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Linear Regression: Coefficient of Determination in Python

Last Update: February 21, 2022

Coefficient of Determination in Python can be estimated using statsmodels package ols function, its summary method and rsquared, rsquared_adj properties found within statsmodels.formula.api module to fit linear regression, print its summary results and estimated coefficients of determination. 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 of multiple determination 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 package statsmodels 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 summary results which include estimated coefficients of multiple determination using its summary method.

In [4]:
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:                Wed, 25 Aug 2021   Prob (F-statistic):           2.95e-55
Time:                        18:41:02   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

[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 of determination using its rsquared and rsquared_adj properties.

In [5]:
Out [5]:
In [6]:
Out [6]:


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[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] Seabold, Skipper, and Josef Perktold. (2010). “statsmodels: Econometric and statistical modeling with python.” Proceedings of the 9th Python in Science Conference.

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