Deck 4: Linear Regression With One Regressor

Question 1

(Requires Appendix material)The sample average of the OLS residuals is

A)some positive number since OLS uses squares.
B)zero.
C)unobservable since the population regression function is unknown.
D)dependent on whether the explanatory variable is mostly positive or negative.

Answer 1

B

Question 2

The following are all least squares assumptions with the exception of:

Answer 2

B

Question 3

The regression $R ^ { 2 }$ is defined as follows:

A) $\frac{E S S}{T S S}$

B) $\frac { R S S } { T S S }$
C) $\frac { \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) \left( X _ { i } - \bar { X } \right) } { \sqrt { \sum _ { i = 1 } ^ { n } \left( Y _ { i } - \bar { Y } \right) ^ { 2 } } \sqrt { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } } }$
D) $\frac { S S R } { n - 2 }$

Answer 3

$\frac{E S S}{T S S}$

Question 4

The slope estimator, $\beta _ { 1 }$ has a smaller standard error, other things equal, if

A) there is more variation in the explanatory variable, X .
B) there is a large variance of the error term, u .
C) the sample size is smaller.
D) $\text { the intercept, } \beta _ { 0 } \text {, is small. }$

Question 5

Binary variables

A)are generally used to control for outliers in your sample.
B)can take on more than two values.
C)exclude certain individuals from your sample.
D)can take on only two values.

Question 6

If the three least squares assumptions hold, then the large sample normal distribution of $\widehat { \beta } _ { 1 }$ is

A) $N \left( 0 , \frac { 1 } { n } \frac { \left. \operatorname { var } \left[ X _ { i } - \mu _ { X } \right) u _ { i } \right] } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$
B) $N \left( \beta _ { 1 } , \frac { 1 } { n } \frac { \left. \operatorname { var } \left( u _ { i } \right) \right] ^ { 2 } } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$
C) $N \left( \beta _ { 1 } , \frac { \sigma _ { u } ^ { 2 } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } } \right.$
D) $N \left( \beta _ { 1 } , \frac { 1 } { n } \frac { \left. \operatorname { var } \left( u _ { i } \right) \right] } { \left[ \operatorname { var } \left( X _ { i } \right) \right] ^ { 2 } } \right)$

Question 7

The OLS residuals, $\hat { u } _ { i } ,$ are defined as follows:

A) $\hat { Y } _ { i } - \widehat { \beta } _ { 0 } - \widehat { \beta } _ { 1 } X _ { i }$
B) $Y _ { i } - \beta _ { 0 } - \beta _ { 1 } X _ { i }$
C) $Y _ { i } - \hat { Y } _ { i }$
D) $\left( Y _ { i } - \bar { Y } \right) ^ { 2 }$

Question 8

The OLS residuals

A)can be calculated using the errors from the regression function.
B)can be calculated by subtracting the fitted values from the actual values.
C)are unknown since we do not know the population regression function.
D)should not be used in practice since they indicate that your regression does not run through all your observations.

Question 9

To obtain the slope estimator using the least squares principle, you divide the

Question 10

When the estimated slope coefficient in the simple regression model,
$\hat { \beta } _ { 1 } ,$ is zero, then

A) $R ^ { 2 } = \bar { Y }$
B) $0 < R ^ { 2 } < 1$
C) $R ^ { 2 } = 0$
D) $R ^ { 2 } > ( S S R / T S S )$

Question 11

(Requires Appendix material) The sample regression line estimated by OLS

A) will always have a slope smaller than the intercept.
B) is exactly the same as the population regression line.
C) cannot have a slope of zero.
D) will always run through the point $( \bar { X } , \bar { Y } )$

Question 12

In the simple linear regression model $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i }$

A) the intercept is typically small and unimportant.
B) $\beta _ { 0 } + \beta _ { 1 } X _ { i }$ represents the population regression function.
C) the absolute value of the slope is typically between 0 and 1 .
D) $\beta _ { 0 } + \beta _ { 1 } X _ { i }$ represents the sample regression function.

Question 13

(Requires Appendix material) Which of the following statements is correct?

A) $T S S = E S S + S S R$
B) $E S S = S S R + T S S$
C) $\text { ESS } > \text { TSS }$
D) $R ^ { 2 } = 1 - ( E S S / T S S )$

Question 14

The normal approximation to the sampling distribution of $\widehat { \beta } _ { 1 }$ is powerful because

A) many explanatory variables in real life are normally distributed.
B) it allows econometricians to develop methods for statistical inference.
C) many other distributions are not symmetric.
D) is implies that OLS is the BLUE estimator for $\beta _ { 1 }$

Question 15

The variance of $Y _ { i }$ is given by

A) $\beta _ { 0 } ^ { 2 } + \beta _ { 1 } ^ { 2 } \operatorname { var } \left( X _ { i } \right) + \operatorname { var } \left( u _ { i } \right)$
B) $\text { the variance of } u _ { i }$
C) $\beta _ { 1 } ^ { 2 } \operatorname { var } \left( X _ { i } \right) + \operatorname { var } \left( u _ { i } \right)$
D) the variance of the residuals.

Question 16

Interpreting the intercept in a sample regression function is

A)not reasonable because you never observe values of the explanatory variables around the origin.
B)reasonable because under certain conditions the estimator is BLUE.
C)reasonable if your sample contains values of Xi around the origin.
D)not reasonable because economists are interested in the effect of a change in X on the change in Y.

Question 17

In the simple linear regression model, the regression slope

A)indicates by how many percent Y increases, given a one percent increase in X.
B)when multiplied with the explanatory variable will give you the predicted Y.
C)indicates by how many units Y increases, given a one unit increase in X.
D)represents the elasticity of Y on X.

Question 18

The OLS estimator is derived by

A)connecting the Yi corresponding to the lowest Xi observation with the Yi corresponding to the highest Xi observation.
B)making sure that the standard error of the regression equals the standard error of the slope estimator.
C)minimizing the sum of absolute residuals.
D)minimizing the sum of squared residuals.

Question 19

The standard error of the regression (SER) is defined as follows

A) $\frac { 1 } { n - 2 } \sum _ { i = 1 } ^ { n } \hat { u } _ { i } ^ { 2 }$
B) S S R
C) $1 - R ^ { 2 }$
D) $\frac { 1 } { n - 1 } \sum _ { i = 1 } ^ { n } \hat { u } _ { i } ^ { 2 }$

Question 20

The reason why estimators have a sampling distribution is that

A)economics is not a precise science.
B)individuals respond differently to incentives.
C)in real life you typically get to sample many times.
D)the values of the explanatory variable and the error term differ across samples.

Question 21

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: a)Enter the data into your regression analysis program (EViews, Stata, Excel, SAS, etc.).
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).

Question 22

In order to calculate the regression you need the TSS and either the SSR or the ESS . The TSS is fairly straightforward to calculate, being just the variation of Y . However, if you had to calculate the SSR or ESS by hand (or in a spreadsheet), you would need all fitted values from the regression function and their deviations from the sample mean, or the residuals. Can you think of a quicker way to calculate the ESS simply using terms you have already used to calculate the slope coefficient?

Question 23

For the simple regression model of Chapter 4, you have been given the following data: (a)Calculate the regression slope and the intercept.

Question 24

In 2001, the Arizona Diamondbacks defeated the New York Yankees in the Baseball
World Series in 7 games.Some players, such as Bautista and Finley for the
Diamondbacks, had a substantially higher batting average during the World Series than
during the regular season.Others, such as Brosius and Jeter for the Yankees, did
substantially poorer.You set out to investigate whether or not the regular season batting
average is a good indicator for the World Series batting average.The results for 11
players who had the most at bats for the two teams are: where Wsavg and Seasavg indicate the batting average during the World Series and the
regular season respectively.
(a)Focusing on the coefficients first, what is your interpretation?

Question 25

(Requires Appendix material) In deriving the OLS estimator, you minimize the sum of squared residuals with respect to the two parameters
. The resulting two equations imply two restrictions that OLS places on the data, namely that and
Show that you get the same formula for the regression slope and the intercept if you impose these two conditions on the sample regression function.

Question 26

Prove that the regression is identical to the square of the correlation coefficient between two variables Y and X . Regression functions are written in a form that suggests causation running from X to Y . Given your proof, does a high regression present supportive evidence of a causal relationship? Can you think of some regression examples where the direction of causality is not clear? Is without a doubt?

Question 27

In the linear regression model, $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { i } + u _ { i } , \beta _ { 0 } + \beta _ { 1 } X _ { i }$
is referred to as

A) the population regression function.
B) the sample regression function.
C) exogenous variation.
D) the right-hand variable or regressor.

Question 28

(Requires Appendix material)Show that the two alternative formulae for the slope given
in your textbook are identical.

Question 29

Sir Francis Galton, a cousin of James Darwin, examined the relationship between the
height of children and their parents towards the end of the 19th 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: where Studenth is the height of students in inches, and Midparh is the average of the
parental heights.(Following Galton's methodology, both variables were adjusted so that
the average female height was equal to the average male height.)
(a)Interpret the estimated coefficients.

Question 30

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: where Earn and Age are measured in dollars and years respectively.
(a)Interpret the results.

Question 31

The baseball team nearest to your home town is, once again, not doing well.Given that
your knowledge of what it takes to win in baseball is vastly superior to that of
management, you want to find out what it takes to win in Major League Baseball (MLB).
You therefore collect the winning percentage of all 30 baseball teams in MLB for 1999
and regress the winning percentage on what you consider the primary determinant for
wins, which is quality pitching (team earned run average).You find the following
information on team performance: (a)What is your expected sign for the regression slope? Will it make sense to interpret the
intercept? If not, should you omit it from your regression and force the regression line
through the origin?

Question 32

To decide whether or not the slope coefficient is large or small,

A)you should analyze the economic importance of a given increase in X.
B)the slope coefficient must be larger than one.
C)the slope coefficient must be statistically significant.
D)you should change the scale of the X variable if the coefficient appears to be too small.

Question 33

Multiplying the dependent variable by 100 and the explanatory variable by 100,000 leaves the

Question 34

(Requires Appendix material)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: (a)Calculate the slope and intercept of the regression and interpret these.

Question 35

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)If you regressed the average growth rate over a time period (1960-1990)on the initial
level of per capita income, what would the sign of the slope have to be to indicate this
type of convergence? Explain.Would this result confirm or reject the prediction of the
neoclassical growth model?

Question 36

You have analyzed the relationship between the weight and height of individuals.
Although you are quite confident about the accuracy of your measurements, you feel that
some of the observations are extreme, say, two standard deviations above and below the
mean.Your therefore decide to disregard these individuals.What consequence will this
have on the standard deviation of the OLS estimator of the slope?

Question 37

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 (nus )for the
years 1980 to 1990.This results in the following regression output: (a)Interpret the results carefully.Is this relationship economically important?

Question 38

Your textbook presented you with the following regression output: (a)How would the slope coefficient change, if you decided one day to measure testscores in
100s, i.e., a testscore of 650 became 6.5? Would this have an effect on your
interpretation?

Question 39

$\mathrm { E } \left( u _ { i } \mid X _ { i } \right) = 0$ says that

A) dividing the error by the explanatory variable results in a zero (on average).
B) the sample regression function residuals are unrelated to the explanatory variable.
C) the sample mean of the Xs is much larger than the sample mean of the errors.
D) the conditional distribution of the error given the explanatory variable has a zero mean.

Question 40

(Requires Calculus)Consider the following model:

Question 41

(Requires Calculus)Consider the following model:

Question 42

Indicate in a scatterplot what the data for your dependent variable and your explanatory variable would look like in a regression with an R² equal to zero. How would this change if the regression R² was equal to one?

Question 43

Assume that there is a change in the units of measurement on both Y and X . The new variables are and What effect will this change have on the regression slope?

Question 44

Assume that there is a change in the units of measurement on X . The new variables Prove that this change in the units of measurement on the explanatory variable has no effect on the intercept in the resulting regression.

Question 45

(Requires Appendix material)Consider the sample regression function

Question 46

Show first that the regression is the square of the sample correlation coefficient. Next, show that the slope of a simple regression of Y on X is only identical to the inverse of the regression slope of X on Y if the regression equals one.

Question 47

The help function for a commonly used spreadsheet program gives the following
definition for the regression slope it estimates: Prove that this formula is the same as the one given in the textbook.

Question 48

(Requires Appendix material) A necessary and sufficient condition to derive the OLS estimator is that the following two conditions hold:
Show that these conditions imply that

Question 49

Consider the sample regression function

Question 50

A peer of yours, who is a major in another social science, says he is not interested in the
regression slope and/or intercept.Instead he only cares about correlations.For example,
in the testscore/student-teacher ratio regression, he claims to get all the information he
needs from the negative correlation coefficient corr(X,Y)=-0.226.What response might
you have for your peer?

Question 51

The OLS slope estimator is not defined if there is no variation in the data for the
explanatory variable.You are interested in estimating a regression relating earnings to
years of schooling.Imagine that you had collected data on earnings for different
individuals, but that all these individuals had completed a college education (16 years of
education).Sketch what the data would look like and explain intuitively why the OLS
coefficient does not exist in this situation.

Question 52

Imagine that you had discovered a relationship that would generate a scatterplot very similar to the relationship , and that you would try to fit a linear regression through your data points. What do you expect the slope coefficient to be? What do you think the value of your regression R² is in this situation? What are the implications from your answers in terms of fitting a linear regression through a non-linear relationship?

Question 53

In order to calculate the slope, the intercept, and the regression for a simple sample regression function, list the five sums of data that you need.

Question 54

Given the amount of money and effort that you have spent on your education, you wonder if it was (is) all worth it. You therefore collect data from the Current Population Survey (CPS) and estimate a linear relationship between earnings and the years of education of individuals. What would be the effect on your regression slope and intercept if you measured earnings in thousands of dollars rather than in dollars? Would the regression be affected? Should statistical inference be dependent on the scale of variables? Discuss.