Download e388_08_Win_Final

Econometrics--Econ 388
Winter 2008, Richard Butler
Final Exam
your name_________________________________________________
Section Problem Points Possible
I 1-20 3 points each
II 21
22
23
24
25
20 points
5 points
5 points
5 points
5 points
III 26
27
28
20 points
20 points
20 points
IV
29
30
20 points
20 points
1
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. 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. null hypothesis2
11. identification problem (in simultaneous equation models)-
12. Wald test -
13. least squares estimation criterion for fitting a linear regression-
14. logit model-
15. dynamically complete models -
16. one-tailed hypothesis test-
17. plim--
18. heteroskedasticity-
19. t-distribution--
20. central limit theorem 3
II. Some Concepts
21. Suppose that two random variables are constructed from rolling a fair dice twice. Define Z to
be a random variable whose value equals the square of the difference in the two rolls (so Z = 0, 1,
4, 9, 16 or 25). Define W to be the random variable whose value equals one if the sum 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=0) (again, no credit unless you show the
right formulas)
E. Are W and Z independent? Why or why not?
4
The next four 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
22. “The law of iterated expectations is a statement about the property of conditional
distributions.”
23. “Let Xˆ  V (V 'V ) 1V ' X where V has the appropriate dimensions. Then Xˆ ' X  Xˆ ' Xˆ .”
5
24. “In a linear regression model (either single or multiple), if the sample means of X are zero
and the sample means of Y is zero, then the intercept will be zero as well.”
25. "A first order autoregressive process, yt   yt 1   t , is both stationary and weakly
dependent if  <1.”
6
III. Some Applications
26. 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
18 as indicated, and the length of the x vector is 5. If the angle between the x vector and the yvector 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).
Y: length= 18
45
X: length=5
7
27. Write STATA programs to make the following tests or estimate the following models
requested below, assuming that the sample variables G and H are endogeneous, and that the
exogeneous variables are C, D, E, and F.
a. Hi  0  1Gi  2 Ci  3 Di  i
hand side of the equation.
Do a Hausman test for endogeneity of G on the right
b. For the same model as in (a), write out the STATA code to test for overidentifying restrictions
on the “extra” identifying variables, E and F.
c. For the same model as in (a), write out the STATA code to estimate the model in (a) by two
stage least squares.
8
III. Some Proofs
28. Show whether there is simultaneous equation bias (right hand side regressors correlated with
the error) in the following particular measurement error framework:
the true model is
Y  X   z   ,
but the variable z (the true value) is measured with error when it is observed, call this observed
value z*, subject to the following relationship
measurement error is
z  z* 
where  is white noise (with the usual independent, zero mean distribution), uncorrelated with
z* and  so that E ( | z*)  0 (and  is uncorrelated with X, z, and z*). Indicate whether or
not there is “simultaneous equation” bias if Y is regressed on X and z* (as always, you are only
graded on your explanation, not on your guess as to the right answer).
9
29. Prove that under the standard assumptions, the OLS estimator for the variance, s2, is an
unbiased estimator for variance of the error term (when the error term is given in the usual linear
model as  in Y  X   ).
10
30. Prove that under the standard assumptions, the OLS estimator
ˆ  ( X ' X ) 1 X ' Y
is the best linear unbiased estimator for the model Y  X   (that is, prove the Gauss-Markov
theorem).
11