1. AUTOCORRELATION 1 Assumption C.5 states that the values of the disturbance term in the observations in the sample are generated independently of each other. Y X Y =  1 +  2 X

2. 2 In the graph above, it is clear that this assumption is not valid. Positive values tend to be followed by positive ones, and negative values by negative ones. Successive values tend to have the same sign. This is described as positive autocorrelation. AUTOCORRELATION Y X Y =  1 +  2 X

3. 3 In this graph, positive values tend to be followed by negative ones, and negative values by positive ones. This is an example of negative autocorrelation. Y =  1 +  2 X AUTOCORRELATION Y X

4. 8 A particularly common type of autocorrelation, at least as an approximation, is first-order autoregressive autocorrelation, usually denoted AR(1) autocorrelation. AUTOCORRELATION First-order autoregressive autocorrelation: AR(1)

5. 8 It is autoregressive, because u t depends on lagged values of itself, and first-order, because it depends only on its previous value. u t also depends on  t, an injection of fresh randomness at time t, often described as the innovation at time t. AUTOCORRELATION First-order autoregressive autocorrelation: AR(1)

6. 8 Here is a more complex example of autoregressive autocorrelation. It is described as fifth- order, and so denoted AR(5), because it depends on lagged values of u t up to the fifth lag. AUTOCORRELATION First-order autoregressive autocorrelation: AR(1) Fifth-order autoregressive autocorrelation: AR(5)

7. 8 The other main type of autocorrelation is moving average autocorrelation, where the disturbance term is a linear combination of the current innovation and a finite number of previous ones. AUTOCORRELATION Third-order moving average autocorrelation: MA(3) First-order autoregressive autocorrelation: AR(1) Fifth-order autoregressive autocorrelation: AR(5)

8. 8 This example is described as third-order moving average autocorrelation, denoted MA(3), because it depends on the three previous innovations as well as the current one. Third-order moving average autocorrelation: MA(3) First-order autoregressive autocorrelation: AR(1) Fifth-order autoregressive autocorrelation: AR(5) AUTOCORRELATION

9. 9 We will now look at examples of the patterns that are generated when the disturbance term is subject to AR(1) autocorrelation. The object is to provide some bench-mark images to help you assess plots of residuals in time series regressions. AUTOCORRELATION

10. 10 We will use 50 independent values of , taken from a normal distribution with 0 mean, and generate series for u using different values of . AUTOCORRELATION

11. 11 We have started with  equal to 0, so there is no autocorrelation. We will increase  progressively in steps of 0.1. AUTOCORRELATION

12. 12 (  = 0.1) AUTOCORRELATION

13. 13 (  = 0.2) AUTOCORRELATION

14. 14 With  equal to 0.3, a pattern of positive autocorrelation is beginning to be apparent. AUTOCORRELATION

15. 15 (  = 0.4) AUTOCORRELATION

16. 16 (  = 0.5) AUTOCORRELATION

17. 17 With  equal to 0.6, it is obvious that u is subject to positive autocorrelation. Positive values tend to be followed by positive ones and negative values by negative ones. AUTOCORRELATION

18. 18 (  = 0.7) AUTOCORRELATION

19. 19 (  = 0.8) AUTOCORRELATION

20. 20 With  equal to 0.9, the sequences of values with the same sign have become long and the tendency to return to 0 has become weak. AUTOCORRELATION

21. 21 The process is now approaching what is known as a random walk, where  is equal to 1 and the process becomes nonstationary. The terms ‘random walk’ and ‘nonstationary’ will be defined in the next chapter. For the time being we will assume |  | < 1. AUTOCORRELATION

22. 22 Next we will look at negative autocorrelation, starting with the same set of 50 independently distributed values of  t. AUTOCORRELATION

23. 23 We will take larger steps this time. AUTOCORRELATION

24. 24 With  equal to –0.6, you can see that positive values tend to be followed by negative ones, and vice versa, more frequently than you would expect as a matter of chance. AUTOCORRELATION

25. 25 Now the pattern of negative autocorrelation is very obvious. AUTOCORRELATION

26. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample: 1959 2003 Included observations: 45 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.005625 0.167903 0.033501 0.9734 LGDPI 1.031918 0.006649 155.1976 0.0000 LGPRHOUS -0.483421 0.041780 -11.57056 0.0000 ============================================================ R-squared 0.998583 Mean dependent var 6.359334 Adjusted R-squared 0.998515 S.D. dependent var 0.437527 S.E. of regression 0.016859 Akaike info criter-5.263574 Sum squared resid 0.011937 Schwarz criterion -5.143130 Log likelihood 121.4304 F-statistic 14797.05 Durbin-Watson stat 0.633113 Prob(F-statistic) 0.000000 ============================================================ 26 Next, we will look at a plot of the residuals of the logarithmic regression of expenditure on housing services on income and relative price. AUTOCORRELATION

27. 27 This is the plot of the residuals of course, not the disturbance term. But if the disturbance term is subject to autocorrelation, then the residuals will be subject to a similar pattern of autocorrelation. AUTOCORRELATION

28. 28 You can see that there is strong evidence of positive autocorrelation. Comparing the graph with the randomly generated patterns, one would say that  is about 0.7 or 0.8. AUTOCORRELATION

29. Copyright Christopher Dougherty 2013 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 12.1 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 20 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2013.03.04