Johns Hopkins University Economics Exam Practice
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Econometrics, Spring 2021
Instructor: U Vasavada
Exam 1
PART ONE: MULTIPLE CHOICE (20 points)
1. The probability density function for a discrete random variable X
a. Is usually normal
b. Is usually uniform
c. Specifies the probability of observing each possible value of X
d. Is a continuous function
2. The expected value of a random variable is
a. The sample mean of the observed data in a given sample
b. The average we would expect to see if we could observe an infinite amount of data
for that variable
c. The weighted sum of all possible values, weighted by the probability of observing
them
d. Both (b) and (c)
e. All of the above
3. The standard deviation of a variable is
a. The same as its variance
b. The square of its variance
c. The square root of its variance
d. Unrelated to its variance
4. Consider the following joint probability density function for X = whether it is
raining, and Y = whether there is a traffic jam on Massachusetts Ave. What is the
conditional probability of getting in a traffic jam, given that it is raining?
a. 5%
No Traffic Jam Traffic Jam
b. 15%
No Rain
0.40
0.40
c. 33%
Rain
0.05
0.15
d. 75%
5. If
a.
b.
c.
d.
X and Y are independent random variables then
They have zero covariance
They have a correlation coefficient of 1
They are linearly related
None of the above
6. Which of the following is NOT one of the basic assumptions of the simple linear model:
a. Cov(X,Y)=0
b. E(u|X)=0
c. The ui are independently and identically distributed
d. We have a random sample: {(Yi, Xi) i=1…n}
7. The vertical distance between the true population regression line and a given data
point is called
a. The error term
b. The residual
c. The standard error
d. The standard deviation
8. The difference between the observed value of y and the predicted or fitted value for
that point is called
a. The error term
b. The residual
c. The standard error
d. The standard deviation
9. The estimated regression line
a. Is identical to the true regression line
b. Is upward sloping
c. Maximizes the sum of squared residuals
d. Passes through the point ( x , y )
10.
Suppose we estimate a regression of Y=family income against X=a binary variable
that takes the value of 1 if the family owns a house and 0 if not. Then the
coefficient on the X variable tells us:
a. The difference in homeownership rates between high and low income families
b. The difference between the average family incomes of homeowners and non-homeowners
c. The effect of an extra dollar of family income on the probability of owning a home
d. None of the above
11.
When we say the OLS estimator is unbiased, we mean that
a. It always gets the right answer
b. It is the most precise linear estimator
c. E( ?? 1)= ?1
d. It is normally distributed in large samples
12.
a.
b.
c.
d.
Heteroskedasticity implies that:
The error terms are correlated with X
The residuals do not have mean zero
The OLS estimator will be biased
The variances of the error terms are not constant across observations
13.
Which of the following does NOT lead to smaller standard errors of our OLS
estimates?
a. Smaller variance of the error term
b. Smaller samples
c. Less correlation between different X-variables
d. X-variables are widely spread out around their means
14.
The difference between ? and
??
a. ? is an estimate of the unknown
b.
??
is
??
is an estimate of the unknown ?
c. the expected value of
d. no difference
??
is always ?
15.
What proportion of the area under the standard normal curve lies in the right-hand
tail above the value of z=1.96?
a. 10%
b. 5%
c. 2.5%
d. 1%
16.
a.
b.
c.
d.
Loosely speaking, statistical significance corresponds to
Large ts and ps
Small ts and ps
Small ts, large ps
Large ts, small ps
17.
a.
b.
c.
d.
The
The
The
The
The
R2 of a simple regression of y on x is equal to
sample correlation coefficient between x and y
square of the sample correlation coefficient between x and y
1-SSE/SST
sum of squared residuals divided by the total sum of squares
18.
When will the omission
associated with X1?
a. If X2 has no effect on
b. If X2 has no effect on
c. If X2 has an effect on
d. If X2 has an effect on
19.
a.
b.
c.
d.
of X2 lead to bias in the estimates of the coefficient
Y
Y
Y
Y
and
and
and
and
is
is
is
is
uncorrelated with X1
correlated with X1
uncorrelated with X1
correlated with X1
Which of the following is true?
An equation with a higher R2 explains a greater proportion of the variance of Y
An equation with a higher R2 is always preferred to one with a lower R2
A high value of R2 means you have eliminated omitted variables bias
R2 = SSTx/SST
20. If a two-sided t-test results in a p-value of 0.06, then the one-sided test will
result in a p-value of:
a. 0.05
b. 0.12
c. 0.03
d. not enough information to answer question
PART TWO: SHORT ANSWERS AND CALCULATIONS
1) Consider the following dataset with 4 observations for X and Y. Answer the questions
below. Use the blank rows and columns in the table for the required intermediate
calculations and label your added rows & columns so I know what you are doing. (5 points)
X
Y
1
15
5
13
8
5
10
3
a) Write the formula for the mean of X.
Calculate MEANS OF X AND Y.
b) Write the formula for the the variance of X.
THE SAMPLE VARIANCE OF Y.
Calculate the SAMPLE VARIANCE OF X AND
c) Write the formula for the SAMPLE COVARIANCE of X and Y and calculate it.
d) Write the formula for the slope of the regression of Y against X and calculate it.
e) Find the intercept of this regression.
2) Consider the following regression predicting birth weight using the variables
listed: (10 points)
variable name
variable label
bwght
birth weight, ounces
cigs
cigs smked per day while preg
faminc
1988 family income, $1000s
motheduc
mother’s yrs of educ
male
=1 if male child
Number of obs =
674
SS
df
MS
Source |
Model | 12066.6671
4 3016.66677
Residual | 305549.726
669 456.726048
————-+—————————–Total | 317616.393
673 471.941149
—————————————————————————–bwght |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
————-+—————————————————————cigs | -.4370689
.1358532
faminc |
.0812292
.050534
motheduc |
.3334542
.3955639
male |
4.538161
1.661007
_cons |
110.1121
4.849262
a) Explain carefully what the intercept and each of four slope parameter estimates means,
including all relevant units.
b) Write the formula for R2 and calculate it for this regression. Explain what R2 tells
us.
c) Write the formula for a 95% confidence interval and calculate it for male only.
d) Write the formula for a t-statistic that tests the null hypothesis Ho:?j=0. Calculate
the t-statistics for cigs, faminc, motheduc, and male. Which of these are
statistically significant at the 1% level (two-tailed), and how do you know?
3) Consider the following formula:
Var ( ? j ) =
?2
SST j (1 ? R 2j )
(5 points)
a) Explain what each term: Var(? j ) , ? 2 , SST j and R j 2means; formulas are optional if your
explanation is good.
b) What is the difference (in words, not equations) between ? 2 and ? 2
c) Suppose the variable xj is highly correlated with many of the other variables in the
regression. Which term on the right hand side of the equation will this affect, and
what will be the effect on the Var(? j ) ?
d)
4) Consider the following three regression results. (I did not print the intercepts)(10
points):
variable name
lwage
tenure
KWW
variable label
natural log of wage
years with current employer
knowledge of world work test score
(i) Regression of log of monthly wage earnings against years of job tenure and KWW test score
??????????????????????????????????????????????????????????????????????????????
lwage |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
?????????????+????????????????????????????????????????????????????????????????
4.67
0.000
.0069994
.0171404
.0025837
tenure |
.0120699
KWW |
.0157515
.0017166
9.18
0.000
.0123827
.0191204
??????????????????????????????????????????????????????????????????????????????
(ii) Regression of log of monthly wage earnings against years of jobe tenure only
??????????????????????????????????????????????????????????????????????????????
lwage |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
?????????????+????????????????????????????????????????????????????????????????
tenure |
.0154222
.0026693
5.78
0.000
.0101836
.0206608
??????????????????????????????????????????????????????????????????????????????
(iii) Regression of KWW score against years of job tenure
??????????????????????????????????????????????????????????????????????????????
Std. Err.
t
P>|t|
[95% Conf. Interval]
KWW |
Coef.
?????????????+????????????????????????????????????????????????????????????????
tenure |
.2128208
.0487803
4.36
0.000
.117089
.3085526
??????????????????????????????????????????????????????????????????????????????
a) Suppose regression (i) represents the TRUE POPULATION model.
of an extra year of job tenure on the wage?
What is the true effect
b) Explain carefully why the effect of job tenure appears to be larger in regression (ii)
than it is in regression (i). Which basic assumption of the OLS model is violated in
regression (ii) and why? (Tell me what that assumption says, not just its number!)
Which equation do you think is more likely to be correct?
c) Write the formula that relates the job tenure parameter estimate in regression (ii) to
the job tenure parameter in regression (i), and do the calculation to confirm that the
formula holds, up to rounding error.
5) Consider a regression of the log(infant mortality rate) in each state against the
variables listed (10 points):
Variable
Meaning
linfmort
Log(Infant mortality rate)
lpcinc
Log(Per capita income)
lphysic
Log(Doctors per 100,000 population}
lpopul
Log(population in 1000s)
DC
=1 for Washington DC
—————————————————————————–linfmort |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
————-+—————————————————————lpcinc | -.2472401
.0928288
-2.66
0.009
-.4314795
-.0630006
lphysic | -.1756327
.0809773
-2.17
0.033
-.3363502
-.0149152
lpopul |
.0628281
.0142942
4.40
0.000
.034458
.0911981
DC |
1.136104
.1305253
8.70
0.000
.8770479
1.395161
_cons |
5.047641
.704294
7.17
0.000
3.649812
6.445469
————-+—————————————————————Here is the variance covariance matrix from this regression:
e(V) |
lpcinc
lphysic
lpopul
DC
_cons
————-+———————————————————–lpcinc | .00861719
lphysic | -.00476487
.00655732
lpopul | .00008304
-.0004003
.00020432
DC | .00303542 -.00681149
.00076755
.01703684
_cons | -.05907059
.01496 -.00034546 -.00001664
.49602999
Calculate the t-statistic that you would use to test the hypothesis that the elasticity
of infant mortality with respect to per capita income is the same as its elasticity with
respect to the number of physicians per population. Interpret the results.
6) Use the following formula and the regression results below to test the joint null
hypothesis that neither mothers nor fathers education has an effect on wages (10
points):
What is the F-statistic, and what is the 1% critical value for that F-stat? What do you
conclude? How can your conclusion be correct, given that neither of the relevant
coefficients is significant at even 5% by itself?
Regression 1)
Source |
SS
df
MS
Number of obs =
722
————-+—————————–F( 8,
713) =
24.84
Model | 27.6422885
8 3.45528607
Prob > F
= 0.0000
Residual | 99.1696272
713 .139087836
R-squared
= 0.2180
————-+—————————–Adj R-squared = 0.2092
Total | 126.811916
721 .175883378
Root MSE
= .37294
—————————————————————————–lwage |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
————-+—————————————————————educ |
.0562629
.007967
7.06
0.000
.0406214
.0719045
exper |
.014224
.0044905
3.17
0.002
.0054077
.0230402
tenure |
.009331
.0029518
3.16
0.002
.0035356
.0151264
age |
.0096873
.0055133
1.76
0.079
-.001137
.0205117
south | -.0807515
.0305296
-2.65
0.008
-.1406902
-.0208128
urban |
.1672097
.0313994
5.33
0.000
.1055634
.228856
meduc |
.0108041
.0061195
1.77
0.078
-.0012103
.0228185
feduc |
.0086483
.0054328
1.59
0.112
-.0020179
.0193145
_cons |
5.184015
.1749834
29.63
0.000
4.84047
5.527559
—————————————————————————–Regression 2)
Source |
SS
df
MS
Number of obs =
722
————-+—————————–F( 6,
715) =
30.89
Model | 26.1070112
6 4.35116853
Prob > F
= 0.0000
Residual | 100.704905
715
.14084602
R-squared
= 0.2059
————-+—————————–Adj R-squared = 0.1992
Total | 126.811916
721 .175883378
Root MSE
= .37529
—————————————————————————–lwage |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
————-+—————————————————————educ |
.066289
.007407
8.95
0.000
.0517469
.0808312
exper |
.0139483
.0045166
3.09
0.002
.0050809
.0228158
tenure |
.0092854
.0029681
3.13
0.002
.0034582
.0151126
age |
.008745
.0055379
1.58
0.115
-.0021275
.0196175
south | -.0968346
.0303248
-3.19
0.001
-.156371
-.0372982
urban |
.17171
.0314958
5.45
0.000
.1098747
.2335453
_cons |
5.288836
.1730791
30.56
0.000
4.949032
5.62864
——————————————————————————
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Explanation & Answer:
2 Parts Exam Practice
Tags:
Economic
Econometrics
probability density function
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