Linden, Mikael Louhelainen, Mika

Tesing for weekday anomaly in inernaional sock index reurns wih non-normal errors Linden, Mikael Louhelainen, Mika ISBN 95-458-44-5 ISSN 458-686X no 4

Tesing for Weekday Anomaly in Inernaional Sock Index Reurns wih Non-Normal Errors Mika Linden* and Mika Louhelainen ** Absrac Empirical research implies ha disribuions of sock indices are non-normal. Boh OLS and MAD esimaion mehods are used o examine weekday anomaly in eigheen inernaional sock exchanges beween 990 and 003. Weekday anomaly is found wih OLS mehod in wo and wih MAD mehod in nine sock exchanges. In shor run a leas one weekday anomaly period exised in every sock exchange. Empirical es disribuions of F-ype es of weekday anomaly under Laplace errors were derived wih simulaions. *) Professor in Economics, Universiy of Joensuu, Deparmen of Business and Economics, Yliopisokau 7, Box, FIN 800. E-mail: mika.linden@joensuu.fi **) PhD suden in Economics, Universiy of Joensuu, Deparmen of Business and Economics, Yliopisokau 7, Box, FIN 800. E-mail: louhelai@cc.joensuu

I. INTRODUCTION Since he early 970s numerous researchers have documened differen seasonaliy anomalies in sock markes. All his research begins wih Fama (970) who presened he efficien marke hypohesis. According o his hypohesis, in efficien sock marke, only he new relevan informaion effec o he sock prices. However many resuls cas doub on his widely acceped hypohesis. One case is he seasonal anomalies. They include he calendar effecs or seasonal paerns such as January-, weekday-, weekend-, urn of he monh- and holiday anomaly. Of he seasonal anomalies, few are as curious as he paern observed in sock reurns across he weekdays. Many researchers have documened ha daily raes of reurns are no saisically equal each oher. This observaion is called he weekday anomaly. The mos ineresing weekday finding is ha a beginning of he week he reurns are ypically lower han in oher weekdays, and hese are on average negaive. Some researchers (e.g. Gibbons and Hess 98) called his observaion as he weekend anomaly. According o efficien marke hypohesis invesors should ake advanage of daily raes of reurn differences. While invesors know hese differences in raes of reurn phenomena sill exiss. This paper examines he weekday anomaly in 8 inernaional sock exchanges in shor and long run periods. In shor run we analyze how he differences in reurns changes in ime. We found ha differences in weekday reurns parallel wih differen sock exchanges. We also compare he shor and he long run resuls o observe possible differences. In he early sudies of weekday anomaly researchers used OLS esimaion mehod o analyze raes of reurn for every weekday. The OLS mehod is reliable when he disribuion of reurns is normal. However disribuions of reurns are usually lepokuric and Laplace disribuion is a more accurae modelling saring poin han normal disribuion (Linden 00). The close connecion of Laplace disribuion wih minimum absolue esimaion (MAD) is exploied in esing of weekday anomaly. The MAD esimaor of reurns model parameers is ML E.g. see for January anomaly (Thaler 987, Haugen and Lakonishok 988, Agrawall and Tandon 994), he weekday anomaly (Agrawall and Tandon 994, Kamara 997), he weekend anomaly (Gibbons and Hess 98, Keim and Sambaum 984, Abraham 994), urn of he monh anomaly (Ariel 987, Lakonishok and Smih 988) and he holiday-anomaly (Ariel 990, Lakonishok and Smih 988)

esimaor under Laplace errors, i.e. he esimaor is asympoically unbiased and efficien compared o OLS esimaor. II. MAD-ESTIMATION AND TESTING OF WEEKDAY ANOMALY MAD esimaion Consider he following regression model for reurns y = ( P P )/ P where P is he daily sock marke price index () y = X + v = T α σ( ) ε,,... where σ > 0, α is he regression parameer vecor for exogenous variables X, and ε is D(0, σ ). v is NI ε IID posiive random variable ha is independen of ε. Basically he model is similar o he produc process suggesed by Taylor (986, p. 70-7) or o he mixure disribuion model where σ ( ) v represens he level of marke aciviy during he rading day. If he densiy of v is () 3 ( ) = exp{ ( ) } gv v v hen he condiional densiy of y given and X is v (3) f( y v, X ) = ( v / σ ) φ{ v ( y αx )/ σ }. The marginal densiy of y given X is p( y X ) = f( y v, X ) g( v ) dv 0 (4) = exp{ z / σ }. σ 3

This is he Laplace (or double exponenial) disribuion for z = y α X. The log likelihood of py ( X; α, σ ) is (5) T ln p( y X ; ασ, ) = Tln( σ) y αx σ T =. = Thus minimum absolue deviaion (MAD) esimaor for α is ML esimaor when he disurbances z = y α X follow he Laplace disribuion. Esimaor for α can be derived wih linear programming mehods (see Taylor 974, Pornoy & Koenker 997) or wih ieraive weighed leas squares (IWLS, see Maddala 977, Schlossmacher 973, Amemiya 985). Phillips (00) shows ha he EM-algorihm for calculaing α is essenially he IWLS. A leas wo differen weigh srucures have been suggesed for EM-IWLS esimaor of α () i ha has form of (6) α i X DX X Dy () = ( ' ) ' where D is he diagonal marix. In he firs alernaive h diagonal elemen of D is given by (7) d o y α( i ) x if y α( i ) x > ε = 0 oherwise where α () i is he esimae obained a he i h ieraion and ε o is some predefined small number, say o ε = 0 6. Alernaively Amemiya (985, p. 78) suggess ha we use (8) o d = min{ y α x, ε }. * ( i ) Though compuing he MAD esimaors is no a major problem, he esing of parameer resricions needs some clarificaion. In his conex he F-es is analyzed in deails. 4

Tesing for weekday effecs wih OLS and MAD under non-normal regression errors Consider following model for weekday effecs in daily reurns y a D a D a D a D a D ε (9) = + + 3 3+ 4 3+ 5 5 + = + Ia ε, =,,3,..., T where I is (5x 5) ideniy marix. Now (0) I I y =. a+ ε. I T = Xa+ I ε. The OLS esimaor reduced o he variance analysis since () y y ' ' aols = ( XI XI ) XI y = y 3 y4 y 5. MAD esimaor in EM-IWLS form is () d y d y ' ' d3 amad = ( XI DXI ) XI Dy = y d4 y d5 y where ' ( X IDX I) i / Sdi = wih differen weekdays (see Eqs. (7) and (8)). i T /5, (,,..,5) = i are he observaion weighs for Sd = d i = 5

Resriced esimaes Considerer following hypohesis H : a = a = a = a = a. (3) 0 3 4 5 The resriced OLS esimaor is now (see Greene & Seaks 99) wih resricions Ra = 0 (4) a = a ( X X ) R'( R( X X ) R) Ra R ' ' ' OLS OLS I I I I OLS. Similarly MAD esimaor has he following form due is IWLS-srucure (5) a = a ( XDX) R'( RXDX ( ) R) Ra R ' ' ' MAD MAD I I I I MAD The hypoheses (6) H0 : ai = aj, i j i, j =,,...,5 are all handled in similar fashion and he OLS- and MAD-esimaions are conduced like above bu he resricion marix is now Rija= 0. F-ype es Under normal errors F es saisics (7) f ( SR SU)( T K) = S J U for he hypohesis H0 : a = a = a3 = a4 = a5 5 f is FT 5 -disribued, and for he hypohesis H : ( 0 ai = aj i j, i, j =,,...,5) i is f. S and S are he resriced and FT 5 R U unresriced residual sums of squares from OLS or MAD esimaion ( K = number of 6

esimaed parameers, and J = dimension of resricion vecor). However when he errors are no normal disribued he F-es saisics f is no F-disribued. We need he correc f - values under non-normaliy. In order o solve his problem we simulaed F-ype es disribuion under H 0 alernaives and under he assumpion ha he errors are disribued as Laplace. Thus we generaed 0.000 replicaions of F-ess from OLS and MAD esimaes of model (8) y = a + ad + a3 3 D3+ a5d5 + ε, =,,..., T D D + a4 wih ε LAPLACE() and a = a = a a = or a = a = ( i j). The used sample 3 = a4 = 5 i j sizes were T = 50, 50, 000 and 5000. Table. gives he 0%, 5% and % criical values for he sandard F-disribuion and he empirical disribuion of F-ype es under : a = a = a3 = a4 = a5 = wih Laplace model 5 H 0 errors esimaed wih OLS and MAD. The resuls show ha MAD esimaor gives disribuion of F-ype es saisics ha has much smaller 0% and 5% criical values compared o sandard F-disribuion under normal errors. The es values are also smaller wih OLS esimaor wih Laplace errors. Thus if he residuals are close o Laplace case he use of sandard F-disribuion leads oo ofen o non-rejecion of hypohesis H : a = a = a = a = a 0 3 4 H a = a =. 0 :. Table. repors similar simulaion resuls for F-ype es under TABLE. 0%, 5% and % criical values for 5 F -disribuion and -ype es wih 5 T 5 FT 5 Laplace model errors esimaed wih OLS and MAD ( H : a = a = a = a = a = ) ) 0 3 4 5 F-disribuion OLS: F-ype es MAD: F-ype es 0% 5% % 0% 5% % 0% 5% % T=50.98.4 3.45.63.04.80.9.74 3.0 T=50.87.5 3.09.56.90.7.5.66 3.04 T=000.85. 3.04.53.87.63.4.58.53 T=5000.84. 3.03.53.86.6.0.56.50 7

TABLE. 0%, 5% and % criical values for F -disribuion and -ype es wih T 5 FT 5 Laplace model errors esimaed wih OLS and MAD ( H : 0 a = a = ). F-disribuion OLS: F-ype es MAD: F-ype es 0% 5% % 0% 5% % 0% 5% % T=50.8 4.04 7.3.69 3.80 6.58. 3.36 6. T=50.73 3.88 6.74.68 3.77 6.5.0 3.33 6.0 T=000.7 3.85 6.66.65 3.75 6.46.9 3. 6.4 T=5000.70 3.84 6.63.6 3.7 6.7.7 3.9 5.76 III. RESULTS Tes model In his secion we repor he OLS- and MAD-esimaion and esing resuls for he model (9) y = ad+ ad + a3d3+ a4d3+ a5d5 + ε, =,,..., T wih hypoheses and H : a = a = a = a = a, 0 3 4 5 H ai aj i j i j : (,,,,...,5) 0 = = where y is weekday rae of reurn and D, D,, D 5 are he dummy variables of he differen weekdays. D is equal o one on every Monday and a he same ime he oher dummy variables are zeros. Likewise, D is equal o one every Tuesday and a he same ime oher dummy variables are zeros and so on. a, a,,a 5 are he coefficiens for each weekday. If we rejec hypohesis H 0, ha is a clear evidence for weekday anomaly in his period. The hypohesis H 0 ess he pair-wise difference beween weekday coefficiens. If we rejec hypohesis for a a, he coefficiens of hose days are saisically significanly H 0 differen. Noe ha H H. i 0 0 j 8

Daa The sudy used eigheen end of he day indices from he beginning of 990s o mid 003. Mos of indices are from he bigges sock exchanges, bu some small sock exchanges are included also like Finland and Denmark. From he Unied Saes, we have hree sock exchanges: Dow Jones, NASDAQ and S&P 500. From he oher counries one sock exchange per counry is included. The daa from hese eigheen sock exchanges conains almos 50.000 observaions (over 500 observaions for each index). All regression included one period lagged endogenous variable if preliminary esing deeced auocorrelaed OLS residuals. Long run resuls This secion describes he saisical resuls of esing he hypohesis ha raes of reurns are equal every weekday in a long run. The ess models were esimaed wih OLS and MAD mehods. OLS and MAD coefficiens of regression are repored in he Table 3. The firs clear observaion is ha he differen esimaion mehods give differen resuls. Coefficiens obained wih OLS mehod equal o mean raes of reurns in differen weekdays. In some cases MAD coefficiens are posiive compared o negaive ones given by OLS mehod or vice versa. For example, for NASDAQ Monday coefficien wih OLS is -0.0039 bu wih MAD 0.0094 is obained. OLS mehod rejecs he hypohesis H : a = a = a = a = a a a 5 % level only for Japan 0 3 4 and Dow Jones. For he oher sixeen socks we canno rejec he hypohesis of equal weekday reurns using sandard F-es disribuion. However MAD mehod rejecs he hypohesis for en sock exchanges (including also Japan and Dow Jones). The ohers are Canada, Hong Kong, Singapore, and Unied Saes wih NASDAQ, Dow Jones and S&P 500, and from Europe Finland, Ialy and Holland. The criical Laplace adjused F-es values are aken from Table. above. Thus MAD esimaion finds more differences in weekday coefficiens han OLS. Noe ha MAD esimaion mehod is more reliable because he residuals of regressions are no normal disribued. Table 3. repors skewness and excess kurosis values for OLS residuals. For normal disribuion boh are zero bu Laplace disribuion has values of zero 5 Ausralia (AORD), Ausria (ATX), Belgium (BFX), Canada (TSE), Denmark (KFX), Finland (HEX) France (CAC), Germany (DAX), Hong Kong (HSI), Ialy (MIBTel), Japan (Nikkei), Holland (AEX), Singapore (STI), Swizerland (SSMI), Unied Kingdom (FTSE) Unied Saes (Dow Jones, NASDAQ and S&P 500) 9

TABLE 3. F-ype ess of weekday anomaly. H0 : a = a = a3 = a4 = a 5. 5% criical values are obained from Table. Bold syle refers o weekday anomaly sock exchanges. Auocorrelaion correced resuls are signed by *. Counry Mehod Monday Tuesday Wednesday Thursday Friday F es Skew. Ex.Kur. Ausralia OLS 0,00035 0,0004-0,0000-0,00004 0,0006 0,4-0,6 8,0 MAD 0,00008 0,00037 0,0000 0,00043 0,0009 0,35 Ausria * OLS 0,00046 0,00054 0,0008-0,0003 0,000 0,54-0,55 4,7 MAD 0,00065 0,00069 0,00047 0,0009 0,0004 0,3 Belgium * OLS 0,000 0,0003-0,00009 0,00037 0,0009 0,7-0,09 5, MAD 0,0008 0,00039 0,00007 0,00033 0,0006 0,4 Canada * OLS 0,00044-0,0003 0,00008 0,00007 0,00053 0,79-0,6 7,34 MAD 0,00094 0,00008 0,0008 0,000 0,00075,83 Denmark * OLS 0,00005 0,0003 0,00033 0,00048 0,0005 0,3-0,3,39 MAD 0,0005 0,0004 0,00078 0,00058 0,00049 0,4 Finland * OLS 0,00033-0,00003-0,00066 0,0064 0,0075,85-0,4 6, MAD 0,00037 0,0007-0,00065 0,000 0,008,8 France OLS -0,000 0,00055-0,000 0,00069 0,0004 0,6-0,03,36 MAD 0,000 0,00063-0,000 0,0007 0,0008 0,77 Germany OLS 0,00073 0,00055-0,00035 0,0005 0,00043 0,5-0,4 3,78 MAD 0,000 0,00088-0,0003 0,0003 0,0007,46 Holland * OLS 0,0037 0,00063-0,00053 0,0007 0,0006,3-0,5 3,9 MAD 0,0096 0,00043-0,0000 0,000 0,00058 3,48 Hong Kong OLS 0,0006 0,00093 0,0005-0,000 0,00044,06 0,8 9,88 MAD -0,00047 0,00084 0,0034-0,00054 0,0034,95 Ialy OLS -0,00048 0,00069-0,00086 0,0005 0,0007,3-0,60 5,79 MAD -0,00007 0,00087-0,00075 0,0009 0,0009 3,04 Japan * OLS -0,003 0,00066-0,00030 0,0005-0,00058 3,7 0,4 3,5 MAD -0,005 0,000-0,00070 0,0004-0,00085 4,0 Singapore OLS 0,000 0,00005 0,00030 0,00056-0,00084,00 0,04,6 MAD -0,0007-0,000 0,00036 0,0000-0,00098, Swizerland OLS 0,000 0,00037 0,00033 0,00044 0,00088 0,34-0,08 4,64 MAD 0,00069 0,00057 0,00043 0,00066 0,00097 0,3 UK OLS 0,0005 0,00044-0,00035 0,00065 0,0006 0,78-0,0,56 MAD 0,0003 0,0004-0,00035 0,00068 0,00048,46 Dow Jones OLS 0,00 0,00037 0,0007-0,000-0,0009,00-0,36 4,8 MAD 0,0078 0,0007 0,00006 0,00003 0,0008 5,69 NASDAQ * OLS -0,00039-0,0007 0,0048 0,006 0,00036,78 0,7 5,83 MAD 0,00094-0,000 0,008 0,000 0,00085 3,30 S&P 500 OLS 0,00079 0,0003 0,0004 0,00006 0,0000 0,57-0,0 3,64 MAD 0,004 0,00003 0,0003-0,00006 0,00039,84 0

and hree. In all cases kurosis values are inside he margins (.36,.6) indicaing ha he Laplace alernaive is more aracive han normal. Figure. repors he esimaed MADcoefficiens in graphical form. The weekday differences are eviden in many counries. Monday Tuesday Wednesday Thursday Friday 0,0050 0,0000 0,0050 0,0000 Rae of Reurn 0,00050 0,00000-0,00050-0,0000-0,0050-0,0000-0,0050. Ausralia Ausria Belgium Canada * Denmark Finland * *) In he sock exchange have weekday anomaly France Germany Holland * Hong Kong Ialy * Japan * Singapore Swizerland UK Down Jones * NASDAQ * S&P 500 * FIGURE. Weekday raes of reurn in 8 sock exchanges Table 4 repors F-ype es values from MAD-esimaion for pair-wise hypohesis H : 0 a (,,,,...,5) i = aj i j i j =. We focus only in hose sock exchanges ha rejeced hypohesis H0 : a = a = a3 = a4 = a 5 above since hypohesis H 0 is nesed in H 0. Thus we repor also OLS resuls for Japan and Dow Jones. Overall endency is ha he beginning of he week differs from he res of he week. Resuls show (see also Table 3. and Figure ) ha here is no jus one weekday whose coefficien is he highes or lowes in every counry. In Dow Jones and S&P 500 indexes and in Holland Monday coefficien is no equal o oher

weekday coefficiens. Monday coefficiens are higher han oher weekday coefficiens in hese indexes. TABLE 4. F-ype es for pair-wise weekday anomaly. H a = a ( i j,.i, j =,,...,5). 5% criical values are obained from Table. Bold syle refers o significan difference. 0 : i j Canada Finland Holland Ialy Japan (OLS) Japan (MAD) Dow Jones (OLS) Dow Jones (MAD) Mon-Tue 6,34 0, 6,70 4,0,66,89,09,33 4,08 7,83 Mon-Wed 4,98,86,5 0,0 6,3 4,4 3,9 6,0,3 5,5 Mon-Thu 6,0 0,7 9,44 5,49 0,4,96 5, 6,43 0,0 8,98 Mon-Fri 0,30 3,58 5,36 4,48 4,0 3,54 6,67,98 0,0 3,85 Tue-Wed 0,09 3,36 0,58 4,8,6,87 0, 0,6 3,00 0, Tue-Thu 0,0 0,5 0,6 0,09 0, 0,00 0,70 0,3 4,7 0,05 Tue-Fri 4,05,06 0,07 0,0,43,59,36 0,00 3,60 0,70 Wed-Thu 0,05 4,78 0,06 6, 0,56,93 0,4 0,00,99 0,46 Wed-Fri,97 0,48,04 5,4 0,3 0,06 0,67 0,8,93 0,4 Thu-Fri 3,79,08 0,59 0,05,5,65 0, 0,35 0,08,09 NASDAQ S&P 500 In Ialy Monday and Wednesday coefficiens are no equal wih coefficien of Tuesday, Thursday and Friday. The Monday and Wednesday coefficiens are lower han oher weekday coefficiens. In Japan, wih boh esimaion mehods, Monday coefficien is srongly negaive and i differs from Tuesday, Wednesday and Thursday coefficiens. Moreover, in MAD esimaion Thursday coefficien differs from Friday coefficien. In Canada Tuesday coefficien differs from Monday, Wednesday and Thursday coefficiens. The reason is ha Tuesday coefficien is negaive. The resuls for Finland and NASDAQ differ from oher counries. In Finland, Wednesday coefficien is no equal wih Thursday and Friday coefficiens. Wednesday coefficien is negaive. Monday coefficien differs also from Friday coefficien. In NASDAQ Tuesday coefficien is no equal wih Monday or any oher weekday because Tuesday coefficien is negaive.

Agrwall and Tandon (994) found weekday anomaly in eigheen counries. They repor ha in every counry Monday or Tuesday coefficiens were he lowes and mos negaive and Friday coefficiens were he highes (see also Kamara 997). Our findings do no suppor hese resuls. The mos remarkable difference in his sudy is ha we found only eigh weekday anomaly socks among he same eigheen sock exchanges. However samples differ in periods. Wih OLS mehod, which hey used, we found only wo weekday anomaly socks. Ineresing finding is ha he reason of seasonaliy is no in Monday s or Tuesday s low coefficien values. Insead Monday coefficien is highes or mos posiive in four of eigh weekday anomaly socks. In addiion we found only from Ialy some evidence of weekend anomaly similar o Abraham and Ikenberry (994). Shor run resuls In he las secion we examined weekday anomaly in differen counries over a 3-year period. Nex we analyze how his seasonaliy changes hrough his period. Daa is analyzed in sequenial over-lapping one-year periods so ha firs period begins in..990 and ends in 3..990. The second period begins in.4.990 and ends 3.3.99. In he same way, we go rough all he daa wih hree monh overlap. Therefore, he maximum coun of periods from one sock exchange is 49. Eigh of eigheen sock exchanges above exhibied he weekday anomaly in long run. The resul was obained wih he robus MAD esimaion mehod. OLS residuals were far from normal having excess kurosis more ypical for Laplace disribuion han normal. Thus we use also in shor run analysis MAD esimaion mehod. Table 5. repors he shor run resuls. Every sock exchange rejecs hypohesis H0 : a = a = a3 = a4 = a 5 0 : a a3 a4 a5 a leas 9 of 49 periods. Indexes from Germany, Ialy, Japan and Unied Saes (Dow Jones and S&P 500) include around 60 % of periods of weekday anomaly. We do no find any counry which has weekday anomaly a every period. Leas seasonaliy is found in Singapore and Ausralia where around 0 % of periods rejec he hypohesis H a = = = =. In shor run mos weekday anomaly periods were found in hose sock exchanges which had weekday anomaly also in long run. There is also one dissimilar case. Germany had shor run seasonaliy while in long period here was no sign of seasonaliy. The reason is following. 3

TABLE 5. Weekday anomaly periods in he differen sock exchange. X means a weekday anomaly period (MAD esimaion, 5% criical es values are aken from Table ) Ausralia Ausria Belgium Canada Denmark Finland France Germany Periods Σ I/90-I/9 X 0 0 0 0 X 0 X 3 II/90-II/9 X 0 X 0 0 0 0 X 0 X 4 III/90-III/9 0 0 X 0 0 0 0 0 0 0 IV/90-IV/9 0 0 X X 0 0 0 0 0 0 0 I/9-I/9 0 0 X 0 X 0 0 X 0 X 0 0 0 4 II/9-II/9 0 X 0 0 0 X X X X X X 0 0 0 7 III/9-III/9 X 0 0 0 0 0 0 0 X 0 0 X X X 5 IV/9-IV/9 X 0 0 0 0 0 X X X 0 0 0 0 0 4 I/9-I/93 0 0 0 0 X 0 X 0 X 0 0 0 X X 5 II/9-II/93 0 X 0 0 0 X 0 0 X 0 0 X X X 6 III/9-III/93 0 X 0 X 0 X 0 X X 0 0 X X X 8 IV/9-IV/93 0 0 0 X 0 X 0 0 X 0 0 0 X X X 6 I/93-I/94 0 X 0 0 X 0 X 0 0 0 X 0 0 X X X 7 II/93-II/94 0 X 0 0 0 X 0 X 0 X 0 X 0 0 X X X 8 III/93-III/94 0 X 0 0 0 0 0 X 0 0 0 X 0 0 X 0 0 4 IV/93-IV/94 0 0 0 0 0 0 0 0 0 0 0 0 X 0 0 X 0 0 I/94-I/95 0 X 0 0 X 0 0 0 0 0 0 X 0 X 0 0 0 0 4 II/94-II/95 0 0 X 0 X 0 0 X 0 0 0 X X 0 0 0 0 0 5 III/94-III/95 0 0 0 0 0 0 0 0 0 0 X 0 0 0 0 0 0 0 IV/94-IV/95 0 0 0 0 X 0 X 0 0 0 X 0 0 0 0 0 0 0 3 I/95-I/96 0 0 0 0 X 0 X X X 0 0 X 0 0 0 0 0 0 5 II/95-II/96 0 X 0 0 X 0 X X X 0 0 X 0 0 0 0 X 0 7 III/95-III/96 0 X 0 X X 0 X X 0 X X X X 0 X 0 0 0 0 IV/95-IV/96 0 X 0 X X X 0 0 0 0 X X 0 0 0 0 0 0 6 I/96-I/97 0 0 0 X 0 X X 0 0 0 X X 0 0 0 0 0 0 5 II/96-II/97 0 0 0 X 0 X 0 X X X X X X 0 0 X 0 X 0 III/96-III/97 0 0 0 X 0 0 0 0 0 0 X X X X 0 X 0 0 6 IV/96-IV/97 X X 0 0 0 0 X X 0 0 X 0 0 0 0 X 0 X 7 I/97-I/98 0 X X 0 0 X X X 0 X X 0 X 0 X X 0 X II/97-II/98 0 X X X 0 X X X X X X X 0 0 X X 0 X 3 III/97-III/98 0 X X X 0 X X X X X 0 X X 0 X X X X 4 IV/97-IV/98 0 X X X X 0 0 X X X 0 X X 0 X X X X 3 I/98-I/99 0 X X X 0 0 0 X 0 X 0 X X X X X X X II/98-II/99 0 0 0 X 0 0 0 X 0 X 0 0 0 X 0 0 X 0 5 III/98-III/99 0 X X X 0 X 0 X 0 0 0 0 0 X 0 X X X 9 IV/98-IV/99 0 X X X X 0 X X 0 X 0 X 0 X X X X X 3 I/99-I/00 X X 0 X 0 X X 0 0 X X X 0 0 X X X X II/99-II/00 X X X X X X X 0 X 0 X X X X X 0 X X 5 III/99-III/00 0 X 0 X X 0 X 0 X X X X X X X 0 X X 3 IV/99-IV/00 0 0 0 0 0 X X X X 0 X X X 0 0 X X X 0 I/00-I/0 X 0 0 0 X X X X 0 X X X 0 0 0 X X X II/00-II/0 X 0 0 X 0 X X X X 0 X X 0 0 X X X X III/00-III/0 0 0 0 X 0 0 X X 0 X 0 0 X 0 X X X X 9 IV/00-IV/0 0 0 0 X 0 0 0 X X X 0 X X 0 X X 0 X 9 I/0-I/0 0 0 0 0 0 0 0 X 0 0 0 0 X 0 0 X 0 X 4 II/0-II/0 0 0 0 0 0 0 0 X 0 X 0 0 0 0 X 0 0 0 3 III/0-III/0 0 0 0 0 0 0 0 0 0 X 0 0 X 0 0 0 0 0 IV/0-IV/0 X 0 0 0 0 X 0 0 X X 0 0 X 0 X X X X 9 I/0-I/03 0 0 0 X 0 0 0 0 0 X 0 0 X 0 X X 0 X 6 Σ 0 8 9 9 9 5 9 0 8 9 8 9 9 350 Holland Hong Kong Ialy Japan Singapore Swizerland UK Dow Jones NASDAQ S&P 500 4

In he early 90s lowes coefficien was found in Wednesday. Conrary o his in middle of decade he highes coefficien was found in Wednesday and lowes were found in he begin and end of he week. In lae 90`s he siuaion changed. The lowes coefficien was in he end of he week and highes in he begin of he week. Japan is disinguishable from he oher sock exchanges in wo ways. The weekday coefficiens are much lower or more negaive han in he oher sock exchanges because of decade of coninuous economic recession. In addiion, he reason for seasonaliy in every period is ha coefficiens of Monday or Friday are he smalles and he highes coefficiens are in he middle of week. In every sock exchange excluding Japan he weekday anomaly appears a Monday. Is coefficien differs from oher weekday coefficiens. However his is he only similariy beween differen sock exchanges in he shor run analysis. Figure. describes how he weekday anomaly periods occur simulaneously in differen sock exchanges. The number of he weekday anomaly periods in sock exchanges rough he analyzed period is no consan. In he early 90s he number of weekday anomaly sock exchanges is around six. In he mid 90s here are only few sock exchanges having weekday anomaly. Jus before he big boom ime in sock markes in he end of 90s he number of weekday anomaly periods sars o rise. This changed in he period beween II/98-II/99. Then only five socks had weekday anomaly. Afer ha number of anomaly periods sars o increase again for a while. Thus here is a posiive relaionship beween number of anomaly socks and marke aciviy. Concerning he moivaion for MAD-esimaion and F-ype es resuls in Table 5. we analyzed also how he excess kurosis of OLS esimaion evolved in ime. Figure 3. repors resuls for he Dow Jones index. Excess kurosis was found in all periods excep during he period I/99-I/00. Similar resuls were found for all oher indexes 5

6 Number of anomaly sock exchanges 4 0 8 6 4 0 I/90-I/9 I/9-I/9 I/9-I/93 I/93-I/94 I/94-I/95 I/95-I/96 I/96-I/97 I/97-I/98 I/98-I/99 I/99-I/00 I/00-I/0 I/0-I/0 I/0-I/03 Time periods FIGURE. Numbers of sock exchanges having he weekday anomaly in he same period. 0 8 Kurosis 6 4 0 4 7 0 3 6 9 5 8 3 34 37 40 43 46 Periods FIGURE 3. Excess kurosis values of he OLS regression residuals in Dow Jones sock reurns index in differen periods. 6

IV. CONCLUSIONS I is a well-known fac ha Monday and Friday have been anomaly days in sock marke worldwide for decades. Numerous sudies have exhibied ha Monday raes of reurn have been lower han Friday reurns. Saisical significan differences in differen weekday raes of reurns are named as weekday anomaly. We examined his weekday anomaly in a new way. To esimae daily raes of reurn in long and shor run we used MAD esimaion mehod besides OLS. The robus MAD mehod was used because i is more efficien when he regression errors show non-normaliy ha is ypical for reurn series. The weekday anomaly was esed wih F-ype ess. The reference es disribuions under non-normaliy, i.e. under Laplace disribuion, were derived wih simulaions. Our sudy covers eigheen sock exchanges in sixeen differen counries during period 990-003. Eigh indexes had a weekday anomaly in he long run when we used MAD esimaion mehod. When OLS mehod was used wih sandard F-es disribuion we found only wo. In four cases he reason was ha Monday coefficien was significanly higher han some oher weekday coefficien. To examine how he weekday anomaly develops and changes, we modified he daa o 49 hree monh over-lapping one-year periods. Only MAD esimaion was used in shor run analysis. We found ha a leas nine weekday anomaly periods in every sock exchange. The mos weekday anomaly periods were found in Germany, Japan, Ialy and he Unied Saes (Dow Jones and S&P 500). I was hard o find similariies in weekday anomalies in differen counries and periods. In some periods weekend anomaly is found only in Ialy. Our sudy did no analyze why raes of reurn in differen weekdays were no equal. Weekday anomaly was no as visible as in previous sudies indicaed even when MAD was used. Acquiring financial benefi from weekday anomaly does no seem o be possible because weekday seasonaliy is unpredicable. The highes or lowes raes of reurn in a specific weekday is no ime invarian. 7

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