print “%14s %15.5f” % (‘skew’, sta[ 4 ])
print “%14s %15.5f” % (‘kurtosis’, sta[ 5 ])
For example, the following shows the function in action, using a flattened version of the
ndarray object containing the log returns. The method flatten returns a 1D array with all
the data given in a multidimensional array:
In [ 10 ]: print_statistics(log_returns.flatten())
Out[10]: statistic value
––––––––––
size 12500000.00000
min -0.15664
max 0.15371
mean 0.00060
std 0.02828
skew 0.00055
kurtosis 0.00085
The data set in this case consists of 12,500,000 data points with the values mainly lying
between +/– 0.15. We would expect annualized values of 0.05 for the mean return and 0.2
for the standard deviation (volatility). The annualized values of the data set come close to
these values, if not matching them perfectly (multiply the mean value by 50 and the
standard deviation by ).
Figure 11-2 compares the frequency distribution of the simulated log returns with the
probability density function (pdf) of the normal distribution given the parameterizations
for r and sigma. The function used is norm.pdf from the scipy.stats sublibrary. There is
obviously quite a good fit:
In [ 11 ]: plt.hist(log_returns.flatten(), bins= 70 , normed=True, label=‘frequency’)
plt.grid(True)
plt.xlabel(‘log-return’)
plt.ylabel(‘frequency’)
x = np.linspace(plt.axis()[ 0 ], plt.axis()[ 1 ])
plt.plot(x, scs.norm.pdf(x, loc=r / M, scale=sigma / np.sqrt(M)),
‘r’, lw=2.0, label=‘pdf’)
plt.legend()
Figure 11-2. Histogram of log returns and normal density function
Comparing a frequency distribution (histogram) with a theoretical pdf is not the only way
to graphically “test” for normality. So-called quantile-quantile plots (qq plots) are also
well suited for this task. Here, sample quantile values are compared to theoretical quantile
values. For normally distributed sample data sets, such a plot might look like Figure 11-3,
with the absolute majority of the quantile values (dots) lying on a straight line:
In [ 12 ]: sm.qqplot(log_returns.flatten()[:: 500 ], line=‘s’)
