Exercises: Comparing ETFs

By Christopher van Hoecke, Maxwell Margenot, and Delaney Mackenzie

Lecture Link :

https://www.quantopian.com/lectures/statistical-moments

https://www.quantopian.com/lectures/hypothesis-testing

IMPORTANT NOTE:

This lecture corresponds to the statistical moments and hypothesis 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.

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Part of the Quantopian Lecture Series:

Key Concepts

t-statistic formula for unequal variances : $ t = \frac{\bar{X}_1 - \bar{X}_2}{(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2})^{1/2}}$

Where $s_1$ and $s_2$ are the standard deviation of set 1 and set 2; and $n_1$ and $n_2$ are the number of observations we have.



In [ ]:

    
# Useful Libraries
import numpy as np
import matplotlib.pyplot as plt
from scipy import stats
import seaborn as sns



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# Useful functions 
def normal_test(X):
    z, pval = stats.normaltest(X)
    if pval < 0.05:
        print 'Values are not normally distributed.'
    else: 
        print 'Values are normally distributed.'
    return

Data



In [ ]:

    
# Get pricing data for an energy (XLE) and industrial (XLI) ETF
xle = get_pricing('XLE', fields = 'price', start_date = '2016-01-01', end_date = '2017-01-01')
xli = get_pricing('XLI', fields = 'price', start_date = '2016-01-01', end_date = '2017-01-01')

# Compute returns
xle_returns = xle.pct_change()[1:]
xli_returns = xli.pct_change()[1:]

Exercise 1 : Hypothesis Testing on Variance.

Plot the histogram of the returns of XLE and XLI
Check to see if each return stream is normally distributed
If the assets are normally distributed, use the F-test to perform a hypothesis test and decide whether they have the two assets have the same variance.
If the assets are not normally distributed, use the Levene test (in the scipy library) to perform a hypothesis test on variance.



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# Histograms of XLE and XLI returns

## Your code goes here



In [ ]:

    
# Checking for normality using function above. 

## Your code goes here



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# Use the levene or the F-test to check hypothesis of variance. 

## Your code goes ehre

Exercise 2 : Hypothesis Testing on Mean.

Since we know that the variances are not equal, we must use Welch's t-test.

Calculate the mean returns of XLE and XLI.
- Find the difference between the two means.
Calculate the standard deviation of the returns of XLE and XLI
Using the formula given above, calculate the t-test statistic (Using $\alpha = 0.05$) for Welch's t-test to test whether the mean returns of XLE and XLI are different.
Consult the Hypothesis Testing Lecture to calculate the p-value for this test. Are the mean returns of XLE and XLI the same?

Now use the t-test function for two independent samples from the scipy library. Compare the results.



In [ ]:

    
# Manually calculating the t-statistic
# Note that the test also requires information about the degrees of freedom
# We will not compute that here

## Your code goes here



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# Alternative form, using the scipy library on python. 

## Your code goes here

Exercise 3 : Skewness

Calculate the mean and median of the two assets
Calculate the skewness using the scipy library



In [ ]:

    
# Calculate the mean and median of xle and xli using the numpy library

## Your code goes here



In [ ]:

    
# Print values of Skewness for xle and xli returns 

## Your code goes here

Exercise 4 : Kurtosis

Check the kurtosis of the two assets, using the scipy library.
Using the seaborn library, plot the distribution of XLE and XLI returns.

Recall:

Kurtosis > 3 is leptokurtic, a highly peaked, narrow deviation from the mean
Kurtosis = 3 is mesokurtic. The most significant mesokurtic distribution is the normal distribution family.
Kurtosis < 3 is platykurtic, a lower-peaked, broad deviation from the mean



In [ ]:

    
# Print value of Kurtosis for xle and xli returns 

## Your code goes here



In [ ]:

    
# Distribution plot of XLE returns in red (for Kurtosis of 1.6). 
# Distribution plot of XLI returns in blue (for Kurtosis of 2.0).

## Your code goes here

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