Introductory Econometrics: A Modern Approach 6th Edition by Jeffrey M Wooldridge

Exercise 2

(i) For a binary response y, let be the proportion of ones in the sample (which is equal to the sample average of the yi .). Let be the percent correctly predicted for the outcome y = 0 and let be the percent correctly predicted for the outcome y = 1. If is the overall percent correctly predicted, show that is a weighted average of and :

(ii) In a sample of 300, suppose that = .70, so that there are 210 outcomes with yi = 1 and 90 with yi = 0. Suppose that the percent correctly predicted when y = 0 is 80, and the percent correctly predicted when y = 1 is 40. Find the overall percent correctly predicted.

Step 1 of 2

(i)

Consider that the variable is a binary response variable with 0 and 1 as the only outcome of the variable.

Consider there are number of observations in the series and is the number of correctly predicted for the outcome and is the number of correctly predicted for the outcome

Consider is the number of actual outcome in the series and is the number of actual outcome

That means

Consider is the sample average of

This implies and

Consider is the percent correctly predicted for the outcome

That means

Consider is the percent correctly predicted for the outcome

That means

Consider is the overall percent correctly predicted

That implies

Thus, proceeding further, the result is: