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Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: consequences of autocorrelation Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 12). [Teaching Resource] © 2012 The Author This version available at: http://learningresources.lse.ac.uk/138/http://learningresources.lse.ac.uk/138/ Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms. http://creativecommons.org/licenses/by-sa/3.0/ http://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

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CONSEQUENCES OF AUTOCORRELATION 1 The consequences of autocorrelation for OLS are similar to those of heteroscedasticity. In general, the regression coefficients remain unbiased, but OLS is inefficient because one can find an alternative regression technique that yields estimators with smaller variances.

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CONSEQUENCES OF AUTOCORRELATION 2 The other main consequence is that autocorrelation causes the standard errors to be estimated wrongly, often being biased downwards. Finally, although in general OLS estimates are unbiased, there is an important special case where they are biased.

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CONSEQUENCES OF AUTOCORRELATION 3 Unbiasedness is easily demonstrated, provided that Assumption C.7 is satisfied. In the case of the simple regression model shown, we have seen that the OLS estimator of the slope coefficient can be decomposed as the second line where the a t are as defined in the third line.

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CONSEQUENCES OF AUTOCORRELATION 4 Now, if Assumption C.7 is satisfied, a t and u t are distributed independently and we can write the expectation of b 2 as shown. At no point have we made any assumption concerning whether u t is, or is not, subject to autocorrelation.

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CONSEQUENCES OF AUTOCORRELATION 5 All that we now require is E(u t ) = 0 and this is easily demonstrated.

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CONSEQUENCES OF AUTOCORRELATION 6 For example, in the case of AR(1) autocorrelation, lagging the process one time period, we have the second line. Substituting for u t–1 in the first equation, we obtain the third.

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CONSEQUENCES OF AUTOCORRELATION 7 Continuing to lag and substitute, we can express u t in terms of current and lagged values of t with diminishing weights. Since, by definition, the expected value of each innovation is zero, the expected value of u t is zero.

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CONSEQUENCES OF AUTOCORRELATION 8 For higher order AR autocorrelation, the demonstration is essentially similar. For moving average autocorrelation, the result is immediate.

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CONSEQUENCES OF AUTOCORRELATION 9 For multiple regression analysis, the demonstration is the same, except that a t is replaced by a t *, where a t * depends on all of the observations on all of the explanatory variables in the model.

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CONSEQUENCES OF AUTOCORRELATION 10 We will not pursue analytically the other consequences of autocorrelation. Suffice to mention that the proof of the Gauss–Markov theorem, which guarantees the efficiency of the OLS estimators, does require no autocorrelation, as do the expressions for the standard errors.

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CONSEQUENCES OF AUTOCORRELATION 11 Now we come to the special case where OLS yields inconsistent estimators if the disturbance term is subject to autocorrelation.

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CONSEQUENCES OF AUTOCORRELATION 12 If the model specification includes a lagged dependent variable, OLS estimators are biased and inconsistent if the disturbance term is subject to autocorrelation. This will be demonstrated for AR(1) autocorrelation and an ADL(1,0) model with one X variable.

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CONSEQUENCES OF AUTOCORRELATION 13 Lagging the ADL(1,0) model by one time period, we obtain the third line. Thus Y t–1 depends on u t–1. As a consequence of the AR(1) autocorrelation u t also depends on u t–1.

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CONSEQUENCES OF AUTOCORRELATION 14 Hence we have a violation of part (1) of Assumption C.7. The explanatory variables, Y t–1, is not distributed independently of the disturbance term. As a consequence, OLS will yield inconsistent estimates.

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CONSEQUENCES OF AUTOCORRELATION 15 This was described as a special case, but actually it is an important one. ADL models are frequently used in time series regressions and autocorrelation is a common problem.

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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 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 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 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. 11.07.25

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