Palgrave Handbook of Econometrics: Applied Econometrics

Efthymios G. Pavlidis, Ivan Paya and David A. Peel 1039

Estimation may be carried out by nonlinear least squares through a grid search

overκ. However, given the need to form an overall objective function from three

sets of residuals, it is easier in the multivariate case to employ maximum like-

lihood estimation. The concentrated log-likelihood function for this system, for

knownκ, is:

L(θ,,,κ)=

−T

2

(

3

{

1 +ln( 2 π)

}

+ln

∣

∣̂

∣

∣), (22.31)

wherê=( 1 /T)TEtEt′is the maximum likelihood estimator of the covariance

matrix (see Davidson and MacKinnon, 1993, p. 316). This is maximized through

a grid search overκwith (22.30) estimated using a full information maximum

likelihood (FIML) estimator at each point in the grid. If̂L(h)is the maximized log-

likelihood conditional on a bandwidth parameterh, the resulting estimator ofκ

may be expressed as:

̂κ=arg max

h∈H

̂L(h),

whereH=[0,max

∣
∣δt
∣
∣]is the range of the grid search. Hypotheses concerning the
parameters can then be tested using an LR statistic defined asλ′= 2 (̂L− ̃L), where
̂Ldenotes the value of the maximized log-likelihood and ̃Ldenotes the maximized

log-likelihood with the relevant restrictions imposed. Empirical marginal signifi-

cance levels for this statistic can be calculated using methods set out in Hansen

(1997) and explained in section 22.2.2. The results reported by Peel and Taylor

were consistent with the conjectures of Keynes (1923) and Einzig (1937). Arbitrage

only occurred when significant deviations from CIP occurred and the adjustment

back to the implied arbitrage bounds was fairly persistent due to the microstruc-

ture faced by arbitragers. Estimation of these threshold (or ESTAR) error correction

mechanisms would be of interest in other areas such as PPP.

In the absence of a new study employing high-quality data of the type employed

by Taylor in the 1980s, it would appear reasonable to assume that restrictions on

arbitrage funds and the ability to make near riskless trades instantaneously implies

that CIP holds within a very small transactions band.

However, before we discuss the uncovered condition there is one feature of the

nonlinear work applied to CIP that needs comment and has applicability to non-

linear models more generally. Whilst the nonlinear assumption is well motivated,

the economic arguments of arbitrage apply to a particular data frequency. For

instance, currently we might expect deviations from the arbitrage bound to occur

near instantaneously. In the 1920s the appropriate period might have been hours,

(possibly days or a week). Unfortunately, if the data frequency available is not

that of the DGP then the estimates of a nonlinear model may generate misleading

results as to the period of dynamic adjustment or the impact of shocks, as discussed

in the context of PPP above. Paya and Peel (2007b) show that systematic sampling

from the true DGP, where they employ everykth observation from the true DGP,

can lead to seriously biased estimates of speeds of adjustment. Estimation of non-

linear models on data sampled at a different frequency to the economic decision

would appear to be a serious problem in the evaluation of nonlinear models.^38