Download e388_10_Spr_Exam1

First Exam: Economics 388, Econometrics
Spring 2010
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) Questions
Section I. Define or explain the following terms (3 points each)
1. 95 percent confidence interval for  j -
2. Does VIF test for perfect collinearity?-
3. show that
N
N
i 1
i 1
 ( yi  y )( xi  x )   ( yi  y ) xi -
4. R-squared -
5. probability significance values (i.e., ‘p-values’)-
6. solve for
are the ith residual, error term from a regression; show
steps assuming the least squares estimator for β is consistent) -
7. orthogonal projection-
8. Var(w) where w is a nx1 vector of random variables-
9. population mean vs. sample mean-
10. maximum likelihood estimation-
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II. Some Concepts
11. Suppose that two random variables are constructed from a flipping a fair coin twice. Define Z to be a
random variable whose value equals the number of heads in two flips (so X = 0, 1 or 2). Define W to be
the random variable whose value equals one if the two flips get the same results (W=1 if the experimental
outcome is either ‘heads, heads,’ or ‘tails, tails’; and W=0 otherwise).
A. Fill in the joint probability density function for the following table (i.e., indicate what the joint
probabilities of each of the outcomes are):
Z=0
Z=1
Z=2
W=0
W=1
B. calculate the marginal probability densities f(Z) and f(W)
C. Calculate E(Z) and V(Z) (no credit unless you show the right formulas).
D. Calculate the conditional probability density f(W|Z=1) (again, no credit unless you show the right
formulas)
E. Are W and Z independent? Why or why not?
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12. Prove that under the usual model assumptions that the least squares estimator, ̂ , is unbiased and has
a covariance matrix equal to  2 ( X ' X ) 1 .
13. Assume that “schooling size” has no negative impact on student performance in standardized math
tests, where the null hypothesis is that enrollment (ln enroll) has no effect on math10 scores and the
alternative hypothesis is that it is not good for math10 scores..
a. What is the mathematical way of stating the null hypothesis and the alternative hypothesis?
b. What is the probability of making a type-II error assuming i) that the critical value of the type I error is
5 percent, and ii) that the true coefficient on the ln(enrollment) variable is
a. –1.5?
b. -.1.8 ?
given the following (with MATH10 is the dependent variable):
VARIABLE
ESTIMATED
NAME
COEFFICIENT
LTOTCOMP
21.155
LSTAFF
3.9800
LENROLL
-1.2680
CONSTANT -207.66
STANDARD
ERROR
4.056
4.190
0.6932
48.70
T-RATIO
404 DF
5.216
0.9500
-1.829
-4.264
PARTIAL STANDARDIZED ELASTICITY
P-VALUE CORR. COEFFICIENT AT MEANS
0.000 0.251
0.3050
9.2493
0.343 0.047
0.0480
0.7600
0.068-0.091
-0.1048
-0.3950
0.000-0.208
0.0000
-8.6143
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14. What does the following regression (taken from your text) test for? Why?
# delimit ;
infile price
assess
bdrms
lotsize
sqrft
colonial lprice
lassess
llotsize lsqrft using "G:\econ388\classrm_data\wooldridge\HPRICE1.RAW", clear;
/*
1. price
price, in dollars
2. assess
assessed value, in dollars
3. bdrms
number of bedrooms
4. lotsize
size of lot, square feet
5. sqrft
size of house, square feet
6. colonial
=1 if home is colonial style
7. lprice
log(price)
8. lassess
log(assess
9. llotsize
log(lotsize)
10. lsqrft
log(sqrft)
*/
summarize;
regress lprice lassess lotsize sqrft bdrms;
test (lassess=1) (lotsize=0) (sqrft=0) (bdrms=0);
Source |
SS
df
MS
Number of obs =
88
-------------+-----------------------------F( 4,
83) =
70.95
Model | 6.20342233
4 1.55085558
Prob > F
= 0.0000
Residual | 1.81419962
83 .021857827
R-squared
= 0.7737
-------------+-----------------------------Adj R-squared = 0.7628
Total | 8.01762195
87 .092156574
Root MSE
= .14784
-----------------------------------------------------------------------------lprice |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------lassess |
.9478747
.1271758
7.45
0.000
.694927
1.200822
lotsize |
1.78e-06
1.67e-06
1.07
0.288
-1.53e-06
5.10e-06
sqrft |
1.19e-06
.0000587
0.02
0.984
-.0001155
.0001179
bdrms |
.0283933
.0222674
1.28
0.206
-.0158956
.0726823
_cons |
.4535164
1.506286
0.30
0.764
-2.542426
3.449459
-----------------------------------------------------------------------------. test (lassess=1) (lotsize=0) (sqrft=0) (bdrms=0);
( 1) lassess = 1
( 2) lotsize = 0
( 3) sqrft = 0
( 4) bdrms = 0
F( 4,
83) =
0.75
Prob > F =
0.5580
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II. 15. a. Prove that the least squares estimator for the variance, s2, is unbiased using matrix algebra.
b. Which OLS assumptions did you use in your proof?
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