How to predict the response variable in a linear model (in Python, using statsmodels)

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Task

If we have a linear model and a value for each explanatory variable, how do we predict the corresponding value of the response variable?

Related tasks:

• How to fit a linear model to two columns of data
• How to fit a multivariate linear model

Solution

Let’s assume that you’ve already built a linear model. We do an example below with fake data, but you can use your own actual data. For more information on the following code, see how to fit a multivariate linear model.

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import pandas as pd
df = pd.DataFrame( {
    'x1' : [ 2,  7,  4,  3, 11, 18,   6, 15,   9,  12],
    'x2' : [ 4,  6, 10,  1, 18, 11,   8, 20,  16,  13],
    'x3' : [11, 16, 20,  6, 14,  8,   5, 23,  13,  10],
    'y'  : [24, 60, 32, 29, 90, 45, 130, 76, 100, 120]
} )

import statsmodels.api as sm
model = sm.OLS( df['y'], sm.add_constant( df[['x1','x2','x3']] ) ).fit()

Let’s say we want to estimate $y$ given that $x_1 = 5$, $x_2 = 12$, and $x_3=50$. We can use the model’s predict() function as shown below, but we must add an entry for the constant term in the model—we can use any value, but we choose 1.

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model.predict( [ 1, 5, 12, 50 ] )

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array([-91.71014402])

For the given values of the explanatory variables, our predicted response variable is $-91.71014402$.

Note that if you want to compute the predicted values for all the data on which the model was trained, simply call model.predict() with no arguments, and it defaults to using the training data.

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model.predict()

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array([ 47.5701159 ,  24.35988296,  42.21531274,  47.27613825,
       110.86526185,  70.03097584,  95.12689978,  70.91290879,
       106.52986696,  91.11263692])

Content last modified on 24 July 2023.

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Contributed by Nathan Carter (ncarter@bentley.edu)