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

Authors: Lema Logamou Seknewna; Peter Mwita Nyamuhanga; Benjamin Kyalo Muema
Journal: Journal of Probability and Statistics
 (2018)

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## 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.


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## Body

## 1. Introduction

Consider a Quantile Autoregressive model,(1)Xt=ατZt+ut,t=1,2,…,where ατ(Zt) is the τth Conditional Quantile Function of Xt given Zt and the innovation ut is assumed to be independent and identically distributed with zero τth quantile and constant scale function; see [1]. A kernel estimator of ατ(Zt) has been determined and its consistency is shown [2]. A bootstrap kernel estimator of ατ(Zt) 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]:(2)Xt=ατZt+ϖτZtεt,t=1,2,…,where ατ(Zt) is the conditional θ-quantile function of Xt given Zt; ϖτ(Zt) is a conditional scale function at τ-level, and εt is independent and identically distributed (i.i.d.) error with zero τ-quantile and unit scale. The function ϖτ(Zt) can be expressed as(3)ϖτZt=λϖZt,where ϖ(Zt) is the so-called volatility found in [4, 5] which are papers of reference on Engle’s ARCH models among many others and λ is a positive constant depending on τ [see [6]]. An example of this kind of function is Autoregressive-Generalized Autoregressive Conditional Heteroscedastic AR(1)-GARCH(1,1),(4)Xt=αt+ϖtet,t=1,2,…,where αt=μ+δXt-1, ϖt=w+αXt-12+βϖt-12, μ∈(-∞,∞), δ<1, β>0, α>0, w>0, α+β<1, and et~i.i.d. with 0 mean and variance 1. Note that αt 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(p)-ARCH(p)-type,(5)Xt=αZt+ϖZtet,t=1,2,…,where Zt=(Xt-1,Xt-2,…,Xt-p) and α(·) and ϖ(·) 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 stationaryα-mixed time series (Xt,Zt), where the Xt∈R and the variate Zt∈Rd are, respectively, At-measurable and At-1-measurable, is considered. For some τ∈(0,1), the conditional τ-quantile of Xt given the past Ft-1 assumed to be determined by Zt is estimated. For simplicity, we assume that Zt=Xt-1∈R throughout the rest of the discussion.We derive a smoothed nonparametric estimator ofϖτ(z) 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 (0,1) and a monotonically increasing function. This is very important to our estimation of the conditional distribution function and its inverse.


## 2. Methods and Estimations

Letf(z) and f(x,z) denote the probability density function (pdf) of Xt and the joint pdf of (Xt,Zt). The dependence between the exogenous Xt and the endogenous variables is described by the following conditional probability density function (CPDF):(6)fx∣z=fx,zfxand the conditional cumulative distribution function (CCDF)(7)Fx∣z=∫-∞xfs∣zds=PX≤x∣Zt=z=EIXt≤x∣Zt=z.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).Table 1
Description of the most used kernel functions.
Kernel functions ExpressionsK(u) r R ( K ) μ 2 ( K ) E f f ( K ) Gaussian 1 2 exp ⁡ - u 2 2 I R ∞ 1 2 2 1 0.2821 Epanechnikov 3 4 1 - u 2 I { u ≤ 1 } 2 3 5 1 5 0.2683  Uniform 1 2 I { u ≤ 1 } 0 1 2 1 3 0.2887 Triangular 1 - u I { u ≤ 1 } 1 Triweight 35 32 1 - u 2 3 I { u ≤ 1 } 6 2 3 1 6 0.2722  Tricube 70 81 ( 1 - u 3 ) 3 I { u ≤ 1 } 9 175 247 35 243 0.2689  Biweight 15 16 1 - u 2 2 I { u ≤ 1 } 4 5 7 1 7 0.2700 Cosine π 4 cos ⁡ π 2 u ∞ π 2 16 - 8 + π 2 π 2 0.2685Assumption 1.
(i)
f(x,z) and f(z) exist.(ii)
For fixed(x,z), 0<F(x∣z)<1 and f(z)>0 are continuous in the neighborhood of z where the estimator is to be estimated.(iii)
The derivativesF(j)(x)=djF(x∣z)/dzj and f(j)(z)=djf(z)/dzj for j=1,2 exist.(iv)
F(x∣z) is a convex function in x for fixed z.(v)
The conditional densityf(x∣z)=dF(x∣z)/dx exists and is continuous in the neighborhood of x.(vi)
fϖτ(z)∣z>0.Assumption 2.
The kernel functionK:Rd→R is(i)
Symmetrical:K(s)=K(-s) with s∈Rd(ii)
Nonnegative and bounded: forΓ<∞, 0<K(s)≤Γ, s∈Rd(iii)
Lipschitz:∃λ>0, mk<∞ such that K(s)-K(t)≤mks-tλ for all s,t∈Rd(iv)
A pdf:∫K(s)ds=1 with ∫RdsK(s)=0.Assumption 3.
The process(Xt,Zt),t=1,2,… is strong mixing with α(s)=os-2-δ, δ>0; see [14, Theorem 1.7].Assumption 4.
The sequence{bn}n∈N of the smoothing parameters is such that bn→0, nbnp→∞ as n→∞ and bn>0.Definition 5 (strong mixing).
LetXt={…,Xt-1,Xt,Xt+1,…} be a stationary time series endowed with σ-algebras At={Xj,-∞<j≤t} and At={Xj,t≤j<∞}. Define α(s) as (8)αs=supA∈At,B∈At+s⁡PA∩B-PAPB.If α(s)→0 as s→∞, then the process is strong mixing.The results in this section are about the case when the Autoregressive part of the model (4) αt,τ=ατ(z)=0 for any τ∈(0,1). We therefore consider the model(9)Xt=ϖτZtεt,t=1,2,….Define the check-function as(10)γτX,μ=γτX-μ=τ-IX-μ≤0X-μ.Here, I is the indicator function. Therefore, γτ is a piece-wise monotone increasing function. γτ(·,·) is a function of any real random variable X with distribution function FX(x)=P(X≤x)=EI{X≤x}, and a real value, μ∈R, is the asymmetric absolute value function whose amount of asymmetry depends on τ; see [15]. In case where Xt is symmetric and τ=1/2, then we have the fact that 2γτ(Xt,μ) is an absolute value function and ϖ0.5(Zt) is the conditional median absolute deviation (CMAD) of Xt. When α became 0 in model (5), we have a purely heteroscedastic ARCH model introduced in [16] and ατ(Zt) for τ>0.5, which, in this particular case, can be seen as a conditional scale function at τ-level.The check-function in (10) is Lipschitz continuous by the following theorem.Theorem 6.
Letγτ be defined as in (10) and (x,σ)∈R2. Then, γτ satisfies the Lipschitz continuity condition:(11)γτx,σ-γτx,σ′≤Mσ-σ′with the Lipschitz constant M=1 and for all σ, σ′.Proof of Theorem6.
See the proof of Lemma 3.1 in [1, p. 74-75].By the next theorem we show clearly why the errorsεt in model (2) are assumed to be zero τ-quantile and unit scale.Theorem 7.
Consider model (5) and the so-called check-function in (10); then, for ϖτ(Zt)∈R+∗,(12)εt=Xt-ατZtϖτZtis zero τ-quantile and unit scale. And the following equations are verifiable:(13)PXt≤ατZt∣Zt=τ,(14)PγτXt,ατZt≤ϖτZt∣Zt=τ.Proof of Theorem7.
Theτth-quantile operator is(15)QτYt=inf⁡μ∈R:PYt≤μ∣Zt≥τwith well-defined properties in [1, p. 9-10]. From model (5), the conditional τ-quantile of Xt is(16)qτZt=QτXt=αZt+ϖZtqτe,where qτe is the τ-quantiles of et. Then, using model (5) and (16), we get(17)Xt-qτZt=ϖZtet-qτe,(18)γτXt,qτZt=ϖZtγτet,qτe.And the τth-quantile of (18) is(19)QτγτXt,qτZt=ϖZtQτγτet,qτe=ϖZtQτe,where Qτe is the τ-quantile of γτet,qτe. Note that, from (17), QτXt-qτ(Zt)=0. The quotient(20)Xt-ατZtQτγτXt,ατZt=et-qτeQτeis zero τ-quantile and unit scale and can be seen as model (2) if εt=(et-qτe)/Qτe, ατ(Zt)=qτ(Zt), and ϖτ(Zt)=QτγτXt,ατZt.
Now, assuming thatεt (independent of Zt) in model (2) is zero τ-quantile, it is equivalent to write(21)Pr⁡εt≤0=Pr⁡εt≤0∣Zt=τ⟹Pr⁡Xt-ατZtϖτZt≤0∣Zt=τ.This proves (13) for ϖτ(z)>0. Also, εt is unit scale, which means(22)Pr⁡γτεt≤1=τ⟹Pr⁡γτXt-ατZtϖτZt≤1∣Zt=τ⟹Pr⁡γτXt-ατZt≤ϖτZt∣Zt=τ.Assumingατ(Zt)=0, the estimator, ϖ^τ(Zt), of the conditional scale function, ϖτ(Zt), is obtained through the minimization of the objective function(23)φz,ϖ=EγτγτXt,ϖ∣Zt=z.Thus, the conditional scale function may be obtained by minimizing φ(z,ϖ) with respect to ϖ; that is,(24)ϖτz=arg⁡minϖ∈R+φz,ϖ,(25)ϖτz=inf⁡μ∈R+∗:Fμ∣z≥τ=F-1τ∣z.The kernel estimator of (24) at Zt=z is given by(26)ϖ^τz=arg⁡minϖ∈R+φ^nz,ϖ.We can express the estimate of φ(z,ϖ) in the random design as it was developed in [17]. Let Yt∗=γτ(γτ(Xt),ϖ) be a nonnegative function of Xt and Y∗=(Y1∗,Y2∗,…,Yn∗) a random vector in R+∗=(0,∞), t=1,2,…,n. In the random design, the conditional expectation (23) can be rewritten as follows:(27)φz,ϖ=EY∗∣Zt=z=∫y∗fy∗∣zdy∗=∫y∗fy∗,zfzdy∗,where f(y∗∣z) represents the conditional pdf of Yt∗=y∗ given Zt=z, f(y∗,z) is the joint pdf of the two random variables Y∗ and Z, and f(z) is the pdf of Zt=z. Using [9, 10] with Kb(u)=b-1Kub-1, a 1-dimensional rescaled kernel with bandwidth b>0, we have the following estimates of f(y∗,z) and f(z) [18]:(28)f^y∗,z=1n∑t=1nKbzZt-zKby∗y∗-Yt∗,f^z=1n∑t=1nKbzZt-z.From the estimations above, φ^(z,ϖ), the estimate of φ(z,ϖ), is(29)φ^nz,ϖ=∫y∗∑t=1nKbzZt-zKby∗y∗-Yt∗∑t=1nKbzZt-zdy∗=∑t=1nKbzZt-z∫y∗Kby∗y∗-Yt∗dy∗∑t=1nKbzZt-z=∑t=1nKbzZt-z∫y∗-Yt∗+Yt∗Kby∗y∗-Yt∗dy∗∑t=1nKbzZt-zand considering the regularity conditions of Kb in Assumption 2 and also the fact that d(y∗-Yt∗)=dy∗, Yt∗∈R+, we have(30)φ^nz,ϖ=∑t=1nKbzZt-zYt∗∑t=1nKbzZt-z=n-1∑t=1nKbzZt-zYt∗f^z,where f^(z) is the estimate of the marginal pdf of Zt at point z and Y∗ can be rewritten as(31)Yt∗=Xtτ-IXt≤0-ϖτ-IXtτ-IXt≤0≤ϖand the derivative of φ^n(z,ϖ) with respect to ϖ is(32)dφ^nz,ϖdϖ=nf^z-1∑t=1nKbzZt-zIXtτ-IXt≤0≤ϖ-τ.The minimizer of (30) is obtained from dφ^n(z,ϖ)/dϖ=0. This leads to the following equation:(33)nf^z-1∑t=1nKbzZt-zIXt∗≤ϖ=τ,where(34)Xt∗=Xtτ-IXt≤0∈R+∗,for all Xt∈R, t=1,2,…. Note that Yt∗=IXt∗≤ϖ in (27). The left part of (33) is a (unsmoothed) conditional cumulative distribution function (CCDF),(35)F^x∗∣z=nf^z-1∑t=1nKbzZt-zIXt∗≤x∗,that needs to be estimated and our estimator is therefore(36)ϖ^τz=inf⁡x∗∈R+:F^x∗∣z≥τ≡F^-1τ∣zwhich is equivalent to F^ϖ^(z)∣z=τ.An algorithm for estimatingF^(x∗∣z) is proposed in the following section. This estimator suffers from the problem of boundary effects as we can see it on Figure 2 due to outliers. We obtain unsmoothed curves of the CCDF because the smoothness is only in the Z direction. A method is proposed by [19] to smooth it in the y. The form of Smoothed Conditional Distribution Estimator is(37)F~x∗∣z=nf^z-1∑t=1nKhz-ZtGx∗-Xt∗h0,where G(·) is an integrated kernel with the smoothing parameter h0 in the X∗ direction. This estimate is smooth rather than the NW which is a jump function in y. To deal with boundary effects, one may think of the Weighted Nadaraya-Watson (WNW) estimate of the CDF discussed in [12, 20], [21, p. 3–18] among others. The WNW estimator’s expression is(38)F~WNWx∗∣z=∑t=1nptz,λKbzZt-zIXt∗≤x∗∑t=1nptz,λKbzZt-zwith conditions ∑t=1npt(z,λ)=1 and Lambda is determined using the Newton-Raphson iteration. Smoothing the CDF does not smooth the estimator in (36).


### 2.1. Algorithm

This algorithm estimates the empirical CCDF,F^(x∗∣z), and its inverse F^-1(τ∣z). Starting with the estimation of the former, the denominator is easy to compute as the estimator of the probability density function of Z as vector of points z.(1)
ObtainXt∗=γτ(Xt), t=1,2,…, for all τ∈(0,1).(2)
Check if eachxt∗ is less than or equal to each observation of the whole sequence x∗=(x1∗,x2∗,…,xn∗)∈Rn. The result determines Ixt∗≤x which can be expressed in (0,1)-matrix of order n×n.(3)
Constructz1∗=min⁡(Z)<z2∗<⋯<zN∗=max⁡(Z) from the sequence of i.i.d random variable Z=(Z1,Z2,…,Zn) with observation z=(z1,z2,…,zn). N is the number of zi∗ from which the probability density function (pdf) of Zt is to be estimated.(4)
Determine the matrix of kernelsK which is(39)K=Kbz1∗-Z1Kbz1∗-Z2⋯Kbz1∗-ZnKbz2∗-Z1Kbz2∗-Z2⋯Kbz2∗-Zn⋮⋮⋮⋮KbzN∗-Z1KbzN∗-Z2⋯KbzN∗-Zn.The row sums ofK over n give the estimator of the pdf of Zt at zi∗, g^(zi∗), i=1,2,…,N. We obtain the matrix of weights W by the ration of K and KIn (element-wise), where In is a matrix of n×n ones. Note that the row sums of W are 1.LetM be the (0,1)-matrix from 2. The estimator of the Conditional Cumulative Distribution Function (CCDF) is(40)F^x∗∣z∗=WM=KM/KIn.


### 2.2. Nadaraya-Watson Smoothing Method

We can makeϖ^τ(z) smooth by using NW regression (one can also use LOWESS (LOcally WEighted Scatter plot Smoother) regression introduced by [22] to smooth the estimator in (36) and it solves the problem of boundary effects). This will provide a smoothed curve at each level τ∈(0,1). We write the regression equation as(41)Yt=ϖτ,sZt+ηtwith Yt=ϖτ(Zt) and ϖτ,s(x)=E[ϖτ(z)∣Zt=z] and the errors {ηi} satisfy E[ηi]=0, V(ηi)=ση2, and cov(ηi)=0 for i≠j. Note that ϖτ,s(x) can be derived using joint pdf f(y,z) as(42)ϖτ,sz=EY∣Z=z=∫yfy,zfzdy,where f(y,z) and f(z) are estimated as in (28).We can perform some transformations on (42) in order to show that it is actually better than the unsmoothed one. By Assumption 1 (iv) and the fact that Fϖτ(z)∣z=τ, we have (43)Fϖτ,sZt∣z=FEϖτz∣Zt=z∣z≤EFϖτz∣z∣Zt=z=FϖτZt∣z=τ.We have used Jensen’s theorem for conditional expectation found in [23] and stated as follows.Theorem 8 (Jensen’s inequality).
For any convex functionl,(44)ElX≥lEX.Proof of Theorem8.
Suppose thatl is differentiable. The function l is convex if(45)lx≥ly+x-yl′x,for  anyx,y.Let x=X and y=E[X]. The inequality l(X)≥l(E[X])+X-EXl′(X) is true for all X and taking its expectation on both sides proves the theorem.This inequality is applicable whenf is a conditional convex function and when E[·] is a conditional expectation. The estimator ϖτ,s(Zt) is also element of the set to which the unsmoothed estimator belongs. This means that Fϖτ,s(Zt)∣z≥τ. The estimator is empirically given by(46)ϖ^τ,sz=∑t=1nKbZt-zyt∑t=1nKbZt-z=∑t=1nKbZt-zϖτZt∑t=1nKbZt-z.


#### 2.2.1. Asymptotic Properties

To show the asymptotic properties of our estimator, we compute its expectation and variance. Assuming the data(Y,Z) is i.i.d, the expectation of the numerator is given by(47)EKbZt-zYt=∬vbKu-zbfu,vdudv=∬vKsfv∣z+sbfz+sbdsdv=∫Ksfz+sb∫vfv∣z=sbdvds=∫Ksfz+sbϖτ,sz+shds.We assume that the first and the second derivatives of ϖτ,s(z) at point Zt=z exist. That is, by Taylor’s expansion of f(z+sb) and ϖτ,s(z+sh) given by(48)fz+sh=fz+f1z1!sbz+f2z2!sbz2+obz2,ϖτ,sz+sh=ϖτ,sz+ϖτ,s1z1!sbz+ϖτ,s2z2!sbz2+obz2,we get(49)EKbZt-zYt=ϖτ,szfz+12b2μ2Kfzϖτ,s2z+f1zϖτ,s1z+f2zϖτ,sz+oh3.Similarly, the expectation of the numerator is(50)EKbZt-z=fz+12b2μ2Kf2z+oh2.For b2 small enough, 1+1/2b2μ2(K)f(2)(z)/f(z)-1≈1-1/2b2μ2(K)f(2)(z)/f(z). Thus,(51)Eϖ^τ,sz≈ϖτ,sz+12b2μ2Kϖτ,s2z+2f1zfzϖτ,s1z.The variance of the numerator, say V(N), is(52)V1n∑t=1nKbZt-zYt=1nb2VKZt-zbYt=1nb2EK2Zt-zbyt2-EKZt-zbYt2≈1nb∬v2K2sfv∣z+sbfz+sbdsdv-o1n=1nb∫K2sfz+sb∫v2fv∣z+sbdvds-o1n≈1nbRKfzση2+ϖτ,s2z.Note that ∫v2f(v∣z+sb)ds≈E[Yt2∣Zt=z]. Similarly, the variance of the denominator, V(D), is V1/n∑t=1nKb(Zt-z)≈1/nbf(z)R(K).The covariance of the numerator and the denominator of the estimator in (46) are given by(53)covN,D=cov1nb∑t=1nKZt-zbYt,1nb∑t=1nKZt-zb=1nb2covKZt-zbYt,KZt-zb=1nb2EK2Zt-zbYt-EKZt-zbYtEKZt-zb≈1nbRKfzϖτ,sz-o1n.The variance of the estimator in (46) is the variance of a ratio of correlated variables that can be calculated using the approximation found in [24]:(54)VND≈ENED2VNEN2+VDED2-2covN,DENED(55)=RKση2nbfz.If Assumption 3 for strong mixing processes holds, then from the Central Limit Theorem (CLT) we have(56)nbϖ^τ,sz-ϖτ,sz-Biasϖ^τ,sz→DN0,RKση2fz.


### 2.3. Asymptotic Normality of QARCH

The CCDF in (35) can be written in the form of an arithmetic mean of a random variable L:(57)F^x∗∣z=1n∑t=1nLtwithLt=KbzZt-zIXt∗≤x∗1/n∑t=1nKbzZt-zand the approximation of the expectation of L is(58)ELt≈EKbzZt-zIXt∗≤x∗E1/n∑t=1nKbzZt-z=ENED[see [24]]. Using the i.i.d assumption over the data, the numerator is(59)EN=1bzEKZt-zbzIXt∗≤x∗=1bz∫∫-∞x∗Ku-zbzfu,vdudv=∫Fx∗∣z+shKsfz+shds.We have used the change of variables s=(u-z)/bz, the definition of the conditional density function turned into f(z+sbz,v)=f(v∣z+sh)f(z+sbz), and Fubuni’s theorem for multiple integrals. Taylor series expansions of F(v∣z+sh) and f(z+sh) yield(60)EN=fzFx∗∣z+bz2μ2Kf1zF1x∗∣z+12f2zFx∗∣z+12fzF2x∗∣z+obz2and, for the denominator, we have(61)ED=fz+12bz2μ2Kf2z+obz2.Thus,(62)ELt≈fzFx∗∣z+bz2μ2Kf1z/fzF1x∗∣z+1/2f2z/fzFx∗∣z+1/2F2x∗∣zfz1+1/2bz2μ2Kf2z/fz=Fx∗∣z+12bz2μ2K2f1zfzF1x∗∣z+F2x∗∣z+obz4.From the assumption that bz→0, the denominator is approximated to 1-bz2μ2(K)f(2)(z)/2f(z). Hence,(63)BiasF^x∗∣z≈12bz2μ2K2f1zfzF1x∗∣z+F2x∗∣z.Some authors assumed that, in this case, the first derivative of the true pdf of Z at point z can be zero [19] as the one for the fixed design and, therefore, the bias can be given by(64)BiasF^x∗∣z≈12bz2μ2KF2x∗∣z.We have(65)VN=V1bzKZt-zbzIXt∗≤x∗=1bz2VKZt-zbzIXt∗≤x∗=1bz2EK2Zt-zbzIXt∗≤x∗-EKZt-zbzIXt∗≤x∗2≈Fx∗∣zfzRKbz-o1,VD=V1n∑t=1nKbzZt-z=1nbz2VKZt-zbz=1nbz2EK2Zt-zbz-EKZt-zbz2≈fzRKnbz-o1n,covN,D=1nbz2covKZt-zbzIXt∗≤x∗,KZt-zbz≈1nbz2EK2Zt-zbzIXt∗≤x∗-o1n≈1nbzFx∗∣zfzRK.Using the same approximation in (54), the variance of F^(x∗∣z) is(66)VLt≈Fx∗∣zRK1-Fx∗∣zbzfzand by the Central Limit Theorem, using Assumption 3 for {(Xt∗,Zt),t=1,2,…},(67)nF^x∗∣z-Fx∗∣z-BiasFx∗∣z→DN0,VLt.Notice that the expectation of F^(x∗∣z) is the same as the one of L and the variance is V(Lt)/n. To show the asymptotic normality of ϖ^τ(z), we use the following theorem.Theorem 9 (delta method).
SupposeF^(x∗∣z) has the asymptotic normal distribution as in (67). Suppose g(·) is a continuous function that has a derivative g(1)(·) at μ=EF^(x∗∣z). Then(68)nbzgF^x∗∣z-gμ→DN0,g1μ2RK1-Fx∗∣zfz.Proof of Theorem9.
The first-order Taylor expansion ofg(·) about the point μ, and evaluated at the random variable F^(x∗∣z), is(69)gF^x∗∣z≈gμ+g1μF^x∗∣z-μand subtracting g(μ) from both sides and multiplying by nb, we get(70)nbgF^x∗∣z-gμ≈nbg1μF^x∗∣z-μwhich tends to N0,g(1)(μ)2R(K)1-F(x∗∣z)/f(z) in distribution.Forg(μ)=F-1(μ∣z), thus, g(1)(μ)=1/fF-1(μ∣z)∣z. In the next section, it is shown that the AMSE (Asymptotic Mean Squared Error) of F^(x∗∣z) is equal to ob4+o1/(nb) which tends to 0 as n→∞ and b→0. This shows the consistency of the CCDF estimate, that is, F^(x∗∣z)→pF(x∗∣z), and we have(71)1fF-1μ∣z∣z→p1fF-1τ∣z∣z=1fϖτz∣zat points x∗’s that satisfy (36). Using again the first-order Taylor expansion, we also have(72)gμ=gFx∗∣z+BiasF^x∗∣z≈gFx∗∣z+BiasF^x∗∣z×g1Fx∗∣z=x∗+BiasF^x∗∣zfx∗∣zfor x∗’s satisfying (36) and replacing F^(ϖτ(z)∣z) by F(ϖ^τ(z)∣z) using the uniqueness assumption of ϖτ(z), (68) becomes(73)nbϖ^τz-ϖτz-Biasϖ^τz→DN0,RKτ1-τfzfϖτz∣z2with Biasϖ^τ(z)=BiasF^(ϖτ(z)∣z)/f(ϖτ(z)∣z)≈1/2f(ϖτ(z)∣z)bz2μ2(K)F(2)(ϖτ(z)∣z).This result can be used to calculate the optimal bandwidth to compute the good estimation of the CSF.


## 2.1. Algorithm

This algorithm estimates the empirical CCDF,F^(x∗∣z), and its inverse F^-1(τ∣z). Starting with the estimation of the former, the denominator is easy to compute as the estimator of the probability density function of Z as vector of points z.(1)
ObtainXt∗=γτ(Xt), t=1,2,…, for all τ∈(0,1).(2)
Check if eachxt∗ is less than or equal to each observation of the whole sequence x∗=(x1∗,x2∗,…,xn∗)∈Rn. The result determines Ixt∗≤x which can be expressed in (0,1)-matrix of order n×n.(3)
Constructz1∗=min⁡(Z)<z2∗<⋯<zN∗=max⁡(Z) from the sequence of i.i.d random variable Z=(Z1,Z2,…,Zn) with observation z=(z1,z2,…,zn). N is the number of zi∗ from which the probability density function (pdf) of Zt is to be estimated.(4)
Determine the matrix of kernelsK which is(39)K=Kbz1∗-Z1Kbz1∗-Z2⋯Kbz1∗-ZnKbz2∗-Z1Kbz2∗-Z2⋯Kbz2∗-Zn⋮⋮⋮⋮KbzN∗-Z1KbzN∗-Z2⋯KbzN∗-Zn.The row sums ofK over n give the estimator of the pdf of Zt at zi∗, g^(zi∗), i=1,2,…,N. We obtain the matrix of weights W by the ration of K and KIn (element-wise), where In is a matrix of n×n ones. Note that the row sums of W are 1.LetM be the (0,1)-matrix from 2. The estimator of the Conditional Cumulative Distribution Function (CCDF) is(40)F^x∗∣z∗=WM=KM/KIn.


## 2.2. Nadaraya-Watson Smoothing Method

We can makeϖ^τ(z) smooth by using NW regression (one can also use LOWESS (LOcally WEighted Scatter plot Smoother) regression introduced by [22] to smooth the estimator in (36) and it solves the problem of boundary effects). This will provide a smoothed curve at each level τ∈(0,1). We write the regression equation as(41)Yt=ϖτ,sZt+ηtwith Yt=ϖτ(Zt) and ϖτ,s(x)=E[ϖτ(z)∣Zt=z] and the errors {ηi} satisfy E[ηi]=0, V(ηi)=ση2, and cov(ηi)=0 for i≠j. Note that ϖτ,s(x) can be derived using joint pdf f(y,z) as(42)ϖτ,sz=EY∣Z=z=∫yfy,zfzdy,where f(y,z) and f(z) are estimated as in (28).We can perform some transformations on (42) in order to show that it is actually better than the unsmoothed one. By Assumption 1 (iv) and the fact that Fϖτ(z)∣z=τ, we have (43)Fϖτ,sZt∣z=FEϖτz∣Zt=z∣z≤EFϖτz∣z∣Zt=z=FϖτZt∣z=τ.We have used Jensen’s theorem for conditional expectation found in [23] and stated as follows.Theorem 8 (Jensen’s inequality).
For any convex functionl,(44)ElX≥lEX.Proof of Theorem8.
Suppose thatl is differentiable. The function l is convex if(45)lx≥ly+x-yl′x,for  anyx,y.Let x=X and y=E[X]. The inequality l(X)≥l(E[X])+X-EXl′(X) is true for all X and taking its expectation on both sides proves the theorem.This inequality is applicable whenf is a conditional convex function and when E[·] is a conditional expectation. The estimator ϖτ,s(Zt) is also element of the set to which the unsmoothed estimator belongs. This means that Fϖτ,s(Zt)∣z≥τ. The estimator is empirically given by(46)ϖ^τ,sz=∑t=1nKbZt-zyt∑t=1nKbZt-z=∑t=1nKbZt-zϖτZt∑t=1nKbZt-z.


### 2.2.1. Asymptotic Properties

To show the asymptotic properties of our estimator, we compute its expectation and variance. Assuming the data(Y,Z) is i.i.d, the expectation of the numerator is given by(47)EKbZt-zYt=∬vbKu-zbfu,vdudv=∬vKsfv∣z+sbfz+sbdsdv=∫Ksfz+sb∫vfv∣z=sbdvds=∫Ksfz+sbϖτ,sz+shds.We assume that the first and the second derivatives of ϖτ,s(z) at point Zt=z exist. That is, by Taylor’s expansion of f(z+sb) and ϖτ,s(z+sh) given by(48)fz+sh=fz+f1z1!sbz+f2z2!sbz2+obz2,ϖτ,sz+sh=ϖτ,sz+ϖτ,s1z1!sbz+ϖτ,s2z2!sbz2+obz2,we get(49)EKbZt-zYt=ϖτ,szfz+12b2μ2Kfzϖτ,s2z+f1zϖτ,s1z+f2zϖτ,sz+oh3.Similarly, the expectation of the numerator is(50)EKbZt-z=fz+12b2μ2Kf2z+oh2.For b2 small enough, 1+1/2b2μ2(K)f(2)(z)/f(z)-1≈1-1/2b2μ2(K)f(2)(z)/f(z). Thus,(51)Eϖ^τ,sz≈ϖτ,sz+12b2μ2Kϖτ,s2z+2f1zfzϖτ,s1z.The variance of the numerator, say V(N), is(52)V1n∑t=1nKbZt-zYt=1nb2VKZt-zbYt=1nb2EK2Zt-zbyt2-EKZt-zbYt2≈1nb∬v2K2sfv∣z+sbfz+sbdsdv-o1n=1nb∫K2sfz+sb∫v2fv∣z+sbdvds-o1n≈1nbRKfzση2+ϖτ,s2z.Note that ∫v2f(v∣z+sb)ds≈E[Yt2∣Zt=z]. Similarly, the variance of the denominator, V(D), is V1/n∑t=1nKb(Zt-z)≈1/nbf(z)R(K).The covariance of the numerator and the denominator of the estimator in (46) are given by(53)covN,D=cov1nb∑t=1nKZt-zbYt,1nb∑t=1nKZt-zb=1nb2covKZt-zbYt,KZt-zb=1nb2EK2Zt-zbYt-EKZt-zbYtEKZt-zb≈1nbRKfzϖτ,sz-o1n.The variance of the estimator in (46) is the variance of a ratio of correlated variables that can be calculated using the approximation found in [24]:(54)VND≈ENED2VNEN2+VDED2-2covN,DENED(55)=RKση2nbfz.If Assumption 3 for strong mixing processes holds, then from the Central Limit Theorem (CLT) we have(56)nbϖ^τ,sz-ϖτ,sz-Biasϖ^τ,sz→DN0,RKση2fz.


## 2.2.1. Asymptotic Properties

To show the asymptotic properties of our estimator, we compute its expectation and variance. Assuming the data(Y,Z) is i.i.d, the expectation of the numerator is given by(47)EKbZt-zYt=∬vbKu-zbfu,vdudv=∬vKsfv∣z+sbfz+sbdsdv=∫Ksfz+sb∫vfv∣z=sbdvds=∫Ksfz+sbϖτ,sz+shds.We assume that the first and the second derivatives of ϖτ,s(z) at point Zt=z exist. That is, by Taylor’s expansion of f(z+sb) and ϖτ,s(z+sh) given by(48)fz+sh=fz+f1z1!sbz+f2z2!sbz2+obz2,ϖτ,sz+sh=ϖτ,sz+ϖτ,s1z1!sbz+ϖτ,s2z2!sbz2+obz2,we get(49)EKbZt-zYt=ϖτ,szfz+12b2μ2Kfzϖτ,s2z+f1zϖτ,s1z+f2zϖτ,sz+oh3.Similarly, the expectation of the numerator is(50)EKbZt-z=fz+12b2μ2Kf2z+oh2.For b2 small enough, 1+1/2b2μ2(K)f(2)(z)/f(z)-1≈1-1/2b2μ2(K)f(2)(z)/f(z). Thus,(51)Eϖ^τ,sz≈ϖτ,sz+12b2μ2Kϖτ,s2z+2f1zfzϖτ,s1z.The variance of the numerator, say V(N), is(52)V1n∑t=1nKbZt-zYt=1nb2VKZt-zbYt=1nb2EK2Zt-zbyt2-EKZt-zbYt2≈1nb∬v2K2sfv∣z+sbfz+sbdsdv-o1n=1nb∫K2sfz+sb∫v2fv∣z+sbdvds-o1n≈1nbRKfzση2+ϖτ,s2z.Note that ∫v2f(v∣z+sb)ds≈E[Yt2∣Zt=z]. Similarly, the variance of the denominator, V(D), is V1/n∑t=1nKb(Zt-z)≈1/nbf(z)R(K).The covariance of the numerator and the denominator of the estimator in (46) are given by(53)covN,D=cov1nb∑t=1nKZt-zbYt,1nb∑t=1nKZt-zb=1nb2covKZt-zbYt,KZt-zb=1nb2EK2Zt-zbYt-EKZt-zbYtEKZt-zb≈1nbRKfzϖτ,sz-o1n.The variance of the estimator in (46) is the variance of a ratio of correlated variables that can be calculated using the approximation found in [24]:(54)VND≈ENED2VNEN2+VDED2-2covN,DENED(55)=RKση2nbfz.If Assumption 3 for strong mixing processes holds, then from the Central Limit Theorem (CLT) we have(56)nbϖ^τ,sz-ϖτ,sz-Biasϖ^τ,sz→DN0,RKση2fz.


## 2.3. Asymptotic Normality of QARCH

The CCDF in (35) can be written in the form of an arithmetic mean of a random variable L:(57)F^x∗∣z=1n∑t=1nLtwithLt=KbzZt-zIXt∗≤x∗1/n∑t=1nKbzZt-zand the approximation of the expectation of L is(58)ELt≈EKbzZt-zIXt∗≤x∗E1/n∑t=1nKbzZt-z=ENED[see [24]]. Using the i.i.d assumption over the data, the numerator is(59)EN=1bzEKZt-zbzIXt∗≤x∗=1bz∫∫-∞x∗Ku-zbzfu,vdudv=∫Fx∗∣z+shKsfz+shds.We have used the change of variables s=(u-z)/bz, the definition of the conditional density function turned into f(z+sbz,v)=f(v∣z+sh)f(z+sbz), and Fubuni’s theorem for multiple integrals. Taylor series expansions of F(v∣z+sh) and f(z+sh) yield(60)EN=fzFx∗∣z+bz2μ2Kf1zF1x∗∣z+12f2zFx∗∣z+12fzF2x∗∣z+obz2and, for the denominator, we have(61)ED=fz+12bz2μ2Kf2z+obz2.Thus,(62)ELt≈fzFx∗∣z+bz2μ2Kf1z/fzF1x∗∣z+1/2f2z/fzFx∗∣z+1/2F2x∗∣zfz1+1/2bz2μ2Kf2z/fz=Fx∗∣z+12bz2μ2K2f1zfzF1x∗∣z+F2x∗∣z+obz4.From the assumption that bz→0, the denominator is approximated to 1-bz2μ2(K)f(2)(z)/2f(z). Hence,(63)BiasF^x∗∣z≈12bz2μ2K2f1zfzF1x∗∣z+F2x∗∣z.Some authors assumed that, in this case, the first derivative of the true pdf of Z at point z can be zero [19] as the one for the fixed design and, therefore, the bias can be given by(64)BiasF^x∗∣z≈12bz2μ2KF2x∗∣z.We have(65)VN=V1bzKZt-zbzIXt∗≤x∗=1bz2VKZt-zbzIXt∗≤x∗=1bz2EK2Zt-zbzIXt∗≤x∗-EKZt-zbzIXt∗≤x∗2≈Fx∗∣zfzRKbz-o1,VD=V1n∑t=1nKbzZt-z=1nbz2VKZt-zbz=1nbz2EK2Zt-zbz-EKZt-zbz2≈fzRKnbz-o1n,covN,D=1nbz2covKZt-zbzIXt∗≤x∗,KZt-zbz≈1nbz2EK2Zt-zbzIXt∗≤x∗-o1n≈1nbzFx∗∣zfzRK.Using the same approximation in (54), the variance of F^(x∗∣z) is(66)VLt≈Fx∗∣zRK1-Fx∗∣zbzfzand by the Central Limit Theorem, using Assumption 3 for {(Xt∗,Zt),t=1,2,…},(67)nF^x∗∣z-Fx∗∣z-BiasFx∗∣z→DN0,VLt.Notice that the expectation of F^(x∗∣z) is the same as the one of L and the variance is V(Lt)/n. To show the asymptotic normality of ϖ^τ(z), we use the following theorem.Theorem 9 (delta method).
SupposeF^(x∗∣z) has the asymptotic normal distribution as in (67). Suppose g(·) is a continuous function that has a derivative g(1)(·) at μ=EF^(x∗∣z). Then(68)nbzgF^x∗∣z-gμ→DN0,g1μ2RK1-Fx∗∣zfz.Proof of Theorem9.
The first-order Taylor expansion ofg(·) about the point μ, and evaluated at the random variable F^(x∗∣z), is(69)gF^x∗∣z≈gμ+g1μF^x∗∣z-μand subtracting g(μ) from both sides and multiplying by nb, we get(70)nbgF^x∗∣z-gμ≈nbg1μF^x∗∣z-μwhich tends to N0,g(1)(μ)2R(K)1-F(x∗∣z)/f(z) in distribution.Forg(μ)=F-1(μ∣z), thus, g(1)(μ)=1/fF-1(μ∣z)∣z. In the next section, it is shown that the AMSE (Asymptotic Mean Squared Error) of F^(x∗∣z) is equal to ob4+o1/(nb) which tends to 0 as n→∞ and b→0. This shows the consistency of the CCDF estimate, that is, F^(x∗∣z)→pF(x∗∣z), and we have(71)1fF-1μ∣z∣z→p1fF-1τ∣z∣z=1fϖτz∣zat points x∗’s that satisfy (36). Using again the first-order Taylor expansion, we also have(72)gμ=gFx∗∣z+BiasF^x∗∣z≈gFx∗∣z+BiasF^x∗∣z×g1Fx∗∣z=x∗+BiasF^x∗∣zfx∗∣zfor x∗’s satisfying (36) and replacing F^(ϖτ(z)∣z) by F(ϖ^τ(z)∣z) using the uniqueness assumption of ϖτ(z), (68) becomes(73)nbϖ^τz-ϖτz-Biasϖ^τz→DN0,RKτ1-τfzfϖτz∣z2with Biasϖ^τ(z)=BiasF^(ϖτ(z)∣z)/f(ϖτ(z)∣z)≈1/2f(ϖτ(z)∣z)bz2μ2(K)F(2)(ϖτ(z)∣z).This result can be used to calculate the optimal bandwidth to compute the good estimation of the CSF.


## 3. Bandwidth Selections

### 3.1. Optimal Bandwidth for Density Estimations

In nonparametric estimations, specially in Kernel Density Estimations, computing a curve of an arbitrary function from the data without guessing the shape in advance requires an adequate choice of the smoothing parameter. The most used method is the “plug-in” method which consists of assigning a pilot bandwidth in order to estimate the derivatives off^(z). We choose the bandwidth that minimizes the AMISE (Asymptotic Mean Integrated Squared Error) below.(74)AMISEf^z=∫Ef^z-fz2dz=∫Ef^z-Ef^z+Biasf^z2dz=∫Ef^z-Ef^z2+Bias2f^zdz=∫Vf^z+Bias2f^zdz=∫RKfznb+14b4μ22Kf2z2dz=RKnb+14b4μ22KRf2z.The general form of the rth derivatives of the AMISE with respect to b was studied in [25], considering that the unknown functions in (74) are also functions of the smoothing parameter.(75)ddzrAMISEf^z=RKrnb2r+1+14b4μ22KRf2+rz.The optimal smoothing parameter minimizing (75) is(76)b∗=2r+1RKrμ22KRf2+rz1/2r+5×n-1/2r+5.Using this result, we came up with the optimal version of optimal bandwidth for CCDF. The aim of derivation of the AMISE in (74) is to get the optimal bandwidth for each f(r) directly. As an example, we consider the Epanechnikov Kernel function in order to compute R(K), μ2(K), and the efficiency of the kernel function given by μ2(K)R(K). Epanechnikov’s kernel function is (77)Ku=341-u2Iu≤1⟹RK=34∫-111-2u2+u4du=35,μ2K=∫-11u2Kudu=∫-11u2-u4du=15and its efficiency is measured by(78)EffK=RKμ2K=3415=0.268which is the smallest of all the other kernel functions.


### 3.2. Optimal Bandwidth for CCDF

The optimal bandwidth for the CCDF estimate is the one that minimizes the AMSE. It is shown below that the AMSE is actually the summation of the variance and the bias of the CCDF estimate. This is useful because the two are linked. When the variance is big, the bias also is big and when the variance is small, the bias is small.(79)AMSEF^x∗∣z=EF^x∗∣z-Fx∗∣z2=EF^x∗∣z-EF^x∗∣z+BiasF^x∗∣z2=EF^x∗∣z-EF^x∗∣z2+BiasF^x∗∣z×EF^x∗∣z-EF^x∗∣z+Bias2F^x∗∣z=VF^x∗∣z+Bias2F^x∗∣z=RKnbzfzFx∗∣z1-Fx∗∣z+b44μ22KF2x∗∣z2which is given by (66) and (64). Therefore,(80)b∗=arg⁡minb>0AMSEF^x∗∣zand d/dbAMSEF^(x∗∣z)=0 leads to(81)b∗=RKFx∗∣z1-Fx∗∣zμ22KfzF2x∗∣z21/5×n-1/5.This result is practically possible by estimating the unknown functions which are dependent on the smoothing parameter. F^(2) is the second derivative of the CCDF from (35) at point Zt=z. The estimator of the rth derivatives of (35) is(82)F^rx∗∣z=drdzr∑t=1nWtzXXt∗≤x∗=∑t=1nWtrzXXt∗≤x∗with(83)Wtz=KZt-z/b∑t=1nKZt-z/b=KZt-z/bnbf^z,the function of weights. Thus, the first derivative is given by(84)Wt1z=1nb2K1Zt-z/bf^z-bKZt-z/bf^1zf1z2=1nb2ABand the second derivative is also(85)Wt2z=1nb2A1B-B1AB2with A(1)=1/bK(2)Zt-z/bf^(z)-bKZt-z/bf^(2)(z) and B(1)=2f^(1)(z)f^(z). Note that the estimation of the CCDF is function of the estimation of the empirical pdf of z. An optimal bandwidth that minimizes the AMISE of f^(z) can also be the one that is optimal for the estimation of the CCDF.Recent findings on the estimation of an optimal bandwidth for KDE (Kernel Density Estimation) are numerous ([25–27]) but the estimation of an optimal smoothing parameter remains irksome due to computation issue and time consuming routines. To do so, we adopt what had been done by [27] to estimate the rth derivatives of the pdf of Zt with respect to z. We extend the idea to estimate the first and the second derivative of the CCDF with respect to z.


## 3.1. Optimal Bandwidth for Density Estimations

In nonparametric estimations, specially in Kernel Density Estimations, computing a curve of an arbitrary function from the data without guessing the shape in advance requires an adequate choice of the smoothing parameter. The most used method is the “plug-in” method which consists of assigning a pilot bandwidth in order to estimate the derivatives off^(z). We choose the bandwidth that minimizes the AMISE (Asymptotic Mean Integrated Squared Error) below.(74)AMISEf^z=∫Ef^z-fz2dz=∫Ef^z-Ef^z+Biasf^z2dz=∫Ef^z-Ef^z2+Bias2f^zdz=∫Vf^z+Bias2f^zdz=∫RKfznb+14b4μ22Kf2z2dz=RKnb+14b4μ22KRf2z.The general form of the rth derivatives of the AMISE with respect to b was studied in [25], considering that the unknown functions in (74) are also functions of the smoothing parameter.(75)ddzrAMISEf^z=RKrnb2r+1+14b4μ22KRf2+rz.The optimal smoothing parameter minimizing (75) is(76)b∗=2r+1RKrμ22KRf2+rz1/2r+5×n-1/2r+5.Using this result, we came up with the optimal version of optimal bandwidth for CCDF. The aim of derivation of the AMISE in (74) is to get the optimal bandwidth for each f(r) directly. As an example, we consider the Epanechnikov Kernel function in order to compute R(K), μ2(K), and the efficiency of the kernel function given by μ2(K)R(K). Epanechnikov’s kernel function is (77)Ku=341-u2Iu≤1⟹RK=34∫-111-2u2+u4du=35,μ2K=∫-11u2Kudu=∫-11u2-u4du=15and its efficiency is measured by(78)EffK=RKμ2K=3415=0.268which is the smallest of all the other kernel functions.


## 3.2. Optimal Bandwidth for CCDF

The optimal bandwidth for the CCDF estimate is the one that minimizes the AMSE. It is shown below that the AMSE is actually the summation of the variance and the bias of the CCDF estimate. This is useful because the two are linked. When the variance is big, the bias also is big and when the variance is small, the bias is small.(79)AMSEF^x∗∣z=EF^x∗∣z-Fx∗∣z2=EF^x∗∣z-EF^x∗∣z+BiasF^x∗∣z2=EF^x∗∣z-EF^x∗∣z2+BiasF^x∗∣z×EF^x∗∣z-EF^x∗∣z+Bias2F^x∗∣z=VF^x∗∣z+Bias2F^x∗∣z=RKnbzfzFx∗∣z1-Fx∗∣z+b44μ22KF2x∗∣z2which is given by (66) and (64). Therefore,(80)b∗=arg⁡minb>0AMSEF^x∗∣zand d/dbAMSEF^(x∗∣z)=0 leads to(81)b∗=RKFx∗∣z1-Fx∗∣zμ22KfzF2x∗∣z21/5×n-1/5.This result is practically possible by estimating the unknown functions which are dependent on the smoothing parameter. F^(2) is the second derivative of the CCDF from (35) at point Zt=z. The estimator of the rth derivatives of (35) is(82)F^rx∗∣z=drdzr∑t=1nWtzXXt∗≤x∗=∑t=1nWtrzXXt∗≤x∗with(83)Wtz=KZt-z/b∑t=1nKZt-z/b=KZt-z/bnbf^z,the function of weights. Thus, the first derivative is given by(84)Wt1z=1nb2K1Zt-z/bf^z-bKZt-z/bf^1zf1z2=1nb2ABand the second derivative is also(85)Wt2z=1nb2A1B-B1AB2with A(1)=1/bK(2)Zt-z/bf^(z)-bKZt-z/bf^(2)(z) and B(1)=2f^(1)(z)f^(z). Note that the estimation of the CCDF is function of the estimation of the empirical pdf of z. An optimal bandwidth that minimizes the AMISE of f^(z) can also be the one that is optimal for the estimation of the CCDF.Recent findings on the estimation of an optimal bandwidth for KDE (Kernel Density Estimation) are numerous ([25–27]) but the estimation of an optimal smoothing parameter remains irksome due to computation issue and time consuming routines. To do so, we adopt what had been done by [27] to estimate the rth derivatives of the pdf of Zt with respect to z. We extend the idea to estimate the first and the second derivative of the CCDF with respect to z.


## 4. Simulation Study

### 4.1. Model Specification

The ARCH(q) models introduced by [16] are widely used in financial applications. An AR(1)-ARCH(1) is a mixed model from an AR(d) and GARCH(p,q) for d=1, p=1, and q=0. In time series, an observation at one time can be correlated with the observations in the previous time. That is,Note that the operator·/· means the element-wise division between matrices.(5)
For each row ofF^(·∣·), find the smallest x∗ such that F^(x∗∣z∗)≥τ,τ∈(0,1).(6)
The quantilesϖ^τ(z) are the x∗’s which satisfy (36). This gives an unsmoothed estimator curve with bad shape at boundaries (see Figure 2).The data to be simulated is given byXt=μ+δXt-1+w+αXt-121/2et, t=1,2,….(i)
autoregressive process of orderp=1,2,…,(86)ARp:Xt=μ+δ1Xt-1+δ2Xt-2+⋯+δpXt-p+et,withεti.i.d,(ii)
autoregressive (p)-General Autoregressive Conditional Heteroscedastic process of order (d=1,2,…;p=1,2,…;q=1,2,…), (87)ARd-GARCHp,q:Xt=∑i=1paiXt-i+ϖtet,witheti.i.d.  andϖt=w+∑i=1pαiui-12+∑i=1qβiϖi-121/2.


### 4.2. Specifications for AR(1)-GARCH(1,1)

#### 4.2.1. Unconditional Expectation

The unconditional expectation is(88)EXt=μ+δEXt-1+Eϖtet=μ+δEXt+EϖtEet.Note that E[Xt]=E[Xt-1] is used to ensure the stationarity of the process. That is, the expectation is therefore given by(89)EXt=μ1-δ.


#### 4.2.2. Unconditional Variance

The unconditional variance of the model is given by the law of total variance(90)VXt=EVXt∣Xt-1+VEXt∣Xt-1(91)=Eϖt2+Vαt.We have(92)Eϖt2=ω+αEXt-12+βEϖt-12.Using the i.i.d. assumption on the sequence of random variables X1,X2,…,Xn, the expected value of Xt2 can be calculated as follows:(93)EXt2=EμXt+δXt-1Xt+ϖtetXt=μEXt+δEXt2=μ21-δ+δμ21-δ2=μ21-δ2,which is independent of time. In another way,(94)EXt2=Eαt2+2αϖtet+ϖt2et2=Eαt2+Eϖt2.Equation (92) becomes(95)Eϖt2=ω+αEαt2+Eϖt2+βEϖt-12=ω+αEαt2+α+βEϖt2stationarity.We obtain(96)Eϖt2=ω+αEαt21-α-β.The expectation of αt2 is given by(97)Eαt2=Eμ+δXt-12=μ2+2μδEXt+δ2EXt2=μ2+2δμ21-δ+δ2μ21-δ2=μ21-δ2.It follows that(98)Eϖt2=ω1-δ2+αμ21-α-β1-δ2and the variance in (91) becomes(99)VXt=ω1-δ2+αμ21-α-β1-δ2+Vμ+δXt-1=ω1-δ2+αμ21-α-β1-δ2+δ2VXt=ω1-δ2+αμ21-α-β1-δ21-δ2.This variance is positive and finite for μ∈R, δ<1, ω>0, α>0, β>0, and α+β<1.


### 4.3. Model Simulation

We simulated the data from (1) with μ=0.5, δ=0.3, for the AR(1) part and w=0.1,α=0.35, for the ARCH(1) and et~i.i.d.N(0,1). The data plot is represented by Figure 1.Figure 1
Plot of the simulated AR(1)-ARCH(1).Figure 2
Conditional scale function estimate atτ=0.75.Our algorithm gives the estimation of the conditional scale function which suffers from boundary effects as it is seen from Figure2. This issue is recurrent while performing Kernel Density Estimations. The reason is that, at the boundaries, g(z) is underestimated because of the minimal number of points [28]. The consistency of our estimator is dependent on this problem of big variations at the boundaries. This increases the Average Squared Error between two different estimations from the same model.


### 4.4. Boundary Correction

To correct the boundary effects, we use the method of box-plot fences proposed by [29] to detect the extreme values that make the estimation too rough at the extremities of the CCDF estimations’ curves. Our estimator, being the inverse of the CCDF, is naturally rough at extremities. Among the Kernel functions, only the Gaussian can handle the sparseness of points at boundaries because its domain is R. The other kernel functions can bring zero at extremities and make the estimation of the CCDF wrong. What we do is to omit the points that are extremely far from the others by the box-plot fences method. The method consist of determining the first and the third quantiles from the Zt’s. Outliers are the points that are located outside the interval(100)Q1-3×Q3-Q1,Q3+3×Q3-Q1,where Q1 and Q3 are the first and the third quantiles. Figure 3 is the representation of Zt and the transformed response variable Xt∗ defined in (34) at level τ=0.75.Figure 3
Scatter plot and outliers detection.The gray points are outliers from (100). We lose some information by deleting them but we get the possibility of performing the estimation of a continuous curve of the CSF. Figure 4 is the estimations of the CSF at levels 0.25, 0.5 (median), 0.75, and 0.9. As we can see on the graphic, despite the optimal bandwidth for the empirical pdf of Zt at point z, we get unsmoothed curves at high level τ>0.5.Figure 4
CSF estimations.The curves represent the estimations of the CSF atτ=0.9,0.75,0.50,0.25 from up to down. As it is seen in Figure 4, some curves are not smooth; that is why the NW method is discussed in Section 2.2 which requires that unsmoothed estimator and the bins z1∗,z2∗,…,zN∗. We obtain Figure 5 which combines the two estimations.Figure 5
Smoothed estimate of the CSF.The next section discusses how precise is our estimation with the optimal bandwidth selection with the calculation of the MASE (Mean Average Squared Errors).


### 4.5. Consistency

The consistency of the estimator can be shown with the calculation of the Mean Average Squared Error providing the quantitative assessment of the accuracy of our estimator. This is a kind of bootstrap method to calculate the average gap betweenm estimated CSFs. The formula is(101)MASEϖ^τz=1n∑j=1n1m∑i=1mϖ^τ,1zi-ϖ^τ,jzi2.Table 2 shows that the estimator of the CSF is more precise at level τ≤0.55 for both the smoothed and the LOWESS versions.Table 2
Mean average squared errors (MASE).
Kern. func. n m = 10 m = 50 m = 100 mase (.25)  Gaussian 250 0.0017 0.0013 0.0010 0.0008 0.0010 0.0009 500 0.0011 0.0008 0.0008 0.0007 0.0007 0.0006 1,000 0.0006 0.0004 0.0005 0.0004 0.0005 0.0004  Epanech 200 0.0012 0.0009 0.0013 0.0011 0.0011 0.0009 500 0.0007 0.0006 0.0010 0.0008 0.0011 0.0009 1,000 0.0006 0.0004 0.0007 0.0005 0.0006 0.0005  Triweight 200 0.0005 0.0005 0.0006 0.0005 0.0006 0.0005 500 0.0006 0.0005 0.0005 0.0004 0.0005 0.0004 1,000 0.0003 0.0002 0.0003 0.0002 0.0004 0.0003  mase (.50)  Gaussian 250 0.0054 0.0045 0.0063 0.0055 0.0051 0.0043 500 0.0036 0.0029 0.0048 0.0042 0.0047 0.0038 1,000 0.0028 0.0022 0.0025 0.0021 0.0028 0.0023  Epanech 200 0.0067 0.0057 0.0105 0.0091 0.0071 0.0060 500 0.0045 0.0036 0.0057 0.0046 0.0042 0.0034 1,000 0.0031 0.0026 0.0041 0.0033 0.0029 0.0023  Triweight 200 0.0008 0.0007 0.0030 0.0026 0.0039 0.0034 500 0.0023 0.0020 0.0021 0.0018 0.0025 0.0021 1,000 0.0019 0.0016 0.0016 0.0013 0.0016 0.0013  mase (.75)  Gaussian 250 0.0234 0.0183 0.0237 0.0197 0.0294 0.0253 500 0.0227 0.0178 0.0223 0.0178 0.0171 0.0132 1,000 0.0099 0.0079 0.0138 0.0110 0.0125 0.0095  Epanech 200 0.0156 0.0123 0.0266 0.0223 0.0274 0.0227 500 0.0184 0.0152 0.0235 0.0189 0.0181 0.0147 1,000 0.0162 0.0130 0.0102 0.0074 0.0136 0.0106  Triweight 200 0.0190 0.0176 0.0145 0.0127 0.0167 0.0150 500 0.0112 0.0099 0.0131 0.0113 0.0097 0.0081 1,000 0.0075 0.0064 0.0073 0.0058 0.0069 0.0056  mase (.90)  Gaussian 250 0.0880 0.0692 0.1180 0.0893 0.0971 0.0770 500 0.0468 0.0377 0.0890 0.0644 0.0742 0.0525 1,000 0.0932 0.0690 0.0491 0.0367 0.0510 0.0365  Epanech 200 0.0664 0.0577 0.1074 0.0866 0.1050 0.0844 500 0.0816 0.0515 0.0827 0.0625 0.0879 0.0617 1,000 0.0740 0.0510 0.0449 0.0315 0.0373 0.0274  Triweight 200 0.0510 0.0434 0.0452 0.0382 0.0467 0.0402 500 0.0453 0.0337 0.0390 0.0333 0.0391 0.0328 1,000 0.0172 0.0133 0.0268 0.0205 0.0267 0.0209


## 4.1. Model Specification

The ARCH(q) models introduced by [16] are widely used in financial applications. An AR(1)-ARCH(1) is a mixed model from an AR(d) and GARCH(p,q) for d=1, p=1, and q=0. In time series, an observation at one time can be correlated with the observations in the previous time. That is,Note that the operator·/· means the element-wise division between matrices.(5)
For each row ofF^(·∣·), find the smallest x∗ such that F^(x∗∣z∗)≥τ,τ∈(0,1).(6)
The quantilesϖ^τ(z) are the x∗’s which satisfy (36). This gives an unsmoothed estimator curve with bad shape at boundaries (see Figure 2).The data to be simulated is given byXt=μ+δXt-1+w+αXt-121/2et, t=1,2,….(i)
autoregressive process of orderp=1,2,…,(86)ARp:Xt=μ+δ1Xt-1+δ2Xt-2+⋯+δpXt-p+et,withεti.i.d,(ii)
autoregressive (p)-General Autoregressive Conditional Heteroscedastic process of order (d=1,2,…;p=1,2,…;q=1,2,…), (87)ARd-GARCHp,q:Xt=∑i=1paiXt-i+ϖtet,witheti.i.d.  andϖt=w+∑i=1pαiui-12+∑i=1qβiϖi-121/2.


## 4.2. Specifications for AR(1)-GARCH(1,1)

### 4.2.1. Unconditional Expectation

The unconditional expectation is(88)EXt=μ+δEXt-1+Eϖtet=μ+δEXt+EϖtEet.Note that E[Xt]=E[Xt-1] is used to ensure the stationarity of the process. That is, the expectation is therefore given by(89)EXt=μ1-δ.


### 4.2.2. Unconditional Variance

The unconditional variance of the model is given by the law of total variance(90)VXt=EVXt∣Xt-1+VEXt∣Xt-1(91)=Eϖt2+Vαt.We have(92)Eϖt2=ω+αEXt-12+βEϖt-12.Using the i.i.d. assumption on the sequence of random variables X1,X2,…,Xn, the expected value of Xt2 can be calculated as follows:(93)EXt2=EμXt+δXt-1Xt+ϖtetXt=μEXt+δEXt2=μ21-δ+δμ21-δ2=μ21-δ2,which is independent of time. In another way,(94)EXt2=Eαt2+2αϖtet+ϖt2et2=Eαt2+Eϖt2.Equation (92) becomes(95)Eϖt2=ω+αEαt2+Eϖt2+βEϖt-12=ω+αEαt2+α+βEϖt2stationarity.We obtain(96)Eϖt2=ω+αEαt21-α-β.The expectation of αt2 is given by(97)Eαt2=Eμ+δXt-12=μ2+2μδEXt+δ2EXt2=μ2+2δμ21-δ+δ2μ21-δ2=μ21-δ2.It follows that(98)Eϖt2=ω1-δ2+αμ21-α-β1-δ2and the variance in (91) becomes(99)VXt=ω1-δ2+αμ21-α-β1-δ2+Vμ+δXt-1=ω1-δ2+αμ21-α-β1-δ2+δ2VXt=ω1-δ2+αμ21-α-β1-δ21-δ2.This variance is positive and finite for μ∈R, δ<1, ω>0, α>0, β>0, and α+β<1.


## 4.2.1. Unconditional Expectation

The unconditional expectation is(88)EXt=μ+δEXt-1+Eϖtet=μ+δEXt+EϖtEet.Note that E[Xt]=E[Xt-1] is used to ensure the stationarity of the process. That is, the expectation is therefore given by(89)EXt=μ1-δ.


## 4.2.2. Unconditional Variance

The unconditional variance of the model is given by the law of total variance(90)VXt=EVXt∣Xt-1+VEXt∣Xt-1(91)=Eϖt2+Vαt.We have(92)Eϖt2=ω+αEXt-12+βEϖt-12.Using the i.i.d. assumption on the sequence of random variables X1,X2,…,Xn, the expected value of Xt2 can be calculated as follows:(93)EXt2=EμXt+δXt-1Xt+ϖtetXt=μEXt+δEXt2=μ21-δ+δμ21-δ2=μ21-δ2,which is independent of time. In another way,(94)EXt2=Eαt2+2αϖtet+ϖt2et2=Eαt2+Eϖt2.Equation (92) becomes(95)Eϖt2=ω+αEαt2+Eϖt2+βEϖt-12=ω+αEαt2+α+βEϖt2stationarity.We obtain(96)Eϖt2=ω+αEαt21-α-β.The expectation of αt2 is given by(97)Eαt2=Eμ+δXt-12=μ2+2μδEXt+δ2EXt2=μ2+2δμ21-δ+δ2μ21-δ2=μ21-δ2.It follows that(98)Eϖt2=ω1-δ2+αμ21-α-β1-δ2and the variance in (91) becomes(99)VXt=ω1-δ2+αμ21-α-β1-δ2+Vμ+δXt-1=ω1-δ2+αμ21-α-β1-δ2+δ2VXt=ω1-δ2+αμ21-α-β1-δ21-δ2.This variance is positive and finite for μ∈R, δ<1, ω>0, α>0, β>0, and α+β<1.


## 4.3. Model Simulation

We simulated the data from (1) with μ=0.5, δ=0.3, for the AR(1) part and w=0.1,α=0.35, for the ARCH(1) and et~i.i.d.N(0,1). The data plot is represented by Figure 1.Figure 1
Plot of the simulated AR(1)-ARCH(1).Figure 2
Conditional scale function estimate atτ=0.75.Our algorithm gives the estimation of the conditional scale function which suffers from boundary effects as it is seen from Figure2. This issue is recurrent while performing Kernel Density Estimations. The reason is that, at the boundaries, g(z) is underestimated because of the minimal number of points [28]. The consistency of our estimator is dependent on this problem of big variations at the boundaries. This increases the Average Squared Error between two different estimations from the same model.


## 4.4. Boundary Correction

To correct the boundary effects, we use the method of box-plot fences proposed by [29] to detect the extreme values that make the estimation too rough at the extremities of the CCDF estimations’ curves. Our estimator, being the inverse of the CCDF, is naturally rough at extremities. Among the Kernel functions, only the Gaussian can handle the sparseness of points at boundaries because its domain is R. The other kernel functions can bring zero at extremities and make the estimation of the CCDF wrong. What we do is to omit the points that are extremely far from the others by the box-plot fences method. The method consist of determining the first and the third quantiles from the Zt’s. Outliers are the points that are located outside the interval(100)Q1-3×Q3-Q1,Q3+3×Q3-Q1,where Q1 and Q3 are the first and the third quantiles. Figure 3 is the representation of Zt and the transformed response variable Xt∗ defined in (34) at level τ=0.75.Figure 3
Scatter plot and outliers detection.The gray points are outliers from (100). We lose some information by deleting them but we get the possibility of performing the estimation of a continuous curve of the CSF. Figure 4 is the estimations of the CSF at levels 0.25, 0.5 (median), 0.75, and 0.9. As we can see on the graphic, despite the optimal bandwidth for the empirical pdf of Zt at point z, we get unsmoothed curves at high level τ>0.5.Figure 4
CSF estimations.The curves represent the estimations of the CSF atτ=0.9,0.75,0.50,0.25 from up to down. As it is seen in Figure 4, some curves are not smooth; that is why the NW method is discussed in Section 2.2 which requires that unsmoothed estimator and the bins z1∗,z2∗,…,zN∗. We obtain Figure 5 which combines the two estimations.Figure 5
Smoothed estimate of the CSF.The next section discusses how precise is our estimation with the optimal bandwidth selection with the calculation of the MASE (Mean Average Squared Errors).


## 4.5. Consistency

The consistency of the estimator can be shown with the calculation of the Mean Average Squared Error providing the quantitative assessment of the accuracy of our estimator. This is a kind of bootstrap method to calculate the average gap betweenm estimated CSFs. The formula is(101)MASEϖ^τz=1n∑j=1n1m∑i=1mϖ^τ,1zi-ϖ^τ,jzi2.Table 2 shows that the estimator of the CSF is more precise at level τ≤0.55 for both the smoothed and the LOWESS versions.Table 2
Mean average squared errors (MASE).
Kern. func. n m = 10 m = 50 m = 100 mase (.25)  Gaussian 250 0.0017 0.0013 0.0010 0.0008 0.0010 0.0009 500 0.0011 0.0008 0.0008 0.0007 0.0007 0.0006 1,000 0.0006 0.0004 0.0005 0.0004 0.0005 0.0004  Epanech 200 0.0012 0.0009 0.0013 0.0011 0.0011 0.0009 500 0.0007 0.0006 0.0010 0.0008 0.0011 0.0009 1,000 0.0006 0.0004 0.0007 0.0005 0.0006 0.0005  Triweight 200 0.0005 0.0005 0.0006 0.0005 0.0006 0.0005 500 0.0006 0.0005 0.0005 0.0004 0.0005 0.0004 1,000 0.0003 0.0002 0.0003 0.0002 0.0004 0.0003  mase (.50)  Gaussian 250 0.0054 0.0045 0.0063 0.0055 0.0051 0.0043 500 0.0036 0.0029 0.0048 0.0042 0.0047 0.0038 1,000 0.0028 0.0022 0.0025 0.0021 0.0028 0.0023  Epanech 200 0.0067 0.0057 0.0105 0.0091 0.0071 0.0060 500 0.0045 0.0036 0.0057 0.0046 0.0042 0.0034 1,000 0.0031 0.0026 0.0041 0.0033 0.0029 0.0023  Triweight 200 0.0008 0.0007 0.0030 0.0026 0.0039 0.0034 500 0.0023 0.0020 0.0021 0.0018 0.0025 0.0021 1,000 0.0019 0.0016 0.0016 0.0013 0.0016 0.0013  mase (.75)  Gaussian 250 0.0234 0.0183 0.0237 0.0197 0.0294 0.0253 500 0.0227 0.0178 0.0223 0.0178 0.0171 0.0132 1,000 0.0099 0.0079 0.0138 0.0110 0.0125 0.0095  Epanech 200 0.0156 0.0123 0.0266 0.0223 0.0274 0.0227 500 0.0184 0.0152 0.0235 0.0189 0.0181 0.0147 1,000 0.0162 0.0130 0.0102 0.0074 0.0136 0.0106  Triweight 200 0.0190 0.0176 0.0145 0.0127 0.0167 0.0150 500 0.0112 0.0099 0.0131 0.0113 0.0097 0.0081 1,000 0.0075 0.0064 0.0073 0.0058 0.0069 0.0056  mase (.90)  Gaussian 250 0.0880 0.0692 0.1180 0.0893 0.0971 0.0770 500 0.0468 0.0377 0.0890 0.0644 0.0742 0.0525 1,000 0.0932 0.0690 0.0491 0.0367 0.0510 0.0365  Epanech 200 0.0664 0.0577 0.1074 0.0866 0.1050 0.0844 500 0.0816 0.0515 0.0827 0.0625 0.0879 0.0617 1,000 0.0740 0.0510 0.0449 0.0315 0.0373 0.0274  Triweight 200 0.0510 0.0434 0.0452 0.0382 0.0467 0.0402 500 0.0453 0.0337 0.0390 0.0333 0.0391 0.0328 1,000 0.0172 0.0133 0.0268 0.0205 0.0267 0.0209


## 5. Conclusion

We have derived an estimator for the conditional scale function in an AR(1)-GARCH(1) and despite the heavy-tail of the data, we could deal with the boundary effect and were able to show the consistency of the estimator through a Monte Carlo study. We assumed that the QAR(1) is known and is zero and, along with the regularity assumptions, we derived the estimator which can be improved in some next papers. The very next paper will focus on the estimation when the QAR(1) is unknown.



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*Source: 4816716-2018-03-12.xml*