Linear regression, also called Ordinary LeastSquares (OLS) Regression, is probably the most commonly used technique in Statistical Learning. It is also the oldest, dating back to the eighteenth century and the work of Carl Friedrich Gauss and AdrienMarie Legendre. It is also one of the easier and more intuitive techniques to understand, and it provides a good basis for learning more advanced concepts and techniques. This post explains how to perform linear regression using the statsmodels Python package. We will discuss the single variable case and defer multiple regression to a future post.
This is part of a series of blog posts to show how to do common statistical learning techniques in Python. We provide only a small amount of background on the concepts and techniques we cover, so if you’d like a more thorough explanation check out Introduction to Statistical Learning or sign up for the free online course by the authors here. If you are just here to learn how to do it in Python skip directly to the examples below.
Statsmodels
Statsmodel is a Python library designed for more statisticallyoriented approaches to data analysis, with an emphasis on econometric analyses. It integrates well with the pandas and numpy libraries we covered in a previous post. It also has built in support for many of the statistical tests to check the quality of the fit and a dedicated set of plotting functions to visualize and diagnose the fit. Scikitlearn also has support for linear regression, including many forms of regularized regression lacking in statsmodels, but it lacks the rich set of statistical tests and diagnostics that have been developed for linear models.
Linear Regression and Ordinary Least Squares
Linear regression is one of the simplest and most commonly used modeling techniques. It makes very strong assumptions about the relationship between the predictor variables (the X) and the response (the Y). It assumes that this relationship takes the form:
(y = beta_0 + beta_1 * x)
Ordinary Least Squares is the simplest and most common estimator in which the two (beta)s are chosen to minimize the square of the distance between the predicted values and the actual values. Even though this model is quite rigid and often does not reflect the true relationship, this still remains a popular approach for several reasons. For one, it is computationally cheap to calculate the coefficients. It is also easier to interpret than more sophisticated models, and in situations where the goal is understanding a simple model in detail, rather than estimating the response well, they can provide insight into what the model captures. Finally, in situations where there is a lot of noise, it may be hard to find the true functional form, so a constrained model can perform quite well compared to a complex model which is more affected by noise.
The resulting model is represented as follows:
(hat{y} = hat{beta}_0 + hat{beta}_1 * x)
Here, the hats on the variables represent the fact that they are estimated from the data we have available. The (beta)s are termed the parameters of the model or the coefficients. (beta_0) is called the constant term or the intercept.
Ordinary Least Squares Using Statsmodels
The statsmodels package provides several different classes that provide different options for linear regression. Getting started with linear regression is quite straightforward with the OLS module.
To start with we load the Longley dataset of US macroeconomic data from the Rdatasets website.
Out[1]:

GNP.deflator 
GNP 
Unemployed 
Armed.Forces 
Population 
Year 
Employed 
1947 
83.0 
234.289 
235.6 
159.0 
107.608 
1947 
60.323 
1948 
88.5 
259.426 
232.5 
145.6 
108.632 
1948 
61.122 
1949 
88.2 
258.054 
368.2 
161.6 
109.773 
1949 
60.171 
1950 
89.5 
284.599 
335.1 
165.0 
110.929 
1950 
61.187 
1951 
96.2 
328.975 
209.9 
309.9 
112.075 
1951 
63.221 
We will use the variable Total Derived Employment ('Employed'
) as our response y
and Gross National Product ('GNP'
) as our predictor X
.
We take the single response variable and store it separately. We also add a constant term so that we fit the intercept of our linear model.
Out[2]:

const 
GNP 
1947 
1 
234.289 
1948 
1 
259.426 
1949 
1 
258.054 
1950 
1 
284.599 
1951 
1 
328.975 
Now we perform the regression of the predictor on the response, using the sm.OLS
class and and its initialization OLS(y, X)
method. This method takes as an input two arraylike objects: X
and y
. In general, X
will either be a numpy array or a pandas data frame with shape (n, p)
where n
is the number of data points and p
is the number of predictors. y
is either a onedimensional numpy array or a pandas series of length n
.
In [3]:
We then need to fit the model by calling the OLS object’s fit()
method. Ignore the warning about the kurtosis test if it appears, we have only 16 examples in our dataset and the test of the kurtosis is valid only if there are more than 20 examples.
In [4]:
est = est.fit()
est.summary()
/usr/local/lib/python2.7/distpackages/scipy/stats/stats.py:1276: UserWarning: kurtosistest only valid for n>=20 ... continuing anyway, n=16
int(n))
Out[4]:
OLS Regression Results
Dep. Variable: 
Employed 
Rsquared: 
0.967 
Model: 
OLS 
Adj. Rsquared: 
0.965 
Method: 
Least Squares 
Fstatistic: 
415.1 
Date: 
Sat, 08 Feb 2014 
Prob (Fstatistic): 
8.36e12 
Time: 
01:28:29 
LogLikelihood: 
14.904 
No. Observations: 
16 
AIC: 
33.81 
Df Residuals: 
14 
BIC: 
35.35 
Df Model: 
1 



coef 
std err 
t 
P>t 
[95.0% Conf. Int.] 
const 
51.8436 
0.681 
76.087 
0.000 
50.382 53.305 
GNP 
0.0348 
0.002 
20.374 
0.000 
0.031 0.038 
Omnibus: 
1.925 
DurbinWatson: 
1.619 
Prob(Omnibus): 
0.382 
JarqueBera (JB): 
1.215 
Skew: 
0.664 
Prob(JB): 
0.545 
Kurtosis: 
2.759 
Cond. No. 
1.66e+03 
After visualizing the relationship we will explain the summary. First, we need the coefficients of the fit.
Out[5]:
const 51.843590
GNP 0.034752
dtype: float64
In [6]:
Out[6]:
[<matplotlib.lines.Line2D at 0x4444350>]
Statsmodels also provides a formulaic interface that will be familiar to users of R. Note that this requires the use of a different api to statsmodels, and the class is now called ols
rather than OLS
. The argument formula
allows you to specify the response and the predictors using the column names of the input data frame data
.
In [7]:
Out[7]:
OLS Regression Results
Dep. Variable: 
Employed 
Rsquared: 
0.967 
Model: 
OLS 
Adj. Rsquared: 
0.965 
Method: 
Least Squares 
Fstatistic: 
415.1 
Date: 
Sat, 08 Feb 2014 
Prob (Fstatistic): 
8.36e12 
Time: 
01:28:29 
LogLikelihood: 
14.904 
No. Observations: 
16 
AIC: 
33.81 
Df Residuals: 
14 
BIC: 
35.35 
Df Model: 
1 



coef 
std err 
t 
P>t 
[95.0% Conf. Int.] 
Intercept 
51.8436 
0.681 
76.087 
0.000 
50.382 53.305 
GNP 
0.0348 
0.002 
20.374 
0.000 
0.031 0.038 
Omnibus: 
1.925 
DurbinWatson: 
1.619 
Prob(Omnibus): 
0.382 
JarqueBera (JB): 
1.215 
Skew: 
0.664 
Prob(JB): 
0.545 
Kurtosis: 
2.759 
Cond. No. 
1.66e+03 
This summary provides quite a lot of information about the fit. The parts of the table we think are the most important are bolded in the description below.
The left part of the first table provides basic information about the model fit:
Element 
Description 
Dep. Variable 
Which variable is the response in the model 
Model 
What model you are using in the fit 
Method 
How the parameters of the model were calculated 
No. Observations 
The number of observations (examples) 
DF Residuals 
Degrees of freedom of the residuals. Number of observations – number of parameters 
DF Model 
Number of parameters in the model (not including the constant term if present) 
The right part of the first table shows the goodness of fit
Element 
Description 
Rsquared 
The coefficient of determination. A statistical measure of how well the regression line approximates the real data points 
Adj. Rsquared 
The above value adjusted based on the number of observations and the degreesoffreedom of the residuals 
Fstatistic 
A measure how significant the fit is. The mean squared error of the model divided by the mean squared error of the residuals 
Prob (Fstatistic) 
The probability that you would get the above statistic, given the null hypothesis that they are unrelated 
Loglikelihood 
The log of the likelihood function. 
AIC 
The Akaike Information Criterion. Adjusts the loglikelihood based on the number of observations and the complexity of the model. 
BIC 
The Bayesian Information Criterion. Similar to the AIC, but has a higher penalty for models with more parameters. 
The second table reports for each of the coefficients

Description 

The name of the term in the model 
coef 
The estimated value of the coefficient 
std err 
The basic standard error of the estimate of the coefficient. More sophisticated errors are also available. 
t 
The tstatistic value. This is a measure of how statistically significant the coefficient is. 
P > t 
Pvalue that the nullhypothesis that the coefficient = 0 is true. If it is less than the confidence level, often 0.05, it indicates that there is a statistically significant relationship between the term and the response. 
[95.0% Conf. Interval] 
The lower and upper values of the 95% confidence interval

Finally, there are several statistical tests to assess the distribution of the residuals
Element 
Description 
Skewness 
A measure of the symmetry of the data about the mean. Normallydistributed errors should be symmetrically distributed about the mean (equal amounts above and below the line). 
Kurtosis 
A measure of the shape of the distribution. Compares the amount of data close to the mean with those far away from the mean (in the tails). 
Omnibus 
D’Angostino’s test. It provides a combined statistical test for the presence of skewness and kurtosis. 
Prob(Omnibus) 
The above statistic turned into a probability 
JarqueBera 
A different test of the skewness and kurtosis 
Prob (JB) 
The above statistic turned into a probability 
DurbinWatson 
A test for the presence of autocorrelation (that the errors are not independent.) Often important in timeseries analysis 
Cond. No 
A test for multicollinearity (if in a fit with multiple parameters, the parameters are related with each other). 
As a final note, if you don’t want to include a constant term in your model, you can exclude it using the minus operator.
In [8]:
Out[8]:
[<matplotlib.lines.Line2D at 0x47eab50>]
But notice that this may not be the best idea… 🙂
Correlation and Causation
Clearly there is a relationship or correlation between GNP and total employment. So does that mean a change in GNP cause a change in total employment? Or does a change in total employment cause a change in GNP? This is a subject we will explore in the next post.
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