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First Exam: Economics 388, Econometrics
Fall 2004 in R. Butler’s class
YOUR NAME:________________________________________
Section I (30 points) Questions 1-10 (3 points each)
Section II (40 points) Questions 11-15 (10 points each)
Section III (30 points) Question 16 (20 points each)
Section I. Define or explain the following terms (3 points each)
1. median-
2. standardized or z-score -
3. type II error -
4. conditional probability density function of y given x-
5. F-test (or “Chow” test)-
6. adjusted R-square -
7. perfect multicollinearity-
8. homoskedasticity -
9. probability significance values (i.e., ‘p-values’)-
10. Var(w) where w is a nx1 vector of random variables-
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II. Some Concepts
11. Indicate whether the following statement is True, False or Uncertain and explain why. You are graded
only on the explanation for your answer. “If we add 2 to every value of the dependent variable (i.e.,
instead of “y” we have “y+2”), the slope coefficient will increase by 2 but the intercept and R-square will
remain unchanged.” (you can answer this either for the simple regression model or the multiple regression
model, you pick)
12. Indicate whether the following statement is True, False or Uncertain and explain why. You are graded
only on the explanation for your answer. “For the population linear regression model, y  X   , the
sample predicted value of y and the residual are independently distributed (uncorrelated). That is,
yˆ  X̂ and ̂ are uncorrelated.”
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13. My son has a pyramid dice, with four sides numbered from 1 to 4. Let W be the random variable
corresponding to number that's on the bottom side when the dice is rolled. If the dice is not fair, but the
probability that the sides with numbers 1 or 3 will occur is one fourth (for each of these events taken
separately) and the likelihood of rolling a 2 is one half, then
a. What is the expected value of the random variable W and what is the variance of W?
b. If we did not know whether the dice were fair or not (i.e., that each outcome was equally probable),
how could we test for that?
14. Hours=number of hours spent on a consulting job (mean=6.7)
Miles=total miles driven to the consulting job (mean=80)
Present=number of presentations given in the course of the consulting job (mean=2.9)
|_ols hours miles present
VARIABLE ESTIMATED STANDARD T-RATIO
PARTIAL STANDARDIZED ELASTICITY
NAME COEFFICIENT ERROR
7 DF
P-VALUE CORR. COEFFICIENT AT MEANS
MILES
0.61135E-01 0.9888E-02 6.182
0.000 0.919
0.7345 0.7300
PRESENT
0.92343
0.2211
4.176
0.004 0.845
0.4962 0.3997
CONSTANT -0.86870
0.9515
-0.9129
0.392-0.326
0.0000 -0.1297
a) Approximately how many minutes does a presentation seem to take?
b) How fast (miles per hour) on average are the presenters traveling to and from their consulting sites?
c) If the value of time for presenters (including overhead) is $200/hour, and if I just want to break even,
what should I charge a customer requiring 2 presentations and a total distance traveled of 70 miles?
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15. Suppose that the population model is yi  0  1 xi  i where  i is distributed as a normal
random variable with a mean of zero and a variance of  2 . Also assume that the population errors are
uncorrelated (independent). Then show for the corresponding sample regression model that
N
y  ˆ0  ˆ1 x where y 
y
i
N
i
and x 
x
i
. That is, the simple regression line always passes
N
N
through the sample means. (HINT: use the normal equations, or at least one of them.)
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16. Suppose that the population model is yi   xi   i where  i is distributed as a normal random
variable with a mean of zero and a variance of  2 . Also assume that the population errors are
uncorrelated (independent). (Note this is a model without an intercept—leave it that way, i.e., it only has
one slope regressor.)
a) Use the orthogonal condition (or calculus) to find the least square estimator for  .
b) Determine whether this estimator is unbiased or not.
c) Derive the variance for this estimator (i.e., the variance for ˆ ).
d) From the Gauss Markov theorem, what can we say about our least square estimator, ˆ ?
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