1. Assignment 3
For these assignment i have chosen the next indices:
A) DAX= DAX measures the performance of the Prime Standard’s German companies in terms
of order book volume and market capitalization.
B) NASDAQ= It is the largest electronic screen-based equity securities trading market in the
United States and second-largest by market capitalization in the world.
C) STI= The Straits Times Index (STI) is a market value-weighted stock market index based on
the stocks of 30 representative companies listed on the Singapore Exchange.
1. Stationarity
A stationary time series is one whose statistical properties such as mean, variance, autocorrelation,
etc. are all constant over time A stationarized series is relatively easy to predict: you simply predict that
its statistical properties will be the same in the future as they have been in the past!
DAX Probability = 0.9559 >0.5 => The series is not stationary
Close Straits Times Probability = 0.9850 >0.5 => The series is not stationary
Close Nasdaq Probability =0.9990 > 0.5 => The series is not stationary
DAX Return Probability = 0.000 <0.5 => The series is stationary
Close Straits Return Times Probability = 0.000 <0.5 => The series is stationary
Close Nasdaq Return Probability =0.000 < 0.5 => The series is stationary
As a conclusion we can see that while the series are not stationary their return is always
stationary.
2. a) Mean = The arithmetic mean is the "standard" average, often simply called the
"mean".
The mean is the arithmetic average of a set of values, or distribution
b) Standard Deviation = It shows how much variation or "dispersion" there is from the
"average" (mean, or expected/budgeted value). A low standard deviation indicates that the data points
tend to be very close to themean, whereas high standard deviation indicates that the data are spread
out over a large range of values. σ = 2
σ =
n
xxi∑ − 2
)(

2. c) Skewness is a measure of the asymmetry of the probability distribution of a real-
valued random variable. The skewness value can be positive or negative, or even undefined.
Qualitatively, a negative skew indicates that the tail on the left side of the probability density function
is longer than the right side and the bulk of the values (including the median) lie to the right of the mean.
A positive skew indicates that the tail on the right side is longer than the left side and the bulk of the
values lie to the left of the mean. A zero value indicates that the values are relatively evenly distributed
on both sides of the mean, typically but not necessarily implying a symmetric distribution. sˆ = 1/n*
∑=
−n
i
i xx
1
3
)(
σ
d) Kurtosis is a measure of the "peakedness" of the probability distribution of a real-
valued random variable. Higher kurtosis means more of the variance is the result of infrequent
extreme deviations, as opposed to frequent modestly sized deviations.
a) Series: RETURN_DAX
Mean -0.0000726
Std. Dev. 0.018587
Skewness -0.316750
Kurtosis 8.864634
Mean Dax Return -0.0000726 The mean of Dax Return beginning with 24 January 2008 to 21
January 2011.
daxσ = 0.018587 The degree of dispersion of DAX’s returns from the mean value is 0.018587.
daxsˆ = -0.316750 The degree of asymmetry of DEX RETURN around its mean is -0.316750. In this
case, the skewness is negative and that indicates a distribution with an asymmetric tail extending
towards more negative values.
k dax = 8.864634 Positive kurtosis indicates a relatively peaked distribution of the returns.
b) Series: RETURN_NASDAQ
Mean -0.000096
Std. Dev. 0.019965

3. Skewness 0.148356
Kurtosis 7.568775
Mean NASDAQ Return= -0.000096 The mean of Dax Return beginning with 24 January 2008 to
21 January 2011.
nasdaqσ = 0.019965The degree of dispersion of NASDAQ’s returns from the mean value is
0.019965.
nasdaqsˆ = 0.148356 The degree of asymmetry of NASDAQ RETURN around its mean is 0.148356.
In this case, the skewness is positive and that indicates that the tail on the right side is longer than the
left side and the bulk of the values lie to the left of the mean.
k nasdaq = 7.568775 Positive kurtosis indicates a relatively peaked distribution of the returns.
c) Series: RETURN_STRAITS
Mean -0.0000998
Std. Dev. 0.016293
Skewness 0.060702
Kurtosis 7.385313
Mean STRAITS Return= -0.0000998 The mean of STRAITS Return beginning with 24 January 2008
to 21 January 2011.
straitsσ = 0.016293 The degree of dispersion of STRAITS’s returns from the mean value is
0.016293.
straitssˆ = 0.060702 The degree of asymmetry of STRAITS RETURN around its mean is 0.060702. In
this case, the skewness is positive and that indicates that the tail on the right side is longer than the left
side and the bulk of the values lie to the left of the mean.

4. k straits = 7.385313 Positive kurtosis indicates a relatively peaked distribution of the returns.
3. From Eviews corelograms I noticed:
a) For Dax Return, autocorrelations are not outside the 95% confidence interval so it has
no lag significance.
b) For Nasdaq Return, autocorrelations are not outside the 95% confidence interval so it
has no lag significance.
c) For Straits Return, the autocorrelation for the 19 lag is higher then 95% confidence
level so it has one lag significance.
d) For Squared Dax Return , autocorrelations for the 5,9,11,16 and 19 log are outside the
95% confidence interval so it displays a positive correlation with the past.
e) For Squared Nasdaq Return , autocorrelations for most of the logs are outside the 95%
confidence interval so it displays a positive correlation with the past.
f) For Squared Straits Return , autocorrelations for the first 16 logs are outside the 95%
confidence interval so it displays a positive correlation with the past.
4. The data was processed using Eviews. The following results have been obtained :
a) Series: DAXRESIDUALS
Mean 0.005560
Std. Dev. 0.011342
Skewness 5.451352
Kurtosis 44.15096

5. Jarque-Bera 50366.01
Mean Dax Residual Return 0.005560 The mean of Dax Residual Return beginning with 24
January 2008 to 21 January 2011.
daxσ = 0.011342 The degree of dispersion of DAX’s Residual returns from the mean value is
0.011342.
daxsˆ = 5.451352 The degree of asymmetry of DEX RETURN around its mean is 5.451352. In this
case, the skewness is negative and that indicates a distribution with an asymmetric tail extending
towards more positive values.
k dax = 50366.01 Positive kurtosis indicates a relatively peaked distribution of the returns.
Jarque-Bera=50366.01 the test indicates that the residuals are not normally distributed
because the value is bigger than 5.99 for the 95% confidence interval.
b) Series: NASDAQRESIDULAS
Mean 0.005769
Std. Dev. 0.010349
Skewness 4.042342
Kurtosis 26.74072
Jarque-Bera 17480.52
Mean NASDAQ Return= 0.005769 The mean of Dax Return beginning with 24 January 2008 to 21
January 2011.
nasdaqσ = 0.010349 The degree of dispersion of NASDAQ’s returns from the mean value is
0.010349.

6. nasdaqsˆ = 4.042342 The degree of asymmetry of NASDAQ RETURN around its mean is 4.042342.
In this case, the skewness is positive and that indicates that the tail on the right side is longer than the
left side and the bulk of the values lie to the left of the mean.
k nasdaq = 26.74072 Positive kurtosis indicates a relatively peaked distribution of the returns.
Jarque-Bera= 17480.52the test indicates that the residuals are not normally distributed because
the value is bigger than 5.99 for the 95% confidence interval.
c) Series: STRAITSRESIDUALS
Mean 0.004665
Std. Dev. 0.008389
Skewness 3.974143
Kurtosis 25.18203
Jarque-Bera 15430.43
Mean STRAITS Return= -0.0000998 The mean of STRAITS Return beginning with 24 January 2008
to 21 January 2011.
straitsσ = 0.016293 The degree of dispersion of STRAITS’s returns from the mean value is
0.016293.
straitssˆ = 0.060702 The degree of asymmetry of STRAITS RETURN around its mean is 0.060702. In
this case, the skewness is positive and that indicates that the tail on the right side is longer than the left
side and the bulk of the values lie to the left of the mean.
k straits = 7.385313 Positive kurtosis indicates a relatively peaked distribution of the returns.
Jarque-Bera= 15430.43 the test indicates that the residuals are not normally distributed
because the value is bigger than 5.99 for the 95% confidence interval.

7. 5. The data was processed using Eviews. The following results have been obtained :
I) Log-returns for frequencies of 5 days
a) Series: DAX_5DAYS_RETURN
Mean 1.001150
Std. Dev. 0.044660
Skewness 1.916070
Kurtosis 13.57932
Mean DAX 5 days Return=1.001150 The mean of Dax 5 days Return beginning with 24 January
2008 to 21 January 2011.
daxσ = 0.044660 The degree of dispersion of Dax 5 days Return from the mean value is
0.044660.
daxsˆ = 1.916070 The degree of asymmetry of Dax 5 days Return around its mean is 1.916070.
In this case, the skewness is negative and that indicates a distribution with an asymmetric tail extending
towards more positive values.
k dax = 13.57932 Positive kurtosis indicates a relatively peaked distribution of the returns.
b) Series: NASDAQ_5DAYS_RETURN
Mean 0.998590
Std. Dev. 0.041086
Skewness 0.706087
Kurtosis 5.651077

8. Mean NASDAQ 5 days Return= 0.998590 The mean of NASDAQ 5 days Return beginning with 24
January 2008 to 21 January 2011.
nasdaqσ = 0.041086 The degree of dispersion of NASDAQ 5 days Return from the mean value is
0.041086.
nasdaqsˆ = 0.706087 The degree of asymmetry of NASDAQ RETURN around its mean is
0.706087. In this case, the skewness is positive and that indicates that the tail on the right side
is longer than the left side and the bulk of the values lie to the left of the mean.
k nasdaq = 5.651077 Positive kurtosis indicates a relatively peaked distribution of the returns.
c) Series: STRAITS_5DAYS_RETURN
Mean 0.999475
Std. Dev. 0.034050
Skewness 0.241597
Kurtosis 5.857704
Mean 5 days STRAITS Return= 0.999475 The mean of 5 days STRAITS Return beginning with 24
January 2008 to 21 January 2011.
straitsσ = 0.034050 The degree of dispersion of 5 days STRAITS Return from the mean value is
0.034050
straitssˆ = 0.241597 The degree of asymmetry of 5 days STRAITS Return around its mean is
0.241597. In this case, the skewness is positive and that indicates that the tail on the right side
is longer than the left side and the bulk of the values lie to the left of the mean.
k straits = 5.857704 Positive kurtosis indicates a relatively peaked distribution of the returns.
II) Log-returns for frequencies of 10 days

9. a) Series: DAX_10DAYS_RETURN
Mean 0.999805
Std. Dev. 0.048112
Skewness 2.717345
Kurtosis 14.08545
Mean DAX 5 days Return= 0.999805 The mean of Dax 10 days Return beginning with 24 January
2008 to 21 January 2011.
daxσ = 0.048112The degree of dispersion of Dax 10 days Return from the mean value is
0.044660.
daxsˆ = 2.717345 The degree of asymmetry of Dax 10 days Return around its mean is 2.717345.
In this case, the skewness is negative and that indicates a distribution with an asymmetric tail extending
towards more positive values.
k dax = 14.08545 Positive kurtosis indicates a relatively peaked distribution of the returns.
b) Series: NASDAQ_10DAYS_RETURN
Mean 0.997335
Std. Dev. 0.053475
Skewness 0.921093
Kurtosis 3.788631
Mean NASDAQ 5 days Return= 0.997335 The mean of NASDAQ 10 days Return beginning with
24 January 2008 to 21 January 2011.
nasdaqσ = 0.053475 The degree of dispersion of NASDAQ 10 days Return from the mean value is
0.053475.
nasdaqsˆ = 0.921093 The degree of asymmetry of NASDAQ RETURN around its mean is
0.921093 . In this case, the skewness is positive and that indicates that the tail on the right side
is longer than the left side and the bulk of the values lie to the left of the mean.

10. k nasdaq = 3.788631 Positive kurtosis indicates a relatively peaked distribution of the returns.
c) Series: STRAITS_10DAYS_RETURN
Mean 0.998625
Std. Dev. 0.046587
Skewness -0.479419
Kurtosis 6.806883
Mean 5 days STRAITS Return= 0.998625 The mean of 10 days STRAITS Return beginning with
24 January 2008 to 21 January 2011.
straitsσ = 0.046587 The degree of dispersion of 10 days STRAITS Return from the mean value is
0.034050
straitssˆ = -0.479419 The degree of asymmetry of 10 days STRAITS Return around its mean is
-0.479419 . In this case, the skewness is negative and that indicates a distribution with an asymmetric tail
extending towards more negative values.
k straits = 6.806883 Positive kurtosis indicates a relatively peaked distribution of the returns.
III) Log-returns for frequencies of 15 days
Series: DAX_15DAYS_RETURN
Mean 1.004933
Std. Dev. 0.082282
Skewness 1.988530

11. Kurtosis 8.554455
Mean DAX 15 days Return= 1.004933 The mean of Dax 10 days Return beginning with 24
January 2008 to 21 January 2011.
daxσ = 0.082282 The degree of dispersion of Dax 10 days Return from the mean value is
0.044660.
daxsˆ = 1.988530The degree of asymmetry of Dax 15 days Return around its mean is 1.988530.
In this case, the skewness is negative and that indicates a distribution with an asymmetric tail extending
towards more positive values.
k dax = 8.554455 Positive kurtosis indicates a relatively peaked distribution of the returns.
Series: NASDAQ_15DAYS_RETURN
Mean 0.998056
Std. Dev. 0.069368
Skewness 1.399526
Kurtosis 6.662932
Mean NASDAQ 15 days Return= 0.998056 The mean of NASDAQ 10 days Return beginning with
24 January 2008 to 21 January 2011.
nasdaqσ = 0.069368 The degree of dispersion of NASDAQ 15 days Return from the mean value is
0.069368.
nasdaqsˆ = 1.399526 The degree of asymmetry of NASDAQ 15 days RETURN around its mean is
1.399526. In this case, the skewness is positive and that indicates that the tail on the right side
is longer than the left side and the bulk of the values lie to the left of the mean.
k nasdaq = 6.662932 Positive kurtosis indicates a relatively peaked distribution of the returns.

12. Series: STRAITS_15DAYS_RETURN
Mean 1.003328
Std. Dev. 0.078096
Skewness 0.944518
Kurtosis 5.322960
Mean 15 days STRAITS Return= 1.003328 The mean of 10 days STRAITS Return beginning with
24 January 2008 to 21 January 2011.
straitsσ = 0.078096 The degree of dispersion of 10 days STRAITS Return from the mean value is
00.078096
straitssˆ = 0.944518The degree of asymmetry of 15 days STRAITS Return around its mean is 0.944518. In
this case, the skewness is positive and that indicates a distribution with an asymmetric tail extending
towards more positive values.
k straits = 5.322960 Positive kurtosis indicates a relatively peaked distribution of the returns.
Comparing the values obtained for DAX return after changing the frequency I observed
that the mean drops at a 10 days frequency and rises again in a 15 days frequency model. The
deviation increases with the decrease of frequency. The skewness and kurtosis rises at a 10 days
frequency and drops again in a 15 days frequency model.
Comparing the values obtained for NASDAQ return after changing the frequency I
observed that the mean drops at a 10 days frequency and rises again in a 15 days frequency
model. The deviation and skewness increases with the decrease of frequency. The and kurtosis
rises at a 10 days frequency and drops again in a 15 days frequency model.
Comparing the values obtained for STRAITS return after changing the frequency I
observed that the mean drops at a 10 days frequency and rises again in a 15 days frequency
model. The deviation and increases with the decrease of frequency. The and kurtosis rises at a
10 days frequency and drops again in a 15 days frequency model while the skewness drops to a
negative value and rises again in the 15 day frequency model.

13. 6)
a) Dependent Variable: RETURN_DAX
Coefficient Std. Error z-Statistic Prob.
Variance Equation
C(1) -0.111533 0.041487 -2.688393 0.0072
C(2) 0.129004 0.023117 5.580383 0.0000
C(3) -0.109685 0.021877 -5.013675 0.0000
C(4) 0.998945 0.003880 257.4412 0.0000
R-squared -0.000015 Mean dependent var -7.26E-05
Adjusted R-squared -0.004547 S.D. dependent var 0.018587
S.E. of regression 0.018629 Akaike info criterion -5.491641
Sum squared resid 0.229747 Schwarz criterion -5.464606
Log likelihood 1832.716 Durbin-Watson stat 2.042920
b)Dependent Variable: RETURN_NASDAQ
Coefficient Std. Error z-Statistic Prob.
Variance Equation
C(1) -0.094950 0.046627 -2.036364 0.0417
C(2) 0.146588 0.029146 5.029482 0.0000
C(3) -0.134664 0.021809 -6.174808 0.0000
C(4) 1.003370 0.004265 235.2335 0.0000
R-squared -0.000023 Mean dependent var -9.60E-05
Adjusted R-squared -0.004548 S.D. dependent var 0.019965

14. S.E. of regression 0.020011 Akaike info criterion -5.516938
Sum squared resid 0.265486 Schwarz criterion -5.489935
Log likelihood 1843.899 Durbin-Watson stat 2.214705
c) Dependent Variable: RETURN_STRAITS
Coefficient Std. Error z-Statistic Prob.
Variance Equation
C(1) -0.158625 0.043395 -3.655374 0.0003
C(2) 0.157902 0.031443 5.021830 0.0000
C(3) -0.047799 0.016566 -2.885446 0.0039
C(4) 0.996002 0.003622 274.9522 0.0000
R-squared -0.000038 Mean dependent var -9.98E-05
Adjusted R-squared -0.004563 S.D. dependent var 0.016293
S.E. of regression 0.016330 Akaike info criterion -5.927856
Sum squared resid 0.176804 Schwarz criterion -5.900852
Log likelihood 1980.940 Durbin-Watson stat 1.979613