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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
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ADAPTIVE EXPECTATIONS
tett uXY 121
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
tett uXY 121
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
tett uXY 121
3
In its simplest form, the dependent variable Yt is related, not to the current value of the explanatory variable, Xt, but to the value anticipated in the next time period, Xe
t+1.
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
4
Xet+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, Xe
t, and the actual outcome, Xt.
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
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
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
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
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
7
Substituting for Xet+1 from the adaptive expectations relationship, we obtain the equation
shown. Unfortunately, there is still an unobservable variable, Xet, on the right side of the
equation.
tettt uXXY 1221
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
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.
tettt uXXY 1221
ett
et XXX 11 1
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
9
Substitute for Xte in the equation for Yt.
tettt uXXY 1221
ett
et XXX 11 1
tetttt uXXXY 1
221221 11
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
10
Lagging and substituting s times in this way, we obtain the equation shown.
tettt uXXY 1221
ett
et XXX 11 1
tetttt uXXXY 1
221221 11
test
sst
s
tttt
uXX
XXXY
1211
2
22
21221
11
...11
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
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.
tettt uXXY 1221
test
sst
s
tttt
uXX
XXXY
1211
2
22
21221
11
...11
ett
et XXX 11 1
tetttt uXXXY 1
221221 11
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
12
ett
et XXX 11 1
The specification is nonlinear in parameters and so we would fit the model using some nonlinear estimation technique.
tetttt uXXXY 1
221221 11
test
sst
s
tttt
uXX
XXXY
1211
2
22
21221
11
...11
tettt uXXY 1221
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
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.
tettt uXXY 1221
1211 tett uXY
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
14
tettt uXXY 1221
From this one obtains an expression for 2Xet.
1211 tett uXY
1112 ttet uYX
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
15
1211 tett uXY
tettt uXXY 1221
1112 ttet uYX
11321
1121
11121
1
11
1
tttt
tttt
ttttt
uuYX
uuYX
uuYXY
1,, 32211
Substituting for 2Xet in the equation for Yt, 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
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
16
1211 tett uXY
tettt uXXY 1221
1112 ttet uYX
11321
1121
11121
1
11
1
tttt
tttt
ttttt
uuYX
uuYX
uuYXY
Note that, apart from the compound disturbance term, it is mathematically the same as that for the partial adjustment model.
1,, 32211where
ADAPTIVE EXPECTATIONS
tett uXY 121 ett
et
et XXXX 1
ett
et XXX 11
17
1211 tett uXY
tettt uXXY 1221
1112 ttet uYX
11321
1121
11121
1
11
1
tttt
tttt
ttttt
uuYX
uuYX
uuYXY
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.
1,, 32211where
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 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
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
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
