• yt-2ma=2 γ xt-3/1+xt-32 , xt-3=1/λPt-3- MA1,t-3/MA1,t-3=1/λP3,t-1- MA3,t-1/MA3,t-1. Here mfund and mma are the minimum probabilities with which the fundamental and moving average forecasting rule are chosen; see Appendix B for details. The parameter set is given by
    Θ={ a>0, σ2>0, R>1, β>0, 0≤ v≤ 1, 0<µ<1, γ>0,
      λ>0, 0≤ η ≤ 1, mfund≥ 0, mma≥ 0, 0 ≤ mfund+mma ≤ 1,
      Cfund≥ Cma ≥ 0 }.
    The difference in probability with which the fundamental and moving average belief are chosen is equal to
    qtfund-qtma=(mfund-mma) + (1-mfund-mma) tanh ⎛
    ⎜
    ⎜
    ⎝
    β
    2
    		( Ftfund-Ftma ) ⎞
    ⎟
    ⎟
    ⎠    		 .
    As can be seen in the above formula, the higher the difference in fitness in favour of the fundamental belief, the higher the difference in probability in favour of the fundamental belief. The fraction of agents who choose the fundamental believe is restricted to be strictly positive qtfund>0, since otherwise there is no solution for the equilibrium price. This condition will be automatically satisfied, even if mfund=0, because of the discrete choice model probabilities which for finite β are always strictly positive2. Furthermore, the following condition should hold for the fraction of total market wealth the moving average traders invest in the risky asset
    qtma ytma<qtfund
    R
    a σ2
    		,
    for otherwise there is also no solution for the equilibrium price.

    Note that

     
    lim
    σ2 ↓ 0
    		P1,t=
     
    lim
    a ↓ 0
    		P1,t =
     
    lim
    qtma ↓ 0
    		P1,t=
     
    lim
    ytma ↓ 0
    		P1,t=
    1
    R
    		Etfund(Pt+1+Dt+1).     (39)
    Hence, if the conditional variance or the risk aversion of the fundamental belief goes to zero, or if the fraction of wealth invested by the moving average belief goes to zero, then the equilibrium price is equal to the discounted value of the expectation of tomorrow's price and dividend of the fundamental belief.
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