Econometrics--Econ 388
Section 3
Winter 2012, 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
5 points
III 26
27
28
20 points
20 points
20 points
IV
20 points
25 points
29
30
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. 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. null hypothesis2
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. 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
21. Suppose we want to test the effect of the Romney “Only-True” stimulus package (“give all
the money to Mitt, and he will spur GNP”) where the null hypothesis is that the Romney
Multiplier is one or greater 𝐻0 : 𝛽1 ≥ 1 vs. the alternative hypothesis that the Romney multiplier
is 0 (for each Mitt Buck Spent, GDP doesn’t change at all) 𝐻𝑎 : 𝛽1 = 0 and we know (the
standard error angel has visited us) that S.E.  j  .3 in either case.
𝐺𝑁𝑃𝑖 = 𝛽0 + 0.5 (𝑀𝑖𝑡𝑡 𝐵𝑢𝑐𝑘𝑠)𝑖
(0.3)
a) Employ the usual 95% confidence interval against a type I error, constructing the critical
cutoff value for the appropriate one-tailed test assuming the null hypothesis (Ho) is true. What is
that critical cutoff value for the percent tail?
b) Can we reject the null hypothesis?
c) What is the size of the type II error given the critical cutoff value in (a), assuming that the
alternative hypothesis is true? (see the table at the end of the test for Z-scores)
4
22. Suppose that for the standard regression model, y  X   , we rescale both the independent
variables and the dependent variable by non-zero variables c0, c1, c2, …, ck by regressing c0yi on
a constant c1, c2x2i, c3x3i,…, ckxki (so there are k regressors, including an intercept which is
measured as c1 instead of 1). In other words, instead of the OLS estimator ˆ  ( X ' X ) 1 X ' Y ,
you do OLS on the transformed data ˆ *  ( X * ' X * ) 1 X * ' Y * , where the transformed data can be
described by:
c1 0 0 0 0 
c0 0 0 0 0 
0 c 0 0 0 
0 c 0 0 0
2
0




c I
*
X *  XC where C   0 0 c3 0 0  and Y  C0Y , C0   0 0 c0 0 0  = 0 .




 0 0 0 ... 0 
 0 0 0 ... 0 
 0 0 0 0 ck 
 0 0 0 0 c0 
th
Prove that the i beta between the transformed data, and the untransformed data, have the
c
following relationship: ˆi*  0 ˆi .
ci
5
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. “Let Xˆ  V (V 'V ) 1V ' X where V has the appropriate dimensions. Then Xˆ ' X  Xˆ ' Xˆ .”
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.”
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 following STATA output, indicate what sort of tests are made (2 tests are specifically
programmed) and what they indicate:
*.
.
.
.
.
.
narr86
pcnv
avgsen
tottime
ptime86
qemp86
# times arrested, 1986
proportion of prior convictions
avg sentence length, mos.
time in prison since 18 (mos.)
mos. in prison during 1986
# quarters employed, 1986;
. regress narr86 pcnv avgsen tottime ptime86 qemp86;
-----------------------------------------------------------------------------narr86 |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------pcnv | -.1512246
.040855
-3.70
0.000
-.2313346
-.0711145
avgsen | -.0070487
.0124122
-0.57
0.570
-.031387
.0172897
tottime |
.0120953
.0095768
1.26
0.207
-.0066833
.030874
ptime86 | -.0392585
.0089166
-4.40
0.000
-.0567425
-.0217745
qemp86 | -.1030909
.0103972
-9.92
0.000
-.1234782
-.0827037
_cons |
.7060607
.0331524
21.30
0.000
.6410542
.7710671
-----------------------------------------------------------------------------. test (avgsen=0) (tottime=0); **FIRST TEST TO BE EXPLAINED********;
F( 2, 2719) =
2.03
Prob > F =
0.1310
. regress narr86 pcnv ptime86 qemp86;
-----------------------------------------------------------------------------narr86 |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------pcnv | -.1499274
.0408653
-3.67
0.000
-.2300576
-.0697973
ptime86 | -.0344199
.008591
-4.01
0.000
-.0512655
-.0175744
qemp86 |
-.104113
.0103877
-10.02
0.000
-.1244816
-.0837445
_cons |
.7117715
.0330066
21.56
0.000
.647051
.776492
-----------------------------------------------------------------------------. predict resids, residuals;
. regress resids pcnv avgsen tottime ptime86 qemp86;
-----------------------------------------------------------------------------resids |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------pcnv | -.0012971
.040855
-0.03
0.975
-.0814072
.0788129
avgsen | -.0070487
.0124122
-0.57
0.570
-.031387
.0172897
tottime |
.0120953
.0095768
1.26
0.207
-.0066833
.030874
ptime86 | -.0048386
.0089166
-0.54
0.587
-.0223226
.0126454
qemp86 |
.0010221
.0103972
0.10
0.922
-.0193652
.0214093
_cons | -.0057108
.0331524
-0.17
0.863
-.0707173
.0592956
-----------------------------------------------------------------------------************ SECOND TEST TO BE EXPLAINED **************;
. gen lm=e(N)*e(r2); . gen test=chi2tail(2,lm);. sum lm test;
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------lm |
2725
4.070729
0
4.070729
4.070729
test |
2725
.1306328
0
.1306328
.1306328
7
27. Data from GPA2.RAW generated the regression below, where
sat=combined SAT score
hsize=size of the individual’s high school graduating class (in hundreds)
hsizesq=square of hsize
female=1 if female, 0 if male
black=1 if black, 0 in non-black
R-SQUARE =
VARIABLE
NAME
HSIZE
HSIZESQ
FEMALE
BLACK
CONSTANT
0.0832
R-SQUARE ADJUSTED =
ESTIMATED STANDARD
COEFFICIENT
ERROR
19.115
3.837
-2.1894
0.5278
-41.608
4.175
-139.29
9.097
1027.0
6.290
T-RATIO
4132 DF
4.982
-4.148
-9.967
-15.31
163.3
0.0823
PARTIAL STANDARDIZED ELASTICITY
P-VALUE CORR. COEFFICIENT AT MEANS
0.000 0.077
0.2381
0.0519
0.000-0.064
-0.1983
-0.0231
0.000-0.153
-0.1485
-0.0182
0.000-0.232
-0.2285
-0.0075
0.000 0.930
0.0000
0.9968
a). Should hsizesq be in the regression? In terms of optimal high school size, what does it
imply?
b). What do the estimated coefficients on the female and black dummy variables indicate (both in
magnitude and statistical importance)?
c). Is this one of those regressions where I should worry about interpreting the constant term?
Why or why not?
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. Derive V(𝛽̂ ), i.e., the variance-covariance matrix for 𝛽̂ = (𝑋 ′ 𝑋)−1 𝑋 ′ 𝑌, given the usual
model assumptions.
2
30. Under the model assumptions, prove that s2 is an unbiased estimator of  for the OLS
regression model, using all the necessary assumptions employed in the proof in class or in the
book, and proving it in the general case using matrix algebra.
10