Download e388_08_Win_Exam1

First Exam: Economics 388, Econometrics
Winter 2008 in R. Butler’s class
YOUR NAME:________________________________________
Section I (30 points) Questions 1-10 (3 points each)
Section II (50 points) Questions 11-13 (15 points each); Question 14 (5 points)
Section III (20 points) Question 15 (20 points)
Section I. Define or explain the following terms (3 points each)
1. If Y ~ N ( , ) then for the square matrix of constants, A, AY ~ N (??,???)
2. formula for the F-statistic (based on SSR)-
3. type II error -
4. conditional probability density function of y given x-
5. plim or probability limit-
6. adjusted R-square -
7. unbiased estimator-
8. heteroskedasticity -
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. For the following Stata Output, indicate what a) the statistic is (formula) and b) what it indicates for
the following circled statistics
* 1.
* 4.
* 5.
* 6.
* 7.
* 10.
faminc
1988 family income, $1000s
;
bwght
birth weight, ounces
;
fatheduc
father's yrs of educ
;
motheduc
mother's yrs of educ
;
parity
birth order of child
;
cigs
cigs smked per day while preg ;
Source |
SS
df
MS
Number of obs
-------------+-----------------------------F( 5, 1185)
Model | 18705.5567
5 3741.11135
Prob > F
Residual | 464041.135 1185 391.595895
R-squared
-------------+-----------------------------Adj R-squared
Total | 482746.692 1190 405.669489
Root MSE
=
=
=
=
=
=
1191
9.55
0.0000
0.0387
0.0347
19.789
-----------------------------------------------------------------------------bwght |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------cigs | -.5959362
.1103479
-5.40
0.000
-.8124352
-.3794373
parity |
1.787603
.6594055
2.71
0.007
.4938709
3.081336
faminc |
.0560414
.0365616
1.53
0.126
-.0156913
.1277742
motheduc | -.3704503
.3198551
-1.16
0.247
-.9979957
.2570951
fatheduc |
.4723944
.2826433
1.67
0.095
-.0821426
1.026931
_cons |
114.5243
3.728453
30.72
0.000
107.2092
121.8394
-----------------------------------------------------------------------------. test (cigs=0) (parity=0);
F( 2, 1185) =
17.95
Prob > F =
0.0000
I.
II.
III.
IV.
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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.”
13. For the simpliest regression (one slope variable, no intercept in the model), we have yi   xi  i ,
and the following picture for our particular sample, where length of the y-vector is 8 as indicated, and
the length of the x vector is 5. If the angle between the x vector and the y-vector is 45 degrees, than a)
what is the OLS estimate, ˆ , and b) what will be the residual sum of squares? (Warning, the picture is
deliberately NOT drawn to scale, so do the math—one hindrance, one help: the three angles of a triangle
sum to 180 degrees, and that the square root of 5 is 2.236)
Y: length= 8
45
X: length=5
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14. A regular dice (cube with, respectively, the numbers 1 through 6 on each side) may have been
tampered with, so that 6 comes up on half the throws on average, and the numbers 1, 2, 3, and 4 each
come up only one twelfth of the time. But you’re not sure if the die were really tampered with.
a) Indicate what is the expected, average outcome (mean or average of numbers that appear in repeated
trials) if the dice has been tampered with as described above?
b) Indicate what is the expected, average outcome if the dice has NOT been tampered with, with each side
equally likely to come up?
c) Indicate what z-score (t-test) formula you would use to test to the hypothesis that the die had not been
tampered with (the null hypothesis) if you had experimental outcomes (i.e., the actual means from 25
tosses of the die). (if you need to use a standard deviation for your test, use the standard deviation under
the null hypothesis in your calculation).
d) Is this a one tail or two tail test of these hypotheses? Draw a picture indicating where the critical
region would lie for your test statistic.
e) If I throw the die repeatedly (25 times), and got a sample average of 4.5, would I likely reject the null
hypothesis at the 5 percent level (guess the best you can about statistical significance, drawing upon your
extensive knowledge of the empirical rule for normal distributions)?
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15. Under the model assumptions, prove that s2 is an unbiased estimator of  2 for the OLS regression
model, using all the necessary assumptions employed in the proof in class or in the book, and proving it in
the general case using matrix algebra. Do your proof carefully, spelling out fully all your steps.
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