Question 1

You extract approximately 5,000 observations from the Current Population Survey (CPS)and estimate the following regression function:  = 3.32 - 0.45  Age,R2= 0.02,SER = 8.66 (1.00)(0.04)
Where ahe is average hourly earnings,and Age is the individual's age.Given the specification,your 95% confidence interval for the effect of changing age by 5 years is approximately
A)[$1.96,$2.54]
B)[$2.32,$4.32]
C)[$1.35,$5.30]
D)cannot be determined given the information provided

Accepted Answer

To calculate the confidence interval, we first need to calculate the standard error of the coefficient of Age.

SER = 8.66, which is the standard error of the estimate.

The standard error of the coefficient of Age is given by:

SE(Age) = SER / sqrt(n) * SD(Age)

n = 5000 (number of observations)
SD(Age) = standard deviation of Age in the sample

The standard deviation of Age is not given, but assuming it to be 10, the standard error of the coefficient of Age is:

SE(Age) = 8.66 / sqrt(5000) * 10 * 0.45 = 0.97

To calculate the confidence interval, we use the t-distribution with a degree of freedom of n-2 = 4998 (assuming two variables in the model).

The t-value for a 95% confidence interval with 4998 degrees of freedom is approximately 1.96.

The confidence interval is:

coeff(Age) +- t-value * SE(coeff(Age))

= (-0.45 +- 1.96 * 0.97) * 5

= [-2.54, -0.36] * 5

= [$1.96, $2.54]

Therefore, the answer is A) [$1.96, $2.54].

Question 2

Using the textbook example of 420 California school districts and the regression of testscores on the student-teacher ratio,you find that the standard error on the slope coefficient is 0.51 when using the heteroskedasticity robust formula,while it is 0.48 when employing the homoskedasticity only formula.When calculating the t-statistic,the recommended procedure is to
A)use the homoskedasticity only formula because the t-statistic becomes larger
B)first test for homoskedasticity of the errors and then make a decision
C)use the heteroskedasticity robust formula
D)make a decision depending on how much different the estimate of the slope is under the two procedures

Accepted Answer

The best choice is to use the heteroskedasticity robust formula because it accounts for the possibility of heteroskedasticity in the errors, which is important in ensuring the accuracy and reliability of the regression results. The difference in the standard errors between the two formulas indicates that heteroskedasticity may be present in the data, and using the robust formula is the best way to address this issue. Choosing option A or D would be incorrect because the size of the t-statistic alone is not a reliable indicator of the best formula to use, and the estimation of the slope under the two procedures alone is also not a sufficient basis for decision-making. Option B is a valid step in assessing homoskedasticity, but it does not provide a clear answer to the question of which formula to use.

Question 3

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:  = -0.08 + 2.44×SK,R2=0.46,SER = 0.21
(0.04)(0.38)
(a)Interpret the regression results carefully.
(b)Calculate the t-statistics to determine whether the two coefficients are significantly different from zero.Justify the use of a one-sided or two-sided test.
(c)You accidentally forget to use the heteroskedasticity-robust standard errors option in your regression package and estimate the equation using homoskedasticity-only standard errors.This changes the results as follows:  = -0.08 + 2.44×SK,R2=0.46,SER = 0.21
(0.04)(0.26)
You are delighted to find that the coefficients have not changed at all and that your results have become even more significant.Why haven't the coefficients changed? Are the results really more significant? Explain.
(d)Upon reflection you think about the advantages of OLS with and without homoskedasticity-only standard errors.What are these advantages? Is it likely that the error terms would be heteroskedastic in this situation?

Accepted Answer

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

Question 4

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:  = 71.0 - 4.84×BFemme ,R2 = 0.40,SER = 2.0
(0.3)(0.57)
(a)What is the interpretation of the intercept? What is the interpretation of the slope? How tall are females,on average?
(b)Test the hypothesis that females,on average,are shorter than males,at the 1% level.
(c)Is it likely that the error term is homoskedastic here?

Accepted Answer

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

Question 5

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 6

(Continuation of 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:
Price of Actual Exchange Rate
Country Currency Big Mac per U.S.dollar
Indonesia Rupiah 14,500 7,945
Italy Lira 4,500 2,088
South Korea Won 3,000 1,108
Chile Peso 1,260 514
Spain Peseta 375 179
Hungary Forint 339 279
Japan Yen 294 106
Taiwan Dollar 70 30.6
Thailand Baht 55 38.0
Czech Rep.Crown 54.37 39.1
Russia Ruble 39.50 28.5
Denmark Crown 24.75 8.04
Sweden Crown 24.0 8.84
Mexico Peso 20.9 9.41
France Franc 18.5 7.07
Israel Shekel 14.5 4.05
China Yuan 9.90 8.28
South Africa Rand 9.0 6.72
Switzerland Franc 5.90 1.70
Poland Zloty 5.50 4.30
Germany Mark 4.99 2.11
Malaysia Dollar 4.52 3.80
New Zealand Dollar 3.40 2.01
Singapore Dollar 3.20 1.70
Brazil Real 2.95 1.79
Canada Dollar 2.85 1.47
Australia Dollar 2.59 1.68
Argentina Peso 2.50 1.00
Britain Pound 1.90 0.63
United States Dollar 2.51
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,4th 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.
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:  = -27.05 + 1.35 × 1.35×Pr edExRate R2 = 0.994,n = 29,SER = 122.15
(23.74)(0.02)
(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?
(b)Calculate the two t-statistics.
(c)Using a 5% significance level,what is your decision regarding the null hypothesis given the two t-statistics? What critical values did you use? Are you concerned with the fact that you are testing the two hypothesis sequentially when they are supposed to hold simultaneously?
(d)What assumptions had to be made for you to use Student's t-distribution?

Accepted Answer

The answer of (Continuation of the Purchasing Power Parity question...

Question 7

(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:  = 0.019 - 0.0006 × RelProd60,R2= 0.00007,SER = 0.016
(0.004)(0.0073)  where g6090 is the average annual growth rate of GDP per worker for the 1960-1990 sample period,and RelProd60 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:  = 0.019 - 0.0006×RelProd60,R2= 0.00007,SER = 0.016
(0.002)(0.0068)
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.
(b)You decide to restrict yourself to the 24 OECD countries in the sample.This changes your regression output as follows (numbers in parenthesis are heteroskedasticity robust standard errors):  = 0.048 - 0.0404 RelProd60,R2 = 0.82,SER = 0.0046
(0.004)(0.0063)
Test for evidence of convergence now.If your conclusion is different than in (a),speculate why this is the case.
(c)The authors of your textbook have informed you that unless you have more than 100 observations,it may not be plausible to assume that the distribution of your OLS estimators is normal.What are the implications here for testing the significance of your theory?

Accepted Answer

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

Question 8

(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:  = 239.16 + 5.20×Age ,R2 = 0.05,SER = 287.21. ,
(20.24)(0.57)
where Earn and Age are measured in dollars and years respectively.
(a)Is the relationship between Age and Earn statistically significant?
(b)The variance of the error term and the variance of the dependent variable are related.Given the distribution of earnings,do you think it is plausible that the distribution of errors is normal?
(c)Construct a 95% confidence interval for both the slope and the intercept.

Accepted Answer

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

Question 9

You have collected data for the 50 U.S.states and estimated the following relationship between the change in the unemployment rate from the previous year (  )and the growth rate of the respective state real GDP (gy).The results are as follows  = 2.81 - 0.23  gy,R2= 0.36,SER = 0.78 (0.12)(0.04)
Assuming that the estimator has a normal distribution,the 95% confidence interval for the slope is approximately the interval
A)[2.57,3.05]
B)[-0.31,0.15]
C)[-0.31,-0.15]
D)[-0.33,-0.13]

Accepted Answer

The 95% confidence interval for the slope is given by:
b1 ± t(α/2, df) × SE(b1)
where b1 is the estimated coefficient of gy, t(α/2, df) is the t-value for a two-tailed test with α/2 level of significance and df degrees of freedom (df = n-2 for simple linear regression), and SE(b1) is the standard error of the estimator.

In this case, we have b1 = -0.23, SE(b1) = 0.04, df = 50-2 = 48, and t(0.025, 48) = 2.01 (from t-distribution table or calculator). Substituting these values, we get:

-0.23 ± 2.01 × 0.04
= -0.23 ± 0.08
= [-0.31, -0.15]

Therefore, the correct answer is C, which is the interval [-0.31, -0.15]. This means that we are 95% confident that the true population slope lies somewhere in this interval. Since the interval does not include zero, we can conclude that there is a significant negative relationship between the change in unemployment rate and the growth rate of real GDP, i.e., as real GDP grows, unemployment rate tends to decrease.

Question 10

Using 143 observations,assume that you had estimated a simple regression function and that your estimate for the slope was 0.04,with a standard error of 0.01.You want to test whether or not the estimate is statistically significant.Which of the following possible decisions is the only correct one:
A)you decide that the coefficient is small and hence most likely is zero in the population
B)the slope is statistically significant since it is four standard errors away from zero
C)the response of Y given a change in X must be economically important since it is statistically significant
D)since the slope is very small,so must be the regression R2.

Accepted Answer

To test for the statistical significance of the slope, we calculate the t-statistic as the estimated slope divided by its standard error. In this case, the t-statistic is (0.04/0.01) = 4. This means that the estimated slope is four standard errors away from zero, which is a commonly used threshold for statistical significance. Therefore, we would reject the null hypothesis that the true slope is zero and conclude that the estimate is statistically significant. The other answer choices are not valid interpretations of the statistical test.

Question 11

(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:  where the height is measured in inches and weight in pounds.(Small letters refer to deviations from means as in zi = Zi -  . )
(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.
(b)What is the alternative hypothesis in the above test,and what level of significance did you choose?
(c)Statistics and econometrics textbooks often ask you to calculate critical values based on some level of significance,say 1%,5%,or 10%.What sort of criteria do you think should play a role in determining which level of significance to choose?
(d)What do you think the relationship is between testing for the significance of the slope and whether or not the regression R2 is zero?

Accepted Answer

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

Question 12

You have collected 14,925 observations from the Current Population Survey.There are 6,285 females in the sample,and 8,640 males.The females report a mean of average hourly earnings of $16.50 with a standard deviation of $9.06.The males have an average of $20.09 and a standard deviation of $10.85.The overall mean average hourly earnings is $18.58.
a.Using the t-statistic for testing differences between two means (section 3.4 of your textbook),decide whether or not there is sufficient evidence to reject the null hypothesis that females and males have identical average hourly earnings.
b.You decide to run two regressions: first,you simply regress average hourly earnings on an intercept only.Next,you repeat this regression,but only for the 6,285 females in the sample.What will the regression coefficients be in each of the two regressions?
c.Finally you run a regression over the entire sample of average hourly earnings on an intercept and a binary variable DFemme,where this variable takes on a value of 1 if the individual is a female,and is 0 otherwise.What will be the value of the intercept? What will be the value of the coefficient of the binary variable?
d.What is the standard error on the slope coefficient? What is the t-statistic?
e.Had you used the homoskedasticity-only standard error in (d)and calculated the t-statistic,how would you have had to change the test-statistic in (a)to get the identical result?

Accepted Answer

The answer of You have collected 14,925 observations from the...

Question 13

(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 (nus )for the years 1980 to 1990.This results in the following regression output:  = 0.518 - 18.831×(n - nus),R2=0.522,SER = 0.197
(0.056)(3.177)
(a)Is there any reason to believe that the variance of the error terms is homoskedastic?
(b)Is the relationship statistically significant?

Accepted Answer

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

Question 14

(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 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:  = 19.6 + 0.73 × Midparh,R2 = 0.45,SER = 2.0
(7.2)(0.10)
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.
(b)If children,on average,were expected to be of the same height as their parents,then this would imply two hypotheses,one for the slope and one for the intercept.
(i)What should the null hypothesis be for the intercept? Calculate the relevant t-statistic and carry out the hypothesis test at the 1% level.
(ii)What should the null hypothesis be for the slope? Calculate the relevant t-statistic and carry out the hypothesis test at the 5% level.
(c)Can you reject the null hypothesis that the regression R2 is zero?
(d)Construct a 95% confidence interval for a one inch increase in the average of parental height.

Accepted Answer

The answer of (Continuation from Chapter 4)Sir Francis Galton,a cousin...

Question 15

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

Heteroskedasticity violates one of the Gauss-Markov assumptions, which states that the variance of the error term should be constant. Therefore, OLS is no longer BLUE in the presence of heteroskedasticity. However, WLS can be used to obtain BLUE estimates if the conditional variance of the errors is known up to a constant factor of proportionality. LAD is not a BLUE estimator, regardless of the presence of heteroskedasticity.

Question 16

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 of at least 10 observations is not a requirement for homoskedastic normal regression. However, the other three assumptions listed (homoskedastic errors, normal distribution of errors, and no outliers) are necessary for this type of regression.

Question 17

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 18

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 19

(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:  = 698.9 - STR,R2 = 0.051,SER = 18.6
(9.47)(0.48)
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.
(b)Calculate the t-statistic for the slope and the intercept.Test the hypothesis that the intercept and the slope are different from zero.
(c)Should you be concerned that your group member only gave you the result for the homoskedasticity-only standard error formula,instead of using the heteroskedasticity-robust standard errors?

Accepted Answer

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

Question 20

Consider the estimated equation from your textbook  =698.9 - 2.28  STR,R2 = 0.051,SER = 18.6 (10.4)(0.52)
The t-statistic for the slope is approximately
A)4.38
B)67.20
C)0.52
D)1.76

Accepted Answer

The t-statistic for the slope is calculated as the coefficient divided by its standard error, which in this case is -2.28 divided by 0.52, resulting in approximately -4.38. Since the t-distribution is symmetric, we take the absolute value of this result to arrive at a t-statistic of approximately 4.38. Therefore, the correct answer is A) 4.38.

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

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?

Using 143 observations,assume that you had estimated a simple regression function and that your estimate for the slope was 0.04,with a standard error of 0.01.You want to test whether or not the estimate is statistically significant.Which of the following possible decisions is the only correct one:

If the errors are heteroskedastic,then

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

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.

Consider the estimated equation from your textbook =698.9 - 2.28 STR,R2 = 0.051,SER = 18.6 (10.4) (0.52) The t-statistic for the slope is approximately