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

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.


Introduction
Consider a Quantile Autoregressive model, where   (  ) is the  th Conditional Quantile Function of   given   and the innovation   is assumed to be independent and identically distributed with zero  th 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 as where (  ) 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   =  +  −1 ,   = √ +  2 −1 +  2 −1 ,  ∈ (−∞, ∞), || < 1,  > 0,  > 0,  > 0,  +  < 1, and   ∼ i.i.d. 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-ARCHmodels, introduced in [7], and their nonparametric generalizations reviewed by [8].For instance, consider a financial time series model of AR()-ARCH()-type,   =  (  ) +  (  )   ,  = 1, 2, . . ., where   = (X −1 ,  −2 , . . .,  − ) 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   ∈ R and the variate   ∈ R  are, respectively, A  -measurable and A −1 -measurable, is considered.For some  ∈ (0, 1), the conditional -quantile of   given the past  −1 assumed to be determined by   is estimated.For simplicity, we assume that   =  −1 ∈ R 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 (0, 1) and a monotonically increasing function.This is very important to our estimation of the conditional distribution function and its inverse.

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).
(ii) For fixed (, ), 0 < ( | ) < 1 and () > 0 are continuous in the neighborhood of  where the estimator is to be estimated.
If () → 0 as  → ∞, then the process is strong mixing.
The results in this section are about the case when the Autoregressive part of the model (4)  , =   () = 0 for any  ∈ (0, 1).We therefore consider the model Define the check-function as Here,  { } is the indicator function.Therefore,   is a piecewise monotone increasing function.  (⋅, ⋅) is a function of any real random variable  with distribution function   () = ( ≤ ) =  {≤} , and a real value,  ∈ R, is the asymmetric absolute value function whose amount of asymmetry depends on ; see [15].In case where   is symmetric and  = 1/2, then we have the fact that 2  (  , ) is an absolute value function and  0.5 (  ) is the conditional median absolute deviation (CMAD) of   .When  became 0 in model ( 5), we have a purely heteroscedastic ARCH model introduced in [16] and   (  ) for  > 0.5, which, in this particular case, can be seen as a conditional scale function at -level.
The check-function in (10) is Lipschitz continuous by the following theorem.Theorem 6.Let   be defined as in (10)  By the next theorem we show clearly why the errors {  } in model ( 2) are assumed to be zero -quantile and unit scale.
Theorem 7. Consider model (5) and the so-called check-function in (10); then, for   (  ) ∈ R * + , is zero -quantile and unit scale.And the following equations are verifiable: Proof of Theorem 7. The  th -quantile operator is with well-defined properties in [1, p. 9-10].From model ( 5), the conditional -quantile of   is where    is the -quantiles of   .Then, using model ( 5) and ( 16), we get And the  th -quantile of ( 18) is where    is the -quantile of   (  ,    ).Note that, from (17),   (  −   (  )) = 0.The quotient is zero -quantile and unit scale and can be seen as model (2) if   = (  −    )/   ,   (  ) =   (  ), and   (  ) =   (  (  ,   (  ))).Now, assuming that   (independent of   ) in model ( 2) is zero -quantile, it is equivalent to write This proves (13) for   () > 0. Also,   is unit scale, which means Pr (  (  ) ≤ 1) =  ⇒ Assuming   (  ) = 0, the estimator, π (  ), of the conditional scale function,   (  ), is obtained through the minimization of the objective function Thus, the conditional scale function may be obtained by minimizing (, ) with respect to ; that is, The kernel estimator of ( 24) at   =  is given by We can express the estimate of (, ) in the random design as it was developed in [17].Let  *  =   (  (  ), ) be a nonnegative function of   and  * = ( * 1 ,  * 2 , . . .,  *  ) a random vector in R * + = (0, ∞),  = 1, 2, . . ., .In the random design, the conditional expectation ( 23) can be rewritten as follows: where ( * | ) represents the conditional pdf of  *  =  * given   = , ( * , ) is the joint pdf of the two random variables  * and , and () is the pdf of   = .Using [9,10] with   () =  −1 ( −1 ), a 1-dimensional rescaled kernel with bandwidth  > 0, we have the following estimates of ( * , ) and () [18]: From the estimations above, φ(, ), the estimate of (, ), is and considering the regularity conditions of   in Assumption 2 and also the fact that ( where f() is the estimate of the marginal pdf of   at point  and  * can be rewritten as and the derivative of φ (, ) with respect to  is The minimizer of (30) is obtained from  φ (, )/ = 0.This leads to the following equation: where for all   ∈ R,  = 1, 2, . ... Note that  *  =  { *  ≤} in (27).The left part of (33) is a (unsmoothed) conditional cumulative distribution function (CCDF), that needs to be estimated and our estimator is therefore An algorithm for estimating F( * | ) 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  direction.A method is proposed by [19] to smooth it in the .The form of Smoothed Conditional Distribution Estimator is where (⋅) is an integrated kernel with the smoothing parameter ℎ 0 in the  * direction.This estimate is smooth rather than the NW which is a jump function in .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 with conditions ∑  =1   (, ) = 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( * | ), and its inverse F−1 ( | ).Starting with the estimation of the former, the denominator is easy to compute as the estimator of the probability density function of  as vector of points .
( ) . (39) The row sums of  over  give the estimator of the pdf of   at  *  , ĝ( *  ),  = 1, 2, . . ., .We obtain the matrix of weights  by the ration of  and I  (element-wise), where I  is a matrix of × ones.Note that the row sums of  are 1.Let  be the (0, 1)-matrix from 2. The estimator of the Conditional Cumulative Distribution Function (CCDF) is (40)
The covariance of the numerator and the denominator of the estimator in (46) are given by cov (, ) ) . ( 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]: (56)
(63) Some authors assumed that, in this case, the first derivative of the true pdf of  at point  can be zero [19] as the one for the fixed design and, therefore, the bias can be given by We have (67) Notice that the expectation of F( * | ) is the same as the one of  and the variance is (  )/.To show the asymptotic normality of π (), we use the following theorem.

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 of f().We choose the bandwidth that minimizes the AMISE (Asymptotic Mean Integrated Squared Error) below.
The general form of the  th derivatives of the AMISE with respect to  was studied in [25], considering that the unknown functions in (74) are also functions of the smoothing parameter.
The optimal smoothing parameter minimizing (75) is 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  () directly.As an example, we consider the Epanechnikov Kernel function in order to compute (),  2 (), and the efficiency of the kernel function given by √ 2 ()().Epanechnikov's kernel function is and its efficiency is measured by which is the smallest of all the other kernel functions.

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.
which is given by (66) and (64).Therefore, and (/)AMSE( F( * | )) = 0 leads to 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   = .The estimator of the  th derivatives of (35) is with the function of weights.Thus, the first derivative is given by and the second derivative is also with  (1) = (1/) (2) ((  − )/) f() − ((  − )/) f(2) () and  (1) = 2 f(1) () f().Note that the estimation of the CCDF is function of the estimation of the empirical pdf of .An optimal bandwidth that minimizes the AMISE of f() 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][26][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  th derivatives of the pdf of   with respect to .We extend the idea to estimate the first and the second derivative of the CCDF with respect to .

Simulation Study
4.1.Model Specification.The ARCH() models introduced by [16] are widely used in financial applications.An AR(1)-ARCH( 1) is a mixed model from an AR() and GARCH(, ) for  = 1,  = 1, and  = 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 of F(⋅ | ⋅), find the smallest  * such that F( * |  * ) ≥ ,  ∈ (0, 1).
The data to be simulated is given by   =  +  −1 + ( + We have Using the i.i.d.assumption on the sequence of random variables  1 ,  2 , . . .,   , the expected value of  2  can be calculated as follows: which is independent of time.In another way,
Our algorithm gives the estimation of the conditional scale function which suffers from boundary effects as it is seen from Figure 2.This issue is recurrent while performing Kernel Density Estimations.The reason is that, at the boundaries, () 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.

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   's.Outliers are the points that are located outside the interval where 1 and 3 are the first and the third quantiles.Figure 3 is the representation of   and the transformed response variable  *  defined in (34) at level  = 0.75.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   at point , we get unsmoothed curves at high level  > 0.5.The curves represent the estimations of the CSF at  = 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  * 1 ,  * 2 , . . .,  *  .We obtain Figure 5 which combines the two estimations.
The next section discusses how precise is our estimation with the optimal bandwidth selection with the calculation of the MASE (Mean Average Squared Errors).
Table 2 shows that the estimator of the CSF is more precise at level  ≤ 0.55 for both the smoothed and the LOWESS versions.

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.

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 between  estimated CSFs.The formula is MASE ( π ()) π,1 (  ) − π, (  ))2 ] .

Table 1 :
Description of the most used kernel functions.