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6.5. THE ‘PESO PROBLEM’ 191The time t+1 rational forecast error iss t+1 − E t (s t+1 ) = λ[δ 0 (p 0t+1 − p 0t )+δ 1 (p 1t+1 − p 1t )]+∆f t+1 − (p 0t δ 0 + p 1t δ 1 )] | {z }E t∆f t+1= λ(δ 1 − δ 0 )[p 1t+1 − p 1t ]+δ 1 + v t+1 − [δ 0 +(δ 1 − δ 0 )p 1t ].The regime probabilities p 1t and the updated probabilities p 1t+1 − p 1tare serially correlated during the learning period. The rational forecasterror therefore contains systematic components and is serially correlated,but the forecast errors are not useful for predicting the futuredepreciation. To determine explicitly the sequence of the agent’s beliefprobabilities, we use,Bayes’ Rule: for events A i ,i =1,...,N that partition the samplespace S, andanyeventB with Prob(B) > 0P(A i |B) =P(A i )P(B|A i )P Nj=1P(A j )P(B|A j ) .To apply Bayes rule to the problem at hand, let news of the possibleregime shift be released at t = 0. Agents begin with the unconditional ⇐(121)probability, p 0 =P(δ = δ 0 ), and p 1 =P(δ = δ 1 ). In the period after theannouncement t = 1, apply Bayes’ Rule by setting B =(∆f 1 ), A 1 = δ 1 ,A 2 = δ 0 to get the updated probabilitiesp 0 P(∆f 1 |δ 0 )p 0,1 =P(δ = δ 0 |∆f 1 )=p 0 P(∆f 1 |δ 0 )+p 1 P(∆f 1 |δ 1 ) . (6.46)As time evolves and observations on ∆f t are acquired, agents updatetheir beliefs according top 0,2 =p 0 P(∆f 2 , ∆f 1 |δ 0 )P(δ 0 |∆f 2 , ∆f 1 )=p 0 P(∆f 2 , ∆f 1 |δ 0 )+p 1 P(∆f 2 , ∆f 1 |δ 1 ) ,p 0,3 =p 0 P(∆f 3 , ∆f 2 , ∆f 1 |δ 0 )P(δ 0 |∆f 3 , ∆f 2 , ∆f 1 )=p 0 P(∆f 3 , ∆f 2 , ∆f 1 |δ 0 )+p 1 P(∆f 3 , ∆f 2 , ∆f 1 |δ 1 ) ,...p 0,T = P(δ 0 |∆f T ,...,∆f 1 )=p 0 P(∆f T ,...,∆f 1 |δ 0 )p 0 P(∆f T ,...,∆f 1 |δ 0 )+p 1 P(∆f T ,...,∆f 1 |δ 1 ) .