Quantile regression was originally introduced to the statistical community by Koenker and Basset ( [46], 1978). Estimating the influence of a regressor vector x on a variable y in evaluating the conditional mean estimated by least squares is only one possible aspect. To learn more about its influence on other features of the distribution of several estimated conditional quantile surfaces may be very valuable. More about this rapidly growing field of quantile regression may be found in Koenkers book (2005). Nonparametric regression with mean models started with two papers by Nadaraya ( [56], 1964) and Watson ( [82], 1964). In the meantime, we observe a vast and rapidly growing literature on local polynomial regression. However, as in the parametric world, these procedures are usually combined with nonparametric estimation of conditional quantile functions. Also, in this area we observe an extensive literature. Like in nonparametric regression, the selection of a bandwidth plays an important role in the estimation procedure for the local neighborhood. The paper intends to give a certain review of this subject and gives few examples for financial time series. There arose also another idea of a nonparametric approach to the quantile regression, similar to that for the nonparametric regression. While within the latter field a large number of contributions to the choice of the bandwidth with the local polynomial regression appeared over the past time, the development ran somewhat differently in the field of the local quantile regression. Abberger ( [1], 1996) improved the cross-validation criterion, which was very popular in early approaches to nonparametric regression. His improvement consists in replacing the squared loss criterion by a check-function as introduced by Koenker and Basset ( [46], 1978). He used the criterion (1.17), introduced and examined it as a weighting function to be a possible approach to the bandwidth choice in the local quantile regression. Whereas Yu and Jones ( [87], 1998) studied the bandwidth selection extensively for the local linear quantile estimation. They treated the local linear check-function minimization according to (1.15) with p = 1 and the local linear double kernel smoothing according to (1.9). They showed how an MSE (mean squared error) optimal bandwidth looks like. The topic of iterative plug-in method to obtain the optimal bandwidth for the local linear regression was studied by Gasser, Kneip, Köhler ( [18], 1991). This method turned out to be very successful in practice. As in local linear regression, bandwidth selection plays a very important role also in local quantile estimation. Many papers treating bandwidth choice in nonparametric regression reappeared in the literature. This topic is less frequently touched in local quantile estimation. Therefore, we have modified this plug-in method to be suitable for the local linear quantile estimation.Our aim was to apply the iterative-plug-in method to the local quantile regression using the R-program to suitable applications in Financial Market data. The plug-in method estimates the Mean Integrated Squared error (MISE), where the unknown terms of the optimal global bandwidth are estimated in several steps. The basic idea of plug-in estimation is to obtain a large sample approximation of MISE, then to minimize the resulting analytical expression with respect to hopt in (4.50) in order to obtain the asymptotically optimal bandwidth ^h, and finally to replace the unknown terms in h^opt in (4.53) by estimators. The study was extended to compare our method with other ones which were employed by Yu and Jones ( [87], 1998) and Abberger [1], 1998).In chapter 1, an overview of the work is presented. In chapter 2, the nonparametric regression is described. We describe the estimation of the conditional density and its first derivatives in the section (2.4). In order to receive an estimation for the conditional distribution function of the quantile regression, we need the theoretical background of Fan, Yao and Tong ( [16],1996). With these tools from chapter 4, we can apply the bandwidth choice to quantile regression. Firstly, we introduce the cross-validation as a possibility of the bandwidth choice for the quantile regression. Thereafter, we present the suggestions of Yu and Jones ( [87], 1998) to apply them to the local linear quantile regression. The authors suggest a rule-of-thumb bandwidth with a double kernel approach, which we try to improve in section (4.3) by deducing a plug-in method for the bandwidth choice by the means of the local linear quantile regression with a double kernel. We obtain the theoretically optimal global bandwidth in the sections (4.4), (4.5) and (4.6). In section (2.4), we present an estimation of the conditional density and its derivative according to Fan, Yao and Tong ( [16], 1996). In sections (2.5) and (2.6), we treat the bandwidth choice for the local linear regression. Firstly, we deduce the MSE-optimal global bandwidth with cross-validation and the plug-in method for local linear polynomial regression according to Härdle ( [25], 1990), Fan and Gijbels ( [13], 1996), Gasser, Kneip and Köhler ( [18], 1991).In chapter 3, we describe the quantile regression. Firstly we explicitly present their problems and introduce the notion of a solution to the quantile regression problem. Then we deal with the asymptotics in the non-parametric general case. The theoretical bases of this chapter have been taken from the article of Koenker and Basset ( [46], 1978), Abberger ( [1], 1996), Koenker [45], Yu and Jones ( [87], 1998). The original work of Koenker and Basset ( [46], 1978) is treated in chapter 3. In section (3.2), we introduce the local linear quantile as the quantile of the conditional distribution function. The asymptotic results are presented in the section (3.4). The local linear check-function and the local linear estimation with double kernel approach are presented in sections (3.5) and (3.6), respectively. The bandwidth selection for the quantile regression is treated in chapter 4. Firstly, the cross-validation method of Abberger ( [1], 1996) is examined and briefly presented. Then the rule-of-thumb is applied with the double exponential distribution replacing the normal distribution. We generalize the iterative Plug-in algorithm for local linear quantile regression with double kernel smoothing in section (4.5). In section (4.6), we present an iterative-Plug-in algorithm for the local linear quantile regression. In section (4.7), we discuss the consistency and the asymptotic normality of the plug-in algorithm. In chapter 5, we apply the algorithm to five-year returns of CASE30 (Cairo and Alexandria stock Exchange), twelve-year returns of DAX100 (DAX100-stock index) and ten-year of SP500-returns (American stock index) in sections (5.1), (5.2) and (5.3), respectively.
