Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

PPHA 421

Midterm Exam

Winter 2024

Due date: Saturday, February 3 at 5:00PM. You must submit via Gradescope on Canvas.

Instructions:

You are free to consult any written resources. However, your final exam is an individual exam. All of your answers should be written in your own words. Cutting and pasting from any document available to you is not allowed. To the extent you rely on course (or other) material, you must paraphrase. You may not discuss anything related to the exam with anyone during the exam period. Should you have a question about the exam, you should post the question on Piazza. While we will work to respond to questions promptly, you should not expect answers to your questions between 7pm and 9am.

Submissions must be typed, converted to a pdf, and uploaded through Gradescope before the deadline. Remember to include your code in the pdf. Exams submitted between 5:01 PM and 6:00 PM on February 3 will receive a 20% penalty. No exams submitted after 6 PM on February 3 will receive credit. You should answer ALL the questions.

A. Analytical problems:

1. Suppose that:

(1) y = Xβ + ε

(2) rank(X) = K

(3) E[ε|X] = 0

(4) E[ε2i|xi] = σ 2h(xi) and E[εiεj |xi , xj ] = 0.

(a) Propose an estimator that will be BLU under assumptions (1) - (4) and denote it ˆβ. You should assume that h(·) > 0 and known.

(b) Suppose you were unable to assume that h(·) is known and unwilling to estimate h(·). Propose an estimator that will be unbiased and consistent under (1) - (4). Denote the estimator ˜β.

2. Assume that the GM assumptions hold:

(1) y = Xβ + ε

(2) rank(X) = K

(3) E[ε|X] = 0

(4) E[εε′|X] = σ2In

The true model is given by

yi = β0 + β1x1i + β2x2i + β3x3i + εi i = 1, . . . , n.

Attention focused on the coefficient β1.

(a) Show that the LS estimator b1 is an unbiased estimator of β1.

Assume that x3 is unobservable, so your choices are to estimate either

yi = β0 + β1x1i + β2x2i + u1i i = 1, . . . , n                  (1)

or

yi = β0 + β1x1i + u2i i = 1, . . . , n.                (2)

(b) Denote the LS estimator of β1 from (1) βˆ 1. Provide an expression for the bias in βˆ1.

(c) Denote the LS estimator of β1 from (2) β˜ 1. Provide an expression for the bias in β˜1.

(d) Are you always better off estimating (1) rather than (2) if your concern is to minimize bias of your estimate of β1? You should assume that x1 is correlated with x3.

B. Computational problems:

For this exercise, use the data set -DahlLochner2012AER.dta- available on Canvas.

1. Estimate the following model

scoreia = β0 + δf amincia + εia

Note that this is a simplified version of the regression from HW1 and HW2. The dependent variable scoreia is defined in HW1 and the other variables are defined below.

2. The model in 1. assumes a constant relationship between income and test scores. An alter-native specification would allow for a parabolic relationship between income and test scores. Formulate and estimate such a model. What is the relationship between test scores and $1000 increase in f aminc? Between this model and the model from 1., which provides a better fit of the data? Explain.

3. The model in 2. assumes a continuous relationship between income and test scores. Formulate and estimate a model that allows for discontinuous jumps at certain thresholds. You should define 4 thresholds. The first threshold should be the 1st quartile of f aminc, the second thresholds should be the 2nd quartile of f aminc, and so fourth.

4. Now add race/ethnicity dummies to your model from 1. Do they belong in the regression? How do they affect the estimates of the income coefficient? What does this say about the correlation between the race/ethnicity dummies and the income variable? Can you verify this in the data?

5. Plot the distribution of score for each of the four income bins defined in 3. Compare the distributions. Do you see any sign of heteroskedasticity in the model in 1.? Explain.