1. 1 We will illustrate the heteroscedasticity theory with a Monte Carlo simulation. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION 1 standard deviation of u Y i = 10 + 2.0X i + u i X i = {5,6,..., 54} u i = X i  i  i ~ N(0,1)

2. 2 Y = 10 + 2X + u, the data for X are the integers from 5 to 54, and u = X , where  is iid N(0,1) (identically and independently distributed, drawn from a normal distribution with zero mean and unit variance). HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION 1 standard deviation of u Y i = 10 + 2.0X i + u i X i = {5,6,..., 54} u i = X i  i  i ~ N(0,1)

3. 3 The blue circles give the nonstochastic component of Y in the observations. The lines give the points one standard deviation of u above and below the nonstochastic component of Y and show how the distribution of Y spreads in the vertical dimension as X increases. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION 1 standard deviation of u Y i = 10 + 2.0X i + u i X i = {5,6,..., 54} u i = X i  i  i ~ N(0,1)

4. 4 The standard deviation of u i is equal to X i. The heteroscedasticity is thus of the type detected by the Goldfeld–Quandt test and it may be eliminated using weighted least squares (WLS). We scale through by X, weighting observation i by multiplying it by 1/X i. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION 1 standard deviation of u Y i = 10 + 2.0X i + u i X i = {5,6,..., 54} u i = X i  i  i ~ N(0,1)

5. 5 The diagram shows the results of estimating the slope coefficient using WLS and OLS for one million samples. 1 million samples HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION Slope coefficient estimated using WLS Y i /X i = 10/X i + 2.0 +  i s.d. = 0.227 Slope coefficient estimated using OLS s.d. = 0.346 Y i = 10 + 2.0X i + u i X i = {5,6,..., 54} u i = X i  i  i ~ N(0,1)

6. 6 Both OLS and WLS are unbiased, but the distribution of the WLS estimates has a smaller variance. This illustrates the fact that OLS is inefficient in the presence of heteroscedasticity. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION 1 million samples Y i = 10 + 2.0X i + u i X i = {5,6,..., 54} u i = X i  i  i ~ N(0,1) Slope coefficient estimated using WLS Y i /X i = 10/X i + 2.0 +  i s.d. = 0.227 Slope coefficient estimated using OLS s.d. = 0.346

7. 7 Heteroscedasticity also causes the OLS standard error to be invalidated. The diagram shows the distributions of the OLS and WLS standard errors of the slope coefficient for the million samples. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION S.e. estimated using WLS Mean of distribution = 0.226 Actual s.d. = 0.227 S.e. estimated using OLS Mean of distribution = 0.319 Actual s.d. = 0.346 1 million samples

8. 8 The standard deviation of the distribution of the WLS estimates of the slope coefficient in the first slide was 0.227, and we can see that the WLS standard errors in the million samples are distributed around this figure. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION S.e. estimated using WLS Mean of distribution = 0.226 Actual s.d. = 0.227 S.e. estimated using OLS Mean of distribution = 0.319 Actual s.d. = 0.346 1 million samples

9. 9 Sometimes the WLS standard error overestimates the underlying standard deviation of the distribution of the slope coefficient, sometimes it underestimates it, but there is no evident bias. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION S.e. estimated using WLS Mean of distribution = 0.226 Actual s.d. = 0.227 S.e. estimated using OLS Mean of distribution = 0.319 Actual s.d. = 0.346 1 million samples

10. 10 The standard deviation of the distribution of the OLS estimates of the slope coefficient in the first slide was 0.346, and we can see that the OLS standard error is downwards biased. The mean of its distribution in the million samples is 0.319. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION S.e. estimated using WLS Mean of distribution = 0.226 Actual s.d. = 0.227 S.e. estimated using OLS Mean of distribution = 0.319 Actual s.d. = 0.346 1 million samples

11. 11 Because the OLS standard error is downwards biased, the OLS t statistic will tend to be higher than it should be. What happens if we use heteroscedasticity-consistent, or robust, standard errors instead? HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION S.e. estimated using WLS Mean of distribution = 0.226 Actual s.d. = 0.227 S.e. estimated using OLS Mean of distribution = 0.319 Actual s.d. = 0.346 1 million samples

12. 12 The mean of the distribution of the robust standard errors is 0.335. This is closer to the true standard deviation, 0.346. But the dispersion of the robust standard errors is larger. Hence, in this case, it is not clear that the robust standard errors are actually more reliable. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION S.e. estimated using WLS Mean of distribution = 0.226 Actual s.d. = 0.227 S.e. estimated using OLS Mean of distribution = 0.319 Actual s.d. = 0.346 1 million samples Robust s.e. estimated using OLS Mean of distribution = 0.335 Actual s.d. = 0.346

13. 13 Robust standard errors are valid only in large samples. For small samples, their properties are unknown and they could actually be more misleading than the ordinary OLS standard errors. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION S.e. estimated using WLS Mean of distribution = 0.226 Actual s.d. = 0.227 S.e. estimated using OLS Mean of distribution = 0.319 Actual s.d. = 0.346 1 million samples Robust s.e. estimated using OLS Mean of distribution = 0.335 Actual s.d. = 0.346

14. 14 We could investigate this In the case of the present model by making the sample larger or smaller than the 50 observations used in each sample in this simulation. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION S.e. estimated using WLS Mean of distribution = 0.226 Actual s.d. = 0.227 S.e. estimated using OLS Mean of distribution = 0.319 Actual s.d. = 0.346 1 million samples Robust s.e. estimated using OLS Mean of distribution = 0.335 Actual s.d. = 0.346

15. Copyright Christopher Dougherty 2012. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section 7.3 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.11.10