3. The data
For the empirical study, estimations and tests of nested hypotheses constructed in the previous section were performed on 54 individual common stocks and the
value-weighted stock index of the Taiwan stock market. The sampling period is from January 1985 to May 1997. The 54 common stocks were selected based on the
criterion that those stocks were listed on the Taiwan stock exchange before the beginning of the sampling period, and never ceased being traded after the beginning
of the sampling period. Weekly stock prices were obtained from the data base published by the Taiwan Economic Journal. Daily data was not used because it is
subject to relatively significant measurement errors, due to the fact that stock prices are measured in discrete units, and some stocks are traded only infrequently. These
may cause fake jumps in stock prices and cause a bias towards admitting the presence of jumps in the model. Moreover, there is a price limit on daily stock
prices for each stock listed on the Taiwan stock exchange. This may, on the other hand, conceal jumps in stock prices and hence cause a bias towards detecting the
presence of jumps in the random process. Thus, daily data was not suitable for the research purpose of this paper. Weekly stock returns continuously compound
returns were calculated as changes in logarithm of weekly prices. Totally, there were 644 weekly data returns for each of the 54 individual stocks and the
value-weighted stock index.
Table 1 shows summaries of statistical characteristics of weekly returns for all sample stocks and portfolios. To save spaces without losing generality, detailed
results of individual stocks are shown only for the Formosa Plastic and the Cathay Life Insurance. For other individual stocks, only the number of stocks that pass a
certain hypothesis test are shown in the table. For individual stocks, it is apparent that the weekly return distributions of almost all stocks have significant non-zero
skewness and high-leveled kurtosis. Normality tests further confirm that stock returns in the Taiwan stock market are not normally distributed. In contract to the
case of Brorsen and Yang 1994, Kim and Kon 1994, and De Santis and Imrohoroglu 1997, all the sample stock returns for the sample period in this study
have their distributions positively skewed. This may be because our sample period contains the historical booming period from 1986 to 1989.
In the case of the value-weighted stock index, the skewness decreases to a trivial level, while the kurtosis remains as high as those for individual stocks. This may
imply that skewness could be diversified through portfolio. To confirm this conjec- ture, we conduct an experiment by constructing random portfolios with different
numbers of individual sample stocks and comparing the statistical characteristics of their returns. Again without losing generality, we only show the result for one of
the ten-stock portfolio cases. As can be seen in Table 1, in the case of the ten-stock portfolio, the skewness decreases to a non-significant level, while the kurtosis
remains as high as those for individual stocks. As for cases of other portfolios, the situations are quite similar to the case of the ten-stock portfolio, except that
skewness decreases further as the number of stocks increases in the portfolio. To the end, in the case of the value-weighted stock index, the skewness for the return
B .-
H .
Lin ,
S .-
K .
Yeh J
. of
Multi .
Fin .
Manag .
10 2000
367 –
395
378 Table 1
Summary statistics for stock returns for the period from January 1985 to May 1997 Cathay Life Insurance
54 individual stocks
f
Value-weighted stock index Ten-stock portfolio
Formosa Plastic Moments
Mean 0.0059
0.0088 0.0049
0.0057 0.0057
0.0670 0.0513
0.0521 0.0559
0.0655 S.D.
0.6870 0.3631 46
0.0366 Skewness
a
0.1772 0.6416
6.3069 5.8485 54
6.6495 Kurtosis
b
6.3605 5.8738
Normality test
c
293.586 360.759
347.211 272.273
54 Autocorrelations for k logs
0.1221 0.0618 34
k = 1 0.1075
0.0949 0.0700
0.1469 0.0573 35
0.1050 0.0572
k = 2 0.1467
0.0596 0.0081 14
k = 3 0.0380
0.0785 −
0.0129 0.0136 32
− 0.0320
− 0.0766
− 0.0610
0.0379 k = 4
0.0387 0.0215 5
0.0505 k = 5
0.0336 0.0499
− 0.0008
0.0011 12 −
0.0299 k = 6
− 0.0566
− 0.0007
0.0394 0.0394
0.0394 S.D.
d
0.0394 0.0394
29.501 31
Ljung–Box test Q6
e
26.675 20.547
9.439 644
644 644
644 644
Number of observations Denotes that the test is statistically significant at the level of 5.
a
Represents tests for hypotheses that the skewness is zero. Critical value is given in Pearson and Hartley 1975.
b
Represents tests for hypotheses that the kurtosis is 3. Critical value is given in Pearson and Hartley 1975.
c
The statistic is T·[skewness
2
6+kurtosis−3
2
24] which is a x
2
distribution with 2 d.f. See Greene 1993. The critical value of the test is 5.99 at the significance level of 5.
d
The standard deviation S.D. of autocorrelations with log p is 1 T.
e
The statistic is Q6 = TT+2
k = 1 6
r
k 2
T−k which is a x
2
distribution with 6 d.f. The critical value of the test is 12.59 at the significance level of 5.
f
Numbers in this column are average values across the 54 individual sample stocks. Numbers in parentheses in this column are the number of stocks out of the 54 individual sample stocks that pass the hypothesis test at a significance level of 5.
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
Kategori