Christopher Dougherty EC220 - Introduction to econometrics (chapter 12) Slideshow: autocorrelation, partial adjustment, and adaptive expectations Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 12). [Teaching Resource] © 2012 The Author This version available at: 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

1 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS This sequence looks at the implications of autocorrelation for the partial adjustment and adaptive expectations models.

2 In the partial adjustment model, the disturbance term in the fitted model is the same as that in the target relationship, except that it has been multiplied by a constant,. AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

3 Thus, if the regression model assumptions are valid in the target relationship, they will also be valid in the fitted relationship. AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

4 The only problem is the finite sample bias caused by using the lagged dependent variable as an explanatory variable, and this is usually disregarded in practice anyway. AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

5 Of course, if the disturbance term in the target relationship is autocorrelated, it will be autocorrelated in the fitted relationship. OLS would yield inconsistent estimates and you should use an AR(1) estimation method instead. AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

In the case of the adaptive expectations model, we derived two alternative regression models. One model expresses Y as a function of current and lagged values of X, enough lags being taken to render negligible the coefficient of the unobservable variable X e t–s+1. 6 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

The disturbance term in the regression model is the same as that in the original model. So if it satisfies the regression model assumptions in the original model it will do so in the regression model, which should be fitted using a standard nonlinear estimation method. 7 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

If it is autocorrelated in the original model, it will be autocorrelated in the regression model. An AR(1) estimation method should be used. 8 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

The other version of the regression model expresses Y as a function of X and lagged Y. The disturbance term is a compound of u t and u t–1. 9 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

Thus if the disturbance term in the original model satisfies the regression model assumptions, the disturbance term in the regression model will be subject to MA(1) autocorrelation (first-order moving average autocorrelation). 10 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

If you compare the composite disturbance terms for observations t and t – 1, you will see that they have a component u t–1 in common. 11 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

The combination of moving-average autocorrelation and the presence of the lagged dependent variable in the regression model causes a violation of Assumption C AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

u t–1 is a component of both Y t–1 and the composite disturbance term. 13 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

Since the current value of the disturbance term is not distributed independently of the current value of one of the explanatory variables, OLS estimates will be biased and inconsistent. Under these conditions, the other regression model should be used instead. 14 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

However, suppose that the disturbance term in the original model were subject to AR(1) autocorrelation. 15 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

Then the composite disturbance term at time t will be as shown. 16 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

It is thus a composite of the innovation in the AR(1) process at time t and u t–1. Now, under reasonable assumptions, both  and should lie between 0 and 1. Hence it is possible that the coefficient of u t–1 may be small enough for the autocorrelation to be negligible. 17 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

If that is the case, OLS could be used to fit the regression model after all. You should, of course, perform a Breusch–Godfrey or Durbin h test to check that there is no (significant) autocorrelation. 18 AUTOCORRELATION, PARTIAL ADJUSTMENT, AND ADAPTIVE EXPECTATIONS

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