Download e388_10_Spr_Final

Econometrics--Econ 388
Spring 2010, Richard Butler
Final Exam
your name_________________________________________________
Section Problem Points Possible
I 1-20 3 points each
II 21
22
23
24
25
10 points
10 points
5 points
5 points
10 points
III 26
27
28
15 points
15 points
25 points
IV
20 points
25 points
29
30
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I. Define or explain the following terms:
1. binary variables-
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. solve for 𝑝𝑙𝑖𝑚(𝜇̂ 𝑖 − 𝜇𝑖 ) =? ? (𝑤ℎ𝑒𝑟𝑒 𝜇̂ 𝑖 , 𝜇𝑖 are the ith residual, error term from a regression;
show steps assuming the least squares estimator for β is consistent) --
6. endogeneous variable-
7. positive definitive matrix-
8. STATA code to regress 𝑌𝑖 𝑜𝑛 𝑋𝑖 𝑎𝑛𝑑 𝑍𝑖 when there is an omitted exogenous variable 𝑈𝑖 by
2SLS-
9. Goldfeld-Quandt test-
10. STATA code to regress 𝑌𝑡 𝑜𝑛 𝐼𝑡 𝑎𝑛𝑑 𝑊𝑡 when the erros are AR(1)2
11. identification problem (in simultaneous equation models)-
12. LaGrange-Multiplier test--
13. least squares estimation criterion for fitting a linear regression-
14. probit model-
15. “estat sum” command in STATA -
16. one-tailed hypothesis test-
17. BLUE (as an estimation criterion) --
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
21. Concerned about the distribution of Emergency Room admissions, the number of ER cases
were regressed on the following trend and dummy variables over a 999 day period:
/*
1. ER
emergency room visits per day
2. trend
time trend (T=1,2,…, 665,666)
3. holiday
=1 if easter, thanksgiving, Christmas; 0 otherwise
4. friday
=1 if Friday; 0 otherwise
5. saturday
=1 if Saturday; 0 otherwise
6. fullmoon
=1 if full moon (±2 days); 0 otherwise
7. newmoon
=1 if new moon (±2 days); 0 otherwise
8. full_sat
= full moon*saturday
*/
regress ER trend holiday Friday Saturday fullmoon newmoon full_sat;
test (fullmoon=0) (full_sat=0);
Source |
SS
df
MS
Number of obs =
999
-------------+-----------------------------F( 7,
992)=
70.95
Model | 6.203422e3
7 1.550855e3
Prob > F
= 0.0000
Residual | 1.814199e3
991 .0218578e3
R-squared
= 0.1737
-------------+-----------------------------Adj R-squared = 0.1628
Total | 8.017621e3
998 .0921565e3
-----------------------------------------------------------------------------ER
|
Coef.
Std. Err.
t
P>|t|
-------------+---------------------------------------------------------------trend
|
.0338747
.0121758
3.06
0.002
holiday |
13.86295
6.44521
2.11
0.012
friday |
6.90981
2.11131
3.27
0.001
saturday |
10.58941
2.11942
5.00
0.000
fullmoon |
2.45453
3.98110
0.62
0.538
newmoon |
6.40593
4.25694
1.50
0.130
full_sat |
9.789933
2.02226
4.84
0.000
_cons |
93.45164
1.55628
60.30
0.000
-----------------------------------------------------------------------------. test (fullmoon=0) (full_sat=0);
( 1) fullmoon = 0
( 2) full_sat = 0
F( 2, 991) =
4.75
Prob > F =
0.0004
a) What do the coefficients indicate?
b) Does ER visits increase when there is a full moon?
4
22. Suppose that two random variables are constructed from rolling a fair dice twice. Define Z
to be a random variable whose value equals the absolute value of the difference in the two rolls
(so Z = 0, 1, 2, 3, 4 or 5). Define W to be the random variable whose value equals one if the sun
of the two rolls is 6 or less (W=1 if the experimental outcome is ‘1,1’ ‘1,2’ ‘1,3’ ‘1,4’ ‘1,5’
….etc.) and W=0 otherwise (the sum of upturned dots on the two rolls is 7 or greater (for
example, rolling a ‘1,6’ or ‘3,4’ or ‘4, 5’ for example).
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
Z=3
Z=4
Z=5
W=1
W=0
B. calculate the marginal probability densities f(Z) and f(W)
C. Calculate E(W) and V(W) (no credit unless you show the right formulas).
D. Calculate the conditional probability density f(W|Z=4) (again, no credit unless you show the
right formulas)
E. Are W and Z independent? Why or why not?
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The next three 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.
23. “In a simultaneous equation system, the reduced form equation can always be estimated by
OLS whereas the structural equation often cannot be estimated by OLS if the structural equation
violates the full rank assumption (assumption II).”
24. “In a linear regression model (either single or multiple), if the sample means of all the
column variables of slope coefficients X are zero (excluding the constant) and the sample mean
of Y is zero, then the intercept will be zero as well.”
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25. "A first order autoregressive process, yt   yt 1   t , is both stationary and weakly
dependent if  <1.”
III. Some Applications
26. Given the usual regression model Yt  X t   t where the population error terms have a
second order moving process: t  0 et  1 et 1  2 et 2 where et is a white noise error term,
independently distributed with zero mean and variance  2 , then a) derive the variancecovariance matrix for  , and b) explain whether the errors are weakly dependent or not.
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27. If you wanted to estimate the slope regressor, β1, in the simple regression model
Yi = β0 + β1Xi + εi
you might decide just to average successive differences in the rise over the run as follows:
=
1 n Y i - Y i -1

n - 1 i=2 X i - X i -1
where Yi-1 means the value of Y from the previous observation (so that Yi-1 = β0 + β1Xi-1 + εi-1). .
Show whether 𝛽̅ is an unbiased estimator of 𝛽1 or not.
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III. Some Proofs
28. For the heteroskedasticity model (and more generally, the generalized least squares):
Y  X 
 ~ N (0, )
where
the generalized least squares (weighted least squares) estimator is
ˆGLS  ( X ' 1 X )1 X ' 1Y
prove that it is the best (minimum variance in the matrix sense), linear, unbiased estimator
among the class of all linear, unbiased estimators for this model (it is the case, and you may
assume that for any nxk matrix N, the matrix is N ' N positive definitive).
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29. Suppose that for the general linear regression model, Y  X    , the modeling
assumptions are satisfied, in particular,  is normally distribution with mean 0 and variance
n
 2 I where I is the n by n identity matrix. Prove that s 2 
 ˆ
i 1
2
i
nk

ˆ ' ˆ
nk
is a consistent
 X'X 
estimator of  2 . (You can assume that plim 
   , a positive definite, symmetric matrix
 n 
for any size n. Also recall the ̂ is the least squares residual,
ˆ  y  X  ( I  X ( X ' X ) 1 X ' ) y  ( I  ( X ( X ' X ) 1 X ' ) .)
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30. We have nine observations in total, three for each of three college educated women (the first
three observations are for Callie, each from a different year; the next three for Bella, and the last
three for Lizzy, each from a different year). Regress their wage rates (Y) on three dummy
variables (with no constant in the model), so that the vector of independent variables (the D
matrix) looks like
1 0 0
1 0 0
1 0 0
0 1 0
D= 0 1 0
0 1 0
0 0 1
0 0 1
(0 0 1)
a) then what is the predicted wages look like, that is, what is: PD Y  D( D ' D) 1 D ' Y  ?
b) what do the residuals look like in this model with just 9 observations, that is, what is
M DY  ( I  PD )Y  ?
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