F-test

F-test is a hypothesis test that assesses whether a set of regression coefficients are jointly different from zero.

Also known asF statistic · joint significance test

ByHoang TruongUpdated June 30, 2026

FrameworkF-test

See it move

Loading infographic...

The side-by-side comparison contrasts a restricted model — in which q slope coefficients are forced to zero, yielding a lower R²_R — against an unrestricted model in which all k coefficients are freely estimated, yielding a higher R²_U. The F-ratio equals (R²_U − R²_R) divided by q, over (1 − R²_U) divided by (n − k − 1); a sufficiently large F leads to rejecting the restrictions, indicating that at least one of the dropped slopes carries genuine explanatory power that the restricted model discards.

Where it fits

SubjectData Analysis & StatisticsAdvancedTopicCommon Significance TestsAdvancedTopicMultiple Regression & InterpretationAdvanced

The formula

LaTeX

F = \frac{(R^2_U - R^2_R) / q}{(1 - R^2_U) / (n - k - 1)}

Variables

$R_{U}^{2}$: R-squared of the unrestricted model
$R_{R}^{2}$: R-squared of the restricted model (restrictions imposed under H₀)
$q$: Number of restrictions being tested
$n$: Sample size
$k$: Number of regressors in the unrestricted model

Reject H₀ if F exceeds the F(q, n − k − 1) critical value; tests joint significance of q coefficients

Check yourself

PracticeCORE

An OLS regression of monthly café revenue on advertising spend and population density produces R² = 0.38 with n = 80 observations and k = 2 regressors. The F-statistic for joint significance is approximately 23.6 (F-critical at 5% with 2 and 77 degrees of freedom ≈ 3.12). What is the correct conclusion?

Select an answer to check your understanding.