Question 1

You have obtained measurements of height in inches of 29 female and 81 male
students (Studenth)at your university.A regression of the height on a constant
and a binary variable (BFemme), which takes a value of one for females and is
zero otherwise, yields the following result: $\begin{aligned}
\widehat{\text { Studenth }}= & 71.0-4.84 \times \text { BFemme }, \quad R^{2}=0.40, S E R=2.0 \
& (0.3)(0.57)
\end{aligned}$ (a)What is the interpretation of the intercept? What is the interpretation of the slope?
How tall are females, on average?

Accepted Answer

The answer of You have obtained measurements of height in...

Question 2

(Requires Appendix)(continuation from Chapter 4).At a recent county fair, you
observed that at one stand people's weight was forecasted, and were surprised by
the accuracy (within a range).Thinking about how the person could have
predicted your weight fairly accurately (despite the fact that she did not know
about your "heavy bones"), you think about how this could have been
accomplished.You remember that medical charts for children contain 5%, 25%,
50%, 75% and 95% lines for a weight/height relationship and decide to conduct
an experiment with 110 of your peers.You collect the data and calculate the
following sums: $$\begin{array} { c } 
\sum _ { i = 1 } ^ { n } Y _ { i } = 17,375 , \sum _ { i = 1 } ^ { n } X _ { i } = 7,665.5 , \\
\sum _ { i = 1 } ^ { n } y _ { i } ^ { 2 } = 94,228.8 , \sum _ { i = 1 } ^ { n } x _ { i } ^ { 2 } = 1,248.9 , \sum _ { i = 1 } ^ { n } x _ { i } y _ { i } = 7,625.9
\end{array}$$ where the height is measured in inches and weight in pounds. (Small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$.) (a)Calculate the homoskedasticity-only standard errors and, using the resulting t-
statistic, perform a test on the null hypothesis that there is no relationship between
height and weight in the population of college students.

Accepted Answer

The answer of (Requires Appendix)(continuation from Chapter 4).At a recent...

Question 3

In the presence of heteroskedasticity, and assuming that the usual least squares assumptions hold, the OLS estimator is
A)efficient
B)BLUE
C)unbiased and consistent
D)unbiased but not consistent

Accepted Answer

Even in the presence of heteroskedasticity, OLS estimator is still unbiased and consistent, but it is no longer efficient or BLUE. Therefore, the best choice is C, which states that the OLS estimator is unbiased and consistent, but it does not make any claims about efficiency or BLUE.

Question 4

The proof that OLS is BLUE requires all of the following assumptions with the exception of: a. the errors are homoskedastic.
b. the errors are normally distributed.
c. $E \left( u _ { i } \mid X _ { i } \right) = 0$.
d. large outliers are unlikely.

Accepted Answer

The answer of The proof that OLS is BLUE requires...

Question 5

The error term is homoskedastic if a. $\operatorname { var } \left( u _ { i } \mid X _ { i } = x \right)$ is constant for $i = 1 , \ldots , n$
b. $\operatorname { var } \left( u _ { i } \mid X _ { i } = x \right)$ depends on $x$
c. $X _ { i }$ is normally distributed
d. there are no outliers.

Accepted Answer

The answer of The error term is homoskedastic if a....

Question 6

The homoskedastic normal regression assumptions are all of the following with the exception of:
A)the errors are homoskedastic.
B)the errors are normally distributed.
C)there are no outliers.
D)there are at least 10 observations.

Accepted Answer

The assumption about having at least 10 observations is not related to homoskedasticity or normality of errors, but rather to the ability to reliably estimate the regression coefficients. Therefore, it is not a homoskedastic normal regression assumption.

Question 7

The effect of decreasing the student-teacher ratio by one is estimated to result in
an improvement of the districtwide score by 2.28 with a standard error of 0.52.
Construct a 90% and 99% confidence interval for the size of the slope coefficient
and the corresponding predicted effect of changing the student-teacher ratio by
one.What is the intuition on why the 99% confidence interval is wider than the
90% confidence interval?

Accepted Answer

The answer of The effect of decreasing the student-teacher ratio...

Question 8

(continuation from Chapter 4) Sir Francis Galton, a cousin of James Darwin, examined the relationship between the height of children and their parents towards the end of the $19 ^ { \text {th } }$ century. It is from this study that the name "regression" originated. You decide to update his findings by collecting data from 110 college students, and estimate the following relationship:
$$\begin{aligned}
\widehat { \text { Studenth } } = & 19.6 + 0.73 \times \text { Midparh, } R ^ { 2 } = 0.45 , S E R = 2.0 \
& ( 7.2 ) ( 0.10 )
\end{aligned}$$ where Studenth is the height of students in inches, and Midparh is the average of
the parental heights.Values in parentheses are heteroskedasticity robust standard
errors.(Following Galton's methodology, both variables were adjusted so that the
average female height was equal to the average male height.)
(a)Test for the statistical significance of the slope coefficient.

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The answer of (continuation from Chapter 4) Sir Francis Galton,...

Question 9

In order to formulate whether or not the alternative hypothesis is one-sided or
two-sided, you need some guidance from economic theory.Choose at least three
examples from economics or other fields where you have a clear idea what the
null hypothesis and the alternative hypothesis for the slope coefficient should be.
Write a brief justification for your answer.

Accepted Answer

The answer of In order to formulate whether or not...

Question 10

Explain carefully the relationship between a confidence interval, a one-sided
hypothesis test, and a two-sided hypothesis test.What is the unit of measurement
of the t-statistic?

Accepted Answer

The answer of Explain carefully the relationship between a confidence...

Question 11

(Requires Appendix material from Chapters 4 and 5)Shortly before you are
making a group presentation on the testscore/student-teacher ratio results, you
realize that one of your peers forgot to type all the relevant information on one of
your slides.Here is what you see: $$\begin{array} { r l r } 
\widehat { \text { TestScore } } = & 698.9 - \text { STR } & R ^ { 2 } = 0.051 , \text { SER } = 18.6 \
& ( 9.47 ) ( 0.48 ) &
\end{array}$$ In addition, your group member explains that he ran the regression in a standard
spreadsheet program, and that, as a result, the standard errors in parenthesis are
homoskedasticity-only standard errors.
(a)Find the value for the slope coefficient.

Accepted Answer

The answer of (Requires Appendix material from Chapters 4 and...

Question 12

(continuation from Chapter 4, number 3)You have obtained a sub-sample of 1744
individuals from the Current Population Survey (CPS)and are interested in the
relationship between weekly earnings and age.The regression, using
heteroskedasticity-robust standard errors, yielded the following result: $\begin{aligned}
\widehat{\text { Earn }}= & 239.16+5.20 \times \text { Age }, R^{2}=0.05, \text { SER }=287.21 ., \
& (20.24)(0.57)
\end{aligned}$
 where Earn and Age are measured in dollars and years respectively.
(a)Is the relationship between Age and Earn statistically significant?

Accepted Answer

The answer of (continuation from Chapter 4, number 3)You have...

Question 13

Carefully discuss the advantages of using heteroskedasticity-robust standard
errors over standard errors calculated under the assumption of homoskedasticity.
Give at least five examples where it is very plausible to assume that the errors
display heteroskedasticity.

Accepted Answer

The answer of Carefully discuss the advantages of using heteroskedasticity-robust...

Question 14

You recall from one of your earlier lectures in macroeconomics that the per capita
income depends on the savings rate of the country: those who save more end up
with a higher standard of living.To test this theory, you collect data from the
Penn World Tables on GDP per worker relative to the United States (RelProd)in
1990 and the average investment share of GDP from 1980-1990 (sK ),
remembering that investment equals saving.The regression results in the
following output: $$\begin{array} { l } 
\widehat { \text { RelProd } } = 0.08 + 2.44 \times s _ { K } , R ^ { 2 } = 0.46 , S E R = 0.21 \
\quad\quad\quad\quad\quad( 0.04 ) ( 0.38 ) \
\end{array}$$ (a)Interpret the regression results carefully.

Accepted Answer

The answer of You recall from one of your earlier...

Question 15

(Continuation from Chapter 4, number 6)The neoclassical growth model predicts
that for identical savings rates and population growth rates, countries should
converge to the per capita income level.This is referred to as the convergence
hypothesis.One way to test for the presence of convergence is to compare the
growth rates over time to the initial starting level.
(a)The results of the regression for 104 countries were as follows: $$\begin{aligned}
\widehat { g 6090 } = & 0.019 - 0.0006 \times \operatorname { RelProd } _ { 60 } , R ^ { 2 } = 0.00007 , S E R = 0.016 \
& ( 0.004 ) ( 0.0073 )
\end{aligned}$$ where $g 6090$ is the average annual growth rate of GDP per worker for the $1960 -$ 1990 sample period, and $\operatorname { RelProd } _ { 60 }$ is GDP per worker relative to the United States in 1960. Numbers in parenthesis are heteroskedasticity robust standard errors. Using the OLS estimator with homoskedasticity-only standard errors, the results
changed as follows: $$\begin{aligned}
\widehat { g 6090 } = & 0.019 - 0.0006 \times \operatorname { RelProd } _ { 60 } , R ^ { 2 } = 0.00007 , S E R = 0.016 \
& ( 0.002 ) ( 0.0068 )
\end{aligned}$$ Why didn't the estimated coefficients change? Given that the standard error of the
slope is now smaller, can you reject the null hypothesis of no beta convergence?
Are the results in the second equation more reliable than the results in the first
equation? Explain.

Accepted Answer

The answer of (Continuation from Chapter 4, number 6)The neoclassical...

Question 16

For the following estimated slope coefficients and their heteroskedasticity robust standard errors, find the  t -statistics for the null hypothesis $H _ { 0 } : \beta _ { 1 } = 0$ Assuming that your sample has more than 100 observations, indicate whether or not you are able to reject the null hypothesis at the  10 %, 5 % , and  1 %  level of a one-sided and two-sided hypothesis.
(a) $\hat { \beta } _ { 1 } = 4.2 , S E \left( \hat { \beta } _ { 1 } \right) = 2.4$
(b) $\hat { \beta } _ { 1 } = 0.5 , S E \left( \hat { \beta } _ { 1 } \right) = 0.37$
(c) $\hat { \beta } _ { 1 } = 0.003 , S E \left( \hat { \beta } _ { 1 } \right) = 0.002$
(d) $\hat { \beta } _ { 1 } = 360 , S E \left( \hat { \beta } _ { 1 } \right) = 300$

Accepted Answer

The answer of For the following estimated slope coefficients and...

Question 17

If the errors are heteroskedastic, then
A)OLS is BLUE
B)WLS is BLUE if the conditional variance of the errors is known up to a constant factor of proportionality
C)LAD is BLUE if the conditional variance of the errors is known up to a constant factor of proportionality
D)OLS is efficient

Accepted Answer

In the case of heteroskedastic errors, OLS is no longer BLUE, and WLS is the preferred method. However, for WLS to be BLUE, the weights used in the regression must be proportional to the inverse of the conditional variance of the errors. If the true conditional variance of the errors is known up to a constant factor of proportionality, then WLS can be used with those weights and will be BLUE. LAD is not preferred in the case of heteroskedastic errors. Therefore, B is the best choice since it acknowledges the need for WLS and the importance of correctly specifying the weights.

Question 18

Below you are asked to decide on whether or not to use a one-sided alternative or
a two-sided alternative hypothesis for the slope coefficient.Briefly justify your
decision.
(a) $$\widehat { q _ { i } ^ { d } } = \widehat { \beta } _ { 0 } + \widehat { \beta } _ { 1 } p _ { i } \text {, where } q ^ { d } \text { is the quantity demanded for a good, and } p \text { is its price. }$$

Accepted Answer

The answer of Below you are asked to decide on...

Question 19

(Continuation from Chapter 4, number 5) You have learned in one of your economics courses that one of the determinants of per capita income (the "Wealth of Nations") is the population growth rate. Furthermore you also found out that the Penn World Tables contain income and population data for 104 countries of the world. To test this theory, you regress the GDP per worker (relative to the United States) in 1990 (RelPersInc) on the difference between the average population growth rate of that country $( n )$ to the U.S. average population growth rate $\left( n _ { u s } \right)$ for the years 1980 to 1990 . This results in the following regression output: $$\begin{aligned}
\widehat { \operatorname { Re } \text { lPersInc } } = & 0.518 - 18.831 \times \left( n - n _ { u s } \right) , R ^ { 2 } = 0.522 , \text { SER } = 0.197 \
& ( 0.056 ) ( 3.177 )
\end{aligned}$$ (a)Is there any reason to believe that the variance of the error terms is
homoskedastic?

Accepted Answer

The answer of (Continuation from Chapter 4, number 5) You...

Question 20

(continuation with the Purchasing Power Parity question from Chapter 4)
The news-magazine The Economist regularly publishes data on the so called Big
Mac index and exchange rates between countries.The data for 30 countries from
the April 29, 2000 issue is listed below:  The concept of purchasing power parity or PPP ("the idea that similar foreign and domestic goods ... should have the same price in terms of the same currency," Abel, A. and B. Bernanke, Macroeconomics, $4 ^ { \text {th } }$ edition, Boston: Addison Wesley, 476) suggests that the ratio of the Big Mac priced in the local currency to the U.S. dollar price should equal the exchange rate between the two countries. 16
After entering the data into your spread sheet program, you calculate the predicted
exchange rate per U.S.dollar by dividing the price of a Big Mac in local currency
by the U.S.price of a Big Mac ($2.51).To test for PPP, you regress the actual
exchange rate on the predicted exchange rate.
The estimated regression is as follows: $$\begin{array} { r l r } 
\widehat { \text { ActualExRate } = }  & - 27.05 + 1.35 \times \operatorname { Pr } \text { edExRate } \quad R ^ { 2 } = 0.994 , n = 29 , S E R = 122.15 \
& \text { (23.74) (0.02) }
\end{array}$$  (a)Your spreadsheet program does not allow you to calculate heteroskedasticity
robust standard errors.Instead, the numbers in parenthesis are homoskedasticity
only standard errors.State the two null hypothesis under which PPP holds.Should
you use a one-tailed or two-tailed alternative hypothesis?

Accepted Answer

The answer of (continuation with the Purchasing Power Parity question...

Quiz 5: Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

In the presence of heteroskedasticity, and assuming that the usual least squares assumptions hold, the OLS estimator is

The proof that OLS is BLUE requires all of the following assumptions with the exception of: a. the errors are homoskedastic. b. the errors are normally distributed. c. $E \left( u _ { i } \mid X _ { i } \right) = 0$ . d. large outliers are unlikely.

The error term is homoskedastic if a. $\operatorname { var } \left( u _ { i } \mid X _ { i } = x \right)$ is constant for $i = 1 , \ldots , n$ b. $\operatorname { var } \left( u _ { i } \mid X _ { i } = x \right)$ depends on $x$ c. $X _ { i }$ is normally distributed d. there are no outliers.

The homoskedastic normal regression assumptions are all of the following with the exception of:

Explain carefully the relationship between a confidence interval, a one-sided hypothesis test, and a two-sided hypothesis test.What is the unit of measurement of the t-statistic?

Carefully discuss the advantages of using heteroskedasticity-robust standard errors over standard errors calculated under the assumption of homoskedasticity. Give at least five examples where it is very plausible to assume that the errors display heteroskedasticity.

If the errors are heteroskedastic, then