The OLS Model Used to Examine the Day-Of-The-Week Effect Is Given By

The OLS model used to examine the day-of-the-week effect is given by:

Rt =  + 1 dummy1 + 2 dummy2 +3 dummy3 +4 dummy4 + e

where e is the error term.

is the return of the benchmark category which is Friday in the analysis

Four dummy variables are created to take care of five day effect in a week. The dummy variables are defined as:

Dummy1 = 1 for Monday, 0 for others

Dummy2 = 1 for Tuesday, 0 for others

Dummy3 = 1 for Wednesday, 0 for others

Dummy4 = 1 for Thursday, 0 for others

To avoid the dummy variable trap we need to use n-1 dummy

The correlogram of log return shows that the log return series is not stationary

Date: 04/27/12 Time: 05:23
Sample: 1 2087
Includedobservations: 2086
Autocorrelation / PartialCorrelation / AC / PAC / Q-Stat / Prob
| | / | | / 1 / 0.005 / 0.005 / 0.0513 / 0.821
| | / | | / 2 / -0.032 / -0.032 / 2.2495 / 0.325
| | / | | / 3 / -0.055 / -0.054 / 8.5202 / 0.036
| | / | | / 4 / 0.064 / 0.064 / 17.102 / 0.002
| | / | | / 5 / -0.056 / -0.060 / 23.582 / 0.000
| | / | | / 6 / 0.000 / 0.002 / 23.582 / 0.001
| | / | | / 7 / 0.009 / 0.013 / 23.761 / 0.001
| | / | | / 8 / 0.030 / 0.019 / 25.639 / 0.001
| | / | | / 9 / -0.057 / -0.050 / 32.460 / 0.000
| | / | | / 10 / -0.007 / -0.007 / 32.572 / 0.000
| | / | | / 11 / 0.020 / 0.018 / 33.371 / 0.000
| | / | | / 12 / 0.057 / 0.049 / 40.220 / 0.000
| | / | | / 13 / 0.022 / 0.031 / 41.199 / 0.000
| | / | | / 14 / 0.008 / 0.008 / 41.350 / 0.000
| | / | | / 15 / -0.023 / -0.020 / 42.456 / 0.000
|* | / |* | / 16 / 0.092 / 0.094 / 60.435 / 0.000
| | / | | / 17 / 0.038 / 0.042 / 63.500 / 0.000
| | / | | / 18 / -0.027 / -0.026 / 65.054 / 0.000
*| | / | | / 19 / -0.070 / -0.057 / 75.244 / 0.000
| | / | | / 20 / -0.013 / -0.025 / 75.609 / 0.000
T

However ADF test shows that the series is stationary because the prob. Is smaller than 0.05.

The ADF test is more reliable then the correlogram so we stick to the last result.

Null Hypothesis: WRETURN has a unit root
Exogenous: Constant
Lag Length: 0 (Automatic based on SIC, MAXLAG=25)
t-Statistic / Prob.*
Augmented Dickey-Fuller test statistic / -45.40720 / 0.0001
Test criticalvalues: / 1% level / -3.433288
5% level / -2.862724
10% level / -2.567447
*MacKinnon (1996) one-sided p-values.
Augmented Dickey-Fuller Test Equation
DependentVariable: D(WRETURN)
Method: LeastSquares
Date: 04/27/12 Time: 05:24
Sample (adjusted): 3 2087
Includedobservations: 2085 afteradjustments
Variable / Coefficient / Std. Error / t-Statistic / Prob.
WRETURN(-1) / -0.995045 / 0.021914 / -45.40720 / 0.0000
C / -0.000266 / 0.000334 / -0.797877 / 0.4250
R-squared / 0.497444 / Meandependentvar / 5.38E-06
AdjustedR-squared / 0.497203 / S.D. dependentvar / 0.021481
S.E. ofregression / 0.015232 / Akaike info criterion / -5.529935
Sum squaredresid / 0.483257 / Schwarzcriterion / -5.524522
Log likelihood / 5766.957 / Hannan-Quinncriter. / -5.527951
F-statistic / 2061.814 / Durbin-Watsonstat / 1.999284
Prob(F-statistic) / 0.000000

To study the effect of day of the week, we first compute the descriptive statistics for individual day which is presented in the following table.

Statistics / Monday / Tuesday / Wednesday / Thursday / Friday
Mean
SD
No. obs. / -0.000513
0.014969
414 / -0.000615
0.014429
421 / 0.000019
0.014717
422 / 0.000016
0.014362
418 / -0.000244
0.01406
414

The table suggests that wednesday has relatively higher average return than rest of the days of a week followed closely by thursday. Also,Monday, Tuesday andFriday have negative average return.

The following table presents the OLS results of the day-of-the-week effects in the Italian stock market for the given period.

DependentVariable: WRETURN
Method: LeastSquares
Date: 04/24/12 Time: 10:43
Sample (adjusted): 2 2087
Includedobservations: 2086 afteradjustments
Variable / Coefficient / Std. Error / t-Statistic / Prob.
C / -0.000244 / 0.000752 / -0.324361 / 0.7457
DUMMY1 / -0.000269 / 0.001061 / -0.253533 / 0.7999
DUMMY2 / -0.000371 / 0.001057 / -0.351587 / 0.7252
DUMMY3 / 0.000263 / 0.001056 / 0.248672 / 0.8036
DUMMY4 / 0.000260 / 0.001058 / 0.245265 / 0.8063
R-squared / 0.000298 / Meandependentvar / -0.000267
AdjustedR-squared / -0.001623 / S.D. dependentvar / 0.015224
S.E. ofregression / 0.015237 / Akaike info criterion / -5.527810
Sum squaredresid / 0.483126 / Schwarzcriterion / -5.514284
Log likelihood / 5770.506 / Hannan-Quinncriter. / -5.522854
F-statistic / 0.155271 / Durbin-Watsonstat / 1.989742
Prob(F-statistic) / 0.960669

Returns of Monday, Tuesday, Wednesday and Thursday can be found out by deducting thecoefficients of these days from the benchmark day, that is, Friday.

The results in the table show that all the p-values of dummy variables are greater than the level of significance 0.01. This suggests that there is no such statistically significant the day of the week effect exist in the market. However, only Wednesday and Thursday have a positive effect on the stock market and rest has positive effect on stock market.

However, return series exhibits autoregressive conditional heteroskedasticity (ARCH) effects. Probability is < 0.05

Heteroskedasticity Test: ARCH
F-statistic / 85.75678 / Prob. F(1,2083) / 0.0000
Obs*R-squared / 82.44488 / Prob. Chi-Square(1) / 0.0000

Test of both types have shown that volatility has been presented.

We now estimate a standard Garch(1,1)

DependentVariable: WRETURN
Method: ML - ARCH (Marquardt) - Normal distribution
Date: 04/27/12 Time: 06:04
Sample (adjusted): 2 2087
Includedobservations: 2086 afteradjustments
Convergenceachievedafter 14 iterations
Presamplevariance: backcast (parameter = 0.7)
GARCH = C(6) + C(7)*RESID(-1)^2 + C(8)*GARCH(-1)
Variable / Coefficient / Std. Error / z-Statistic / Prob.
C / 0.000479 / 0.000438 / 1.093089 / 0.2744
DUMMY1 / -0.000157 / 0.000570 / -0.275228 / 0.7831
DUMMY2 / -0.000468 / 0.000618 / -0.757113 / 0.4490
DUMMY3 / 0.000610 / 0.000617 / 0.989907 / 0.3222
DUMMY4 / -0.000242 / 0.000600 / -0.403649 / 0.6865
VarianceEquation
C / 1.28E-06 / 2.93E-07 / 4.387526 / 0.0000
RESID(-1)^2 / 0.096829 / 0.008961 / 10.80500 / 0.0000
GARCH(-1) / 0.899973 / 0.009487 / 94.86517 / 0.0000
R-squared / -0.002129 / Meandependentvar / -0.000267
AdjustedR-squared / -0.005505 / S.D. dependentvar / 0.015224
S.E. ofregression / 0.015266 / Akaike info criterion / -6.110635
Sum squaredresid / 0.484300 / Schwarzcriterion / -6.088994
Log likelihood / 6381.393 / Hannan-Quinncriter. / -6.102706
Durbin-Watsonstat / 1.985844

ARCH LM test also indicate that there is no ARCH effect in residual now

There’s no pattern in the residual. However none of the coefficient are significant at 5% level

which indicate that there is no week end effect.

Heteroskedasticity Test: ARCH
F-statistic / 2.399517 / Prob. F(1,2083) / 0.1215
Obs*R-squared / 2.399058 / Prob. Chi-Square(1) / 0.1214

EGARCH

DependentVariable: WRETURN
Method: ML - ARCH (Marquardt) - Normal distribution
Date: 04/29/12 Time: 13:22
Sample (adjusted): 2 2087
Includedobservations: 2086 afteradjustments
Convergenceachievedafter 19 iterations
Presamplevariance: backcast (parameter = 0.7)
LOG(GARCH) = C(6) + C(7)*ABS(RESID(-1)/@SQRT(GARCH(-1))) + C(8)
*RESID(-1)/@SQRT(GARCH(-1)) + C(9)*LOG(GARCH(-1))
Variable / Coefficient / Std. Error / z-Statistic / Prob.
C / 4.36E-05 / 0.000410 / 0.106363 / 0.9153
DUMMY1 / 4.13E-05 / 0.000558 / 0.073967 / 0.9410
DUMMY2 / -0.000414 / 0.000570 / -0.727269 / 0.4671
DUMMY3 / 0.000671 / 0.000584 / 1.148681 / 0.2507
DUMMY4 / -0.000231 / 0.000568 / -0.407233 / 0.6838
VarianceEquation
C(6) / -0.231089 / 0.026852 / -8.605951 / 0.0000
C(7) / 0.124280 / 0.016167 / 7.687044 / 0.0000
C(8) / -0.109773 / 0.008859 / -12.39078 / 0.0000
C(9) / 0.985017 / 0.002071 / 475.7227 / 0.0000
R-squared / -0.000589 / Meandependentvar / -0.000267
AdjustedR-squared / -0.004443 / S.D. dependentvar / 0.015224
S.E. ofregression / 0.015258 / Akaike info criterion / -6.149474
Sum squaredresid / 0.483555 / Schwarzcriterion / -6.125128
Log likelihood / 6422.902 / Hannan-Quinncriter. / -6.140554
Durbin-Watsonstat / 1.989074