"Frontmatter". In: Analysis of Financial Time Series

NONLINEAR MODELS 141indicating that small bandwidths reproduce the data. Second, if h →∞, thenˆm(x t ) →∑ Tt=1K h (0)y t∑ Tt=1K h (0)= 1 TT∑y t =ȳ,t=1suggesting that large bandwidths lead to an oversmoothed curve—the sample mean.In general, the bandwidth function h acts as follows. If h is very small, then theweights focus on a few observations that are in the neighborhood around each x t .If h is very large, then the weights will spread over larger neighborhoods of x t .Consequently, the choice of h plays an important role in kernel regression. This isthe well-known problem of bandwidth selection in kernel regression.4.1.5.2 Bandwidth SelectionThere are several approaches for bandwidth selection; see Härdle (1990). The firstapproach is the plug-in method, which is based on the asymptotic expansion of themean squared error (MSE) for kernel smoothersE[ ˆm(x) − m(x)] 2 ,where m(.) is the true function. Under some regularity conditions, one can derivethe optimal bandwidth that minimizes the MSE. The optimal bandwidth typicallydepends on several unknown quantities that must be estimated from the data withsome preliminary smoothing. Several iterations are often needed to obtain a reasonableestimate of the optimal bandwidth. In practice, the choice of preliminarysmoothing can become a problem.The second approach to bandwidth selection is the leave-one-out cross-validation.First, one observation (x j , y j ) is left out. The remaining T − 1 data points are usedto obtain the following smoother at x jˆm h, j (x j ) = 1T − 1∑w t (x j )y t ,which is an estimate of y j . Second, perform Step-1 for j = 1,...,T and define thefunctiont̸= jCV(h) = 1 TT∑[y j −ˆm h, j (x j )] 2 W(x j ),j=1where W(.) is a non-negative weight function that can be used to down-weight theboundary points if necessary. Decreasing the weights assigned to data points close tothe boundary is needed because those points often have fewer neighboring observations.The function CV(h) is called the cross-validation function because it validatesthe ability of the smoother to predict {y t }t=1 T . One chooses the bandwidth h thatminimizes the CV(.) function.