Palgrave Handbook of Econometrics: Applied Econometrics

1034 The Econometrics of Exchange Rates

speeds of mean-reversion to be much faster than when it is assumed constant, with
the half-life of shocks to the real exchange rate being less than two years.
However, Paya and Peel (2006a) also highlight the possibility of spurious rela-
tionships in nonlinear models if standard critical values are considered as valid
when a persistent variable or vector of variables(xt)are included as proxies for the
equilibrium level of the real exchange rate in the ESTAR model:

yt=α+δxt+exp

(

−γ(yt− 1 −α−δxt− 1 )^2

)∑p

i= 1

βi(yt−i−α−δxt−i)+ut. (22.16)

The bootstrap methodology is used to provide a better finite sample approxima-
tion to the distribution of a particular estimator in cases where classical asymptotic
theory might not yield a reliable guide.^23 If the true DGP admits heteroskedastic-
ity of unknown form, it cannot be replicated in the bootstrap DGP. The bootstrap
method called the wild bootstrap solves this problem by using the following
procedure (see, e.g., Wu, 1986; Mammen, 1993; Davidson and Flachaire, 2001).^24
The null hypothesis is that the coefficients (δ) on the proxy variables for the

equilibrium real exchange rate are zero. Accordingly, an “artificial”series foryt(̂ybt)
is simulated using previously estimated coefficients of the ESTAR model (22.16)
and setting the coefficients of the equilibrium determinants(δ)equal to zero:

̂yib=̂α+exp

(

−̂γ(yt− 1 −̂α)^2

)∑p

i= 1

βˆi(yt−i−̂α)+ubi, (22.17)

where thei=1,...,Bare replications. The residualsubi are obtained from boot-
strapping the estimated residuals(̂ut)obtained from the ESTAR model (22.16)
which includes the equilibrium determinants.^25 In other words, every replication
employs the actual residuals from regression (22.16) and creates a new series of

residuals (ubi) based on̂utas follows:

ubi=̂ut (^) i,
where (^) iis drawn from the following two-point distribution:
(^) i=
⎧
⎪⎨
⎪⎩
−(
√
5 − 1 )/ 2 with probabilityp=
( 1 +
√
5 )
2
√
5
,
(
√
5 + 1 )/ 2 with probability( 1 −p).
The (^) iare mutually independent drawings from a distribution independent of the
original data. The distribution has the properties thatE(
i)=0,E(
i^2 )=1, and
E(
^3 i)=1.^26 A consequence of these properties is that any heteroskedasticity in
the estimated residuals (̂ut)is preserved in the newly created residuals,ubi.^27
This procedure provides an empirical distribution for̂δand their associated stan-
dard errors. The idea in usingBreplications is to determine the appropriatet-values
andF-statistic so we do not reject the null of̂δ=0. These critical values can then
be used to determine whether the estimates of̂δreject the null or not. Paya and
Peel (2006a) find that the hypothesis that the real dollar–sterling rate follows an
ESTAR process with time-varying equilibrium proxied by productivity differentials
and/or wealth cannot be rejected at the usual significance level.^28