distribution almost decays to a trivial level. This primary evidence implies that skewness might be diversified away, while the kurtosis is not diversifiable through
the portfolio. Finally, autocorrelations exist in many of the individual stock return series, and
in all stock portfolios and the value-weighted stock index return series. This is consistent with the results of Kim and Kon 1994 which showed that the significant
autocorrelation coefficients for the indexes are most likely a result of the nonsyn- chronous trading effect.
4. Empirical analysis
Tables 2 – 11 show the estimated coefficients for various models as well as tests of various hypotheses for each individual stock and portfolio return. To save space,
we only show the detailed results for two individual stocks, the ten-stock portfolio, and the value-weighted stock index. We summarize the results for the 54 individual
stocks in Tables 6 and 7.
4
.
1
. The case of indi6idual stocks Table 2 and Table 3 are estimation results for the Formosa Plastic. In the
stationary normal model Section 2.1.1, the estimated parameters and the statisti- cal tests are consistent with those in Table 1. The skewness and kurtosis tests show
the return distribution is not normal. The insignificant Ljung – Box Q12 test for the standardized residual implies that autocorrelation does not exist in the stock
return series. On the other hand, the Ljung – Box Q12 test is significant for the squared standardized residual implying that autoregressive heteroscedasticity might
exist in the stock return series. The result of Section 2.1.2 shows that the GARCH parameters are statistically significant. After including the GARCH process, the
volatility parameter s is significantly lower compared to that of Section 2.1.1, meaning that a significant part of the volatility can be explained by the GARCH
model. Moreover, there is a negative leverage effect on the return volatility in the case of Formosa Plastic. These GARCH factors account for a significant part of
the return volatility, which can be judged from the low Ljung – Box Q12 statistics for the squared standardized residual in Section 2.1.2. Both of the two ARMA
parameters are not significant revealing that there is no autocorrelation existing in the return process, which is consistent with previous results. As a whole, the
ARMA – GARCH model with the conditional normal distribution is significant according to the x
2
test and the Schwarz criterion, while it is still unable to explain the skewness and excess kurtosis of the stock return distribution.
Since the conditional normal distribution seems unable to describe stock returns, next we use the Student-t distribution. The result for Section 2.2.1 shows the d.f.
parameter for the model is 2.4 meaning the observed high kurtosis may be partly captured by the parameter
3
. Although the model is superior to Sections 2.1.1 and
3
In order to explain the observed kurtosis relative to the stationary normal, the d.f. parameter for the Student-t model should be in the range from 2 to 10.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
380 Table 2
Parameters estimation for various models Formosa Plastic Parameters for models
Models GARCH
Leverage effect Diffusion
JumpStudent tmixed normal ARMA
m
j
m
2
s
j
s
2
a b
f c
d mm
1
ss
1
ln 0.0559
0.0057 1
a
0.0022 0.0015
0.1918 −
0.1809 0.6798
0.3708 −
0.2356 2
b
0.0046 0.0171
0.0117 0.1001
0.1116 0.0420
0.0790 0.0789
0.0016 0.0023
0.0779 2.4098
3
c
0.2543 0.0185
0.0015 0.0594
0.2953 0.1602
0.1918 4
d
0.0857 0.0021
0.0591 3.5458
0.0545 0.0522
0.1832 0.1031
0.1898 0.0031
0.0539 0.0087
0.0688 0.0006
0.0254 0.5139
0.0098 5
e
0.0017 0.0021
0.1037 0.0050
0.0068 0.3896
− 0.3817
0.0875 −
0.0107 0.0278
0.0395 −
0.0011 6
f
0.0064 0.6690
0.0272 0.0176
0.0120 0.3322
0.3799 0.7275
0.0038 1.0155
0.0383 0.2923
0.0090 0.0009
0.0851 0.0263
0.6372 7
g
0.0137 0.0058
0.0498 0.0061
0.0018 0.0019
0.0596 −
0.0020 0.1895
− 0.1812
0.5576 0.3522
− 0.0676
0.0000 0.9699
0.0856 8
h
0.0010 0.0128
0.1508 0.1667
0.0559 0.0774
0.0999 0.0034
0.0140 0.0286
a
R
t
= m
t
+ h
t
Z
t
, m
t
= m
, h
t
= s
2
. Z
t
follows a standard normal distribution.
b
R
t
= m
t
+ h
t
Z
t
, m
t
= m+a
R
t−1
+ bo
t−1
, h
t
= s
2
+ f
h
t−1
+ co
t−1 2
+ d
·max0,−o
t−1 2
.
c
R
t
= m
t
+ h
t
T
t
, m
t
= m
, h
t
= s
2
. T
t
follows a Student-t distribution.
d
R
t
= m
t
+ h
t
T
t
, m
t
= m+a
R
t−1
+ bo
t−1
, h
t
= s
2
+ f
h
t−1
+ co
t−1 2
+ d
·max0,−o
t−1 2
.
e
R
t
= m
t
+ h
t
P
t
, m
t
= m+lm
J
, h
t
= s
2
+ l
m
J 2
+ s
J 2
. P
t
follows a mixed Poisson–normal distribution.
f
R
t
= m
t
+ h
t
P
t
, m
t
= m+lm
J
+ a
R
t−1
+ bo
t−1
, h
t
= s
2
+ l
m
J 2
+ s
J 2
+fh
t−1
+ co
t−1 2
+ d
·max0,−o
t−1 2
.
g
R
t
= m
t
+ h
t
M
t
, m
t
= m
j
, h
t
= s
j 2
. M
t
follows a mixed normal distribution.
h
R
t
= m
t
+ h
t
M
t
, m
t
=
j = 1 L
l
j
m
j
+ a
R
t−1
+ bo
t−1
, h
t
=
j = 1 L
l
j
m
j 2
+ s
j 2
−
j = 1 L
l
j
m
j 2
+ f
h
t−1
+ co
t−1 2
+ d
·max0,−o
t−1 2
. Denotes that the test is statistically significant at the level of 5. Numbers in parentheses are standard deviations for a parameter estimation.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
381 Table 3
Statistical tests for various models Formosa Plastic l
R, u Standardized residual
Squared standardized Models
x
2
test Schwarz
residual criterion
h
Normality test Ljung–Box
Ljung–Box Q12
i
Excess kurtosis Skewness
Q12
i
347.211 11.5232
146.7104 1
942.232 –
937.761 0.6416
3.3605 3.1303
297.380 5.8546
3.8633 0.5665
x
1 2
981.116 2
1003.725 =
122.986
,a
1008.370 3.3605
347.211 11.5232
146.7104 –
998.671 3
0.6416 3.8505
1011.013 462.941
0.4788 0.0004
x
1 2
= 5.286
985.142 0.7788
4 3.3605
347.211 11.5232
146.7104 0.6416
5 1002.727
1018.892 x
2 2
= 153.32
,b
x
1 2
= 38.554
3.4576 369.957
10.3557 143.9673
0.6768 x
3 2
6 1038.169
1005.853 =
68.888
,c
x
4 2
= 191.874
,d
x
5 2
0.6416 3.3605
347.211 11.5232
146.7104 1017.013
7 1000.848
= 149.562
,e
x
1 2
= 96.572
1032.984 0.6809
3.5703 391.808
5.8308 4.8721
x
6 2
1065.299 8
= 123.148
,f
x
7 2
= 246.134
,g a
x
1 2
is the statistic for the hypothesis test H :a = b = f = c = d = 0 with the corresponding alternative Sections 2.1.2, 2.2.2 and 2.3.2 or Section 2.4.2 is
true. It is a x
2
distribution with 5 d.f.
b
x
2 2
is the statistic for the hypothesis test H :l = m
J
= s
J
= 0 with the alternative that Section 2.3.1 is true. It is a x
2
distribution with 3 d.f.
c
x
3 2
is the statistic for the hypothesis test H :l = m
J
= s
J
= 0 with the alternative that Section 2.3.2 is true. It is a x
2
distribution with 3 d.f.
d
x
4 2
is the statistic for the hypothesis test H :l = m
J
= s
J
= a = b = f = c = d =
0 with the alternative that Section 2.3.2 is true. It is a x
2
distribution with 8 d.f.
e
x
5 2
is the statistic for the hypothesis test H :l = m
2
= s
2
= 0 with the alternative that Section 2.4.1 is true. It is a x
2
distribution with 3 d.f.
f
x
6 2
is the statistic for the hypothesis test H :l = m
2
= s
2
= 0 with the alternative that Section 2.4.2 is true. It is a x
2
distribution with 3 d.f.
g
x
7 2
is the statistic for the hypothesis test H :l = m
2
= s
2
= a = b = f = c = d =
0 with the alternative that Section 2.4.2 is true. It is a x
2
distribution with 8 d.f.
h
The value of the Schwarz criterion is lR, u−
1 2
p·ln T, where p is the number of parameters estimated in the model.
i
The statistic is Q12 = TT+2
k = 1 12
[r
k 2
T−k] which is a x
2
distribution with 12 d.f. The critical value of the test is 12.59 at the significance level of 5.
Denotes that the test is statistically significant at the level of 5.
2.1.2 based on the Schwarz criterion, it is unable to explain the conditional heterscedasticity in the return series. Section 2.2.2 is the combination of the
ARMA – GARCH and the Student-t model. The GARCH parameters in the model are not significant. The chi-square test x
1 2
and the Schwarz criterion confirm that Section 2.2.2 is not significant relative to Section 2.2.1. If the Student-t-GARCH
model generates the sample data, the standardized residuals should be an i.i.d. Student-t distribution. While, as shown in Table 3, the skewness for the standard-
ized residuals either for Section 2.2.1 or Section 2.2.2 is still significantly positive, implying that the conditional distribution of stock returns may not be the Student-t
distribution. This is consistent with Brorsen and Yang 1994. Also, as noted by Vlaar and Palm 1993, symmetric distributions such as the normal, Student-t or
generalized error distribution are unlikely to give appropriate results.
To deal with the skewness in stock return distribution, we examine several mixtures of normal distribution models. Section 2.3.1 is the Poisson – normal
jump-diffusion model. According to Table 3, the model is superior to the previous model. The chi-square test x
2 2
shows the model is better than the stationary normal model. The Schwarz criterion shows it is better than any other previous
models. The mean parameter for jump magnitude m
J
is positive consistent with positive skewness of stock return distribution. With the jump component added to
the simple diffusion model Section 2.3.1, the value of s decreases further, and the value of m diminishes to a non-significant level. This means that the jump in the
stock price explains most of the mean and some portion of the volatility of stock returns. In Section 2.3.1, the mean and variance of stock returns can be found by
ER
t
= m + lm
J
and VarR
t
= s
2
+ l
s
J 2
+ m
J 2
, which would be equivalent to the mean and variance obtained in Section 2.1.1. The jump intensity parameter l in
Section 2.3.1 is estimated as 0.5139, which can be explained as, on average, a jump in stock returns happens every 2 weeks or so. The mean parameter for jump
magnitude m
J
is estimated as 0.0098, which is significantly positive, consistent with the positive skewness in the return distribution.
Unfortunately, Section 2.3.1 can not explain the conditional heteroscedasticity in the return process. Section 2.3.2 combines the ARMA – GARCH model and the
Poisson – normal jump-diffusion model. Since both jump and GARCH components have been found in stock returns, the observed leptokurtosis in stock returns can be
explained by either of the two models. The question arises as to which of the two processes provides a superior description of the data. The result shows the GARCH
parameters are small and only one is significant compared to those from Section 2.1.2. Thus in Section 2.3.2, only a small part of conditional heterscedasitcity can
be explained. This can be confirmed by the Ljung – Box test for the square standardized residual in Table 3. Overall Section 2.3.2 is still a’posteriori more
probable than the previous models according the Schwarz criterion.
In the GARCH jump-diffusion model, Section 2.3.2, the GARCH parameters are small compared to those of Jorion 1988. The volatility parameter of the jump part
is lower because it is partly explained by GARCH parameters. On the other hand, the volatility parameter of the diffusion part is at the same level as in Section 2.3.1
4
.
4
This is a general case for almost all of the sample individual stock and portfolio returns. This can be found in Table 6.
This implies that in the GARCH jump-diffusion model, only the jump volatility is related to the conditional heteroscedasticity. This makes sense since volatility
clustering happens when jumps happen in the return process. The results are very different from those of Jorion 1988 and others. Their models only allow the
GARCH process in the diffusion part. Thus they accredit the conditional het- eroscedasticity to the diffusion part only. It is obviously not appropriate for the
case of our study, since most of the conditional heteroscedasticity is associated with the jump part. Our model can consider the GARCH process for both the diffusion
part and jump part and to tell which component is more related to GARCH. Our empirical results show the disadvantage of the Jorion 1988 model in that it is
unable to explain the GARCH phenomenon in the jump process.
One of the probable reasons why the GARCH parameters are not significant in Section 2.3.2 is its complication in structure. In the Jorion 1988 model, the
GARCH process is only in the diffusion part and the jump component is additional to the diffusion GARCH model. In Section 2.3.2, the GARCH is dependent upon
the mixture of the jump component and the diffusion part. Presumably, the kurtosis of stock returns is largely explained by the Poisson jump diffusion model, thus
leaving the GARCH factor non-significant. Moreover, since the probability density function of a Poisson jump diffusion distribution involves infinite summation,
truncation errors as noted by Vlaar and Palm 1993 matter very much in Section 2.3.2. This complication may cause estimation errors resulting in non-significant
GARCH parameters in Section 2.3.2.
Section 2.4.1 is the mixed – normal model that is similar to the Bernoulli jump model suggested by Ball and Torous 1983. This model has an advantage in its
simplicity in structure. The probability density function can have an exact form rather than involving infinite summation as in the Poisson jump-diffusion model.
The result shows the estimated parameters in Section 2.4.1 are all significant with the exception of one of the mean parameters. As Vlaar and Palm 1993 and Kim
and Kon 1994 noted, the mixture of three normal distributions may be preferred. For parsimonious purposes, we only use the mixture of two normal distributions.
This is especially desirable when we combine the mixture of normal distributions and the GARCH model. In Section 2.4.1, m
1
and m
2
are significantly different consistent with non-zero skewness in the return distribution. l is significantly
different from zero consistent with the high kurtosis in the return distribution. The mean and variance of stock returns can be found by ER
t
= lm
1
+ 1 − lm
2
and VarR
t
= ls
1 2
+ 1 − ls
2 2
, which would be equivalent to the mean and variance obtained in the Section 2.1.1. Section 2.4.2 shows that the combination of the
GARCH model and the mixed – normal model is the best model according to x
2
tests and the Schwarz criterion. In Section 2.4.2, the two volatility parameters s
1
and s
2
are significantly lower than those of Section 2.4.1. This implies that a significant part of the volatility of the mixed – normal process can be explained by
the large and significant GARCH parameters. Our results also differ from those of Vlaar and Palm 1993. Vlaar and Palm 1993 concluded that the GARCH results
for the normal – Poisson mixture and normal – Bernoulli mixture were very similar. Our model is different from theirs in that we allow GARCH phenomena in the
whole normal – jump mixture, while their model, similar to Jorion 1988, only allows the GARCH phenomenon in the diffusion – normal component.
Why the GARCH parameters for the Student-t distribution in Section 2.2.2 and jump-diffusion process in Section 2.3.2 are insignificant is that they may absorb a
large part of heteroscedasticity, explaining most of the kurtosis in the unconditional stock distribution.
Table 4 and Table 5 show the results for another individual stock, Cathay Life Ins. The results are similar to those of the Formosa Plastic except that most ARMA
parameters are significant, which is consistent with the results in Table 1 showing autocorrelation exists in the return series of the Cathay Life Ins. stock. Other
detailed explanations are not repeated here.
As to other sample stocks, Table 6 and Table 7 summarizes the results for the 54 individual stocks. Most of the sample stocks show results similar to those of
Taiwan Plastic and Cathay Life Ins. Almost all sample stocks have their x
2
tests significant at the level of 5 in any case. This implies that the normal distribution
is not appropriate for individual stock returns. And the ARMA – GARCH model is significant under any of the four distributions assumed. Moreover, according to the
Schwarz criterion, 39 out of the 54 sample stocks have their SC largest for Section 2.4.2, meaning that the mixed – normal – GARCH is the most probable model for
individual stock returns. In addition, the leverage effect is not significant for most of the individual stocks.
4
.
2
. The case of stock portfolios Table 8 and Table 9 show the estimated coefficients and hypothesis tests for the
ten-stock portfolio returns
5
. The results are different from those of individual stocks in some respects. First with respect to their jump behaviors in Section 2.3.1
and Section 2.3.2, the jump intensity parameter l is, in general, smaller than that for most of the individual stocks, although it is still statistically significant. This
might be because diversification can eliminate some jump risks in the portfolio. Moreover, with the jump component included in the model, the mean for the
diffusion model m does not change a lot, while the mean of jump size m
J
becomes non-significant from zero. Thus although the jump component persists, the jump
becomes symmetric, and the mean of the jump magnitude is zero in a stock portfolio. In other words, the jump risk cannot be fully diversified, while the
skewness is diversifiable through the portfolio.
Table 10 and Table 11 show the results for the value-weighted stock index. The results are quite similar to those of the ten-stock portfolio. From the results of the
stock portfolio, one can conclude that that in the Taiwan stock market, the jump and the GARCH components persist, while the jump size becomes symmetric in
stock portfolios. In other words, although kurtosis cannot be fully diversified, skewness is diversifiable through the portfolio.
5
This is one of our random portfolios from repeated experiments. To save space, other results for ten-stock, 30-stock, 50-stock portfolio are available from the author for interested readers upon requests.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
385 Table 4
Parameters estimation for various models Cathay Life Ins.
a
Models Parameters for models
GARCH Leverage effect
ARMA JumpStudent tmixed normal
Diffusion a
b f
c d
s
j
s
2
mm
1
ss
1
m
j
m
2
ln 0.0088
0.0655 1
0.0025 0.0018
0.8368 −
0.7521 0.6028
0.3343 −
0.0796 2
0.0013 0.0222
0.0010 0.1163
0.1470 0.0621
0.0720 0.0724
0.0222 0.0018
0.1098 2.2478
3 0.1816
0.0298 0.0015
4 0.0643
0.0021 0.0097
0.1674 0.1831
0.4556 0.0649
2.6186 0.0376
0.0518 0.8237
0.5190 1.3101
0.0021 0.2428
0.0465 0.0779
− 0.0024
0.0274 0.5460
0.0205 5
0.0018 0.0023
0.0922 0.0059
0.0068 0.6506
− 0.6254
0.1631 −
0.0067 0.0078
0.0534 −
0.0035 6
0.0110 0.3025
0.0211 0.0033
0.0010 0.0174
0.0192 0.0193
0.0088 0.0054
0.0018 0.0363
0.0021 0.0267
0.0958 7
− 0.0025
0.0285 0.6116
0.0057 0.0485
0.0070 0.0019
0.0022 0.0685
− 0.0054
0.5362 −
0.4503 0.2514
0.4852 −
0.1056 0.0000
0.7765 0.0323
8 0.0107
0.0071 0.2230
0.2406 0.0640
0.0995 0.1334
0.0019 0.0045
0.0572
a
See notes in Table 2.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
386
Table 5 Statistical tests for various models Cathay Life Ins.
a
Models Standardized residual
Squared standardized x
2
test l
R, u Schwarz
residual criterion
Normality test Ljung–Box
Ljung–Box Q12 Excess kurtosis
Skewness Q12
839.683 2.8738
272.273 40.1609
208.5069 833.217
1 0.6870
2.0217 188.267
11.1598 15.8973
0.8557 899.811
2 922.421
x
1 2
= 165.476
272.273 40.1609
208.5069 3
906.346 896.647
0.6870 2.8738
167.550 29.4607
71.9340 1.9789
0.7629 4
900.653 x
1 2
= 40.294
926493 2.8738
928.371 272.273
40.1609 208.5069
x
2 2
= 69.376
912.206 0.6870
5 x
1 2
= 50.702
3.7705 953.722
415.817 26.6647
204.1013 x
3 2
= 62.602
921.407 0.5656
6 x
4 2
= 120.078
272.273 40.1609
208.5069 0.6870
2.8738 908.340
x
5 2
= 61.644
924.505 7
x
1 2
= 82.348
185.523 15.5569
35.7187 2.0921
8 0.7964
965.679 x
6 2
= 86.516
933.365 x
7 2
= 251.992
a
See notes in Table 3.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
387 Table 6
Parameters estimation for various models 54 individual stocks
a
Models Parameters for models
Diffusion GARCH
Leverage ARMA
JumpStudent tmixed normal effect
a b
f c
d s
j
s
2
mm
1
ss
1
m
j
m
2
ln 0.0059
b
0.0060 1
50
c
54 0.3261
− 0.2753
0.6862 0.2536
− 0.0393
2 0.0034
0.0192 11
32 35
41 38
12 54
0.0030 0.0913
3.2133 3
52 53
49 0.0847
4 0.0762
0.0026 0.3095
0.1628 0.0777
0.0697 2.9238
28 31
25 21
9 9
52 53
0.0683 −
0.0013 0.0280
0.8348 0.0089
5 9
53 52
38 54
0.4969 −
0.4476 0.1087
− 0.0163
0.0549 0.0505
− 0.0013
6 0.0072
0.5728 0.0335
52 9
26 28
27 29
11 54
52 38
0.0142 0.0948
7 0.0327
− 0.0004
0.5707 54
52 38
49 53
0.0345 0.0171
0.2492 −
0.2072 0.5064
0.2978 −
0.0140 0.0303
0.0760 0.0204
8 38
54 33
31 42
38 13
53 43
48
a
See notes in Table 2.
b
Numbers in this table without brackets are the average values of the estimated coefficients for the 54 individual stocks.
c
Numbers in this table within brackets are the number of stocks that exhibit statistical significance for the estimated coefficients at a significance level of 5.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
388
Table 7 Statistical tests for various models 54 individual stocks
a
Standardized residual x
2
test Squared standardized
Schwarz l
R, u Models
criterion residual
Skewness Excess kurtosis
Normality test Ljung–Box
Ljung–Box Q12 Q12
823.03 0.3647
829.50
b
1 5.8506
46
c
54 54
36 52
[0] [0]
d
1.9761 0.4621
879.0787 2
901.6883 x
1 2
: 54 46
[3] 52
6 7
[4] 43
2.8506 875.6714
865.9722 0.3647
3 54
36 52
54 46
[6] [5]
4 849.2544
x
1 2
: 43 823.4148
0.5561 63.3415
54 11
26 52
[0] 48
[1] 5.8506
887.57 x
2 2
: 53 871.41
0.3647 5
46 54
54 36
52 [0]
[2] 0.7960
10.0540 6
907.6779 x
1 2
: 52 875.3627
54 9
23 54
52 [2]
x
3 2
: 48 x
4 2
: 54 [3]
0.3647 2.8506
x
5 2
: 54 7
866.5971 882.7625
54 46
52 54
46 [0]
[0] 2.1325
921.3983 x
1 2
: 54 889.0831
0.4735 8
49 54
54 4
6 x
6 2
: 54 x
7 2
: 54 [39]
[43]
a
See notes in Table 3.
b
Numbers in this table without brackets are the average values of the estimated statistics for the 54 individual stocks.
c
Numbers within the bracket are the number of stocks that exhibit statistical significance for the estimated coefficients at a significance level of 5.
d
Numbers within the bracket [] are the number of stocks that have the largest statistics among the 54 individual stocks.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
389 Table 8
Parameters estimation for various models ten-stock portfolio
a
Models Parameters for models
GARCH Leverage effect
ARMA JumpStudent tmixed normal
Diffusion a
b f
c d
s
j
s
2
mm
1
ss
1
m
j
m
2
ln 0.0057
0.0521 1
0.0021 0.0015
0.7020 −
0.6627 0.6515
0.3162 −
0.0017 2
0.0018 0.0123
0.0017 0.2758
0.2902 0.0489
0.0654 0.0823
0.0016 0.0055
0.0557 3.2827
3 0.4766
0.0048 0.0017
4 0.0943
0.0051 0.0099
0.1495 0.0996
0.4984 0.0581
2.8829 0.0384
0.4057 0.0459
0.0484 0.0975
0.0019 0.0074
0.0023 0.0750
0.0051 0.0320
0.3017 0.0019
5 0.0018
0.0026 0.0986
0.0073 0.0107
0.5337 −
0.5007 0.1240
− 0.0334
0.0828 0.0500
0.0030 6
0.0050 0.0429
0.0403 0.0105
0.0013 0.0164
0.0149 0.0243
0.0008 0.0158
0.0017 0.0209
0.0066 0.0075
0.0893 7
0.0052 0.0326
0.7596 0.0085
0.0575 0.0081
0.0019 0.0023
0.0152 0.0063
0.6566 −
0.6087 0.6270
0.3023 0.0483
0.0481 0.0726
0.0019 8
0.0019 0.0054
0.2557 0.2691
0.0544 0.0719
0.0973 0.0145
0.0152 0.0323
a
See notes in Table 2.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
390
Table 9 Statistical tests for various models ten-stock portfolio
a
Standardized residual x
2
test Squared standardized
Schwarz l
R, u Models
criterion residual
Excess kurtosis Normality test
Ljung–Box Ljung–Box Q12
Skewness Q12
360.759 29.3196
468.8059 976.717
0.1772 3.6495
983.184 1
1098.854 0.9465
25.590 13.5815
7.5544 x
1 2
= 231.340
1076.245 2
0.1202 360.759
3 29.3196
1036.327 468.8059
1026.628 0.1772
3.6495 158.767
19.0939 193.6974
2.3354 x
1 2
= 15.942
4 1052.269
0.3401 1026.430
0.1772 x
2 2
= 118.36
3.6495 360.759
29.3196 468.8059
1042.364 1026.198
5 x
1 2
= 35.512
1027.805 0.3374
4.5634 571.013
41.6763 2.2709
1060.120 x
3 2
= –
6 x
4 2
= 153.872
1021.885 0.1772
3.6495 360.759
29.3196 468.8059
7 x
5 2
= 109.734
1038.051 x
1 2
= 132.978
0.9990 28.753
12.6904 7.8728
1072.226 x
6 2
= 11.372
0.1356 8
1104.540 x
7 2
= 242.712
a
See notes in Table 3.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
391 Table 10
Parameters estimation for various models value-weighted stock index
a
Models Parameters for models
GARCH Leverage effect
ARMA JumpStudent tmixed normal
Diffusion a
b f
c d
s
j
s
2
mm
1
ss
1
m
j
m
2
ln 0.0049
0.0513 1
0.0020 0.0014
0.6980 −
0.6137 0.6662
0.2894 0.0017
2 0.0015
0.0126 0.0006
0.1303 0.1338
0.0525 0.0743
0.0135 0.0019
0.0050 0.0548
3.3258 3
0.5399 0.0049
0.0016 4
0.1558 0.0048
− 0.0288
0.9430 −
0.0789 0.1648
0.0940 2.5751
0.0350 0.0369
0.1598 0.0072
0.1155 0.0017
0.0321 0.0068
0.0743 0.0042
0.0325 0.2882
0.0021 5
0.0020 0.0027
0.1057 0.0076
0.0114 0.0404
0.1575 0.2035
− 0.0441
0.0585 0.0598
0.0048 6
0.0051 0.1639
0.0488 0.0212
0.0034 0.0656
0.0661 0.1092
0.0068 0.0822
0.0038 0.1084
0.0160 0.0069
0.0878 7
0.0043 0.0329
0.7646 0.0090
0.0630 0.0082
0.0020 0.0024
0.0102 0.0019
0.6599 −
0.5751 0.6691
0.2797 0.0122
0.0526 0.0493
0.0017 8
0.0012 0.0044
0.1720 0.1884
0.0464 0.0592
0.0750 0.0135
0.0137 0.0175
a
See notes in Table 2.
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
392
Table 11 Statistical tests for various models value-weighted stock index
a
Standardized residual x
2
test Squared standardized
Schwarz l
R, u Models
criterion residual
Excess kurtosis Normality test
Ljung–Box Q12 Ljung–Box Q12 Skewness
293.586 35.1577
469.9958 0.0366
1 3.3069
986.364 992.830
1101.545 1.3693
50.419 12.9823
8.9016 x
1 2
= 217.430
1078.935 2
− 0.0315
293.586 3
35.1577 1042.326
469.9958 1032.627
0.0366 3.3069
277.413 56.9653
492.918 3.1975
x
1 2
= 119.502
4 1102.077
− 0.1691
1076.206 0.0366
x
2 2
= 110.486
3.3069 293.586
35.1577 469.9958
1048.073 1031.908
5 x
1 2
= 35.634
1033.550 −
0.1056 3.6689
362.396 67.6949
24.1971 1065.890
x
3 2
= –
6 x
4 2
= 146.42
1027.710 0.0366
3.3069 293.586
35.1577 469.9958
7 x
5 2
= 102.09
1043.875 x
1 2
= 137.136
1.4389 55.655
13.4078 8.2411
1080.128 x
6 2
= 21.796
− 0.0303
8 1112.443
x
7 2
= 239.226
a
See notes in Table 3.
In the portfolio cases, skewness is not significant as shown in Table 1. The tendency of decreasing skewness as the size of the portfolio increases, implies skewness may
be diversifiable. If the portfolio return is specified as the Poisson jump-diffusion process, as shown in Table 8 and Table 9, the mean parameter m
J
of the jump magnitude becomes insignificant for both Section 2.3.1 and Section 2.3.2, meaning
symmetric jumps exist in the portfolio return process. Similarly, if the portfolio return is specified as a mixed-normal process, as we can see in Table 8 and Table 10, the
mean parameters m
1
and m
2
for both Section 2.4.1 and Section 2.4.2 are not different in magnitude. These results are consistent with the non-significant skewness in the
portfolio return distribution. Moreover, according to Table 9 and Table 11, the Schwarz criterion and chi-square tests x
3 2
and x
6 2
for Section 2.3.2 and Section 2.4.2 show that Section 2.1.2, the GARCH – normal model may be a reasonable model for
portfolio return distribution.
5. Conclusion
