# How to compute a confidence interval for a regression coefficient (in Python, using statsmodels)

## Task

Say we have a linear regression model, either single variable or multivariate. How do we compute a confidence interval for the coefficient of one of the explanatory variables in the model?

Related tasks:

- How to compute a confidence interval for a mean difference (matched pairs)
- How to compute a confidence interval for a population mean
- How to compute a confidence interval for a single population variance
- How to compute a confidence interval for the difference between two means when both population variances are known
- How to compute a confidence interval for the difference between two means when population variances are unknown
- How to compute a confidence interval for the difference between two proportions
- How to compute a confidence interval for the expected value of a response variable
- How to compute a confidence interval for the population proportion
- How to compute a confidence interval for the ratio of two population variances

## Solution

We’ll assume that you have fit a single linear model to your data, as in the code below, which uses fake example data. You can replace it with your actual data.

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import statsmodels.api as sm
xs = [ 34, 9, 78, 60, 22, 45, 83, 59, 25 ]
ys = [ 126, 347, 298, 309, 450, 187, 266, 385, 400 ]
xs = sm.add_constant( xs )
model = sm.OLS( ys, xs )
results = model.fit()

We can use Python’s `conf_int()`

function to find the confidence interval for the model coefficients. You can change the `alpha`

parameter to specify a different significance level. Note that if you have a multiple regression model, it will make confidence intervals for all of the coefficient values.

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results.conf_int( alpha=0.05 )

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array([[172.63807531, 535.52642049],
[ -4.49196063, 2.47393542]])

Each list in the array represents the 95% confidence interval for the corresponding coefficient in the model beginning with the intercept and each regression coefficient thereafter. Accordingly, the 95% confidence interval for the regression coefficient is $[-4.49196063,2.47393542]$.

Content last modified on 24 July 2023.

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Contributed by Andrew Quagliaroli (aquagliaroli@falcon.bentley.edu)