Sir Francis Galton (1822-1911), an Anthropologist and Cousin of Charles $\widehat { \text { studenth } } = 19.6 + 0.73 \times \text { Midparh, } R ^ { 2 } = 0.45 , S E R = 2.0$

Essay

Sir Francis Galton (1822-1911), an anthropologist and cousin of Charles Darwin, created the term regression. In his article "Regression towards Mediocrity in Hereditary Stature," Galton compared the height of children to that of their parents, using a sample of 930 adult children and 205 couples. In essence he found that tall (short)parents will have tall (short)offspring, but that the children will not be quite as tall (short)as their parents, on average. Hence there is regression towards the mean, or as Galton referred to it, mediocrity. This result is obviously a fallacy if you attempted to infer behavior over time since, if true, the variance of height in humans would shrink over generations. This is not the case.
(a)To research this result, you collect data from 110 college students and estimate the following relationship: $\widehat { \text { studenth } } = 19.6 + 0.73 \times \text { Midparh, } R ^ { 2 } = 0.45 , S E R = 2.0$
$\quad$$\quad$$\quad$$\quad$$\quad$ (7.2) $( 0.10 )$
where Studenth is the height of students in inches and Midparh is the average of the parental heights. Values in parentheses are heteroskedasticity-robust standard errors. Sketching this regression line together with the 45 degree line, explain why the above results suggest "regression to the mean" or "mean reversion."
(b)Researching the medical literature, you find that height depends, to a large extent, on one gene ("phog")and on environmental influences. Let us assume that parents and offspring have the same invariant (over time)gene and that actual height is therefore measured with error in the following sense, $\tilde { X _ { i , 0 } } = X _ { i } + v _ { i , 0 } \text { and } \tilde { X _ { i , p } } = X _ { i } + w _ { i , p }$ where $\bar { X }$ is measured height, X is the height given through the gene, v and w are environmental influences, and the subscripts o and p stand for offspring and parents, respectively. Let the environmental influences be independent from each other and from the gene.
Subtracting the measured height of offspring from the height of parents, what sort of population regression function do you expect?
(c)How would you test for the two restrictions implicit in the population regression function in (b)? Can you tell from the results in (a)whether or not the restrictions hold?
(d)Proceeding in a similar way to the proof in your textbook, you can show that $\hat { \beta } _ { 1 }\xrightarrow{p} 1 - \frac { \sigma _ { v } ^ { 2 } } { \sigma _ { X } ^ { 2 } + \sigma _ { v } ^ { 2 } }$ for the situation in (b). Discuss under what conditions you will find a slope closer to one for the height comparison. Under what conditions will you find a slope closer to zero?
(e)Can you think of other examples where Galton's Fallacy might apply?