A Reexamination of Diffusion Estimators With …orfe.princeton.edu/~jqfan/papers/01/timehomo.pdf ·...

A Reexamination of Diffusion Estimators WithApplications to Financial Model Validation

Jianqing Fan and Chunming Zhang

Time-homogeneous diffusion models have been widely used for describing the stochastic dynamics of the underlying economic variablesRecently Stanton proposed drift and diffusion estimators based on a higher-order approximation scheme and kernel regression method Heclaimed that ldquohigher order approximations must outperform lower order approximationsrdquo and concluded nonlinearity in the instantaneousreturn function of short-term interest rates To examine the impact of higher-order approximations we develop general and explicitformulas for the asymptotic behavior of both drift and diffusion estimators We show that these estimators will reduce the numericalapproximation errors in asymptotic biases but their asymptotic variances escalate nearly exponentially with the order of approximationSimulation studies also con rm our asymptotic results This variance in ation problem arises not only from nonparametric tting butalso from parametric tting Stantonrsquos work also postulates the interesting question of whether the short-term rate drift is nonlinear Basedon empirical simulation studies Chapman and Pearson suggested that the nonlinearity might be spurious due partially to the boundaryeffect of kernel regression This prompts us to use the local linear t based on the rst-order approximation proposed by Fan and Yaoto ameliorate the boundary effect and to construct formal tests of parametric nancial models against the nonparametric alternativesOur simulation results show that the local linear method indeed outperforms the kernel approach Furthermore our nonparametricldquogeneralized likelihood ratio testsrdquo are indeed versatile and powerful in detecting nonparametric alternatives Using this formal testingprocedure we show that the evidence against the linear drift of the short-term interest rates is weak whereas evidence against a familyof popular models for the volatility function is very strong Application to Standard amp Poor 500 data is also illustrated

KEY WORDS Goodness of t Local polynomial regression Markovian Stochastic differential equation Variance estimation

1 INTRODUCTION

Consider the problem of estimating the drift function Œ4cent5and diffusion function lsquo 4cent5 for a continuous-time diffusionprocess 8Xt1 a micro t micro T9 following the stochastic differentialequation

dXtD Œ4Xt5 dt Clsquo 4Xt5 dWt1 (1)

where 8Wt1 a micro t micro T 9 is a standard one-dimensional Brow-nian motion Suf cient conditions due to Itocirc imposed onŒ4cent5 and lsquo 4cent5 for the existence uniqueness and a measur-able Markov process of the diffusion solution 8Xt9 havebeen given by for example Wong (1971) and Kloeden andPlaten (1992) Further regularity conditions for the station-arity of 8Xt9 have been established by Banon (1978) Thistime-homogeneous diffusion model has been widely used fordescribing the stochastic dynamics of the underlying economicvariables of many well-known single-factor nancial modelsExamples include the geometric Brownian motion (GBM)

dXtD ŒXt dt ClsquoXt dWt1 (2)

by Osborne (1959) for modeling stock price and models

VAS 2 dXtD 40

C 1Xt5 dt Clsquo dWt1 (3)

CIR SR 2 dXtD 40

C 1Xt5 dt ClsquoX1=2t dWt1 (4)

CIR VR 2 dXtDlsquoX3=2

t dWt1 (5)

Jianqing Fan is Professor Department of Statistics University of North Car-olina Chapel Hill NC and Professor of Statistics Department of StatisticsChinese University of Hong Kong (E-mail jfanstatuncedu) Fanrsquos researchwas partially supported by National Science Foundation grants DMS-0196041and DMS-0204329 a Research Grants Council direct grant and the ResearchGrants Council grant CUHK429900P from the Hong Kong SAR ChunmingZhang is Assistant Professor Department of Statistics University of Wiscon-sin Madison WI 53706 (E-mail cmzhangstatwiscedu) The authors thankthe editor the associate editor and the referee for their constructive commentsand suggestions that have led to signi cant improvement of this article

and

CKLS 2 dXtD 40

C 1Xt5 dt ClsquoXƒt dWt1 (6)

by Vasicek (1977) Cox Ingersoll and Ross (1985) CoxIngersoll and Ross (1980) and Chan Karolyi Longstaff andSanders (1992) for modeling interest rate dynamics

Current research including parametric approaches to esti-mating Œ4cent5 and lsquo 4cent5 has been surveyed by Stanton (1997)To relax model assumptions and reduce possible model-ing biases nonparametric regression techniques have recentlybeen studied in this area Pham (1981) and Prakasa Rao (1985)proposed nonparametric drift estimators Ar (1995 1998)showed that the NadarayandashWatson (N-W) kernel estimator ofdrift is uniformly strongly consistent under ergodic conditionsand reached the same conclusion for the kernel regressionestimate of the diffusion function Fan and Yao (1998) usedlocal linear regression to the squared residuals for estimatinglsquo 24cent5 and showed that the proposed approach is ef cient Aiumlt-Sahalia (1996) proposed a semiparametric procedure for esti-mating the diffusion function under the parametric speci ca-tion of the drift function Jiang and Knight (1997) developeda nonparametric kernel estimator for the diffusion functionand then derived a consistent nonparametric drift estimatorUsing an in nitesimal generator and Taylor series expansionStanton (1997) constructed the rst- second- and third-orderapproximation formulas for Œ4cent5 and lsquo 4cent5 and further claimedthe superiority of higher-order approximations These formu-las contain unknown conditional expectations estimated byN-W kernel regression Stantonrsquos approach can estimate thediffusion functionlsquo 4cent5 separately without knowing or estimat-ing Œ4cent5 a priori This feature makes his method simple andattractive

copy 2003 American Statistical AssociationJournal of the American Statistical Association

March 2003 Vol 98 No 461 Theory and MethodsDOI 101198016214503388619157

118

Fan and Zhang Diffusion Estimators in Financial Models 119

Stantonrsquos approach has some problems Chapman andPearson (2000) studied the nite-sample properties of Stan-tonrsquos estimator By applying his procedure to simulated sam-ple paths of a squared-root diffusion they found that Stantonrsquosestimator produces spurious nonlinearity when the underly-ing drift function is truly linear Chapman and Pearson nicelyconcluded that the ldquomean reversionrdquo and small sample at theboundary create arti cial patterns of nonlinearity displayednoticeably near the boundary regions Meanwhile two sensi-ble questions naturally arise (1) Do higher-order approxima-tions outperform their lower-order counterparts and (2) Arethere any reasonable and formal procedures that help deter-mine whether the observed nonlinearity in the drift is real ordue to chance variation

In an attempt to answer the rst question on the order ofapproximations we derive explicitly the formulas of a higher-order approximation scheme that generalizes Stantonrsquos ideaWe then compute explicitly the asymptotic variances of non-parametric estimators based on higher-order approximationsA striking result from our asymptotic study is that higher-order approximations will reduce the numerical approximationerrors in asymptotic biases but escalate (nearly exponentially)the asymptotic variances This variance in ation phenomenonis not only an artifact of nonparametric ttingmdashit also appliesto parametric modeling The issue of a trade-off between biasreduction and variance increment is made explicit in Theo-rem 4 (Sec 2)

Stantonrsquos work raises some other interesting issues Is thedrift function in the short-term rate model nonlinear Or moregenerally does a parametric model t a given set of economicor nancial data An example of this is whether models (2)ndash(6) adequately t short-term rate data Chapman and Pearson(2000) suggested that the nonlinearity of the drift functionmight be spurious Their method is based on simulated datafrom diffusion models with a linear drift function and eval-uates whether the estimated drift looks linear This graphicalprocedure is useful but informal To set up formal statisti-cal tests an alternative hypothesis (model) is needed Becausewe usually do not have strong preference for alternative com-peting models the nonparametric model (1) serves as a nat-ural candidate The hypothesis testing problem becomes oneof testing a parametric (or semiparametric) null hypothesisagainst a nonparametric alternative The latter half of this arti-cle is thus devoted to model validation There we extend theidea of the generalized likelihood ratio (GLR) statistic devel-oped by Fan Zhang and Zhang (2001) and apply it to thetime-homogeneous diffusion models Our simulation resultsshow that GLR tests are indeed powerful and give the correcttest size They provide useful tools for assessing the validityof various models in economics and nance

The remainder of the article is organized as follows InSection 2 we discuss the distributional properties of Stan-tonrsquos drift and diffusion estimators and also derive explicitexpressions of asymptotic biases and variances for higher-order approximations To justify our analyses on empiricalgrounds we report on simulations in Section 3 In Section 4we propose model validation methods using the GLR testbased on the rst-order approximation combined with the locallinear estimation Simulations of the GLR test and real data

applications are also demonstrated In Section 5 we brie ysummarize our conclusions We collect outlines of the proofsin the Appendix

2 HIGHER-ORDER APPROXIMATIONS

This section begins with a description of Stantonrsquosapproach Although his initial construction is based solely onthe rst- second- and third-order approximations we canbuild with some extra effort a more general framework thatgives us the exibility to examine the impact of higher-orderapproximations

21 Conditional Means and Conditional Variancesof Higher-Order Differences

Following Stantonrsquos notations under appropriate conditionson Œ4cent51lsquo 4cent5 and an arbitrary bivariate function f 4cent1 cent5 theconditional expectation Et8f 4XtCatilde1 t C atilde59 can be expressedin the form of a Taylor series expansion

Et8f 4XtCatilde1 t C atilde59 D f 4Xt1 t5C notf 4Xt1 t5atilde C 1

2not2f4Xt1 t5atilde2

C cent cent centC 1nWnot

nf 4Xt1 t5atilden C O4atildenC151

as time increment atilde 0 Here the symbol Et denotes the con-ditional expectation given Xt and the in nitesimal generatornot of the process 8Xt9 is de ned by

notf 4x1 t5 D limrsquot

E8f 4Xrsquo 1 rsquo5mdashXtD x9 ƒ f 4x1 t5

rsquo ƒ t

D iexclf 4x1 t5

iexcltC iexclf 4x1 t5

iexclxŒ4x5

C 1

2

iexcl2f 4x1 t5

iexcl2xlsquo 24x5 (7)

(see Oslashksendal 1985 for more details) Thus the rst-orderapproximation formula for the target function notf 4Xt1 t5 isgiven by

atildeƒ1Et8f 4XtCatilde1 t C atilde5 ƒ f 4Xt1 t59 D notf4Xt1 t5C O4atilde50 (8)

In particular setting f 4x1 t5 D x (or f4x1 t5 D x ƒ Xt) givesnotf 4x1 t5 D Œ4x5 likewise taking f4x1 t5 D 4x ƒ Xt5

2 impliesnotf 4x1 t5 D 24x ƒ Xt5Œ4x5 Clsquo 24x5 which at x D Xt equalslsquo 24Xt5 Hence these two special functions f 4cent1 cent5 can exactlyrecover Œ4Xt5 and lsquo 24Xt5 In such cases estimating theleft side of (8) by the N-W kernel method leads to Stan-tonrsquos estimates for Œ4x5 and lsquo 24x5 based on the rst-orderapproximation

Higher-order approximations (or differences) can beachieved through a linear combination of terms on the leftside of (8) More precisely for any xed integer k para 1 anysequence of constants 8ak1j1 j D 11 1 k9 and any discretelyobserved time steps jatilde for j D 11 1 k we consider the

120 Journal of the American Statistical Association March 2003

following linear combination

atildeƒ1kX

jD1

ak1jEt8f 4XtCjatilde1 t C jatilde5 ƒ f4Xt1 t59

D(

kX

jD1

jak1j

)

notf 4Xt1 t5C(

kX

jD1

j2ak1j

)not2f4Xt1 t5

2atilde

C cent cent centC(

kX

jD1

jkak1j

)notkf4Xt1 t5

kW atildekƒ1

C(

kX

jD1

jkC1ak1j

)notkC1f4Xt1 t5

4k C 15Watildek C O4atildekC150

It is readily seen that a kth order approximation scheme

atildeƒ1kX

jD1

ak1jEt8f 4XtCjatilde1 t C jatilde5 ƒ f 4Xt1 t59

D notf 4Xt1 t5C O4atildek51

is obtained by choosing coef cients 8ak1j9kjD1 to satisfy the

system of equations

8gtgtgtgtlt

gtgtgtgt

PkjD1 jak1j

D 1Pk

jD1 j2ak1jD 0000

PkjD1 jkak1j

D 00

(9)

The general form of the solutions 8ak1j1 j D 11 1 k9 ispresented in Theorem 1 the proof of which is given in theAppendix Apparently with orders k D 1121 3 the values of8ak1j1 j D 11 1 k9 coincide with those derived by Stan-ton (1997)mdashnamely 819 for k D 11 821ƒ1=29 for k D 2 and831 ƒ3=21 1=39 for k D 3

Theorem 1 For each xed integer k para 1 the unique solu-tions to the system of (9) are given by

ak1jD 4ƒ15jC1

sup3k

j

acutej1 j D 11 1 k0 (10)

Furthermore with these choices of 8ak1j9kjD1 we have

kX

jD1

jkC1ak1jD 4ƒ15kC1kW0

Therefore using the foregoing unique solutions 4ak111 1ak1k5 we obtain for notf 4Xt1 t5 a general form of the kth orderapproximation formula

atildeƒ1kX

jD1

ak1jEt8f 4XtCjatilde1 t C jatilde5ƒ f4Xt1 t591 (11)

with the approximation error term expressed as

4ƒ15kC1 notkC1f4Xt1 t5

4k C 15atildek C O4atildekC150 (12)

Equations (11) and (12) imply that

atildeƒ1kX

jD1

ak1jEt4XtCjatildeƒXt5

DŒ4Xt5Cmicro4ƒ15kC1 notkC1f14Xt1t5

4kC15atildek CO4atildekC15

para(13)

with the choice f14x1 t5 D x and that

atildeƒ1kX

jD1

ak1jEt4XtCjatildeƒ Xt5

2

D lsquo 24Xt5 Cmicro


4k C 15atildek C O4atildekC15

para(14)

with the choice f24x1 t5 D 4x ƒ Xt52 From (14) one can sim-

ply take the square root operation to obtain the kth-orderapproximation formula for the function lsquo 4Xt5 such that

lsquo 4Xt5 DAgrave

atildeƒ1kX

jD1


2

Aacute1=2

C O4atildek50 (15)

In addition for each of the choices f`4x1 t51 ` D 112 the termnotkC1f`4Xt1 t5 does not vanish and is independent of the timevariable t Therefore the resulting numerical approximationerrors for Œ4cent51lsquo 24cent5 and lsquo 4cent5 maintain for any integer k para 1the same convergence rates O4atildek5 to 0 Simulation com-parisons of the rst three order approximations with the truedrift and diffusion functions for the processes (3) and (4)were demonstrated in tables IndashIV of Stanton (1997) whereasnumerical comparisons conducted for the interest rate datawere shown in his gures 4ndash7 along with the pointwise 95con dence bands based only on the rst-order approximation

With the kth-order approximation formulas available forŒ4cent5 and lsquo 24cent5 the involved conditional expectations remainto be estimated Given the initial calendar time point t0 andtime series data 8Xt0Ciatilde1 i D 11 1 n9 observed at equallyspaced time points our rst step is to form 4n ƒ k5 pairs ofsynthetic data

sup3Xt0Ciatilde1atildeƒ1

kX

jD1

ak1j8Xt0C4iCj5atildeƒ Xt0Ciatilde9

acutesup2 4X uuml

iatilde1 Y uumliatilde51

i D 11 1 n ƒ k1 (16)

for estimating Œ4cent5 together with


kX

jD1


acutesup2 4X uuml

iatilde1Z uumliatilde51

i D 11 1 n ƒ k1 (17)

for estimating lsquo 24cent5 Our second step is to use appropriatepointwise nonparametric regression estimators OŒ11atilde4x05 andOŒ21atilde4x05 for estimating the conditional expectations

E4Y uumliatilde

mdashX uumliatilde

D x05 D Œ4x05 C O4atildek5 and

E4Z uumliatilde

mdashX uumliatilde

D x05 Dlsquo 24x05C O4atildek51 (18)

from (13) and (14)


Table 1 Variance Inlsquo ation Factors Using Higher-Order Differences

Order k

1 2 3 4 5 6 7 8 9 10

V1(k) 1000 2050 4083 9025 18095 42068 105049 281065 798001 21364063V2(k) 1000 3000 8000 21066 61050 183040 570066 11837028 61076025 201527022

There are many nonparametric methods for estimating theconditional expectations in (18) the N-W estimator is thesimplest It can be improved by local polynomial techniques(Fan and Gijbels 1996) Therefore our subsequent analyticaldiscussions are concentrated on OŒ11atilde4x05 and OŒ21atilde4x05 for aninterior point x0 via the qth-degree local polynomial estima-tion 4q para 05 the N-W estimator corresponds to the local con-stant method with degree q D 0 We now brie y describe thetechnique for estimating E4Y uuml

iatildemdashX uuml

iatildeD x05 By a Taylor series

expansion a smooth function m4x5 D E4Y uumliatilde

mdashX uumliatilde

D x5 with xlocated in a neighborhood of x0 can be locally approximatedby a qth-degree polynomial that is

m4x5 ordm m4x05 C 4x ƒ x05m04x05 C cent cent centC 4x ƒ x05

q m4q54x05=qW0

Denote the coef cient vector by Acirc4x05 D 4m4x051m04x051 1m4q54x05=qW5T D 4sbquo01sbquo11 1sbquoq5T Then the local poly-nomial estimator OAcirc4x05 of the qth degree is determined bythe minimizer of the residual sum of squares between Y uuml

iatilde andthe local model on m4X uuml

iatilde5 weighted by the distance of X uumliatilde

from the tting point x0 Formally OAcirc4x05 minimizes the objec-tive function

nƒkX

iD1

8Y uumliatilde

ƒ sbquo0ƒ 4X uuml

iatildeƒ x05sbquo1

ƒ cent cent centƒ 4X uumliatilde

ƒ x05qsbquoq92Kh4X uuml

iatildeƒ x05 (19)

over values of Acirc4x05 where Kh4cent5 D K4cent=h5=h Here K4cent5 andh are referred to as the kernel function and the bandwidth(or smoothing parameter) The rst component of the vectorOAcirc4x05 gives OŒ11atilde4x05 the qth degree local polynomial estimateof E4Y uuml

iatildemdashX uuml

iatildeD x05 A similar procedure can be applied to

obtain the qth degree local polynomial estimate OŒ21atilde4x05 ofE4Z uuml

iatildemdashX uuml

iatildeD x05 For practical application Fan and Gijbels

(1996) recommended the use of local linear t (q D 1)Because any nonparametric regression procedure is in

essence a weighted average of local data its performancealways depends on the local variation namely the conditionalvariance For our current applications based on the syntheticdata the corresponding conditional variances are

lsquo 211atilde4x05 D var4Y uuml

iatildemdashX uuml

iatildeD x05 and

lsquo 221atilde4x05 D var4Z uuml

iatildemdashX uuml

iatildeD x050 (20)

Theorem 2 proved in the Appendix summarizes the mag-nitudes of lsquo 2

11atilde4x05 and lsquo 221atilde4x05 Note that some regularity

conditions (see eg Wong 1971 chapter 4 prop 41) puton Œ4cent51lsquo 4cent5 and Xt0

for the unique existence and Markovprocess of 8Xt9 in (1) are always assumed implicitly in Theo-rems 2 and 4

Theorem 2 Assume that 8Xt9 is a Markov process LetA11k and A21k be k k matrices with 4i1 j5th entry equal to

min4i1 j5 and min4i21 j25 and let Aacutek be a k 1 vector the jthelement of which is given in (10) Denote V14k5 D AacuteT

k A11kAacutek

and V24k5 D AacuteTk A21kAacutek Then as atilde 0 the conditional vari-

ance of the kth-order difference formula for Œ4x05 is given by

lsquo 211atilde4x05 Dlsquo 24x05V14k5atildeƒ181 C O4atilde591 (21)

whereas the conditional variance of the kth order differenceformula for lsquo 24x05 is given by

lsquo 221atilde4x05 D 2lsquo 44x05V24k581 C O4atilde590 (22)

The factors V14k5 and V24k5 re ect the premium that higher-order approximations must pay For this reason we call themthe variance ination factors for using higher-order approxi-mations To provide some numerical impression Table 1 sum-marizes the numerical values of V14k5 and V24k5 for approx-imations of orders up to the 10th For visual assessmentFigure 1 contains plots of log8V14k59 and log8V24k59 versusorder k The overall impacts of higher-order approximationson variance in ation are striking

It is also notable from Table 1 and Figure 1 that the variancein ation factors grow nearly exponentially fast as the order k

increases This relation can indeed be veri ed analytically asshown in the following theorem

Theorem 3 (a) For k para 1 the factor V14k5 in (21) isbounded below by

k2 ƒ 3k ƒ 2k4k C 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C 34k C 152

ordm 4k

1=2k5=21

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

Figure 1 Theoretical Values of logVj ( k) Versus Order k The fac-tors Vj (k) are given in Theorem 2 where j D 1 (--) refers to drift Œ( cent)and j D 2 (- amp- - ) refers to squared diffusion lsquo 2( cent)


and bounded above by

5k2 ƒ k ƒ 2k4k C 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C 6k C 54k C 152

ordm 5 4k

1=2k5=20

(b) For k gt 1 the factor V24k5 in (22) is given by

V24k5 Diexcl

2k

k

centƒ 4k C 15

k ƒ 1ordm 4k

1=2k3=20

22 Asymptotic Behaviorof Nonparametric Estimators

The asymptotic bias and variance of the pointwise drift esti-mator OŒ11atilde4x05 and the squared diffusion function estimatorOŒ21atilde4x05 based on the kth order approximation scheme andthe qth degree local polynomial tting are presented in The-orem 4 The results demonstrate that higher-order differencesresult in reductions of the asymptotic bias while translatingthe variance in ation into the asymptotic variance of the asso-ciated nonparametric drift and diffusion estimators

We rst introduce some notations and de nitions Setf14x1 t5 D x1 f24x1 t5 D 4x ƒ Xt5

21ŒjD

RujK4u5 du1 j

DRujK24u5 du1 e1 D 41101 105T 1 S D 4ŒiCjƒ25i1jD11 1qC11

S uuml D 4iCjƒ25i1jD11 1qC11 cqD 4ŒqC11 1Œ2qC15

T and QcqD

4ŒqC21 1Œ2qC25T For integers ` gt 0 let p`4ymdashx5 denote the

conditional probability density of Xt0C4`C15atilde given Xt0Catilde

Theorem 4 Let 8Xt0Ciatilde1 i D 11 1 n ƒ k9 be a sequenceof observations on a stationary Markov process with abounded continuous density p4cent5 Assume that p`4ymdashx5 is con-tinuous in the variables 4y1x5 and is bounded by a constantindependent of ` The sequence 8Xt0Ciatilde1 i D 11 1 nƒ k9 sat-is es the stationarity conditions of Banon 419785 and theG2 condition of Rosenblatt 419705 on the transition operatorAssume that the kernel K is a bounded symmetric probabilitydensity function with bounded support Suppose that x0 is anygiven point in the interior of the support of p where p4x05 gt

01lsquo 24x05 gt 0 and that Œ4qC154cent5 and 4lsquo 254qC154cent5 are contin-uous in a neighborhood of x0 Put lsquo 2

1 4x03 k5 D lsquo 24x05V14k5

and lsquo 22 4x03 k5 D 2lsquo 44x05V24k5 Let n ˆ such that h 0

and nh ˆ and atilde 0 then at any time t D t0C iatilde1 i D

11 1 n ƒ k

(a) The asymptotic bias of OŒ11atilde4x05 for odd degrees q isgiven by

4ƒ15kC1 notkC1f14x01 t5


C eT1 Sƒ1cq

Œ4qC154x05

4q C 15WhqC1 C oP 4hqC151 (23)

whereas for even degrees q the last two terms in 4235 become

eT1 Sƒ1Qcq

4q C 25WcopyŒ4qC254x05 C 4q C 25Œ4qC154x05

p04x05=p4x05ordfhqC2 C oP4hqC251 (24)

provided that p04cent5 and Œ4qC254cent5 are continuous in a neighbor-hood of x0 and nh3 ˆ Assume further that h D O4atilde1=25then the asymptotic variance is

4nhatilde5ƒ1eT1 Sƒ1S uuml Sƒ1e1lsquo

21 4x03 k5=p4x0581C o41590 (25)

(b) The asymptotic bias of OŒ21atilde4x05 for odd degrees q isgiven by



C eT1 Sƒ1cq

4lsquo 254qC154x05

4q C 15WhqC1 C oP4hqC151 (26)


eT1 Sƒ1Qcq

4q C 25Wcopy4lsquo 254qC254x05 C 4q C 254lsquo 254qC154x05


provided that p04cent5 and 4lsquo 254qC254cent5 are continuous in a neigh-borhood of x0 and nh3 ˆ Assume further that h D O4atilde1=45then the asymptotic variance is

4nh5ƒ1eT1 Sƒ1S uuml Sƒ1e1lsquo

22 4x03 k5=p4x0581 C o41590 (28)

It is clearly observed from (23) that the bias of OŒ11atilde4x05 iscomposed of a numerical approximation error expressed byE4Y uuml

iatildemdashX uuml

iatildeD x05 ƒ Œ4x05 in addition to the usual nonparam-

etric estimation bias OŒ11atilde4x05 ƒ E4Y uumliatilde

mdashX uumliatilde

D x05 Results of(23) and (24) indicate that for the kernel estimator used byStanton (1997) the leading term of its asymptotic bias is


4k C 15atildek

C Œ2

2h28Œ004x05 C 2Œ04x05p

04x05=p4x0591 (29)

whereas for the local linear method the second term becomes2ƒ1Œ2h

2Œ004x05 A similar comparison can be made forOŒ21atilde4x05

Remark 1 As shown by Banon and Nguyen (1981lemma 21) a stationary Markov process satisfying a cer-tain mixing condition namely the G2 condition of Rosen-blatt (1970) is asymptotically uncorrelated (Rosenblatt 1971)Therefore the ldquobig-block and small-blockrdquo arguments simi-lar to those used by Fan and Gijbels (1996 theorem 61) canbe incorporated to show the asymptotic normality of OŒ11atilde4x05

and OŒ21atilde4x05 The lengthy details are omitted here

Remark 2 The conclusions of Theorems 2 and 3 do notdepend on the stationarity condition The stationarity conditionin Theorem 4 is imposed to facilitate technical manipulationsit is not a necessary condition The stationarity condition pos-sibly can be relaxed


3 SIMULATIONS

Realistically we do not know whether the stationary Marko-vian assumption remains valid for nancial data recorded atdiscrete time points We also do not know whether the asymp-totic results re ect reality Nevertheless we can still carry outthe drift and diffusion estimations using higher-order approx-imations and nonparametric regression techniques This willenable us to assess empirically how our asymptotic results arere ected in nite samples Our simulation studies show thefact that the variance in ation due to higher-order approxima-tions is re ected in nite samples

31 CoxndashIngersollndashRoss Squared-Root Diffusion

As a rst illustration we consider the well-known CoxndashIngersollndashRoss (CIR) model for interest rate term structure

dXtD Š4ˆ ƒ Xt5 dt ClsquoX1=2

t dWt1 t para t01 (30)

where the spot rate Xt moves around its long-run equilib-rium level ˆ at speed Š When the condition 2Šˆ paralsquo 2 holdsthis process is shown to be positive and stationary Providedthat the time step size atilde is small we can use the discrete-time order 10 strong approximation scheme given in (314) ofKloeden Platen Schurz and Soslashrensen (1996) In this exam-ple the scheme takes the form

XtiC1ordm Xti

C 8Š4ˆ ƒ Xti5 ƒ 4ƒ1lsquo 29atilde

C 2ƒ1lsquoh8Xti

C 4Šˆ ƒ ŠXtiƒ 4ƒ1lsquo 25atilde

Clsquo 4Xti51=2

C ˜i

patilde91=2

C C 4Xti51=2

C

i˜i

patilde1 (31)

for 1 micro i micro n ƒ 1 where ˜i

iidsup1 N 40115 and xC D max4x105Alternatively one might use the transition density properties ofthe process (see Cox et al 1985) That is given Xt

D x at thecurrent time t the variable 2cXs at the future time s has a non-central chi-squared distribution with degrees of freedom 2q C2and noncentrality parameter 2u where q D 2Šˆ=lsquo 2 ƒ 11 u DcxeƒŠ4sƒt5 and c D 2Š

lsquo 281ƒeƒŠ4sƒt59 The initial value of Xt0

can begenerated from the steady-state gamma distribution of 8Xt9with the probability density p4y5 D mdash=acirc 45yƒ1eƒmdashy where D 2Šˆ=lsquo 2 and mdash D 2Š=lsquo 2 For each simulation experimentwe generate a sample path of length 10000 and computebased on the synthetic data [see (16) and (17)] Stantonrsquos ker-nel drift estimate OŒ11atilde4x05 and the squared diffusion estimateOŒ21atilde4x05 We replicate the experiments 1000 times and calcu-late the sample variances of 8 OŒ11atilde4x059 and 8 OŒ21atilde4x059 acrossthese 1000 simulations respectively

Choices of kernel function depend purely on individualpreferences Throughout our numerical work in this arti-cle we use the Epanechnikov kernel de ned by K4u5 D3=441ƒu25I4mdashumdash micro 15 where I 4cent5 stands for the indicator func-tion For a given kernel function the choice of an effectivebandwidth parameter is very important to the performanceof a nonparametric regression estimator It is often selectedthrough either visual inspection of the resulting smooths ora data-driven technique Popular data-dependent approaches

include cross-validation (Allen 1974 Stone 1974) general-ized cross-validation (Wahba 1977) the preasymptotic sub-stitution method (Fan and Gijbels 1995) the plug-in method(Ruppert Sheather and Wand 1995) and the empirical biasmethod (Ruppert 1997) These techniques provide various use-ful means for automatic bandwidth selection but involve inten-sive computation and extra effort to program A more detailedlook at these methods regarding theoretical properties andimplementations was given by Fan and Gijbels (1996) Alter-natively a simple rule of thumb bandwidth formula such as

h D constant std48X uumlatilde1 1 X uuml

4nƒk5atilde95 nƒ1=51 (32)

also can be used To show the occurrence of variance in a-tion with order k by nite-sample simulation an appropriatechoice of bandwidth is constant-valued and independent of keven though the optimal bandwidth may depend on k For thepurpose of illustration we set h D 0004 in this example Otherchoices of bandwidth have also been tried and the results havebeen similar

In our implementation the values of the model parame-ters are cited from Chapman and Pearson (2000) that is Š D0214591 ˆ D 0085711lsquo D 007830 and atilde D 1=250 To differ-entiate the effects of the higher-order approximation schemefrom the boundary effects of the kernel estimator we focuson an interior state point x0

D 01 The natural logarithms ofthe simulated variance ratios of OŒ11atilde4015 and OŒ21atilde4015 basedon higher-order difference to those of their rst-order coun-terparts are displayed in Figure 2 where plot (a) is based onsample paths generated from the conditional chi-squared dis-tribution and plot (b) results from the discretization scheme(31) Meanwhile for the purpose of comparison we alsopresent in plots (a0) and (b0) the corresponding results bylocal linear estimation All plots mimic (except in amplitude)our theoretical results shown in Figure 1

32 Geometric Brownian Motion

We include another familiar example of geometric Brown-ian motion determined by

dXtD 4Œ C 2ƒ1lsquo 25Xt dt ClsquoXt dWt1 0 micro t micro T 0 (33)

Apparently from its construction both the drift and diffu-sion are linear and thus 8Xt9 is Markovian (see Wong 1971prop 41) but the technical assumption of stationarity is vio-lated This model is incorporated to illustrate that the conclu-sion of Theorem 4 extends to more general diffusion processes

For (33) we simulate in time interval 601T 7 with T D 10the corresponding approximate process with parameters Œ D0087 and lsquo D 0178 starting at X0

D 1 We choose the order 10scheme

XtiC1ordm Xti

C 4Œ C 2ƒ1lsquo 25XtiatildeClsquoXti

˜i

patilde

C 2ƒ1lsquo 2Xti4˜2

iƒ 15atilde (34)

given in (35) of Kloeden et al (1996) Alternatively wecould directly use the explicit solution Xt

D X0 exp8Œt Clsquo Wt9

for (33) For both schemes 1000 sample paths of length 1000are generated The bandwidth parameter h D 004 is used for


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

V j(k)

(a) KERNEL ESTIMATION (CHISQUARED DISTRIBUTION)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

Vj(k

)

(b) KERNEL ESTIMATION (DISCRETIZATION)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k

log

Vj(

k)

(arsquo) LOCAL LINEAR ESTIMATION (CHISQUARED DISTRIBUTION)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

(brsquo) LOCAL LINEAR ESTIMATION (DISCRETIZATION)

Figure 2 Simulated Values of logVj (k) Versus Order k for CIR Model dXt D 21459( 08571 - Xt ) dt + 07830X 1=2t dWt The index j D 1 (--)

refers to the drift estimator OŒ1 atilde( 1) j D 2 (- amp- - ) refers to the squared diffusion estimator OŒ2 atilde( 1) Plots (a) and (a0) are based on the same setsof sample paths generated by the noncentral chi-squared distribution whereas plots (b) and (b0) are based on the same sets of sample pathsgenerated by the discretization scheme (31)

local smoothing Again this number serves for the sake ofillustration For the same reason stated in the previous exam-ple we restrict attention to the state value x0

D 100 simplybecause more data points fall within its local region Figure 3displays similar types of plots as those shown in Figure 2For comparison plots (a) and (a0) are based on data generatedfrom the exact solution and plots (b) and (b0) depend on thediscretization scheme (34) Again all plots in Figure 3 sup-port our theoretical results in Figure 1 although we used asmaller sample size and lower sampling frequency than thosein the preceding example of the CIR model

33 Local Linear Fit Boundary Correction

Overall the foregoing simulation studies present convinc-ing evidence that at least for models similar to those twotypes the higher-order approximations substantially amplifyvariances As discussed in Section 2 this phenomenon alwaysoccurs regardless of the method used for nonparametric

regression It is well known that the kernel regression esti-mator can create boundary biases In contrast the local linearestimator enjoys the theoretical advantages of design adapta-tion automatic boundary correction and minimax ef ciency(see Fan and Gijbels 1996 for further details) This naturallyleads us to substitute kernel estimation by local linear estima-tion A similar application of local linear t to the rst-orderapproximation of continuous-time diffusion models was usedby Fan and Yao (1998) who also suggested correcting thedrift term before the variance estimation

To examine the performance of local linear estimation ofdiffusion models we revisit the CIR square-root diffusionmodel discussed in Section 31 We adopt the same valuesof model parameters Š1ˆ and lsquo to generate with weeklyfrequency sample paths of length 5000 using the (noncen-tral chi-squared) transition density To conduct kernel andlocal linear ts based on the rst-order synthetic data a scaleconstant 6 is used in the empirical bandwidth formula (32)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

log

V j(k)

log

V j(k)

log

V j(k)

(a) KERNEL ESTIMATION (EXACT SOLUTION)

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k

(arsquo) LOCAL LINEAR ESTIMATION (EXACT SOLUTION)

1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k


Figure 3 Simulated Values of logVj ( k) Versus Order k for Geometric Brownian Motion dX t D ( 087 + 1782=2)Xt dt + 178Xt dWt The indexj D 1 (--) refers to the drift estimator OŒ1atilde( 10) j D 2 (- amp- - ) refers to the squared diffusion estimator OŒ2 atilde( 10) Plots (a) and (a0) are based onthe same sets of sample paths generated from the exact solution Xt D X0 exp087t + 178Wt whereas plots (b) and (b0) are based on the samesets of sample paths generated by the discretization scheme (34)

For individual simulated trajectories we compared the esti-mated drift and diffusion for which we observed that inmost cases the local linear approach is superior to the kernelmethod In fact according to Fan (1992) the local linear thas a better bias-correction property than the kernel methodThus as the bandwidth gets larger the outperformance ofthe local linear t over the kernel method can become evenmore dramatic In contrast the sample ranges of 8Xt9 varyconsiderably across different simulations Extremely high lev-els of those states x (eg 20) rarely occur in reality orare visited in practical simulations To conduct more sensiblecomparisons we simulate 101 sample paths with range inter-val copy D 60031 0157 The drift and diffusion are estimated foreach realization and the 25th and 75th percentiles (dashedcurves) and the median (dash dotted curves) of the estimatesover the 101 realizations are presented in Figure 4 Similargraphs using discretization schemes such as (31) are omitted

here For the volatility estimates we nd that the local lin-ear method achieves more gains in alleviating the impact ofldquoboundary effectsrdquo than the kernel counterpart The same con-clusion applies to estimation of the drift function The widerbands of the interquartile ranges of the drift estimates com-pared to those of the diffusion estimates can be easily under-stood from Theorem 4 which states that the estimates of driftare more variable than the estimates of diffusion Furthermorethis necessitates the importance of developing formal proce-dures for model validation

4 MODEL VALIDATION

Model diagnosis plays an important role in examining therelevance of speci c assumptions underlying the modelingprocess and in identifying unusual features of the data thatmay in uence conclusions Despite a wide variety of well-known parametric models imposed on the short-term interest


005 01 015ndash003

ndash002

ndash001

0

001

002Kernel Estimate of Drift

005 01 015

0015

002

0025

003

Kernel Estimate of Diffusion

005 01 015ndash003

ndash002

ndash001

0

001

002Local Linear Estimate of Drift

005 01 015

0015

002

0025

003

Local Linear Estimate of Diffusion

Figure 4 Estimated Drift and Diffusion Functions for CIR Model dXt D 21459( 08571 - Xt ) dt + 07830X1=2t dWt The solid curves are the true

functions the dashed-dotted curves denote the medians of the estimates and the dashed curves correspond to the 25th and 75th samplepercentiles of the estimates over simulated data (101 replications) The sample paths are generated by the transitional noncentral chi-squareddistribution

rates and stock price indices relatively little is known abouthow these models capture the actual stochastic dynamics ofthe underlying processes Among them a majority of the use-ful models have been studied and compared in terms of theirrelative performances under a uni ed parametric framework

dXtD 4 C sbquoXt5 dt ClsquoXƒ

t dWt1 (35)

in Chan et al (1992) The generalized method of moments ofHansen (1982) is frequently used to estimate the parametersHowever the question frequently arises whether model (35)itself correctly captures the stochastic dynamics of a given setof economic data To address this issue we need an alternativefamily of stochastic models Nonparametric models offer avery nice solution to this problem Depending on the cases andthe natures of model validation the alternative nonparametricmodels can be of the form

dXtD Œ4Xt5 dt ClsquoX

ƒt dWt1 (36)

dXtD 4 C sbquoXt5 dt Clsquo 4Xt5 dWt1 (37)

or the more generic model (1) which places no particularrestriction on either the structural shift or volatility Thesekinds of hypothesis testing problems often arise in nancialmodeling

In this section we rst describe approaches used for esti-mating parameters of models (35)ndash(37) To testify againstthese models (null hypotheses) we treat model (1) as our alter-native hypothesis We propose new hypothesis-testing proce-dures based on the ldquogeneralized likelihood ratiordquo by Fan et al(2001) and demonstrate the explanatory power and versatilityof the GLR tests by simulations and two sets of real data

41 Parametric Estimation

For ease of exposition we proceed from the parametricmodel (35) Given discretely sampled observations 8Xti

1 i D11 1 n9 from this model denote atildei

D tiC1 ƒ ti and YtiD

XtiC1ƒXti

for 1 micro i micro nƒ1 Then the parameters 1 sbquo1lsquo andƒ can be estimated through a discrete-time speci cation

Ytiordm 4 C sbquoXti

5atildeiClsquoXƒ

ti˜i

patildei1 i D 11 1 n ƒ 11 (38)

where ˜i

iidsup1 N 40115 Three steps summarize the estimationprocedure

Step I Pretend that model 4385 is homoscedastic andobtain the least squares estimates of 41sbquo5 denoted by4 O 4151 Osbquo4155

Step II Let OetiD 8Yti

ƒ 4 O 415 C Osbquo415Xti5atildei9=atilde1=2

i whichtransforms model 4385 into

log4 Oe2ti5 ordm log4lsquo 25 C ƒ log4X2

ti5 C log4˜2

i 51

i D 11 1 nƒ 10 (39)

Obtain least squares estimates 4 Olsquo 4151 Oƒ4155 of 4lsquo 1 ƒ5 aftersubtracting E8log4Z259 ordm ƒ10270362845 from both sides ofmodel (39) where Z sup1 N 40115

Step III (optional) Substitute 4 Olsquo 4151 Oƒ4155 into model(38) and get weighted least squares estimates of 41 sbquo5denoted by 4 O4251 Osbquo4255 Meanwhile get updated estimates4 Olsquo 4251 Oƒ4255 at step II

This approach can be exibly modi ed For instance the dif-fusion parameters lsquo and ƒ in model (36) could be estimated


Table 2 Parameter Estimates and Standard Errors ( in brackets) for the CIR Model dXt D ( + sbquoXt ) dt + lsquo Xƒ

t dWt Where D 0183925sbquo D - 21459lsquo D 0783 and ƒ D 5

n O (1) O (2) Osbquo(1) Osbquo(2) Olsquo (1) Olsquo (2) Olsquo (3) Oƒ (1) Oƒ (2) Oƒ (3)

5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)

directly from step II except for setting Oetiin (39) to 8Yti

ƒOŒ4Xti

5atildei9=atilde1=2i where OŒ4Xti

5 is estimated nonparametricallyby the local linear method Call 4 Olsquo 4351 Oƒ 4355 the resulting esti-mators Estimation of the drift parameters of model (37) canbe accomplished by similar adjustment

To assess the ef ciency of the parametric estimators4 O 4`51 Osbquo4`51 Olsquo 4`51 Oƒ4`551 ` D 11 2 and 4 Olsquo 4351 Oƒ4355 we generatewith weekly frequency and by the transition density pathwisesamples of lengths 5000 and 10000 from the CIR modeldXt

D 400183925ƒ 021459Xt5 dt C 00783X1=2t dWt The sam-

ple means and standard errors of these estimates over 1000samples are reported in Table 2 Obviously lsquo and ƒ can beestimated far more ef ciently than and sbquo This is directlyattributed to the lower magnitude of signal compared with thatof stochastic noise in (35) or (38) Also the improvementsof the weighted least squares estimators over the unweightedestimators are negligible This is why we leave step III asoptional

42 Generalized Likelihood Ratio Test

Interest rate volatility plays a key role in valuing contingentclaims and hedging interest rate risks For the sake of brevitywe describe how to test model (36) against the nonparametricalternative (1) namely the following testing problem

H0 2lsquo 4Xt5 D lsquoXƒt vs H1 2 lsquo 4Xt5 6DlsquoXƒ

t 0

Let bEtiD 8Yti

ƒ OŒ4Xti5atildei9=atilde1=2

i and Y415ti

D log4bE2ti5 Then sim-

ilar to (38) and (39) we have approximately

bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti

ordm log8lsquo 24Xti59 C log4˜2

i 51 i D 11 1 n ƒ 10 (40)

This transforms the test originally for (36) into that for

H0 2 log8lsquo 24Xt59 D log4lsquo 25 C ƒ log4X2t 5 versus

H1 2 log8lsquo 24Xt59 6D log4lsquo 25 C ƒ log4X2t 51 (41)

that is testing the linear relationship of the bivariate data84Xti

1 Y415ti

5nƒ1iD1 9 Under the null hypothesis in (41) let Olsquo and Oƒ

be the parameter estimates outlined in Section 41 Under thealternative model (1) let Olsquo 4cent5 be the estimated diffusion func-tion based on the local linear approach The GLR test statisticproposed by Fan et al (2001) is given by

lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)

where RSS0 and RSS1 [depending on h through Olsquo 4cent5] repre-sent the residual sums of squares of model (40) under the nulland alternative hypotheses in (41) Under H0 there will be lit-tle difference in size between RSS0 and RSS1 whereas underthe alternative RSS0 should become systematically larger thanRSS1 and the GLR statistic thus will tend to take large posi-tive values Hence a high value of the test statistic lsaquon4h5 indi-cates that the null hypothesis should be rejected This proce-dure can similarly be applied to testing other forms of drift ordiffusion functions

In the nonparametric regression model with independentdata Fan et al (2001) showed the Wilks type of result thatrKlsaquon4h5 under certain types of null hypotheses is asymp-totically distributed as 2

dn4h5 Here the normalizing constant

is rKD 4Kƒ2ƒ1K uuml K5405R

4Kƒ2ƒ1K uuml K524t5dt the degrees of freedom is dn4h5 D

rKcKmdashigravemdashhƒ1 with cK

D 4K ƒ2ƒ1K uuml K5405 and mdashigravemdash measuresthe length of the support of the regressor variable In the samepaper it was shown that lsaquon is asymptotically equivalent to aquadratic form

PniD1

PnjD1 Wijn4Ri1Rj5 in which the variables

8Ri9 are independent Although the GLR statistic applied toour current setup (40) involves more complicated stochasticerrors and requires more detailed technical justi cations webelieve that a similar Wilks type of result continues to holdunder the null hypothesis in (41) This is due to the fact thatthe quadratic form is a special case of Hoeffdingrsquos U statisticProbabilistic limit theorems (limit law convergence rate) on U

statistics and von Mises statistics for weakly dependent pro-cesses are available (see Denker and Keller 1983) Thereforewith dependent 8Ri9 it is technically feasible to work out thelimiting distribution of lsaquon Indeed we have conducted sub-stantial simulations that provide stark evidence to support thisclaim However rigorous justi cations are beyond the scopeof this article

43 Power Calculation

One advantage of nonparametric regression is attributed toits exibility in model assumptions This broadens the scopeof applications As a result nonparametric tests while gain-ing signi cant exibility may result in loss of power com-pared with the parametric counterparts when the parametricassumptions provide a suitable description of the true patternTo gauge the level and power of our proposed GLR test weconduct the following simulation studies

First we compute the empirical critical values of theGLR statistics under each form of the following typical nullhypotheses

H415

0 2 Œ4Xt5 D 0C sbquo0Xt1 lsquo 4Xt5 D c01 (43)

H425

0 2 Œ4Xt5 D 0C sbquo0Xt1 lsquo 4Xt5 D c1X

05t 1 (44)


H435

0 2 Œ4Xt5 D 01 lsquo 4Xt5 D c2X105t 1 (45)

and

H445

0 2 Œ4Xt5 D 0 C sbquo0Xt1 lsquo 4Xt5 D lsquoXƒt 1 (46)

against the nonparametric alternative (1) Here we set 0 D000739 and sbquo0

D ƒ011798 which result from the weightedleast squares estimates of the 3-month interest rate data(described at the beginning of Sec 44) The constants c0

D0012721 c1 D 005596 and c2 D 090114 are put in (43) (44)and (45) to match the average height of the local linear esti-mates of volatility while the parameters lsquo and ƒ in (46) areunknown We have generated with weekly frequency 1000pathwise samples of length 2400 from each of the four hypo-thetical models starting at an initial value of 013 the rstobservation of the interest rate data In such instances we usethe scheme (314) of Kloeden et al (1996) for models (44)and (46) and use their scheme (35) for models (43) and (45)To simulate realizations from model (46) we take the param-etrically tted diffusion function for which the weighted leastsquares estimates Olsquo D 0071258 and Oƒ D 072957 are obtainedfrom the interest rate data

To perform the GLR test combined with the local lin-ear approach we adopt the empirical formula for band-width For simplicity three different scales of bandwidth hj

D105jƒ1h01 j D 11213 are also considered to evaluate simul-taneously the impact of bandwidth choice on the test Thesebandwidths are roughly viewed as ldquosmallerrdquo ldquojust rightrdquo andldquobiggerrdquo In particular we use

h0 D 4 std48Xt11 Xt2

1 1Xtn95 nƒ2=91 (47)

where 8Xti1 i D 11 1 n9 denotes the simulated sample path

and the rate nƒ2=9 was shown by Fan et al (2001) to be theasymptotically optimal rate of bandwidth such that the GLRtest can detect alternatives converging to the null at the optimalrate for nonparametric testing To expedite the computationwe evaluate the local linear ts at 200 grid points distributedevenly on the ranges of the simulated samples and then takelinear interpolation to obtain the estimates at all of the 2400data points The results of the quantiles are summarized in

Table 3 100(1 - ) th Percentiles of Test Statistics lsaquon( h j ) j D 12 3Under Models H( `)

0 ` D 12 34

Percentile

Null Test statistic D 001 D 0025 D 005 D 010

H (1)0 lsaquon (h1) 12706 10904 8507 6602

lsaquon (h2) 11903 10508 8500 6504lsaquon (h3) 12107 9400 7801 6500

H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion

Figure 5 Comparison of Volatility Curves Under Null Hypotheses(44)ndash(46) The dashed line is c0 the solid line is c1X 5

t the dotted lineis c2X15

t the dash-dotted line is lsquo X ƒ

t The constants are c0 D 01272c1 D 05596 c2 D 90114 lsquo D 071258 and ƒ D 72957

Table 3 As can be seen the empirical critical values of lsaquon4hj5

do not depend sensitively on the true parameter values of thenull models although they should depend on the choice ofbandwidth and signi cance level

Second to examine the power of the GLR test statisticslsaquon4hj51 j D 1121 3 we consider testing for CIR model (44)against the nonparametric alternative (1) We evaluate thepower of the tests at a nominal level 5 based on 400datasets simulated from the speci c models H

4`5

0 1 ` D 11 21314Figure 5 depicts how far apart the volatility functions 01272090114X105

t and 0071258X 072957t deviate from the hypothetical

volatility function 005596X 05t Thus the GLR tests as shown in

Table 4 are powerful in detecting slight departures from thenull in addition to keeping the right size

44 Testing Commonly Used Short Rate Models

The Treasury bill (T-bill) dataset for our study consistsof 2400 weekly observations covering the period January 81954ndashDecember 31 1999 US Treasury bill secondary mar-ket rates are the averages of the bid rates quoted on a bankdiscount basis by a sample of primary dealers who report tothe Federal Reserve Bank of New York The rates reported arebased on quotes at the of cial close of the US governmentsecurities market for each business day Figure 6 shows theestimated drift and volatility curves based on a local lin-ear approach The estimated drift function exhibits strongnonlinearities at the right boundary region also the estimatedvolatility curve looks like a CIR VR form

Table 4 Simulated Rejection Rates Against Models H ( `)0 ` D 1 23 4

Rejection rate

Test statistic H(1)0 H(2)

0 H(3)0 H(4)

0

lsaquon(h1) 06175 00525 100000 09525lsaquon(h2) 06125 00450 100000 09575lsaquon(h3) 06300 00375 100000 09475


0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)

Figure 6 Estimated Drift (a) and Volatility (b) of Short Rate Estimated drift and volatility functions based on a local linear approach calculatedusing weekly data January 8 1954ndashDecember 31 1999 The bandwidths are h j D 15j - 1h0 j D 12 3 where h0 D 01984 is calculated fromformula (47) (mdash h1 - - - h2 cent cent cent cent h3)

We rst address the issue raised by Chapman and Pearson(2000) of whether the short-rate drift is actually nonlinearwhich becomes tantamount to testing model (37) versusmodel (1) Due to the presence of a larger magnitude of noisedistinguishing the pattern of the signal component from therandom-error component becomes very challenging DespiteChapman and Pearsonrsquos full coverage and great efforts inexplaining the seemingly nonlinear drift function there arestill no convincing procedures for formally justifying whetherthe observed deviation from linearity indicates signi cantdeparture from model (37) With the aid of the powerful GLRtest we can compute the associated p value based on a regres-sion bootstrap method for approximating the empirical nulldistributions of the GLR test statistics A complete procedurecomprises the following steps

Step 1 For the original T-bill data 8Xti1 i D 11 1 n9

denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1

iD1 9 obtain leastsquares estimates 4 O1 Osbquo5 and RSS0

D Pnƒ1iD1 8Yti

=atilde ƒ O ƒOsbquoXti

92 Use a local linear approach with bandwidth h toobtain OŒ4Xti

51 Olsquo 4Xti5 and RSS14h5 D Pnƒ1

iD1 8Yti=atilde ƒ OŒ4Xti

592Compute the observed value of the test statistic lsaquon3 obs4h5 Dnƒ1

2 log RSS0

RSS14h5 Get the standardized residuals Oeti

D Ytiƒ OŒ4Xti

5atilde

Olsquo 4Xti5atilde1=2

Step 2 Obtain the bootstrap residuals 8Oe4b5ti

1 i D 11 1n ƒ 19 via sampling randomly and with replacement from8Oetj

1 j D 11 1 n ƒ 19 and de ne the bootstrap responses

Y4b5ti

D 4 O C OsbquoXti5atildeC Olsquo 4Xti

5atilde1=2 Oe4b5ti

Use the bootstrap sample84Xti

1 Y4b5ti

5nƒ1iD1 9 to get the bootstrap test statistic lsaquo4b5

n 4h5

Table 5 Testing Linear Drift Function for T-Bill Short Rate

Test statistic Bootstrap p value Rejection rate

lsaquon (h1) 0141 006lsaquon (h2) 0104 011lsaquon (h3) 0092 009

Step 3 Repeat step 2 many times (indexed by super-scripts b D 11 111000 say) and compute the proportion oftimes that 8lsaquo4b5

n 4h59 exceeds lsaquon3 obs4h5 This yields the p valueof the observed GLR test statistic

Using this bootstrap procedure we obtain the p value of theGLR test for model (37) against model (1) shown in the sec-ond column of Table 5 with three different bandwidths 8hj9

as in Section 43 Thus there is no strong evidence against thenull hypothesis of linear drift Our proposed test provides for-mal proofs to reinforce the ndings of Chapman and Pearson(2000)

We also apply similar procedures for assessing the ade-quacy of some previously established hypotheses regardingthe variance nature in particular competing forms (2)ndash(6)for volatility functions The associated p values are displayedin Table 6 Surprisingly strong evidence indicates that theseassumptions on the volatility function cannot be validated byour GLR tests This is consistent with the results reported byGallant and Long (1997)

To calibrate the GLR testrsquos ability to correctly reject nullhypotheses we simulate 100 datasets each containing 2400observations from the CIR squared root model (44) Basedon the level 5 critical values of the foregoing bootstrappednull distributions a decision on whether or not to reject the

Table 6 Testing Forms of Volatility Function for T-Bill Short Rate

Test statistic GBM VAS CIR SR CIR VR CKLS

Bootstrap p valuelsaquon (h1) 0000 0000 0000 0000 0000lsaquon (h2) 0000 0000 0000 0000 0000lsaquon (h3) 0000 0000 0002 0000 0015

Rejection ratelsaquon (h1) 1 1 008 1 008lsaquon (h2) 1 1 004 1 006lsaquon (h3) 1 1 004 1 003


4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)

Figure 7 Estimated Drift (a) and Volatility (b) of the SampP 500 Index Estimated drift and volatility functions based on a local linear approachcalculated using daily data January 4 1971ndashApril 8 1998 The bandwidths are h j D 15j - 1h0 j D 123 where h0 D 4019 is calculated fromformula (47) (mdash h1 - - - h2 cent cent cent cent h3)

null hypothesis of linear drift can be made with respect toeach sample The proportion of rejections across 100 samplesis presented in the third column of Table 5 Similar resultsconcerning volatility functions are listed in Table 6 Thereforeboth Table 5 and Table 6 strengthen the assertion that ourbootstrap procedures are powerful in correctly accepting orrejecting the null hypotheses

45 Testing Models for Standard amp Poor 500 Index

In addition to the interest rate application we investigatethe signi cance of structural shifts of Standard amp Poor (SampP)500 data from previously studied models This dataset con-tains 6890 daily observations on the SampP composite priceindex for January 4 1971ndashApril 8 1998 Following the con-ventional practice in nance research we rst take the loga-rithmic transformation of the price index The estimated driftand volatility based on a local linear approach are displayed inFigure 7 and the associated bootstrap p values are presentedin Tables 7 and 8 Clearly there is no strong evidence againstthe hypothesis on the linear drift For the volatility functionour test suggests that the GBM and CIR VR models do not tthe logarithm of the index Furthermore our test also indicatesthat the VAS CIR SR and CKLS models cannot be validatedbased on the test statistics lsaquon4hj5 for j D 11 21 3 together

Table 7 Testing Linear Drift Function for Logarithmsof the SampP 500 Index

Test statistic Bootstrap p value

lsaquon (h1) 0814lsaquon (h2) 0554lsaquon (h3) 0582

5 CONCLUSION

Stanton (1997) proposed drift and diffusion estimatorsbased on a higher-order approximation scheme and a non-parametric kernel estimation He claimed (p 1982) that ldquothehigher the order of the approximation the faster it will con-verge to the true drift and diffusion of the process givenin equation (1) as we observe the variable Xt at ner and ner time intervals Eventually if we can sample arbitrar-ily often higher order approximations must outperform lowerorder approximationsrdquo and reiterated (p 1983) that ldquoeven withdaily or weekly data we can achieve gains by using higherorder approximations compared with the traditional rst orderdiscretizationsrdquo Actually these claims are correct but some-what misleading They ignore the variance in ation in sta-tistical estimation due to higher-order approximation Thisvariance in ation phenomenon is not an artifact of nonpara-metric tting it also applies to parametric models With thetool of asymptotic analysis we show that higher-order approx-imations bene t from reducing the numerical approximationerror within asymptotic bias a statement correctly made byStanton (1997) but nevertheless they are penalized by anasymptotic variance escalating nearly exponentially with theorder of the approximations This shadows the higher-orderapproximation scheme This phenomenon can be accountedfor by the stochastic nature of the Taylor series expansion

Table 8 Testing Forms of Volatility Function for Logarithmsof the SampP 500 Index

Bootstrap p value


lsaquon(h1) 0 0000 0000 0 0031lsaquon(h2) 0 0295 0004 0 0418lsaquon(h3) 0 0491 0204 0 0576


in (8) accumulated with the linear combination of higher-orderdifferences (11) Caution should be taken when using higher-order formulas This bias and variance trade-off phenomenonyields general and insightful understandings of the estimatorsIt also provides useful guidance for determining an optimalstrategy for order of approximation as well as proposing pos-sibly more ef cient estimators

Encouragingly by using the local linear approach spu-rious ldquoboundary effectsrdquo from Stantonrsquos kernel estimationare ameliorated especially for estimating diffusion functionsThis local linear estimation approach could also be incor-porated with the GLR statistic to test a wide variety ofparametric time-homogeneous diffusion models and also toformally check nonlinearity of the short-rate drift Our simu-lation shows that our procedures are indeed powerful and havenearly the correct size of the test The procedures are usefulfor verifying various models in nance and economics

APPENDIX PROOF OF THEOREMS

A1 Proof of Theorem 1

Using the matrix notation the system of equations in (9) can bewritten as Ax D b where

A D

2

66664

1 2 cent cent cent j cent cent cent k

1 22 cent cent cent j2 cent cent cent k2

000000

0 0 0000

0 0 0000

1 2k cent cent cent jk cent cent cent kk

3

77775and b D

2

66664

1

0000

0

3

777750

Thus the solution x D 4x11 1 xk5T is uniquely determined by

x D mdashAmdashƒ1Auuml b1 (A1)

where A uuml and mdashAmdash denote the adjoint matrix and the determinant ofthe matrix A that is x is the rst column of Aƒ1 Applying theproperty of the Vandermonde matrix we see that the determinant ofthe matrix A is

mdashAmdash D 2 3 cent cent cent k

shyshyshyshyshyshyshyshyshyshy

1 1 cent cent cent 1 cent cent cent 1


000000

0 0 0000

0 0 0000

1 2kƒ1 cent cent cent jkƒ1 cent cent cent kkƒ1


D kWY Y

1microl1ltl2microk

4l2 ƒ l151

and that the jth entry in the rst column of matrix A uuml is

A uuml 4j1 15 D 4ƒ15jC1 4kW52

j2

Y Y

1microl1 ltl2microkl1 6Dj1 l2 6Dj

4l2 ƒ l150

Hence in (A1) the solutions xj1 j D 11 1 k can be simpli ed as

xj D 4ƒ15jC1 4kW52

j2

Q Q1microl1 ltl2microkl1 6Dj1 l2 6Dj

4l2 ƒ l15

kWQ Q1microl1ltl2microk4l2 ƒ l15

D 4ƒ15jC1kWj24j ƒ15W 4k ƒ j5W

D 4ƒ15jC1

sup3k

j

acutej0

This proves the rst statement We now prove the second statementThe proof is based on the recursion relation which we now derive

For any 1 micro j micro k1iexcl

k

j

centj D

iexclkƒ1jƒ1

centk which when applied to the rst

statement results in

kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0

Using the binomial expansion for the factor 4j C 15kƒ1 and exchang-ing the order of summations we obtain

kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0

This together with (9) yields

kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0

The conclusion follows from the foregoing inductive formula


Before we derive the asymptotic variances in Theorem 2 we needthe following lemma

Lemma A1 Assume the same regularity conditions on 8Xt9 asin Theorem 2 For each xed x0 as atilde 0 it holds that

E84XtCatildeƒXt5mdashXt

D x09 D Œ4x05atildeCO4atilde251 (A2)

E84XtCatildeƒXt5

2mdashXtD x09 Dlsquo 24x05atildeC O4atilde251 (A3)

E84XtCatilde ƒXt53mdashXt D x09

D 3lsquo 24x058Œ4x05 C2ƒ14lsquo 2504x059atilde2 CO4atilde351 (A4)

E84XtCatildeƒXt5

4mdashXtD x09 D 3lsquo 44x05atilde2 CO4atilde351 (A5)

E84XtCatildeƒ Xt5Œ4XtCatilde5mdashXt

D x09

D 8Œ24x05C Œ04x05lsquo24x059atildeC O4atilde251 (A6)

E84XtCatildeƒ Xt5

2lsquo 24XtCatilde5mdashXtD x09 Dlsquo 44x05atildeC O4atilde251 (A7)

and

E84XtCatilde ƒXt53Œ4XtCatilde5mdashXt D x09 D O4atilde250 (A8)

Proof To show results (A2)ndash(A8) we choose the correspond-ing functions f14x1 t5 D 4x ƒ Xt51f24x1 t5 D 4x ƒ Xt5

21 f34x1 t5 D4x ƒ Xt5

31 f44x1 t5 D 4x ƒ Xt541 f54x1 t5 D 4x ƒ Xt5Œ4x51f64x1 t5 D

4xƒXt52lsquo 24x5 and f74x1 t5 D 4xƒXt5

3Œ4x5 Straightforward calcu-lations applying the differential operator not de ned by (7) give the


following relations

notf14x1 t5 D Œ4x51

not2f14x1 t5 D Œ04x5Œ4x5C 2ƒ1Œ004x5lsquo 24x51

notf24x1 t5 D 24x ƒXt5Œ4x5 Clsquo 24x51

not2f24x1 t5 D 82Œ4x5C 24x ƒXt5Œ04x5 C 4lsquo 2504x59Œ4x5

C2ƒ184Œ04x5 C24x ƒ Xt5Œ004x5C 4lsquo 25004x59lsquo 24x53

notf34x1 t5 D 34x ƒXt52Œ4x5 C34x ƒ Xt5lsquo

24x51

not2f34x1 t5 D 864x ƒXt5Œ4x5 C34x ƒ Xt52Œ04x5 C3lsquo 24x5

C 34x ƒXt54lsquo2504x59Œ4x5 C2ƒ1lsquo 24x5

86Œ4x5 C124x ƒXt5Œ04x5 C34x ƒ Xt5

2Œ004x5

C64lsquo 2504x5 C34x ƒXt54lsquo25004x591

notf44x1 t5 D 44x ƒXt53Œ4x5 C64x ƒ Xt5

2lsquo 24x51

not2f44x1 t5 D 8124x ƒXt52Œ4x5C 44x ƒXt5

3Œ04x5

C 124x ƒXt5lsquo24x5C 64x ƒXt5

24lsquo 2504x59Œ4x5

C2ƒ18244x ƒXt5Œ4x5 C244x ƒXt52Œ04x5

C44x ƒ Xt53Œ004x5 C12lsquo 24x5

C244x ƒXt54lsquo2504x5

C64x ƒXt524lsquo 25004x59lsquo 24x53

and

notf54x1 t5 D 8Œ4x5 C 4x ƒ Xt5Œ04x59Œ4x5

C2ƒ18Œ04x5 CŒ04x5 C 4x ƒXt5Œ004x59lsquo 24x51

notf64x1 t5 D 824x ƒ Xt5lsquo24x5 C 4x ƒXt5


C2ƒ182lsquo 24x5C 44x ƒXt54lsquo2504x5

C 4x ƒXt524lsquo 25004x59lsquo 24x51

notf74x1 t5 D 834x ƒXt52Œ4x5 C 4x ƒ Xt5

3Œ04x59Œ4x5

C2ƒ1864x ƒ Xt5Œ4x5 C64x ƒ Xt52Œ04x5

C 4x ƒ Xt53Œ004x59lsquo 24x50

The proof of Lemma A1 is completed by using a Taylor seriesexpansion in (8)

To show Theorem 2 we start by considering the conditional vari-ance of the drift estimator Write t D t0 C àtilde for any xed index` D 11 1 nƒk throughout the following derivations From the def-initions in (16) and (20) we have

lsquo 211 atilde4x05

D atildeƒ2

X

1microjmicrok

a2k1 jvar84XtCjatilde

ƒ Xt5mdashXtD x09 C2

X X

1microiltjmicrok

ak1 iak1 j

cov4XtCiatilde ƒx01 XtCjatilde ƒx0mdashXt D x05

0 (A9)

For j para 1 (A2) and (A3) imply that

var84XtCjatilde ƒXt5mdashXt D x09

D E84XtCjatildeƒ Xt5

2mdashXtD x09 ƒ 6E84XtCjatilde

ƒ Xt5mdashXtD x0972

Dlsquo 24x05jatildeC O4atilde250 (A10)

For 1 micro i lt j micro k combining the Markov property of 8Xt1 t para 09 with(A2) (A3) and (A6) we have

E84XtCiatilde ƒ x054XtCjatilde ƒx05mdashXt D x09

D E64XtCiatildeƒx05E84XtCjatilde

ƒx05mdashXtCiatilde9mdashXtD x07

4Markovian property5

D E64XtCiatildeƒx0584XtCiatilde

ƒ x05C Œ4XtCiatilde54j ƒ i5atilde

C O4atilde259mdashXt D x07

D E84XtCiatildeƒx052 C 4XtCiatilde

ƒx05Œ4XtCiatilde54j ƒ i5atilde

C 4XtCiatildeƒx05O4atilde25mdashXt

D x09

Dlsquo 24x05iatildeCO4atilde250 (A11)

We also obtain according to (A2) that

E84XtCiatildeƒx05mdashXt

D x09E84XtCjatildeƒx05mdashXt

D x09

D 8Œ4x05iatilde CO4atilde2598Œ4x05jatildeC O4atilde259 D O4atilde250 (A12)

The expression (21) follows readily from the combination of (A9)(A10) (A11) and (A12)

We now consider the conditional variance of the squared diffusionestimator In the same vein from equations (17) and (20) we have


D atildeƒ2

X

1microjmicrok

a2k1 jvar84XtCjatilde ƒXt5

2mdashXt D x09 C2X X

1microiltjmicrok

ak1 iak1 j

cov844XtCiatilde ƒ x0521 4XtCjatilde ƒ x0525mdashXt D x09

0 (A13)


var84XtCjatildeƒ Xt5

2mdashXtD x09

D E84XtCjatildeƒXt5


ƒXt52mdashXt

D x0972

D 2lsquo 44x054jatilde52 CO4atilde350 (A14)


E84XtCiatildeƒx0524XtCjatilde

ƒx052mdashXtD x09


ƒ x052mdashXtCiatilde9mdashXt

D x07



ƒx052 C 424XtCiatildeƒx05Œ4XtCiatilde5

Clsquo 24XtCiatilde554j ƒ i5atildeCO4atilde359mdashXtD x07

D E84XtCiatildeƒx05

4 C24XtCiatildeƒx05

3Œ4XtCiatilde54j ƒ i5atilde

C 4XtCiatildeƒx052lsquo 24XtCiatilde54j ƒ i5atildeCO4atilde35mdashXt

D x09

D 3lsquo 44x054iatilde52 CO4atilde35 Clsquo 44x054iatilde54j ƒ i5atildeCO4atilde35

D 2lsquo 44x054iatilde52 Clsquo 44x05ijatilde2 C O4atilde350 (A15)

We also obtain from (A3) that



D x09

D 8lsquo 24x05iatildeCO4atilde2598lsquo 24x05jatilde CO4atilde259

Dlsquo 44x05ijatilde2 CO4atilde350 (A16)


The equality (22) follows directly from the combination of (A13)(A14) (A15) and (A16)


The proofs in this section are based on some combinatorial rela-tions Let ƒ D limnˆ8

PnkD1 kƒ1 ƒ log4n59 ordm 0577216 be the Eulerrsquos

constant and ndash4z5 D acirc 04z5=acirc4z5 be the Psi function where acirc4z5 DR ˆ0 uzƒ1eƒu du for z gt 0 First we consider part (a) With the aid of

Mathematica we obtain the identities

kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W

84kC 15W92C 4kC1acirc43=2 C k5

4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W

84kC 15W92C 22kC3acirc43=2 Ck5

4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k

acuteCndash4k C150 (A19)

Consequently putting ak1 j D 4ƒ15jC1iexcl

k

j

cent=j and simplifying the right

sides of (A17) and (A18) we have

kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)

Applying (A19) and the identity ndash4n5 D Pnƒ1jD1 jƒ1 ƒƒ which holds

for any integer n para 2 we deduce

XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)

Hence (21) (A9) and (A22) together with inequalities (A20) and(A21) ensure that V14k5 has a lower bound

k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)

The conclusion follows from applying Stirlingrsquos formula nW D42 n51=24n=e5n exp8ˆ=412n59 for some 0 lt ˆ lt 1 to the rst domi-nating terms of (A23) and (A24)

Next we consider part (b) For k para 1 it follows directly that

kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)

Again with the aid of Mathematica we obtain the identity that fork gt 1 and 2 micro j micro k

jƒ1X

iD1

4ƒ15iC1

sup3k

i

acutei D 4ƒ15jjacirc 4k5

acirc 4j5acirc4kƒ j C15ƒ 4ƒ15jacirc4k ƒ15

acirc 4j5acirc4k ƒ j51 (A26)

which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)

The conclusion (b) follows from (22) (A13) (A25) (A27) andStirlingrsquos formula


It suf ces to consider only Part (1) similar treatmentsapply to Part (2) We denote a generic constant by C LetX D 44X uuml

iatildeƒ x05

j5iD11 1nƒk3jD01 1q1 y D 4Y uumlatilde 1 1 Y uuml

4nƒk5atilde5T 1 W Ddiag8Kh4X uuml

iatildeƒ x051 i D 11 1 n ƒ k9 and m D 4E4Y uuml

atildemdashX uuml

atilde51 1E4Y uuml

4nƒk5atildemdashX uuml

4nƒk5atilde55T Denote Sn

D XT WX and TnD XT Wy Then by

(19) we can write OAcirc4x05 D Sƒ1n Tn and thus

OAcirc4x05 ƒAcirc4x05 D Sƒ1n XT W8mƒ XAcirc4x059 CSƒ1

n XT W4y ƒm51

sup2 b C t0

We rst establish the asymptotic behavior of the bias vector b D4b01 b11 1 bq5T Set Zn1 `

D Kh4X uumlàtilde

ƒ x054Xuumlàtilde

ƒ x05j and Sn1 jDPnƒk

`D1 Zn1 ` then SnD 4Sn1 iCjƒ25i1 jD11 1 qC1 A Taylor expansion leads

to the expression

b D Sƒ1n 8sbquoqC14Sn1 qC11 1 Sn1 2qC15T CsbquoqC24Sn1 qC21 1 Sn1 2qC25T

C oP 4nhqC25H191 (A28)

with a 4q C 15 4q C 15 matrix H D diag411 h1 1 hq5 and a 4q C15 1 vector 1 D 411 115T To derive the asymptotic form of bwe need only apply the expression

Sn1 j D nhj8p4x05Œj Chp04x05ŒjC1 C OP 4an591 (A29)

where an D h2 C 4nh5ƒ1=2 Equation (A29) can be obtained via pro-cedures similar to those of Fan and Gijbels (1996 thm 31) How-ever to verify the term OP 4an5 in our current context we need todo the variance calculation for Sn1j which is different than that ofFan and Gijbels To this end using the assumption on the transitiondensity we rst obtain

mdashcov4Zn1 11Zn1 `C15mdash micro Ch2j81C o41590 (A30)

Recall for a bounded real-valued Borel measurable function g thetransition probability operator acute ` of the process 8X uuml

iatilde1 i D 11 1n ƒ k9 is de ned by

4acute `g54x5 D E8g4X uuml4`C15atilde5mdashX uuml

atildeD x90

By the G2 condition of Rosenblatt (1970) there exists a constant 2 40115 for acute such that for g4cent5 D Kh4centƒ x054cent ƒx05j ƒE8Kh4centƒx054centƒ x05j9 we have

mdashcov4Zn1 11Zn1 `C15mdash D mdashE8g4X uumlatilde5acute `g4X uuml

atilde59mdash

micro ˜g4X uumlatilde5˜2ãcute `g4X uuml

atilde5˜2

micro ˜g4X uumlatilde5˜2

2mdashacute `mdash2

micro Ch2jƒ1`1 (A31)


where mdashacute `mdash2 D supg2g 6DE4g5ãcute `gƒE4g5˜2

˜gƒE4g5˜2 and E stands for expectation

with respect to the stationary density p4cent5 Now select an integer dn

so that dn ˆ and dnh 0 (eg dn D hƒ1=2) then (A30) and(A31) give

nƒkƒ1X

`D1

mdashcov4Zn1 11Zn1 `C15mdash Dsup3 dnX

`D1

Cnƒkƒ1X

`DdnC1

acutemdashcov4Zn1 11Zn1 `C15mdash

D o4h2jƒ150 (A32)

This along with the stationarity assumption yield

var4Sn1 j5 D 4nƒ k5var4Zn1 15 C 2nƒkƒ1X

`D1

4nƒ k ƒ `5cov4Zn1 11 Zn1 `C15

D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk

acutecov4Zn1 11 Zn1 `C15

para1

from whence (A29) is obtainedThe asymptotic bias expression in (23) then results from the

decomposition

OŒ11atilde4x05 ƒŒ4x05 Dcopy

OŒ11atilde4x05ƒ E4Y uumliatilde

mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590

On the right side we see that OŒ11atilde4x05 ƒ E4Y uumliatilde

mdashX uumliatilde

D x05 D b0by (13) we see that E4Y uuml

iatildemdashX uuml

iatildeD x05 ƒ Œ4x05 D 4ƒ15kC1

notkC1f1 4x0 1t0Ciatilde5

4kC15atildek CO4atildekC15 This completes the proof of (23)

Next consider the asymptotic variance of OŒ11atilde4x05 By (A29)t D pƒ14x05Hƒ1Sƒ1u81 C oP 4159 where u D nƒ1Hƒ1XT W4y ƒ m5For any constant vector c de ne

QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde

mdashX uumliatilde59Ch4X uuml

iatildeƒx051

where C4x5 D PpjD0 cj xjK4x5 and Ch4x5 D C4x=h5=h Set vn1`

D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml

iatilde59Ch4X uumliatilde

ƒx05 Then direct calculations give that

var4vn115 D 4hatilde5ƒ1lsquo 21 4x03 k5p4x05cT S uuml c81 Co41590 (A33)

Similar procedures to those used in (A30)ndash(A32) lead to

nƒkƒ1X

`D1

mdashcov4vn111 vn1`C15mdash micro dnh2atildeƒ2 C hatildeƒ2nƒkƒ1X

`DdnC1

` D o4hatildeƒ251

which combined with (A33) and the assumption on h imply thatvar4u5 D 4nhatilde5ƒ1lsquo 2

1 4x03 k5p4x05Suuml 81 Co4159 and therefore (25)

[Received November 2000 Revised February 2002]

REFERENCES

Aiumlt-Sahalia Y (1996) ldquoNonparametric Pricing of Interest Rate DerivativeSecuritiesrdquo Econometrica 64 527ndash560

Allen D M (1974) ldquoThe Relationship Between Variable and Data Augmen-tation and a Method of Predictionrdquo Technometrics 16 125ndash127

Ar M (1995) ldquoNon-Parametric Drift Estimation from Ergodic SamplesrdquoJournal of Nonparametric Statistics 5 381ndash389

(1998) ldquoNon-Parametric Variance Estimation from Ergodic Sam-plesrdquo Scandinavia Journal of Statistics 25 225ndash234

Banon G (1978) ldquoNonparametric Identi cation for Diffusion ProcessesrdquoSIAM Journal of Control and Optimization 16 380ndash395

Banon G and Nguyen H T (1981) ldquoRecursive Estimation in DiffusionModelsrdquo SIAM Journal of Control and Optimization 19 676ndash685

Chan K C Karolyi A G Longstaff F A and Sanders A B (1992) ldquoAnEmpirical Comparison of Alternative Models of the Short-Term InterestRaterdquo Journal of Finance 47 1209ndash1227

Chapman D A and Pearson N D (2000) ldquoIs the Short Rate Drift ActuallyNonlinearrdquo Journal of Finance 55 355ndash388

Cox J C Ingersoll J E and Ross S A (1980) ldquoAn Analysis of VariableRate Loan Contractsrdquo Journal of Finance 35 389ndash403

(1985) ldquoA Theory of the Term Structure of Interest Ratesrdquo Econo-metrica 53 385ndash407

Denker M and Keller G (1983) ldquoOn U Statistics and VMisesrsquos Statis-tics for Weakly Dependent Processesrdquo Z Wahrscheinlichkeitstheorie verwGebiete 64 505ndash522

Fan J (1992) ldquoDesign-Adaptive Nonparametric Regressionrdquo Journal of theAmerican Statistical Association 87 998ndash1004

Fan J and Gijbels I (1995) ldquoData-Driven Bandwidth Selection in LocalPolynomial Fitting Variable Bandwidth and Spatial Adaptationrdquo Journalof the Royal Statistical Society Ser B 57 371ndash394

(1996) Local Polynomial Modeling and Its Applications LondonChapman and Hall

Fan J and Yao Q W (1998) ldquoEf cient Estimation of Conditional VarianceFunctions in Stochastic Regressionrdquo Biometrika 85 645ndash660

Fan J Zhang C M and Zhang J (2001) ldquoGeneralized Likelihood RatioStatistics and Wilks Phenomenonrdquo The Annals of Statistics 29 153ndash193

Gallant A R and Long J R (1997) ldquoEstimating Stochastic Differ-ential Equations Ef ciently by Minimum Chi-Squaredrdquo Biometrika 84125ndash141

Hansen L P (1982) ldquoLarge Sample Properties of Generalized Method ofMoments Estimatorsrdquo Econometrica 50 1029ndash1054

Jiang G J and Knight J L (1997) ldquoA Nonparametric Approach to theEstimation of Diffusion Processes With an Application to a Short-TermInterest Rate Modelrdquo Econometric Theory 13 615ndash645

Kloeden P E and Platen E (1992) Numerical Solution of Stochastic Dif-ferential Equations Berlin Springer-Verlag

Kloeden P E Platen E Schurz H and Soslashrensen M (1996) ldquoOn Effects ofDiscretization on Estimators of Drift Parameters for Diffusion ProcessesrdquoJournal of Applied Probability 33 1061ndash1076

Oslashksendal B (1985) Stochastic Differential Equations An Introduction WithApplications New York Springer-Verlag

Osborne M F M (1959) ldquoBrownian Motion in the Stock Marketrdquo Opera-tions Research 7 145ndash173

Pham D T (1981) ldquoNonparametric Estimation of the Drift Coef cient inthe Diffusion Equationrdquo Mathematische Operationsforschung und StatistikSeries Statistics 12 61ndash73

Prakasa Rao B L S (1985) ldquoEstimation of the Drift for Diffusion ProcessrdquoStatistics 16 263ndash275

Rosenblatt M (1970) ldquoDensity Estimates and Markov Sequencesrdquo in Non-parametric Techniques in Statistical Inferences ed M Puri LondonCambridge University Press pp 199ndash210

(1971) Markov Processes Structure and Asymptotic BehaviorNew York Springer-Verlag

Ruppert D (1997) ldquoEmpirical-Bias Bandwidths for Local Polynomial Non-parametric Regression and Density Estimationrdquo Journal of the AmericanStatistical Association 92 1049ndash1062

Ruppert D Sheather S J and Wand M P (1995) ldquoAn Effective BandwidthSelector for Local Least Squares Regressionrdquo Journal of the AmericanStatistical Association 90 1257ndash1270

Stanton R (1997) ldquoA Nonparametric Model of Term Structure Dynamicsand the Market Price of Interest Rate Riskrdquo Journal of Finance 52 1973ndash2002

Stone M (1974) ldquoCross-Validatory Choice and Assessment of StatisticalPredictionsrdquo (with discussion) Journal of the Royal Statistical Society SerB 36 111ndash147

Vasicek O A (1977) ldquoAn Equilibrium Characterization of the Term Struc-turerdquo Journal of Financial Economics 5 177ndash188

Wahba G (1977) ldquoA Survey of Some Smoothing Problems and theMethod of Generalized Cross-validation for Solving themrdquo in Appli-cations of Statistics ed P R Krishnaiah Amsterdam North-Hollandpp 507ndash523

Wong E (1971) Stochastic Processes in Information and Dynamical Sys-tems New York McGraw-Hill


Stantonrsquos approach has some problems Chapman andPearson (2000) studied the nite-sample properties of Stan-tonrsquos estimator By applying his procedure to simulated sam-ple paths of a squared-root diffusion they found that Stantonrsquosestimator produces spurious nonlinearity when the underly-ing drift function is truly linear Chapman and Pearson nicelyconcluded that the ldquomean reversionrdquo and small sample at theboundary create arti cial patterns of nonlinearity displayednoticeably near the boundary regions Meanwhile two sensi-ble questions naturally arise (1) Do higher-order approxima-tions outperform their lower-order counterparts and (2) Arethere any reasonable and formal procedures that help deter-mine whether the observed nonlinearity in the drift is real ordue to chance variation

In an attempt to answer the rst question on the order ofapproximations we derive explicitly the formulas of a higher-order approximation scheme that generalizes Stantonrsquos ideaWe then compute explicitly the asymptotic variances of non-parametric estimators based on higher-order approximationsA striking result from our asymptotic study is that higher-order approximations will reduce the numerical approximationerrors in asymptotic biases but escalate (nearly exponentially)the asymptotic variances This variance in ation phenomenonis not only an artifact of nonparametric ttingmdashit also appliesto parametric modeling The issue of a trade-off between biasreduction and variance increment is made explicit in Theo-rem 4 (Sec 2)

Stantonrsquos work raises some other interesting issues Is thedrift function in the short-term rate model nonlinear Or moregenerally does a parametric model t a given set of economicor nancial data An example of this is whether models (2)ndash(6) adequately t short-term rate data Chapman and Pearson(2000) suggested that the nonlinearity of the drift functionmight be spurious Their method is based on simulated datafrom diffusion models with a linear drift function and eval-uates whether the estimated drift looks linear This graphicalprocedure is useful but informal To set up formal statisti-cal tests an alternative hypothesis (model) is needed Becausewe usually do not have strong preference for alternative com-peting models the nonparametric model (1) serves as a nat-ural candidate The hypothesis testing problem becomes oneof testing a parametric (or semiparametric) null hypothesisagainst a nonparametric alternative The latter half of this arti-cle is thus devoted to model validation There we extend theidea of the generalized likelihood ratio (GLR) statistic devel-oped by Fan Zhang and Zhang (2001) and apply it to thetime-homogeneous diffusion models Our simulation resultsshow that GLR tests are indeed powerful and give the correcttest size They provide useful tools for assessing the validityof various models in economics and nance

The remainder of the article is organized as follows InSection 2 we discuss the distributional properties of Stan-tonrsquos drift and diffusion estimators and also derive explicitexpressions of asymptotic biases and variances for higher-order approximations To justify our analyses on empiricalgrounds we report on simulations in Section 3 In Section 4we propose model validation methods using the GLR testbased on the rst-order approximation combined with the locallinear estimation Simulations of the GLR test and real data

applications are also demonstrated In Section 5 we brie ysummarize our conclusions We collect outlines of the proofsin the Appendix

2 HIGHER-ORDER APPROXIMATIONS

This section begins with a description of Stantonrsquosapproach Although his initial construction is based solely onthe rst- second- and third-order approximations we canbuild with some extra effort a more general framework thatgives us the exibility to examine the impact of higher-orderapproximations

21 Conditional Means and Conditional Variancesof Higher-Order Differences

Following Stantonrsquos notations under appropriate conditionson Œ4cent51lsquo 4cent5 and an arbitrary bivariate function f 4cent1 cent5 theconditional expectation Et8f 4XtCatilde1 t C atilde59 can be expressedin the form of a Taylor series expansion

Et8f 4XtCatilde1 t C atilde59 D f 4Xt1 t5C notf 4Xt1 t5atilde C 1

2not2f4Xt1 t5atilde2

C cent cent centC 1nWnot

nf 4Xt1 t5atilden C O4atildenC151

as time increment atilde 0 Here the symbol Et denotes the con-ditional expectation given Xt and the in nitesimal generatornot of the process 8Xt9 is de ned by

notf 4x1 t5 D limrsquot

E8f 4Xrsquo 1 rsquo5mdashXtD x9 ƒ f 4x1 t5

rsquo ƒ t

D iexclf 4x1 t5

iexcltC iexclf 4x1 t5

iexclxŒ4x5

C 1

2

iexcl2f 4x1 t5

iexcl2xlsquo 24x5 (7)

(see Oslashksendal 1985 for more details) Thus the rst-orderapproximation formula for the target function notf 4Xt1 t5 isgiven by

atildeƒ1Et8f 4XtCatilde1 t C atilde5 ƒ f 4Xt1 t59 D notf4Xt1 t5C O4atilde50 (8)

In particular setting f 4x1 t5 D x (or f4x1 t5 D x ƒ Xt) givesnotf 4x1 t5 D Œ4x5 likewise taking f4x1 t5 D 4x ƒ Xt5

2 impliesnotf 4x1 t5 D 24x ƒ Xt5Œ4x5 Clsquo 24x5 which at x D Xt equalslsquo 24Xt5 Hence these two special functions f 4cent1 cent5 can exactlyrecover Œ4Xt5 and lsquo 24Xt5 In such cases estimating theleft side of (8) by the N-W kernel method leads to Stan-tonrsquos estimates for Œ4x5 and lsquo 24x5 based on the rst-orderapproximation

Higher-order approximations (or differences) can beachieved through a linear combination of terms on the leftside of (8) More precisely for any xed integer k para 1 anysequence of constants 8ak1j1 j D 11 1 k9 and any discretelyobserved time steps jatilde for j D 11 1 k we consider the



atildeƒ1kX

jD1


D(

kX

jD1

jak1j

)

notf 4Xt1 t5C(

kX

jD1

j2ak1j

)not2f4Xt1 t5

2atilde

C cent cent centC(

kX

jD1

jkak1j

)notkf4Xt1 t5

kW atildekƒ1

C(

kX

jD1

jkC1ak1j

)notkC1f4Xt1 t5



atildeƒ1kX

jD1




system of equations

8gtgtgtgtlt

gtgtgtgt

PkjD1 jak1j

D 1Pk

jD1 j2ak1jD 0000

PkjD1 jkak1j

D 00

(9)



ak1jD 4ƒ15jC1

sup3k

j

acutej1 j D 11 1 k0 (10)


kX

jD1



atildeƒ1kX

jD1






atildeƒ1kX

jD1




para(13)


atildeƒ1kX

jD1


2




para(14)



lsquo 4Xt5 DAgrave

atildeƒ1kX

jD1


2

Aacute1=2

C O4atildek50 (15)




kX

jD1


acutesup2 4X uuml


i D 11 1 n ƒ k1 (16)



kX

jD1


acutesup2 4X uuml


i D 11 1 n ƒ k1 (17)


E4Y uumliatilde

mdashX uumliatilde


E4Z uumliatilde

mdashX uumliatilde


from (13) and (14)



Order k

1 2 3 4 5 6 7 8 9 10

V1(k) 1000 2050 4083 9025 18095 42068 105049 281065 798001 21364063V2(k) 1000 3000 8000 21066 61050 183040 570066 11837028 61076025 201527022


iatildemdashX uuml



mdashX uumliatilde



q m4q54x05=qW0





nƒkX

iD1

8Y uumliatilde

ƒ sbquo0ƒ 4X uuml

iatildeƒ x05sbquo1



iatildeƒ x05 (19)


iatildemdashX uuml



iatildemdashX uuml





iatildemdashX uuml

iatildeD x05 and


iatildemdashX uuml

iatildeD x050 (20)







k A11kAacutek










k2 ƒ 3k ƒ 2k4k C 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C 34k C 152

ordm 4k

1=2k5=21

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)




5k2 ƒ k ƒ 2k4k C 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C 6k C 54k C 152

ordm 5 4k

1=2k5=20


V24k5 Diexcl

2k

k

centƒ 4k C 15

k ƒ 1ordm 4k

1=2k3=20




21ŒjD

RujK4u5 du1 j



T and QcqD





1 4x03 k5 D lsquo 24x05V14k5



11 1 n ƒ k




C eT1 Sƒ1cq

Œ4qC154x05



eT1 Sƒ1Qcq





21 4x03 k5=p4x0581C o41590 (25)




C eT1 Sƒ1cq

4lsquo 254qC154x05



eT1 Sƒ1Qcq





22 4x03 k5=p4x0581 C o41590 (28)


iatildemdashX uuml



mdashX uumliatilde



4k C 15atildek

C Œ2

2h28Œ004x05 C 2Œ04x05p

04x05=p4x0591 (29)







3 SIMULATIONS







XtiC1ordm Xti


C 2ƒ1lsquoh8Xti


Clsquo 4Xti51=2

C ˜i

patilde91=2

C C 4Xti51=2

C

i˜i

patilde1 (31)



















XtiC1ordm Xti


˜i

patilde


iƒ 15atilde (34)





1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

Vj(k

)


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k

log

Vj(

k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)











1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

log

V j(k)

log

V j(k)

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k





4 MODEL VALIDATION



005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003


005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003






t dWt1 (35)



ƒt dWt1 (36)








XtiC1ƒXti



5atildeiClsquoXƒ

ti˜i


where ˜i







ti5 C log4˜2

i 51

i D 11 1 nƒ 10 (39)








5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)


ƒOŒ4Xti









t 0

Let bEtiD 8Yti


i and Y415ti



bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti


i 51 i D 11 1 n ƒ 10 (40)





1 Y415ti



lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)








PniD1







H415


H425


05t 1 (44)


H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES





































atildeƒ1kX

jD1


D(

kX

jD1

jak1j

)

notf 4Xt1 t5C(

kX

jD1

j2ak1j

)not2f4Xt1 t5

2atilde

C cent cent centC(

kX

jD1

jkak1j

)notkf4Xt1 t5

kW atildekƒ1

C(

kX

jD1

jkC1ak1j

)notkC1f4Xt1 t5



atildeƒ1kX

jD1




system of equations

8gtgtgtgtlt

gtgtgtgt

PkjD1 jak1j

D 1Pk

jD1 j2ak1jD 0000

PkjD1 jkak1j

D 00

(9)



ak1jD 4ƒ15jC1

sup3k

j

acutej1 j D 11 1 k0 (10)


kX

jD1



atildeƒ1kX

jD1






atildeƒ1kX

jD1




para(13)


atildeƒ1kX

jD1


2




para(14)



lsquo 4Xt5 DAgrave

atildeƒ1kX

jD1


2

Aacute1=2

C O4atildek50 (15)




kX

jD1


acutesup2 4X uuml


i D 11 1 n ƒ k1 (16)



kX

jD1


acutesup2 4X uuml


i D 11 1 n ƒ k1 (17)


E4Y uumliatilde

mdashX uumliatilde


E4Z uumliatilde

mdashX uumliatilde


from (13) and (14)



Order k

1 2 3 4 5 6 7 8 9 10

V1(k) 1000 2050 4083 9025 18095 42068 105049 281065 798001 21364063V2(k) 1000 3000 8000 21066 61050 183040 570066 11837028 61076025 201527022


iatildemdashX uuml



mdashX uumliatilde



q m4q54x05=qW0





nƒkX

iD1

8Y uumliatilde

ƒ sbquo0ƒ 4X uuml

iatildeƒ x05sbquo1



iatildeƒ x05 (19)


iatildemdashX uuml



iatildemdashX uuml





iatildemdashX uuml

iatildeD x05 and


iatildemdashX uuml

iatildeD x050 (20)







k A11kAacutek










k2 ƒ 3k ƒ 2k4k C 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C 34k C 152

ordm 4k

1=2k5=21

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)




5k2 ƒ k ƒ 2k4k C 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C 6k C 54k C 152

ordm 5 4k

1=2k5=20


V24k5 Diexcl

2k

k

centƒ 4k C 15

k ƒ 1ordm 4k

1=2k3=20




21ŒjD

RujK4u5 du1 j



T and QcqD





1 4x03 k5 D lsquo 24x05V14k5



11 1 n ƒ k




C eT1 Sƒ1cq

Œ4qC154x05



eT1 Sƒ1Qcq





21 4x03 k5=p4x0581C o41590 (25)




C eT1 Sƒ1cq

4lsquo 254qC154x05



eT1 Sƒ1Qcq





22 4x03 k5=p4x0581 C o41590 (28)


iatildemdashX uuml



mdashX uumliatilde



4k C 15atildek

C Œ2

2h28Œ004x05 C 2Œ04x05p

04x05=p4x0591 (29)







3 SIMULATIONS







XtiC1ordm Xti


C 2ƒ1lsquoh8Xti


Clsquo 4Xti51=2

C ˜i

patilde91=2

C C 4Xti51=2

C

i˜i

patilde1 (31)



















XtiC1ordm Xti


˜i

patilde


iƒ 15atilde (34)





1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

Vj(k

)


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k

log

Vj(

k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)











1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

log

V j(k)

log

V j(k)

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k





4 MODEL VALIDATION



005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003


005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003






t dWt1 (35)



ƒt dWt1 (36)








XtiC1ƒXti



5atildeiClsquoXƒ

ti˜i


where ˜i







ti5 C log4˜2

i 51

i D 11 1 nƒ 10 (39)








5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)


ƒOŒ4Xti









t 0

Let bEtiD 8Yti


i and Y415ti



bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti


i 51 i D 11 1 n ƒ 10 (40)





1 Y415ti



lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)








PniD1







H415


H425


05t 1 (44)


H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES





































Order k

1 2 3 4 5 6 7 8 9 10

V1(k) 1000 2050 4083 9025 18095 42068 105049 281065 798001 21364063V2(k) 1000 3000 8000 21066 61050 183040 570066 11837028 61076025 201527022


iatildemdashX uuml



mdashX uumliatilde



q m4q54x05=qW0





nƒkX

iD1

8Y uumliatilde

ƒ sbquo0ƒ 4X uuml

iatildeƒ x05sbquo1



iatildeƒ x05 (19)


iatildemdashX uuml



iatildemdashX uuml





iatildemdashX uuml

iatildeD x05 and


iatildemdashX uuml

iatildeD x050 (20)







k A11kAacutek










k2 ƒ 3k ƒ 2k4k C 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C 34k C 152

ordm 4k

1=2k5=21

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)




5k2 ƒ k ƒ 2k4k C 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C 6k C 54k C 152

ordm 5 4k

1=2k5=20


V24k5 Diexcl

2k

k

centƒ 4k C 15

k ƒ 1ordm 4k

1=2k3=20




21ŒjD

RujK4u5 du1 j



T and QcqD





1 4x03 k5 D lsquo 24x05V14k5



11 1 n ƒ k




C eT1 Sƒ1cq

Œ4qC154x05



eT1 Sƒ1Qcq





21 4x03 k5=p4x0581C o41590 (25)




C eT1 Sƒ1cq

4lsquo 254qC154x05



eT1 Sƒ1Qcq





22 4x03 k5=p4x0581 C o41590 (28)


iatildemdashX uuml



mdashX uumliatilde



4k C 15atildek

C Œ2

2h28Œ004x05 C 2Œ04x05p

04x05=p4x0591 (29)







3 SIMULATIONS







XtiC1ordm Xti


C 2ƒ1lsquoh8Xti


Clsquo 4Xti51=2

C ˜i

patilde91=2

C C 4Xti51=2

C

i˜i

patilde1 (31)



















XtiC1ordm Xti


˜i

patilde


iƒ 15atilde (34)





1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

Vj(k

)


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k

log

Vj(

k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)











1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

log

V j(k)

log

V j(k)

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k





4 MODEL VALIDATION



005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003


005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003






t dWt1 (35)



ƒt dWt1 (36)








XtiC1ƒXti



5atildeiClsquoXƒ

ti˜i


where ˜i







ti5 C log4˜2

i 51

i D 11 1 nƒ 10 (39)








5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)


ƒOŒ4Xti









t 0

Let bEtiD 8Yti


i and Y415ti



bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti


i 51 i D 11 1 n ƒ 10 (40)





1 Y415ti



lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)








PniD1







H415


H425


05t 1 (44)


H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES





































5k2 ƒ k ƒ 2k4k C 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C 6k C 54k C 152

ordm 5 4k

1=2k5=20


V24k5 Diexcl

2k

k

centƒ 4k C 15

k ƒ 1ordm 4k

1=2k3=20




21ŒjD

RujK4u5 du1 j



T and QcqD





1 4x03 k5 D lsquo 24x05V14k5



11 1 n ƒ k




C eT1 Sƒ1cq

Œ4qC154x05



eT1 Sƒ1Qcq





21 4x03 k5=p4x0581C o41590 (25)




C eT1 Sƒ1cq

4lsquo 254qC154x05



eT1 Sƒ1Qcq





22 4x03 k5=p4x0581 C o41590 (28)


iatildemdashX uuml



mdashX uumliatilde



4k C 15atildek

C Œ2

2h28Œ004x05 C 2Œ04x05p

04x05=p4x0591 (29)







3 SIMULATIONS







XtiC1ordm Xti


C 2ƒ1lsquoh8Xti


Clsquo 4Xti51=2

C ˜i

patilde91=2

C C 4Xti51=2

C

i˜i

patilde1 (31)



















XtiC1ordm Xti


˜i

patilde


iƒ 15atilde (34)





1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

Vj(k

)


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k

log

Vj(

k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)











1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

log

V j(k)

log

V j(k)

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k





4 MODEL VALIDATION



005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003


005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003






t dWt1 (35)



ƒt dWt1 (36)








XtiC1ƒXti



5atildeiClsquoXƒ

ti˜i


where ˜i







ti5 C log4˜2

i 51

i D 11 1 nƒ 10 (39)








5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)


ƒOŒ4Xti









t 0

Let bEtiD 8Yti


i and Y415ti



bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti


i 51 i D 11 1 n ƒ 10 (40)





1 Y415ti



lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)








PniD1







H415


H425


05t 1 (44)


H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES




































3 SIMULATIONS







XtiC1ordm Xti


C 2ƒ1lsquoh8Xti


Clsquo 4Xti51=2

C ˜i

patilde91=2

C C 4Xti51=2

C

i˜i

patilde1 (31)



















XtiC1ordm Xti


˜i

patilde


iƒ 15atilde (34)





1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

Vj(k

)


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k

log

Vj(

k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)











1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

log

V j(k)

log

V j(k)

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k





4 MODEL VALIDATION



005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003


005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003






t dWt1 (35)



ƒt dWt1 (36)








XtiC1ƒXti



5atildeiClsquoXƒ

ti˜i


where ˜i







ti5 C log4˜2

i 51

i D 11 1 nƒ 10 (39)








5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)


ƒOŒ4Xti









t 0

Let bEtiD 8Yti


i and Y415ti



bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti


i 51 i D 11 1 n ƒ 10 (40)





1 Y415ti



lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)








PniD1







H415


H425


05t 1 (44)


H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES




































1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

Order k

log

Vj(k

)


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k

log

Vj(

k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)











1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

log

V j(k)

log

V j(k)

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k





4 MODEL VALIDATION



005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003


005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003






t dWt1 (35)



ƒt dWt1 (36)








XtiC1ƒXti



5atildeiClsquoXƒ

ti˜i


where ˜i







ti5 C log4˜2

i 51

i D 11 1 nƒ 10 (39)








5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)


ƒOŒ4Xti









t 0

Let bEtiD 8Yti


i and Y415ti



bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti


i 51 i D 11 1 n ƒ 10 (40)





1 Y415ti



lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)








PniD1







H415


H425


05t 1 (44)


H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES




































1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k

log

V j(k)

log

V j(k)

log

V j(k)

log

V j(k)


1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9

10

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k


1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

Order k





4 MODEL VALIDATION



005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003


005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003






t dWt1 (35)



ƒt dWt1 (36)








XtiC1ƒXti



5atildeiClsquoXƒ

ti˜i


where ˜i







ti5 C log4˜2

i 51

i D 11 1 nƒ 10 (39)








5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)


ƒOŒ4Xti









t 0

Let bEtiD 8Yti


i and Y415ti



bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti


i 51 i D 11 1 n ƒ 10 (40)





1 Y415ti



lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)








PniD1







H415


H425


05t 1 (44)


H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES




































005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003


005 01 015ndash003

ndash002

ndash001

0

001


005 01 015

0015

002

0025

003






t dWt1 (35)



ƒt dWt1 (36)








XtiC1ƒXti



5atildeiClsquoXƒ

ti˜i


where ˜i







ti5 C log4˜2

i 51

i D 11 1 nƒ 10 (39)








5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)


ƒOŒ4Xti









t 0

Let bEtiD 8Yti


i and Y415ti



bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti


i 51 i D 11 1 n ƒ 10 (40)





1 Y415ti



lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)








PniD1







H415


H425


05t 1 (44)


H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES







































5000 00224 00217 ƒ02620 ƒ02534 00782 00783 00781 04979 04983 04976(072) (065) (8047) (7076) (082) (081) (081) (4005) (4002) (4000)

10000 00205 00200 ƒ02385 ƒ02328 00778 00779 00777 04971 04974 04968(045) (040) (5044) (4086) (056) (057) (055) (2083) (2084) (2078)


ƒOŒ4Xti









t 0

Let bEtiD 8Yti


i and Y415ti



bEtiordmlsquo 4Xti

5 ˜i1 i D 11 1 nƒ 1

and

Y415ti


i 51 i D 11 1 n ƒ 10 (40)





1 Y415ti



lsaquon4h5 D nƒ 1

2log

RSS0

RSS14h51 (42)








PniD1







H415


H425


05t 1 (44)


H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES




































H435


and

H445








1 1Xtn95 nƒ2=91 (47)




0 ` D 12 34

Percentile


H (1)0 lsaquon (h1) 12706 10904 8507 6602


H (2)0 lsaquon (h1) 13204 11406 9203 7409


H (3)0 lsaquon (h1) 13207 10908 9100 7007


H (4)0 lsaquon (h1) 11905 10203 8303 6506


0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

007

Short Rate

Vol

atili

ty F

unct

ion








4`5








Rejection rate


0 H(3)0 H(4)

0



0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES




































0 002 004 006 008 01 012 014 016 018ndash035

ndash03

ndash025

ndash02

ndash015

ndash01

ndash005

0

005

Short Rate

Drif

t

0 002 004 006 008 01 012 014 016 0180

001

002

003

004

005

006

Short Rate

Vol

atili

ty

(a) (b)




denote YtiD XtiC1

ƒ Xti From 84Xti

1 Yti5nƒ1


D Pnƒ1iD1 8Yti






2 log RSS0


D Ytiƒ OŒ4Xti

5atilde





Y4b5ti


5atilde1=2 Oe4b5ti


1 Y4b5ti


n 4h5















4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES




































4 45 5 55 6 65 7 750

001

002

003

004

005

006

007

008

log(Index)

Drif

t

4 45 5 55 6 65 7 75004

005

006

007

008

009

01

011

012

013

014

log(Index)

Vola

tility

(a) (b)








5 CONCLUSION



Bootstrap p value









A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES









































A D

2

66664



000000

0 0 0000

0 0 0000


3

77775and b D

2

66664

1

0000

0

3

777750








000000

0 0 0000

0 0 0000



D kWY Y

1microl1ltl2microk

4l2 ƒ l151



j2

Y Y


4l2 ƒ l150


xj D 4ƒ15jC1 4kW52

j2


4l2 ƒ l15



D 4ƒ15jC1

sup3k

j

acutej0



k

j

centj D

iexclkƒ1jƒ1



kX

jD1

jkC1ak1 j DkX

jD1

4ƒ15jC1

sup3k

j

acutejk

D ƒk

micro4ƒ15C

kƒ1X

jD1

4ƒ15jC1

sup3k ƒ 1

j

acute4j C 15kƒ1

para0


kX

jD1

jkC1ak1 j D ƒk

micro4ƒ15 C

kƒ1X

lD0

sup3k ƒ1

l

acutekƒ1X

jD1

j lC1akƒ11 j

para0


kX

jD1

jkC1ak1 jD ƒk

micro4ƒ15 C1 C

kƒ1X

jD1

jkakƒ11 j

para

D ƒkkƒ1X

jD1

jkakƒ11 j 0







E84XtCatildeƒXt5




E84XtCatildeƒXt5



D x09


E84XtCatildeƒ Xt5


and



21 f34x1 t5 D4x ƒ Xt5





following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES




































following relations







24x51




2Œ004x5



2lsquo 24x51


3Œ04x5







and








3Œ04x59Œ4x5






D atildeƒ2

X

1microjmicrok



X X

1microiltjmicrok

ak1 iak1 j


0 (A9)


















D x09





D x09





D atildeƒ2

X

1microjmicrok



1microiltjmicrok

ak1 iak1 j


0 (A13)



2mdashXtD x09



ƒXt52mdashXt

D x0972




ƒx052mdashXtD x09



D x07









D x09






D x09










kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

iD1

8Y uumliatilde

ƒE4Y uumliatilde


iatildeƒx051


D8Y uuml

iatildeƒ E4Y uuml

iatildemdashX uuml





nƒkƒ1X

`D1


`DdnC1

` D o4hatildeƒ251




REFERENCES










































kX

jD1

sup3k

j

acute2 4j C 25

4j C152D 42k C15W


4k C153 1=2kWƒ 2k2 C4k C 3

4k C1521

(A17)

kX

jD1

sup3k

j

acute2 4j C 35

4j C152D 42k C15W


4k C153 1=2kWƒ 3k2 C6k C5

4k C1521

(A18)

and

kX

jD2

(jƒ1X

iD1

4ƒ15iC1

sup3k

i

acute)4ƒ15jC1

sup3k

j

acutej

D 1 Cƒk

kƒ 1

k

sup32k

k



k

j



kX

jD1

ja2k1 j gt

kX

jD1

sup3k

j

2 4j C 25

4j C152D 42k C 154k C35

4kC 153

sup32k

k

acuteƒ 2k2 C4k C3

4k C 152

(A20)

and

kX

jD1

ja2k1j micro

kX

jD1

sup3k

j

2 4j C35

4jC152D 42kC154kC55

4kC153

sup32k

k

acuteƒ 3k2 C6kC5

4kC1520

(A21)



XX

1microiltjmicrok

iak1 iak1 jD 1

kC

kX

jD1

1j

ƒ 1k

sup32k

k

acute0 (A22)


k2 ƒ3k ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 2k2 C 4k C34kC 152

(A23)

and an upper bound

5k2 ƒk ƒ2k4kC 153

sup32k

k

acuteC 2

kC 2

kX

jD1

1j

ƒ 3k2 C6k C54k C152

0 (A24)



kX

jD1

j2a2k1 j

Dsup3

2k

k

acuteƒ 10 (A25)


jƒ1X

iD1

4ƒ15iC1

sup3k

i




which implies that

X X

1microiltjmicrok

i2ak1 iak1 jD 1

k ƒ1

kX

jD2

sup3k ƒ1

j

acutesup3k

j

acuteƒ

kX

jD2

sup3k ƒ1k ƒ j

acutesup3k

j

acute

D ƒiexcl2kƒ1

k

cent4k ƒ25 C1

k ƒ10 (A27)




iatildeƒ x05




atildemdashX uuml

atilde51 1E4Y uuml






n XT W4y ƒm51

sup2 b C t0


D Kh4X uumlàtilde

ƒ x054Xuumlàtilde



to the expression










atildeD x90



atilde59mdash


atilde5˜2


2mdashacute `mdash2







nƒkƒ1X

`D1


`D1

Cnƒkƒ1X

`DdnC1


D o4h2jƒ150 (A32)



`D1


D nh2jƒ1

microp4x052j

C o415C 2hƒ42jƒ15

nƒkƒ1X

`D1

sup31 ƒ `

n ƒk


para1


decomposition



mdashX uumliatilde

D x05ordf

ƒ 8E4Y uumliatilde

mdashX uumliatilde

D x05ƒ Œ4x0590


mdashX uumliatilde


iatildemdashX uuml





QnD cT u D 1

n

nƒkX

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Date post:	07-Sep-2018
Category:	Documents
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A Reexamination of Diffusion Estimators With …orfe.princeton.edu/~jqfan/papers/01/timehomo.pdf ·...

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