EC410-9.24

EC410-9.24



EC410-9.24

Part 1

Model

The general model for the given situation is

Cb = (0.2)(Gb) + (0.8)(Pb)

Where,

Cb = the share of the combined population that is bald

Gb = the share of Green People who are bald

Pb = the share of Purple People who are bald

The combined share for years 2000 and 2008 will be

Cb, 2000 = (0.2)(Gb, 2000) + (0.8)(Pb, 2000) = 0.30

Cb, 2008 = (0.2)(Gb, 2008) + (0.8)(Pb, 2008) = 0.70

The difference of the years will be

(0.2)(Gb, 2008 - Gb, 2000) + (0.8)(Pb, 2008 - Pb, 2000) = 0.40

Suppose that Gb, 2008 = 1 (ie 100%) and Gb, 2000 = 0. Then:

(0.2)(1) + (0.8)(Pb, 2008 - Pb, 2000) = 0.40

0.2 + (0.8)(Pb, 2008 - Pb, 2000) = 0.40

The above equation shows that Green people only account for at most half of the overall change. This is because Green People make up a small share of the total. If the increase in baldness among the Green People is less than 100%, then their contribution to the total is less than a half.

Define a as Green People's share of the total, so that (1- a) must be Purple People's share of the total. Different years are indexed by year t and year t+1. Then:

Cb, t = ?(Gb, t) + (1? ?)(Pb, t)

Cb, t+1 = ??Gb, t+1) + (1? ?)(Pb, t+1)

Cb, t+1 ??Cb, t = ??Gb, t+1 ? Gb, t) + (1? ?)( Pb, t+1 - Pb, t)

This framework can then be used to test any conjectures about how much each group could have contributed to a change in the combined average.

Empirics

The main issue in Andrew's reasoning is that it is a simple selection bias. Andrew has not considered many buildings in the sample. He sees virtually all the buildings that have been built in the last 30 years, but only a ...