1. HETEROSCEDASTICITY 1 This sequence relates to Assumption A.4 of the regression model assumptions and introduces the topic of heteroscedasticity. This relates to the distribution of the disturbance term in a regression model. 11 X Y =  1 +  2 X Y X3X3 X5X5 X4X4 X1X1 X2X2

2. 11 X Y 2 We will discuss it in the context of the regression model Y =  1 +  2 X + u. To keep the diagram uncluttered, we will suppose that we have a sample of only five observations, the X values of which are as shown. X3X3 X5X5 X4X4 X1X1 X2X2 HETEROSCEDASTICITY

3. 3 If there were no disturbance term in the model, the observations would lie on the line as shown. HETEROSCEDASTICITY 11 X Y =  1 +  2 X Y X3X3 X5X5 X4X4 X1X1 X2X2

4. 11 X Y 4 Now we take account of the effect of the disturbance term. It will displace each observation in the vertical dimension, since it modifies the value of Y without affecting X. X3X3 X5X5 X4X4 X1X1 X2X2 HETEROSCEDASTICITY

5. 5 The disturbance term in each observation is hypothesized to be drawn randomly from a given distribution. In the diagram, three assumptions are being made. HETEROSCEDASTICITY 11 X Y =  1 +  2 X Y X3X3 X5X5 X4X4 X1X1 X2X2

6. 6 One is that the expected value of u in each observation is 0 (Assumption A.3). The second is that the distribution in each observation is normal (Assumption A.6). We are not concerned with either of these and we will assume them to be true. HETEROSCEDASTICITY 11 X Y =  1 +  2 X Y X3X3 X5X5 X4X4 X1X1 X2X2

7. 7 The third, Assumption A.4, is that the variance of the distribution of the disturbance term is the same for each observation. In the present case, that means that the normal distributions shown all have the same variance. HETEROSCEDASTICITY 11 X Y =  1 +  2 X Y X3X3 X5X5 X4X4 X1X1 X2X2

8. 8 If Assumption A.4 is satisfied, the disturbance term is said to be homoscedastic (Greek for same scattering). HETEROSCEDASTICITY 11 X Y =  1 +  2 X Y X3X3 X5X5 X4X4 X1X1 X2X2

9. 9 Each observation is then potentially (before the sample is drawn) an equally reliable guide to the location of the line Y =  1 +  2 X. HETEROSCEDASTICITY 11 X Y =  1 +  2 X Y X3X3 X5X5 X4X4 X1X1 X2X2

10. 10 Once the sample has been drawn, some observations will lie closer to the line than others, but we have no way of anticipating in advance which ones these will be. 11 X Y =  1 +  2 X Y X3X3 X5X5 X4X4 X1X1 X2X2 HETEROSCEDASTICITY

11. 11 Now consider the situation illustrated by the diagram above. The distribution of u associated with each observation still has expected value 0 and is normal. However Assumption A.4 is violated and the variance is no longer constant. X3X3 X5X5 X4X4 X1X1 X2X2 11 X Y =  1 +  2 X Y HETEROSCEDASTICITY

12. 12 Obviously, observations where u has low variance, like that for X 1, will tend to be better guides to the underlying relationship than those like that for X 5, where it has a relatively high variance. X3X3 X5X5 X4X4 X1X1 X2X2 11 X Y =  1 +  2 X Y HETEROSCEDASTICITY

13. 13 When the distribution is not the same for each observation, the disturbance term is said to be subject to heteroscedasticity. HETEROSCEDASTICITY X3X3 X5X5 X4X4 X1X1 X2X2 11 X Y =  1 +  2 X Y

14. 14 There are two major consequences of heteroscedasticity. One is that the standard errors of the regression coefficients are estimated wrongly and the t tests (and F test) are invalid. HETEROSCEDASTICITY X3X3 X5X5 X4X4 X1X1 X2X2 11 X Y =  1 +  2 X Y

15. 15 The other is that OLS is an inefficient estimation technique. An alternative technique which gives relatively high weight to the relatively low-variance observations should tend to yield more accurate estimates. HETEROSCEDASTICITY X3X3 X5X5 X4X4 X1X1 X2X2 11 X Y =  1 +  2 X Y

16. In the scatter diagram manufacturing output is plotted against GDP, both measured in US$ million, for 30 countries for 1997. The data are from the UNIDO Yearbook. The sample is restricted to countries with GDP at least $10 billion and GDP per capita at least $2000. 16 HETEROSCEDASTICITY

17. The scatter diagram is dominated by the observations for Japan and the USA and it is difficult to detect any kind of pattern. 17 Japan USA HETEROSCEDASTICITY

18. However it those two countries are dropped and the scatter diagram rescaled, a clear picture of heteroscedasticity emerges. 18 HETEROSCEDASTICITY

19. The reason for the heteroscedasticity is that variations in the size of the manufacturing sector around the trend relationship increase with the size of GDP. 19 South Korea Mexico HETEROSCEDASTICITY

20. South Korea and Mexico are both countries with relatively large GDP. The manufacturing sector is relatively important in South Korea, so its observation is far above the trend line. The opposite was the case for Mexico, at least in 1997. 20 HETEROSCEDASTICITY South Korea Mexico

21. Singapore and Greece are another pair of countries with relatively large and small manufacturing sectors. However, because the GDP of both countries is small, their variations from the trend relationship are also small. 21 Singapore Greece HETEROSCEDASTICITY

22. 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 7.1 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