Deck 7: Hypothesis Tests and Confidence Intervals in Multiple Regression
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Deck 7: Hypothesis Tests and Confidence Intervals in Multiple Regression
1
All of the following are correct formulae for the homoskedasticity-only F-statistic, with the exception of
1

B
2
When there are two coefficients, the resulting confidence sets are
A)rectangles.
B)ellipses.
C)squares.
D)trapezoids.
A)rectangles.
B)ellipses.
C)squares.
D)trapezoids.
B
3
The following linear hypothesis can be tested using the F-test with the exception of 

A
4
When testing the null hypothesis that two regression slopes are zero simultaneously, then you cannot reject the null hypothesis at the 5% level, if the ellipse contains the point 

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5
The overall regression F-statistic tests the null hypothesis that
A)all slope coefficients are zero.
B)all slope coefficients and the intercept are zero.
C)the intercept in the regression and at least one, but not all, of the slope coefficients is zero.
D)the slope coefficient of the variable of interest is zero, but that the other slope coefficients are not.
A)all slope coefficients are zero.
B)all slope coefficients and the intercept are zero.
C)the intercept in the regression and at least one, but not all, of the slope coefficients is zero.
D)the slope coefficient of the variable of interest is zero, but that the other slope coefficients are not.
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6
Let be 0.4366 and 0.4149 respectively. The difference between the unrestricted and the restricted model is that you have imposed two restrictions. There are 420 observations. The F -statistic in this case is
A) 4.61
B) 8.01
C) 10.34
D) 7.71
A) 4.61
B) 8.01
C) 10.34
D) 7.71
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7
A 95% confidence set for two or more coefficients is a set that contains
A)the sample values of these coefficients in 95% of randomly drawn samples.
B)integer values only.
C)the same values as the 95% confidence intervals constructed for the coefficients.
D)the population values of these coefficients in 95% of randomly drawn samples.
A)the sample values of these coefficients in 95% of randomly drawn samples.
B)integer values only.
C)the same values as the 95% confidence intervals constructed for the coefficients.
D)the population values of these coefficients in 95% of randomly drawn samples.
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8
In the multiple regression model, the t-statistic for testing that the slope is significantly different from zero is calculated 

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9
The homoskedasticity-only F-statistic is given by the following formula 

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10
The formula for the standard error of the regression coefficient, when moving from one explanatory variable to two explanatory variables,
A)stays the same.
B)changes, unless the second explanatory variable is a binary variable.
C)changes.
D)changes, unless you test for a null hypothesis that the addition regression coefficient is zero.
A)stays the same.
B)changes, unless the second explanatory variable is a binary variable.
C)changes.
D)changes, unless you test for a null hypothesis that the addition regression coefficient is zero.
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11
If you reject a joint null hypothesis using the F-test in a multiple hypothesis setting, then
A)a series of t-tests may or may not give you the same conclusion.
B)the regression is always significant.
C)all of the hypotheses are always simultaneously rejected.
D)the F-statistic must be negative.
A)a series of t-tests may or may not give you the same conclusion.
B)the regression is always significant.
C)all of the hypotheses are always simultaneously rejected.
D)the F-statistic must be negative.
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12
If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal distribution you can
A)safely assume that your regression results are significant.
B)reject the null hypothesis.
C)reject the assumption that the error terms are homoskedastic.
D)conclude that most of the actual values are very close to the regression line.
A)safely assume that your regression results are significant.
B)reject the null hypothesis.
C)reject the assumption that the error terms are homoskedastic.
D)conclude that most of the actual values are very close to the regression line.
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13
When testing joint hypothesis, you should
A)use t-statistics for each hypothesis and reject the null hypothesis is all of the restrictions fail.
B)use the F-statistic and reject all the hypothesis if the statistic exceeds the critical value.
C)use t-statistics for each hypothesis and reject the null hypothesis once the statistic exceeds the critical value for a single hypothesis.
D)use the F-statistics and reject at least one of the hypothesis if the statistic exceeds the critical value.
A)use t-statistics for each hypothesis and reject the null hypothesis is all of the restrictions fail.
B)use the F-statistic and reject all the hypothesis if the statistic exceeds the critical value.
C)use t-statistics for each hypothesis and reject the null hypothesis once the statistic exceeds the critical value for a single hypothesis.
D)use the F-statistics and reject at least one of the hypothesis if the statistic exceeds the critical value.
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14
The homoskedasticity-only F-statistic is given by the following formula 

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15
If you wanted to test, using a 5 % significance level, whether or not a specific slope coefficient is equal to one, then you should
A) subtract 1 from the estimated coefficient, divide the difference by the standard error, and check if the resulting ratio is larger than 1.96 .
B) add and subtract 1.96 from the slope and check if that interval includes 1 .
C) see if the slope coefficient is between 0.95 and 1.05 .
D) check if the adjusted is close to 1 .
A) subtract 1 from the estimated coefficient, divide the difference by the standard error, and check if the resulting ratio is larger than 1.96 .
B) add and subtract 1.96 from the slope and check if that interval includes 1 .
C) see if the slope coefficient is between 0.95 and 1.05 .
D) check if the adjusted is close to 1 .
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16
All of the following are examples of joint hypotheses on multiple regression coefficients, with the exception of 

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17
When your multiple regression function includes a single omitted variable regressor, then
A)use a two-sided alternative hypothesis to check the influence of all included variables.
B)the estimator for your included regressors will be biased if at least one of the included variables is correlated with the omitted variable.
C)the estimator for your included regressors will always be biased.
D)lower the critical value to 1.645 from 1.96 in a two-sided alternative hypothesis to test the significance of the coefficients of the included variables.
A)use a two-sided alternative hypothesis to check the influence of all included variables.
B)the estimator for your included regressors will be biased if at least one of the included variables is correlated with the omitted variable.
C)the estimator for your included regressors will always be biased.
D)lower the critical value to 1.645 from 1.96 in a two-sided alternative hypothesis to test the significance of the coefficients of the included variables.
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18
To test joint linear hypotheses in the multiple regression model, you need to 

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19
The confidence interval for a single coefficient in a multiple regression 

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20
For a single restriction (q=1) , the F -statistic
A) is the square root of the t -statistic.
B) has a critical value of 1.96 .
C) will be negative.
D) is the square of the t -statistic.
A) is the square root of the t -statistic.
B) has a critical value of 1.96 .
C) will be negative.
D) is the square of the t -statistic.
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21
The OLS estimators of the coefficients in multiple regression will have omitted variable bias 

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22


1% level?
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23
A subsample from the Current Population Survey is taken, on weekly earnings of
individuals, their age, and their gender.You have read in the news that women make 70
cents to the $1 that men earn.To test this hypothesis, you first regress earnings on a
constant and a binary variable, which takes on a value of 1 for females and is 0 otherwise.
The results were:
(a)Perform a difference in means test and indicate whether or not the difference in the mean
salaries is significantly different.Justify your choice of a one-sided or two-sided
alternative test.Are these results evidence enough to argue that there is discrimination
against females? Why or why not? Is it likely that the errors are normally distributed in
this case? If not, does that present a problem to your test?
individuals, their age, and their gender.You have read in the news that women make 70
cents to the $1 that men earn.To test this hypothesis, you first regress earnings on a
constant and a binary variable, which takes on a value of 1 for females and is 0 otherwise.
The results were:

salaries is significantly different.Justify your choice of a one-sided or two-sided
alternative test.Are these results evidence enough to argue that there is discrimination
against females? Why or why not? Is it likely that the errors are normally distributed in
this case? If not, does that present a problem to your test?
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24
In the multiple regression model with two explanatory variables
the U.S.)in 1990.The various sums needed to calculate the OLS estimates are given
below:
The heteroskedasticity-robust standard errors of the two slope coefficients are 1.99 (for
population growth)and 0.23 (for the saving rate).Calculate the 95% confidence interval
for both coefficients.How many standard deviations are the coefficients away from zero?


below:

population growth)and 0.23 (for the saving rate).Calculate the 95% confidence interval
for both coefficients.How many standard deviations are the coefficients away from zero?
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25
If the estimates of the coefficients of interest change substantially across specifications,
A)then this can be expected from sample variation.
B)then you should change the scale of the variables to make the changes appear to be smaller.
C)then this often provides evidence that the original specification had omitted variable bias.
D)then choose the specification for which your coefficient of interest is most significant.
A)then this can be expected from sample variation.
B)then you should change the scale of the variables to make the changes appear to be smaller.
C)then this often provides evidence that the original specification had omitted variable bias.
D)then choose the specification for which your coefficient of interest is most significant.
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26
The cost of attending your college has once again gone up.Although you have been told
that education is investment in human capital, which carries a return of roughly 10% a
year, you (and your parents)are not pleased.One of the administrators at your
university/college does not make the situation better by telling you that you pay more
because the reputation of your institution is better than that of others.To investigate this
hypothesis, you collect data randomly for 100 national universities and liberal arts
colleges from the 2000-2001 U.S.News and World Report annual rankings.Next you
perform the following regression
where Cost is Tuition, Fees, Room and Board in dollars, Reputation is the index used in
U.S.News and World Report (based on a survey of university presidents and chief
academic officers), which ranges from 1 ("marginal")to 5 ("distinguished"), Size is the
number of undergraduate students, and Dpriv, Dlibart, and Dreligion are binary variables
indicating whether the institution is private, a liberal arts college, and has a religious
affiliation.The numbers in parentheses are heteroskedasticity-robust standard errors.
(a)Indicate whether or not the coefficients are significantly different from zero.
that education is investment in human capital, which carries a return of roughly 10% a
year, you (and your parents)are not pleased.One of the administrators at your
university/college does not make the situation better by telling you that you pay more
because the reputation of your institution is better than that of others.To investigate this
hypothesis, you collect data randomly for 100 national universities and liberal arts
colleges from the 2000-2001 U.S.News and World Report annual rankings.Next you
perform the following regression

U.S.News and World Report (based on a survey of university presidents and chief
academic officers), which ranges from 1 ("marginal")to 5 ("distinguished"), Size is the
number of undergraduate students, and Dpriv, Dlibart, and Dreligion are binary variables
indicating whether the institution is private, a liberal arts college, and has a religious
affiliation.The numbers in parentheses are heteroskedasticity-robust standard errors.
(a)Indicate whether or not the coefficients are significantly different from zero.
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27
Set up the null hypothesis and alternative hypothesis carefully for the following cases:
(a)k = 4, test for all coefficients other than the intercept to be zero
(a)k = 4, test for all coefficients other than the intercept to be zero
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28
You have collected data from Major League Baseball (MLB)to find the determinants of
winning.You have a general idea that both good pitching and strong hitting are needed to do
well.However, you do not know how much each of these contributes separately.To
investigate this problem, you collect data for all MLB during 1999 season.Your strategy is to
first regress the winning percentage on pitching quality ("Team ERA"), second to regress the
same variable on some measure of hitting ("OPS - On-base Plus Slugging percentage"), and
third to regress the winning percentage on both.
(a)Use the t-statistic to test for the statistical significance of the coefficient.
winning.You have a general idea that both good pitching and strong hitting are needed to do
well.However, you do not know how much each of these contributes separately.To
investigate this problem, you collect data for all MLB during 1999 season.Your strategy is to
first regress the winning percentage on pitching quality ("Team ERA"), second to regress the
same variable on some measure of hitting ("OPS - On-base Plus Slugging percentage"), and
third to regress the winning percentage on both.


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29
At a mathematical level, if the two conditions for omitted variable bias are satisfied, then 

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30
Attendance at sports events depends on various factors.Teams typically do not change
ticket prices from game to game to attract more spectators to less attractive games.
However, there are other marketing tools used, such as fireworks, free hats, etc., for this
purpose.You work as a consultant for a sports team, the Los Angeles Dodgers, to help
them forecast attendance, so that they can potentially devise strategies for price
discrimination.After collecting data over two years for every one of the 162 home games
of the 2000 and 2001 season, you run the following regression:
where Attend is announced stadium attendance, Temperat it the average temperature on
game day, DodgNetWin are the net wins of the Dodgers before the game (wins-losses),
OppNetWin is the opposing team's net wins at the end of the previous season, and
DFSaSu, Drain, D150m, Ddiv, and D2001 are binary variables, taking a value of 1 if the
game was played on a weekend, it rained during that day, the opposing team was within a
150 mile radius, the opposing team plays in the same division as the Dodgers, and the
game was played during 2001, respectively.Numbers in parentheses are
heteroskedasticity- robust standard errors.
(a)Are the slope coefficients statistically significant?
ticket prices from game to game to attract more spectators to less attractive games.
However, there are other marketing tools used, such as fireworks, free hats, etc., for this
purpose.You work as a consultant for a sports team, the Los Angeles Dodgers, to help
them forecast attendance, so that they can potentially devise strategies for price
discrimination.After collecting data over two years for every one of the 162 home games
of the 2000 and 2001 season, you run the following regression:

game day, DodgNetWin are the net wins of the Dodgers before the game (wins-losses),
OppNetWin is the opposing team's net wins at the end of the previous season, and
DFSaSu, Drain, D150m, Ddiv, and D2001 are binary variables, taking a value of 1 if the
game was played on a weekend, it rained during that day, the opposing team was within a
150 mile radius, the opposing team plays in the same division as the Dodgers, and the
game was played during 2001, respectively.Numbers in parentheses are
heteroskedasticity- robust standard errors.
(a)Are the slope coefficients statistically significant?
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31
Females, on average, are shorter and weigh less than males.One of your friends, who is a
pre-med student, tells you that in addition, females will weigh less for a given height.To
test this hypothesis, you collect height and weight of 29 female and 81 male students at
your university.A regression of the weight on a constant, height, and a binary variable,
which takes a value of one for females and is zero otherwise, yields the following result:
where Studentw is weight measured in pounds and Height is measured in inches
(heteroskedasticity-robust standard errors in parentheses).
Calculate t-statistics and carry out the hypothesis test that females weigh the same as
males, on average, for a given height, using a 10% significance level.What is the
alternative hypothesis? What is the p-value? What critical value did you use?
pre-med student, tells you that in addition, females will weigh less for a given height.To
test this hypothesis, you collect height and weight of 29 female and 81 male students at
your university.A regression of the weight on a constant, height, and a binary variable,
which takes a value of one for females and is zero otherwise, yields the following result:

(heteroskedasticity-robust standard errors in parentheses).
Calculate t-statistics and carry out the hypothesis test that females weigh the same as
males, on average, for a given height, using a 10% significance level.What is the
alternative hypothesis? What is the p-value? What critical value did you use?
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32
All of the following are true, with the exception of one condition: 

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33
In the process of collecting weight and height data from 29 female and 81 male students
at your university, you also asked the students for the number of siblings they have.
Although it was not quite clear to you initially what you would use that variable for, you
construct a new theory that suggests that children who have more siblings come from
poorer families and will have to share the food on the table.Although a friend tells you
that this theory does not pass the "straight-face" test, you decide to hypothesize that peers
with many siblings will weigh less, on average, for a given height.In addition, you
believe that the muscle/fat tissue composition of male bodies suggests that females will
weigh less, on average, for a given height.To test these theories, you perform the
following regression:
where Studentw is in pounds, Height is in inches, Female takes a value of 1 for females
and is 0 otherwise, Sibs is the number of siblings (heteroskedasticity-robust standard
errors in parentheses).
(a)Carrying out hypotheses tests using the relevant t-statistics to test your two claims
separately, is there strong evidence in favor of your hypotheses? Is it appropriate to use
two separate tests in this situation?
at your university, you also asked the students for the number of siblings they have.
Although it was not quite clear to you initially what you would use that variable for, you
construct a new theory that suggests that children who have more siblings come from
poorer families and will have to share the food on the table.Although a friend tells you
that this theory does not pass the "straight-face" test, you decide to hypothesize that peers
with many siblings will weigh less, on average, for a given height.In addition, you
believe that the muscle/fat tissue composition of male bodies suggests that females will
weigh less, on average, for a given height.To test these theories, you perform the
following regression:

and is 0 otherwise, Sibs is the number of siblings (heteroskedasticity-robust standard
errors in parentheses).
(a)Carrying out hypotheses tests using the relevant t-statistics to test your two claims
separately, is there strong evidence in favor of your hypotheses? Is it appropriate to use
two separate tests in this situation?
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34
The general answer to the question of choosing the scale of the variables is
A)dependent on you whim.
B)to make the regression results easy to read and to interpret.
C)to ensure that the regression coefficients always lie between -1 and 1.
D)irrelevant because regardless of the scale of the variable, the regression coefficient is unaffected.
A)dependent on you whim.
B)to make the regression results easy to read and to interpret.
C)to ensure that the regression coefficients always lie between -1 and 1.
D)irrelevant because regardless of the scale of the variable, the regression coefficient is unaffected.
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35

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36
The administration of your university/college is thinking about implementing a policy of
coed floors only in dormitories.Currently there are only single gender floors.One reason
behind such a policy might be to generate an atmosphere of better "understanding"
between the sexes.The Dean of Students (DoS)has decided to investigate if such a
behavior results in more "togetherness" by attempting to find the determinants of the
gender composition at the dinner table in your main dining hall, and in that of a
neighboring university, which only allows for coed floors in their dorms.The survey
includes 176 students, 63 from your university/college, and 113 from a neighboring
institution.
The Dean's first problem is how to define gender composition.To begin with, the survey
excludes single persons' tables, since the study is to focus on group behavior.The Dean
also eliminates sports teams from the analysis, since a large number of single-gender
students will sit at the same table.Finally, the Dean decides to only analyze tables with
three or more students, since she worries about "couples" distorting the results.The Dean
finally settles for the following specification of the dependent variable:
After considering various explanatory variables, the Dean settles for an initial list of
eight, and estimates the following relationship, using heteroskedasticity-robust standard
errors (this Dean obviously has taken an econometrics course earlier in her career and/or
has an able research assistant):
where Size is the number of persons at the table minus 3, DCoed is a binary variable,
which takes on the value of 1 if you live on a coed floor, DFemme is a binary variable,
which is 1 for females and zero otherwise, DRoommate is a binary variable which equals
1 if the person at the table has a roommate and is zero otherwise, DAthlete is a binary
variable which is 1 if the person at the table is a member of an athletic varsity team,
DCons is a variable which measures the political tendency of the person at the table on a
seven-point scale, ranging from 1 being "liberal" to 7 being "conservative," SAT is the
SAT score of the person at the table measured on a seven-point scale, ranging from 1 for
the category "900-1000" to 7 for the category "1510 and above," and increasing by one
for 100 point increases, and SibOther is the number of siblings from the opposite gender
in the family the person at the table grew up with.
(a)Indicate which of the coefficients are statistically significant.
coed floors only in dormitories.Currently there are only single gender floors.One reason
behind such a policy might be to generate an atmosphere of better "understanding"
between the sexes.The Dean of Students (DoS)has decided to investigate if such a
behavior results in more "togetherness" by attempting to find the determinants of the
gender composition at the dinner table in your main dining hall, and in that of a
neighboring university, which only allows for coed floors in their dorms.The survey
includes 176 students, 63 from your university/college, and 113 from a neighboring
institution.
The Dean's first problem is how to define gender composition.To begin with, the survey
excludes single persons' tables, since the study is to focus on group behavior.The Dean
also eliminates sports teams from the analysis, since a large number of single-gender
students will sit at the same table.Finally, the Dean decides to only analyze tables with
three or more students, since she worries about "couples" distorting the results.The Dean
finally settles for the following specification of the dependent variable:

eight, and estimates the following relationship, using heteroskedasticity-robust standard
errors (this Dean obviously has taken an econometrics course earlier in her career and/or
has an able research assistant):

which takes on the value of 1 if you live on a coed floor, DFemme is a binary variable,
which is 1 for females and zero otherwise, DRoommate is a binary variable which equals
1 if the person at the table has a roommate and is zero otherwise, DAthlete is a binary
variable which is 1 if the person at the table is a member of an athletic varsity team,
DCons is a variable which measures the political tendency of the person at the table on a
seven-point scale, ranging from 1 being "liberal" to 7 being "conservative," SAT is the
SAT score of the person at the table measured on a seven-point scale, ranging from 1 for
the category "900-1000" to 7 for the category "1510 and above," and increasing by one
for 100 point increases, and SibOther is the number of siblings from the opposite gender
in the family the person at the table grew up with.
(a)Indicate which of the coefficients are statistically significant.
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37
You have collected data for 104 countries to address the difficult questions of the
determinants for differences in the standard of living among the countries of the world.
You recall from your macroeconomics lectures that the neoclassical growth model
suggests that output per worker (per capita income)levels are determined by, among
others, the saving rate and population growth rate.To test the predictions of this growth
model, you run the following regression:
where RelPersInc is GDP per worker relative to the United States, n is the average
population growth rate, 1980-1990, and sK is the average investment share of GDP from
1960 to1990 (remember investment equals saving).Numbers in parentheses are for
heteroskedasticity-robust standard errors.
(a)Calculate the t-statistics and test whether or not each of the population parameters are
significantly different from zero.
determinants for differences in the standard of living among the countries of the world.
You recall from your macroeconomics lectures that the neoclassical growth model
suggests that output per worker (per capita income)levels are determined by, among
others, the saving rate and population growth rate.To test the predictions of this growth
model, you run the following regression:

population growth rate, 1980-1990, and sK is the average investment share of GDP from
1960 to1990 (remember investment equals saving).Numbers in parentheses are for
heteroskedasticity-robust standard errors.
(a)Calculate the t-statistics and test whether or not each of the population parameters are
significantly different from zero.
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38
Explain carefully why testing joint hypotheses simultaneously, using the F-statistic, does
not necessarily yield the same conclusion as testing them sequentially ("one at a time"
method), using a series of t-statistics.
not necessarily yield the same conclusion as testing them sequentially ("one at a time"
method), using a series of t-statistics.
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39
Prove that


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40
Consider a situation where economic theory suggests that you impose certain restrictions
on your estimated multiple regression function.These may involve the equality of
parameters, such as the returns to education and on the job training in earnings functions,
or the sum of coefficients, such as constant returns to scale in a production function.To
test the validity of your restrictions, you have your statistical package calculate the
corresponding F-statistic.Find the critical value from the F-distribution at the 5% and 1%
level, and comment whether or not you will reject the null hypothesis in each of the
following cases.
(a)number of observations: 152; number of restrictions: 3; F-statistic: 3.21
on your estimated multiple regression function.These may involve the equality of
parameters, such as the returns to education and on the job training in earnings functions,
or the sum of coefficients, such as constant returns to scale in a production function.To
test the validity of your restrictions, you have your statistical package calculate the
corresponding F-statistic.Find the critical value from the F-distribution at the 5% and 1%
level, and comment whether or not you will reject the null hypothesis in each of the
following cases.
(a)number of observations: 152; number of restrictions: 3; F-statistic: 3.21
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41
The homoskedasticity only F-statistic is given by the formula



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42
You have estimated the following regression to explain hourly wages, using a sample of
250 individuals:
250 individuals:

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43
Consider the following regression output for an unrestricted and a restricted model.
Calculate the homoskedasticity only F-statistic and determine whether the null hypothesis
can be rejected at the 5% significance level.


can be rejected at the 5% significance level.
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44

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45
Trying to remember the formula for the homoskedasticity-only F-statistic, you forgot
whether you subtract the restricted SSR from the unrestricted SSR or the other way
around.Your professor has provided you with a table containing critical values for the F
distribution.How can this be of help?
whether you subtract the restricted SSR from the unrestricted SSR or the other way
around.Your professor has provided you with a table containing critical values for the F
distribution.How can this be of help?
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46

two confidence intervals for the respective coefficients.For each case where the ellipse
does not coincide in area with the corresponding rectangle, indicate what your decision
would be if you relied on the two confidence intervals vs.the ellipse generated by the F-
statistic.
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47
You have estimated the following regression to explain hourly wages, using a sample of
250 individuals:
250 individuals:

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48
Consider the following two models to explain testscores.
:
Explain why you cannot use the F-test in this situation to discriminate between Model 1
and Model 2.


and Model 2.
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49
Adding the Percent of English Speakers (PctEL)to the Student Teacher Ratio (STR)in
your textbook reduced the coefficient for STR from 2.28 to 1.10 with a standard error of
0.43.Construct a 90% and 99% confidence interval to test the hypothesis that the
coefficient of STR is 2.28.
your textbook reduced the coefficient for STR from 2.28 to 1.10 with a standard error of
0.43.Construct a 90% and 99% confidence interval to test the hypothesis that the
coefficient of STR is 2.28.
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50
Consider the following Cobb-Douglas production function 
(where Y is output, A is the level of technology, K is the capital stock, and L is the labor force), which has been linearized here (by using logarithms) to look as follows:

Assuming that the errors are heteroskedastic, you want to test for constant returns to scale. Using a t -statistic and "Approach #2," how would you proceed.

(where Y is output, A is the level of technology, K is the capital stock, and L is the labor force), which has been linearized here (by using logarithms) to look as follows:

Assuming that the errors are heteroskedastic, you want to test for constant returns to scale. Using a t -statistic and "Approach #2," how would you proceed.
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