Question 1

Sports economics typically looks at winning percentages of sports teams as one of
various outputs, and estimates production functions by analyzing the relationship
between the winning percentage and inputs.In Major League Baseball (MLB), the
determinants of winning are quality pitching and batting.All 30 MLB teams for the 1999
season.Pitching quality is approximated by "Team Earned Run Average" (ERA), and
hitting quality by "On Base Plus Slugging Percentage" (OPS). Summary of the Distribution of Winning Percentage, On Base Plus Slugging Percentage, and Team Earned Run Average for MLB in 1999

Your regression output is:
$$\begin{aligned}
\widehat { \text { Winpct } = } = & - 0.19 - 0.099 \times \text { teamera } + 1.490 \times \text { ops } , R ^ { 2 } = 0.92 , \text { SER } = 0.02 . \
& ( 0.08 ) ( 0.008 ) \quad\quad\quad\quad\quad\quad\quad ( 0.126 )
\end{aligned}$$ (a)Interpret the regression.Are the results statistically significant and important?

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The answer of Sports economics typically looks at winning percentages...

Question 2

Females, it is said, make 70 cents to the dollar in the United States.To investigate this
phenomenon, you collect data on weekly earnings from 1,744 individuals, 850 females
and 894 males.Next, you calculate their average weekly earnings and find that the
females in your sample earned $346.98, while the males made $517.70.
(a)Calculate the female earnings in percent of the male earnings.How would you test
whether or not this difference is statistically significant? Give two approaches.

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The answer of Females, it is said, make 70 cents...

Question 3

Assume that you had data for a cross-section of 100 households with data on
consumption and personal disposable income.If you fit a linear regression function
regressing consumption on disposable income, what prior expectations do you have about
the slope and the intercept? The slope of this regression function is called the "marginal
propensity to consume." If, instead, you fit a log-log model, then what is the
interpretation of the slope? Do you have any prior expectation about its size?

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The answer of Assume that you had data for a...

Question 4

The figure shows is a plot and a fitted linear regression line of the age-earnings profile of
1,744 individuals, taken from the Current Population Survey.  (a)Describe the problems in predicting earnings using the fitted line.What would the pattern
of the residuals look like for the age category under 40?

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The answer of The figure shows is a plot and...

Question 5

Many countries that experience hyperinflation do not have market-determined interest
rates.As a result, some authors have substituted future inflation rates into money demand
equations of the following type as a proxy: $$m = \beta _ { 0 } \times ( 1 + \Delta \ln P ) ^ { \beta _ { 1 } } \times e ^ { u }$$
( $m$ is real money, and $P$ is the consumer price index).
Income is typically omitted since movements in it are dwarfed by money growth and the inflation rate. Authors have then interpreted $\beta _ { 1 }$ as the "semi-elasticity" of the inflation rate. Do you see any problems with this interpretation?

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The answer of Many countries that experience hyperinflation do not...

Question 6

In nonlinear models, the expected change in the dependent variable for a change in one of the explanatory variables is given by a. $\quad \Delta Y = f \left( X _ { 1 } + \Delta X _ { 1 } , X _ { 2 } , \ldots , X _ { k } \right)$.
b. $\Delta Y = f \left( X _ { 1 } + \Delta X _ { 1 } , X _ { 2 } + \Delta X _ { 2 } , \ldots , X _ { k } + \Delta X _ { k } \right) - f \left( X _ { 1 } , X _ { 2 } , \ldots X _ { k } \right)$.
c. $\Delta Y = f \left( X _ { 1 } + \Delta X _ { 1 } , X _ { 2 } , \ldots , X _ { k } \right) - f \left( X _ { 1 } , X _ { 2 } , \ldots X _ { k } \right)$.
d. $\quad \Delta Y = f \left( X _ { 1 } + X _ { 1 } , X _ { 2 } , \ldots , X _ { k } \right) - f \left( X _ { 1 } , X _ { 2 } , \ldots X _ { k } \right)$

Accepted Answer

The answer of In nonlinear models, the expected change in...

Question 7

In the model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 } + \beta _ { 2 } X _ { 2 } + \beta _ { 3 } \left( X _ { 1 } \times X _ { 2 } \right) + u _ { i }$ the expected effect $\frac { \Delta Y } { \Delta X _ { 1 } }$ is

Accepted Answer

A)  $\beta _ { 1 } + \beta _ { 3 } X _ { 2 }$ 
B)  $\beta _ { 1 } .$ 
C)  $\beta _ { 1 } + \beta _ { 3 }$ 
D)  $\beta _ { 1 } + \beta _ { 3 } X _ { 1 }$

Question 8

(Requires Calculus)Show that for the log-log model the slope coefficient is the
elasticity.

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The answer of (Requires Calculus)Show that for the log-log model...

Question 9

Choose at least three different nonlinear functional forms of a single independent
variable and sketch the relationship between the dependent and independent variable.

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The answer of Choose at least three different nonlinear functional...

Question 10

The binary variable interaction regression

Accepted Answer

A) can only be applied when there are two binary variables, but not three or more. 
B) is the same as testing for differences in means. 
C) cannot be used with logarithmic regression functions because ln(0)is not defined. 
D) allows the effect of changing one of the binary independent variables to depend on the value of the other binary variable.

Question 11

The exponential function a. is the inverse of the natural logarithm function.
b. does not play an important role in modeling nonlinear regression functions in econometrics.
c. can be written as $\exp \left( e ^ { x } \right)$.
d. is $e ^ { x }$, where $e$ is $3.1415 \ldots$.

Accepted Answer

The answer of The exponential function a. is the inverse...

Question 12

To investigate whether or not there is discrimination against a sub-group of individuals,
you regress the log of earnings on determining variables, such as education, work
experience, etc., and a binary variable which takes on the value of one for individuals in
that sub-group and is zero otherwise.You consider two possible specifications.First you
run two separate regressions, one for the observations that include the sub-group and one
for the others.Second, you run a single regression, but allow for a binary variable to
appear in the regression.Your professor suggests that the second equation is better for the
task at hand, as long as you allow for a shift in both the intercept and the slopes.Explain
her reasoning.

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The answer of To investigate whether or not there is...

Question 13

In the log-log model, the slope coefficient indicates a. the effect that a unit change in $X$ has on $Y$.
b. the elasticity of $Y$ with respect to $X$.
c. $\Delta Y / \Delta X$.
d. $\frac { \Delta Y } { \Delta X } \times \frac { Y } { X }$.

Accepted Answer

The answer of In the log-log model, the slope coefficient...

Question 14

The interpretation of the slope coefficient in the model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } \ln \left( X _ { i } \right) + u _ { i }$
is as follows:

Accepted Answer

A)  a  1 %  change in  X  is associated with a $\beta _ { 1 } \%$ change in  Y . 
B)  a  1 %  change in  X  is associated with a change in  Y  of  0.01  $\beta _ { 1 }$ 
C)  a change in  X  by one unit is associated with a 100 $\beta _ { 1 } \%$ change in  Y . 
D)  a change in  X  by one unit is associated with a $\beta _ { 1 }$ change in  Y .

Question 15

Including an interaction term between two independent variables, $X _ { 1 } \text { and } X _ { 2 } \text {, }$ allows for the following, except that:

Accepted Answer

A)  the interaction term lets the effect on  Y  of a change in  $X _ { 1 }$ depend on the value of $X _ { 2 }$ 
B)  the interaction term coefficient is the effect of a unit increase in  $X _ { 1 }$  and  $X _ { 2 }$  above and beyond the sum of the individual effects of a unit increase in the two variables alone. 
C)  the interaction term coefficient is the effect of a unit increase in $\sqrt { \left( X _ { 1 } \times X _ { 2 } \right) }$ 
D)  the interaction term lets the effect on  Y  of a change in  $X _ { 2 }$ depend on the value of $X _ { 1 }$

Question 16

To test whether or not the population regression function is linear rather than a polynomial of order r, a. check whether the regression $R ^ { 2 }$ for the polynomial regression is higher than that of the linear regression.
b. compare the TSS from both regressions.
c. look at the pattern of the coefficients: if they change from positive to negative to positive, etc., then the polynomial regression should be used.
d. use the test of $( r - 1 )$ restrictions using the $F$-statistic.

Accepted Answer

The answer of To test whether or not the population...

Question 17

Indicate whether or not you can linearize the regression functions below so that OLS
estimation methods can be applied:
(a) $Y _ { i } = e ^ { \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i } }$

Accepted Answer

The answer of Indicate whether or not you can linearize...

Question 18

You have learned that earnings functions are one of the most investigated relationships in
economics.These typically relate the logarithm of earnings to a series of explanatory
variables such as education, work experience, gender, race, etc.
(a)Why do you think that researchers have preferred a log-linear specification over a linear
specification? In addition to the interpretation of the slope coefficients, also think about
the distribution of the error term.

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The answer of You have learned that earnings functions are...

Question 19

A polynomial regression model is specified as: a. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 2 } X _ { i } ^ { 2 } + \cdots + \beta _ { r } X _ { i } ^ { r } + u _ { i }$.
b. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 1 } ^ { 2 } X _ { i } + \cdots + \beta _ { 1 } ^ { r } X _ { i } + u _ { i }$.
c. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + \beta _ { 2 } Y _ { i } ^ { 2 } + \cdots + \beta _ { r } Y _ { i } ^ { r } + u _ { i }$.
d. $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 } + \beta _ { 3 } \left( X _ { 1 i } \times X _ { 2 i } \right) + u _ { i }$.

Accepted Answer

The answer of A polynomial regression model is specified as:...

Question 20

To decide whether $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X + u _ { i } \text { or } \ln \left( Y _ { i } \right) = \beta _ { 0 } + \beta _ { 1 } X + u _ { i }$
fits the data better, you cannot consult the regression R2 because

Accepted Answer

A)   ln (Y)  may be negative for  0<Y<1 . 
B)  the TSS are not measured in the same units between the two models. 
C)  the slope no longer indicates the effect of a unit change of  X  on  Y  in the log-linear model. 
D)  the regression $R ^ { 2 }$  can be greater than one in the second model.

Exam 8: Nonlinear Regression Functions

In the model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 } + \beta _ { 2 } X _ { 2 } + \beta _ { 3 } \left( X _ { 1 } \times X _ { 2 } \right) + u _ { i }$ the expected effect $\frac { \Delta Y } { \Delta X _ { 1 } }$ is

(Requires Calculus)Show that for the log-log model the slope coefficient is the elasticity.

Choose at least three different nonlinear functional forms of a single independent variable and sketch the relationship between the dependent and independent variable.

The binary variable interaction regression

The exponential function a. is the inverse of the natural logarithm function. b. does not play an important role in modeling nonlinear regression functions in econometrics. c. can be written as $\exp \left( e ^ { x } \right)$ . d. is $e ^ { x }$ , where $e$ is $3.1415 \ldots$ .

In the log-log model, the slope coefficient indicates a. the effect that a unit change in $X$ has on $Y$ . b. the elasticity of $Y$ with respect to $X$ . c. $\Delta Y / \Delta X$ . d. $\frac { \Delta Y } { \Delta X } \times \frac { Y } { X }$ .

The interpretation of the slope coefficient in the model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } \ln \left( X _ { i } \right) + u _ { i }$ is as follows:

Including an interaction term between two independent variables, $X _ { 1 } \text { and } X _ { 2 } \text {, }$ allows for the following, except that:

Indicate whether or not you can linearize the regression functions below so that OLS estimation methods can be applied: (a) $Y _ { i } = e ^ { \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i } }$

To decide whether $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X + u _ { i } \text { or } \ln \left( Y _ { i } \right) = \beta _ { 0 } + \beta _ { 1 } X + u _ { i }$ fits the data better, you cannot consult the regression R² because

Exam 8: Nonlinear Regression Functions

(Requires Calculus)Show that for the log-log model the slope coefficient is the elasticity.

Choose at least three different nonlinear functional forms of a single independent variable and sketch the relationship between the dependent and independent variable.

The binary variable interaction regression

In the log-log model, the slope coefficient indicates a. the effect that a unit change in XXX has on YYY . b. the elasticity of YYY with respect to XXX . c. ΔY/ΔX\Delta Y / \Delta XΔY/ΔX . d. ΔYΔX×YX\frac { \Delta Y } { \Delta X } \times \frac { Y } { X }ΔXΔY​×XY​ .

The interpretation of the slope coefficient in the model Yi=β0+β1ln⁡(Xi) +uiY _ { i } = \beta _ { 0 } + \beta _ { 1 } \ln \left( X _ { i } \right) + u _ { i }Yi​=β0​+β1​ln(Xi​) +ui​ is as follows:

Including an interaction term between two independent variables, X1 and X2, X _ { 1 } \text { and } X _ { 2 } \text {, }X1​ and X2​, allows for the following, except that:

Indicate whether or not you can linearize the regression functions below so that OLS estimation methods can be applied: (a) Yi=eβ0+β1Xi+uiY _ { i } = e ^ { \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i } }Yi​=eβ0​+β1​Xi​+ui​

In the log-log model, the slope coefficient indicates a. the effect that a unit change in $X$ has on $Y$ . b. the elasticity of $Y$ with respect to $X$ . c. $\Delta Y / \Delta X$ . d. $\frac { \Delta Y } { \Delta X } \times \frac { Y } { X }$ .

The interpretation of the slope coefficient in the model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } \ln \left( X _ { i } \right) + u _ { i }$ is as follows:

Including an interaction term between two independent variables, $X _ { 1 } \text { and } X _ { 2 } \text {, }$ allows for the following, except that:

Indicate whether or not you can linearize the regression functions below so that OLS estimation methods can be applied: (a) $Y _ { i } = e ^ { \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i } }$