By Christopher Fenaroli and Delaney Mackenzie
https://www.quantopian.com/lectures/confidence-intervals
This lecture corresponds to the Confidence Intervals 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:
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def generate_autocorrelated_data(theta, mu, sigma, N):
X = np.zeros((N, 1))
for t in range(1, N):
X[t] = theta * X[t-1] + np.random.normal(mu, sigma)
return X
def newey_west_SE(data):
ind = range(0, len(data))
ind = sm.add_constant(ind)
model = regression.linear_model.OLS(data, ind).fit(cov_type='HAC',cov_kwds={'maxlags':1})
return model.bse[0]
def newey_west_matrix(data):
ind = range(0, len(data))
ind = sm.add_constant(ind)
model = regression.linear_model.OLS(data, ind).fit()
return sw.cov_hac(model)
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# Useful Libraries
import numpy as np
import seaborn as sns
from scipy import stats
import matplotlib.pyplot as plt
from statsmodels.stats.stattools import jarque_bera
import statsmodels.stats.sandwich_covariance as sw
from statsmodels import regression
import statsmodels.api as sm
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np.random.seed(11)
POPULATION_MU = 105
POPULATION_SIGMA = 20
sample_size = 50
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sample = np.random.normal(POPULATION_MU, POPULATION_SIGMA, sample_size)
#Your code goes here
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#Your code goes here
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#Your code goes here
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#Your code goes here
Assuming our interval was correctly calculated and that the underlying data was independent, if we take many samples and make many 95% confidence intervals, the intervals will contain the true mean 95% of the time. Run 1000 samples and measure how many of their confidence intervals actually contain the true mean.
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n = 1000
correct = 0
samples = [np.random.normal(loc=POPULATION_MU, scale=POPULATION_SIGMA, size=sample_size) for i in range(n)]
#Your code goes here
Assuming our samples are independent, the distribution of the sample means should be normally distributed, regardless of the underlying distribution.
Draw 500 samples of size sample_size from the same normal distribution from question 1, plot the means of each of the samples, and check to see if the distribution of the sample means is normal using the jarque_bera function (see here more information on the Jarque-Bera test)
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n = 500
normal_samples = [np.mean(np.random.normal(loc=POPULATION_MU, scale=POPULATION_SIGMA, size=sample_size)) for i in range(n)]
#Your code goes here
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n = 500
expo_samples = [np.mean(np.random.exponential(POPULATION_MU, sample_size)) for i in range(n)]
#Your code goes here
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n = 500
autocorrelated_samples = [(generate_autocorrelated_data(0.5, 0, 1, sample_size) + POPULATION_MU) for i in range(n)]
autocorrelated_means = [np.mean(autocorrelated_samples[i]) for i in range(n)]
#Your code goes here
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n = 500
autocorrelated_samples = [(generate_autocorrelated_data(0.5, 0, 1, sample_size) + POPULATION_MU) for i in range(n)]
autocorrelated_stds = [np.std(autocorrelated_samples[i]) for i in range(n)]
#Your code goes here
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n = 100
small_size = 3
correct = 0
samples = [np.random.normal(loc=POPULATION_MU, scale=POPULATION_SIGMA, size=small_size) for i in range(n)]
#Your code goes here
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n = 100
small_size = 5
correct = 0
samples = [np.random.normal(loc=POPULATION_MU, scale=POPULATION_SIGMA, size=small_size) for i in range(n)]
#Your code goes here
Run 100 samples of the following autocorrelated distribution and measure how many of their 95% confidence intervals actually contain the true mean. (Use the helper function generate_autocorrelated_data(theta, noise_mu, noise_sigma, sample_size) to generate the samples)
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n = 100
correct = 0
theta = 0.5
noise_mu = 0
noise_sigma = 1
#Your code goes here
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n = 100
correct = 0
#Your code goes here
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X = generate_autocorrelated_data(theta, noise_mu, noise_sigma, sample_size) + POPULATION_MU
#Your code goes here
Run 100 samples of the following autocorrelated distribution, this time accounting for the autocorrelation by using a Newey-West correction on the standard error, and measure how many of their 95% confidence intervals actually contain the true mean to see if the correction works. (Use the helper function newey_west_SE to find the corrected standard error)
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n = 100
correct = 0
#Your code goes here
Congratulations on completing the Confidence Intervals 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.
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