1.. (8 points) Suppose that instead of fitting the regression model Y X i ii =+ + aß µ , you instead fit the model Y Z i ii =+ + ?d e where 10 2 Z X i i = + . How are the least-squares estimators ?ˆ and ˆd for this second model related to the corresponding LS estimators aˆ and ß ˆ of the linear model? (In other words, obtain ?ˆ and ˆd as a function of aˆ and ß ˆ ). 2. (12 points) Using a random sample of 670 individuals for the population of people in the workforce in 1976, we want to estimate the impact of education on wages. Let wage denote hourly wage in 1976 U.S. dollars and let educ denote years of schooling. We obtain the following OLS regression line: wage = -0.54 + 0.54educ. How do you interpret the slope of this regression line? What is the expected difference in the hourly wage between a worker that finished four years of college and a worker with finished high school? What is the predicted wage for a person with one year of education? Does that make sense? If it is not, what is the name of this problem in econometrics? How do we deal with it? 3. (30 points) Suppose you are interested in the effect of skipping classes on college GPA, and collect a sample of economic variables from 400 college students to analyze the problem. Included in your data are college GPA on a four-point scale (COLGPA), high school GPA on a four-point scale (HSGPA), achievement test score (ATS), and the average number of Economics 122B lectures missed per week (SKIP). Running a regression of the dependent variable COLGPA on the other explanatory variables including a constant (and homoskedastic errors) yields: Economics 122B Midterm 1 October 27, 209 Page 2 of 7 A. (5 points) Determine which of the coefficient estimates are (individually) significantly different from zero at a 5% level. B. (8 points) Test at a 5% level the null hypothesis H0: ß3 = -0.2 against the alternative H1: ß3< -0.1, where ß3 is the (true) coefficient on SKIP. C. (9 points) Test the null hypothesis that all coefficients except the constant are zero at a 5% level. You can use the following table of the F distribution with large (>50) second degree of freedom. (First) Degree of Freedom Significance level 10% 5% 1% 3 2.08 2.60 3.78 4 1.94 2.37 3.32 5 1.85 2.21 3.02 D. (8 points) Suppose the Economics Department was contemplating a policy to make attendance at lectures mandatory (perhaps by putting a high penalty on skipping classes). Can you use the regression results above to predict the impact on students’ college GPA?
This is a closed book exam, except that you can have one 2-sided sheet of notes. Please
be sure to put your name and UCI ID# in the upper right corner of each sheet you
turn in, and don’t staple your sheets together since we will be scanning them.
Please completely fill out the Rapid Return Cover sheet – including your assigned
seat number. Do not leave the exam room until you have turned in your exam and
your name has been checked off the class roster by the instructor or TA. Do not
write your answers on this exam –put them on separate sheets of paper. Please use
black or dark blue ink or pencil. Be sure to show your work for partial credit. There
are 100 points possible, and the points for each sub-question are in parenthesis at the
beginning of the question.
PART 1 (50 points)
1.. (8 points) Suppose that instead of fitting the regression model Y X i ii =+ + αβ µ , you
instead fit the model Y Z i ii =+ + γδ ε where 10 2 Z X i i = + . How are the least-squares
estimators γˆ and ˆδ for this second model related to the corresponding LS estimators
αˆ and β
ˆ of the linear model? (In other words, obtain γˆ and ˆδ as a function of αˆ
and β
ˆ ).
2. (12 points) Using a random sample of 670 individuals for the population of people in
the workforce in 1976, we want to estimate the impact of education on wages. Let wage
denote hourly wage in 1976 U.S. dollars and let educ denote years of schooling. We
obtain the following OLS regression line: wage = -0.54 + 0.54educ. How do you interpret
the slope of this regression line? What is the expected difference in the hourly wage
between a worker that finished four years of college and a worker with finished high
school? What is the predicted wage for a person with one year of education? Does that
make sense? If it is not, what is the name of this problem in econometrics? How do we
deal with it?
3. (30 points) Suppose you are interested in the effect of skipping classes on college GPA,
and collect a sample of economic variables from 400 college students to analyze the
problem. Included in your data are college GPA on a four-point scale (COLGPA), high
school GPA on a four-point scale (HSGPA), achievement test score (ATS), and the
average number of Economics 122B lectures missed per week (SKIP). Running a
regression of the dependent variable COLGPA on the other explanatory variables
including a constant (and homoskedastic errors) yields:
Economics 122B Midterm 1 October 27, 209
Page 2 of 7
A. (5 points) Determine which of the coefficient estimates are (individually)
significantly different from zero at a 5% level.
B. (8 points) Test at a 5% level the null hypothesis H0: β3 = -0.2 against the
alternative H1: β3< -0.1, where β3 is the (true) coefficient on SKIP.
C. (9 points) Test the null hypothesis that all coefficients except the constant are
zero at a 5% level. You can use the following table of the F distribution with
large (>50) second degree of freedom.
(First) Degree of
Freedom
Significance level
10% 5% 1%
3 2.08 2.60 3.78
4 1.94 2.37 3.32
5 1.85 2.21 3.02
D. (8 points) Suppose the Economics Department was contemplating a policy to
make attendance at lectures mandatory (perhaps by putting a high penalty on
skipping classes). Can you use the regression results above to predict the
impact on students’ college GPA?
Variable Coefficients Std. Error t-Statistic
HSGPA 0.41 0.09 4.38
ATS 0.02 0.01 1.36
SKIP -0.08 0.03 3.19
Constant 1.39 0.33 4.21
R-squared 0.234
F-statistic 4
Economics 122B Midterm 1 October 27, 209
Page 3 of 7
PART 2 (50 points)
The dataset used in this question contains data on 180 economics journals for the year
2000. The variable descriptions are as follows:
logoclc – log of the number of library subscription
loglibcit – log of the library subscription price per citation.
logage – log of the age of the journal in years (current year at the time of study-year in
which journal was founded).
logchars – log of the total number of characters.
Answer the following questions based on this dataset:
A. (10 points) A researcher believes that effect of a one percent increase in journal
age on the dependant variable is twice the effect of a one percent increase in the
total number of characters. If the value of loglibcit = 2, do you agree with him?
(Hint: Use the following regression and the following covariance matrix to
answer the question and α = 0.05). Make sure to state clearly the null and
alternative hypothesis being tested and what do you conclude from the test (not
only if you reject or fail to reject the null but also a one line summary of the result
obtained from the test).
Dependent Variable: LOGOCLC
Method: Least Squares
Included observations: 180
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
LOGLIBCIT -0.940186 0.162515 -5.785237 0.0000
LOGLIBCIT^2 0.009160 0.019204 0.476996 0.6340
LOGAGE 0.373395 0.117644 3.173945 0.0018
LOGAGE*LOGLIBCIT 0.155586 0.051957 2.994545 0.0031
LOGCHARS 0.228874 0.096737 2.365954 0.0191
C 3.420021 0.373764 9.150211 0.0000
R-squared 0.634339 Mean dependent var 4.740388
Adjusted R-squared 0.623831 S.D. dependent var 1.123617
0
Variance-covariance matrix for the regression given in part “a”
LOGLIBCIT LOGLIBCIT^2 LOGAGE
LOGAGE*LOG
LIBCIT LOGCHARS C
LOGLIBCIT 0.026411 -0.001410 -0.001169 -0.008070 0.000630 0.002743
LOGLIBCIT^2 -0.001410 0.000369 0.000344 0.000624 0.000136 -0.001801
LOGAGE -0.001169 0.000344 0.013840 0.001376 -0.001788 -0.042005
LOGAGE*LOGLIBCIT -0.008070 0.000624 0.001376 0.002699 0.000131 -0.004567
LOGCHARS 0.000630 0.000136 -0.001788 0.000131 0.009358 -0.003394
C 0.002743 -0.001801 -0.042005 -0.004567 -0.003394 0.139700
Economics 122B Midterm 1 October 27, 209
Page 4 of 7
B. (12 points) What is the effect of a percent increase in variable loglibcit on the
predicted value of the dependant variable? Also give a 90% confidence band for this
change in predicted value of the dependant variable. Use the regression output in part “a”
and covariance matrix in part “b” of this question. Evaluate the expression at the
following initial values: loglibcit=1 and logage=3. You can either calculate the
confidence band, or give the formula and he EVIEWS command to calculate the relevant
standard error.
C. (18 points) For each of the following three “wald” test outputs, state clearly the null
and alternative hypothesis being tested and what do you conclude from the test (not only
if you reject or fail to reject the null but also a one line summary of the result obtained
from the test).
Output 1:
Wald Test:
Equation: EQ02
Test Statistic Value df Probability
F-statistic 47.23754 (3, 174) 0.0000
Chi-square 141.7126 3 0.0000
Null Hypothesis Summary:
Normalized Restriction (= 0) Value Std. Err.
C(1) -0.940186 0.162515
C(2) 0.009160 0.019204
C(4) 0.155586 0.051957
Output 2:
Wald Test:
Equation: EQ02
Test Statistic Value df Probability
F-statistic 7.774586 (2, 174) 0.0006
Chi-square 15.54917 2 0.0004
Null Hypothesis Summary:
Normalized Restriction (= 0) Value Std. Err.
C(3) 0.373395 0.117644
C(4) 0.155586 0.051957
Output 3:
Wald Test:
Equation: EQ02
Economics 122B Midterm 1 October 27, 209
Page 5 of 7
Test Statistic Value df Probability
F-statistic 6.078750 (2, 174) 0.0028
Chi-square 12.15750 2 0.0023
Null Hypothesis Summary:
Normalized Restriction (= 0) Value Std. Err.
C(2) 0.009160 0.019204
C(4) 0.155586 0.051957
D. (10 points) Using the following regression output and the “wald” test result, name and
explain in short the test being conducted. Using this test, what do you conclude about the
internal validity of the model given in part “a” of this question? (Note: LOGOCLCHAT
is the predicted value of the dependant variable obtained using the model given in part
“a” of this question)
Dependent Variable: LOGOCLC
Method: Least Squares
Included observations: 180
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable Coefficient Std. Error t-Statistic Prob.
LOGLIBCIT 0.127024 2.570019 0.049425 0.9606
LOGLIBCIT^2 0.067784 0.045726 1.482390 0.1401
LOGAGE -0.093627 1.060707 -0.088268 0.9298
LOGAGE*LOGLIBCIT -0.047224 0.422661 -0.111731 0.9112
LOGCHARS -0.002401 0.639098 -0.003757 0.9970
LOGOCLCHAT^2 0.496093 0.677863 0.731849 0.4653
LOGOCLCHAT^3 -0.054289 0.055873 -0.971655 0.3326
C -0.291233 5.786715 -0.050328 0.9599
R-squared 0.640316 Mean dependent var 4.740388
Adjusted R-squared 0.625677 S.D. dependent var 1.123617
Wald Test:
Equation: EQ02
Test Statistic Value df Probability
F-statistic 1.059666 (2, 172) 0.3488
Chi-square 2.119331 2 0.3466
Null Hypothesis Summary:
Normalized Restriction (= 0) Value Std. Err.
C(6) 0.496093 0.677863
C(7) -0.054289 0.055873
Economics 122B Midterm 1 October 27, 209
Page 6 of 7
2. (27 points) Consider three stocks A, B and C costing $100 each. The annual returns on
the three stocks have mean $5 and variance $10.
a. (9 points) Suppose that the returns on the three stocks are i.i.d. Find the means
and variance of the returns on Portfolio I, consisting of 3 units of A, and Portfolio
II, consisting of 1 units each of A, B and C?
b. (9 points) Suppose the returns from A and B have a correlation coefficient of -0.8
but they are uncorrelated with returns from C. Find the means and variances of
the returns on the two portfolios.
c. (9 points) Suppose the returns from A, B, and C are perfectly correlated (each pair
have a correlation =1). Find the means and variances of the returns on the two
portfolios. Is there any benefit to diversification in this case?
3 (28 points). The following table shows the results of fitting a linear regression model
of starting annual salaries on a constant, GPA (4 point scale), and a variable (Metrics =1)
indicating whether the recent economics graduate took an econometrics course for a
random sample of 50 recent economics graduates from a large state university. Note
that econometrics was not required at this university.
Dependent Variable: SALARY
Method: Least Squares
Date: 02/04/07 Time: 14:21
Sample: 1 50
Included observations: 50
Variable Coefficient Std. Error t-Statistic Prob.
C 24199.65 1078.423 22.43985 0.0000
GPA 1643.267 352.2898 4.664531 0.0000
METRICS 5033.085 456.3073 11.03003 0.0000
R-squared 0.736879 Mean dependent var 30430.92
Adjusted R-squared 0.725682 S.D. dependent var 2739.002
S.E. of regression 1434.561 Akaike info criterion 17.43323
Sum squared resid 96724389 Schwarz criterion 17.54795
Log likelihood -432.8308 F-statistic 65.81256
Durbin-Watson stat 1.798742 Prob(F-statistic) 0.000000
a. (9 points) Can these results be used to predict what would happen if this university
made econometrics a required course for economics students? (hint: consider the
possibility of sample selection bias).
Economics 122B Midterm 1 October 27, 209
Page 7 of 7
b. (10 points) Suppose that econometrics is a very hard class, and the instructor is a very
hard grader. What is the predicted change in starting salary for a student who has just
taken econometrics, and this resulted in decreasing his GPA by .5 points. Give a 90%
confidence interval for this prediction. The covariance matrix of the estimated
parameters are:
C GPA METRICS
C 1162997. -370462.7 -124113.6
GPA -370462.7 124108.1 22428.11
METRICS -124113.6 22428.11 208216.3
c. (9 points) Note that the standard errors in the above regression are calculated assuming
homoskedasticity. Explain why we typically use heteroskedastic-consistent standard
errors.
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