Modeling the wage distribution

Part I: (15pts) Consider modeling the wage distribution using a continuous distribution
with the following PDF:
f(x; θ) = (
0 x < 1 θ xθ+1 x ≥ 1. (1) (i) Suppose that θ > 1. Prove that
E[X] = θ
θ − 1
. (2)
(ii) Based on (i), find an expression for the method of moments estimator ˜θ of θ. wage.dta
contains data on wage (in $10,000/year) across individuals. Report your method of moments
estimate of θ.
(iii) Derive the log-likelihood function. Find an expression for the maximum likelihood estimator ˆθ of θ.
(iv) Using wage.dta, report your maximum likelihood estimate of θ.
(v) Based on (1), derive the cumulative distribution function (CDF). Using the CDF and
your estimate from (iv), estimate the proportion of individuals whose wage is above 10 (in
$10,000/year). Repeat this exercise using your estimate from (ii). Summarize your findings.
Part II: (30pts) (Selected questions will be graded.)
Work on the following questions from Chapters 7 & 8 of the textbook.
Chapter 7: 10, 16, 24, 26
Chapter 8: 4, 8, 10

Sample Solution