Estimating Conditional Quantiles

for Financial Time Series

by Bootstrapping and Subsampling Methods�

Thanasis Stengos and Ling Yang

Department of Economics, University of GuelphGuelph, Ontario, Canada, N1G 2W1

April 29th, 2006

Abstract

Value at Risk (VaR) has become one of the most commonlyused measures of risk for �nancial risk management. Econometri-cally, a suitable conditional quantile model can provide accurateestimation for this purpose. However due to the special depen-dence features of �nancial time series, a classical econometricmethodology does not lend itself for this purpose .In this paper, the main objective is to combine bootstrap

technique with nonparametric methodology to estimate the con-ditional quantile for �nancial time series. Three newly developedbootstrap based methods (nonparametric wild bootstrap, blockbootstrapping and subsampling) are adopted, and local linearnonparametric estimation is then used for estimation. Movingblock bootstrapping is applied to generate the con�dence inter-vals for the conditional quantile estimates. The performance ofthe models is evaluated by means of Monte Carlo simulations.

�This is the �rst draft of this paper. Any comment is welcome.

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1 Introduction

Conditional quantile regression, also simply called quantile regressionhas been widely used in economics and �nance, see Koenker and Bassett(1978). It can be generally denoted as:

Qt = f (Xt; �) + ut; t = 1; 2; :::; n (1)

The traditional assumption for ut is that it follows an i.i.d. (indepen-dent and identically distributed) process, something that often fails withreal economic data. This is especially true when the data are exhibitingvarious forms of dependence and conditional heteroskedasticity at thesame time, see Portnoy (1991). This problem is particularly importantin �nancial time series, as they present special features and challengesfor traditional estimators. The most obvious ones are serial correlationand conditional heteroskedasticity as stock returns are not independentand they display clustering over time.Value at Risk (VaR) is de�ned as the maximum potential change in

the value of a portfolio of �nancial instruments at a given probabilityover a certain horizon. Mathematically, it can be described as:

P (Xt < ajXt�1; Xt�2; :::) = � (2)

where a is the VaR under interest. Statistically, it is the quantileof any underlying distribution. As denoted above, it is a conditionalquantile given past information, where � is the given probability. In theestimation of VaR, � takes small values, such as 1% or 5%.VaR corresponds to the market risk, which estimates the uncertainty

of future earnings, due to changes in market conditions. Formally, aVaR calculation aims at making a statement that �the probability thatwe shall lose more than "a" dollars in the next days is 1%(or 5%)�.In words, it is a number that indicates how much a �nancial institu-tion can expect to lose with probability � over a given time horizon.VaR reduces the (market) risk associated with any portfolio to just onenumber, that is the loss associated with a given probability. The usualway of estimating the Value at Risk is to estimate the cumulative den-sity functions by either a parametric or a nonparametric method, andthen use the � quantile as the VaR value of interest, see Cai (2002), Liand Racine (2005). Since VaR is one application of estimating quantilesgiven some information, it would be interesting to employ conditionalquantile techniques to see if it will be a good candidate for the samepurpose. Although various methodologies have been used in estimatingthe VaR, conditional quantiles have rarely been applied. The main dif-ference between conditional quantile and other VaR estimation methods

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is that, for a conditional quantile method, a VaR curve rather than asingle number is obtainable.There are many di¤erent methodologies that have been developed

for the purpose of estimating conditional quantiles consistently. Toovercome shortcomings from parametric conditional quantile estima-tion methods, nonparametric techniques have been adopted recently,see Gourieroux and Monfort (1996) and Yu and Jones (1998)). Lin-ear quantile regression, a quantile estimation method with a parametricform for the systematic part of the econometric model is typically basedon the assumption of Gaussian errors. Local polynomial as in Yu andJones (1998) improves the parametric quantile estimation of Koenkerand Bassett (1978) as it performs better when the errors are not nor-mal, see Constinot, Roncalli and Teiletche(2000). However, the mainpractice in the literature so far, irrespective of the quantile regressionmodel used, is to still assume that the errors are i..i.d. In this paper, thelocal polynomial method, particularly local linear techniques are com-bined with di¤erent bootstrap methods to estimate VaR. We try to o¤era more general way of estimating conditional quantiles for �nancial timeseries without making any assumptions about the error distribution ordata dependence.The essential point of the bootstrap method is to provide a frame-

work to estimate both the statistic and its distribution without puttingunrealistic or unveri�able assumptions about the DGP (data generat-ing process). Another attraction of the bootstrap is that it is easy toapply even when the statistic is very complex to compute. Under cer-tain circumstances, the bootstrap distribution enables us to make moreaccurate inferences than the asymptotic approximation.Efron�s (1979) bootstrap (or the simple bootstrap) may not be capa-

ble to mimic the properties of �nancial time series data. This is because,the simple bootstrap focuses on estimated residuals, assuming that theyare i.i.d. with mean zero. Secondly, heteroskedasticity presents an ad-ditional problem, especially for �nancial time series, even if the mean ofestimated residuals is zero. If heteroskedasticity is not treated properly,the distribution of estimators and thus inferences will be incorrect. Inthis paper, various bootstrap based methods are combined with nonpara-metric techniques to estimate conditional quantiles. Note, that extremeconditional quantiles are di¢ cult to estimate accurately, since there isless information available for this purpose as compared with the estima-tion of the sample mean or median.We apply, three estimation strategies. The �rst, is a combination of

classic time series analysis technique with a more generalized nonpara-metric wild bootstrap method. The nonparametric conditional quantile

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method is then applied on the bootstrap samples. The second one aban-dons parametric regression and applies nonparametric conditional quan-tile estimation on the block-bootstrapped data. Finally, a subsamplingmethod is used and a nonparametric conditional quantile estimation isestimated after the data are subsampled. Furthermore, moving blockbootstrapping techniques are used to generate con�dence intervals forall three estimators.The paper is organized as follows. In section 2, the three techniques

used in the estimation of the conditional quantiles are described in detail.In section 3 we discuss con�dence interval estimation and in section 4 wepresent the simulation results. Finally, section 5 concludes the paper.In the appendix we present some additional details about con�denceinterval estimation.

2 Bootstrap and subsampling approximations of es-timating conditional quantiles for �nancial timeseries data

Before proceeding with the description of the di¤erent estimation meth-ods we need to �rst present some general notation that is used in thediscussion that follows.LetXt = fX1; X2; :::; Xng denote a �nite stretch of random variables,

observed from a weakly dependent process fXigi2Z inRd . Let Qt be anylevel-1 regression model, speci�cally here we consider the conditionalquantile of Xt given Xt�1. The estimation of Qt is based on last period�sinformation. i.e.

Qt = q (Xt�1) + vt, vt~d�0; �2

�(3)

where d is some distribution with mean 0 and unconditional variance�2. In theory q can take any form, either parametric or nonparametric.We focus on the nonparametric form of q. We adopt the local linearregression model proposed by Yu and Jones (1998) to estimate the 1%and 5% conditional quantile for �nancial time series. The local linearregression model can be written as:

Qt(XtjX = xt�1 + h) = q + a � h+ vt, vt~d�0; �2

�(4)

We want to estimate the constant term q when h ! 0 (the band-width). The problem now is to solve

min(q;a)2R2

nXt=1

wt�� (xt � a (xt�1 � xt)� q) (5)

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where wt is the weighting fuction associated with local linear quantileregression. It is also possible to include a quadratic form in the regressionform. Here we only use local linear analysis. The choice of bandwidthh is discussed in detail in Yu and Jones (1998) and Ruppert, Sheatherand Wand (1995). We use the method that was proposed in Ruppert,Sheather and Wand (1995) to calculate the optimal bandwidth.Let 'n denote level�2 parameters describing the properties of bQt; the

estimate of Qt; such as bias, variance or even the distribution of bQt. Insection 3 we discuss how to obtain con�dence intervals of bQt. Hence,'ndenotes distribution characteristics of bQt.2.1 ARIMA model combined with SCM bootstrap

methodIf the simple bootstrap is applied without modi�cation to a time series,the resampled data will not preserve the properties of the original dataset and thus inconsistent statistical results will be obtained.Adesi et al (1999) have used a bootstrap procedure to compute the

distribution of �nancial time series, and thus to obtain VaR estimates.They �t the volatility of �nancial time series with a particular para-metric model and resample from the standardized residuals using theestimated conditional standard deviations from a GARCH (parametric)volatility model. In a subsequent study, Adesi et al. (2001) comparethis method with the traditional bootstrap and analyze a speci�c sit-uation with an option in the stock portfolio. Pascual and Ruiz (2003)conduct Monte Carlo simulations by using Adesi�s method and concludethat the estimated VaR are very close to the actual values. The Adesi etal. (1999) model and Pascual and Ruiz (2003) simulations are based onvery speci�c numerical examples, speci�cally a real DGP of a GARCH(1,1) model, which is the same model used to �t the volatility in theirsimulations. However, their result that this parametric approach o¤ersconsistent estimates are based on using the same DGP as the modelused in estimation. If for example serial dependence and conditionalheteroskedasticity beyond what is captured by the GARCH(1,1) are ig-nored, the bootstrap based on the parametric GARCH(1,1) model wouldbe inconsistent. Therefore, Pascual and Ruiz (2003)�s approach cannotbe generalized.A feasible way of implementing the nonparametric wild bootstrap

analysis was proposed by Gozalo (1997) to deal with independent butnot necessarily identically distributed, or heteroskedastic, data. TheSCM (Smooth Conditional Moment) bootstrap allows for the presenceof heteroskedasticity in cross sectional data and it is a data based non-parametric bootstrapping method. However, for time series data there

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is no evidence regarding the use of the SCM method. It is also worthnoting that the i..i.d. (or simple) bootstrap and the wild bootstrap arespecial cases of the SCM bootstrap under stronger assumptions aboutthe data. If the data follow an i..i.d. or i.n.i.d. DGP, SCM turns out tobe one of the two special bootstrap methods, either the i.i.d. bootstrapor the parametric wild bootstrap. Also because it is a nonparametricbootstrapping method, it avoids the restrictions imposed by parametricmethodologies.In the present paper, �rst we remove the serial correlation from the

�nancial time series data using a certain parametric model but we leavethe conditional heteroskedasticity intact. Subsequently, we use the SCMbootstrap to analyze the conditional heteroskedasticity left in residuals.The two methods combined o¤er a model-free estimation approach aswe combine a parametric serial correlation model with a nonparametricwild bootstrap technique.Speci�cally, an ARMA (1,1) model is �rst used to �t the �nancial

time series. This is because serial correlation can be removed by anARMA �t successfully for weakly dependent time series. We estimatethe following model:

xt = �xt�1 + �"t�1 + "t; "t~inid (6)

Here i.n.i.d. denotes errors that are independent but not necessarilyidentically distributed. To render the errors i.i.d., we need to removeboth serial correlation and any volatility clusters that would be present.ARMA can solve the serial correlation problem but not volatility cluster-ing. To remove volatility clustering we use the SCM bootstrap method.The procedure works by �rst mimicking the conditional distribution ofXt given Xt�1 which is assumed to change in a smooth fashion. In par-ticular, we assume that the �rst three moments of the conditional errordistribution to be smooth functions of Xt�1 . The SCM bootstrap takesthis into account, while still allowing for the possibility that each residualhas a di¤erent conditional distribution.After estimating the ARMA model, residuals are obtained as:

but = xt � b�xt�1 � b�but�1 (7)

Let b�2t;h2 (xt�1)and b�3t;h3 (xt�1) denote consistent nonparametric esti-mates of �2t (xt�1)and �

3t (xt�1) , respectively, based on fXt�1; Xtg. Non-

parametric estimation here can rely on some smoother such as a kernelwith suitable choices of bandwidths h2 and h3 respectively. The kernel

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estimators are as follows:

b�2t;h2 (xt�1) = n�1 nXt=1

Wh2;t (xt�1) bu2t ; b�3t;h3 (xt�1) = n�1 nXt=1

Wh3;t (xt�1) bu3t(8)

where Wh;t (xt�1) = Kh (xt�1 �Xt�1) = bfh (xt�1) ;bfh (xt�1) = n�1Pnt=1Kh (xt�1 �Xt�1) is the kernel function, which

possesses the properties of being symmetric, integrating to one and alsobeing absolutely integrable. The bandwidth parameter(s) are sequencesh is such that h ! 0 and nhd ! 1 as n ! 1. Picking a suitablebandwidth h is done by cross validation. The simple constant kernelregression method is used in Gozalo (1997). In this paper we use locallinear estimation instead of the simple constant kernel estimator on theresiduals obtained from (7). The reason for that is that local linearestimation is better behaved at the tails of the distribution than thesimple constant kernel estimator.The next step is to apply the nonparametric wild bootstrap.(a) For each index t, randomly draw (with replacement) the boot-

strap residual u�t from an arbitrary distribution bF st such that for Z~bF st ,E bF st Z = 0 (9)

E bF st Z2 = b�2 (Xt�1) (10)

E bF st Z3 = b�3 (Xt�1) (11)

As suggested by Gozalo (1997), the distribution bF st could be cho-sen as a discrete distribution that puts mass on two points, that isbF st = pt�at + (1� pt) �bt, , where �x denotes a probability measure thatputs mass one at x, pt 2 [0; 1], then solving the system results in at =�b�3t (Xt�1)� Tt

�=�2b�2t (Xt�1)

�, bt =

�b�3t (Xt�1) + Tt�=�2b�2t (Xt�1)

�, and

pt = (1=2)�1 +

�b�3t (Xt�1) =Tt��, where Tt =

q�b�3t (Xt�1)�2+ 4

�b�2t (Xt�1)�3

There is the possibility that E bF st Z 6= 0 because the model chosen isa misspeci�ed parametric model. Then (9) should be replaced by

E bF st Z = bvt (xt�1) (12)

where bvt (xt�1) denotes any consistent nonparametric estimation, suchas kernel or local polynomial estimation, of E(ut jXt�1) (based on theresiduals but of the misspeci�ed model) with bandwidth hv. The SCMbootstrap steps consisting of the moment with but, such as (10) and (11),should be replaced by the centered residuals but � bvhv (xt�1). This is oneof the advantages of the SCM bootstrap as discussed before. Here we

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use local linear method to estimate (12) and then modify (10) and (11)accordingly.(b)The new set of bootstrapped data is generated as

nXt�1; X

�t =

bXt + u�t

o. At this point, a local linear conditional quantile regression method, asdiscussed above from Yu and Jones (1998), is applied to estimate the1% and 5% conditional quantile by using fXt�1; X

�t g. The bandwidth

h, is picked following their recommendation.(c) Repeat process (a) to (b) B times. Then for each Xt�1, take the

median of B estimated conditional quantiles to be the SCM bootstrap-ping counterpart. The idea behind it is that, due to the possibility ofmisspeci�cation, the mean value sometimes is distorted by the extremevalue of estimated residuals and thus in�uences the accuracy of the es-timates. Hence, we use the median instead of the mean of the residualsmoments conditional on Xt�1.

2.2 Block bootstrap method to estimate conditionalquantile for �nancial time series

An alternative way of investigating the properties of conditional quan-tiles of �nancial time series is the block bootstrap. Historically in theapplication of the original bootstrap methodology, Efron (1979) derivedthe bootstrap within a context of an i.i.d. framework by focusing onresiduals of some general regression model. In that case, residuals areresampled, but not the original observations. It has been shown thatthe simple (or i..i.d.) bootstrap behaves well in many situations, whichinclude linear regression, see Freedman (1981), Freedman (1984), Wu(1986), Liu (1988) for example, autoregressive time series, see Efron andTibhirani (1986), Bose (1988), and nonparametric regression and non-parametric kernel spectral estimation, see Franke and Hardle (1992).However, the success of this approach is based on the reliability of theregression model.A more general approach is to apply resampling on the original data

sequence by considering blocks of data rather than single data pointsas in the i.i.d. setup. If we have a m-dependent sequence and stilluse the i.i.d.bootstrap method to estimate the parameters of interest,that will yield inconsistent estimates. The motivation behind the blockbootstrap is that within each block the dependence structure of the un-derlying model is preserved and if the block size is allowed to tend toin�nity as the sample size increases, asymptotically one will obtain cor-rect inferences. In such a way, the data are divided into blocks, and theseblocks are then resampled. Carlstein (1986) proposed a non-overlappingblock bootstrap. As compared with resampling a single observation at

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a time, the dependent structure of the original observations is preservedwithin each block. Kunsch (1989) and Liu and Singh (1992) introducedthe Moving Block Bootstrap (MBB ) which employs overlapping blocks.The moving block bootstrap method divides the data into overlappingblocks of �xed length and resampling is done with replacement fromthese blocks. Any block bootstrap technique can be applied to depen-dent data without any parametric model assumptions.In both Carlstein (1986) and Kunsch (1989) bootstrap blocks of �xed

length are resampled. However, the newly generated pseudo-time seriesmight not be stationary as it may have been initially. To �x this problemPolitis and Romano (1994a) proposed the stationary block bootstrap(SBB). The SBB method is to resample blocks of data with randomlengths, applied to stationary time series. The whole process could bedescribed as:1. Wrap the data fX1; :::; XNg around a circle.2. Let i1; i2; :::; be i.i.d. draws from a uniform distribution on the set

f1; 2; :::; Ng; these are the starting points of the new blocks.3. Let b1; b2; :::; be i.i.d. draws from some distribution Fb (�) that

depends on a parameter b; these are the block sizes.4. Construct a bootstrap pseudo-series X�

1 ; :::; X�N . The starting

observation for the �rst block is Xi1 and the block size is b1. Thusthe �rst block contains observations Xi1 ; :::; Xi1+b1�1. Then we moveon to the next b2 observations in the pseudo-time series, which includesXi2 ; :::; Xi2+b2�1. This process is stopped once N observations in thepseudo-time series have been generated.5. Finally, we focus on the �rst N observations of the bootstrapped

series and construct the bootstrap sample statistics. Local linear tech-niques are employed here to estimate the conditional quantiles.6. Repeat the process B times, take the median level of B estimated

conditional quantiles for each point as the SBB conditional quantile.In general the approach here is designed to use MBB or SBB to

preserve the dependence structure of the time series, while the nonpara-metric methodology is applied to solve the heteroskedasticity problemafterwards. It is worth noting that as the main problem with nonpara-metric estimation is to pick a suitable bandwidth h, the key point forblock bootstrapping technique is to �nd the appropriate block size.If the distribution Fb is a unit mass on the positive integer b; the

procedure described above becomes the CBB (circular block bootstrap)of Politis and Romano (1992). It is an asymptotically equivalent vari-ation of the MBB (moving block bootstrap) of Kunsch (1989) and Liuand Singh (1992). The only di¤erence is that in CBB, the original dataare packed circularly but not in MBB. By using this Fb, the block size is

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�xed. The problem of using a �xed block bootstrap, as mentioned above,is that it may produce a nonstationary time series after bootstrappingeven if the original series is stationary. The SBB (stationary bootstrap)of Politis and Romano (1994a) uses a Geometric distribution with meanequal to the real number b. This distribution is also used in our pa-per. The requirement for SBB bock size selection is less restricted ascompared with MBB. It only requires that when b!1 when n!1,while n=b ! 1 as n ! 1. We also use an experimental method topick b. When n is small, a smaller value of b is used. A larger valueof b is associated with a larger number of observations. It seems thatmore dependent data should be associated with a larger b to have thestrong dependence preserved. In that case, a pre-test procedure could beemployed before b is decided, whereas stronger dependence would leadto a bigger value of b.

2.3 SubsamplingAnother way of estimating the conditional quantile is to adopt the sub-sampling method proposed by Politis and Romano (1994b). The sub-sampling process can be described as:(a) de�ne Yi to be the subsample (Xi; Xi+1; ::; Xi+bn+1), for i = 1; :::q

and q = n � bn + 1; Yi consists of bn consecutive observations from theX1; :::; Xn sequence, and the order of the observations is preserved.(b) Within each subsample, use the local linear method to estimate

the conditional quantile.(c) For each Xi, for i = 1; :::; n, take the median of all the corre-

sponding estimated conditional quantiles from various subsamples.The di¤erence between the bootstrap and subsampling is that sub-

sampling takes samples without replacement of size b from the originalsample of size n, with b much smaller than n. Subsamples. are them-selves samples of size b from the true unknown distribution F of theoriginal sample. Efron�s bootstrap on the other hand takes sampleswith replacement of size n from the original sample of size n from theempirical distribution bFn associated with the original sample.The advantage of subsampling as compared with the bootstrap can

be summarized as:Firstly, as the underlying sample distribution is unknown, each sub-

sample as a part of the original series is naturally thought to mimic thisdistribution. It then seems reasonable to expect that one can gain in-formation about the sampling distribution of a statistic by evaluating iton all subseries, or "subsamples".Secondly, there are fewer assumptions needed in order to apply this

procedure, see Politis and Romano (1997) who investigated the issue of

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heteroskedasticity that may arise in time series by applying subsamplingmethods.Bertail et al (2004) gives the theoretical proof of consistency. They

pointed out that for the asymptotic approximation to be useful in a�nite-sample situation, b not only has to be �xed, but it has to be smallas compared to n; so that b=n is small. They also remark that for aconstant b, the consistency is true under few assumptions. Both Politisand Romano (1994b) and Bertail et al (2004) produce simulation resultsto examine the robustness of subsampling when applied to �nancial timeseries. Simulations in Politis and Romano (1994b) focus on univariatemean estimation and its variance for heteroskedastic time series. Bertailet al (2004) explore subsampling estimation in estimating the VaR forsome stock portfolios. In our paper, we combine subsampling methodwith conditional quantile techniques to assess its performance.

3 Block bootstrap estimation for con�dence inter-vals

Since the bootstrap is a widely used technique to estimate the proper-ties of an unknown distribution, it is reasonable to use it in order toconstruct con�dence intervals of the conditional quantiles. We use theMBB method to do that. Fitenberger (1997) used MBB for the purposeof inference in parametric quantile regressions and he proved that MBBprovides a valid asymptotic procedure for linear quantile regressions.In our paper, MBB is combined with local linear quantile regressiontechnique to generate the point-wise con�dence intervals for all threeestimation methods presented above.The MBB literature points out that the behavior of block bootstrap

estimation critically depends on the block size. In a recent paper Lahiri,Furukawa and Lee (2003) propose a plug-in method for selecting theoptimal block length of MBB estimation of con�dence interval. Theoptimal block size l0n can be showed to be:

bl0n = �2 bB2nrbvn� 1r+2

n1

r+2 (13)

where r = 2 for this purpose according to discussion in Lahiri, Fu-rukawa and Lee (2003) . In the appendix we analyze the above expressionin more detail.

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4 Monte-Carlo simulation result

4.1 DGP�s and simulation settingsBelow we present the framework of our simulation analysis. We gener-ate a Generalized Autoregressive Conditional Heteroskedastic (GARCH)model that combines both an autoregressive and heteroskedastic struc-ture. The GARCH model is generated according to the following equa-tions:

yt = "t�t (14)

�2t = 0:1275 + 0:16264 � y2t�1 + (0:76� 0:16264)�2t�1 (15)

We also consider the non-stationary case, where we generate

yt = "t�t (16)

�2t = 0:53102592 + 0:16264 � y2t�1 + (1� 0:16264)�2t�1 (17)

This is the Integrated GARCH (IGARCH) model which has beenused extensively in empirical �nancial economics. We can see that thesum of two coe¢ cients in front of the lagged return and the laggedvolatility term equals to one, but it is less than one in the simple GARCHmodel.An explosive time series model is rarely considered in the literature.

An IGARCHmodel can be transformed into an explosive GARCHmodelwith a little modi�cation.

yt = "t�t (18)

�2t = 0:53102592 + 0:16264 � y2t�1 + (1:05� 0:16264)�2t�1 (19)

where the parameters are such that the roots of the characteristicpolynomial all lie in the region of the complex plane. Note that unlikethe stationary case, the error variables "t�s in (18) are not required tohave zero mean.Lahiri (2003) examined the performance of the bootstrap in esti-

mating the coe¢ cients of an explosive autoregressive model, under aModel-Based bootstrap framework It is shown that consistency of theestimator depends greatly on the initial values of the estimation model.The parameters p and q for all 3 GARCH(p; q) related models are

picked to be p = q = 1. Sample with observations of 50, 100 and 250are investigated separately for each DGP.1

1At this stage, we have 500 replications done for when n = 50. Due to the timerequirements of the simulation when n goes up, 200 replications have been done forn = 100 and n = 500.

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4.2 Simulation ResultsThe Mean Squared Error (MSE ) for each estimator is listed in Table1-3 for the 1% and 5% conditional quantiles for di¤erent DGP�s andsample sizes. The MSE results for the three methods are as follows.When the time series is stationary, both the SCM nonparametric wildbootstrap (SCM-NWB) and SBB attains a smallerMSE. When the 1%conditional quantile is under investigation, SCM-NWB always performsbetter. MSE attained by SBB is less than that from SCM-NWB if weconsider the 5% conditional quantile. When we look at the nonstationarycase, SBB always outperforms the other two methods, for both the 1% and 5% conditional quantile estimates. For the nonstationary timeseries (IGARCH) the MSE0s from SCM-NWB are larger as comparedwith SBB and subsampling. When the time series is explosive, neitherSCM-NWB nor SBB attains small MSE0s. In that case the smallestMSE is achieved by subsampling for both the 1% and 5% conditionalquantile estimates. When the sample size is small and the time seriesis explosive, it is not easy to judge if any of these three methodologieso¤ers a better choice for conditional quatnile estimation. Even for thesubsampling method, we do not observe the convex shape of the MSEfunction when subsample size increases. When the sample size is larger,theMSE keeps decreasing. For small samples and explosive time series,conditional quantile estimation is hard to achieve reliable results by anyof the three techniques discussed. When the sample size increases, wedo observe certain subsampling size which minimizes MSE. In otherwords, the convex shape of the MSE is obtained when sample size islarge enough, as when n=250 where the optimal subsample size for 1%and 5% conditional quantile estimation is around 20 and 15 respectively2.Two types of coverage rates are calculated by using the critical val-

ues estimated from the MBB process. If we use two sided critical values(at a 95% con�dence interval), the coverage rate is not good enough.Generally speaking, the coverage rate is higher for the 5% conditionalquantile than for the 1% conditional quantile. The coverage rate is alsohigher for the stationary time series process than for the nonstationaryand explosive cases. The distribution of the conditional quantile has anunclear shape, especially for the extreme lower quantile level. The dis-tribution of the conditional quantile seems to be skewed severely to oneside and thus leads to inaccurate inferences using a two-sided con�denceinterval. In addition to the two-sided con�dence interval, we establishan one-sided 95% con�dence interval to see if the coverage rate will im-

2We do not discuss how to pick the optimal subsample size in this paper. We usean empirical way of �nding the subsample size which results in the smallest MSE.

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prove. We notice that the undercoverage of the two-sided con�denceintervals is largely improved when the one-sided con�dence interval isapplied, especially for the stationary time series process3. The under-coverage is also diminished for nonstationary process and the explosivetime series, but to a lesser extent than the stationary case. Generallyspeaking, when the sample size is bigger, the coverage rate is close tothe con�dence interval level.To check the stability of the three conditional quantile estimators, we

generate another set of measurements called "rejection rates" by usingthe MBB con�dence intervals. As discussed above, the one sided MBBcon�dence interval outperforms the two-sided one in calculating the cov-erage rates. We also calculate rejection rates by using an one-sided con-�dence interval. The results show that SBB method provides the moststable estimators among all three methodologies. The SCM-NWB be-haves well when the time series is stationary but it is very volatile for thenonstationary and explosive cases. The rejection rates for subsamplingmethod decrease when the subsample size increases. It is natural toexpect that the bigger the subsample size, the more information is uti-lized. However, the higher rejection rates for the subsampling estimatorindicates that it overestimates the real conditional quantiles, especiallyfor the lowest quantiles such as 1%.

5 Conclusion

In this paper we apply three di¤erent types of bootstrap based method-ologies to the estimation of the extreme lower conditional quantiles of�nancial time series data, which exhibit dependence and heteroskedas-ticity simultaneously. Estimation of such quantiles is valuable for riskmanagement in �nancial economics. The conditional quantile is inter-preted as the VaR in risk management. The SCM bootstrap is originallyproposed to be applied to cross sectional data to tackle problems of het-eroskedasticity. In this paper the SCM is combined with time seriesanalysis techniques to estimate point-wise conditional quantiles. SBBhas the advantage of preserving the dependence properties of �nancialtime series. Subsampling is a newly developed technique to estimatethe sample distribution properties of statistics of interest, even thoughit has not yet been applied to estimate point-wise conditional quantiles.We analyze the performance of the three estimators by means of MonteCarlo simulations. The results show that both SCM-NWB and SBBperforms well when the time series is stationary, with SCM-NWB per-

3As the bootstrap combined with nonparametric method is very time consumingfor large sample size, here for n=250, the result is only for 100 replications. Morework is being currently undertaken to obtain more accurate coverage rates.

14

forming better for the 1% and SBB for the 5% conditional quantile. Forthe nonstationary or explosive cases, SCM-NWB does not work well atall, while SBB still does reasonably well for the nonstationary case butnot so for the explosive case. Results from the subsampling methoddepend on the sample size and the choice of the subsample size. Sub-sampling performs better when the sample size is large enough and italso outperforms the other methods for the explosive case. Finally, MBBestimation provides better critical values for one-side than two-side con-�dence intervals.

15

6 Appendix

The most important point turns out to be how to choose the constantterm in front of the term (13). Consistent estimation of bB is equivalentto consistent estimation of the bias of the con�dence interval estimate,and this is what is called a level 3 parameter. A consistent estimation ofbv is actually an estimate of the variance of the con�dence interval, also alevel 3 parameter. As shown in Lahiri, Furukawa and Lee (2003), for es-timating bB consistently, we need to combine two moving block bootstrapestimators suitably. Estimation of the level 3 parameter associated withthe variance part employ the Jackknife-After-Bootstrap (JAB) methodof Efron (1992) and Lahiri (2002).For constructing the bias estimator, it is suggested that a consistent

estimator of Bias (b'n (l1)) may be constructed as\BIASn � \BIASn (l1) = 2 (b'n (l1)� b'n (2l1)) (20)

with 1 < l1 < n1

r=2 as n!1where l1is equivalent to the optimal block size bl0n for bias estimation.b'n (l1) and b'n (2l1) denote MBB estimation of the statistic b'n of interest

with block size l1 and 2l1 respectively.A speci�c choice of fl1ngn�1 will be suggested later for the plug-in

estimator bl0n. Note that, as pointed out earlier, the estimator \BIASn isbased on only two block bootstrap estimators of 'n and may be com-puted using only one level of resampling.The JAB estimator is applied to estimate the variance of the con-

ditional quantile. The JAB method was proposed by Efron (1992) forassessing the accuracy of bootstrap estimators based on the i.i.d. boot-strap for independent data. A modi�ed version of the method for blockbootstrap estimators in the case of dependent data was proposed byLahiri (2002). The JAB method for dependent data applies a version ofthe block jackknife method to a block bootstrap estimators.Let b'n � b'n (l) be the MBB estimator of a level 2 parameter 'n

based on (overlapping) blocks of size l from �n = fX1; :::; Xng. LetBi = fXi; :::; Xi+l�1g, i = 1; :::; N (with N = n � l + 1 ) denote thecollection of all overlapping blocks contained in �n that are used forde�ning the MBB estimator b'n . Also, let m be an integer such thatm�1 + n�1m = o(1) as n ! 1. Note that the MBB estimator b'n (l) isde�ned in terms of the �basic building blocks�Bi�s. Hence, instead ofdeleting blocks of original observations fXi; :::; Xi+m�1g, as done in thetraditional jackknife method, the JAB method of Lahiri(2002) de�nesthe jackknife point-values by deleting m blocks of Bi�s.

16

Since there are N observed blocks of length l, we can de�neM = N�m+1 many JAB point-values corresponding to the bootstrap estimatorb'n, by deleting the overlapping �blocks of blocks� fBi; :::; Bi+m�1g ofsize m for i = 1; :::;M . Let I0i = f1; :::; Ng n fi; :::; i+m� 1g ; i =1; :::;M . To de�ne the ith JAB point-value b'(i)n � b'(i)n (l), we needto resample b = [n=l] blocks randomly and with replacement from thereduced collection fBj : j 2 I0i g and construct the MBB estimator of 'using these resampled blocks.Let ��(i)n denote the resampled data obtained, and the JAB point-

value b'(i)n is given by applying the functional to the conditional distrib-ution4 as b'(i)n = '

� bGn;j� (21)

.The JAB variance estimator of b'n is calculated as

[V ARJAB (b'n) = m

N �m � 1M

MXi=1

�e'(i)n � b'n�2 (22)

where e'(i)n = m�1�Nb'n � (N �M) e'(i)n � denotes the ith JAB pseudo-

value corresponding to b'n.At last, the nonparametric plug-in procedure depends on the choice of

the smoothing parameter l1, and on the JAB �blocks of blocks�deletionparameter m.It urns out that a reasonable choice of l1 is of the form (Lahiri 2003)

l1 = C3n1

r+4 (23)

where r is as in (13), and C3 is a population parameter. As for theother smoothing parameter, Lahiri (2202a) suggests that a reasonablechoice of the JAB parameter m is given by

m = C4n1=3l

2=31 (24)

Results of Lahiri, Furukawa and Lee (2003) show that the choiceC3 = 1, and C4 = 0 yields good results for the distribution estimation.

4Any consistent estimation of conditional distribution can be used. Here theKernel estimation of conditional quantile is applied to accelerate the computationalspeed.

17

Table1: MSE Results (n=50)GARCH IGARCH EXGARCH

1% 5% 1% 5% 1% 5%

SCN NWB 0.3346 1.2503 18.8987 150.3998 51934.693 479427.25

SBB 0.4229 0.6400 6.1716 8.2818 69.5833 46.0783

SUBb=5

b=10

1.0837

0.7619

0.9880

0.8813

8.8657

6.5781

11.1784

9.0050

46.0657

29.8019

62.7585

44.6788

Table2: MSE Results (n=100)GARCH IGARCH EXGARCH

1% 5% 1% 5% 1% 5%

SCM NWB 0.1769 0.1073 8.1400 6.8938 120.2032 54.6070

SBB 0.2582 0.0847 5.2373 1.2635 355.2175 78.4289

SUB

b=5

b=10

b=15

b=20

1.0798

0.7227

0.5760

0.5046

0.5228

0.3474

0.2884

0.2568

8.9119

6.4281

5.9256

6.1285

4.3448

3.2881

3.3556

3.2535

82.1050

50.5279

40.6582

38.4194

36.2196

20.4718

18.4396

21.8834

Table3: MSE Results (n=250)GARCH IGARCH EXGARCH

1% 5% 1% 5% 1% 5%

SCM NWB 0.1180 0.0778 12.3060 8.9033 323.0285 85.7410

SBB 0.2199 0.0698 4.5787 1.1186 392.0752 80.3799

SUB

b=5

b=10

b=15

b=20

b=25

b=30

b=35

b=40

b=45

b=50

1.0939

0.7461

0.6075

0.5398

0.4932

0.4597

0.4310

0.4099

0.3876

0.3737

0.5285

0.3573

0.3006

0.2728

0.2456

0.2213

0.1969

0.1755

0.1560

0.1420

9.0096

6.5792

6.4559

8.3123

8.6078

9.4632

9.7028

9.1515

9.1906

9.4552

4.3686

3.3858

3.6156

4.0421

4.0444

3.6051

3.1607

2.8308

2.5973

2.4162

136.5012

85.4326

67.2817

61.8222

63.9086

76.3885

107.9911

137.7513

162.3414

183.5969

59.3294

33.7098

29.6585

33.7339

37.5328

42.0563

48.0100

58.2327

56.0048

62.6051

18

Table 4: Coverage Rate by MBB Two-Side Con�denceIntervals

Sample S ize n=50 n=100 n=250

Quantile Level 1% 5% 1% 5% 1% 5%

GARCH 0.6064 0.8026 0.8263 0.9317 0.8548 0.9014

IGARCH 0.5924 0.7542 0.7150 0.8096 0.6696 0.7352

EXGARCH 0.4879 0.6959 0.7136 0.7764 0.6084 0.6650

Table 5: Coverage Rate by MBB One-Side Con�denceInterval

Sample S ize n=50 n=100 n=250

Quantile Level 1% 5% 1% 5% 1% 5%

GARCH 0.9598 0.9438 0.9702 0.9817 0.9031 0.9351

IGARCH 0.8839 0.8971 0.8486 0.8902 0.7383 0.8141

EXGARCH 0.9221 0.9144 0.9334 0.9266 0.7118 0.7693

19

Table 6: Rejection Rate by MBB Two-Side Con�denceIntervals (n=50)

GARCH IGARCH EXGARCH

1% 5% 1% 5% 1% 5%

SCM NWB 0.1738 0.1330 0.3552 0.3179 0.7680 0.7315

SBB 0.0206 0.0214 0.0454 0.0408 0.0410 0.0374

SUBb=5

b=10

0.5428

0.3043

0.4438

0.2068

0.5777

0.3525

0.4745

0.2523

0.5549

0.3435

0.4626

0.2433

Tabel 7: Rejection Rate by MBB Two-Side Con�denceIntervals (n=100)

GARCH IGARCH EXGARCH

1% 5% 1% 5% 1% 5%

SCM NWB 0.1038 0.0669 0.5623 0.5812 0.2811 0.3300

SBB 0.0011 0.0023 0.0197 0.0166 0.1389 0.0603

SUB

b=5

b=10

b=15

b=20

0.6987

0.5075

0.3455

0.2286

0.5527

0.3140

0.1768

0.1085

0.7364

0.5699

0.4245

0.3063

0.6033

0.3943

0.2609

0.1837

0.5567

0.3643

0.2486

0.1857

0.4736

0.2977

0.2173

0.1847

Table 8: Rejection Rate by MBB Two-Side Con�denceIntervals (n=250)

GARCH IGARCH EXGARCH

1% 5% 1% 5% 1% 5%

SCM NWB 0.1109 0.1054 0.6626 0.7256 0.2981 0.2759

SBB 0.0192 0.0212 0.2628 0.2526 0.0775 0.0656

SUB

b=5

b=10

b=15

b=20

b=25

b=30

b=35

b=40

b=45

b=50

0.8502

0.7347

0.6147

0.5083

0.4241

0.3493

0.2830

0.2343

0.1905

0.1568

0.7194

0.5394

0.4062

0.3357

0.2773

0.2380

0.2016

0.1668

0.1344

0.1108

0.8635

0.7507

0.6489

0.5611

0.4771

0.4139

0.3496

0.2982

0.2529

0.2093

0.7460

0.5905

0.4808

0.4036

0.3469

0.3158

0.2867

0.2548

0.2239

0.1920

0.8025

0.7038

0.6266

0.5541

0.4888

0.4357

0.3891

0.3452

0.3055

0.2694

0.6884

0.5521

0.4667

0.4112

0.3667

0.3345

0.3100

0.2804

0.2502

0.2194

20

Table 9: Rejection Rate by MBB One-Side Con�denceIntervals (n=50)

n=50 GARCH IGARCH EXGARCH

1% 5% 1% 5% 1% 5%

SCM NWB 0.0870 0.0501 0.0801 0.0411 0.0104 0.0054

SBB 0.0165 0.0157 0.0371 0.0298 0.0314 0.0238

SUBb=5

b=10

0.5371

0.2942

0.4338

0.1837

0.5715

0.3403

0.4632

0.2206

0.5464

0.3325

0.4498

0.2150

Table 10: Rejection Rate by MBB One-Side Con�denceIntervals (n=100)

GARCH IGARCH EXGARCH

1% 5% 1% 5% 1% 5%

SCM NWB 0.0592 0.0169 0.4402 0.4193 0.2811 0.3300

SBB 0.0010 0.0022 0.0104 0.0091 0.0061 0.0101

SUB

b=5

b=10

b=15

b=20

0.6969

0.5046

0.3423

0.2248

0.5462

0.2927

0.1501

0.0802

0.7326

0.5623

0.4119

0.2897

0.5888

0.3519

0.2045

0.1211

0.5457

0.3433

0.2186

0.1468

0.4497

0.2437

0.1408

0.0894

Table 11: Rejection Rate by MBB One-Side Con�denceIntervals (n=250)

GARCH IGARCH EXGARCH

1% 5% 1% 5% 1% 5%

SCM NWB 0.0862 0.0478 0.5369 0.5396 0.2871 0.2721

SBB 0.0180 0.0184 0.1145 0.1166 0.0717 0.0576

SUB

b=5

b=10

b=15

b=20

b=25

b=30

b=35

b=40

b=45

b=50

0.8488

0.7298

0.6063

0.4958

0.4088

0.3314

0.2660

0.2185

0.1760

0.1430

0.7016

0.4882

0.3318

0.2395

0.1720

0.1359

0.1147

0.0975

0.0801

0.0660

0.8597

0.7393

0.6275

0.5296

0.4381

0.3680

0.3014

0.2480

0.2002

0.1568

0.7215

0.5213

0.3766

0.2765

0.2072

0.1731

0.1531

0.1331

0.1148

0.1014

0.7985

0.6954

0.6125

0.5347

0.4653

0.4080

0.3562

0.3083

0.2661

0.2270

0.6692

0.5047

0.3974

0.3226

0.2654

0.2284

0.2081

0.1861

0.1625

0.1366

21

Figure 1

X(i­1)

1% Q

uantiles

­1.0 ­0.5 0.0 0.5 1.0

­2.0­1.5

­1.0­0.5

1% Conditional Quatile Estimation for GARCH Modle (n=50)

True ValueSCMSBBSUBC.I

Figure 2

X(i­1)

1% Q

uantiles

­1.0 ­0.5 0.0 0.5 1.0

­2.5­2.0

­1.5­1.0

­0.5

1% Conditional Quatile Estimation for GARCH Modle (n=100)

True ValueSCMSBBSUBC.I

22

Figure 3

X(i­1)

1% Q

uantiles

­1.0 ­0.5 0.0 0.5 1.0

­3.0­2.5

­2.0­1.5

­1.0

1% Conditional Quatile Estimation for GARCH Modle (n=250)

True ValueSCMSBBSUBC.I

23

Figure 4

X(i­1)

5% Q

uantiles

­1.0 ­0.5 0.0 0.5 1.0

­2.0­1.5

­1.0­0.5

0.0

5% Conditional Quatile Estimation for GARCH Modle (n=50)

True ValueSCMSBBSUBC.I.

Figure 5

X(i­1)

5% Q

uantiles

­1.0 ­0.5 0.0 0.5 1.0

­2.0­1.5

­1.0­0.5

0.0

5% Conditional Quatile Estimation for GARCH Modle (n=100)

True ValueSCMSBBSUBC.I.

24

Figure 6

X(i­1)

5% Q

uantiles

­1.0 ­0.5 0.0 0.5 1.0

­2.0­1.5

­1.0­0.5

5% Conditional Quatile Estimation for GARCH Modle (n=250)

True ValueSCMSBBSUBC.I.

25

Figure 7

X(i­1)

1% Q

uantiles

­3 ­2 ­1 0 1 2 3

­12­10

­8­6

­4­2

1% Conditional Quatile Estimation for IGARCH Modle (n=50)

True ValueSCMSBBSUBC.I

Figure 8

X(i­1)

1% Q

uantiles

­3 ­2 ­1 0 1 2 3

­12­10

­8­6

­4­2

1% Conditional Quatile Estimation for IGARCH Modle (n=100)

True ValueSCMSBBSUBC.I

26

Figure 9

X(i­1)

1% Q

uantiles

­3 ­2 ­1 0 1 2 3

­12­10

­8­6

­4­2

1% Conditional Quatile Estimation for IGARCH Modle (n=250)

True ValueSCMSBBSUBC.I

27

Figure 10

X(i­1)

5% Q

uantiles

­3 ­2 ­1 0 1 2 3

­10­8

­6­4

­20

5% Conditional Quatile Estimation for IGARCH Modle (n=50)

True ValueSCMSBBSUBC.I.

Figure 11

X(i­1)

5% Q

uantiles

­3 ­2 ­1 0 1 2 3

­10­8

­6­4

­20

5% Conditional Quatile Estimation for IGARCH Modle (n=100)

True ValueSCMSBBSUBC.I.

28

Figure 12

X(i­1)

5% Q

uantiles

­3 ­2 ­1 0 1 2 3

­10­8

­6­4

­2

5% Conditional Quatile Estimation for IGARCH Modle (n=250)

True ValueSCMSBBSUBC.I.

29

Figure 13

X(i­1)

1% Q

uantiles

­6 ­4 ­2 0 2 4 6

­60­50

­40­30

­20­10

1% Conditional Quatile Estimation for EX­GARCH Modle (n=50)

True ValueSCMSBBSUBC.I

Figure 14

X(i­1)

1% Q

uantiles

­5 0 5

­30­25

­20­15

­10­5

1% Conditional Quatile Estimation for EX­GARCH Modle (n=100)

True ValueSCMSBBSUBC.I

30

Figure 15

X(i­1)

1% Q

uantiles

­10 ­5 0 5 10

­70­60

­50­40

­30­20

­10

1% Conditional Quatile Estimation for Ex­GARCH Modle (n=250)

True ValueSCMSBBSUBC.I

31

Figure 16

X(i­1)

5% Q

uantiles

­6 ­4 ­2 0 2 4 6

­50­40

­30­20

­100

5% Conditional Quatile Estimation for Ex­GARCH Modle (n=50)

True ValueSCMSBBSUBC.I.

Figure 17

X(i­1)

5% Q

uantiles

­5 0 5

­25­20

­15­10

­5

5% Conditional Quatile Estimation for Ex­GARCH Modle (n=100)

True ValueSCMSBBSUBC.I.

32

Figure 18

X(i­1)

5% Q

uantiles

­10 ­5 0 5 10

­30­20

­10

5% Conditional Quatile Estimation for ExGARCH Modle (n=250)

True ValueSCMSBBSUBC.I.

33

References

[1] Barone_Adesi, G., Giannopoulos, K. and Vosper, L. (1999) "VaRWithout Correlations for Non-linear Portfolios" , Journal o f Fu-tures Markets, 19, 583-602

[2] Barone_Adesi, G., Giannopoulos, K. (2001) "Non-parametric VaRTechniques. Myths and Realities", Economic Notes, 30, 167-181

[3] Bertail, P. C. Haefke, D.N. Politis and H.White (2004) "A Subsam-pling Approach to Estimating the Distribution of Diverging Statis-tics with Applications to Assessing Financial Market Risks", Jour-nal of Econometrics, 120, 295-326

[4] Bose, A. (1988), "Edgeworth correction by bootstrap in autoregres-sions", The Annuals of Statistics, 16, 1709-1722

[5] Cai, Z. (2002), "Regession quantiles for time series", EconometricTheory, 18, 169-192

[6] Carlstein, E. (1986), "The use of Subseries Methods for Estimatingthe Variance of a General Statistic from a Stationary Time Series",The Annuals of Statistics, 14, 1171-1179

[7] Constinot, A., Roncalli, T., and Teiletche, J. (2000), "Revisitingthe dependence between �nancial markets with copalas", Groupede Recherche Operationnelle, Credit Lyonnais, Working paper

[8] Efron(1979), "Bootstrap methods: Another look at the jacknife",The Annals of Statistics 7, 1-26

[9] Efron, B. and Tibshirani, R. (1986), "Bootstrap and Other Resam-pling Plans", SIAM, Philadelphia

[10] Efronm B. (1992), "The Jackknife, the Bootstrap and Other Re-sampling Plans", SIAM, Philadephia.

[11] Fitzenberger, B. (1997), "The moving blocks bootstrap and robustinference for linear least squares and quantile regressions", Journalof Econometrics, 82, 235-287

[12] Franke, J. and Hardle, W. (1992), "On Bootstrapping kernel spec-tral estimates", The Annuals of Statistics�20, 121-145

[13] Freeman, D. A. (1981), "Bootstrapping Regression Models", TheAnnuals of Statistics 9, 1218-1228

[14] Freeman, D. A. (1984), "On Bootstrapping Two-stage Least-sqaureEstimates in Stationay Linear Models"�, The Annuals of Statistics12, 827-842

[15] Gourieroux, C and Monfort, A. (1996), Statistics and EconometricModels, Cambridge University Press, Cambridge.

[16] Gozalo, P. (1997), "Nonparametric Bootstrap Analysis with Appli-cations to Demographic E¤ects in Demand Functions", Journal ofEconometrics, 81, 357-393

[17] Hall, P. (1985), "Resampling a converage pattern", Stochastic

34

Processes and Their Applications, Springer_Verlag, Berlin; NewYork

[18] Koenker, R. and Bassett, G. (1978), "Regression Quantiles", Econo-metrica, 46, 33-50.

[19] Kunsch, H.R. (1989), "The Jackknife and the Bootstrap for GeneralStationary observations", The Annuals of Statistics, 17, 1217-1241

[20] Lahiri, S.N.(2002), " On the Jackknife after Bootstrap Methodfor Dependent Data and Its Consistency Properties", EconometricTheory 18, 79-98

[21] Lahiri, S.N. (2003), "Resampling Methods for Dependent Data",Springer-Verlag, Berlin.

[22] Lahiri, S.N., Furukawa, K. and Lee, Y-D. (2003), " A Nonparamet-ric Plug-in Rule for Selecting the optimal Block Length for BlockBootstrap Methods", Preprint, Department of Statistics, Iowa StateUniversity, Ames, IA

[23] Li, Q. and Racine, J. (2005), "Nonparametric Estimation of Con-ditional CDF and Quantile Functions with Mixed Categorical andContinuous Data", CESG 2005.

[24] Liu, R.Y. (1988), "Bootstrap procedures under some non-iid mod-els", Annals of Statistics, 16, 1696-1708.

[25] Liu, R.Y. and Singh, K. (1992), "Moving Blocks Jackknife andBootstrap Capture Weak Dependence" in R. Lepage and L.Billard,eds, Exploring the Limits of the Bootstrap, Wiley, New York, 225-248

[26] Pascual, L. and Ruiz, E. (2003), "Bootstrapping Financial TimeSeries", Journal of Economic Syrveys, 16, 271-300

[27] Politis, D.N. and Romano, J.P. (1992)," A Circular Block Resam-pling Procedure for Stationay Data", in R. Lepage and L.Billard,eds, Exploring the Limits of the Bootstrap, Wiley, New York, 263-270

[28] Politis, D.N. and Romano, J.P. (1994a), "The stationary boot-strap", Journal of the American Statistical Association, 89, 1303-1313

[29] Politis, D.N. and Romano, J.P. (1994b),"Large Sample Con�denceRegions Based on Subsamples under Minimal Assumptions", An-nuals of Statistics, 22, 2031-2050

[30] Politis, D.N., Romano, J.P. and M. Wolf (1997), "Subsampling forHeteroskedastic Time Series", Journal of Econometrics, 81, 281-317

[31] Portnoy (1991), "Asymptotics for Least Absolute Deviation Regres-sion Estimators", Econometric Theory, 7, 186-199

[32] Ruppert, D., S.J. Sheather and M.P.Wand (1995), "An E¤ective

35

Bandwidth Selector for Local Least Square Regression", Journal ofthe American Statistical Association, 90, 1257-1270

[33] Wu, C.F.J. (1986), "Jackknife, Bootstrap and Other ResamplingMothods in Regression Analysis", The Annuals of Statistics, 14,1261-1295

[34] Yu, K. and M.C. Jones (1998), "Local Linear Quantile Regression",Journal of the American Statistical Association, 93, 228-237

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