Download e388_08_Spr_Exam1

First Exam: Economics 388, Econometrics
Spring 2008 in R. Butler’s class
YOUR NAME:________________________________________
Section I (30 points) Questions 1-10 (3 points each)
Section II (40 points) Questions 11-14 (10 points each)
Section III (30 points) Question 15 (30 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. show that
N
N
i 1
i 1
 ( yi  y )( xi  x )   ( yi  y ) xi -
3. type II error -
4. conditional probability density function of y given x-
5. omitted variable bias (say left out Z in a regression of Y on X)-
6. adjusted R-square -
7. unbiased estimator-
8. homoskedasticity -
9. orthogonal projection-
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. faminc
1988 family income, $1000s
;
* 4. bwght
birth weight, ounces
;
* 5. fatheduc
father's yrs of educ
;
* 6. motheduc
mother's yrs of educ
;
* 7. parity
birth order of child
;
* 10. cigs
cigs smked per day while preg ;
. reg bwght cigs parity faminc motheduc
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
------------------------------------------------------------------------------
This output above is right (all correct), but now we modify a variable with STATA code (see just below)
and the output below contains some errors (has been changed by Coach from what originally printed);
. gen bwght2 = bwght + 2;
. regress bwght2 cigs parity faminc motheduc fatheduc;
Source |
SS
df
MS
-------------+-----------------------------Model | 18705.5567
5 3741.11135
Residual | 464041.135 1185 391.595895
-------------+-----------------------------Total | 482746.692 1190 405.669489
Number of obs
F( 5, 1185)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
1191
9.55
0.0000
0.0514
0.0501
19.789
-----------------------------------------------------------------------------bwght2 |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------cigs | -2.595936
.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
31.25
0.000
109.2092
123.8394
------------------------------------------------------------------------------
List all the errors in the output printed just above:
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12. You are told that X is normally distributed with mean of 8 and variance of 4, and Y is normally
distributed with mean of 0 and variance of 16. X and Y are independently distributed. The what are
distributions for the following variables:
a. W1 = 3X + 2
b. W2 = X + 4Y
c. W3 = [(X-8)/2]2 + [(Y-0)/4]2
d. W4 =
[( X  8) / 2]2
[(Y  0]/ 4]2
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. 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, 2 or 3 will occur is one sixth (for each of these events taken
separately) 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),
and decided to test for that by repeatedly (25 times) throwing the die, and got a sample average of 2.0,
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 the least squares estimator is BLUE (that is, prove the
Gauss Markov theorem).
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