Journal of Probability and Statistics

Volume 2018, Article ID 4816716, 13 pages

https://doi.org/10.1155/2018/4816716

## Smoothed Conditional Scale Function Estimation in AR(1)-ARCH(1) Processes

^{1}Department of Mathematics, Pan African University Institute for Basic Sciences, Technology and Innovation, P.O. Box 62000, Nairobi 00200, Kenya^{2}Department of Mathematics, Machakos University, P.O. Box 136, Machakos 90100, Kenya^{3}Department of Statistics and Actuarial Sciences, Jomo Kenyatta University of Agriculture and Technology, P.O. Box 62000, Nairobi 00200, Kenya

Correspondence should be addressed to Lema Logamou Seknewna; ek.ca.taukj.stneduts@amel.uomagol

Received 27 November 2017; Revised 11 January 2018; Accepted 24 January 2018; Published 12 March 2018

Academic Editor: Aera Thavaneswaran

Copyright © 2018 Lema Logamou Seknewna et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The estimation of the Smoothed Conditional Scale Function for time series was taken out under the conditional heteroscedastic innovations by imitating the kernel smoothing in nonparametric QAR-QARCH scheme. The estimation was taken out based on the quantile regression methodology proposed by Koenker and Bassett. And the proof of the asymptotic properties of the Conditional Scale Function estimator for this type of process was given and its consistency was shown.

#### 1. Introduction

Consider a Quantile Autoregressive model,where is the Conditional Quantile Function of given and the innovation is assumed to be independent and identically distributed with zero quantile and constant scale function; see [1]. A kernel estimator of has been determined and its consistency is shown [2]. A bootstrap kernel estimator of was determined and shown to be consistent [3]. This research will extend [3] by assuming that the innovations follow Quantile Autoregressive Conditional Heteroscedastic process similar to Autoregressive-Quantile Autoregressive Conditional Heteroscedastic process proposed in [1]:where is the conditional -quantile function of given ; is a conditional scale function at -level, and is independent and identically distributed (i.i.d.) error with zero -quantile and unit scale. The function can be expressed aswhere is the so-called volatility found in [4, 5] which are papers of reference on Engle’s ARCH models among many others and is a positive constant depending on [see [6]]. An example of this kind of function is Autoregressive-Generalized Autoregressive Conditional Heteroscedastic AR(1)-GARCH(1,1),where , , , , , , , , and with 0 mean and variance 1. Note that may also be an ARMA (see [7]). The specifications for model (4) are given in Section 4.2.

Considering other financial time series models, the model (1) can be seen as a robust generalization of AR-ARCH-models, introduced in [7], and their nonparametric generalizations reviewed by [8]. For instance, consider a financial time series model of AR()-ARCH()-type,where and and are arbitrary functions representing, respectively, the conditional mean and conditional variance of the process.

The focus of this paper is to determine a smoothed estimator of the conditional scale function (CSF) and its asymptotic properties. This study is essential since volatility is inherent in many areas, for example, hydrology, finance, and weather. The volatility needs to be estimated robustly even when the moments of distribution do not exist.

A partitioned stationary -mixed time series , where the and the variate are, respectively, -measurable and -measurable, is considered. For some , the conditional -quantile of given the past assumed to be determined by is estimated. For simplicity, we assume that throughout the rest of the discussion.

We derive a smoothed nonparametric estimator of and show its consistency using standard estimate of Nadaraya [9]-Watson [10] type. This estimate is obtained from the estimate of the conditional scale function in [11] which is a type of estimator that has some disadvantages of not being adaptive and having some boundary effects but can be fixed by well-known techniques ([12]). It is though a constrained estimator in and a monotonically increasing function. This is very important to our estimation of the conditional distribution function and its inverse.

#### 2. Methods and Estimations

Let and denote the probability density function (pdf) of and the joint pdf of . The dependence between the exogenous and the endogenous variables is described by the following conditional probability density function (CPDF):and the conditional cumulative distribution function (CCDF)The estimation of the conditional scale function is derived through the CCDF. However, the following assumptions and definitions (these assumptions are commonly used for kernel density estimation (KDE), bias reduction [13], asymptotic properties, and normality proof) are necessary (see Table 1).