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

Question 1

The construction of the t-statistic for a one- and a two-sided hypothesis

A)depends on the critical value from the appropriate distribution
B)is the same.
C)is different since the critical value must be 1.645 for the one-sided hypothesis, but 1.96 for the two-sided hypothesis (using a 5% probability
For the Type I error).
D)uses ±1.96 for the two-sided test, but only +1.96 for the one-sided test.

Answer 1

B

Question 2

The confidence interval for the sample regression function slope

A)can be used to conduct a test about a hypothesized population regression function slope.
B)can be used to compare the value of the slope relative to that of the intercept.
C)adds and subtracts 1.96 from the slope.
D)allows you to make statements about the economic importance of your estimate.

Answer 2

A

Question 3

The t-statistic is calculated by dividing

A)the OLS estimator by its standard error.
B)the slope by the standard deviation of the explanatory variable.
C)the estimator minus its hypothesized value by the standard error of the estimator.
D)the slope by 1.96.

Answer 3

C

Question 4

Imagine that you were told that the t -statistic for the slope coefficient of the regression line $\widehat { \text { TestScore } } = 698.9 - 2.28 \times S T R$ was 4.38 . What are the units of measurement for the t -statistic?

A) points of the test score
B) number of students per teacher
C) $\frac{\text { TestScore }}{S T R}$

D) standard deviations

Question 5

A binary variable is often called a

A)dummy variable
B)dependent variable
C)residual
D)power of a test

Question 6

When estimating a demand function for a good where quantity demanded is a linear function of the price, you should

A)not include an intercept because the price of the good is never zero.
B)use a one-sided alternative hypothesis to check the influence of price on quantity.
C)use a two-sided alternative hypothesis to check the influence of price on quantity.
D)reject the idea that price determines demand unless the coefficient is at least 1.96.

Question 7

Consider the following regression line: $\widehat { \text { TestScore } } = 698.9 - 2.28 \times \text { STR }$
You are told that the t -statistic on the slope coefficient is 4.38 . What is the standard error of the slope coefficient?

A) 0.52
B) 1.96
C) -1.96
D) 4.38

Question 8

The p-value for a one-sided left-tail test

Question 9

The 95% confidence interval for the predicted effect of a general change in X is

Question 10

Finding a small value of the p-value (e.g.less than 5%)

A)indicates evidence in favor of the null hypothesis.
B)implies that the t-statistic is less than 1.96.
C)indicates evidence in against the null hypothesis.
D)will only happen roughly one in twenty samples.

Question 11

The only difference between a one- and two-sided hypothesis test is

A)the null hypothesis.
B)dependent on the sample size n.
C)the sign of the slope coefficient.
D)how you interpret the t-statistic.

Question 12

With heteroskedastic errors, the weighted least squares estimator is BLUE.You should use OLS with heteroskedasticity-robust standard errors because

A)this method is simpler.
B)the exact form of the conditional variance is rarely known.
C)the Gauss-Markov theorem holds.
D)your spreadsheet program does not have a command for weighted least squares.

Question 13

Heteroskedasticity means that

A)homogeneity cannot be assumed automatically for the model.
B)the variance of the error term is not constant.
C)the observed units have different preferences.
D)agents are not all rational.

Question 14

If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal distribution, you can

A)reject the null hypothesis.
B)safely assume that your regression results are significant.
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.

Question 15

The 95% confidence interval for $\beta _ { 1 }$ is the interval

A) $\left( \beta _ { 1 } - 1.96 S E \left( \beta _ { 1 } \right) , \beta _ { 1 } + 1.96 S E \left( \beta _ { 1 } \right) \right)$
B) $\left( \widehat { \beta } _ { 1 } - 1.645 \operatorname { SE } \left( \widehat { \beta } _ { 1 } \right) , \widehat { \beta } _ { 1 } + 1.645 \operatorname { SE } \left( \widehat { \beta } _ { 1 } \right) \right)$
C) $\left( \widehat { \beta } _ { 1 } - 1.96 \operatorname { SE } \left( \widehat { \beta } _ { 1 } \right) , \widehat { \beta } _ { 1 } + 1.96 \operatorname { SE } \left( \widehat { \beta } _ { 1 } \right) \right)$
D) $\left( \widehat { \beta } _ { 1 } - 1.96 , \widehat { \beta } _ { 1 } + 1.96 \right)$

Question 16

The homoskedasticity-only estimator of the variamce of $\widehat { \beta } _ { 1 }$ is

A) $\frac { s _ { \hat { u } } ^ { 2 } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$
B) $\frac{s_{\hat{u}}}{\sqrt{\sum_{i=1}^{n}\left(X_{i}-\bar{X}\right)^{2}}}$

C) $\frac { s _ { \hat { u } } ^ { 2 } } { \sum _ { i = 1 } ^ { n } X _ { i } ^ { 2 } - \bar { X } }$
D) $\frac { 1 } { n } \times \frac { \frac { 1 } { n - 2 } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } \hat { u } _ { i } ^ { 2 } } { \left[ \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } \right] ^ { 2 } }$

Question 17

One of the following steps is not required as a step to test for the null hypothesis:

Question 18

In general, the t-statistic has the following form:

Question 19

Under the least squares assumptions (zero conditional mean for the error term, Xi and Yi being i.i.d., and Xi and ui having finite fourth moments), the OLS estimator
For the slope and intercept

Question 20

The 95 % confidence interval for $\beta _ { 0 }$ is the interval

A) $\left( \beta _ { 0 } - 1.96 S E \left( \beta _ { 0 } \right) , \beta _ { 0 } + 1.96 S E \left( \beta _ { 0 } \right) \right)$
B) $\left( \widehat { \beta } _ { 0 } - 1.645 \operatorname { SE } \left( \widehat { \beta } _ { 0 } \right) , \widehat { \beta } _ { 0 } + 1.645 \operatorname { SE } \left( \widehat { \beta } _ { 0 } \right) \right)$
C) $\left( \widehat { \beta } _ { 0 } - 1.96 \operatorname { SE } \left( \widehat { \beta } _ { 0 } \right) , \widehat { \beta } _ { 0 } + 1.96 \operatorname { SE } \left( \widehat { \beta } _ { 0 } \right) \right)$
D) $\left( \widehat { \beta } _ { 0 } - 1.96 , \widehat { \beta } _ { 0 } + 1.96 \right)$

Question 21

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: (a)What is the interpretation of the intercept? What is the interpretation of the slope?
How tall are females, on average?

Question 22

(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: (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.

Question 23

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

Question 24

The proof that OLS is BLUE requires all of the following assumptions with the exception of:

Question 25

The error term is homoskedastic if

Question 26

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.

Question 27

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?

Question 28

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.

Question 29

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.

Question 30

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?

Question 31

(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: 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.

Question 32

(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: where Earn and Age are measured in dollars and years respectively.
(a)Is the relationship between Age and Earn statistically significant?

Question 33

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.

Question 34

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: (a)Interpret the regression results carefully.

Question 35

(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: Using the OLS estimator with homoskedasticity-only standard errors, the results
changed as follows: 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.

Question 36

For the following estimated slope coefficients and their heteroskedasticity robust standard errors, find the t -statistics for the null hypothesis 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)
(b)
(c)
(d)

Question 37

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

Question 38

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)

Question 39

(a)Is there any reason to believe that the variance of the error terms is
homoskedastic?

Question 40

(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: 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: (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?

Question 41

Changing the units of measurement obviously will have an effect on the slope of your regression function. For example, let
Then it is easy
but tedious to show that Given this result, how do you think
the standard errors and the regression will change?

Question 42

(Requires Appendix material) Your textbook shows that OLS is a linear estimator
https://d2lvgg3v3hfg70.cloudfront.net/TB34225555/.
For OLS to be conditionally unbiased, the following two conditions must hold:

Show that this is the case.

Question 43

You are delighted to see that this program has already calculated p-values for you.
However, a peer of yours points out that the correct p-value should be 0.4562.
Who is right?

Question 44

Your textbook discussed the regression model when X is a binary variable

Question 45

Your textbook states that under certain restrictive conditions, the t- statistic has a
Student t-distribution with n-2 degrees of freedom.The loss of two degrees of
freedom is the result of OLS forcing two restrictions onto the data.What are these
two conditions, and when did you impose them onto the data set in your
derivation of the OLS estimator?

Question 46

22

Question 47

In many of the cases discussed in your textbook, you test for the significance of
the slope at the 5% level.What is the size of the test? What is the power of the
test? Why is the probability of committing a Type II error so large here?

Question 48

Consider the sample regression function
The table below lists estimates for the slope and the variance of the slope estimator In each case calculate the p -value for the null hypothesis of and a two-tailed alternative hypothesis. Indicate in which case you would reject the null hypothesis at the 5 % significance level.

Question 49

Assume that the homoskedastic normal regression assumption hold.Using the
Student t-distribution, find the critical value for the following situation:

Question 50

Consider the following two models involving binary variables as explanatory
variables:

Question 51

Your textbook discussed the regression model when X is a binary variable

Question 52

In the regression through the origin model the OLS estimator is
Prove that the estimator is a linear function of and prove
that it is conditionally unbiased.

Question 53

Assume that your population regression function is i.e., a regression through the origin (no intercept).Under the homoskedastic
normal regression assumptions, the t-statistic will have a Student t distribution
with n-1 degrees of freedom, not n-2 degrees of freedom, as was the case in
Chapter 5 of your textbook.Explain.Do you think that the residuals will still sum
to zero for this case?