ECON2515 · Intermediate Applied Econometrics Ii
Testing More Than One Parameter: the F-test
This is where ECON 2515 inference stops asking about one coefficient and starts testing several restrictions at once — does this block of variables jointly belong, does the model explain anything at all, is a combination of coefficients equal to some value? The engine is a comparison of two nested models: the full unrestricted model and a smaller restricted model that imposes H₀, compared through their residual sums of squares in F = [(SSR_R − SSR_U)/q] / [SSR_U/(n − k − 1)]. It is a one-tailed test on F(q, n − k − 1): reject only when F is large. The recurring exam traps are miscounting the restrictions q, getting the two degrees of freedom wrong, and dropping a block of variables just because their individual t-statistics look small.
What this chapter covers
- 011. Joint hypotheses — testing q restrictions simultaneously (e.g. β₂ = β₃ = 0), which a single t-test cannot do
- 022. Restricted vs unrestricted models — impose H₀ to build the restricted model; it always fits worse, so SSR_R ≥ SSR_U
- 033. The F-statistic — F = [(SSR_R − SSR_U)/q] / [SSR_U/(n − k − 1)], a signal-to-noise ratio of the drop in fit
- 044. Degrees of freedom — numerator q (number of '=' signs in H₀), denominator n − k − 1 (k = slopes in the unrestricted model)
- 055. The decision rule — one-tailed: reject H₀ if F > F* = Fα(q, n − k − 1); F ≥ 0 always, so there is no lower tail
- 066. Overall significance — H₀: all slopes = 0, restricted model is intercept-only; this is the printout's 'Prob > F'
- 077. Linear-combination tests — substitute the null into the model to rebuild the restricted regression, then form F
- 088. The t² = F link — for a single restriction (q = 1) the F-test equals the square of the two-sided t-test
Joint F-test: do two region dummies belong in a wage model?
- +2State the joint hypothesis and count restrictions. H₀: β₃ = β₄ = 0 (south and urban jointly irrelevant) vs H₁: at least one ≠ 0. There are two '=' signs, so the number of restrictions q = 2.
- +1Get the degrees of freedom. The unrestricted model has k = 4 slope coefficients, and n = 88, so the denominator df = n − k − 1 = 88 − 4 − 1 = 83. The statistic is distributed F(q, n − k − 1) = F(2, 83).
- +3Form the F-statistic. F = [(SSR_R − SSR_U)/q] / [SSR_U/(n − k − 1)] = [(279 − 249)/2] / [249/83] = (30/2) / 3.0 = 15 / 3.0 = 5.0.
- +2Apply the one-tailed decision rule. Reject H₀ if F > F* = 3.11. Since 5.0 > 3.11, reject H₀: the two region dummies are jointly significant and should stay in the model — even though each was individually insignificant, because that pattern is exactly what the joint F is built to catch.
Key terms
- Joint hypothesis
- A null that places two or more restrictions on the coefficients at once, e.g. H₀: β₂ = β₃ = 0 or H₀: β₃ + 2β₄ = 1. A single t-test can only handle one restriction, so joint hypotheses require the F-test.
- Unrestricted model
- The full regression with all the coefficients free, giving the smaller residual sum of squares SSR_U. It is the benchmark fit that the restricted model is compared against, and its residual df n − k − 1 goes in the F denominator.
- Restricted model
- The regression after imposing the null H₀ (dropping variables, or substituting a linear combination). With fewer free parameters it can only fit as well or worse, so SSR_R ≥ SSR_U always.
- F-statistic
- F = [(SSR_R − SSR_U)/q] / [SSR_U/(n − k − 1)], the drop in fit per restriction divided by the residual variance. It is distributed F(q, n − k − 1) under H₀ and is always non-negative.
- Number of restrictions (q)
- How many separate conditions the null imposes — count the '=' signs in H₀. It is the numerator degrees of freedom, and is not the same as the number of regressors in the model.
- Overall significance
- The joint test H₀: β₁ = β₂ = … = βₖ = 0 (every slope zero at once). The restricted model is intercept-only, so F = (SSE/k) / (SSR/(n − k − 1)); it is the 'F(k, n − k − 1)' and 'Prob > F' printed at the top of regression output.
- Linear restriction
- A null written as an equation among coefficients, e.g. H₀: β₃ + 2β₄x₀ = 1. You test it by substituting the equality into the model to build a transformed restricted regression, getting SSR_R, then forming F (here with q = 1).
- t² = F identity
- For a single restriction (q = 1) the F-test and the two-sided t-test are equivalent: the F-statistic equals the square of the corresponding t, and their p-values match. F generalises the t-test to more than one restriction.
Testing More Than One Parameter: the F-test FAQ
When do I use an F-test instead of a t-test?
Use a t-test for a single restriction on a single coefficient (is β₂ = 0? is β₂ > 3?). Use an F-test whenever the null involves more than one restriction at once — 'do x₂ and x₃ jointly belong?', 'does the model explain anything overall?', or 'is β₃ + 2β₄ equal to 1?'. If there is only one restriction, the two agree exactly because t² = F.
What exactly is q, and how do I count it?
q is the number of restrictions in the null — literally the number of '=' signs. H₀: β₂ = β₃ = 0 imposes two restrictions, so q = 2; H₀: β₂ = β₃ = β₄ = 0 has q = 3. A common mistake is to set q to the number of variables in the model; q counts the conditions in H₀, and it is also the numerator degrees of freedom.
Which SSR goes on top, and why is the F never negative?
The numerator is SSR_R − SSR_U (restricted minus unrestricted) and the denominator uses SSR_U. Because removing variables or imposing a restriction can never improve the least-squares fit, the restricted model always has the larger residual sum of squares, so SSR_R ≥ SSR_U and the numerator is non-negative. A negative 'F' means you swapped the restricted and unrestricted models.
Two variables each have insignificant t-stats — can I just drop them both?
Not on the t-stats alone. When two regressors are correlated with each other (multicollinearity), each individual t can be small even though together they explain a lot of variation. Run the joint F-test first: if it rejects, the block is jointly significant and should stay, even though no single t looked convincing. This is one of the main reasons the F-test exists.
How do I test the overall significance of a regression?
Test H₀: β₁ = β₂ = … = βₖ = 0 (all slopes zero). The restricted model is just the intercept, so the statistic becomes F = (SSE/k) / (SSR/(n − k − 1)) on F(k, n − k − 1). Software prints this automatically as the 'F(k, n − k − 1)' line with a 'Prob > F'; a tiny Prob > F rejects the null and says the regressors jointly explain y — though it does not tell you which individual ones matter.
How do I test a null that is a combination of coefficients, like β₃ + 2β₄ = 1?
You cannot get SSR_R by deleting a column. Substitute the equality into the model — solve for one coefficient (β₃ = 1 − 2β₄), plug it back in and rearrange so the known part moves to the dependent variable — then estimate that transformed restricted regression to obtain SSR_R. Form the usual F with q = 1 (one '=' sign), and you can cross-check because t² = F for a single restriction.
Exam move
Treat every F-test question as a fixed four-move routine: write the null and count q, name the unrestricted and restricted models and their SSRs, compute F = [(SSR_R − SSR_U)/q] / [SSR_U/(n − k − 1)], then compare to F* from the provided F-table at df (q, n − k − 1). Practise the three flavours the exam reuses — a joint exclusion of a block of variables, overall significance (all slopes zero, read off the 'Prob > F'), and a linear-combination test where you must rebuild the restricted model by substituting the null. Drill the degrees of freedom out loud before looking anything up, because the wrong table entry flips borderline decisions, and always sanity-check that SSR_R is the larger SSR so your numerator is positive. Finally, memorise that the test is one-tailed (reject only when F > F*) and that t² = F for a single restriction, and rehearse the reasoning that individually insignificant variables can still be jointly significant — it is both a favourite MCQ and the interpretive sentence that earns the last marks in a worked answer.
