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Testing Multiple Restrictions with the F Test

The F test evaluates several restrictions at once by comparing a restricted and an unrestricted model, F=(SSRr−SSRur)/qSSRur/(n−k−1)F=\dfrac{(SSR_r-SSR_{ur})/q}{SSR_{ur}/(n-k-1)}, where qq is the number of restrictions. A large FF means dropping those regressors raises the residual sum of squares enough to reject the joint null. Separate t tests cannot replace the F test, because they ignore the correlation between the estimates and distort the joint error rate.

Why it matters

Sometimes the question is whether a group of variables matters together, like all the experience terms or every industry dummy. The F test asks how much fit you lose by deleting them as a block. Checking each one with its own t test can miss joint significance, especially when the regressors are correlated.

Formulas

F statistic (SSR form)
F=(SSRr−SSRur)/qSSRur/(n−k−1)F=\frac{(SSR_r-SSR_{ur})/q}{SSR_{ur}/(n-k-1)}
qq restrictions in the numerator; (n−k−1)(n-k-1) denominator degrees of freedom.
Overall significance F
F=R2/k(1−R2)/(n−k−1)F=\frac{R^2/k}{(1-R^2)/(n-k-1)}
Tests whether all slopes are jointly zero, using the model R2R^2.

Worked examples

Scenario

You ask whether exper and tenure jointly matter in `regress lwage educ exper tenure`.

Solution

After the regression, `test exper tenure` reports an F statistic for H0:βexper=βtenure=0H_0:\beta_{\mathrm{exper}}=\beta_{\mathrm{tenure}}=0. A large F with a small p-value rejects the joint null, so the experience block belongs even if one term looks weak alone.

NoteStata’s `test` command builds the F statistic for you; the regression header F tests overall significance.

Common mistakes

  • ✗Thinking two individually insignificant variables cannot be jointly significant. Correlated regressors can be jointly significant even when each t test fails.
  • ✗Running many separate t tests as a substitute for one F test. That inflates the overall Type I error and ignores the joint distribution.
  • ✗Believing the F test can compare models with different dependent variables or non-nested specifications. It requires nested models with the same yy.
  • ✗Reading a significant overall F as proof every regressor matters. It only says at least one slope is nonzero.

Revision bullets

  • •F test: F=[(SSRr−SSRur)/q] / [SSRur/(n−k−1)]F=[(SSR_r-SSR_{ur})/q]\,/\,[SSR_{ur}/(n-k-1)] for qq joint restrictions.
  • •Large F means the restrictions cost too much fit, so reject the joint null.
  • •Separate t tests cannot test joint hypotheses; use the F test.
  • •The overall F tests whether all slopes are jointly zero.
  • •The F test needs nested models with the same dependent variable.

Quick check

The F statistic in the SSR form compares:

Why can you not test a joint hypothesis on several coefficients with separate t tests?

Connected topics

R-squaredCLM normalityt testConf. intervalsPoly & Inter.Dummies

Sources

  1. Wooldridge, Introductory Econometrics, Ch. 4
    Wooldridge (2019), Introductory Econometrics: A Modern Approach, 7th ed., Sec. 4.5 (multiple restrictions, F test)
  2. Greene, Econometric Analysis
    Greene (2018), Econometric Analysis, 8th ed., Ch. 5
How to cite this page
Dr. Phil's Quant Lab. (2026). Testing Multiple Restrictions with the F Test. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-f-test
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