Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1051By iterating forward we obtain:
st=
1
1 +λEt⎛
⎝
∑∞j= 0(
λ
1 +λ)j
(mt+j−m∗t+j+yt+j−γ(wt+j−w∗t+j))⎞
⎠+λ
1 +λ
Et⎛
⎝∑∞j= 0(
λ
1 +λ)j
rpt+j⎞
⎠. (22.65)In this case,b=λ/( 1 +λ)anda′ 2 xt=rpt. If we assume, further, that PPP and UIP
hold, then the exchange rate model simplifies to:
st=
1
1 +λEt⎛
⎝∑∞j= 0(
λ
1 +λ)j
ft+j⎞
⎠, (22.66)whereft=a′ 1 xt=
(
mt−m∗t)
−(
wt−w∗t).^57 However, it is well known that the
above equation is a single solution from a potentially infinite set. Lettingsf,tdenote
the fundamental solution, the rational expectations solutions to (22.64) are given
by:
st=sf,t+Bt, (22.67)
where:
EtBt+ 1 =( 1 +b)
b
Bt. (22.68)The termBtis the speculative bubble, which has to follow the form given by
(22.68).^58 We can also write the solution for the bubble as (Salge, 1997):
Bt=
Mt
λt. (22.69)
From (22.69) and (22.68):
EtBt+ 1 =EtMt+ 1
λt+^1=
1
λMt
λt, (22.70)so that:
EtMt+ 1 =Mt, (22.71)
implying thatMtis a martingale process. A more general form of a stochastic
martingale process is given by:
Mt=ρtMt− 1 +ut, (22.72)whereEtρt+ 1 =1,Et−jut=0,j =1,...,n,EtρtMt =0, andEtutMt=0. In
general, a bubble can depend on its lagged values, called a Markovian bubble, on an
extraneous process “M,” which follows a martingale process, or on fundamentals,
called an intrinsic bubble (Froot and Obstfeld, 1991).^59
Bubbles, if they exist, can of course pop and a variety of forms of rational bubbles
that exhibit this property and are consistent with (22.68) have been proposed (see,