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4 In our case, the starting point should be the model with all the lagged variables. DYNAMIC MODEL SPECIFICATION General model with lagged variables Static.

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Presentation on theme: "4 In our case, the starting point should be the model with all the lagged variables. DYNAMIC MODEL SPECIFICATION General model with lagged variables Static."— Presentation transcript:

1 4 In our case, the starting point should be the model with all the lagged variables. DYNAMIC MODEL SPECIFICATION General model with lagged variables Static model AR(1) model Model with lagged dependent variable

2 5 Having fitted it, we might be able to simplify it to the static model, if the lagged variables individually and as a group do not have significant explanatory power. DYNAMIC MODEL SPECIFICATION General model with lagged variables Static model AR(1) model Model with lagged dependent variable

3 6 General model with lagged variables Static model AR(1) model Model with lagged dependent variable DYNAMIC MODEL SPECIFICATION If the lagged variables do have significant explanatory power, we could perform a common factor test and see if we could simplify the model to an AR(1) specification.

4 7 General model with lagged variables Static model AR(1) model Model with lagged dependent variable Sometimes we may find that a model with a lagged dependent variable is an adequate dynamic specification, if the other lagged variables lack significant explanatory power. DYNAMIC MODEL SPECIFICATION

5 4 In our case, the starting point should be the model with all the lagged variables. DYNAMIC MODEL SPECIFICATION Model with lagged variables AR(p)DL(q)ADL(p,q)

6 5 The model includes lags of the explained variable. DYNAMIC MODEL SPECIFICATION General model with lagged variables AR(p)DL(q)ADL(p,q)

7 6 General model with lagged variables AR(p)DL(q)ADL(p,q) DYNAMIC MODEL SPECIFICATION The model includes lags of the independent variable.

8 7 General model with lagged variables AR(p)DL(q)ADL(p,q) The model includes lags of the dependent and independent variables. DYNAMIC MODEL SPECIFICATION

9 2 In the graph above, 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

10 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

11 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)

12 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)

13 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)

14 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

15 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

16 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

17 11 We have started with  equal to 0, so there is no autocorrelation. We will increase  progressively in steps of 0.1. AUTOCORRELATION

18 12 (  = 0.1) AUTOCORRELATION

19 13 (  = 0.2) AUTOCORRELATION

20 14 With  equal to 0.3, a pattern of positive autocorrelation is beginning to be apparent. AUTOCORRELATION

21 15 (  = 0.4) AUTOCORRELATION

22 16 (  = 0.5) AUTOCORRELATION

23 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

24 18 (  = 0.7) AUTOCORRELATION

25 19 (  = 0.8) AUTOCORRELATION

26 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

27 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

28 22 Next we will look at negative autocorrelation, starting with the same set of 50 independently distributed values of  t. AUTOCORRELATION

29 23 We will take larger steps this time. AUTOCORRELATION

30 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

31 25 Now the pattern of negative autocorrelation is very obvious. AUTOCORRELATION

32 20 Under the null hypothesis of no autocorrelation, nR 2 has a chi-squared distribution with q degrees of freedom. Breusch–Godfrey test Test statistic: nR 2, distributed as  2 (q) TESTS FOR AUTOCORRELATION I: BREUSCH–GODFREY TEST

33 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

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