Running the First Regression in Python¶

Suppose this is your first time to write the code. Perhaps, you want to run a simple regression using two series of asset prices to fin the equity beta. Let's use a step-by-step approach to complete the task.

Step 1: Download two assets' prices from the web
Step 2: Put them onto a matrix form
Step 3: Run the OLS
Step 4: Plot data

Step 1: Download data¶

We will use yahoo finance package (https://pypi.org/project/yfinance/) to download Yahoo Finance data from the web. We need to (1) install and (2) import this package.

In [ ]:

!pip install yfinance   # to install, remove # and run the cell
import yfinance as yf    # to import

In [3]:

# download
mystock = yf.download("TSLA", start="2011-01-01", end="2022-05-31", interval='1mo')['Adj Close'].rename('TSLA')
index = yf.download("SPY", start="2011-01-01", end="2022-05-31", interval='1mo')['Adj Close'].rename('SPY')

[*********************100%***********************]  1 of 1 completed
[*********************100%***********************]  1 of 1 completed

Step 2: Put two time series onto a matrix¶

We need pandas module, so let's install and import it. https://pandas.pydata.org/

In [4]:

#!pip install pandas   # Actually, you have this alread when you isntalled Anaconda.
import pandas as pd

In [5]:

# combine two asset prices onto one matrix called pandas dataframe
data = pd.concat([mystock, index], axis=1)

# drop missing observations
data2 = data.dropna()

# compute monthly returns and drop the first observation
data3 = data2.pct_change().dropna()
data3

Out[5]:

	TSLA	SPY
Date
2011-02-01	-0.008714	0.034737
2011-03-01	0.161574	-0.004206
2011-04-01	-0.005405	0.033431
2011-05-01	0.092029	-0.011214
2011-06-01	-0.033510	-0.021720
...	...	...
2022-01-01	-0.113609	-0.049413
2022-02-01	-0.070768	-0.029517
2022-03-01	0.238009	0.034377
2022-04-01	-0.191945	-0.084935
2022-05-01	-0.129197	0.002257

136 rows × 2 columns

Step 3: Run OLS¶

We need to install and import statsmodels module. https://www.statsmodels.org/stable/index.html

In [6]:

#!pip install statsmodels
import statsmodels.formula.api as smf
import statsmodels.api as sm

In [7]:

# run OLS
formula = 'TSLA ~ SPY'                      # set dep var and indep var
results = smf.ols(formula, data3).fit()     # run OLS
print(results.summary())                    # print  

                            OLS Regression Results                            
==============================================================================
Dep. Variable:                   TSLA   R-squared:                       0.159
Model:                            OLS   Adj. R-squared:                  0.153
Method:                 Least Squares   F-statistic:                     25.35
Date:                Fri, 26 Aug 2022   Prob (F-statistic):           1.51e-06
Time:                        06:44:56   Log-Likelihood:                 54.624
No. Observations:                 136   AIC:                            -105.2
Df Residuals:                     134   BIC:                            -99.42
Df Model:                           1                                         
Covariance Type:            nonrobust                                         
==============================================================================
                 coef    std err          t      P>|t|      [0.025      0.975]
------------------------------------------------------------------------------
Intercept      0.0319      0.015      2.198      0.030       0.003       0.061
SPY            1.7553      0.349      5.035      0.000       1.066       2.445
==============================================================================
Omnibus:                       43.835   Durbin-Watson:                   1.592
Prob(Omnibus):                  0.000   Jarque-Bera (JB):              109.887
Skew:                           1.285   Prob(JB):                     1.37e-24
Kurtosis:                       6.576   Cond. No.                         24.9
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.

beta of TSLA = 1.7553¶

Step 4: Plot the result¶

We need to install and import matplotlib module. https://matplotlib.org/

In [8]:

#!pip install matplotlib                            #again, if you installed Anaconda, you have this already.
import matplotlib.pyplot as plt

fig, ax=plt.subplots(figsize=(10,6))
fig = sm.graphics.plot_partregress_grid(results, fig=fig)

Extra 1: using scipy module, we can get the same beta!¶

In [9]:

#!pip install scipy
from scipy import stats

beta,alpha,r_value,p_value,std_err = stats.linregress(data3['SPY'],data3["TSLA"])

print(beta.round(4))
print(alpha.round(4))
print(r_value.round(2))
print(p_value.round(4))

Extra 2: using a beta formula, we can get the same beta.¶

$$¶

\beta_{tsla} = \frac{\sigma_{tsla,spy}}{\sigma_{spy}^2} $$

In [38]:

#  find covariance matrix
cov = data3.cov() * 12
print(cov)
print('\n')     # to give a space 
print(round(cov.iloc[0,1]/cov.iloc[1,1], 4))

          TSLA       SPY
TSLA  0.376962  0.034166
SPY   0.034166  0.019465


1.7553

Extra3: using linear algebra, we can get the same beta.¶

Need to install numpy and import it. You probably have this alreay. So skip installation. Just import it. https://numpy.org/

In [32]:

# warnings are annoying, so I include below to supress them. You do not need to do this.
import warnings
warnings.simplefilter(action='ignore', category=FutureWarning)

In [34]:

import numpy as np

X = data3['SPY']
y = data3['TSLA']
X_ols = sm.add_constant(X)   # add a constant vector 
#print(X_ols)

# compute beta using matrix operation
beta = np.linalg.inv(X_ols.T.dot(X_ols)).dot(X_ols.T.dot(y))
print(round(beta[1], 4))

1.7553

In [ ]: