AIMs: Define and differentiate between homoskedasticity and heteroskedasticity. Describe the implications of homoskedasticity and heteroskedasticity. Explain the Gauss-Markov Theorem and its limitations, and alternatives to the OLS.
Questions:
218.1. We want to regress hourly Earnings (the regressand) against years of Education (the regressor) based on the following OLS regression model: Earnings(i) = B(0) + B(1)*Education(i) + u(i), where u(i) is the error term. After we run the regression, which of the following statement MOST NEARLY demonstrates homoskedasticity?
a. Education(i) is not a linear function of any other regressor
b. Earnings(i) is independent of Education(i)
c. The variance of the error, u(i), is independent of Education(i)
d. The error term has a conditional mean of zero, E[u(i) | Education(i)] = 0
218.2. Assume we have confirmed that all three of Stock & Watson's assumptions are true for our OLS linear regression model; i.e., the error term has a mean of zero conditional on the regressor; the [X(i),Y(i)] observations are i.i.d. random draws; and large outliers are unlikely. Our OLS regression model is: Y(i) = B(0) + B(1)*X(i) + u(i). Each of the following is true EXCEPT for:
a. Whether the errors are homo- or heteroskedastic, the OLS estimators are are unbiased, consistent and asymptotically normal
b. If the errors are heteroskedastic, we can compute heteroskedasticity-robust standard errors
c. If it is true that, in addition to the three assumptions above, that the errors are homoskedastic, then our OLS estimator for B(1) is BLUE
d. As heteroskedasticity is a special case of homoskedasticity, and given that homoskedasticity is more most prevalent, the safest practice is to employ homoskedasticity-robust standard errors
218.3. You presented a regression model to your boss, the Chief Risk Officer (CRO). She is a certified FRM so you know that she knows statistics, although she laments the decision to replace rigorous Gujarati with a softer, gentler Stock & Watson. She queries you on the dataset and your regression, and you admit to two realities: First, the error term is heteroskedastic. Second, there are many extreme outliers in the dataset. Your boss makes the following assertions:
I. "It is okay, for our purposes, that the error term is heteroskedastic: the slope (B1) estimator remains efficient and BLUE."
II. "Since we have many extreme outliers, the least absolute deviations (LAD) is a viable alternative to OLS, because its estimators may be more efficient (i.e., have smaller variances)"
Which of your boss' statements is (are) true?
a. Neither
b. I. only
c. II. only
d. Both are true
Answers:
Questions:
218.1. We want to regress hourly Earnings (the regressand) against years of Education (the regressor) based on the following OLS regression model: Earnings(i) = B(0) + B(1)*Education(i) + u(i), where u(i) is the error term. After we run the regression, which of the following statement MOST NEARLY demonstrates homoskedasticity?
a. Education(i) is not a linear function of any other regressor
b. Earnings(i) is independent of Education(i)
c. The variance of the error, u(i), is independent of Education(i)
d. The error term has a conditional mean of zero, E[u(i) | Education(i)] = 0
218.2. Assume we have confirmed that all three of Stock & Watson's assumptions are true for our OLS linear regression model; i.e., the error term has a mean of zero conditional on the regressor; the [X(i),Y(i)] observations are i.i.d. random draws; and large outliers are unlikely. Our OLS regression model is: Y(i) = B(0) + B(1)*X(i) + u(i). Each of the following is true EXCEPT for:
a. Whether the errors are homo- or heteroskedastic, the OLS estimators are are unbiased, consistent and asymptotically normal
b. If the errors are heteroskedastic, we can compute heteroskedasticity-robust standard errors
c. If it is true that, in addition to the three assumptions above, that the errors are homoskedastic, then our OLS estimator for B(1) is BLUE
d. As heteroskedasticity is a special case of homoskedasticity, and given that homoskedasticity is more most prevalent, the safest practice is to employ homoskedasticity-robust standard errors
218.3. You presented a regression model to your boss, the Chief Risk Officer (CRO). She is a certified FRM so you know that she knows statistics, although she laments the decision to replace rigorous Gujarati with a softer, gentler Stock & Watson. She queries you on the dataset and your regression, and you admit to two realities: First, the error term is heteroskedastic. Second, there are many extreme outliers in the dataset. Your boss makes the following assertions:
I. "It is okay, for our purposes, that the error term is heteroskedastic: the slope (B1) estimator remains efficient and BLUE."
II. "Since we have many extreme outliers, the least absolute deviations (LAD) is a viable alternative to OLS, because its estimators may be more efficient (i.e., have smaller variances)"
Which of your boss' statements is (are) true?
a. Neither
b. I. only
c. II. only
d. Both are true
Answers:
