First Exam: Economics 388, Econometrics
Winter 2013 in R. Butler’s class
YOUR NAME:________________________________________
Section I (30 points) Questions 1-10 (3 points each)
Section II (50 points) Questions 11,12 (15points each); Question 13,14 (10 points each)
Section III (20 points) Question 15 (20 points)
Section I. Define or explain the following terms (3 points each)
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1-3. Let C= [
0
−1
]
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1. Trace of C--
2. Calculate B=C’C
3. show C’C from above (or in general) is a positive definite matrix-
4. law of large numbers-
5. what to probability significance values printed out in the stata output represent?
6. estate vif;
7. what is the “xi:” suffix do in xi: reg salary male educ i.occupation;
8. heteroskedasticity-
9. constant elasticity model specification
10. central limit theorem as applied to 𝛽̂ , the least square estimator--
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II. Some Concepts
11. Suppose that a 4 sided dice with numbers 1, 2, 3, and 4 are etched one each of the four surfaces
respectively. The dice is fair in that each number is equally probably to be on the bottom. Define two
random variables for two throws of the dice (let i=number on the bottom in the first throw, and j=number
on the bottom of the second throw), and let
W= i + j (the sum of the two throws)
Z= | i – j| (the absolute value of the difference in the two throws)
a. chart out the joint distribution of W and Z (the likelihood that W takes one feasible number and Z takes
another feasible number)
b. what is the marginal distribution of Z?
What is its expected value?
c. Is W independent from Z? Show why or why not?
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12. a. Explain what the following test statement in STATA does (salary = annual salary income; educ =
educational attainment; OJT = formal years of on the job training; male= male carbon-based biped):
regress salary educ OJT male;
test (educ=OJT);
b. Derive an alternative way to test the hypothesis in “a” that does NOT involve a test statement (but may
involve the creation of an additional variable) using all the necessary STATA code, assuming the
variables “salary educ OJT male” variables are already defined:
Explain what statistic from this regression provides the test of the null hypothesis given in “a”?
13. Suppose birthweight (bwght in pounds) is regressed on cigs (mom’s average cigarettes per day during
pregnancy), parity (birth order of child), faminc (annual family income), momeduc (mother’s years of
schooling completed), and dadeduc (father’s years of schooling completed).
With all the variables, suppose that the SSR (sum of squares of residuals) is 100. Dropping the momeduc
and dadeduc variables from the model, and rerunning the regression, the SSR is 105. It there are 1006
observations in the regression, are the parental education variables jointly significant in this example? No
or yes by themselves get 0 credit. (The table at the end of the test may be of help.) if you don’t show
your work, you also don’t get credit.
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14. Derive the omitted variable bias (for 𝛽̂ , using the OLS estimator) when the true model is:
𝑌 = 𝑋𝛽 + 𝑍𝛼 + 𝜇
But we leave out the Z variable (and its effect, 𝛼) and estimate the model:
𝑌 = 𝑋𝛽 + 𝜇
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15. For the population linear regression model (n observations with k explanatory variables including the
intercept),
Y  X ,
with the usual model assumptions, do the following:
1. explain what the ‘orthogonality’ condition (what it is, and why it is sensible).
2. state the orthogonality condition (also called the normal equations) in matrix terms.
3. derive the OLS estimator for 𝛽 from the orthogonality conditions.
4. Prove that your estimator, call it 𝛽̂ , is unbiased.
5. derive the variance-covariance matrix for 𝛽̂ , under the model assumptions.
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