Ols Regression Results For Day Of The Week

Published Date: 02 Nov 2017

The regression results testing for the presence of day-of-the-week effect are shown in Table VII.

As reported in Panel A, all the coefficients of SEMDEX are insignificant at the 5% level except for that of Friday. This shows a positive Friday effect meaning that Fridays tend to be always positive. As it can be seen the return on Fridays are the highest. According to Lenkkeri et al. (2006), Fridays are known to experience positive returns. The coefficient value of 0.000502, suggests that on average, SEMDEX returns are around 0.005% higher on Fridays than the average for other days of the week. Apart from the Friday effect, the result indicates that the average daily returns for SEMDEX are independent for the other days of the week. However we conclude that SEMDEX is weak-form inefficient because of the Friday effect. On the other hand, all the t-statistics for DEMEX are negative and statistically insignificant for all days of the week. This means that no day-of-the-week effect is observed for that index.

Furthermore, when a dummy is included for the crisis, as reported in Panel B, the results change considerably and is better explained, proven by the increased adjusted R-squared. The crisis variable is noted to be significant for DEMEX. Remarkably, this explains 0.027% of the variations in the daily returns of DEMEX. However, the coefficients of the other days of the week variables are proven to be statistically insignificant, hence suggesting that return from the other days of the week are independent. Despite, after accounting for the volatility caused by the crisis in the stock market, both SEMDEX and DEMEX are at the conclusion that they are weak-form efficient.

Table VII: OLS Regression Results for Day of the Week Effect

Panel A shows the results for the regression which was run on the daily log returns by applying the formula for the two indices over the period January 2007 to October 2012. Panel B shows the regression results which was run for the same sample data and over the same time period, but including a dummy variable representing the crisis period. The equation used was

Panel A: Results of Regression without the ‘Crisis’ Dummy variable

SEMDEX

DEMEX

Monday

-0.0000674

-0.000111

(-0.29)

(-0.827)

Tuesday

0.00000179

-0.0000765

(0.00774)

(-0.572)

Wednesday

0.000056

0.000236

(0.241)

(1.763)

Thursday

-0.0000297

-0.000118

(-0.128)

(-0.882)

Friday

0.000502

0.000172

(2.16)*

(1.282)

R-squared

0.00265

0.00421

Adjusted R-squared

0.000022

0.00159

Coefficients are given in each cell followed by t-ratios in parentheses; * and ** denote significance at the 5% and 1%, levels respectively.

Panel B: Results of Regression with the ‘Crisis’ Dummy variable

SEMDEX

DEMEX

Monday

-0.000240

-0.000273

(-0.901)

(-1.78)

Tuesday

-0.000171

-0.000239

(-0.641)

(-1.557)

Wednesday

-0.000117

0.0000726

(-0.44)

(0.473)

Thursday

-0.000203

-0.000281

(-0.76)

(-1.829)

Friday

0.000329

0.00000884

(1.233)

(0.0575)

Crisis

0.000281

0.000265

(1.319)

(2.156)*

R-squared

0.00379

0.00726

Adjusted R-squared

0.000509

0.00398

Coefficients are given in each cell followed by t-ratios in parentheses; * and ** denote significance at the 5% and 1%, levels respectively.

5.7 OLS Regression Results for Month of the Year Effect

The regression results testing for the presence of the month-of-the-year effect are shown in Table VIII. As reported in Panel A, none of the coefficients are significant at the 5% level. Economically, this means that no such anomaly is observed in those indices.

Likewise, the regression was run a second time with the inclusion of a variable for the event of the crisis. The crisis variable has reported to be significant for SEMDEX, with a negative coefficient of -2.2%. However SEMDEX did not show any seasonal effects therefore it can be concluded being weak-form efficient. On the other hand DEMEX exhibited a positive month-of-the-year effect. It reported a positive significant coefficient for June at 1.4% therefore we can conclude that DEMEX has a positive month of the year effect. This may be due to the disclosure of interim reports around that period of the year causing a June effect.

Table VIII: OLS Regression Results for Month of the Year Effect

Panel A shows the results for the regression which was run on the monthly log returns by applying the formula for the two indices over the period January 2007 to October 2012. Panel B shows the regression results which was run for the same sample data and over the same time period, but including a dummy variable representing the crisis period. The equation used was

Panel A: Results of Regression without the ‘Crisis’ Dummy variable

SEMDEX

DEMEX

January

0.00461

-0.00166

(0.444)

(-0.270)

February

-0.0168

-0.000874

(-1.622)

(-0.143)

March

0.00861

-0.0000977

(0.83)

(-0.016)

April

0.00815

0.00166

(0.786)

(0.272)

May

0.00637

-0.000111

(0.614)

(-0.0181)

June

0.00114

0.0122

(1.0979)

(1.991)

July

0.000104

-0.00426

(0.0101)

(-0.696)

August

-0.00907

-0.00582

(-0.875)

(-0.949)

September

0.00814

0.00174

(0.785)

(0.283)

October

-0.00117

0.000476

(-0.1128)

(0.0777)

November

-0.00694

-0.00945

(-0.611)

(-1.407)

December

0.00737

0.00399

(0.649)

(0.595)

R-squared

0.115

0.12

Adjusted R-squared

-0.0524

-0.0465

Coefficients are given in each cell followed by t-ratios in parentheses; * and ** denote significance at the 5% and 1%, levels respectively.

Panel A: Results of Regression with the ‘Crisis’ Dummy variable

SEMDEX

DEMEX

January

0.012

0.00216

(1.214)

(0.361)

February

-0.0094

0.00294

(-0.948)

(0.492)

March

0.016

0.00372

(1.617)

(0.622)

April

0.0119

0.00357

(1.222)

(0.61)

May

0.0101

0.0018

(1.0389)

(0.307)

June

0.0151

0.0141

(1.555)

(2.409)*

July

0.00382

-0.00235

(0.393)

(-0.402)

August

-0.00536

-0.00391

(-0.552)

(-0.667)

September

0.0119

0.00365

(1.221)

(0.623)

October

0.00254

0.00239

(0.262)

(0.407)

November

-0.00249

-0.00715

(-0.234)

(-1.114)

December

0.0118

0.00628

(1.11)

(0.978)

Crisis

-0.0223

-0.0115

(-3.19)**

(-2.72)

R-squared

0.249

0.221

Adjusted R-squared

0.0914

0.0574

Coefficients are given in each cell followed by t-ratios in parentheses; * and ** denote significance at the 5% and 1%, levels respectively.

Part-II:-Volatility-Modelling

5.8 Diagnosing-for-ARCH-Effects

From Table IX, it is shown that the residuals from the OLS regression, used to diagnose ARCH effects for the daily returns, reports significant autocorrelation for SEMDEX at all lags while for DEMEX autocorrelation is significant at the 1st and 3rd lag only. Additionally, the Ljung-Box test proves to be significant at all lags for both indices.

Conversely, these ARCH effects are reduced when they are modelled under regressions that take into account volatility. This holds true under the GARCH model, some autocorrelation remains up to lag 1 for SEMDEX. However, given this autocorrelation is low, the selection of a GARCH-(1,1) is-quite-satisfactory.

Likewise, the LM-ARCH test proves to be significant under the OLS estimation but disappears under the GARCH except for SEMDEX. Additionally, the Jarque-Bera statistics is rejected under all models, concluding that the residuals do not follow normal distribution, but it improves under GARCH models. Therefore, the GARCH model assesses the day-of-the-week effect better than the linear estimation whereby several misspecifications are found.

Observed in Table X, the month-of-the-year models show different results. The residuals from the OLS regression, used to diagnose ARCH effects for the daily returns, exhibits significant autocorrelation at lag 2 for SEMDEX and the Ljung-Box test also proves to be jointly significant at all lags. However the ARCH effects are reduced under the GARCH model.

On the other hand, DEMEX reports insignificant autocorrelation and the Ljung-Box test prove the same. Likewise, the LM-ARCH test proves to be insignificant under both the OLS estimation as well as GARCH for both SEMDEX and DEMEX. Importantly, normality of the residuals is respected under the GARCH modelling approach for both indices.

TABLE IX: Diagnosing ARCH effects under different regression approaches for the day-of-the-week

This table shows the results of ARCH tests performed on the residuals under two different model estimation methods. The tests included will help to assess the suitability of each methodology in modelling for the day-of-the-week. Presence of significant ARCH effects are denoted by an * tested at the 5% significance level which will undermine the validity of the coefficient estimates of that model. AC stands for autocorrelation and PAC stands for partial autocorrelation.

OLS

GARCH

Lag

ACF

PACF

PROB

ACF

PACF

PROB

Autocorrelation of

1

0.426*

0.426

0.000

0.111*

0.111

0.000

Squared standardized

2

0.178*

-0.004

0.000

0.020

0.008

0.000

SEMDEX

Residuals

3

0.146*

0.087

0.000

-0.043

-0.047

0.000

4

0.086*

-0.008

0.000

-0.022

-0.012

0.000

5

0.076*

0.040

0.000

-0.023

-0.018

0.000

Heteroskedasticity

F-Statistics

Prob

F-Statistics

Prob

Test: ARCH

336.768

0.000

19.0265

0.000

Normality test of

Jarque-Bera

JB-stat

Jarque-Bera

JB-stat

Residuals

18368.25

0.000

1021.897

0.000

OLS

GARCH

Lag

ACF

PACF

PROB

ACF

PACF

PROB

Autocorrelation of

1

0.226*

0.226

0.000

0.008

0.008

0.741

Squared standardized

2

0.051

-0.001

0.000

-0.028

-0.028

0.531

DEMEX

Residuals

3

0.067*

0.059

0.000

0.002

0.003

0.735

4

-0.000

-0.030

0.000

-0.035

-0.036

0.530

5

0.040

0.047

0.000

0.008

0.009

0.658

Heteroskedasticity

F-Statistics

Prob

F-Statistics

Prob

Test: ARCH

81.884

0.000

0.109

0.742

Normality test of

Jarque-Bera

JB-stat

Jarque-Bera

JB-stat

Residuals

5553.236

0.000

2981.730

0.000

TABLE X: Diagnosing ARCH effects under different regression approaches for the month-of-the-year

This table shows the results of ARCH tests performed on the residuals under two different model estimation methods. The tests included will help to assess the suitability of each methodology in modelling for the month-of-the-year. Presence of significant ARCH effects are denoted by an * tested at the 5% significance level which will undermine the validity of the coefficient estimates of that model. AC stands for autocorrelation and PAC stands for partial autocorrelation.

OLS

GARCH

Lag

ACF

PACF

PROB

ACF

PACF

PROB

Autocorrelation of

1

0.236

0.236

0.048

0.152

0.152

0.194

Squared standardized

2

0.300*

0.259

0.006

-0.128

-0.155

0.234

SEMDEX

Residuals

3

0.268

0.175

0.001

0.236

0.295

0.069

4

0.145

0.000

0.002

0.404*

0.323

0.001

5

0.159

0.031

0.002

0.064

0.033

0.001

Heteroskedasticity

F-Statistics

Prob

F-Statistics

Prob

Test: ARCH

3.791

0.0559

1.601

0.21

Normality test of

Jarque-Bera

JB-stat

Jarque-Bera

JB-stat

Residuals

50.362

0.000

1.769

0.413

OLS

GARCH

Lag

ACF

PACF

PROB

ACF

PACF

PROB

Autocorrelation of

1

0.030

0.030

0.805

-0.039

-0.039

0.737

Squared standardized

2

-0.078

-0.079

0.781

-0.086

-0.088

0.716

DEMEX

Residuals

3

0.070

0.075

0.838

0.189

0.183

0.341

4

-0.013

-0.025

0.930

0.111

0.121

0.369

5

0.034

0.048

0.967

-0.004

0.038

0.510

Heteroskedasticity

F-Statistics

Prob

F-Statistics

Prob

Test: ARCH

0.0561

0.814

0.105

0.746

Normality test of

Jarque-Bera

JB-stat

Jarque-Bera

JB-stat

Residuals

19.729

0.000052

1.02

0.6

5.9 Modified-GARCH-Model

Day-of-the-Week-Effect

The results are presented in Table XI and two regressions were run, one with the crisis variable and one without it. It can be seen that the log-likelihood of the regression increases when the crisis variable is included.

Surprisingly, the Friday effect noted for SEMDEX has disappeared under the volatility model and is showing significant negative return on Monday instead, indicating that the returns on Monday are lowest than the other days. This pattern of negative Monday returns is consistent to Coutts et al. (2000). Remarkably, no such anomaly was reported for DEMEX under OLS as well as the GARCH model.

However when the dummy for crisis is included in the model, both SEMDEX and DEMEX showed considerable day-of-the-week effect, unlike reported under the OLS model. SEMDEX has significant positive returns on Wednesday, Thursday and Friday as well with a negative significant return for the crisis. On the other hand, DEMEX exhibits significant negative return on Monday, Tuesday, Thursday and a positive significant return for the crisis.

Considering the variance equation, the ARCH and GARCH coefficients are significant at 5% level for all constituents proving that the GARCH model is valid. The sum of these two coefficients, being nearly 1, reports the momentum effects that existed during the crisis. It is vital to understand that when the sum of the significant coefficients on the lagged squared error-(ARCH) and lagged conditional variance-(GARCH) is close to 1, it implies that there is volatility clustering-whereby shocks-will-be persistent (Cont, 2005). Furthermore, both SEMDEX and DEMEX exhibit volatility on the different days of the week. However, the volatility is very small, being close to zero.

The highest volatility occurs on Thursday for SEMDEX while for that of DEMEX it occurs on Wednesday when the crisis variable is not included. Furthermore, the lowest volatility occurs on Thursday for DEMEX.

Likewise when the crisis dummy is introduced, the highest volatility is reported on Monday for both SEMDEX and DEMEX. The lowest volatility is reported on Thursday for SEMDEX and on Tuesday for DEMEX. The crisis variable is proved to be negatively significant for both SEMDEX and DEMEX.

Table XI: GARCH Regression Results for Day of the Week

On this table, the column X shows the results for the regression which was run on the daily log returns by applying the mean formula and variance formula for the two indices over the period January 2007 to October 2012. Column X* shows the regression results which was run for the same sample data and over the same time period, but including a dummy variable representing the crisis period. * = significant at the 5% level.

Panel A: Results of GARCH Regression without the ‘Crisis’ Dummy variable

SEMDEX DEMEX

X

X*

X

X*

Mean

Monday

-0.000264*

0.000131

-0.0000606

-0.000326*

Tuesday

0.0000229

0.000306

-0.0002

-0.000390*

Wednesday

0.000128

0.000364**

0.000157

-0.0000707

Thursday

0.0000103

0.000393*

-0.00014

-0.000353*

Friday

0.000212

0.000503*

0.0000681

-0.00012

Crisis

-0.000415*

0.000303*

Variance

Constant

0.000000190

0.0000032*

0.00000217*

0.00000339*

Tuesday

0.0000000425

0.000000183

-0.00000102*

-0.00000107*

Wednesday

0.000000713

0.000000501

0.000000647**

0.00000132*

Thursday

-0.000000806**

-0.000000941*

-0.00000159*

-0.00000138*

Friday

-0.000000175

0.000000392

-0.0000017*

-0.00000141*

Crisis

-0.00000259*

-0.00000151*

ARCH

0.242*

0.54*

0.305*

0.253*

GARCH

0.792*

0.462*

0.469*

0.402*

Log Likelihood

6768.533

6814.771

7205.936

7237.321

Month-of-the-Year-Effect

As reported in table XII, it is clear the DEMEX exhibits negative month of the year effect which is consistent with the OLS result. However, some results not shown under OLS model is that there is significant positive return in April and December while significant negative return in August for SEMDEX.

On the other hand, when the crisis dummy is included DEMEX showed a positive significant return for August. While SEMDEX had again a significant positive return in April and December while significant negative return in August. Moreover, no volatility is experienced for any indices as expected for low frequency data.

Table XII: GARCH Regression results for Month-of-the-year

On this table, the column X shows the results for the regression which was run on the daily log returns by applying the mean formula and variance formula for the two indices over the period January 2007 to October 2012. Column X* shows the regression results which was run for the same sample data and over the same time period, but including a dummy variable representing the crisis period. * = significant at the 5% level.

Panel A: Results of GARCH Regression

SEMDEX DEMEX

X

X*

X

X*

Mean

January

0.00830

0.0134

-0.000687

0.000942

February

-0.0102

-0.000761

0.00175

0.00366

March

-0.000633

0.00231

-0.00330

-0.00165

April

0.00697**

0.0108*

0.00102

0.00363

May

-0.00156

-0.00195

-0.000698

0.000766

June

0.00838

0.0109

0.0126

0.0139

July

-0.000745

0.00153

-0.00668

-0.00367

August

-0.0103*

-0.00917*

-0.00629

-0.00631*

September

0.0106

0.0110

-0.000604

-0.000162

October

0.00561

0.00406

0.00263

0.00481

November

0.00144

0.000512

-0.00358

-0.00455

December

0.00883*

0.00799*

0.00386

0.00453

Crisis

-0.0168

-0.00809

Variance

Constant

0.000453

0.000387

0.000169

0.000145

February

-0.000176

-0.000173

-0.000167

-0.000150

March

-0.000639

-0.000501

-0.000222

-0.000184

April

-0.000503

-0.000449

-0.0000999

-0.000108

May

-0.000333

-0.000275

-0.0000755

-0.0000158

June

-0.000385

-0.000314

-0.0000788

-0.0000638

July

-0.000352

-0.000325

-0.000136

-0.000148

August

-0.000543

-0.000452

-0.000250

-0.000208

September

-0.000130

-0.000197

-0.000159

-0.000127

October

-0.0000266

0.000118

0.000174

0.0000564

November

-0.000685

-0.000705

-0.000368

-0.000253

December

-0.000542

-0.000416

-0.000122

-0.000119

Crisis

0.000151

0.0000771

ARCH

0.186

0.168

0.0975

0.0862

GARCH

0.545

0.521

0.576

0.581

Log Likelihood

185.746

191.167

218.995

219.419

6. CONCLUSION

This study attempts to test the findings of Roberts (1959) who concluded that stocks follow a random walk. In order to test the weak form of EMH the autocorrelation testing, variance ratio tests, run test and calendar effects testing, made under OLS regression as well as the GARCH were used. Two indices namely the DEMEX and SEMDEX are tested by using both daily and monthly return data for the period from 1st January 2007 to 31st October 2012.

The empirical evidence obtained from these studies is mixed. While some studies show empirical results that reject the null hypothesis of weak form market efficiency, the others report evidence to support the weak form of EMH.

The autocorrelation test shows evidence to reject the null hypothesis of random walk for all studied series using both daily and monthly returns. Additionally, the run test indicates that the null hypothesis of random walk is rejected for daily returns of SEMDEX and DEMEX while not rejected for the monthly series. The results of the Lo and MacKinley’s variance ratio test under both homoscedasticity and heteroscedaticity assumptions for both SEMDEX and DEMEX fails to support weak-form-efficiency for the daily returns. Additionally, when the monthly returns are used, random walk under the assumption of homoscedasticity is rejected for both indices however under the heteroscedasticity variance ratio test only DEMEX rejected random walk while SEMDEX proved to be weak form efficient.

Validated under both a daily perspective as well as on a monthly one, the returns of DEMEX has consistently proven to follow a random walk while SEMDEX has shown the contrary. It showcased a day-of-the-week effect as well as a month-of-the-year effect. SEMDEX showed evidence that it does follow a random walk when tested using monthly data but these effects vanished under the GARCH-M monthly perspective. Also under OLS, it had positive significant Friday effect while under the GARCH-M model it displayed positive Monday effect.

However when the crisis dummy was introduced under OLS, DEMEX which earlier proved to be weak-form efficient now exhibits positive return of around 1.4% in June. While under the GARCH-M, it showed positive significant returns in August. On the other hand, under the GARCH-M SEMDEX reported positive return on Wednesday, Thursday and Friday. Moreover it exhibited positive return in April and December while negative return in August. Both SEMDEX and DEMEX may have exhibited such trend due to the crisis where fear and lack of confidence among investors were governing the market.

These results will help investors earning abnormal profits by devising a trading rule to exploit those detected anomalies (Lim et al., 2004). Nevertheless, several reasons may cause an investor not successfully reap profits from exploiting these calendar effects. Firstly, transaction costs might be more than the potential gain and thus making the transaction not profitable especially if it is small. Secondly, there may be reasons external to market such as the timing of public announcement of interest rate changes which result in the uncertainty as to whether the calendar effects will materialize.

6.1 Limitations of the study

Due to lack of resources and time, only the two indices were used to see whether they were weak-form efficient or not.

Also the GARCH family could have been used, however only the M-GARCH was used.

An analysis of the different sectors constituting of the SEM could have been tested for weak-form-efficiency.