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CFA Level II Exam · Evaluating Regression Model Fit and Interpreting Model Results

F-Test and Joint Hypothesis Tests in Multiple Regression

Updated 6 October 2026 · Fact-checked

The F-test checks whether several slope coefficients are all zero at once. From the ANOVA table, F = MSR ÷ MSE. For a subset test, compare restricted and unrestricted models: F = [(SSE_R − SSE_U) ÷ q] ÷ [SSE_U ÷ (n − k − 1)]. Reject the null if F exceeds the critical value.

Understand F-test and Joint Hypothesis Tests

A t-test looks at one slope coefficient. But a model has several slopes, and sometimes you need to ask a joint question: do all of them equal zero, or do a few of them equal zero together? The F-test answers that. It tests the null hypothesis as one combined statement, not as separate tests.

The idea is simple. Take the unrestricted model, which has all the variables. Then take the restricted model, which drops the variables that the null says are zero. If those dropped variables really add nothing, removing them barely raises the unexplained error (SSE). If they matter, SSE jumps. The F-statistic measures that jump relative to the error left in the full model.

For the test that all slopes equal zero, you do not need two models. The ANOVA table already has it. The restricted model there has only an intercept, so its SSE equals SST. F = MSR ÷ MSE, where MSR = RSS ÷ k and MSE = SSE ÷ (n − k − 1). Here k is the number of slope coefficients in the full model, and RSS is the regression (explained) sum of squares.

The F-test is always one-tailed, right tail. A large F means the variables explain a lot, so you reject the null. Degrees of freedom are numerator q (the number of restrictions) and denominator n − k − 1 (from the unrestricted model).

Why not just run many t-tests? Individual t-tests do not control the joint error rate, and with correlated variables they can mislead. With multicollinearity, you can see a significant F but insignificant t-statistics for every slope. That is a classic sign. The F-test says the variables together matter, even if you cannot separate their individual effects.

Key formulas to remember

F-statistic from ANOVA (all slopes zero)
F = MSR ÷ MSE = (RSS ÷ k) ÷ (SSE ÷ (n − k − 1))
H0: all slope coefficients = 0. Degrees of freedom: k and n − k − 1. One-tailed, right tail.
ANOVA components
SST = RSS + SSE; MSR = RSS ÷ k; MSE = SSE ÷ (n − k − 1)
SST is total variation, RSS is explained, SSE is unexplained. df: SST n − 1, RSS k, SSE n − k − 1.
F-test for a subset (restricted vs unrestricted)
F = [(SSE_R − SSE_U) ÷ q] ÷ [SSE_U ÷ (n − k − 1)]
q = number of restrictions (slopes set to zero). k = slopes in the unrestricted model. df: q and n − k − 1.
F in terms of R-squared
F = [(R²_U − R²_R) ÷ q] ÷ [(1 − R²_U) ÷ (n − k − 1)]
Valid when both models use the same dependent variable. For the all-slopes test, R²_R = 0.
Decision rule
Reject H0 if F > critical F(q, n − k − 1)
Rejecting means at least one tested slope is not zero.
Link between t and F with one restriction
F = t²
Holds when q = 1 and the test is two-sided.

How to solve F-test and Joint Hypothesis Tests questions

Use this method for any F-test question. Read the vignette for which variables are in each model and what the null says.

  1. 1Write the null. For the overall test, all slopes = 0. For a subset test, only the named slopes = 0. The alternative is that at least one is not zero.
  2. 2Count q, the number of restrictions, which is the number of slopes the null sets to zero. Find k and n for the unrestricted model.
  3. 3Find the data. For the overall test, take RSS and SSE (or MSR and MSE) from the ANOVA table. For a subset test, take SSE or R-squared from both models.
  4. 4Compute the F-statistic with the right formula. Check that the denominator uses SSE_U and n − k − 1 from the unrestricted model.
  5. 5Get degrees of freedom: numerator q, denominator n − k − 1. Read the critical value from the exhibit at the stated significance level.
  6. 6Compare. If F is greater than the critical value, reject H0. Right tail only.
  7. 7State the conclusion in words, and note any link to t-statistics or multicollinearity if the question asks.

Quickest way: Fast F-test from the ANOVA table

When to use it: Use when the vignette gives an ANOVA table or sums of squares and asks for F or a conclusion.

  1. If you have MSR and MSE, divide them. Done.
  2. If you have RSS and SSE, divide RSS by k and SSE by n − k − 1, then divide the results.
  3. If you have only R², use F = (R² ÷ k) ÷ ((1 − R²) ÷ (n − k − 1)).
  4. For subset tests, the numerator is the SSE increase per restriction. The denominator is the full model's MSE.
  5. Compare with the critical value. If F is clearly large, such as far above 3 or 4 with reasonable degrees of freedom, expect rejection, but always check the exhibit.

Common mistakes in F-test and Joint Hypothesis Tests

  • Using n − k instead of n − k − 1 in the denominator degrees of freedom.

    Students forget the intercept uses one degree of freedom.

    Fix: Always write n − k − 1, where k counts slope coefficients only.

  • Taking SSE from the restricted model in the denominator of the subset F-test.

    Both SSEs appear in the formula and they get mixed up.

    Fix: The denominator is always the unrestricted SSE divided by its degrees of freedom.

  • Using the number of variables kept, not the number dropped, as q.

    The word 'restricted' is confused with what remains in the model.

    Fix: q equals the number of slopes the null forces to zero, which is the difference in variables between the two models.

  • Treating the test as two-tailed.

    Students carry over habits from t-tests.

    Fix: The F-test is one-tailed. Only a large F rejects H0.

  • Concluding that every variable is significant after rejecting H0.

    The conclusion is read as 'all slopes differ from zero'.

    Fix: Rejection means at least one slope is not zero. Use t-tests to see which.

  • Assuming an insignificant F always follows insignificant t-statistics, or the reverse.

    Students ignore multicollinearity.

    Fix: A significant F with insignificant t-statistics points to multicollinearity. An insignificant F means you cannot reject that all slopes are zero.

Worked examples

Example 1

An analyst regresses a fund's return on 3 factors using 43 monthly observations. The ANOVA table shows RSS = 120 and SSE = 80. The critical F value at 5% for (3, 39) degrees of freedom is 2.85. (1) State the null. (2) Compute the F-statistic. (3) What is the conclusion?

Show the solution
  1. Null: all three slope coefficients equal zero. Alternative: at least one is not zero.
  2. k = 3, n = 43, so SSE degrees of freedom = 43 − 3 − 1 = 39.
  3. MSR = 120 ÷ 3 = 40.
  4. MSE = 80 ÷ 39 = 2.0513.
  5. F = 40 ÷ 2.0513 = 19.5.
  6. 19.5 is greater than 2.85, so reject the null.

Answer: F ≈ 19.5 with (3, 39) df. Reject H0: at least one factor slope differs from zero.

Example 2

An analyst models a bond's yield change using 5 variables with 66 observations. The unrestricted model has SSE = 50. She tests whether the last 2 variables add explanatory power. The restricted model without them has SSE = 58. The critical F at 5% for (2, 60) degrees of freedom is 3.15. (1) Compute F. (2) State the conclusion.

Show the solution
  1. k = 5, n = 66, so denominator df = 66 − 5 − 1 = 60. q = 2.
  2. Numerator: (SSE_R − SSE_U) ÷ q = (58 − 50) ÷ 2 = 4.
  3. Denominator: SSE_U ÷ 60 = 50 ÷ 60 = 0.8333.
  4. F = 4 ÷ 0.8333 = 4.8.
  5. 4.8 is greater than 3.15, so reject the null that both slopes are zero.

Answer: F = 4.8. Reject H0: at least one of the two variables has a non-zero slope, so the unrestricted model is justified.

Exam tips

  • Read the vignette for k and n of the unrestricted model first. Most arithmetic errors come from wrong degrees of freedom.
  • If the question gives R-squared values for both models, use the R² form of the F-test. It is faster than finding SSE.
  • When you see a high R² and a significant F but insignificant t-statistics, expect the answer to be multicollinearity.
  • Check whether the F-test is overall or for a subset. The overall test uses the ANOVA table directly.
  • With no penalty for wrong answers, never leave a question blank. Eliminate options that use a left-tail or two-tailed rule.

F-test 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.

F-test 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 slope coefficient at a time. An F-test checks a joint hypothesis, such as all slopes being zero or a subset being zero. With one restriction, F equals t squared.

How do I calculate the F-statistic from an ANOVA table?

Divide the regression sum of squares by k to get MSR. Divide the error sum of squares by n − k − 1 to get MSE. Then F = MSR ÷ MSE.

What is the restricted versus unrestricted model F-test formula?

F = [(SSE_R − SSE_U) ÷ q] ÷ [SSE_U ÷ (n − k − 1)]. The unrestricted model has all variables. The restricted model drops the q variables that the null sets to zero.

Can the F-test be significant when no t-statistic is significant?

Yes. This usually signals multicollinearity. The variables together explain the dependent variable, but they are too correlated for the model to assign effects to each one.