1068 The Econometrics of Exchange Rates
- The order of autoregression is chosen through inspection of the PACF function ofy∗t.
- In the ESTAR model (22.15), if the error variance is not reported as standardized but
has a standard error ofse, it is necessary to multiply the estimated speed of adjustment
parameterγby(se^2 )for comparison purposes.
- Under the null hypothesis{ H 0 :γ =0,ytis generated as an RW with initial values
yi
} 0
i=−max{d,p}=0 and sample size set equal to the observed sample size. The under-
lying noise processutisNID(0,s^2 ). The value ofsis chosen to be equal to the standard
error of each estimated model. Regression (22.15) is estimated using the respectivedval-
ues and thet-ratios are stored for the estimates ofγ. This is repeated 10,000 times and the
critical values are obtained from the upper empirical quantiles since the empirical distri-
butions are not symmetric and interest rests on the one-tail alternativeγ>0. See Paya
and Peel (2006b) for a discussion of possible bias in the estimatedγfor small samples.
- We know from the transition functionF(·)in ESTAR models that adjustment is time-
varying and depends on the size of the deviation. However, for comparison purposes, we
need a value of speed in time periods. The generalized impulse response function (GIRF)
introduced by Koopet al.(1996) successfully confronts the challenges that arise in defining
impulse responses for nonlinear models and is defined as:
GIRF̂h(h,δ,ωt− 1 )=E(yt+h|ut=δ,ωt− 1 )−E(yt+h|ut=0,ωt− 1 ),
whereh=1, 2,...denotes horizon,utis a random shock of sizeδoccurring at timet,
ωt− 1 is defined by all sets{y∗ 1 +i,...,y∗d+i}Ti=− 0 d−^1 , andt− 1 is a random variable defining
possible history sets. Since analytic expressions for the conditional expectations involved
in the expression above are not available forh>1, Gallantet al.(1993) and Koopet al.
(1996) used stochastic simulation to approximate it (see Venetiset al., 2007, for an ana-
lytical expression for the “naive” impulse response function of an ESTAR model). Taylor
and Peel (2000) conduct GIRF analysis on the deviations of real exchange rates from mon-
etary fundamentals and Tayloret al.(2001) use impulse response functions to gauge how
long shocks survive in real exchange rate nonlinear models. The half-life of shocks is
dramatically shorter than that obtained or implied by linear models.
- Analytic results are available on the impact of temporal aggregation on a linear series (e.g.,
Rossana and Seater, 1995) but not for nonlinear series.
- For instance, MacKinnon and White (1985) showed that in finite samples the White
HCCME can be seriously biased.
- In Paya and Peel (2006a) the real dollar–sterling exchange rate series spans several
exchange rate regimes, and within this context, a parametric form may not adequately
capture the conditional heteroskedasticity in the data (see Gonçalves and Kilian, 2003).
- Wild resampling typically underestimates the variance of the parent distribution. This can
be remedied by replacing the observed residuals with “leveraged” residuals 1 ̂−utht, where
htis the leverage for theith residual estimated from the parametric model (see Davidson
and Hinkley, 1997).
- An alternative wild bootstrap has the following distribution:εi=1 withp=0.5; and
εi=−1 withp=0.5 (see Davidson and Flachaire, 2001).
- The wild bootstrap matches the moments of the observed error distribution around the
estimated regression function at each design point(̂yb). Liu (1988) and Mammen (1993)
show that the asymptotic distribution of the wild bootstrap statistics are the same as the
statistics they try to mimic.
- For the cases where the time-varying equilibrium is defined by deterministic components,
such as dummies and time trends, see Paya and Peel (2003, 2004).
- Kapetanios (1999) considers the properties of model selection using information criteria
in the context of nonlinear threshold models: in particular, Akaike information criterion
(AIC), Schwarz criterion (SC), Hannan–Quinn (HQ) criterion, the generalized information
criterion (GIC) and the informational complexity criterion (ICOMP). In a Monte Carlo
