Last Update: February 21, 2022
Linear Regression: Coefficients Analysis in Python can be done using statsmodels package ols function and summary method found within statsmodels.formula.api module for analyzing linear relationship between one dependent variable and two or more independent variables. It is also used for evaluating whether adding independent variables individually improved linear regression model. Main parameters within ols function are formula with “y ~ x1 + … + xp” model description string and data with data frame object including model variables.
As example, we can estimate coefficients table from multiple linear regression of house price explained by its lot size and number of bedrooms using data included within AER R package HousePrices object [1].
First, we import statsmodels package for data downloading and model fitting [2].
In [1]:
import statsmodels.api as sm
import statsmodels.formula.api as smf
Second, we create houseprices data object using get_rdataset function and display first five rows and three columns of data using print function and head data frame method to view its structure.
In [2]:
houseprices = sm.datasets.get_rdataset(dataname="HousePrices", package="AER", cache=True).data
print(houseprices.iloc[:, 0:3].head())
Out [2]:
price lotsize bedrooms
0 42000.0 5850 3
1 38500.0 4000 2
2 49500.0 3060 3
3 60500.0 6650 3
4 61000.0 6360 2
Third, we fit model with ols function using variables within houseprices data object and store outcome within mlr object. Within ols function, parameter formula = “price ~ lotsize + bedrooms” fits model where house price is explained by its lot size and number of bedrooms.
In [3]:
mlr = smf.ols(formula="price ~ lotsize + bedrooms", data=houseprices).fit()
Fourth, we can print mlr model estimated coefficients table using its summary method.
In [4]:
print(mlr.summary())
Out [4]:
OLS Regression Results
==============================================================================
Dep. Variable: price R-squared: 0.370
Model: OLS Adj. R-squared: 0.368
Method: Least Squares F-statistic: 159.6
Date: Thu, 16 Dec 2021 Prob (F-statistic): 2.95e-55
Time: 14:41:02 Log-Likelihood: -6213.1
No. Observations: 546 AIC: 1.243e+04
Df Residuals: 543 BIC: 1.245e+04
Df Model: 2
Covariance Type: nonrobust
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 5612.5997 4102.819 1.368 0.172 -2446.741 1.37e+04
lotsize 6.0530 0.424 14.265 0.000 5.219 6.887
bedrooms 1.057e+04 1247.676 8.470 0.000 8116.488 1.3e+04
==============================================================================
Omnibus: 77.789 Durbin-Watson: 1.193
Prob(Omnibus): 0.000 Jarque-Bera (JB): 146.854
Skew: 0.833 Prob(JB): 1.29e-32
Kurtosis: 4.919 Cond. No. 2.60e+04
==============================================================================
Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
[2] The condition number is large, 2.6e+04. This might indicate that there are
strong multicollinearity or other numerical problems.
Fifth, we can also print mlr model estimated coefficients table individually using its summary method and selecting its second tables attribute.
In [5]:
print(mlr.summary().tables[1])
Out [5]:
==============================================================================
coef std err t P>|t| [0.025 0.975]
------------------------------------------------------------------------------
Intercept 5612.5997 4102.819 1.368 0.172 -2446.741 1.37e+04
lotsize 6.0530 0.424 14.265 0.000 5.219 6.887
bedrooms 1.057e+04 1247.676 8.470 0.000 8116.488 1.3e+04
==============================================================================
Courses
My online courses are hosted at Teachable website.
For more details on this concept, you can view my Linear Regression in Python Course.
References
[1] Data Description: Sales prices of houses sold in the city of Windsor, Canada, during July, August and September, 1987.
Original Source: Anglin, P., and Gencay, R. (1996). Semiparametric Estimation of a Hedonic Price Function. Journal of Applied Econometrics, 11, 633–648.
[2] statsmodels Python package: Seabold, Skipper, and Josef Perktold. (2010). “statsmodels: Econometric and statistical modeling with python.” Proceedings of the 9th Python in Science Conference.