# Example of Multiple Linear Regression in Python

In this guide, you’ll see how to perform multiple linear regression in Python using both sklearn and statsmodels.

## The Example

In the following example, you’ll see how to perform multiple linear regression for a fictitious economy, where the index_price is the dependent variable, and the 2 independent/input variables are:

• interest_rate
• unemployment_rate

Please note that you’ll have to validate that a linear relationship exists between the dependent variable and the independent variable/s (more on that under the checking for linearity section).

To start, here is the full dataset that includes the dependent and independent variables:

`import pandas as pddata = {"year": [2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016],        "month": [12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1],        "interest_rate": [2.75, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.25, 2.25, 2.25, 2, 2, 2, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75],        "unemployment_rate": [5.3, 5.3, 5.3, 5.3, 5.4, 5.6, 5.5, 5.5, 5.5, 5.6, 5.7, 5.9, 6, 5.9, 5.8, 6.1, 6.2, 6.1, 6.1, 6.1, 5.9, 6.2, 6.2, 6.1],        "index_price": [1464, 1394, 1357, 1293, 1256, 1254, 1234, 1195, 1159, 1167, 1130, 1075, 1047, 965, 943, 958, 971, 949, 884, 866, 876, 822, 704, 719]        }df = pd.DataFrame(data)print(df)`

The result:

``````    year  month  interest_rate  unemployment_rate  index_price
0   2017     12           2.75                5.3         1464
1   2017     11           2.50                5.3         1394
2   2017     10           2.50                5.3         1357
3   2017      9           2.50                5.3         1293
4   2017      8           2.50                5.4         1256
5   2017      7           2.50                5.6         1254
6   2017      6           2.50                5.5         1234
7   2017      5           2.25                5.5         1195
8   2017      4           2.25                5.5         1159
9   2017      3           2.25                5.6         1167
10  2017      2           2.00                5.7         1130
11  2017      1           2.00                5.9         1075
12  2016     12           2.00                6.0         1047
13  2016     11           1.75                5.9          965
14  2016     10           1.75                5.8          943
15  2016      9           1.75                6.1          958
16  2016      8           1.75                6.2          971
17  2016      7           1.75                6.1          949
18  2016      6           1.75                6.1          884
19  2016      5           1.75                6.1          866
20  2016      4           1.75                5.9          876
21  2016      3           1.75                6.2          822
22  2016      2           1.75                6.2          704
23  2016      1           1.75                6.1          719``````

## Checking for Linearity

Next, check that a linear relationship exists between the:

• index_price (dependent variable) and interest_rate (independent variable)
• index_price (dependent variable) and unemployment_rate (independent variable)

You can use scatter diagrams to perform a quick linearity check (utilizing the Matplotlib library).

To plot the relationship between the index_price and the interest_rate:

`import pandas as pdimport matplotlib.pyplot as pltdata = {"year": [2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016],        "month": [12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1],        "interest_rate": [2.75, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.25, 2.25, 2.25, 2, 2, 2, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75],        "unemployment_rate": [5.3, 5.3, 5.3, 5.3, 5.4, 5.6, 5.5, 5.5, 5.5, 5.6, 5.7, 5.9, 6, 5.9, 5.8, 6.1, 6.2, 6.1, 6.1, 6.1, 5.9, 6.2, 6.2, 6.1],        "index_price": [1464, 1394, 1357, 1293, 1256, 1254, 1234, 1195, 1159, 1167, 1130, 1075, 1047, 965, 943, 958, 971, 949, 884, 866, 876, 822, 704, 719]        }df = pd.DataFrame(data)plt.scatter(df["interest_rate"], df["index_price"], color="red")plt.title("Index Price Vs Interest Rate", fontsize=14)plt.xlabel("Interest Rate", fontsize=14)plt.ylabel("Index Price", fontsize=14)plt.grid(True)plt.show()`

Notice that indeed a linear relationship exists between the index_price and the interest_rate. Specifically, when interest rates go up, the index price also goes up.

To plot the relationship between the index_price and the unemployment_rate:

`import pandas as pdimport matplotlib.pyplot as plt  data = {"year": [2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016],        "month": [12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1],        "interest_rate": [2.75, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.25, 2.25, 2.25, 2, 2, 2, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75],        "unemployment_rate": [5.3, 5.3, 5.3, 5.3, 5.4, 5.6, 5.5, 5.5, 5.5, 5.6, 5.7, 5.9, 6, 5.9, 5.8, 6.1, 6.2, 6.1, 6.1, 6.1, 5.9, 6.2, 6.2, 6.1],        "index_price": [1464, 1394, 1357, 1293, 1256, 1254, 1234, 1195, 1159, 1167, 1130, 1075, 1047, 965, 943, 958, 971, 949, 884, 866, 876, 822, 704, 719]        }df = pd.DataFrame(data) plt.scatter(df["unemployment_rate"], df["index_price"], color="green")plt.title("Index Price Vs Unemployment Rate", fontsize=14)plt.xlabel("Unemployment Rate", fontsize=14)plt.ylabel("Index Price", fontsize=14)plt.grid(True)plt.show()`

You’ll notice that a linear relationship also exists between the index_price and the unemployment_rate – when the unemployment rates go up, the index price goes down (here you still have a linear relationship, but with a negative slope).

## Performing the Multiple Linear Regression in Python

Once you added the data into Python, you may use either sklearn or statsmodels to get the regression results. Either method would work, but you’ll see both methods for illustration purposes.

First, install the sklearn package:

`pip install scikit-learn`

Then, install the statsmodels package:

`pip install statsmodels`

Here is the complete code to perform the multiple linear regression in Python:

`import pandas as pdfrom sklearn import linear_modelimport statsmodels.api as smdata = {"year": [2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2017, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016, 2016],        "month": [12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1],        "interest_rate": [2.75, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5, 2.25, 2.25, 2.25, 2, 2, 2, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75, 1.75],        "unemployment_rate": [5.3, 5.3, 5.3, 5.3, 5.4, 5.6, 5.5, 5.5, 5.5, 5.6, 5.7, 5.9, 6, 5.9, 5.8, 6.1, 6.2, 6.1, 6.1, 6.1, 5.9, 6.2, 6.2, 6.1],        "index_price": [1464, 1394, 1357, 1293, 1256, 1254, 1234, 1195, 1159, 1167, 1130, 1075, 1047, 965, 943, 958, 971, 949, 884, 866, 876, 822, 704, 719]        }df = pd.DataFrame(data)x = df[["interest_rate", "unemployment_rate"]]y = df["index_price"]# Using sklearnregression = linear_model.LinearRegression()regression.fit(x, y)predictions_sklearn = regression.predict(x)print("Intercept: \n", regression.intercept_)print("Coefficients: \n", regression.coef_)# Using statsmodelsx = sm.add_constant(x)model = sm.OLS(y, x).fit()predictions_statsmodels = model.predict(x)summary = model.summary()print(summary)`

Once you run the code in Python, you’ll observe two parts:

### (1) The first part shows the output generated by sklearn:

``````Intercept:
1798.4039776258564
Coefficients:
[ 345.54008701 -250.14657137]``````

This output includes the intercept and coefficients. You can use this information to build the multiple linear regression equation as follows:

index_price = (intercept) + (interest_rate coef)*X1 + (unemployment_rate coef)*X2

And once you plug the numbers:

index_price = (1798.4040) + (345.5401)*X1 + (-250.1466)*X2

### (2) The second part displays a comprehensive table with statistical info generated by statsmodels.

This information can provide additional insights about the model used (such as the fit of the model, standard errors, etc):

``````                            OLS Regression Results
==============================================================================
Dep. Variable:            index_price   R-squared:                       0.898
Method:                 Least Squares   F-statistic:                     92.07
Date:                Sat, 30 Jul 2022   Prob (F-statistic):           4.04e-11
Time:                        13:47:01   Log-Likelihood:                -134.61
No. Observations:                  24   AIC:                             275.2
Df Residuals:                      21   BIC:                             278.8
Df Model:                           2
Covariance Type:            nonrobust
=====================================================================================
coef    std err          t      P>|t|      [0.025      0.975]
-------------------------------------------------------------------------------------
const              1798.4040    899.248      2.000      0.059     -71.685    3668.493
interest_rate       345.5401    111.367      3.103      0.005     113.940     577.140
unemployment_rate  -250.1466    117.950     -2.121      0.046    -495.437      -4.856
==============================================================================
Omnibus:                        2.691   Durbin-Watson:                   0.530
Prob(Omnibus):                  0.260   Jarque-Bera (JB):                1.551
Skew:                          -0.612   Prob(JB):                        0.461
Kurtosis:                       3.226   Cond. No.                         394.
==============================================================================``````

Notice that the coefficients captured in this table (highlighted in yellow) match with the coefficients generated by sklearn.

You got consistent results by applying both sklearn and statsmodels.

## Conclusion

Linear regression is often used in Machine Learning. You have seen some examples of how to perform multiple linear regression in Python using both sklearn and statsmodels.

Before applying linear regression models, make sure to check that a linear relationship exists between the dependent variable (i.e., what you are trying to predict) and the independent variable/s (i.e., the input variable/s).