Many Python packages such as SciPy come with several variants of regression functions. In particular, the statsmodels package is a complement to SciPy with descriptive statistics and estimation of statistical models. The official page for statsmodels is http://statsmodels.sourceforge.net/.

In this example, we will use the ols function of the statsmodels module to perform an ordinary least squares regression and view its summary.

Let's assume that you have implemented an APT model with seven factors that return the values of Y. Consider the following set of data collected over 9 time periods, to . X1 to X7 are independent variables observed at each period. The regression problem is therefore structured as:.

A simple ordinary least squares regression on values of X and Y can be performed with the following code:

""" Least squares regression with statsmodels """
import numpy as np
import statsmodels.api as sm

# Generate some sample data
num_periods = 9
all_values = np.array([np.random.random(8)
                       for i in range(num_periods)])

# Filter the data
y_values = all_values[:, 0]  # First column values as Y
x_values = all_values[:, 1:]  # All other values as X

x_values = sm.add_constant(x_values)  # Include the intercept
results = sm.OLS(y_values, x_values).fit()  # Regress and fit the model

Let's view the detailed statistics of the regression:

>>> print results.summary()

The OLS regression results will output a pretty long table of statistical information. However, our interest lies in the particular section that gives us the coefficients of our APT model:

============================================= 
                 coef    std err          t       
--------------------------------------------- 
const          0.5224      0.825      0.633     
x1             0.0683      0.246      0.277     
x2             0.1455      1.010      0.144      
x3            -0.2451      0.330     -0.744      
x4             0.5907      0.830      0.712      
x5            -0.3252      0.256     -1.271      
x6            -0.2375      0.788     -0.301      
x7            -0.1880      0.703     -0.267 

Similarly, we can use the params function to display our coefficients of interest:

>>> print results.params
[ 0.52243605  0.06827488  0.14550665 -0.24508947  0.5907154   -0.32515442  -0.23751989  -0.18795065]

Both the function calls produce the same coefficient values for the APT model in the same order.