Question

You have fit a regression model with two regressors to a data set that has 20 observations. The total sum of squares is 1000 and the model sum of squares is 750.(a) What is the value of R2 for this model?(b) What is the adjusted R2 for this model?(c) What is the value of the F-statistic for testing the significance of regression? What conclusions would you draw about this model if α = 0.05? What if α = 0.01?(d) Suppose that you add a third regressor to the model and as a result, the model sum of squares is now 785. Does it seem to you that adding this factor has improved the model?

Answer 1

0.75

0.7205882

25.5

Result is significant at α = 0.01 and α = 0.05

Model improved

Explanation:

Given that:

Number of observations (n) = 20

Total sum of squares (SST) = 1000

Model sum of squares (SSR) = 750

1) R² = SSR / SST = 750 / 1000 = 0.75

2.)

Adjusted R² = [(SST - SSR) /(n-k-1)] / (SST ÷ (n - 1))

k = number of regressors = 2

Adj R² = 1 - ((1000 - 750) / (20-2-1)) / (1000 / (20 - 1))

1 - 0.2794117 = 0.7205882

3.) Fstat = (SSR / k) / ((SST - SSR) / (n - k-1))

= (750 /2) / ((1000 - 750) / (20 - 2 - 1))

= 25.5

4.) At α = 0.05

Fα,k,(n - k-1) = F0.05, 2, (20 - 2 - 1) = F0.05,2, 17 = 3.5915 (f distribution calculator)

Fstat > F0.05, 2, (20 - 2 - 1)

25.5 > 3.5915 (Hence result is significant at α = 0.05

At α = 0.01

Fα,k,(n - k-1) = F0.01, 2, (20 - 2 - 1) = F0.01,2, 17 = 6.112 (f distribution calculator)

Fstat > F0.01, 2, (20 - 2 - 1)

25.5 > 6.112 (Hence result is significant at α = 0.01

Adjusted R² if a 3rd regressors is added : k = 3

Adjusted R² = [(SST - SSR) /(n-k-1)] / (SST ÷ (n - 1))

k = number of regressors = 3

SSR = 785

Adj R² = 1 - ((1000 - 785) / (20-3-1)) / (1000 / (20 - 1))

1 - 0.2553125 = 0.7446875

Adjusted R² value is now 0.7446875 which is greater than with 2 regressors,. Hence, adding a third regressors improved the model.