International macroe.. - Free

6.5. THE ‘PESO PROBLEM’ 189on E t (s t+1 ) which is a function of the currently available informationset, I t .Sincef t is the only exogenous variable and the model is linear,it is reasonable to conjecture that the solution has forms t = π 0 + π 1 f t . (6.35)Now you need to determine the coefficients π 0 and π 1 that make (6.35)the solution. From (6.34), the one-period ahead forecast of the fundamentalsis, E t f t+1 = δ 0 + f t . If (6.35) is the solution, you can advancetime by one period and take the conditional expectation as of date t togetE t (s t+1 )=π 0 + π 1 (δ 0 + f t ). (6.36)Substitute (6.35) and (6.36) into (6.33) to obtainπ 0 + π 1 f t = γf t + ψ(π 0 + π 1 δ 0 + π 1 f t ). (6.37)In order for (6.37) to be a solution, the coefficients on the constant andon f t on both sides must be equal. Upon equating coefficients, you seethat the equation holds only if π 0 = λδ 0 and π 1 = 1. The no bubblessolution for the exchange rate when the fundamentals follow a randomwalk with drift δ 0 is therefores t = λδ 0 + f t . (6.38)A possible regime shift. Now suppose that market participants are toldat date t 0 that the drift of the process governing the fundamentals mayhave increased to δ 1 > δ 0 . Agents attach a probabilityp 0t =Prob(δ = δ 0 |I t ) that there has been no regime change and aprobability p 1t =Prob(δ = δ 1 |I t ) that there has been a regime changewhere I t is the information set available to agents at date t. Agents usenew information as it becomes available to update their beliefs aboutthe true drift. At time t, they form expectations of the future values ofthe fundamental according toE t (f t+1 ) = p 0t E(δ 0 + v t + f t )+p 1t E(δ 1 + v t + f t )= p 0t δ 0 + p 1t δ 1 + f t . (6.39)