15.5 ExploRing the Data


15.5.1 Typical Spectral Shape

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 $ f$. These results are consistent with $ 0 < f_y(0) < \infty$, and $ {(1 - L)}^d y_t =x_t$.

Figure 15.1: Spectral density for the returns computed from DBS50 index
\includegraphics[scale=.55]{Figure_1_dbs50_density}

Figure 15.2: Periodogram for the returns computed from DBS50 index
\includegraphics[scale=.55]{Figure_2_dbs50_period}

Figure 15.3: Spectral density for the first difference of the returns computed from DBS50 index
\includegraphics[scale=.55]{Figure_3_dbs50_diff_density}

Figure 15.4: Periodogram for the first difference of the returns computed from DBS50 index
\includegraphics[scale=.55]{Figure_4_dbs50_diff_period}


15.5.2 Typical Distribution: Mean, Variance, Skewness and Kurtosis

We use the command 25370 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,] " "

Figure 15.5: Histogram of returns of DBS50
\includegraphics[scale=.55]{Figure_5}

Figure 15.6: Comparison with Normal distribution of returns of DBS50
\includegraphics[scale=.55]{Figure_6}