You run a regression predicting annual salary using years_of_education and years_of_experience. Both variables are highly correlated and both have individually insignificant t-statistics (p > 0.10). What should you conclude?
ADrop both variables — individually insignificant coefficients mean the variables add no explanatory power
BDrop the less significant variable and re-run; the remaining variable may become significant
CRun an F-test first — correlated variables can be jointly significant even when individually insignificant
DBoth variables are statistically redundant, so keep only their average as a single predictor
When regressors are correlated (multicollinear), OLS inflates individual standard errors — each variable appears insignificant not because it lacks explanatory power, but because the model can't attribute the shared variance to one variable versus the other. The F-test evaluates whether the two variables jointly reduce unexplained variance. A high F-statistic with insignificant individual t-statistics is the signature pattern of multicollinearity. Dropping both based only on t-tests would discard genuine predictive content.
Question 2 Multiple Choice
The F-statistic for overall model significance equals (RSS_restricted − RSS_unrestricted)/q ÷ RSS_unrestricted/(n−k−1). What does the term q represent in this formula?
AThe number of observations in the sample
BThe number of restrictions being tested — here, the number of slope coefficients jointly set to zero
CThe ratio of explained to unexplained variance in the full model
DThe degrees of freedom penalty for each additional regressor
q is the number of restrictions imposed under the null hypothesis. For the overall F-test, the null sets all slope coefficients to zero, so q equals the number of slope coefficients (k). For a partial F-test of a subset of variables, q equals how many you're testing. The F-distribution requires two degree-of-freedom parameters — (q, n−k−1) — because both the number of restrictions and the full model's residual degrees of freedom affect the reference distribution.
Question 3 True / False
If most individual slope coefficients in a regression have p-values above 0.05, the overall F-test for joint significance will also fail to reject the null hypothesis.
TTrue
FFalse
Answer: False
This is the central misconception the F-test corrects. Under multicollinearity, individual t-tests have inflated standard errors that mask each coefficient's contribution. But the F-test evaluates the joint reduction in unexplained variance, which can be substantial even when no single variable looks significant on its own. A model can have a highly significant overall F-statistic alongside uniformly insignificant individual t-statistics — a pattern that tells you the variables together matter but their individual contributions can't be separately identified given the correlation structure.
Question 4 True / False
When testing a single linear restriction (q = 1), the F-statistic equals the square of the corresponding t-statistic.
TTrue
FFalse
Answer: True
This is a direct mathematical relationship: F(1, df) = t²(df). It serves as a useful sanity check — if you run an F-test on a single coefficient and compare it to the t-test for that same coefficient, you should get F = t². The two tests give identical p-values for a single restriction. This connection also helps build intuition: the F-test is a generalization of the t-test to multiple simultaneous restrictions, collapsing to the familiar t-test when only one restriction is being tested.
Question 5 Short Answer
Why can't you simply run multiple t-tests — one for each coefficient — to determine whether a set of variables is jointly significant?
Think about your answer, then reveal below.
Model answer: Running multiple t-tests inflates the Type I error rate. If each test has a 5% false positive rate, the probability of at least one false rejection across k independent tests is 1 − 0.95^k, which grows rapidly with k. The F-test controls this by testing all restrictions simultaneously under a single null hypothesis, maintaining a correct overall false positive rate. Additionally, individual t-tests cannot detect the case where correlated variables are jointly but not individually significant — the F-test is designed specifically for this.
This is why the F-test exists: it is not redundant with the set of individual t-tests but answers a different question. The question 'is at least one of these variables significant?' answered by scanning t-tests has uncontrolled error. The question 'do these variables jointly explain variance beyond what would be expected by chance?' is what the F-test answers cleanly.