FRM Exam Part I · Regression with Multiple Explanatory Variables

Joint Hypothesis Tests and the F-Statistic in Regression

Q: What is the difference between a t-test and an F-test in regression?

A t-test checks one restriction, usually one coefficient. An F-test checks several restrictions at once. With one restriction, F equals t squared and both lead to the same conclusion.

Q: What are the degrees of freedom for the F-statistic?

The numerator is q, the number of restrictions. The denominator is n − k − 1, taken from the unrestricted model.

Q: Can variables be jointly significant but individually insignificant?

Yes. When regressors are highly correlated, standard errors inflate and t-statistics shrink. The F-test can still reject the joint null.

Q: What does the F-statistic in regression output test?

It tests the null that all slope coefficients are zero, so the model has no explanatory power beyond the intercept. A large F rejects that null.

Updated 11 October 2026 · Fact-checked

A joint hypothesis test checks several coefficient restrictions at once. You fit an unrestricted model and a restricted model, then compute F = [(SSR_r − SSR_u) ÷ q] ÷ [SSR_u ÷ (n − k − 1)]. If F exceeds the critical value with (q, n − k − 1) degrees of freedom, you reject the joint null.

Understand Joint Hypothesis Tests and the F-Statistic

A t-test checks one coefficient at a time. Often you want to know whether a group of variables matters together. For example: are the coefficients on two macro factors both zero? A joint hypothesis states several restrictions in one null, such as β2 = 0 and β3 = 0.

You cannot just run two t-tests and combine them. Each test has its own error rate, and the estimated coefficients are usually correlated. Two variables can each look insignificant by t-test, yet be significant together. This happens often when regressors are highly correlated (multicollinearity), because the standard errors are inflated.

The F-test compares two models. The unrestricted model includes all variables. The restricted model imposes the null, for example by dropping the variables whose coefficients are set to zero. If the restrictions are true, forcing them should barely raise the sum of squared residuals (SSR). If SSR rises a lot, the restrictions are rejected.

The F-statistic measures that rise in SSR per restriction, relative to the unrestricted model's error variance. It follows an F distribution with q numerator and n − k − 1 denominator degrees of freedom, where q is the number of restrictions and k is the number of slope coefficients in the unrestricted model. The test is one-sided: you reject only for large F.

The usual regression output reports an F-statistic for the null that all slope coefficients are zero. That is just a special case, with q = k.

Key formulas to remember

F-statistic (SSR form): F = [(SSR_restricted − SSR_unrestricted) ÷ q] ÷ [SSR_unrestricted ÷ (n − k − 1)]; q = number of restrictions; k = slope coefficients in the unrestricted model; n = observations.
F-statistic (R² form): F = [(R²_unrestricted − R²_restricted) ÷ q] ÷ [(1 − R²_unrestricted) ÷ (n − k − 1)]; Valid only when both models have the same dependent variable, so TSS is the same.
Overall significance F-test: F = (ESS ÷ k) ÷ (SSR ÷ (n − k − 1)); Null: all slope coefficients equal zero. Degrees of freedom (k, n − k − 1).
Degrees of freedom: numerator = q; denominator = n − k − 1; Denominator uses the unrestricted model.
Decision rule: Reject H0 if F > F critical (q, n − k − 1); One-tailed, upper tail only.
Single restriction link: F = t² when q = 1; A one-restriction F-test gives the same conclusion as a two-sided t-test.

How to solve Joint Hypothesis Tests and the F-Statistic questions

Use this method for any joint hypothesis question.

1Write the null: list every restriction, such as β2 = 0 and β3 = 0. Count them to get q.
2Identify the unrestricted model and count its slope coefficients k. Find n − k − 1.
3Get SSR for both models, or R² for both if the dependent variable is the same.
4Compute F with the SSR form (or R² form) using the formula.
5State the degrees of freedom (q, n − k − 1) and find the critical value, or compare the p-value with the significance level.
6Reject the null if F is larger than the critical value. Conclude that at least one restriction fails.
7If you fail to reject, say the variables are not jointly significant, which supports the restricted model.

Quickest way: Fast F-test from SSR or R²

When to use it: When the question gives both SSRs or both R² values and a critical value or asks for the F number.

Compute the change: SSR_r − SSR_u (or R²_u − R²_r).
Divide it by q.
Divide the unrestricted SSR (or 1 − R²_u) by n − k − 1.
Divide the first result by the second to get F.
Compare with the critical value given. Larger F means reject.

Common mistakes in Joint Hypothesis Tests and the F-Statistic

Using the restricted model's degrees of freedom in the denominator.
The restricted model has fewer parameters, so it looks like the natural choice.
Fix: The denominator is always n − k − 1 from the unrestricted model.
Swapping SSR_r and SSR_u, giving a negative F.
Students forget which model has more variables.
Fix: Restricted SSR is never smaller than unrestricted SSR. F must be non-negative; recheck if not.
Using the R² form when the dependent variables differ.
The R² shortcut looks easier.
Fix: Use the R² form only if both models share the same dependent variable. Otherwise use SSR.
Concluding that all variables are significant after rejecting the null.
Rejection sounds like a verdict on every coefficient.
Fix: Rejection means at least one restriction is false, not that all are.
Running separate t-tests instead of an F-test when regressors are correlated.
t-statistics are shown on every output.
Fix: Individually insignificant t-stats can hide joint significance. Use the F-test for joint claims.
Setting q to the number of variables in the model rather than the number of restrictions.
Confusing k with q.
Fix: Count only the equalities in the null. Dropping 2 variables means q = 2.

Worked examples

Example 1

A regression of monthly fund returns on 4 factors uses n = 63 observations. The unrestricted SSR is 120. Testing H0: β3 = β4 = 0, the restricted model (2 factors) has SSR 138. Compute F and decide at 5% given the critical value F(2, 58) ≈ 3.15.

Show the solution

q = 2 restrictions; k = 4; n − k − 1 = 63 − 4 − 1 = 58.
Numerator: (138 − 120) ÷ 2 = 18 ÷ 2 = 9.
Denominator: 120 ÷ 58 = 2.0690.
F = 9 ÷ 2.0690 = 4.35.
Compare: 4.35 > 3.15, so reject H0.

Answer: F ≈ 4.35, which exceeds 3.15. Reject the null: factors 3 and 4 are jointly significant.

Example 2

Using the same dependent variable, an unrestricted model with 3 slope coefficients and n = 44 has R² = 0.40. Testing H0: β2 = β3 = 0, the restricted model has R² = 0.34. Compute F.

Show the solution

q = 2; k = 3; n − k − 1 = 44 − 3 − 1 = 40.
Numerator: (0.40 − 0.34) ÷ 2 = 0.06 ÷ 2 = 0.03.
Denominator: (1 − 0.40) ÷ 40 = 0.60 ÷ 40 = 0.015.
F = 0.03 ÷ 0.015 = 2.0.
The 5% critical value for F(2, 40) is about 3.23. Since 2.0 < 3.23, do not reject.

Answer: F = 2.0, below the critical value of about 3.23. You fail to reject: the two variables are not jointly significant at 5%.

Exam tips

Always check which model has the larger SSR. The restricted one does. This lets you spot arithmetic slips quickly.
Read carefully whether the question gives SSR, R² or ESS. Pick the matching formula and the right degrees of freedom.
Questions often ask why a joint F-test rejects while individual t-tests do not. The answer is correlation among regressors (multicollinearity).
Remember that with q = 1, F = t². Use this to cross-check a single-restriction question.
Count q from the null hypothesis, not from the number of regressors.

Practice questions from Regression with Multiple Explanatory Variables

Joint Hypothesis Tests and the F-Statistic in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Joint Hypothesis Tests and the F-Statistic: frequently asked questions

What is the difference between a t-test and an F-test in regression?

A t-test checks one restriction, usually one coefficient. An F-test checks several restrictions at once. With one restriction, F equals t squared and both lead to the same conclusion.

What are the degrees of freedom for the F-statistic?

The numerator is q, the number of restrictions. The denominator is n − k − 1, taken from the unrestricted model.

Can variables be jointly significant but individually insignificant?

Yes. When regressors are highly correlated, standard errors inflate and t-statistics shrink. The F-test can still reject the joint null.

What does the F-statistic in regression output test?

It tests the null that all slope coefficients are zero, so the model has no explanatory power beyond the intercept. A large F rejects that null.