1. EC220 - Introduction to econometrics (chapter 7)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 7)
Slideshow: heteroscedasticity: Monte Carlo illustration
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 7). [Teaching Resource]
© 2012 The Author
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Available in LSE Learning Resources Online: May 2012
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2. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
1 standard
deviation of u
We will illustrate the heteroscedasticity theory with a Monte Carlo simulation in which Y = X, the data for X are the integers from 5 to 54, and u = Xe, where e is iid N(0,1) (identically and independently distributed, drawn from a normal distribution with zero mean and unit variance). 1

3. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
1 standard
deviation of u
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. 2

4. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
1 standard
deviation of u
The standard deviation of ui is equal to Xi. 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/Xi. 3

5. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
1 million samples
Slope coefficient estimated using WLS Yi /Xi = 10/Xi ei
s.d. = 0.227
Slope coefficient estimated using OLS
s.d. = 0.346
The diagram shows the results of estimating the slope coefficient using WLS and OLS for one million samples. 4

6. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
1 million samples
Slope coefficient estimated using WLS Yi /Xi = 10/Xi ei
s.d. = 0.227
Slope coefficient estimated using OLS
s.d. = 0.346
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. 5

7. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
S.e. estimated using WLS
Mean of distribution = 0.226
Actual s.d. = 0.227
1 million samples
S.e. estimated using OLS
Mean of distribution = 0.319
Actual s.d. = 0.346
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. 6

8. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
S.e. estimated using WLS
Mean of distribution = 0.226
Actual s.d. = 0.227
1 million samples
S.e. estimated using OLS
Mean of distribution = 0.319
Actual s.d. = 0.346
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. 7

9. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
S.e. estimated using WLS
Mean of distribution = 0.226
Actual s.d. = 0.227
1 million samples
S.e. estimated using OLS
Mean of distribution = 0.319
Actual s.d. = 0.346
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. 8

10. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
S.e. estimated using WLS
Mean of distribution = 0.226
Actual s.d. = 0.227
1 million samples
S.e. estimated using OLS
Mean of distribution = 0.319
Actual s.d. = 0.346
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 9

11. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
S.e. estimated using WLS
Mean of distribution = 0.226
Actual s.d. = 0.227
1 million samples
S.e. estimated using OLS
Mean of distribution = 0.319
Actual s.d. = 0.346
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? 10

12. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
S.e. estimated using WLS
Mean of distribution = 0.226
Actual s.d. = 0.227
1 million samples
S.e. estimated using OLS
Mean of distribution = 0.319
Actual s.d. = 0.346
Robust s.e. estimated using OLS
Mean of distribution = 0.335
Actual s.d. = 0.346
The mean of the distribution of the robust standard errors is This is closer to the true standard deviation, 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. 11

13. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
S.e. estimated using WLS
Mean of distribution = 0.226
Actual s.d. = 0.227
1 million samples
S.e. estimated using OLS
Mean of distribution = 0.319
Actual s.d. = 0.346
Robust s.e. estimated using OLS
Mean of distribution = 0.335
Actual s.d. = 0.346
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. 12

14. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION
Yi = Xi + ui Xi = {5,6, ..., 54} ui = Xiei ei ~ N(0,1)
S.e. estimated using WLS
Mean of distribution = 0.226
Actual s.d. = 0.227
1 million samples
S.e. estimated using OLS
Mean of distribution = 0.319
Actual s.d. = 0.346
Robust s.e. estimated using OLS
Mean of distribution = 0.335
Actual s.d. = 0.346
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. 13

15. Copyright Christopher Dougherty 2011.
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
Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course
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20 Elements of Econometrics