Maximum Likelihood Estimation for Normal and Exponential Models
Notebook Quantopian lectures
Summary
This tutorial introduces maximum likelihood estimation through normal and exponential distributions. For a normal sample, it derives estimates for the mean and standard deviation and compares them with library estimates. For an exponential sample, it explains the parameter convention used by the numerical library and estimates the distribution’s scale from the sample average. Plots of fitted probability densities provide a visual comparison with simulated observations.
The final example applies a normal fit to daily returns for a single equity, then uses the Jarque–Bera test to examine whether the return sample is consistent with normality. The tutorial emphasizes that fitting a distribution alone does not establish that the distribution is appropriate; goodness-of-fit should be checked. The demonstrations are introductory, relying on simulated samples and a limited historical return example. A fitted normal distribution may not capture important features of financial returns, and the document does not establish that the example’s returns are normal or that the fitted parameters will remain stable.
Key ideas
- Maximum likelihood chooses distribution parameters that make the observed sample most plausible under the model.
- For a normal sample, the maximum likelihood estimates use the sample mean and a standard deviation calculated with the observation count as divisor.
- For the exponential distribution convention shown, the estimated scale is the sample average.
- A fitted density can be compared visually with observed data, but visual agreement is not a formal validation.
- The Jarque–Bera test can assess normality; fitting a normal distribution does not itself show that returns are normal.
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Full text
# Maximum Likelihood Estimates (MLEs)
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***
[Quant Finance Lectures (adapted Quantopian Lectures)](Introduction.ipynb) › Lecture 13 - Maximum Likelihood Estimation
***
# Maximum Likelihood Estimates (MLEs)
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:
1. Compute the MLE for a normal distribution.
2. Compute the MLE for an exponential distribution.
3. Fit a normal distribution to asset returns using MLE.
First we need to import some libraries
```python
import math
import matplotlib.pyplot as plt
import numpy as np
import scipy
import scipy.stats
```
## Normal Distribution
We'll start by sampling some data from a normal distribution.
```python
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
$$\hat\mu = \frac{1}{T}\sum_{t=1}^{T} x_t$$
$$\hat\sigma = \sqrt{\frac{1}{T}\sum_{t=1}^{T}{(x_t - \hat\mu)^2}}$$
```python
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`
```python
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))
```
Now let's estimate both parameters at once with scipy's built in `fit()` function.
```python
mu, std = scipy.stats.norm.fit(X)
print("mu estimate:", str(mu))
print("std estimate:", str(std))
```
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`.
```python
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', ]);
```
## Exponential Distribution
Let's do the same thing, but for the exponential distribution. We'll start by sampling some data.
```python
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
$$\hat\lambda = \frac{\sum_{t=1}^{T} x_t}{T}$$
```python
def exp_lamda_MLE(X):
T = len(X)
s = sum(X)
return s/T
```
```python
print("lambda estimate:", str(exp_lamda_MLE(X)))
```
```python
# 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)
```
```python
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', ]);
```
## MLE for Asset Returns
Now we'll fetch some real returns and try to fit a normal distribution to them using MLE.
```python
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.
```python
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.
```python
from statsmodels.stats.stattools import jarque_bera
jarque_bera(returns)
```
```python
jarque_bera(np.random.normal(0, 1, 100))
```
---
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