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ECON 4230/6230: Econometrics (Fall 2021)
发布时间:2022-12-13
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Exam 1 (100 points)
ECON 4230/6230: Econometrics (Fall 2021)
Question 1 (20 points): Review of Statistics and Probability (approx. 10 minutes)
(a) Let X and Y be random variables with zero means and finite second moments. Denote the variances of X and Y by x(2) and
y(2) , respectively, and the covariance by
xy . What restriction is needed to ensure that X + Y and X − Y are uncorrelated?
(b) Let X1 , X2 and X3 be independently distributed as normal random variables such that X1 and X2 follow Normal (0, 1) and X3 follows Normal (2, 9). Find E(5 + 2X$ + (3 + X2 )X3(2)) .
Question 2 (40 points): Simple Linear Regression (approx. 25 minutes)
Professor Denteh ran an experiment to measure the effect of time pressure on final exam scores. He gives each of the 10 students the same exam, but some students have 50 minutes to complete the exam while others have 80 minutes. Each student is assigned to one of the exam times by the toss of a fair coin.
Let Y denote the exam score (between 0 and 100s) and X be the indicator of the assigned exam time. That is, X=1 for those assigned to 50 minutes and X=0 for those assigned to 80 minutes.
Here’s what you know about this sample of students. The sigma notation denotes summation. , X = 4 , X 2 = 4
, Y = 509 , Y2 = 25,935 , XY = 196
Consider the regression model given by Y = & +
1X + u .
a) Do you think E(u |X) = 0 is satisfied in this regression model. Why or why not?
b) Use the method of least squares to estimate 0 and
1 in the above regression. Interpret both coefficient estimates.
c) Verify that the coefficient estimate, ! , equals a scaled measure of the correlation between Y and X. That is, verify that
1 =
Corr(X, Y) .
d) Given that ∑i($)$
i 2 = 2.8333, where
i represents the residuals, calculate the R2 .
Question 3 (40 points): Multiple Linear Regression (approx. 15 minutes)
Let math10 denote the percentage of 10th graders at a high school receiving a passing score on a standardized math exam. We are interested in estimating the effect of per-student spending, denoted by expend (measured in logarithm) on math performance from the model
matℎ10 = p" + p1 log(expend) + p2poverty + u ,
where poverty is the percentage of students living in poverty. Assume that the above model satisfies the Gauss-Markov assumptions.
a) Suppose that poverty is unobserved to you. If you only regress matℎ10 on log(expend) , what is the likely bias in 1 from this regression? State any assumptions you make to answer this question.
b) Suppose you observe a variable called lnchprg which denotes the percentage of students who are eligible for a federally funded lunch program. Is lnchprg a sensible proxy for poverty? Explain why or why not.1
c) Assume you estimate the regression model
matℎ10 = p& + p1 log(expend) + p2 lncℎprg + u,
where we obtain the following regression output,
mUatℎ10 = − 20.36 + 6.23log(expend) − 0.305 lncℎprg + u
n = 408, R2 = 0.180
Interpret the coefficient on lncℎprg .
d) From (c), suppose that we regress lnchprg on log(expend) and obtain the output below:
lnrg = 160.809 − 16.201log(expend)
n = 408, R2 = 0.037
Use this information to obtain in the following simple regression given by
matℎ10 = p& + p1 log(expend) + u .
Is biased? If so, what is the direction of the bias?