Consider the simple regression model Y = 0 + 1 x + u and let z be a binary instrumental variable for x. Use (15.10)

Consider the simple regression model
Y = β0 + β1 x + u
and let z be a binary instrumental variable for x. Use (15.10) to show that the IV estimator 1, can be written as

where 0 and 0 are the sample averages of yi and xi over the part of the sample with zi = 0, and where 1 and 1 are the sample averages of yi and xi over the part of the sample with zi = 1. This estimator, known as a grouping estimator, was first suggested by Wald (1940).

B,= (9,-5MR, - *),