Exercises: Statistical Moments and Normality Testing - Answer Key¶
Lecture Link :¶
https://www.quantopian.com/lectures/statistical-moments
IMPORTANT NOTE:¶
This lecture corresponds to the Statistical Moments and Normality Testing lecture, which is part of the Quantopian lecture series. This homework expects you to rely heavily on the code presented in the corresponding lecture. Please copy and paste regularly from that lecture when starting to work on the problems, as trying to do them from scratch will likely be too difficult.
Part of the Quantopian Lecture Series:
# Useful Libraries
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd
import scipy.stats as stats
from statsmodels.stats.stattools import jarque_bera
xs2 = np.linspace(stats.gamma.ppf(0.01, 0.7, loc=-1), stats.gamma.ppf(0.99, 0.7, loc=-1), 150) + 1
X = stats.gamma.pdf(xs2, 1.5)
#Your code goes here
skew = stats.skew(X)
plt.plot(xs2, X)
print 'Skew:', skew
if skew > 0:
print 'The distribution is positively skewed'
elif skew < 0:
print 'The distribution is negatively skewed'
else:
print 'The distribution is symmetric'
b. Real Example¶
Use the results from the stats.skew function to determine the skew of the returns of NFLX and use it to make a conclusion about the symmetry of the stock's returns.
start = '2015-01-01'
end = '2016-01-01'
pricing = get_pricing('NFLX', fields='price', start_date=start, end_date=end)
returns = pricing.pct_change()[1:]
#Your code goes here
skew = stats.skew(returns)
plt.hist(returns, 30)
print 'Skew:', skew
print 'The returns of NFLX have a strong positive skew, meaning their volatility is characterized by frequent small changes in price with interspersed large upticks.'
xs = np.linspace(-6,6, 300) + 2
Y = stats.cosine.pdf(xs)
#Your code goes here
plt.plot(xs, Y)
print 'Excess kurtosis of Y:', (stats.kurtosis(Y))
print 'Because the excess kurtosis is negative, Y is platykurtic. Platykurtic distributions cluster around the mean, so large values in either direction are less likely'
b. Real Example¶
Use the results from the stats.kurtosis function to determine the kurtosis of the returns of NFLX and use it to make a conclusion about the volatility of the stock's price.
start = '2015-01-01'
end = '2016-01-01'
pricing = get_pricing('NFLX', fields='price', start_date=start, end_date=end)
returns = pricing.pct_change()[1:]
#Your code goes here
kurt = stats.kurtosis(returns)
plt.hist(returns, 30)
print 'Kurtosis:', kurt
print 'The historical returns of NFLX are strongly leptokurtic. Because of a leptokurtic distribution`s fatter tails, small changes in prices happen less often and large changes are more common. This makes the stock a riskier investment.'
xs2 = np.linspace(stats.lognorm.ppf(0.01, 0.7, loc=-.1), stats.lognorm.ppf(0.99, 0.7, loc=-.1), 150)
lognorm = stats.lognorm.pdf(xs2, 0.4)
Z = lognorm/2 + lognorm[::-1]
#Your code goes here
skew = stats.skew(Z)
print 'Skew:', skew
if skew > 0:
print 'The distribution is positively skewed'
elif skew < 0:
print 'The distribution is negatively skewed'
else:
print 'The distribution is symmetric'
b. Jarque-Bera Calibration¶
Ensure that the jarque-bera function is calibrated by running it on many trials of simulated data and ensuring that the sample probability that the test returns a result under the p-value is equal to the p-value.
N = 1000
M = 1000
pvalues = np.ndarray((N))
for i in range(N):
# Draw M samples from a normal distribution
X = np.random.normal(0, 1, M);
_, pvalue, _, _ = jarque_bera(X)
pvalues[i] = pvalue
num_significant = len(pvalues[pvalues < 0.05])
#Your code goes here
print float(num_significant) / N
print 'Our answer is around 5%, which is what we would expect for a cutoff of 5% and a correctly-calibrated Jarque-Bera test.'
c. Jarque-Bera Test¶
Use the Jarque-Bera function to determine the normality of Z.
#Your code goes here
_, pvalue, _, _ = jarque_bera(Z)
print pvalue
if pvalue > 0.05:
print 'The returns are likely normal.'
else:
print 'The returns are likely not normal.'
d. Skewness and Normality¶
Plot Z and observe that skewness is not informative unless the underlying distribution is somewhat normal.
#Your code goes here
plt.plot(Z)
print 'The positive skew found in part a would have led us to believe values are concentrated below the mean and a tail extends to the right, however this is not the case. Because Z is bimodal, we can make no conclusions based on the skewness value alone. In order for skewness to be useful, the underlying distribution must be somewhat normal'
start = '2014-01-01'
end = '2016-01-01'
pricing = get_pricing('AMC', fields='price', start_date=start, end_date=end)
returns = pricing.pct_change()[1:]
#Your code goes here
print 'The returns of AMC from 2014 through 2016 are unimodal and vaguely normal, so a skewness measure would be relevant.'
plt.hist(returns, 30);
b. Test for Skew¶
Find the skew of the historical returns of AMC between 2014 to 2016.
start = '2014-01-01'
end = '2016-01-01'
pricing = get_pricing('AMC', fields='price', start_date=start, end_date=end)
returns = pricing.pct_change()[1:]
#Your code goes here
print 'Skew of AMC:', stats.skew(returns)
c. Out of Sample Test¶
Find the skew of the historical retunrs of AMC from the first half of 2016 to determine if the skew from part b holds outside of the original sample.
start = '2016-01-01'
end = '2016-07-01'
out_pricing = get_pricing('AMC', fields='price', start_date=start, end_date=end)
out_returns = out_pricing.pct_change()[1:]
#Your code goes here
print 'Skew of AMC:', stats.skew(out_returns)
print 'The negative skew of AMC between 2014 and 2016 did not hold outside of the orignal sample, meaning the skew of AMC might be volatile and not reliable enough for predictions about future behavior.'
d. Rolling Skew¶
Plot the rolling skew of AMC using the pd.rolling_skew function.
AMC = get_pricing('AMC', fields='price', start_date='2015-01-01', end_date='2017-01-01')
#Your code goes here
rolling_skew = AMC.rolling(window=60,center=False).skew()
plt.plot(rolling_skew)
plt.xlabel('Day')
plt.ylabel('60-day Rolling Skew')
print "This confirms our result from part c, that the skew is too volatile to use it to make predictions outside of the sample."
Congratulations on completing the Statistical Moments and Normality Testing exercises!
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