代写Economics 3120: Applied Econometrics Summer 2025 Exercise #2帮做R程序

Economics 3120: Applied Econometrics

Summer 2025

Exercise #2

This exercise is designed to review the key parts of the regression part of the course.  It is open book, open note, I just ask that you do it on your own.

Please answer the questions below.  When done, scan your answers to pdf (preferred) or take photos of them (less preferred) and email them to me as an attachment.  Watch the size of the attachment – if it is too large then the mail system will reject it.  In that case you should send me separate emails with smaller attachments.  Please include your name and netid in the file name of the attachment.

1.         (40 points) Consider the regression model:  Yi  = β0  + β 1Xi1  + … + βkXik + εi  .

a.         State the minimization problem solved by the least squares estimators b0, b1, … , bk ? (10 points)

b.         What are the three properties of the residuals that are implications of the normal equations (i.e., the first order conditions of the least squares minimization problem)? (30 points)

[Hint:  You may find the handout “Statistics as Economics” helpful here.]

2.          (100 points) You are interested in the following model for a short run production function:

log Y = β0  + β 1X1 + u

where Y = output in tons and X1 = kilowatts of electricity.

a.         Interpret the intercept and the slope coefficient of the model. (10 points)

[Hint:  look at the handout on interpretation of intercepts and basic models with logs and levels]

b.         Suppose you changed the units of measurement of the energy input from kilowatts to megawatts (one megawatt = 1000 kilowatts).  What will happen to the two coefficients and R2? (30 points)

c.         Suppose you changed the units of measurement of output, before taking its log, from tons to pounds (one ton = 2000 pounds). What will happen to the two coefficients and R2? (30 points)

[Hint:  Units changes do not change the fundamental relationship.]

Unfortunately, your research assistant is not good at following instructions, and they estimated the following model instead:

log Y = α0 + α1 log X1 + u

d.         What computations would you make, using the α's, to try to get at β 1? (10 points)

Some people never learn …  This time they estimates the following model:

Y = γ0 + γ1 X1 + u

e.         What computations would you make, using the γ's, to try to get at β 1? (10 points)

One last time:  You hire a new research assistant, and she comes back with the following model:

Y = δ0 + δ 1 log X1  + u

f.         What computations would you make, using the δ's, to try to get at β 1? (10 points) [Hint:  You may find the handout “Basic Models with Logs and Levels” helpful here.]

3.         (40 points) You are trying to learn whether a training program leads to higher post-training wage rates.  You compare the average wage rates for workers who went to training to the average wage rates for workers who did not go to training.  Suppose that the workers with the worst skills are those who went to training, and that better skills lead to higher wage rates.  Will   the comparison above accurately give the impact of training?  Why or why not?  If the answer is  “not” will that comparison over- or under-estimate the effect of going to training, and why?  If you find that the workers who went to training have lower wage rates on average, does that imply that the training hurts workers?  Why or why not?

[Hint:  think of the comparison between the simple regression which only has training on the RHS and the multiple regression which has training and skill on the RHS.  Now use the relationship between the simple and multiple regression coefficients from the handout to answer the question.  You can refer to “Notes on Multiple Regression” for more details.]

Appendix 1 at the end contains statistical results from analyzing 8,646 households drawn from the Consumer Expenditure Survey.  The first panel contains variable definitions and descriptive statistics.  The next panels contain regression results for various regression of food expenditure  and family size using levels and logs.

4.         (40 points)

a.         What is the mean log food expenditure?  (5)

b.         exponentiate your answer to part a.  What do you get?  (5)

c.         What is the mean food expenditure?  (5)

d.         Which one is larger, your answer to part b or part c?  (5)

e.         Why do you get different answers to parts b and c?  (20)

5.         (120 points)  For each of the four regression models on the second and third page of the appendix, what is the numerical value for marginal impact of an additional person on annual food expenditure, the rate of return on family size, and the elasticity of food expenditure with respect to family size?  If needed, evaluate food expenditure and family size at their sample means. SHOW YOUR WORK.

6.         (30 points)  For model (2) in the appendix, what would be the intercept, the slope coefficient, and the R2 if we had measured the food expenditure in dollars rather than 1000’s of dollars, and then took the logarithm?  SHOW YOUR WORK.

7.         (30 points)  For model (3) in the appendix, what would be the intercept, the slope coefficient, and the R2  if we had measured family size in scores  before taking its logarithm?  (1 score = 20 people) SHOW YOUR WORK.