Figures 15.1 and 15.2 are plots of the periodogram and spectral density for
the returns
computed from DBS50 index. We can see that as frequency approaches zero, the
spectral density estimate displayed in Figure 15.2 increases rapidly.
Granger (1966) has
observed that this is a ``typical spectral shape'' of many observed economic time
series. Figures 15.3 and 15.4 are plots of the periodogram and spectral density for the first
difference of the returns computed from DBS50 index. Taking the first difference of
the series, we now observe that the spectral density estimate is zero at zero frequency
and it increases with . These results are consistent with
, and
.
We use the command
descriptive
to obtain the summary statistics of
DBS50 returns.
We observe that the returns distribution is a ``typical thicker-tail and asymmetric''
distribution of many observed financial time series (Campbell, Lo, and Mackinlay; 1997, Chapter 7).
The daily return has extremely high sample kurtosis of 50. This is
a clear sign of thicker tails or leptokurtic. The skewness estimate is -1.87. If one
believes in the finite higher moments, then using fat-tailed distributions are
consistent
with the empirical observation.
Figure 15.5 plots the
histogram and Figure 15.6 will give an idea of the degree of deviation
from normal distribution.
Contents of desc [ 1,] " " [ 2,] "=========================================================" [ 3,] " Variable z" [ 4,] "=========================================================" [ 5,] " " [ 6,] " Mean 0.0104578" [ 7,] " Std.Error 0.589856 Variance 0.34793" [ 8,] " " [ 9,] " Minimum -12.0418 Maximum 6.37548" [10,] " Range 18.4172" [11,] " " [12,] " Lowest cases Highest cases " [13,] " 1935: -12.0418 4688: 3.45313" [14,] " 1934: -6.11557 4509: 3.91006" [15,] " 1937: -5.96022 4498: 4.03957" [16,] " 1469: -4.72795 1936: 6.26283" [17,] " 2430: -4.27287 2684: 6.37548" [18,] " " [19,] " Median 0.004296" [20,] " 25% Quartile -0.239372 75% Quartile 0.272342" [21,] " " [22,] " Skewness -1.87137 Kurtosis 50.9346" [23,] " " [24,] " Observations 4740" [25,] " Distinct observations 4705" [26,] " " [27,] " Total number of {-Inf,Inf,NaN} 0" [28,] " " [29,] "=========================================================" [30,] " "