1. EC220 - Introduction to econometrics (chapter 12)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 12)
Slideshow: autocorrelation
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 12). [Teaching Resource]
© 2012 The Author
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Available in LSE Learning Resources Online: May 2012
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2. AUTOCORRELATION Y Y = b1 + b2X X
Assumption C.5 states that the values of the disturbance term in the observations in the sample are generated independently of each other. 1

3. AUTOCORRELATION Y Y = b1 + b2X X
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. 2

4. AUTOCORRELATION Y Y = b1 + b2X X
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. 3

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

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

7. First-order autoregressive autocorrelation: AR(1)
Fifth-order autoregressive autocorrelation: AR(5)
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 ut up to the fifth lag. 8

8. First-order autoregressive autocorrelation: AR(1)
Fifth-order autoregressive autocorrelation: AR(5)
Third-order moving average autocorrelation: MA(3)
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. 8

9. First-order autoregressive autocorrelation: AR(1)
Fifth-order autoregressive autocorrelation: AR(5)
Third-order moving average autocorrelation: MA(3)
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. 8

10. AUTOCORRELATION
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. 9

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

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

13. AUTOCORRELATION 12

14. AUTOCORRELATION 13

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

16. AUTOCORRELATION 15

17. AUTOCORRELATION 16

18. AUTOCORRELATION
With r 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. 17

19. AUTOCORRELATION 18

20. AUTOCORRELATION 19

21. AUTOCORRELATION
With r 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. 20

22. AUTOCORRELATION
The process is now approaching what is known as a random walk, where r 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 | r | < 1. 21

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

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

25. AUTOCORRELATION
With r 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. 24

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

27. AUTOCORRELATION
============================================================
Dependent Variable: LGHOUS
Method: Least Squares
Sample:
Included observations: 45
Variable Coefficient Std. Error t-Statistic Prob. C
LGDPI
LGPRHOUS
R-squared Mean dependent var
Adjusted R-squared S.D. dependent var
S.E. of regression Akaike info criter
Sum squared resid Schwarz criterion
Log likelihood F-statistic
Durbin-Watson stat Prob(F-statistic)
Next, we will look at a plot of the residuals of the logarithmic regression of expenditure on housing services on income and relative price. 26

28. AUTOCORRELATION
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. 27

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

30. Copyright Christopher Dougherty 2011.
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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
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
EC212 Introduction to Econometrics
or the University of London International Programmes distance learning course
20 Elements of Econometrics