FRM Exam Part I · Linear Regression

Multiple Regression and Joint Hypothesis Tests (F-Test)

Updated 11 October 2026 · Fact-checked

Multiple regression explains one dependent variable using several explanatory variables. Each slope is a partial effect, holding the others fixed. To test several coefficients together, use the F-statistic: compare a restricted and an unrestricted model, F = [(SSR_r − SSR_u) ÷ q] ÷ [SSR_u ÷ (n − k − 1)], and reject if F exceeds the critical value.

Understand Multiple Regression and Joint Hypothesis Tests

Multiple regression extends simple regression by using k explanatory variables: Y = β0 + β1X1 + β2X2 + ... + βkXk + ε. You estimate the coefficients with ordinary least squares (OLS), which picks the values that minimise the sum of squared residuals (SSR).

Each slope is a partial slope coefficient. β1 is the expected change in Y for a one-unit rise in X1, holding all other included variables constant. This is why the slope on X1 can change when you add or drop another variable. If X1 and X2 are correlated, the simple-regression slope on X1 mixes in some of X2's effect. The partial slope removes it.

A t-test checks one coefficient at a time. The null is usually H0: βj = 0. You compute t = (estimate − hypothesised value) ÷ standard error, with n − k − 1 degrees of freedom.

A joint hypothesis makes several claims at once, such as H0: β2 = β3 = 0. Running separate t-tests does not work, because the estimates are correlated and the overall error rate is wrong. The F-test handles it. You fit the model twice: the unrestricted model (all variables) and the restricted model (the H0 restrictions imposed, for example the variables dropped). If imposing the restrictions raises SSR a lot, the restrictions are rejected.

The special case where H0 says all slope coefficients are zero is the overall F-test (test of model significance). The F-test is always one-tailed: only large values of F lead you to reject. Because F measures the loss of fit from restricting, it can be significant even when no individual t-statistic is, which typically happens under multicollinearity.

Key formulas to remember

Multiple regression model: Y = β0 + β1X1 + β2X2 + ... + βkXk + ε; k is the number of slope coefficients, excluding the intercept. Each βj is a partial effect.
t-statistic for one coefficient: t = (β̂j − β_H0) ÷ SE(β̂j), df = n − k − 1; Usually β_H0 = 0. Two-tailed test unless the alternative is directional.
F-statistic (restricted vs unrestricted, SSR form): F = [(SSR_r − SSR_u) ÷ q] ÷ [SSR_u ÷ (n − k − 1)]; q is the number of restrictions. k is the number of slopes in the unrestricted model. Degrees of freedom are q and n − k − 1.
F-statistic (R² form): F = [(R²_u − R²_r) ÷ q] ÷ [(1 − R²_u) ÷ (n − k − 1)]; Valid only when both models have the same dependent variable.
Overall F-statistic: F = (ESS ÷ k) ÷ (SSR ÷ (n − k − 1)) = (R² ÷ k) ÷ ((1 − R²) ÷ (n − k − 1)); Tests H0: all slope coefficients equal zero. Here q = k and the restricted model has only an intercept.
Adjusted R²: Adjusted R² = 1 − (1 − R²) × (n − 1) ÷ (n − k − 1); Unlike R², it can fall when you add a variable that adds little.
Link between t and F: F = t² when q = 1; A joint test with one restriction is the same as a two-tailed t-test.
Sum of squares identity: TSS = ESS + SSR; R² = ESS ÷ TSS; SSR here is the sum of squared residuals. Check which label the question uses.

How to solve Multiple Regression and Joint Hypothesis Tests questions

Use this routine for any question on multiple regression, partial slopes or joint tests.

1Write down n, k and the number of restrictions q. Compute the degrees of freedom: n − k − 1 for the unrestricted model.
2State H0 and H1. For a joint test, H0 lists every restriction, for example β2 = β3 = 0. H1 says at least one does not hold.
3Choose the test. One coefficient: t-test. Two or more coefficients together, or the whole model: F-test.
4Collect the inputs: SSR_u and SSR_r, or R²_u and R²_r, or ESS and SSR for the overall F. Convert between forms if needed.
5Compute the statistic with the matching formula, keeping the numerator and denominator degrees of freedom straight.
6Compare with the critical value (F with q and n − k − 1 df, upper tail) or use the p-value. Reject H0 if F is larger than critical or p is below the significance level.
7State the conclusion in words. For interpretation questions, say 'holding the other variables constant' and give the units.

Quickest way: Fast F-test from R² or SSR

When to use it: Any question that gives two models, two R² values, or ESS and SSR, and asks whether variables are jointly significant.

Find q by counting the variables dropped or the equalities in H0.
Find the denominator df as n − k − 1 using k from the full model.
Plug into the SSR form or the R² form. Do the numerator and denominator separately, then divide.
Sanity check: if q = 1, F should equal the square of the t-statistic.
Compare with the given critical value. If the F is only a little above or below it, recheck your df before answering.

Common mistakes in Multiple Regression and Joint Hypothesis Tests

Testing a joint hypothesis with several separate t-tests.
The t-test is familiar, so it feels natural to run it on each coefficient.
Fix: Use the F-test whenever H0 involves more than one restriction. Individual t-tests ignore the correlation between the estimates.
Using n − k as the denominator degrees of freedom.
Students forget the intercept uses up one degree of freedom.
Fix: Always use n − k − 1, where k counts slope coefficients only.
Confusing k with q in the numerator.
Both are counts of variables. In the overall F-test they are equal, which hides the difference.
Fix: q is the number of restrictions being tested. k is the number of slopes in the full model. Divide the change in fit by q.
Reading a partial slope as the effect of X1 alone.
Simple regression habits carry over.
Fix: Say 'holding the other regressors constant'. The slope is an effect with the other included variables fixed, not a raw association.
Assuming a higher R² means the extra variable belongs in the model.
R² never falls when you add a regressor, even a useless one.
Fix: Use adjusted R² or an F/t-test to judge whether the added variables are worthwhile.
Concluding the model is useless because no t-statistic is significant while the overall F is highly significant.
Students expect the tests to agree.
Fix: With highly correlated regressors, individual standard errors inflate but the variables are jointly informative. Trust the F-test for joint significance and suspect multicollinearity.

Worked examples

Example 1

A regression of a fund's monthly excess return on three factors uses n = 60 observations. The output shows ESS = 240 and SSR = 180. Compute the overall F-statistic. Which is closest? A) 22.22 B) 24.89 C) 26.67 D) 80.00

Show the solution

k = 3 and n = 60, so the denominator df = n − k − 1 = 56.
Formula: F = (ESS ÷ k) ÷ (SSR ÷ (n − k − 1)).
Numerator = 240 ÷ 3 = 80.
Denominator = 180 ÷ 56 = 3.2143.
F = 80 ÷ 3.2143 = 24.89.
Cross-check with R²: TSS = 420, R² = 240 ÷ 420 = 0.5714. F = (0.5714 ÷ 3) ÷ (0.4286 ÷ 56) = 0.19048 ÷ 0.007653 = 24.89.
The critical F with (3, 56) df at 5% is about 2.8, so F is far above it and the model is jointly significant.

Answer: B) 24.89. Reject H0 that all three slopes are zero.

Example 2

An analyst regresses a bond's yield change on four variables using n = 44 observations. The unrestricted model has SSR = 120. She tests H0: β3 = β4 = 0 by dropping those two variables, and the restricted model has SSR = 150. The 5% critical F value with (2, 39) df is about 3.24. What is the conclusion?

Show the solution

Unrestricted model: k = 4, so n − k − 1 = 44 − 4 − 1 = 39. Restrictions q = 2.
Formula: F = [(SSR_r − SSR_u) ÷ q] ÷ [SSR_u ÷ (n − k − 1)].
Numerator = (150 − 120) ÷ 2 = 15.
Denominator = 120 ÷ 39 = 3.0769.
F = 15 ÷ 3.0769 = 4.875.
Compare: 4.875 is greater than 3.24, so reject H0.
Interpretation: dropping β3 and β4 worsens fit by more than chance would explain, so at least one of the two variables matters. The F-test does not say which one.

Answer: F = 4.875, above the 3.24 critical value. Reject H0: the two variables are jointly significant at the 5% level.

Exam tips

Count carefully: q is restrictions, k is slopes in the full model, and the denominator df is n − k − 1. Many wrong options are built from these slips.
If a question gives R² for two models, use the R² form of F, but only if both models share the same dependent variable.
Spot the multicollinearity pattern: a significant overall F with insignificant individual t-statistics.
For interpretation questions, pick the option that includes 'holding other variables constant'.
F = t² only for a single restriction. Use it as a quick check or shortcut, not for joint tests with q ≥ 2.

Practice questions from Linear Regression

Multiple Regression and Joint Hypothesis Tests in other exams

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

Multiple Regression and Joint Hypothesis Tests: frequently asked questions

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

A t-test checks one coefficient, such as H0: β1 = 0. An F-test checks two or more restrictions together, or all slopes at once. With one restriction, F equals t squared and both give the same conclusion for a two-tailed test.

How do I calculate the F-statistic in multiple regression?

For the overall test, F = (ESS ÷ k) ÷ (SSR ÷ (n − k − 1)). For a joint test on a subset of variables, F = [(SSR_r − SSR_u) ÷ q] ÷ [SSR_u ÷ (n − k − 1)]. Compare it with the critical value for q and n − k − 1 degrees of freedom.

How do I interpret a partial slope coefficient?

It is the expected change in the dependent variable for a one-unit increase in that regressor, with all other regressors in the model held constant. It can differ from the slope you get in a simple regression on that variable alone.

Why is the F-test one-tailed?

F measures how much fit is lost by imposing the restrictions, so it cannot be negative. Only large values count as evidence against H0. You therefore reject only when F exceeds the upper-tail critical value.

Can the overall F-test be significant when every t-test is not?

Yes. This usually happens when regressors are highly correlated, which inflates the standard errors of individual coefficients. The variables are still jointly useful, so the F-test shows significance while the t-tests do not.