F-Test & Model Significance

Question 1

what is the F-Test?

Answer 1

It is used in multiple regression to test whether the regression model as a whole is statistically significant

The F-test checks if multiple coefficients are jointly significant

Question 2

What is the Null hypothesis (H0) in an F test?

Answer 2

all the regression coefficients (except the intercept) are zero

(e.g the independent variables do not explain Y)

Question 3

What is the Alternative Hypothesis in an F-Test?

Answer 3

At least one of the regression coefficients is not zero, making the model useful

It means that the regression model explains some variation in Y and is statistically significant

Question 4

What is the formula for the F-statistic?

Answer 4

F= (SSE/k) / (SSR/n-k-1)

SSE = sum of squared explained variation
SSR = sum of squared residuals
k = Number of independent variables
n = sample size

The higher the F-statistic means the model explains a lot of variation in Y

Question 5

What is the decision rule for the F test?

Answer 5

1) compare the F-statistic to a critical value from the F-table

2) if F is greater than the critical value = reject H0 (model is significant)

3) if F is smaller = fail to reject H0 (model is not significant)

Question 6

work through an example of the F Test:

Answer 6

Suppose we estimate the following multiple regression model:

wage=𝛽0+𝛽1(education)+𝛽2(experience)+𝛽3(gender)+𝑢

We get:

SSE=450
SSR=150
k=3 (3 independent variables: education, experience, gender)
n=50 (sample size)

1) Compute the F-Statistic

𝐹=(450/3) / (150/50−3−1)

𝐹=150/3.19

= 47.02

Step 2: Compare to Critical Value

From the F-table (at α=0.05), the critical value for k=3, n−k−1=46 is approximately 2.8.

Since 47.02 > 2.8, we reject
𝐻0 → the regression model is statistically significant.