"Frontmatter". In: Analysis of Financial Time Series

NONLINEAR MODELS 139average of y t instead of the simple average to estimate m(x). The weight should belarger for those Y t with X t close to x and smaller for those Y t with X t far away fromx. Mathematically, the estimate of m(x) for a given x can be written asˆm(x) = 1 TT∑w t (x)y t , (4.18)t=1where the weights w t (x) are larger for those y t with x t close to x and smaller forthose y t with x t far away from x.From Eq. (4.18), the estimate ˆm(x) is simply a local weighted average withweights determined by two factors. The first factor is the distance measure (i.e., thedistance between x t and x). The second factor is the assignment of weight for a givendistance. Different ways to determine the distance between x t and x andtoassignthe weight using the distance give rise to different nonparametric methods. In whatfollows, we discuss the commonly used kernel regression and local linear regressionmethods.4.1.5.1 Kernel RegressionKernel regression is perhaps the most commonly used nonparametric method insmoothing. The weights here are determined by a kernel, which is typically a probabilitydensity function, is denoted by K (x), and satisfiesK (x) ≥ 0,∫K (z) dz = 1.However, to increase the flexibility in distance measure, one often rescales the kernelusing a variable h > 0, which is referred to as the bandwidth. The rescaled kernelbecomesK h (x) = 1 ∫h K (x/h), K h (z) dz = 1. (4.19)The weight function can now be defined asw t (x) =1TK h (x − x t )∑ Tt=1K h (x − x t ) , (4.20)where the denominator is a normalization constant that makes the smoother adaptiveto the local intensity of the X variable and ensures the weights sum to T . PluggingEq. (4.20) into the smoothing formula (4.18), we have the well-known Nadaraya–Watson kernel estimatorˆm(x) = 1 TT∑w t (x)y t =t=1∑ Tt=1K h (x − x t )y t∑ Tt=1K h (x − x t ) ; (4.21)