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Autocorrelation: Remedies

Autocorrelation: Remedies. Aims and Learning Objectives. By the end of this session students should be able to: Use the generalised least squares procedure to deal with autocorrelation Understand the various ways  can be estimated Describe other ways of dealing with the

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Autocorrelation: Remedies

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  1. Autocorrelation: Remedies

  2. Aims and Learning Objectives • By the end of this session students should be able to: • Use the generalised least squares procedure to deal • with autocorrelation • Understand the various ways  can be estimated • Describe other ways of dealing with the • autocorrelation problem

  3. Introduction We said in Lecture 13 that when the errors are correlated the OLS estimators are inefficient (they are LUE rather than BLUE) In order to remedy the situation we need to know something about the nature of the interdependence between the disturbance terms

  4. Regression Model Yt = 1 + 2X2t + 3X3t + Ut Cov (Ut, Ut-s) or E(Ut, Ut-s) = 0 No autocorrelation: Cov (Ut, Ut-s)  0 or E(Ut, Ut-s)  0 Autocorrelation:

  5. Remedies • Respecification: Include lagged variables and • dummies (particularly if working with seasonal • data) • Generalised Least Squares • Newey-West Robust Standard Errors

  6. Generalised Least Squares AR(1) : Ut = Ut1 + t substitute in for Ut Yt = 1 + 2X2t + 3X3t + Ut Yt = 1 + 2X2t +3X3t +Ut1 + t Now we need to “get rid” of Ut1 (continued)

  7. Yt = 1 + 2X2t +3X3t +Ut1 + t Yt = 1 + 2X2t + 3X3t + Ut lag the errors once Ut = Yt1 - 2X2t3X3t Ut1 = Yt1 1 - 2X2t-1- 3X3t-1 Yt = 1 + 2X2t +3X3t + Yt11 - 2X2t-13X3t-1+ t (continued)

  8. Yt = 1 + 2X2t +3X3t + Yt11 - 2X2t-13X3t-1+ t

  9. Problems estimating this model: 1. One observation is used up in creating the transformed (lagged) variables leaving only (n1) observations for estimating the model. 2. The value of  is not known. We must find some way to estimate it.

  10. Estimating Unknown  Value • Estimated  from OLS residuals • Estimated  from Durbin-Watson d Statistic • Cochrane-Orcutt Method for estimating 

  11. et = et1 + t et = Yt - 1 - 2X2t - 3X3t Estimating  from OLS residuals First, use least squares to estimate the model: ^ ^ ^ Yt = 1 + 2X2t + 3X3t + et The residuals from this estimation are: ^ ^ ^ Next, estimate the following by least squares: Use this  to run (estimated) GLS by substituting it into

  12. ^ d  2(1) Estimating  from Durbin-Watson d Statistic Recall from lecture 13: Therefore Use this  to run (estimated) GLS by substituting it into

  13. et = et1 + t Estimating  Using the Cochrane-Orcutt Procedure More accurate method for obtaining  Step 1: estimate the regression and obtain the residuals Step 2: estimate Step 3: Use this  to run (estimated) GLS by substituting it into ^

  14. Estimating  Using the Cochrane-Orcutt Procedure Step 4: The previous steps are repeated (iterated) until further iteration results in little change in  (we say it has converged) ^ This involves substituting the values of obtained in step 3 into the original regression (estimated in step 1)

  15. Other Remedies • Re-specification: • Including other variables and their lags may remove • autocorrelation arising from misspecification • Newey-West Robust Standard Errors: • Focuses on adjusting the standard errors • of the estimates • (These procedures can be implemented in Microfit)

  16. Summary In this lecture we have: 1. Discussed how the GLS procedure can be used to remove problems associated with autocorrelated disturbances 2. Discussed practical ways of estimating  3. Outlined alternative methods for dealing with autocorrelation

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