Solutions_Midterm1-prep_306
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Pennsylvania State University *
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306
Subject
Economics
Date
Apr 29, 2024
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Pages
4
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Name:
Email ID:
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ECON306, Fall 2022
Solutions: Study Material for Midterm #1
1
Multiple Choice
1) If you find that your regression results are ˆ
y
i
= 15
.
32 + 0
.
00
x
i
then
a)
R
2
= 0
X
b) 0
< R
2
<
1
c)
R
2
=
¯
Y
d)
R
2
>
SSR
TSS
2) The slope estimator,
ˆ
β
1
, has a smaller standard error, all else equal, If
a) the sample size is larger
X
b) the intercept,
ˆ
β
0
is smaller in absolute value
c) there is less variation in the explanatory variable, X
d) there is a large variance of the error term, ˆ
u
i
3) If you’re trying to predict the price of a car based on its weight, then a positive value of the error
term means that
a) the weight is higher than predicted
b) the car is more expensive than predicted
X
c) the
R
2
is small
d) the slope of the regression line is positive
4) If you have GDP information for every country in 2021, this is an example of
a) experimental data
b) cross-sectional data
X
c) time series data
d) panel data
Solutions: ECON 306, Study Material for Midterm #1
Page 2 of 4
2
Short Answer
11. What are the 3 Gauss-Markov conditions and how do they relate to the assumptions of OLS?
You may write them “in math” or “in English.”
The Gauss-Markov conditions are:
GM1)
E
(
u
i
|
X
1
, ..., X
n
) = 0:
u
i
has a conditional mean of 0
GM2)
var
(
u
i
|
X
1
, ..., X
n
) =
σ
2
u
,
0
< σ
2
u
<
∞
:
u
i
has a constant variance (homoskedastic)
GM3)
E
(
u
i
u
j
|
X
1
, ..., X
n
) = 0
, i
6
=
j
. the errors are uncorrelated for different observations
If the OLS conditions hold, then the additional assumption of homoskedasticity will allow the
G-M conditions to hold.
In the regression below, sleep represents the number of minutes of sleep at night per week and
age is measured in years. Use the Stata output provided to answer the following questions
. summarize age sleep
Variable |
Obs
Mean
Std. dev.
Min
Max
-------------+---------------------------------------------------------
age |
706
38.81586
11.34264
23
65
sleep |
706
3266.356
444.4134
755
4695
. regress sleep age
Source |
SS
df
MS
Number of obs
=
706
-------------+----------------------------------
F(1, 704)
=
5.80
Model |
1137207.85
1
1137207.85
Prob > F
=
0.0163
Residual |
138102628
704
196168.506
R-squared
=
0.0082
-------------+----------------------------------
Adj R-squared
=
0.0068
Total |
139239836
705
197503.313
Root MSE
=
442.91
------------------------------------------------------------------------------
sleep | Coefficient
Std. err.
t
P>|t|
[95% conf. interval]
-------------+----------------------------------------------------------------
age |
3.540881
1.470639
2.41
0.016
.6535177
6.428244
_cons |
3128.913
59.46811
52.61
0.000
3012.157
3245.669
------------------------------------------------------------------------------
12. Interpret the intercept in the context of this regression. Explain why the interpretation is or is
not meaningful.
You would get 3128.913 minutes of sleep per week if you are 0 years old. This is not helpful
because newborn babies have way different sleep patterns. Notice that the minimum age in the
data set is 23 years.
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49
236
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255
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Ý
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Focus
88
B
E
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(0-1)
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(b) Choose the right option.
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E(x – x)2 = 236
E(y - y)2 = 1,978
= -153
%3D
%3D
Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer.
a
-0.648
b
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0.265
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3
7
12
14
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=
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%
n
0
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=
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In
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In
is
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the value of the investment will reach
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DEPENDENT VARIABLE: Y
OBSERVATIONS: 15
VARIABLE
INTERCEPT
Multiple Choice
O
X
R-SQUARE
0.6010
PARAMETER
ESTIMATE
412.18
0.6358
F-RATIO
19.58
STANDARD
ERROR
102.54
0.1765
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0.0001
T-RATIO P-VALUE
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0.0032
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4.02
3.60
X increases by 0.1765 units when Yincreases by one unit.
X increases by 0.6358 units when Y increases by one unit.
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Y increases by 0.6358 units when X increases by one unit.
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Y
60
3.1
61
3.6
62
3.8
63
4
65
4.1
To Find,
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The numbers in parentheses below each estimated coefficient are the estimated standard errors. A detailed description of the variables used in the data set is available here
.
Suppose you wanted to test the hypothesis that BDR equals zero. That is,
H0: BDR=0 vs H1: BDR≠0
Report the t-statistic for this test. The t-statistic is ________
(Round your response to three decimal places)
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1. Suppose output (Q) is related to labor (L) and capital (K) in the following nonlinear way:
Q = albKc
When taking log to this equation, it is transformed into a linear
LnQ = Ina + b In(L) + c Ln (K)
One hundred twenty-three observations are used to obtain the following regression results:
Dependant
Variable:
Observations:
Variable
Intercept
L
K
Q
123
5.5215
Parameter Standard
Estimate
error
0.650
R-square
0.350
0.7547
0.9750
0.2950
0.1450
F-ratio
184.56
t-ratio
5.66
2.20
2.41
p-value on F
0.00001
p-value
0.0001
0.0295
0.0173
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True or False
For a linear regression model including only an intercept, the OLS estimator of that intercept is equal to the sample mean of the independent variable.
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You are the manager of a firm that produces a vegetable cooking oil in Ghana. In order to make
informed decision, you engaged an economist to estimate the demand equation for your
product. Using data from 25 supermarkets around the country for the month of February, 2021,
the estimated linear regression result for your product is shown in the table below:
Variable
Constant
Parameter Estimates
Standard error
-164.0
20.24
Price of vegetable cooking oil (P,)
Price of palm oil (P,)
Per capita Income ()
-3.50
1.55
2,50
0.28
0.45
0.52
R-squared
0.8672
Adjusted R-squared
0.8132
F-statistic
15.6893
a) Suppose the average price of 3 gallons of vegetable cooking oil is GH¢12, price of 2
gallons of palm oil is GH¢60, the per capita income of Ghana is GH¢420.
i. Write down the estimated demand equation for your firm's product and interpret
the parameter estimates.
ii. Detemine the quantity of vegetable cooking oil sold.
Estimate the own price elasticity of demand and state the type of demand
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2. Consider the following estimated regression equation (standard errors in parentheses):
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(0.05) (1.00)
R² = 0.5
i.
ii.
iii.
where
A
Yi = the corn yield (bushels/ha) in year t
Ft = fertilizer intensity (pounds/ha) in year t
Rt = rainfall (inches) in year t
Interpret the meaning of the intercept.
Suppose you are told that the true value of BF (coefficient on fertilizer intensity) is
known to be 0.20. Does this show that the estimate is biased? Why or why not?
Suppose you were told that the equation does not meet all the classical assumptions
and, therefore, the OLS estimator used is not BLUE. Does this mean that the true BR
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1. Suppose that you have following data:
Variable
Description
CEO salary measured in thousands of $
Firm's sale measure in millions ofS
Return on equity in percent
Salary
sales
roe
*Return on equity is a measure of financial performance calculated by dividing net
income by shareholders' equity.
Your estimated regression is given by
log (salary) = 4.322 + 0.276 log(sale) + 0.0215roe - 0.0008roe?, R = 282, n = 209.
(324) (0.033)
(0.0129)
(0.00026)
a) Is the effect of all independent variables statistically equal to 0?
b) Interpret the coefficient on log(sale).
c) Interpret the effect of roe on log(salary).
• Without more information, your interpretation of the effect of roe on
log(salary) should include answers to these sub-question.
Should the roe be included in this model?
il.
Comment on relationship between roe and log(salary): is it U-shaped or inverse
U-shaped?
What is the turning point? How would you interpret this point?
Plot log(salary) vs roe.
v.
ii.
iv.
Compute predicted value…
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Suppose you are the manager of a firm that produces good X in Ghana. In order to make informed
decision, you engaged an economist to estimate the demand equation for your product. Using data
from 30 supermarkets around the country for the month of April, 2021, the estimated linear
regression result for your product is shown in the table below:
Variable
Parameter Estimates
Standard error
Constant
-164.0
20.24
Price of good X (P,)
Price of good Y (P,)
-3.50
1.55
2.50
0.28
Per capita Income (/)
0.45
0.52
R-squared
0.8672
Adjusted R-squared
0.8132
F-statistic
15.6893
a) Suppose the average price of 3 units of good X is GH¢12, price of 2 units of goodY is
GH¢60, the per capita income of Ghana is GH¢420.
1.
Write down the estimated demand equation for your firm's product and interpret
the parameter estimates.
Determine the quantity of good X sold.
Estimate the own price elasticity of demand and state the type of demand curve
11.
111.
your firm has?
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- Regression analysis was applied between $ sales (y) and $ advertising (r) across all the branches of a major international corporation. The following regression function was obtained. ŷ = 5000 + 7.25r (a) Predict the amount for sales where the advertising amount is $ 1,000,000.00. (b) If the advertising budgets of two branches of the corporation differ by $30,000, then what will be the predicted difference in their sales?arrow_forward1. For a regression model y = XB + u where u is N(0, o?1), y is nx1 matrix, X is nxp matrix, B is px1 matrix and u is nx1 matrix, a. derive the estimators B using the method of least squaresarrow_forward11. Which of the following statements is not true about multicollinearity? (a) Perfect multicollinearity will prevent you from being able to estimate a linear regression model. (b) Imperfect mulitcollinearity affects the individual t-statistics of the regressors. (c) Multicollinearity is defined as a linear relationship between different independent variables. (d) Imperfect multicollinearity affects model validity of the model. (e) The least squares estimators are unbiased in the presence of imperfect multicollinearity.arrow_forward
- Consider the output here from a regression in R. What is 3₂? Coefficients: Estimate (Intercept) 1.708 5.404 -1.478 9.531 X1 X2 X3 Std. Error 0.555 2.792 0.6 2.758arrow_forwardDetermine the mode choice (personal vehicle or bus system) for the following regression model: Utility Function: Umode – (8.333 x 10-4)*(Access time in sec) – (6.667 x 10-4)*(Wait Time in sec) – (5.00 x 10-4)*(Riding time in sec) – (1.40)*(Cost, $) PARAMETER PERSONAL VEHICLE CITY BUS SYSTEM MODE CONSTANT -0.01 -0.07 ACCESS TIME (SECS) 300 600 WAITING TIME (SECS) 0 900 RIDING TIME (SECS) 1,500 6,000 COST (DOLLARS) $1.50 $1.00arrow_forward2. Which of the following types of regressions will always have a binary outcome variable? (A) Probit (B) Difference-in-differences (C) Regression discontinuity (D) (A) and (B) will both have binary outcome variablesarrow_forward
- 1. If in a simple linear regression, SST = 315 and the sample correlation coefficient between your dependent and independent variable is 0.96, then the value of SSE is equal to? a. 24.696 b. 290.304 c. 302.4 d. 12.6 e. 0.9216arrow_forwardA linear regression model for the revenue data for a company is R=27.1t+203 where R is total annual revenue and t is time since 1/31/02 in years. 12 months 12 months 12 months Billions of Dollars Revenue Gross Profit 12 months 12 months ending 1/31/02ending 1/31/03ending 1/31/04 ending 1/31/05 ending 1/31/06 500- 201 49 236 54 255 60 500- 277 65 (A) Draw a scatter plot of the data and a graph of the model on the same axes. OA. B. O.C. KICB Q 2 316 72 500- oo D. 500- Q Garrow_forwardWater is being poured into a large, cone-shaped cistern. The volume of water, measured in cm³, is reported at different time intervals, measured in seconds. A regression analysis was completed and is displayed in the computer output. Regression Analysis: cuberoot (Volume) versus Time Predictor Coef SE Coef Constant -0.006 0.00017 -35.294 0.000 Time 0.640 0.000018 35512.6 0.000 s=0.030 R-Sq=1.000 R-sq (adj)=1.000 What is the equation of the least-squares regression line? Volume = 0.640 - 0.006(Time) Volume = 0.640 - 0.006(Time) Volume = -0.006 + 0.640(Time) Volume = - 0.006 + 0.640(Time?)arrow_forward
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