Test (2009) 18: 455–457DOI 10.1007/s11749-009-0162-x

D I S C U S S I O N

Comments on: A review on empirical likelihoodmethods for regression

Carlos Velasco

Published online: 4 November 2009© Sociedad de Estadística e Investigación Operativa 2009

I first wish to congratulate the authors for this insightful and inspiring review onEmpirical Likelihood (EL) based inference for regression models. The authors areable to show that EL ideas can be applied to a variety of methods that cover a widerange of fields of application. In this comment I would like to stress how some ofthe main ideas put forward in this review can also be extended to time series data,possibly involving some data transformation.

One of the most celebrated and useful properties of EL methods is that of auto stu-dentization, which leads to nonparametric versions of Wilk’s theorem. In regressionproblems within an iid setting this accounts for conditional heteroskedasticity ad-justment, but in time series contexts, including autoregressions, correcting for serialcorrelation can be also necessary for valid asymptotic inference. However, this neednot be always the case, as in Chen et al. (2003) goodness-of-fit test of parametric timeseries regression described in Sect. 7, whose asymptotic analysis only relies on theconditional variance. This property can be explained because asymptotic propertiesof nonparametric kernel regression estimates only depend on marginal properties ofthe vector (XT

t , Yt ), not on the infinite-dimensional distribution of the correspondingstochastic process. Furthermore, parameter estimation plays no role in the test due tosmoothing. It is possible that these features can be exploited to develop EL methodsin other non- and semiparametric contexts involving dependent data.

However, in many time series parametric problems autocorrelation is a critical is-sue, and in the EL literature, there have appeared three main proposals to achievethe automatic standardization property. All these lines of research are based on trans-

This comment refers to the invited paper available at: doi:10.1007/s11749-009-0159-5.

C. Velasco (�)Department of Economics, Universidad Carlos III de Madrid, Madrid, Spaine-mail: carlos.velasco@uc3m.es

456 C. Velasco

forming the original data a priori, so that they, or the individual contributions to thenew estimating equations, become approximately independent.

The first approach is based on blocking of data as proposed by Kitamura (1997),following similar proposals in the bootstrap literature (e.g., Künsch 1989). For that,define integers L and M that depend on n, so that M → ∞, M = o(n1/2), L = O(M)

as n → ∞, and L ≤ M. Let Bi , i ∈ N, be a block of M consecutive observa-tions (Z(i−1)L+1, . . . ,Z(i−1)L+M). M is the window width, and L is the separa-tion between starting points of different blocks. Then Q = [(n − M)/L] + 1 isthe number of blocks in the sample, which now is the effective number of datapoints to construct the EL. This is achieved through a function of the observedblocks Ti = φM(Bi) that replaces the original estimating functions, Zt = Zt(β) ={Yt − m(Xt ;β)}∂m(Xt ;β)/∂β . φM is typically the average of the M original esti-mating function observations inside a block, and Ti = M−1 ∑M

t=1 Z(i−1)L+t . Besidesasymptotic chi-squared distribution for the blockwise EL ratio statistics, it can alsobe showed that they are also Barlett correctable under regularity conditions.

The second approach uses kernel-smoothed moment conditions (or estimatingfunctions) instead of data windowing, as was proposed in Kitamura and Stutzer(1997). Thus the original moments are replaced by

Ztn(β) = S−1n

t−1∑

s=t−n

K

(s

Sn

)

Zt−s(β),

where Sn is a bandwidth parameter, and K is a real-valued kernel function. See alsoGuggenberger and Smith (2008) and Otsu (2006). In this case n observations aremaintained, and all the observations are used in each new smoothed data point butreceive different weighting in terms of the time distance according with Sn and K.

This generalizes the blocking approach and also leads to asymptotic chi-squared ELratios. In both cases the windowing or reweighting prior to the computation of theEL permits the internal studentization that in usual parametric inference is performedthrough Newey and West (1994) type of estimates.

The third proposal works with data in the frequency domain, as proposed by Monti(1997) for dealing with Whittle’s estimates of linear models. Nordman and Lahiri(2006) extended this idea to more general spectral estimating equations and depen-dence schemes. For a linear regression model Yt = XT

t β + εt , with εt possibly notindependent, we could use the transformed model

wY (λj ) = wX(λj )T β + wε(λj )

in terms of the discrete Fourier transform, wZ(λj ) = (2πn)−1/2 ∑nt=1 Zt exp(itλj ),

j = 1, . . . , [n/2], where the pseudo data are approximately independent, but het-eroskedastic. However, some useful properties of methods based on this frequencydomain data might require Gaussianity or strict exogeneity conditions, so that wX

and wε are effectively orthogonal and Zj = {wY (λj )−wX(λj )T β0}wX(λj ) are zero

mean under the null. These methods then might be less robust than ones based onblocking and smoothing, but in turn the latter depend on the choice of bandwidthparameters, the study of the sensitivity of EL inference to such parameters in generalframeworks being an important topic for future research.

Comments on: A review on empirical likelihood methods 457

References

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