1052 The Econometrics of Exchange Rates
e.g., Blanchard and Watson, 1982; Evans, 1991). The bubble proposed by Evans
takes the form:
Bt+ 1 =⎧
⎨
⎩λ[−^1 Bt (^) t+ 1 ifBt≤k,
δ+π−^1 λ−^1 φt+ 1
(
Bt−λδ
)]
(^) t+ 1 ifBt>k,
(22.73)
whereEt (^) t+ 1 =1, andφt+ 1 takes the value one or zero with probabilitiesπor 1−π.
Taking expectations of the second equation in (22.73) we have that:
EtBt+ 1 =π
[
δ+π−^1 λ−^11
(
Bt−λδ
)]
+( 1 −π)δ=
Bt
λ
. (22.74)
In this case, when the Evans Markovian bubble exceeds the value ofkthen it grows
at a faster rate till it pops to a value ofδ. In expectation the bubble is explosive,
which implies thatstin (22.67) is explosive regardless of the integration properties
of the fundamental. There appears to be no reason why an intrinsic bubble that
pops could also not be specified – though as yet no research has investigated such
bubbles.
22.5.2 Testing and evidence
Though the majority of applications are based on stock price data, Evans (1986)
and Meese (1986) are two applications to exchange rates.^60 The empirical tests to
date on exchange rates are inconclusive as to the existence of bubbles. However, in
an important new development in testing for bubbles, Phillipset al.(2006) set out
a unit root testing procedure that has good power characteristics in finite samples
and enables dating the origination and the collapse of bubbles.^61 Their study is
based on the presumption that bubbles can be identified by way of manifestation
of explosive characteristics in the data. This can be achieved by estimating the
regression equation:
st=μ+φst− 1 +∑Jj= 1ξjst−j+εs,t, εs,t∼NID(0,σs^2 ), (22.75)and testing the null hypothesis of a unit root,H 0 :φ=1 against the alternativeH 1 :
φ>1. Phillipset al.(2006) propose two tests, a right-side ADF test and a sup test,
based on the recursive estimation of (22.75). Recursive estimation is implemented
by fitting (22.75) for a fraction of the sample, sayr 0 , and sequentially increasing this
fraction by including successive observations. Under the null the corresponding
test statistics, denoted byADFrand supr∈[r 0 ,1]ADFr, are:
ADFr⇒∫r0WdW
∫r0W^2, (22.76)