Joint Significance
Can predictors be statistically insignificant individually but jointly significant, and what tests do you use? Give two reasons why this can happen.
Answer
Yes they can be.
Tests:
Individual significance: use \(t\)-tests for each coefficient \(H_0:\beta_j=0\).
Joint significance: use an \(F\)-test for \(H_0:\beta_1=\beta_2=\cdots=\beta_p=0\).
Why can this happen?
Multicollinearity: Predictors are highly correlated, so each coefficient’s standard error is large \(\Rightarrow\) \(t\)-stats are small hence large p-values (statistical insignificance). But together they explain variation in \(Y\), so the joint \(F\)-test can still be large.
Shared signal spread across predictors: The true effect is distributed across several predictors; each one alone looks weak, but combined they matter.