The Adaptive Expectation model (AE) provides a theoretical underpinning to the purely algebraic process of the Koyck model. It provides a fairly simple means of modelling expectations in economic theory whilst postulating a mode of behaviour upon the part of economic agents which seems eminently sensible. The belief that people learn from experience is obviously a more sensible starting point than the implicit assumption that they are totally devoid of memory, characteristic of static expectations thesis. Moreover, the assertion that more distant experiences exert a lesser effect than more recent experience would accord with common sense and would appear to be amply confirmed by simple observation
Assume Yt=b0+b1Xt* +ut (1) Where Xt* is the expected value of a variable
- How are expectations formulated?
Xt* - X*t-1=γ (Xt - X*t-1) Adaptive expectations.
Xt* = X*t-1 + γ (Xt - X*t-1)
Where γ (such that 0<γ<1) is known as the coefficient of expectation.
The rationale behind adaptive expectation argues “economic agents will adapt their expectations in the light of past experiences and that in particular they will learn from their mistakes” (Shaw cited in Gujarati, 2009 p.630).
The hypothesis of adaptive expectation contends that expectations are based only on the past values of the variable Xt
More specifically it states that, expectations are revised each period by a fraction γ of the gap between current actual value and the previous expected value.
From equation (2) we have
Xt* = Xt-1* + γ (Xt - Xt-1*)
Xt* = γ Xt + (1- γ )Xt-1* (3) Therefore if γ=0, it means that the expectations are static, while if γ=1 the expectations are realised immediately
Substituting (3) into (1)
(4)
Now lag (1) one period
And multiply by (1- γ)
(5)
Subtract (5) from (4)
Yt= γ β0+ β1γ Xt+ (1- γ) Yt-1+ vt (6)
vt= μt-(1- γ) μt-1