By Christopher van Hoecke and Maxwell Margenot
https://www.quantopian.com/lectures/hypothesis-testing
This lecture corresponds to the 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.
When you feel comfortable with the topics presented here, see if you can create an algorithm that qualifies for the Quantopian Contest. Participants are evaluated on their ability to produce risk-constrained alpha and the top 10 contest participants are awarded cash prizes on a daily basis.
https://www.quantopian.com/contest
Part of the Quantopian Lecture Series:
In [ ]:
# Useful Libraries
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import t
import scipy.stats
Using the techniques laid out in lecture, verify if we can state that the returns of TSLA are greater than 0.
In [ ]:
prices1 = get_pricing('TSLA', start_date = '2015-01-01', end_date = '2016-01-01', fields = 'price')
returns_sample_tsla = prices1.pct_change()[1:]
print 'Tesla return sample mean', returns_sample_tsla.mean()
print 'Tesla return sample standard deviation', returns_sample_tsla.std()
print 'Tesla return sample size', len(returns_sample_tsla)
Write your hypotheses here:
In [ ]:
# Testing
## Your code goes here
## Sample mean difference:
## Z- Statistic:
print 't-statistic is:', test_stat
## Finding the p-value for one tail test
print 'p-value is: ', p_val
Using the techniques laid out in lecture, verify if we can state that the returns of TSLA are equal to 0.
In [ ]:
## Your code goes here
## Sample mean difference:
## Z- Statistic:
print 't-statistic is:', test_stat
## Finding the p-value for one tail test
print 'p-value is: ', p_val
Find the critical values associated with $\alpha = 1\%, 5\%, 10\%$ and graph the rejection regions on a plot for a two tailed test.
Useful formulas: $$ f = 1 - \frac{\alpha}{2} $$
In order to find the z-value associated with each f value use the z-table here.
You can read more about how to read z-tables here
In [ ]:
## Your code goes here
# For alpha = 10%
alpha = 0.1
f =
print 'alpha = 10%: f = ', f
# For alpha = 5%
alpha = 0.05
f =
print 'alpha = 5%: f = ', f
# For alpha = 1%
alpha = 0.01
f =
print 'alpha = 1%: f = ', f
In [ ]:
# Plot a standard normal distribution and mark the critical regions with shading
x = np.linspace(-3, 3, 100)
norm_pdf = lambda x: (1/np.sqrt(2 * np.pi)) * np.exp(-x * x / 2)
y = norm_pdf(x)
fig, ax = plt.subplots(1, 1, sharex=True)
ax.plot(x, y)
# Value for alpha = 1%
ax.fill_between(x, 0, y, where = x > ## Your code goes here
, label = 'alpha = 10%')
ax.fill_between(x, 0, y, where = x < ) ## Your code goes here
# Value for alpha = 5%
ax.fill_between(x, 0, y, where = x > ## Your code goes here
, color = 'red', label = 'alpha = 5%')
ax.fill_between(x, 0, y, where = x < ## Your code goes here
, color = 'red')
#Value for alpha = 10%
ax.fill_between(x, 0, y, where = x > ## Your code goes here
, facecolor='green', label = 'alpha = 1%')
ax.fill_between(x, 0, y, where = x < ## Your code goes here
, facecolor='green')
plt.title('Rejection regions for a two-tailed hypothesis test at 90%, 95%, 99% confidence')
plt.xlabel('x')
plt.ylabel('p(x)')
plt.legend();
In [ ]:
# Calculating Critical Values probability
alpha = 0.1
f = ## Your code goes here
print f
In [ ]:
data = get_pricing('SPY', start_date = '2016-01-01', end_date = '2017-01-01', fields = 'price')
returns_sample = data.pct_change()[1:]
# Running the T-test.
n = len(returns_sample)
test_statistic = ## Your code goes here
print 't test statistic: ', test_statistic
In [ ]:
# Running p-value test.
alpha = 0.1
p_val = ## Your code goes here
print 'p-value is: ', p_val
note: one formula for t involves equal variance, the other does not. Use the right one given the information above
Write your hypotheses here:
In [ ]:
# Data Collection
alpha = 0.1
symbol_list = ['XLF', 'MCD']
start = '2015-01-01'
end = '2016-01-01'
pricing_sample = get_pricing(symbol_list, start_date = start, end_date = end, fields='price')
pricing_sample.columns = map(lambda x: x.symbol, pricing_sample.columns)
returns_sample = pricing_sample.pct_change()[1:]
# Sample mean values
mu_xlf, mu_gs = returns_sample.mean()
s_xlf, s_gs = returns_sample.std()
n_xlf = len(returns_sample['XLF'])
n_gs = len(returns_sample['MCD'])
test_statistic = ## Your code goes here
df = ## Your code goes here
print 't test statistic: ', test_statistic
print 'Degrees of freedom (modified): ', df
print 'p-value: ', ## Your code goes here
Write your hypotheses here:
In [ ]:
# Data
symbol_list = ['XLF', 'MCD']
start = "2015-01-01"
end = "2016-01-01"
pricing_sample = get_pricing(symbol_list, start_date = start, end_date = end, fields = 'price')
pricing_sample.columns = map(lambda x: x.symbol, pricing_sample.columns)
returns_sample = pricing_sample.pct_change()[1:]
# Take returns from above, MCD and XLF, and compare their variances
## Your code goes here
print 'XLF standard deviation is: ', xlf_std_dev
print 'MCD standard deviation is: ', mcd_std_dev
# Calculate F-test with MCD.std > XLF.std
## Your code goes here
print "F Test statistic: ", test_statistic
#degree of freedom
df1 = ## Your code goes here
df2 = ## Your code goe here
print df1
print df2
# Calculate critical values.
from scipy.stats import f
upper_crit_value = f.ppf(0.975, df1, df2)
lower_crit_value = f.ppf(0.025, df1, df2)
print 'Upper critical value at a = 0.05 with df1 = {0} and df2 = {1}: '.format(df1, df2), upper_crit_value
print 'Lower critical value at a = 0.05 with df1 = {0} and df2 = {1}: '.format(df1, df2), lower_crit_value
Congratulations on completing the Hypothesis Testing exercises!
As you learn more about writing trading models and the Quantopian platform, enter the daily Quantopian Contest. Your strategy will be evaluated for a cash prize every day.
Start by going through the Writing a Contest Algorithm tutorial.
This presentation is for informational purposes only and does not constitute an offer to sell, a solicitation to buy, or a recommendation for any security; nor does it constitute an offer to provide investment advisory or other services by Quantopian, Inc. ("Quantopian"). Nothing contained herein constitutes investment advice or offers any opinion with respect to the suitability of any security, and any views expressed herein should not be taken as advice to buy, sell, or hold any security or as an endorsement of any security or company. In preparing the information contained herein, Quantopian, Inc. has not taken into account the investment needs, objectives, and financial circumstances of any particular investor. Any views expressed and data illustrated herein were prepared based upon information, believed to be reliable, available to Quantopian, Inc. at the time of publication. Quantopian makes no guarantees as to their accuracy or completeness. All information is subject to change and may quickly become unreliable for various reasons, including changes in market conditions or economic circumstances.