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Econometrics--Econ 388
Winter 2010, Richard Butler
Final Exam
your name_________________________________________________
Section Problem Points Possible
I 1-20 3 points each
II 21
22
23
24
25
26
15 points
10 points
10 points
10 points
10 points
10 points
III 27
30 points
IV
30 points
15 points
28
29
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I. Define or explain the following terms:
1. inverse of a matrix-2. for the regression Yi  0  1Wi  2 X i  3 Zi  i Write out the STATA code to do a
Hausman test for endogeneity of W on the right hand side of the equation (X and Z are
exogeneous, as are the excluded variables T, U).
3. Write out the STATA code to estimate the model in question 2 with two-stage least squares
estimators using exogeneous variables X, Z, T, U:
4. write out the STATA code for second version of the White test (where predicted variances are
checked against the predicted value of Y and predicted value of Y-squared using a LM test)-
5. STATA code for the linear probability model where I regress a married dummy on age, educ,
and male, controlling for heteroskedasticity -
6. law of large numbers-
7. cointegration of two time series, wt and vt--
8. dummy variable trap-
9. give the reason “iterations” are printed out for probit estimates but not regular regression
estimates-
10. method of moment estimators –
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11. maximum likelihood estimation criterion -
12. logistic regression model -
13. random walk -
14. structural vs. reduced form equations -
15. Breusch-Pagan test-
16. t-ratio or t-statistic for an estimated regression beta-
17. type I vs. type II errors in statistics -
18. Durbin-Watson test -
19. orthogonal projection -
20. formula for the 95% prediction interval for yT (value of yt next year) when the true model is
yt  xt   t when the usual assumptions hold-3
II. Some Concepts
21. Suppose that the joint distribution for random variables x, y is given as
f ( x, y)  .6 x .41 x .3 y .521 y 2 xy
for values x=0, 1 and y=0, 1.
A. What are the joint probabilities f(x=0,y=0), f(x=0,y=1), f(x=1,y=0), and f(x=1,y=1)?
B. calculate the marginal probability densities f(x) and f(y)
C. Calculate E(x) and V(x) (no credit unless you show the right formulas).
D. Calculate the conditional probability density f(y|x=0) (again, no credit unless you show the
right formulas)
E. Are x and y independent? Why or why not?
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22. Peter Trueheart regressed wages on SEX (1=male, 2=female) and got the results
𝑊𝑎𝑔𝑒𝑠𝑖 = 𝛽̂0 + 𝛽̂1 𝑆𝐸𝑋𝑖 + 𝜇̂ 𝑖 , when he realized his error and replaced SEX with a dummy
variable for FEMALE (1=female, 0=male), he got the new estimates
𝑊𝑎𝑔𝑒𝑠𝑖 = 𝛼̂0 + 𝛼̂1 𝐹𝐸𝑀𝐴𝐿𝐸𝑖 + 𝜀̂𝑖
What is the exact mathematical relationship between the alphas and the betas, and how is 𝜇̂ 𝑖
mathematically related to 𝜀̂𝑖 ?
23. Calculate the expenditure (quarterly expenses are given by E) elasticity with respect to
income (annual consumer income is given by I) when the coefficients from the expenditure
regression are given by
𝐸𝑖 = 1256 + .065 𝐼𝑖 + 𝜇̂ 𝑖
And where average quarterly expenditures are $2000 and average annual income is $20,000.
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24. Is there a correlated regressor problem (a right hand side variable violates regression
assumption number III) when we leave out an important determinant of Y (say a Z variable) that
is uncorrelated with all the included right hand side variables (the Xs, so Z is uncorrelated with
the Xs)? Why or why not?
25. Explain what the following test does in this panel data set that relates individual wages to
their socio-demographic information, occupation, and year wages were observed:
*nr = equals the unique identifier for each individual in this panel data set;
********* need to avoid lagging between people **************************;
** so the bysort only generates it for those with the same id, namely *******;
** the same nr values. *******************************************;
bysort nr: gen laglwage = lwage[_n-1];
regress lwage laglwage black hisp educ exper expersq married union occ1
occ2 occ4 occ5 occ6 occ7 occ8 occ9
d82
d83
d84
d85 d86 ;
predict resids, residuals;
bysort nr: gen lag_resids = resids[_n-1]; *need to avoid lagging between people;
list nr year resids lag_resids;
regress lwage laglwage black hisp educ exper expersq married union occ1
occ2 occ4 occ5 occ6 occ7 occ8 occ9
d82
d83
d84
d85 d86
lag_resids;
(what does lag_resids test do in this regression)?
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26. Let { et : t , -1, 0, 1, 2, …}be a sequence of independent, identically distributed random
variables with mean zero and variance  2 . Define a stochastic process by
yt  et  (1/ 3)et 1  (1/ 2)et 2 for t= 1, 2, 3, ….
a) find the mean, variance, and covariances (find covariances for Cov(yt, yt-h) for h=1,2,3 and 4)
for the process yt.
b) Is this process stationary? explain
c) Is this process weakly dependent? explain
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III. Some Bigger Proofs.
27. Corrections for
a) Heteroskedasticity,
b) First Order Autoregressive Correlation, or
c) Measurement Error (using Instrumental Variables),
can all be viewed as regressing TY on TX for the following linear model: Y  X   
In each of the three cases, indicate what the T matrix is that is pre-multiplied through the model
to make the respective correction (make clear whatever additional variables or parameters you
introduce represent).
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28. For the heteroskedasticity model (and more generally, the generalized least squares):
where
Y  X 
 ~ N (0, )
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. Given the usual assumptions about the n by k matrix of instrumental variables, Z, prove that
the instrumental variable estimator is consistent:
Z 'X
i.e., prove that for ˆIV  (Z ' X )1 Z 'Y that plim ˆIV   . (You may assume that
is a
n
positive definite matrix with finite elements for any value of n, the sample size).
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