Palgrave Handbook of Econometrics: Applied Econometrics

Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1031

Teräsvirta (1998) conclude that conditional heteroskedasticity may result in severe
size distortions and that the robust version of Granger and Teräsvirta (1993) appears
to have very low power.^12 Pavlidiset al.(2007) investigate the performance of
possible alternatives to improve the properties (size and power) of linearity LM
tests using heteroskedasticity consistent covariance matrix estimators (HCCME)
and the wild bootstrap.^13 They show that in the case of the LM linearity tests,
the fixed-design wild bootstrap appears to improve tests both in terms of size and
size-adjusted power.
However, besides the functional form of the real exchange rate, a major con-
cern in the PPP literature is that real exchange rates follow a RW. Kapetanioset
al.(2003a) (KSS) develop a test with a linear unit root null against the alterna-
tive of a stationary ESTAR. Their test is also based on a Taylor approximation of
the nonlinear autoregressive model. For simplicity, assumingp=1,d=1,β 1 =1,
φ 1 =−β 1 ,c=0, then (22.3) becomes:

yt=yt− 1 +

[

1 −exp

(

−γy^2 t− 1

)]

(−yt− 1 )+ut. (22.9)

Using the first-order Taylor expansion (22.6) in this particular case:

yt=yt− 1 −δ(yt− 1 )(yt− 1 )^2 ⇒yt=δy^3 t− 1 +ut. (22.10)

Under the null hypothesis of linearity,H 0 :γ=0,yt=ut. KSS also consider the
more general case where model (22.9) includes deterministic components. To ease
notation, letyt∗=yt−cˆ′xtwherext=(1,t), andcˆ′denotes least squares estimates.
Then we can rewrite equation (22.10) as:

y∗t=

∑p

j= 1

ajy∗t−j+δyt∗−^31 +ut, (22.11)

where lags of the dependent variable address the issue of possible error autocorre-
lation. Testing forδ=0 againstδ<0 corresponds to testing the null hypothesis.
Thet-statistic for the null of a linear unit root is given by:

tNL(cˆ′)=

δˆ

s.e(δ)ˆ

, (22.12)

wheres.e(δ)ˆ denotes the standard error ofδˆ. The asymptotic distribution of (22.12)
is not standard but converges weakly to a complicated functional of Brownian
motions.^14 Asymptotic critical values for thetNL(ˆc′)statistics have been tabulated
via stochastic simulation. KSS use quarterly data for bilateral real exchange rates
for 11 OECD countries against the dollar covering the period 1957–98. While the
ADF test was unable to reject the null of a unit root in any of the rates, the KSS test
rejected the null in favor of an ESTAR in six cases, thus giving support to PPP.^15
The linearity tests reviewed so far test the null of linear stationarity or a linear
unit root process against a globally stationary nonlinear process in levels. Harvey
and Leybourne (2007) (HL) develop a testing procedure for the null of linearity