© Copyright Quantopian Inc.
© Modifications Copyright QuantRocket LLC
Licensed under the Creative Commons Attribution 4.0.
© Copyright Quantopian Inc.
© Modifications Copyright QuantRocket LLC
Licensed under the Creative Commons Attribution 4.0.
By Delaney Granizo-Mackenzie and Andrei Kirilenko developed as part of the Masters of Finance curriculum at MIT Sloan.
In this tutorial notebook, we'll do the following things:
First we need to import some libraries
import math
import matplotlib.pyplot as plt
import numpy as np
import scipy
import scipy.stats
We'll start by sampling some data from a normal distribution.
TRUE_MEAN = 40
TRUE_STD = 10
X = np.random.normal(TRUE_MEAN, TRUE_STD, 1000)
Now we'll define functions that, given our data, will compute the MLE for the $\mu$ and $\sigma$ parameters of the normal distribution.
Recall that
$$\hat\mu = \frac{1}{T}\sum_{t=1}^{T} x_t$$$$\hat\sigma = \sqrt{\frac{1}{T}\sum_{t=1}^{T}{(x_t - \hat\mu)^2}}$$def normal_mu_MLE(X):
# Get the number of observations
T = len(X)
# Sum the observations
s = sum(X)
return 1.0/T * s
def normal_sigma_MLE(X):
T = len(X)
# Get the mu MLE
mu = normal_mu_MLE(X)
# Sum the square of the differences
s = sum( np.power((X - mu), 2) )
# Compute sigma^2
sigma_squared = 1.0/T * s
return math.sqrt(sigma_squared)
Now let's try our functions out on our sample data and see how they compare to the built-in np.mean and np.std
print("Mean Estimation")
print(normal_mu_MLE(X))
print(np.mean(X))
print("Standard Deviation Estimation")
print(normal_sigma_MLE(X))
print(np.std(X))
Mean Estimation 40.10868841891504 40.10868841891503 Standard Deviation Estimation 9.967034665881854 9.967034665881858
Now let's estimate both parameters at once with scipy's built in fit() function.
mu, std = scipy.stats.norm.fit(X)
print("mu estimate:", str(mu))
print("std estimate:", str(std))
mu estimate: 40.10868841891503 std estimate: 9.967034665881858
Now let's plot the distribution PDF along with the data to see how well it fits. We can do that by accessing the pdf provided in scipy.stats.norm.pdf.
pdf = scipy.stats.norm.pdf
# We would like to plot our data along an x-axis ranging from 0-80 with 80 intervals
# (increments of 1)
x = np.linspace(0, 80, 80)
plt.hist(X, bins=x, density='true')
plt.plot(pdf(x, loc=mu, scale=std))
plt.xlabel('Value')
plt.ylabel('Observed Frequency')
plt.legend(['Fitted Distribution PDF', 'Observed Data', ]);
Let's do the same thing, but for the exponential distribution. We'll start by sampling some data.
TRUE_LAMBDA = 5
X = np.random.exponential(TRUE_LAMBDA, 1000)
numpy defines the exponential distribution as $$\frac{1}{\lambda}e^{-\frac{x}{\lambda}}$$
So we need to invert the MLE from the lecture notes. There it is
$$\hat\lambda = \frac{T}{\sum_{t=1}^{T} x_t}$$Here it's just the reciprocal, so
$$\hat\lambda = \frac{\sum_{t=1}^{T} x_t}{T}$$def exp_lamda_MLE(X):
T = len(X)
s = sum(X)
return s/T
print("lambda estimate:", str(exp_lamda_MLE(X)))
lambda estimate: 5.239531351645476
# The scipy version of the exponential distribution has a location parameter
# that can skew the distribution. We ignore this by fixing the location
# parameter to 0 with floc=0
_, l = scipy.stats.expon.fit(X, floc=0)
pdf = scipy.stats.expon.pdf
x = range(0, 80)
plt.hist(X, bins=x, density='true')
plt.plot(pdf(x, scale=l))
plt.xlabel('Value')
plt.ylabel('Observed Frequency')
plt.legend(['Fitted Distribution PDF', 'Observed Data', ]);
Now we'll fetch some real returns and try to fit a normal distribution to them using MLE.
from quantrocket.master import get_securities
from quantrocket import get_prices
aapl_sid = get_securities(symbols="AAPL", vendors='usstock').index[0]
prices = get_prices('usstock-free-1min', data_frequency='daily', sids=aapl_sid, fields='Close', start_date='2014-01-01', end_date='2015-01-01')
prices = prices.loc['Close'][aapl_sid]
# This will give us the number of dollars returned each day
absolute_returns = np.diff(prices)
# This will give us the percentage return over the last day's value
# the [:-1] notation gives us all but the last item in the array
# We do this because there are no returns on the final price in the array.
returns = absolute_returns/prices[:-1]
Let's use scipy's fit function to get the $\mu$ and $\sigma$ MLEs.
mu, std = scipy.stats.norm.fit(returns)
pdf = scipy.stats.norm.pdf
x = np.linspace(-1,1, num=100)
h = plt.hist(returns, bins=x, density='true')
l = plt.plot(x, pdf(x, loc=mu, scale=std))
Of course, this fit is meaningless unless we've tested that they obey a normal distribution first. We can test this using the Jarque-Bera normality test. The Jarque-Bera test will reject the hypothesis of a normal distribution if the p-value is under a c.
from statsmodels.stats.stattools import jarque_bera
jarque_bera(returns)
(866.6489261082853, 6.450254076465886e-189, -0.14337787515083608, 12.098604328561175)
jarque_bera(np.random.normal(0, 1, 100))
(1.844145388244916, 0.3976938884502871, 0.2366822260779503, 2.532537714233625)
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