Question 1

You extract approximately 5,000 observations from the Current Population Survey (CPS)and estimate the following regression function: $\widehat { \text { ahe } }$ = 3.32 - 0.45 $\times$ 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

Accepted Answer

The effect of changing age by 5 years is -0.45 * 5 = -2.25. The standard error for the age coefficient is 0.04, so for 5 years, it is 0.04 * 5 = 0.20. The 95% confidence interval is -2.25 ± 1.96 * 0.20, which is approximately [-2.65, -1.85]. However, since the question asks for the absolute value, the interval is [1.85, 2.65], which rounds to [1.96, 2.54].

Question 2

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 ta}_{kes 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_-_statist_ic? _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 3

If the errors are heteroskedastic, then

Accepted Answer

Heteroskedasticity means that the variance of the error terms is not constant across all observations. In this case, OLS is inefficient and produces biased estimators. Weighted Least Squares (WLS) is one possible solution, where the weights are chosen based on the inverse of the conditional variance function of the errors. However, this requires knowledge of the functional form of the variance function, which is often unknown. If the variance function is known up to a constant factor of proportionality, then WLS is BLUE. LAD is not a suitable method for correcting heteroskedasticity.

Question 4

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 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

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 7

(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: $\widehat {\text { TestScore }}$ = 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 8

Consider the estimated equation from your textbook $\widehat {\text { TestScore }}$ =698.9 - 2.28 $\times$ STR, R² = 0.051, SER = 18.6 (10.4)(0.52) The t-statistic for the slope is approximately

Accepted Answer

The t-statistic is calculated by dividing the coefficient by its standard error. Here, the slope coefficient is -2.28 and its standard error is 0.52. So, the t-statistic is -2.28 / 0.52 = -4.38.

Question 9

(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: $\widehat { g 6090 }$ = 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: $\widehat { g 6090 }$ = 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): $\widehat { g 6090 }$ = 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...

Question 10

(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. (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 (Continuation from Chapter 4)At a recent county...

Question 11

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

Accepted Answer

The recommended procedure is to use the heteroskedasticity robust formula as it provides more accurate standard errors when there is heteroskedasticity present in the data. As the standard error on the slope coefficient is higher when using the heteroskedasticity robust formula, it is likely that heteroskedasticity is present in the data, and thus using the heteroskedasticity robust formula is appropriate. The other options (A, B, and D) do not provide a sound rationale for their choice.

Question 12

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

Accepted Answer

The assumption that there are at least 10 observations is not specific to the homoskedastic normal regression assumptions but rather a general guideline for regression modeling. Therefore, it is not an assumption specific to homoskedastic normal regression. The other three choices are all assumptions specific to homoskedastic normal regression.

Question 13

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:

Accepted Answer

The correct decision is that the slope is statistically significant since it is four standard errors away from zero. This is determined by dividing the estimated coefficient (0.04) by its standard error (0.01) to get a t-statistic of 4. Therefore, we reject the null hypothesis that the coefficient is zero and conclude that there is evidence of a non-zero relationship between X and Y in the population. The other choices are incorrect or irrelevant to the statistical significance of the slope estimate.

Question 14

(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 (n_us )for the years 1980 to 1990. This results in the following regression output: $\widehat {\text { RelPersInc }}$ = 0.518 - 18.831×(n - n_us), 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...

Question 15

(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^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: $\widehat {\text { Studenth }}$ = 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...

Question 16

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 ( $\widehat{\Delta u r}$ )and the growth rate of the respective state real GDP (g_y). The results are as follows $\widehat{\Delta u r}$ = 2.81 - 0.23 $\times$ g_y, 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

Accepted Answer

The 95% confidence interval for the slope coefficient (-0.23) is calculated using the standard error of the estimate (0.04) and the critical value for a 95% confidence level from the normal distribution (approximately 1.96). The interval is computed as -0.23 ± 1.96*0.04, which results in approximately [-0.31, -0.15].

Question 17

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 (S_K), remembering that investment equals saving. The regression results in the following output: $\widehat {\text { RelProd }}$ = -0.08 + 2.44×S_K, 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: $\widehat {\text { RelProd }}$ = -0.08 + 2.44×S_K, 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 18

(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: $\widehat {E a m }$ = 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...

Question 19

(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: 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^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. 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: $\widehat{\text { ActualExRate }}$ = -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 20

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: $\widehat {\text { Studenth }}$ = 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...

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

If the errors are heteroskedastic, then

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.

Consider the estimated equation from your textbook $\widehat {\text { TestScore }}$ =698.9 - 2.28 $\times$ STR, R² = 0.051, SER = 18.6 (10.4) (0.52) The t-statistic for the slope is approximately

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

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

If the errors are heteroskedastic, then

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.

Consider the estimated equation from your textbook TestScore ^\widehat {\text { TestScore }} TestScore =698.9 - 2.28 ×\times× STR, R2 = 0.051, SER = 18.6 (10.4) (0.52) The t-statistic for the slope is approximately

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

Consider the estimated equation from your textbook $\widehat {\text { TestScore }}$ =698.9 - 2.28 $\times$ STR, R² = 0.051, SER = 18.6 (10.4) (0.52) The t-statistic for the slope is approximately