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

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?

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?

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

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?}

Consider the sample regression function $\hat { Y }$ _i = $\hat{\beta} _ { 0 }$ + $\hat{\beta} _ { 1 }$ X_i. The table below lists estimates for the slope ( $\hat{\beta }_ { 1 }$ )and the variance of the slope estimator ( $\hat { \sigma } ^ { 2 } { \hat { \beta } 1 }$ ). In each case calculate the p-value for the null hypothesis of ?₁ = 0 and a two-tailed alternative hypothesis. Indicate in which case you would reject the null hypothesis at the 5% significance level. -1.76 0.0025 2.85 -0.00014 0.37 0.000003 117.5 0.0000013

(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.

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?

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

Consider the following two models involving binary variables as explanatory variables: $\widehat { \text { Wage } }$ = $\widehat { \beta 0 }$ + $\widehat { \beta 1 }$ DFemme and $\widehat { \text { Wage } }$ = $\widehat { \phi _ { 1 } }$ DFemme + $\widehat { \phi _ { 2 } }$ Male where Wage is the hourly wage rate, DFemme is a binary variable that is equal to 1 if the person is a female, and 0 if the person is a male. Male = 1 - DFemme. Even though you have not learned about regression functions with two explanatory variables (or regressions without an intercept), assume that you had estimated both models, i.e., you obtained the estimates for the regression coefficients. What is the predicted wage for a male in the two models? What is the predicted wage for a female in the two models? What is the relationship between the ? s and the ?s? Why would you prefer one model over the other?

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?

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

(Requires Appendix material and Calculus)Equation (5.36)in your textbook derives the conditional variance for any old conditionally unbiased estimator $\beta$ ₁ to be var( $\beta$ 1 X₁, ..., X_n)= $\sigma _ { u } ^ { 2 } \sum _ { i = 1 } ^ { n } a _ { i } ^ { 2 }$ where the conditions for conditional unbiasedness are $\sum _ { i = 1 } ^ { n } a _ { i }$ = 0 and $\sum _ { i = 1 } ^ { n } a _ { i } X _ { i }$ = 1. As an alternative to the BLUE proof presented in your textbook, you recall from one of your calculus courses that you could minimize the variance subject to the two constraints, thereby making the variance as small as possible while the constraints are holding. Show that in doing so you get the OLS weights $\hat { a } _ { i }$ (You may assume that X₁,..., X_n are nonrandom (fixed over repeated samples).)

Using data from the Current Population Survey, you estimate the following relationship between _{average hourly earnings (ahe)and the number of years of education (educ):} = -4.58 + 1.71 educ The heteroskedasticity-robust standard error on the slope is (0.03). Calculate the 95% confidence interval for the slope. Repeat the exercise using the 90% and then the 99% confidence interval. Can you reject the null hypothesis that the slope coefficient is zero in the population?

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

The error term is homoskedastic if

If the errors are heteroskedastic, then

(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: =17,375,=7,665.5, =94,228.8,=1,248.9,=7,625.9 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?

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

(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.

(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?