribuions of hese sock marke index daases are ofen noiceably skewed. In addiion, a new daa ransformaion, also proposed by Srandberg and Iglewicz (01), is considered. This ransformaion is a modified measure of percen change (MMPC), similar o a sock reurn. In our sock index daa analysis sudy, he proposed ransformaion was he only considered ransformaion ha led o consisenly rejecing he null hypohesis wih high power for all considered ess and sock marke indices.. Descripion of Tess Our moivaing applicaion is daily closing values from sock marke index daa, which consis of large number of observaions. Large sample daases are becoming more common in oher saisical areas, such as regression and linear and nonlinear models. There are many ess ha can be used o deermine wheher a series, such as daily changes in index daa, consiss of a random sample. These ess include boh classical saisics mehods and mehods popular in he economic and finance lieraure, such as variance raio ess. In his paper we compare he proposed es wih he radiional Durbin and Wason (1950) es and he mos referenced of he variance raio ess found in he lieraure, Lo and MacKinlay (1988)..1 Srandberg and Iglewicz (01) The es by Srandberg and Iglewicz (01) is based on a sraighforward condiional binomial probabiliy and es saisic ha converges in disribuion o he sandard normal disribuion. I ess for randomness of a ime series based on a model ha assumes i.i.d. random observaions. Consider a ime series, Y 1, Y,..., Y N consising of N observaions, under he null hypohesis Y (1) where ε is i.i.d.(0, ), Var [ ] and for all. Noe ha no addiional disribuional assumpions are made on ε in (1). Le Y = {Y 1, Y, Y 3,., Y N }. Noice ha while we are dealing wih a series, Y is a se. Consider he subses Y* ={Y : Y R} and Y** ={Y : Y R}, where R is he simple inerval [Y {0.05}, Y {0.975} ] and Y {p} is he p h sample percenile. Noe ha Y** is he complemen of Y*. Oher regions for R may be useful, bu here we only consider he inerval [Y {0.05}, Y {0.975} ]. Concenraing on observaions ouside upper and lower perceniles allows his es o focus on ail observaions, which are of special ineres in sock marke invesigaions, raher han on he enire daase. While his choice of percenile works well in erms of providing he appropriae significance level, noe ha i is no direcly relaed o he common significance level of α = Assume ha he null hypohesis, ha hese observaions consiue a random sample, is rue. Le N(Y**) be he number of elemens of Y** and P(Y Y**) = π. We can esimae π by ˆ = N(Y**)/N (Srandberg and Iglewicz 01). Nex subdivide he N observaions ino M consecuive inervals each having an equal number of K observaions, such ha M N / K inervals, where is he floor (larges ineger) funcion and K is an ineger ha is small relaive o N. If N/K M 0, 908

3 Business and Economic Saisics Secion JSM 01 hen using M inervals leaves ou K(N/K M) = b observaions. We hen recommend ignoring he firs b observaions in he ime series. These b observaions are considered an incomplee group of size less han K. Since N is large relaive o K, having b 0 should no be a problem. For our moivaing example, we consider complee weeks consising of K = 5 days; hen N/K is an ineger. For simpliciy assume b = 0. For i = 1,,, K, j = 1,..., M, denoe Y (j) i = Y K(j-1) + i as he i h observaion in he j h inerval. For j K ( j) = 1,..., M, le W j = I( Y i Y**), where I is he indicaor funcion. Under he i1 null hypohesis ha Y 1,..., Y N are i.i.d., we have W 1,..., W M i.i.d. Binomial (K, ). I follows ha K 1 K (1 ) P( W j 1 W j 0), j = 1,..., M. () K 1 (1 ) We denoe he condiional probabiliy in () by D. This suggess we consider all inervals M M such ha W j > 0. Le L = IW j 0and L 1 = IW j j1 j1 1. Thus in L ou of he M inervals, we have W j greaer han 0, or a leas one observaion wihin he j h inerval belongs o he se Y**. Similarly here are L 1 ou of he M inervals, where exacly one observaion belongs o he se Y**. Noice ha he j h inerval wih W j = 1 mus also saisfy W j > 0. Thus we can rewrie L 1 as L L 1 = IW j 1 W j 0. Under he null hypohesis, i follows ha W j 1 Wj 0 j1 a Bernoulli random variable wih success probabiliy D = W j 1 W 0 I is P. Consequenly, L 1 follows Binominal (L, D). For large L, by cenral limi heorem, we L1 L D d know N0,1 where d means converge in disribuion. For D 1 D L moderae L, he binomial approximaion can be improved by using a correcion facor wih modified es saisic L1 L D ch L Z (3) D(1 D) L L1 1 if D where L c H and is he useful correcion facor needed o preven he L 1 1 if D L L es from being oo conservaive (Srandberg and Iglewicz 01). In his sudy we use c = 0.50 and K = 5 o mimic sock marke daa wih full 5 day rading weeks.. Lo and MacKinlay (1988) As an alernaive, we considered he popular variance raio es of Lo and MacKinlay (1988). Under he null hypohesis, he relaion beween observaions is a random walk, Y 1 (4) Y where Y is he value of a reurn a ime, is an unknown arbirary drif parameer and ε is a disurbance erm wih Var(ε ) =. A random walk assumes he condiional mean j 909

4 Business and Economic Saisics Secion JSM 01 and variance are linear in ime, a condiion ha may no always be reasonable when considering sock marke daa. Alhough, Lo and MacKinlay developed wo variance raio ess, M 1 and M, our simulaion invesigaion, using he vres package of R (Kim 010), showed ha he simpler es, M 1, performs slighly beer han M. As a resul, our simulaion resuls will only be reporing for M 1. Under he null hypohesis M 1 assumes ε are i.i.d. N(0, σ ε ). The es saisic for M 1 is VR( k) 1 M 1( k) (5) 1 ( k) where (k 1)( k 1) ( k), 3kN 1 wih k( N k 1)(1 kn ) N ˆ ( k) 1 VR( k), ˆ ( k) m ˆ ( Y Y (1) 1 m for k 1, and ˆ N N 1 Y Y k 1 k k ˆ), such ha when VR(k) = 1 observaions are serially uncorrelaed; likewise when VR(k) 1some auocorrelaions beween observaions exis (Lo and MacKinlay 1988, Charles and Darné 009). Noe ha VR(k) is a variance raio ha compares he variance of he k-period reurn wih k imes he variance of he one-period reurn. In his sudy he common value of k = is used. Under he null hypohesis, M 1 follows asympoically he sandard normal disribuion. However he sampling disribuion of his variance raio es saisic is known o be skewed (Chen and Deo 006, Charles and Darné 009)..3 Durbin and Wason (1950) A well known radiional es of randomness was inroduced by Durbin and Wason (1950). They es for non-randomness in he residuals of an ordinary leas squares regression equaion wih he es saisic, d N ( e e ) 1, (6) N e 1 where e is he h residual and N is he number of observaions. This es assumes i.i.d. N(0, σ ) errors. The es saisic, d, is asympoically normal wih mean of and variance of 4/N (Harvey 1990). When daa are posiively (negaively) serially correlaed, d will have a value ha ends o zero (four). Alhough his es was designed for residuals of leas squares regression, i can be used o es ime series daa such as Y 1, Y, Y 3,..., Y N, by subsiuing Y = e in (6), assuming he assumpion of i.i.d. N(0, σ ) is valid. In some applicaions, he resriced condiions of normaliy and serial correlaion are limiing. When daa are heavily skewed, his es does no mee he size requiremen (Ali and Sharma 1993). 3. Sock Marke Example In his secion we evaluae randomness of daily closing values of sock marke indices using he Dow Jones Indusrial Average (DJIA), he Sandards & Poor's 500 (S&P 500), and he Naional Associaion of Securiies Dealers Auomaed Quoaion Sysem (Nasdaq). We only consider daa for full 5 day weeks, wih daa ending on December 18, 009. These daa were also analyzed by Srandberg and Iglewicz (01, 013). We 910

5 Business and Economic Saisics Secion JSM 01 consider hree differen ransformaions, which include wo common ransformaions and a newly proposed ransformaion by Srandberg and Iglewicz (01). Consider he following ransformaions: lag 1 closing daily sock marke index differences, Y Y -1 ; Y ( i) daily percenage change defined as where Y (i) is he daily closing price Y 1 ( i) for index i on day (Cizeau, Poers and Bouchaud 001, Bandyopadhyay, Biswas and Mukherjee 008, Mukherjee and Bandyopadhyay 011), i = 1,, 3, for DJIA, S&P 500, and Nasdaq, respecively; he modified measure of percen change, MMPC, Y ( i) where MA is a delayed moving average of q observaions such ha ( i) MA( q: q1) q 1 Y jq MA, were, as suggesed by Srandberg and Iglewicz (01), q = 10. I ( q: q1) q is reasonable o es wheher hese ransformed observaions consiue a random sample. Resuls are included in Table 1. To assis in he readabiliy of Table 1, afer each es saisics * is added if significance is a he 10% level, ** if significance is a he 5% level and *** if significance is a he 1% level. Exremely high es saisic values conain only *** wih blanks for associaed numbers. Table 1: Tes Saisics for Sock Marke Indices Applicaion Resuls DJIA SI DW M 1 Y - Y ***.11*** -7.44*** 100((Y / Y -1 ) -1) *** * 100((Y /MA)-1) *** 0.07*** *** S&P 500 Y - Y ***.11*** g-6.3*** 100((Y / Y -1 ) -1) *** 1.94** 3.00*** 100((Y /MA)-1) *** 0.08*** *** Nasdaq Y - Y *** *** 100((Y / Y -1 ) -1) *** 1.87*** 5.68*** 100((Y /MA)-1) -6.34*** 0.06*** 89.*** The es saisics show *** if significan a he 1% level, ** if significan a he 5% level and * if significan a he 10% level. Tes saisics greaer han 100 or less han -100 are no shown since hey are highly significan a he 1% level: SI = Srandberg and Iglewicz (01) wih c = 0.5, DW = Durbin and Wason (1950), M 1 = Lo and MacKinlay (1988) wih k = In Table 1 noice ha here are some very high absolue es saisics resuling in p- values below Therefore exac p-values are no included in Table 1 because we do no claim ha level of accuracy. In his able es saisics differ beween ransformaions 911

6 Business and Economic Saisics Secion JSM 01 and indices, wih he excepion of Srandberg and Iglewicz, SI, which is significan a he 1% level for all indices and ransformaions. I is possible ha sock marke index daa are mosly moderaely sable wih periods of ime where values are insable. These unsable periods of ime may resul in far more oulier values or values ouside he inerval [Y {0.05}, Y {0.975} ]. In cases like his, SI shows high power o rejec he null hypohesis of random daa. Furhermore, Z values for SI are all negaive, while Z values for M 1 are ofen posiive. The negaive Z values from SI indicae ha more han expeced muliple unusual ransformed values end o occur wihin weeks conaining a leas one unusual ransformed value. The posiive Z values for M 1 indicae ha he variance of wo ime periods is higher han expeced. Resuls from he Durbin and Wason es, DW, generally indicae ha ransformed indices values are posiiviy correlaed his occurs when d in (6) <.00. Among ransformaions, only he MMPC resuls in consisenly rejecing he null hypohesis a he 1% significan level for all ess, while he oher wo ransformaions show inconsisen rejecion levels among ess and indices. I is ineresing o observe ha for DW and M 1 resuls can differ depending on he ransformaion used ye es resuls based on he MMPC ransformaion are all significan a he 1% level. Differences beween he firs wo ransformaions include no only changes in significan levels, bu also changes in inerpreaions - since for M 1 es saisics changes in value from negaive o posiive and for DW es saisics change from d <.00 o d >.00, where d is defined as in (6). In summary, he only es ha shows consisen resuls for each considered ransformaion is SI, while M 1 and DW show inconsisen resuls among ransformaions. In addiion, he only ransformaion ha shows consisen resuls among individual ess for each considered daase is he MMPC ransformaion while all oher ransformaions had inconsisen resuls among he considered indices. In summary, when dealing wih financial daa, he es used and choice of ransformaion can play a key role in resuling conclusions. 4. Simulaion Resuls In his Secion we invesigae each es over a diverse group of disribuions and daa models. We will view hese ess as compeiive, even hough M 1 is based on a random walk model, hus no sensiive o changes in variance, while he oher considered ess are based on a random sample null hypohesis. Since in pracice mos praciioners will no know he rue disribuion of heir daa, mehods ha perform well for a variey of disribuions are preferred. We will consider random samples of N = 300, and 10,000 observaions and will use he g- and -h disribuional family (Tukey 1977, Marinez and Iglewicz 1984, Hoaglin 1985) o obain or approximae he sandard normal, Z, disribuion, he Suden s disribuion wih 3 degrees of freedom, 3, and he chi-square disribuion wih 4 degrees of freedom, 4. The 4 observaions are sandardized before performing each es. These disribuions are considered null cases since hey are known disribuions expeced o preserve he es size. 91

7 Business and Economic Saisics Secion JSM 01 We also consider some alernaive daa models. We considered a model where 30% of observaions follow f 1 (y), hen 40% follow f (y), and he remaining 30% follow f 3 (y). A model wih consan mean and changing variance is creaed by leing, f 1 (y) be N(0, 0.75 ), f (y) be N(0, 1), and f 3 (y) be N(0, 1.5 ), while a model wih consan variance and changing mean is creaed by leing f 1 (y) be N(-, 1), f (y) be N(0, 1), and f 3 (y) be N(, 1). In addiion, four correlaed cases, C1, C, C3 and C4 are considered. For hese cases, moivaed by sock marke weekly daa, observaions are simulaed from complee rading weeks where weeks wih marke closures or holidays are no included. Each correlaed model is designed such ha 90% of hese weeks have observaions ha are i.i.d. N(0,1) and a correlaion srucure exiss for he remaining 10% of hese weeks. Consider a ypical member of he 10% correlaed weeks. Le Y M be he Monday value coming from N(0, ). The oher generaed values are Y j, j = T, W, Th, F. For C1, Y T = ρy M + ε where ε is N(0,1) and he remaining hree days are from i.i.d. N(0, 1). For C, Y F = ρy M + ε and he remaining hree days are from i.i.d. N(0, 1). For C3, Y j = ρy M + ε, where j is randomly chosen from T, W, Th, F and he remaining hree days are from i.i.d. N(0, 1). In C4 he correlaion srucure is presen over wo weeks, such ha Y M in he firs week, Y 1 M, has a value from N(0, ). Two days are correlaed observaions, such ha Y j = ρy M + ε, where j is randomly chosen wice wih replacemen from T 1, W 1, Th 1, F 1, M, T, W, Th, F, he remaining days are generaed from i.i.d. N(0, 1). In all four correlaed cases ρ = More null and alernaive cases are discussed in Srandberg and Iglewicz (01, 013). Table summarizes he simulaion resuls. For each case 10,000 replicaions are simulaed and he rejecion rae is he percenage of ess ou of 10,000 where p-value < For he null cases, since he significance level is se a 5.00%, we expec rejecion raes o be close o 5.00%. Wih 10,000 replicaions, he sandard error of a 0.05 proporion of rejecions, under he null hypohesis, is herefore rejecion raes in he range of %, 5.43% are expeced. All considered null cases are wihin his range and herefore are reasonably close o he desired 5.00% rejecion rae excep for SI when N is small. When N is small SI is slighly conservaive. For he alernaive cases, as expeced, greaer power is generally seen as N increases; however power can differ considerably across alernaives. When N is small all considered ess, excluding SI, have high power o deec changes in mean, while all ess have low power for oher cases. As N becomes large, he SI is he only es wih high power for all alernaive cases. Only one es, SI, is able o show considerable power agains changing variances especially when N is large, while rejecion raes for M 1 show low power. This is no surprising, as he null hypohesis for M 1 consiss of a random walk, hus oleraing changes in variance. If deecing changes in variance is a concern, he SI es should be used. The SI es is able o show high power for all he correlaed cases wih large N, while M 1 and DW have modes o low power for C, C3 and C4. When N is large all ess show high power for C1 and modes power as N decreases. In summary, when N is large, SI is he only considered es wih high power for all correlaed cases. This es works well irrespecive of he correlaion srucure following a highly volaile day, wihou a priori knowledge of fuure dependencies, while ess wih se auocorrelaion srucures may no always be able o reain power in such siuaions. 913

8 Business and Economic Saisics Secion JSM 01 Table : Simulaion Rejecion Raes Comparisons SI DW M 1 N = 300 Z 4.1% 5.1% 5.18% % 4.94% 4.85% X 4 Sandardized 4.% 4.78% 4.90% Changing Variance 15.83% 6.87% 6.53% Changing Mean 13.81% % % C %.81% 18.40% C 16.84% 6.8% 6.34% C3 17.1% 7.50% 6.79% C4 8.18% 7.01% 6.11% N = 10,000 Z 5.9% 4.86% 5.1% % 4.93% 5.07% X 4 Sandardized 4.91% 4.9% 4.96% Changing Variance 95.3% 6.89% 7.43% Changing Mean 99.4% % % C % % % C 99.85% 6.73% 7.10% C % 37.63% 36.15% C % 6.80% 5.40% SI = Srandberg and Iglewicz (01) es wih c = 0.5, DW = Durbin and Wason (1950) es, M 1 = he Lo and MacKinlay (1988) wih k = 5. Conclusion In his paper we summarize and compare hree ess for deecing randomness in ime series daa, including comparisons using sock marke index daa. One of our considered ess, SI is unique because i is based on a simple es saisic ha surprisingly works well for varied null and alernaive daa choices wih minimal disribuional assumpions. Since his proposed new mehod is no based on auocorrelaions or relaed measures, i is shown o have high power, for larger sample sizes, o deec correlaed srucures no only beween consecuive observaions bu also over longer lags. Of he sudied ess, only he SI es has power o deec changes in variance and all four correlaed cases when N is large. Through simulaions, i is also shown ha SI mees he size requiremens for a variey of null cases. For our sock marke index daa analysis we review and compare hree ransformaions and demonsrae ha he choice of ransformaion can have a noiceable effec on es resuls. Only he MMPC ransformaion resuls in consisen rejecion of he null hypohesis wih high power for all ess and indices. While he SI es is he only es ha is able o srongly rejec he null hypohesis for all hree sudied indices and considered 914

9 Business and Economic Saisics Secion JSM 01 ransformaions. We noe wih ineres ha for he oher considered ess, resuls can differ depending on he ransformaion used. The only es ha does no show evidence of his concern is SI. When deciding among ess, consideraion mus no only be given o meeing size requiremens for he null hypohesis, bu also he possible alernaive hypoheses choices as power can differ considerably across alernaives. In our simulaion sudy we considered a number of alernaive cases o show he advanages and disadvanages of each of he sudied ess. In summary, we believe ha hese comparisons and resuls will be helpful, especially when dealing wih financial daa and ess of randomness. Acknowledgemens The auhors would like o hank Drs. Pallavi Chiuri, and Yuexiao Dong, boh from he Fox School of Business a Temple Universiy, Philadelphia PA, for heir ime and helpful suggesions. They would also like o like o hank he referees and Edior from Communicaions in Saisics- Simulaion and Compuaion for heir helpful commens. References Ali, M. M., and Sharma, S. C. (1993), Robusness o Nonnormaliy of he Durbin-Wason Tes for Auocorrelaions, Journal of Economerics, 57, Bandyopadhyay, U., Biswas, A., and Mukherjee, A. (008). Conrolling Type-I Error Rae in Monioring Srucural Changes Using Parially Sequenial Procedures. Communicaion in Saisics Simulaion and Compuaion 37, 3: Charles, A., and Darné, O. (009), Variance-Raio Tess of Random Walk: An Overview, Journal of Economic Surveys, 3, Chen, W. W., and Deo, R. S. (006), The Variance Raio Saisic a Large Horizons, Economeric Theory,, Chu, C. J., Sinchcombe, M., and Whie, H. (1996). Monioring Srucural Change, Economerica 64: Cizeau, P., Poers, M., and Bouchaud, J. (001), Correlaion Srucure of Exreme Sock Reurns, Quaniaive Finance, 1, 17-. Durbin, J., and Wason, G. S. (1950), Tesing for Serial Correlaion in Leas Squares Regression: I, Biomerika, 37, Harvey, A. C. (1990), The Economeric Analysis of Time Series ( nd ed.), Cambridge, MA: MIT Press. Hoaglin, D. C. (1985), Summarizing Shape Numerically: The g-and-h Disribuions, in Exploring Daa Tables, Trends, and Shapes: Robus and Exploraory Techniques, Hoaglin, D. C., Moseller, F., Tukey, J. W. (eds.), Hoboken, NJ: John Wiley & Sons, Inc., pp Kim, J.H. (010), vres: Variance Raio ess and oher ess for Maringale Difference Hypohesis. R package version hp://cran.r-projec.org/package=vres (accessed July 14, 011). Lo, A. W., and MacKinlay, A. C. (1988), Sock Marke Prices Do No Follow Random Walks: Evidence from a Simple Specificaion Tes, The Review of Financial Sudies, 1, Marinez, J., and Iglewicz, B. (1984), Some Properies of he Tukey g and h Family of Disribuions, Communicaions in Saisics-Theory and Mehods, 13,

10 Business and Economic Saisics Secion JSM 01 Mukherjee, A., and Bandyopadhyay, U. (011). Some Parially Sequenial Nonparameric Tess for Deecing Linear Trend. Journal of Saisical Planning and Inference 8: Srandberg, A.G., and Iglewicz, B. (01), A Nonparameric Tes for Deviaion from Randomness wih Applicaions o Sock Marke Index Daa. Communicaions in Saisics- Simulaion and Compuaion, o Appear. Srandberg, A.G., and Iglewicz, B. (013), Deecing Randomness: A Review of Exising Tess wih New Comparisons. Communicaions in Saisics- Simulaion and Compuaion, o Appear. Tukey, J. W. (1977), Exploraory Daa Analysis, Reading, MA: Addison-Wesley. 916