Expectation Formation and Endogenous Fluctuations in Aggregate Demand

tion up to T periods into the past. Analysis of past data allows economic agents
to identify predictors. These predictors are in turn used to design priors. Priors
are obtained by applying predictors to past data and assuming that true errors
are distributed according to the distribution formed by residuals. For notational
simplicity let the underlying dynamic equation be expressed as

^ψ (^{1 -} ¾²) _e /

1+ ψq2 ^{EtXt + 1}

where X_t = X_t — -γψ-.

Let f (x_t,x_t-1,...,x_t-κ+^ be a predictor of X_t+1 obtained on past data
{x_t,x_t-∙∣ ,...,x_t-τ∣. Specifically, let f () to be of a simple linear form with
geometric weights

^f (^xt^,xt-i ^,∙∙∙^,xt-κ+ι) = ∑2=1^{α (i) x}t-i+1^{,                 (36)}

where a(i~) = a (1) ω^l-1 for i ∈ {2,..., K} , ω is a constant and the weights sum
up to one. Moreover, let

■^j = ^xt+i-j ^- ∑f=ι^α(i)xt-j-_i+ι^, J ∈ {^1,∙∙^{,t - K}}

be the errors obtained on past data with the predictor. The prior is formed
using the predictor of x_t+1 and assuming that potential errors are distributed
according to the distribution of the residuals. Under these assumption the
evolution of variable x_t follows the process⁵

^xt + 1 = ^-ψ/+ ψq^ ^=1^{a (i) x}t-i+1^{,                (37)}

where the quality of signals q_t² is given by equation (31.)

As shown by Chiarella and He [11] the dynamic properties of equation (37)
depend on the magnitude of the coefficients. Specifically a sufficient condition
for lack of stability is given by

2 ^ψ gɪ a (1) > 1+ '     q2>.

1 + ψqt²              1 + ψq2

Observe that were convergence to occur then the quality of signals would be
eventually arbitrarily close to zero and the above condition would simplify to
2ψα (1) > 1 + ψ. Moreover, for ω small enough the condition is necessarily
satisfied. Therefore, convergence cannot occur. The figure (6) presents sample
dynamics when economic agents form expectations according to (36).

⁵Note that the predictor can be biased. If this were the case it would be natural to modify
the predictor with the bias _τ -_κ ∑J^-iK εt₊j _-1. Note, however, that if convergence were to
occur the bias would be arbitrarily close to zero, hence, it would become negligible and its
presence would not change the results.

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