SAMPLE MIDTERM EXAM
ECON 120B
ECONOMETRICS
Table: Standard Normal Distribution Table
Large-sample critical values for the t-statistic from the Standard Normal Distribution
|
|
Significance Level
|
|
|
10%
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5%
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1%
|
|
2-sided test (=)
|
1.64
|
1.96
|
2.58
|
|
1-sided test (>)
|
1.28
|
1.64
|
2.33
|
|
1-sided test (<)
|
-1.28
|
-1.64
|
-2.33
|
Useful notes:
(1) Point totals for each question are given in brackets. The actual exam may have more or
fewer points. Use the point totals for the questions to help you allocate your time effectively.
(2) The actual midterm may have multiple choice questions, definition questions or a combination of these types of questions.
(3) The actual midterm will have one or more short answer questions. Make sure to show your work and explain your steps for full credit.
(4) The actual midterm will be shorter. I made the sample midterm longer to give you more examples of the types of questions which might appear on the exam.
Question 1 [18 points]
Circle the correct answer to each multiple choice question.
a) The population variance of the random variable X equals:
A)
B) for a continuous random variable
C)
D)
b) If β1 denotes the OLS estimator of the slope coefficient, E(β1) is:
A) A random variable
B) An estimate
C) An estimator
D) None of the above
c) The least squares estimator of the slope coefficient in the linear regression model with one regressor is the:
A) sample covariance of X and Y divided by the sample variance of Y
B) sample variance of X divided by the sample covariance of X and Y
C) sample covariance of X and Y divided by the sample variance of X
D) population covariance of X and Y divided by the population variance of Y
Question 2 [12 points]
a) Define what it means for an estimator to be unbiased.
b) Define the least squares estimator of the slope coefficient in the linear regression model (with one regressor) in terms of sample variances and sample covariances.
Question 3 [30 points]
You are interested in the causal effect of attending the midterm review session on a student’s midterm exam score. The following table shows the potential exam scores, actual exam score, and whether the student attended the review for 5 students. yi denotes the actual exam score for person i, Di is an indicator for whether person i attended the review session, and the potential outcome notation is the same as used in class.
|
|
Seojun
|
Torben
|
Jin
|
Susan
|
Marcus
|
|
yi0
|
77
|
91
|
92
|
84
|
87
|
|
yi1
|
85
|
95
|
92
|
89
|
93
|
|
yi
|
85
|
95
|
92
|
84
|
93
|
|
Di
|
1
|
1
|
0
|
0
|
1
|
a) What is the causal effect of attending the review session on Torben’s exam score?
b) Define the average effect of treatment on the treated (ATT).
c) What is the ATT for this population of students?
d) Suppose you only have access to the variables yi and Di . What is the mean difference in exam scores between those who attended the review session and those who did not?
e) Decompose the difference in part d into the true causal effect of attending the review session for the treated group and the selection bias. Write out the decomposition as a formula before plugging in any numbers for full credit.
Question 4 [30 points]
Adult males are taller, on average, than adult females. You suggest to your little sister that she collect data on height and gender of children in the 6th grade in her school district (which has hundreds of 6th graders) as part of her science project. She collects a simple random sample with the help of district. The accompanying table shows her findings.
Height of 6th grade boys and girls, in inches
|
Boys
|
Girls
|
|
-
Yb
|
sb
|
Nb
|
-
Y
|
sg
|
Ng
|
|
57.8
|
3.9
|
55
|
58.4
|
4.2
|
57
|
a) Suppose you want to test whether 6th grade boys and girls in the school district are the same height. Write down the null and alternative hypotheses.
b) Calculate the difference in mean heights and the standard error of this difference.
c) Calculate the appropriate t-statistic for the test you wrote down in part (a). Do you reject the null hypothesis at the 1% significance level?
d) What is the asymptotic distribution of yb? Be specific; don’t just state the distribution’s name.
e) What result or theorem are you appealing to for your answer in part (d)?
Question 5 [20 points]
The sample covariance between the random variables X and Y is:
Prove that this is identical to the following alternative specification:
Question 6 [30 points]
You download a dataset of 1,744 individuals from the Current Population Survey, which you are told is a simple random sample of the entire U.S. population. You are interested in the relationship between weekly earnings and age. OLS regression yields the following result:
yi = 239. 16 + 5.20xi
R2 = 0.05, RMSE = 287.21
where yi is predicted weekly earnings in dollars and xi is age in years.
a) Interpret the estimated intercept. Does a literal interpretation make sense here?
b) Interpret the regression R2 .
c) How much weekly earnings would the regression model predict a 32 year old woman would earn?