Application of sequential nonparametric confidence bands in finance

In a nonparametric setting, the functional form of the relationship between the response variable and the associated predictor variables is assumed to be unknown when data is fitted to the model. Non-parametric regression models can be used for the same types of applications such as estimation, prediction, calibration, and optimization that traditional regression models are used for. The main aim of nonparametric regression is to highlight an important structure in the data without any assumptions about the shape of an underlying regression function. Hence the nonparametric approach allows the data to speak for itself. Applications of sequential procedures to a nonparametric regression model at a given point are considered.

The primary goal of sequential analysis is to achieve a given accuracy by using the smallest possible sample sizes. These sequential procedures allow an experimenter to make decisions based on the smallest number of observations without compromising accuracy. In the nonparametric regression model with a random design based on independent and identically distributed pairs of observations (X ,Y ), where the regression function m(x) is given bym(x) = E(Y X = x), estimation of the Nadaraya-Watson kernel estimator (m (x)) NW and local linear kernel estimator (m (x)) LL for the curve m(x) is considered. In order to obtain asymptotic confidence intervals form(x), two stage sequential procedure is used under which some asymptotic properties of Nadaraya-Watson and local linear estimators have been obtained.

The proposed methodology is first tested with the help of simulated data from linear and nonlinear functions. Encouraged by the preliminary findings from simulation results, the proposed method is applied to estimate the nonparametric regression curve of CAPM.