Christopher Dougherty EC220 - Introduction to econometrics (chapter 11) Slideshow: adaptive expectations Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 11). [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

ADAPTIVE EXPECTATIONS 1 The dynamics in the partial adjustment model are attributable to inertia, the drag of the past. Another, completely opposite, source of dynamics, is the effect of anticipations.

ADAPTIVE EXPECTATIONS 2 On the basis of information currently available, agents—individuals, households, enterprises—form expectations about the future values of key variable and adapt their plans accordingly.

ADAPTIVE EXPECTATIONS 3 In its simplest form, the dependent variable Y t is related, not to the current value of the explanatory variable, X t, but to the value anticipated in the next time period, X e t+1.

ADAPTIVE EXPECTATIONS 4 X e t+1 in general will be subjective and unobservable. To make the model operational, we hypothesize that expectations are updated in response to the discrepancy between what had been anticipated for the current time period, X e t, and the actual outcome, X t.

ADAPTIVE EXPECTATIONS 5 As in the partial adjustment model, may be interpreted as a speed of adjustment and should lie between 0 and 1. We can rewrite the adaptive expectations relationship as shown.

ADAPTIVE EXPECTATIONS 6 This indicates that, according to this model, the expected level of X in the next time period is a weighted average of what had been expected for the current time period and the actual outcome for the current time period.

ADAPTIVE EXPECTATIONS 7 Substituting for X e t+1 from the adaptive expectations relationship, we obtain the equation shown. Unfortunately, there is still an unobservable variable, X e t, on the right side of the equation.

ADAPTIVE EXPECTATIONS 8 There are two ways of dealing with this problem. One involves repeated lagging and substitution. If the adaptive expectations process is true for time period t, it is true for time period t–1.

ADAPTIVE EXPECTATIONS 9 Substitute for X t e in the equation for Y t.

ADAPTIVE EXPECTATIONS 10 Lagging and substituting s times in this way, we obtain the equation shown.

ADAPTIVE EXPECTATIONS 11 We are assuming that 0 < ≤ 1. It follows that 0 ≤ 1 – < 1 and hence that (1 – ) s tends to zero as s becomes large. Hence, for sufficiently large s, we can drop the unobservable final term without incurring serious omitted variable bias.

ADAPTIVE EXPECTATIONS 12 The specification is nonlinear in parameters and so we would fit the model using some nonlinear estimation technique.

ADAPTIVE EXPECTATIONS 13 The other way of dealing with the unobservable term proceeds as follows. If the original model is valid for time period t, it is also valid for time period t – 1.

ADAPTIVE EXPECTATIONS 14 From this one obtains an expression for  2 X e t.

ADAPTIVE EXPECTATIONS 15 Substituting for  2 X e t in the equation for Y t, one obtains a model in ADL(1,0) form. The model is now entirely in terms of observable variables and is therefore operational. where

ADAPTIVE EXPECTATIONS 16 Note that, apart from the compound disturbance term, it is mathematically the same as that for the partial adjustment model. where

ADAPTIVE EXPECTATIONS 17 Hence, if one fitted the model to a sample of data, it would be difficult to tell whether the underlying process were partial adjustment or adaptive expectations, despite the fact that the approaches are opposite in spirit. This is an example of observational equivalence of two theories. where

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 11.4 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