1. Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: eliminating AR(1) 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/

2. ELIMINATING AR(1) AUTOCORRELATION 1 This sequence shows how AR(1) autocorrelation can be eliminated from a regression model. The AR(1) process is the equation at the top right. We will start with the simple regression model, top left.

3. ELIMINATING AR(1) AUTOCORRELATION 2 If the regression model is valid at time t, it is also valid at time t–1. For reasons that will become obvious in a moment, we have multiplied through the second equation by .

4. ELIMINATING AR(1) AUTOCORRELATION 3 We now subtract the second equation from the first.

5. ELIMINATING AR(1) AUTOCORRELATION 4 The disturbance term now reduces to  t, the innovation at time t in the AR(1) process. By assumption, this is independently distributed, so the problem of autocorrelation has been eliminated.

6. ELIMINATING AR(1) AUTOCORRELATION 5 There is one minor problem. The revised specification involves a nonlinear restriction. The coefficient of X t–1 is minus the product of the coefficients of X t and Y t–1.

7. ELIMINATING AR(1) AUTOCORRELATION 6 This means that we should not try to fit the equation using ordinary least squares. OLS would not take account of the restriction and so we would end up with conflicting estimates of the parameters.

8. ELIMINATING AR(1) AUTOCORRELATION 7 For example, we might obtain the equation shown. From it we could deduce estimates of 0.5 for  and 0.8 for  2. But these numbers would be incompatible with the estimate of 0.6 for  2 .

9. ELIMINATING AR(1) AUTOCORRELATION 8 We therefore need to use a nonlinear estimation technique. Before doing this, we will extend the model to multiple regression with two explanatory variables.

10. ELIMINATING AR(1) AUTOCORRELATION 9 The procedure is the same. Write the model a second time, lagged one time period, and multiply through by .

11. ELIMINATING AR(1) AUTOCORRELATION 10 Subtract the second equation from the first.

12. ELIMINATING AR(1) AUTOCORRELATION 11 Again, we obtain a model that is free from autocorrelation.

13. ELIMINATING AR(1) AUTOCORRELATION 12 Now there are two restrictions. One involves the coefficients of Y t–1, X 2t, and X 2t–1.

14. ELIMINATING AR(1) AUTOCORRELATION 13 The other involves the coefficients of Y t–1, X 3t, and X 3t–1.

15. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================ 14 ELIMINATING AR(1) AUTOCORRELATION Here is the output for a logarithmic regression of expenditure on housing services on income and price, assuming an AR(1) process, using EViews.

16. 15 ELIMINATING AR(1) AUTOCORRELATION EViews allows two ways of specifying a regression equation. One is to list the variables, starting with the dependent variable, continuing with C for the intercept, and finishing with a list of the explanatory variables. This is fine for linear regressions. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================

17. 16 ELIMINATING AR(1) AUTOCORRELATION The other method is to write the model as an equation, referring to the parameters as C(1), C(2), etc. This is what you should do when fitting a nonlinear model, such as the present one. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================

18. 17 ELIMINATING AR(1) AUTOCORRELATION Here  1 has been denoted C(1). ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================

19. 18 ELIMINATING AR(1) AUTOCORRELATION , the coefficient of the lagged dependent variable, has been denoted C(2). It is also a component of the intercept in this model. The estimate of , 0.72, is quite high. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================

20. 19 ELIMINATING AR(1) AUTOCORRELATION  2, the coefficient of income, has been denoted C(3). The estimate is close to the OLS estimate, 1.03. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================

21. 20 ELIMINATING AR(1) AUTOCORRELATION The coefficient of lagged income must then be specified as –C(2)*C(3). ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================

22. 21 ELIMINATING AR(1) AUTOCORRELATION  3, the coefficient of price, has been denoted C(4). The estimate is the same as the OLS estimate, –0.48, at least to two decimal places. (This is a bit of a coincidence.) ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================

23. 22 ELIMINATING AR(1) AUTOCORRELATION The coefficient of lagged price must then be specified as –C(2)*C(4). ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================

24. 23 ELIMINATING AR(1) AUTOCORRELATION The only problem with this method of fitting the AR(1) model is that specifying the model in equation form is a tedious task and it is easy to make mistakes. ============================================================ Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 Durbin-Watson stat 1.901081 ============================================================

25. Dependent Variable: LGHOUS Method: Least Squares Sample(adjusted): 1960 2003 Included observations: 44 after adjusting endpoints Convergence achieved after 21 iterations ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.154815 0.354989 0.436111 0.6651 LGDPI 1.011295 0.021830 46.32642 0.0000 LGPRHOUS -0.478070 0.091594 -5.219437 0.0000 AR(1) 0.719102 0.115689 6.215836 0.0000 ============================================================ R-squared 0.999205 Mean dependent var 6.379059 Adjusted R-squared 0.999145 S.D. dependent var 0.421861 S.E. of regression 0.012333 Akaike info criter-5.866567 Sum squared resid 0.006084 Schwarz criterion -5.704368 Log likelihood 133.0645 F-statistic 16757.24 Durbin-Watson stat 1.901081 Prob(F-statistic) 0.000000 ============================================================ 24 ELIMINATING AR(1) AUTOCORRELATION Since the AR(1) specification is a common one, most serious regression applications provide some short-cut for specifying it easily. In the case of EViews, AR(1) estimation is invoked by adding AR(1) to the list of explanatory variables.

26. ============================================================= Dependent Variable: LGHOUS LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.154815 0.354989 0.436111 0.6651 LGDPI 1.011295 0.021830 46.32642 0.0000 LGPRHOUS -0.478070 0.091594 -5.219437 0.0000 AR(1) 0.719102 0.115689 6.215836 0.0000 ============================================================ 25 ELIMINATING AR(1) AUTOCORRELATION The constant is an estimate of  1.

27. 26 ELIMINATING AR(1) AUTOCORRELATION The income coefficient is the estimate of the elasticity with respect to current income.. ============================================================= Dependent Variable: LGHOUS LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.154815 0.354989 0.436111 0.6651 LGDPI 1.011295 0.021830 46.32642 0.0000 LGPRHOUS -0.478070 0.091594 -5.219437 0.0000 AR(1) 0.719102 0.115689 6.215836 0.0000 ============================================================

28. 27 ELIMINATING AR(1) AUTOCORRELATION The price coefficient is the estimate of the elasticity with respect to current price. ============================================================= Dependent Variable: LGHOUS LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.154815 0.354989 0.436111 0.6651 LGDPI 1.011295 0.021830 46.32642 0.0000 LGPRHOUS -0.478070 0.091594 -5.219437 0.0000 AR(1) 0.719102 0.115689 6.215836 0.0000 ============================================================

29. 28 ELIMINATING AR(1) AUTOCORRELATION The coefficient of AR(1) is an estimate of . ============================================================= Dependent Variable: LGHOUS LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.154815 0.354989 0.436111 0.6651 LGDPI 1.011295 0.021830 46.32642 0.0000 LGPRHOUS -0.478070 0.091594 -5.219437 0.0000 AR(1) 0.719102 0.115689 6.215836 0.0000 ============================================================

30. 29 ELIMINATING AR(1) AUTOCORRELATION The coefficients of lagged income and lagged price are not reported because they are implicit in the estimates of ,  2, and  3. ============================================================= Dependent Variable: LGHOUS LGHOUS=C(1)*(1-C(2))+C(2)*LGHOUS(-1)+C(3)*LGDPI-C(2)*C(3) *LGDPI(-1)+C(4)*LGPRHOUS-C(2)*C(4)*LGPRHOUS(-1) ============================================================ Coefficient Std. Error t-Statistic Prob. ============================================================ C(1) 0.154815 0.354989 0.436111 0.6651 C(2) 0.719102 0.115689 6.215836 0.0000 C(3) 1.011295 0.021830 46.32641 0.0000 C(4) -0.478070 0.091594 -5.219436 0.0000 ============================================================ Variable Coefficient Std. Error t-Statistic Prob. ============================================================ C 0.154815 0.354989 0.436111 0.6651 LGDPI 1.011295 0.021830 46.32642 0.0000 LGPRHOUS -0.478070 0.091594 -5.219437 0.0000 AR(1) 0.719102 0.115689 6.215836 0.0000 ============================================================

31. 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 12.3 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