a) Download necessary data from http://www.equality-of-opportunity.org/data/ and replicate
the national intergenerational mobility plot for the US in my lecture 1. Make sure to include both
conditional expectation dots and OLS regression lines (fitted values).
b) Let x = 0, 1, …, 100 be a parent income rank in the national income distribution and Yx be the
average child income rank among households where the parent income rank is x. For your plot in
(a), what is the mathematical expression of each conditional expectation dot in terms of Yx and x?
What is the mathematical definition of the slope of the OLS regression line? What about its intercept?
c) Your plot in (a) shows the line of fitted values
Yˆ
x,OLS ≡ aˆOLS + ˆbOLSx,
where (ˆaOLS,
ˆbOLS) are OLS regression coefficients. Pick any other values a
∗ 6= ˆaOLS and b
∗ 6= ˆbOLS,
and form the associated linear approximation Y
∗
x ≡ a
∗ + b
∗X. Compare the approximation performance of Yˆ
x,OLS and Y
∗
x by computing mean-squared approximation errors, Σ
N
i=1(Yx − Yˆ
x,OLS)
2/N
and Σ
N
i=1(Yx − Y
∗
x
)
2/N. Which is a better approximation in terms of the mean-squared approximation errors? Which approximation would you like to you? Why?
d) Consider the following alternative approximation to Yx:
Y
∗∗
x ≡ 0.5 × Yx + 0.5(ˆaOLS + ˆbOLSx).
1
Compare the approximation performance of Yˆ
x,OLS and Y
∗∗
x by computing mean-squared approximation errors, Σ
N
i=1(Yx − Yˆ
x,OLS)
2/N and Σ
N
i=1(Yx − Y
∗∗
x
)
2/N. Which is a better approximation in
terms of the mean-squared approximation errors? Which approximation would you like to you? Why?
e) Pick the commuting zone where you live and another commuting zone of your choice (e.g. the place
where you are born, a place where you lived or you’d like to live). Find out the two commuting
zones’ “relative mobility” and “absolute mobility” measures. What does each measure say about
intergenerational income mobility in the two commuting zones you pick?
Sample Solution
