Econometrics--Econ 388
Winter 2013, Richard Butler
Final Exam
your name_________________________________________________
Section Problem Points Possible
I 1-20 3 points each
II 21
22
23
24
25
5 points
5 points
10 points
10 points
20 points
III 26
27
28
20 points
20 points
30 points
IV
10 points
10 points
29
30
1
I. Define or explain the following terms:
1. bootstrapping-
2. The prediction error for YT, i.e., the variance of a forecast value of y given a specific value of
the regressor vector, XT (from YT  X T ˆ  T )-
3. formula for VIF test for collinearity--
4. structural vs. reduced form parameters in simultaneous equations-
5. dummy variable trap -
6. endogeneous variable-
7. maximum likelihood estimation criteria-
8. F-test-
9. Goldfeld-Quandt test-
10. impure heteroskedasticity (that would lead to biased 𝛽̂ )-2
11. necessary condition (order condition) for identification in simultaneous equation models-
2 0
12-14. Let C= [
]
−1 1
12. Trace of C-13. Calculate B=C’C
14. show C’C from above (or in general) is a positive definite matrix-
15. dynamically complete models -
16. one-tailed hypothesis test-
17. model corresponding to “prais y x1 x2 x3;” procedure in STATA --
18. show that
N
N
i 1
i 1
 ( yi  y )( xi  x )   ( yi  y ) xi --
19. probability significance values (i.e., ‘p-values’)-
20. central limit theorem 3
II. Some Concepts
The next six questions consist of statements that are True, False, or Uncertain (Sometimes True).
You are graded solely on the basis of your explanation in your answer
21. “As long as the time series are covariance stationary and weakly dependent, then the usual
way we calculate the R-squared will consistently estimate the population R-squared.”
22. “For the model, 𝜇𝑡 = 𝜌𝜇𝑡−1 + 𝑒𝑡 , regressing the residuals on the lagged value of the residuals
will always provide a consistent estimate of 𝜌, regardless of the null hypothesis.”
4
23. “In estimating a causal model of how skipping class affects the final exam score, an intrepid
econometrician estimates the following model:
𝑠𝑐𝑜𝑟𝑒 = 𝛽0 + 𝛽1 𝑠𝑘𝑖𝑝𝑝𝑒𝑑 + 𝜇.
If skipping class was related to important omitted variables, then commuting distance from a
student’s residence to the class may be a good instrumental variable.”
24. “Two stage least squares is employed only for simultaneous equation models with just one
omitted (from the equation being estimated) exogenous variable.”
5
25. 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?
6
III. Some Applications
26. Data from ATTEND.RAW generated the regression below, where
# delimit ; /* attend
classes attended out of 32
priGPA
cumulative GPA prior to term
ACT
ACT score
atndrte
percent classes attended
stndfnl
(final - mean)/sd */
infile attend termgpa priGPA ACT final atndrte hwrte frosh soph skipped
using
"D:\BYUclasses_2013\econ388\classrm_data\wooldridge\ATTEND.RAW", clear;
gen priGPAsq=priGPA*priGPA;
gen ACTsq=ACT*ACT;
gen priGPA_atn=priGPA*atndrte;
regress stndfnl atndrte priGPA priGPAsq ACT ACTsq priGPA_atn;
estat sum; WITH THESE RESULTS
Source |
SS
df
MS
-------------+-----------------------------Model | 152.001001
6 25.3335002
Residual |
512.76244
673 .761905557
-------------+-----------------------------Total | 664.763441
679
.97903305
Number of obs
F( 6,
673)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
stndfnl
680
33.25
0.0000
0.2287
0.2218
.87287
-----------------------------------------------------------------------------stndfnl |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------atndrte | -.0067129
.0102321
-0.66
0.512
-.0268035
.0133777
priGPA |
-1.62854
.4810025
-3.39
0.001
-2.572986
-.6840938
priGPAsq |
.2959046
.1010495
2.93
0.004
.0974945
.4943147
ACT | -.1280394
.098492
-1.30
0.194
-.3214279
.0653492
ACTsq |
.0045334
.0021764
2.08
0.038
.00026
.0088068
priGPA_atn |
.0055859
.0043174
1.29
0.196
-.0028913
.0140631
_cons |
2.050293
1.360319
1.51
0.132
-.6206864
4.721272
-----------------------------------------------------------------------------------------------------------------------------------------Variable |
Mean
Std. Dev.
Min
Max
-------------+----------------------------------------------stndfnl |
.0296589
.989461
-3.30882
2.78361
atndrte |
81.70956
17.04699
6.25
100
priGPA |
2.586775
.5447141
.857
3.93
priGPAsq |
6.987682
2.892395
.734449
15.4449
ACT |
22.51029
3.490768
13
32
ACTsq |
518.8809
158.8418
169
1024
priGPA_atn |
215.3252
72.10833
14.0625
393
a). how does attendance effect final class standing (stndfnl)?
b) how does ACT score affect final class standing (stndfnl)?
7
27. Prediction, and the calculation of the associated prediction confidence intervals, are
important for business applications of regression analysis.
a) explain intuitively how the following code gets a predicted value for wages and the associated
prediction error, using our perennial FHSS professor wages and teaching experience as the
example. Be as explicit as possible.
input obs salary experience male_dummy pred_dummy;
1 45000 2 1 0;
2 60600 7 0 0;
3 70000 10 1 0;
4 85000 18 1 0;
5 50800 6 0 0;
6 64000 8 1 0;
7 62500 8 0 0;
8 87000 15 1 0;
9 92000 25 0 0;
10 89500 22 1 0;
11 0
12 0 -1;
end;
reg salary experience pred_dummy;
b) Using the OLS criteria, prove that the intuitive strategy above works—that is, show the
estimated coefficients for slope and intercept are unchanged with the addition of the “preddummy” variable, and that the predicted value for experience=12 is exactly derived.
8
III. Some Proofs
𝑋′𝑋
28. Under the usual assumptions, including 𝑛 converges to a matrix of finite elements as n goes
to infinity, prove that the OLS estimator is consistent for the usual model: 𝑌 = 𝑋𝛽 + 𝜇 .
9
29. Describe the basics of obtaining feasible generalized least squares (also call weighted least
squares) for the linear regression model when the errors are AR(1): 𝜇𝑡 = 𝜌𝜇𝑡−1 + 𝑒𝑡
30. Derive the correct variance-covariance matrix for 𝛽̂ , OLS estimator, when there is really
heteroskedasticity but we don’t do weighted least squares. That is, what covariance matrix is
generated by the “robust” option in STATA.
10