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.
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Part of the Quantopian Lecture Series:
In [1]:
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)
In [2]:
# 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
In [3]:
np.random.seed(11)
POPULATION_MU = 105
POPULATION_SIGMA = 20
sample_size = 50
In [4]:
sample = np.random.normal(POPULATION_MU, POPULATION_SIGMA, sample_size)
#Your code goes here
Mean = np.mean(sample)
print "Mean:", Mean
In [5]:
#Your code goes here
SD = np.std(sample)
print "Standard Deviation:", SD
In [6]:
#Your code goes here
SE = SD / np.sqrt(sample_size)
print "Standard Error:", SE
In [7]:
#Your code goes here
print "95% Confidence Interval:", (-1.96 * SE + Mean, 1.96 * SE + Mean)
print "90% Confidence Interval:", (-1.64 * SE + Mean, 1.64 * SE + Mean)
print "80% Confidence Interval:", (-1.28 * SE + Mean, 1.28 * SE + Mean)
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.
In [8]:
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
for i in range(n):
sample_mean = np.mean(samples[i])
sample_SE = stats.sem(samples[i])
if ((POPULATION_MU >= -1.96 * sample_SE + sample_mean) and (POPULATION_MU <= 1.96 * sample_SE + sample_mean)):
correct += 1;
print "Expected Result:", .95 * 1000
print "Actual Result", correct
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)
In [9]:
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
plt.hist(normal_samples, 10)
_, pvalue, _, _ = jarque_bera(normal_samples)
print pvalue
if pvalue > 0.05:
print 'The distribution of sample means is likely normal.'
else:
print 'The distribution of sample means is likely not normal.'
In [10]:
n = 500
expo_samples = [np.mean(np.random.exponential(POPULATION_MU, sample_size)) for i in range(n)]
#Your code goes here
plt.hist(expo_samples, 10)
_, pvalue, _, _ = jarque_bera(expo_samples)
print pvalue
if pvalue > 0.05:
print 'The distribution of sample means is likely normal, despite the underlying distribution being non-normal (exponential).'
else:
print 'The distribution of sample means is likely not normal.'
In [11]:
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
plt.hist(autocorrelated_means, 10)
_, pvalue, _, _ = jarque_bera(autocorrelated_means)
print pvalue
if pvalue > 0.05:
print 'The distribution of sample means is likely normal, despite an autocorrelated underlying distribution.'
else:
print 'The distribution of sample means is likely not normal.'
In [12]:
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
plt.hist(autocorrelated_stds, 10)
_, pvalue, _, _ = jarque_bera(autocorrelated_stds)
print pvalue
if pvalue > 0.05:
print 'The distribution of sample standard deviations is likely normal.'
else:
print 'The distribution of sample standard deviations is likely not normal, due to the autocorrelated underlying distribution and the different assumptions for the CLT for means and for standard deviations.'
In [13]:
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
for i in range(n):
sample_mean = np.mean(samples[i])
sample_SE = stats.sem(samples[i])
if ((POPULATION_MU >= -1.96 * sample_SE + sample_mean) and (POPULATION_MU <= 1.96 * sample_SE + sample_mean)):
correct += 1
print "Expected Result:", .95 * n
print "Actual Result:", correct
print "Due to the small sample size, the actual number of confidence intervals containing the population mean is much lower than what we would expect given a correctly calibrated interval."
In [14]:
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
for i in range(n):
sample_mean = np.mean(samples[i])
sample_SE = stats.sem(samples[i])
h = sample_SE * stats.t.ppf((1+0.95) / 2, len(samples[i])-1)
if ((POPULATION_MU >= sample_mean - h) and (POPULATION_MU <= sample_mean + h)):
correct += 1
print "Expected Result:", .95 * n
print "Actual Result:", correct
print "After using the t-distribution to correct for the smaller sample size, the actual number of confidence intervals containing the population mean is about what we expected."
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)
In [15]:
n = 100
correct = 0
theta = 0.5
noise_mu = 0
noise_sigma = 1
#Your code goes here
for i in range(n):
X = generate_autocorrelated_data(theta, noise_mu, noise_sigma, sample_size) + POPULATION_MU
sample_mean = np.mean(X)
sample_SE = np.std(X) / np.sqrt(sample_size)
if ((POPULATION_MU >= -1.96 * sample_SE + sample_mean) and (POPULATION_MU <= 1.96 * sample_SE + sample_mean)):
correct += 1
print "Expected Result:", .95 * n
print "Actual Result:", correct
print "Because the underlying data was autocorrelated, the actual number of confidence intervals containing the population mean is much lower than what we expected."
In [16]:
n = 100
correct = 0
#Your code goes here
for i in range(n):
X = generate_autocorrelated_data(theta, noise_mu, noise_sigma, sample_size) + POPULATION_MU
sample_mean = np.mean(X)
sample_SE = np.std(X) / np.sqrt(sample_size)
h = sample_SE * stats.t.ppf((1+0.95) / 2, len(X)-1)
if ((POPULATION_MU >= sample_mean - h) and (POPULATION_MU <= sample_mean + h)):
correct += 1
print "Expected Result:", .95 * n
print "Actual Result:", correct
print "We did not see a significant improvement in the actual number of confidence intervals containing the population mean. This is because a t-distribution only corrects for small sample sizes, not autocorrelation."
In [17]:
X = generate_autocorrelated_data(theta, noise_mu, noise_sigma, sample_size) + POPULATION_MU
#Your code goes here
print newey_west_matrix(X)
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)
In [18]:
n = 100
correct = 0
#Your code goes here
for i in range(n):
X = generate_autocorrelated_data(theta, noise_mu, noise_sigma, sample_size) + POPULATION_MU
sample_mean = np.mean(X)
sample_SE = newey_west_SE(X)
if ((POPULATION_MU >= -1.96 * sample_SE + sample_mean) and (POPULATION_MU <= 1.96 * sample_SE + sample_mean)):
correct += 1
print "New Standard Error:", sample_SE
print "Expected Result:", .95 * n
print "Actual Result:", correct
print "After accounting for autocorrelation by finding a Newey-West standard error, the actual number of confidence intervals containing the population mean is about what we expected."
Congratulations on completing the Confidence Intervals exercises!
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