F-Test for Joint Significance – Student
Notes
Understanding the F-Test for Joint Significance
The F-test for joint significance is used in multiple regression analysis to determine whether
a group of explanatory variables adds meaningful explanatory power to a model.
When do we use it?
Suppose we have a regression model, and we want to test whether a set of variables (e.g.,
calendar month and whether there's a sporting event) significantly improves the model.
These variables may not be significant individually, but together they might be important.
What are we testing?
We compare two models:
- Restricted model: excludes the group of variables we’re testing.
- Unrestricted model: includes the group of variables.
We test the hypotheses:
- H₀: All the coefficients of the additional variables are zero (they have no effect).
- H₁: At least one coefficient of the additional variables is not zero (they do have an effect).
F-test statistic formula
F = ((R²ᵤ - R²ᵣ) / k) / ((1 - R²ᵤ) / (n - k - 1))
Where:
- R²ᵤ: R-squared of the unrestricted model
- R²ᵣ: R-squared of the restricted model
- k: Number of variables being tested (added in the unrestricted model)
- n: Total number of observations
This formula measures the improvement in fit per added variable (numerator) compared to
the unexplained variance per degree of freedom (denominator).
Decision Rule
1. Calculate the F-statistic.
2. Compare it to the critical value from the F-distribution with degrees of freedom (k, n - k 1).
3. If F_calculated > F_critical, we reject H₀.
That means the added variables are jointly significant and should be kept in the model.
Why is this important?
- It helps us avoid omitting important variables.
- It ensures our model includes variables that genuinely improve the explanatory power.
- It helps in model selection: should we include more variables or keep the model simpler?
Example Recap (from your question):
- Restricted model: R² = 0.36
- Unrestricted model: R² = 0.51
- Sample size: 104
- Variables tested: 2
Calculated F ≈ 15.46, critical value ≈ 3.09
Since 15.46 > 3.09 ⇒ Reject H₀
Conclusion: The new variables (calendar month & sporting event) are jointly significant at
the 5% level.