FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS 1 We saw in the previous sequence that AR(1) autocorrelation could be eliminated by a simple manipulation of the model. The regression model is nonlinear in parameters, but that now presents no problem for fitting it.

2 However, in the early days of computing, nonlinear estimation was not so simple and it was avoided whenever possible. The Cochrane–Orcutt iterative procedure was an ingenious method of using linear regression analysis to fit this nonlinear model. FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS

3 It is of no practical interest now, but you may see references to it occasionally. This sequence explains how it worked. FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS

4 We return to line 3 and note that the model can be rewritten as shown with appropriate definitions. We now have a simple regression model free from autocorrelation. FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS

5 However, to construct the artificial variables Y t and X t, we need an estimate of . We obtain one using the residuals. If the disturbance term is generated by an AR(1) process, e t will be related to e t–1 by a similar process. ~~ FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS

6 The first step is to regress the original model, using OLS. FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS 1.Regress Y t on X t using OLS 2.Calculate e t = Y t – b 1 – b 2 X t and regress e t on e t–1 to obtain an estimate of . 3.Calculate Y t and X t and regress Y t on X t to obtain revised estimates b 1 and b 2. Return to (2) and continue until convergence.

7 We save the residuals and regress e t on e t–1. The slope coefficient will be an estimate of . FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS 1.Regress Y t on X t using OLS 2.Calculate e t = Y t – b 1 – b 2 X t and regress e t on e t–1 to obtain an estimate of . 3.Calculate Y t and X t and regress Y t on X t to obtain revised estimates b 1 and b 2. Return to (2) and continue until convergence.

8 We then calculate the artificial variables Y t and X t and regress Y t on X t. The slope coefficient will be an estimate of  2 and an estimate of  1 can be derived from the intercept. ~~~~ FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS ~~~~ 1.Regress Y t on X t using OLS 2.Calculate e t = Y t – b 1 – b 2 X t and regress e t on e t–1 to obtain an estimate of . 3.Calculate Y t and X t and regress Y t on X t to obtain revised estimates b 1 and b 2. Return to (2) and continue until convergence.

9 We return to Step 2, recalculate the residuals, and regress e t on e t-1 again to obtain a better estimate of . FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS 1.Regress Y t on X t using OLS 2.Calculate e t = Y t – b 1 – b 2 X t and regress e t on e t–1 to obtain an estimate of . 3.Calculate Y t and X t and regress Y t on X t to obtain revised estimates b 1 and b 2. Return to (2) and continue until convergence. ~~~~

10 We then return to Step 3, and keep alternating between Step 2 and Step 3 until convergence is obtained. Thus the nonlinear model could be fitted using linear regression analysis. ~ FOOTNOTE: THE COCHRANE–ORCUTT ITERATIVE PROCESS 1.Regress Y t on X t using OLS 2.Calculate e t = Y t – b 1 – b 2 X t and regress e t on e t–1 to obtain an estimate of . 3.Calculate Y t and X t and regress Y t on X t to obtain revised estimates b 1 and b 2. Return to (2) and continue until convergence. ~~~

Copyright Christopher Dougherty 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 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