short-term interest rate dynamics: a spatial approach

Journal of Financial Economics 65 (2002) 73–110

Short-term interest rate dynamics:a spatial approach$

Federico M. Bandi

Graduate School of Business, The University of Chicago, 1101 East 58th Street, Chicago, IL 60637, USA

Received 16 August 1999; received in revised form 27 February 2001

Abstract

We use new fully functional methods to describe and study the dynamics of the short-term

interest rate process in continuous-time. The suggested procedure exploits the spatial

properties, embodied in the local time process, of the diffusion of interest, and is robust

against deviations from stationarity.

Our results indicate that the misspecification of a standard constant elasticity of variance

model with linear mean-reverting drift cannot be attributed to the nonlinear behavior of the

infinitesimal first moment of the short-term interest rate process at high rates. Rather, it

should be attributed to the martingale nature of the process over most of its empirical range

(i.e., between 3% and about 15%).

r 2002 Elsevier Science B.V. All rights reserved.

JEL classification: C14; C22; G12

Keywords: Local time; Nonparametric estimation; Short-term interest rate

1. Introduction

A large body of recent literature devotes itself to the estimation of the short-terminterest rate process. There are several reasons to take an interest in this issue. First,

$I am grateful to Heber Farnsworth, Anders Karlsson, Konstantyn Tyurin, and, especially, the

members of my thesis committee at Yale University, Peter C.B. Phillips, Nagpurnanand Prabhala, and

Chris Sims, for helpful discussions. I thank Yacine A.ıt-Sahalia for providing his data. The comments of an

anonymous referee led to substantial improvement of the paper. I also wish to thank seminar participants

at Berkeley, Chicago, Duke, Lehman Brothers, the London Business School, New York University,

Princeton, Rochester, Yale, and the 1999 WFA meetings at Santa Monica. Financial support from

Mediocredito Centrale and the Alfred P. Sloan Foundation is gratefully acknowledged.

E-mail address: [email protected] (F.M. Bandi).

0304-405X/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 4 0 5 X ( 0 2 ) 0 0 1 3 5 - 6

it is particularly relevant given the role of the short-term interest rate as a keyeconomic variable linking real and monetary phenomena. Second, interest ratemodel specification (often in continuous-time) has implications for the pricing offixed-income securities and derivatives. Finally, interest rate levels constitute atraditional benchmark to evaluate asset pricing, due to the fact that expectedequilibrium returns are defined in terms of excess returns relative to the risk-free rate.

In continuous-time finance, the dynamic evolution of the spot interest rate processis usually driven by a Markov stochastic differential equation. Stochastic differentialequations are completely described by two functions, the drift and the diffusionfunction. Parametric approaches to the estimation of these two functions yieldcontradictory results. A.ıt-Sahalia (1996b), for example, suggests a semiparametricprocedure to discriminate among alternative parametric specifications. He rejectsevery conventional one-factor model of the short rate but some recent evidenceshows that his procedure has distorted size and low power in finite samples (Pritsker,1998). Fully nonparametric methods have been developed but they either rely on theexistence of a time-invariant marginal density for the underlying process (Jiang,1998; Jiang and Knight, 1997) or stationarity is assumed despite robustness todeviations from it (Stanton, 1997).

In this work, we implement a unified approach to the estimation of the drift andthe diffusion function of the short-term interest rate process based on a newestimation procedure recently proposed by Bandi and Phillips (1998) (BP,henceforth). We use functional methods. Minimal requirements are placed on thedata-generating mechanism allowing for both stationary and nonstationary systems.Cross-restrictions on the functional forms of the drift and diffusion function (as inA.ıt-Sahalia, 1996a, b; Jiang, 1998; Jiang and Knight, 1997) are not imposed, nor isthe existence of a time-invariant marginal data density either required or assumed(Stanton, 1997). In consequence, the new approach is robust against deviations fromstationarity. The available data are taken to be a set of discrete sample observations.Econometric estimation proceeds by constructing refined sample analogues ofunknown drift and diffusion function.

The proposed methodology has several important features. First, as mentionedearlier, despite a flurry of theoretical contributions (see Duffie, 1992, for example),empirical results do not offer complete support for any specific parametrization.Given the importance of the short-term riskless rate in valuing and hedging a broadarray of fixed-income contingent claims, fully nonparametric methods areparticularly suitable to avoid potential misspecifications.

Second, since the drift is theoretically harder to identify than the diffusion term(see, e.g., A.ıt-Sahalia, 1996a; BP, 1998; Jiang and Knight, 1997), a unified andcomplete asymptotic theory for both estimated functions is crucial in fixed-incomepricing. In effect, the drift of the underlying short-rate process plays a role inassessing the value of fixed-income securities even under the no-arbitrage restrictionsimposed by martingale pricing.

Third, the evidence on the stationarity of the short-rate process is quiteambiguous. Preliminary unit-root tests either fail to reject the null of nonstationarityor deliver results very close to the rejection threshold. This observation explains why

F.M. Bandi / Journal of Financial Economics 65 (2002) 73–11074

high-frequency spot interest rate series are often modelled as nonstationary processesin macroeconomics (A.ıt-Sahalia, 1996b). In continuous-time empirical finance,stationarity is generally assumed up front to assist in developing a completeestimation theory. Some researchers have provided plausible ex post justifications forthis assumption based on estimated drift and diffusion functions. A.ıt-Sahalia (1996b)suggests that the spot rate can be locally nonstationary over the range of the processcorresponding to a drift very close to zero. Nevertheless, a nonlinear mean-revertingdrift at the edges of the range of the process can be sufficient to pull the series backinto its middle region and determine global stationarity. Conley et al. (1997) suggestvolatility-induced stationarity. The infinitesimal first moment can even be positive athigh rates, but increasing volatility can be sufficient to import stationarity into theseries.

Due to the mixed a priori empirical evidence, estimation methods relying onstationarity can yield imprecise inference and suggest misleading conclusions. Inconsequence, we do not make the assumption of stationarity in this work but insteadassume recurrence. That is, we require the continuous trajectory of the process tovisit any set in its range an infinite number of times over time almost surely.Recurrence is less restrictive than stationarity (i.e., recurrent processes do not have tobe stationary) and, indeed, makes economic sense because we expect interest rates toreturn to the values in their range over and over again.

We are interested in estimating the drift and the diffusion function at each point inthe range of the sample interest rate process, so the density of the observations thereplays a role in the operation of the asymptotics. This information is contained in theestimated local time of the spot interest rate process. The local time can be defined asfollows (classical references are Chung and Williams, 1990; Karatzas and Shreve,1991; Revuz and Yor, 1994):

Definition 1. If Xt is a continuous semimartingale, then there exists a nondecreasingstochastic process (nondecreasing in t; that is) LX ðt; aÞ; called the local time of X at a:This process is defined, almost surely, as

LX ðt; aÞ ¼ lime-0

1

e

Z t

0

1½a;aþe½ðXsÞ d½X �s: ð1Þ

LX ðt; aÞ represents the amount of time that the process Xt spends in the vicinity ofthe point a: Time is measured in units of the quadratic variation process (½X �t). Theseare information units as they represent the amount of information that is beingaccumulated about the process.

Consider, for consistency with our framework, a diffusion process Xt withinfinitesimal second moment s2ðXtÞ: LX ðt; aÞ reduces to

LX ðt; aÞ ¼ lime-0

1

e

Z t

0

1½a;aþe½ðXsÞs2ðXsÞ ds: ð2Þ

F.M. Bandi / Journal of Financial Economics 65 (2002) 73–110 75

If we now divide through by s2ðaÞ; we obtain

%LX ðt; aÞ ¼1

s2ðaÞLX ðt; aÞ ¼

1

s2ðaÞlime-0

1

e

Z t

0

1½a;aþe½ðXsÞs2ðXsÞ ds

� �: ð3Þ

%LX ðt; aÞ can be called ‘chronological local time’ (see Phillips and Park (1997), for asimilar notion in the Brownian motion case). This formulation gives aninterpretation of the local time in terms of amount of time, in real time units now,spent by the process in the spatial neighborhood of a point. Also, this definitionshows the sense in which the local time, even though random in nature, is analogousto a probability density. In fact, it provides meaningful quantitative informationabout the locational features of the process, in just the same way as a probabilitydensity function can be used to characterize stationary time series.

Below, we interpret the local time process of the short-term interest rate process asa series of spatial densities. Additionally, we show how to consistently estimatespatial densities using nonparametric density-like kernel estimators. Our inference isbased on a complete asymptotic theory for spatial densities of diffusion processesthat are potentially nonstationary solutions to possibly nonlinear stochasticdifferential equations.

Furthermore, we define functionals of spatial densities, such as spatial hazardrates (Phillips, 1998). Spatial hazard rates can be interpreted as spatial analogues totraditional hazard rates obtained from spatial densities rather than from time-invariant marginal distribution functions, i.e.,

%HX ðt; aÞ ¼%LX ðt; aÞR

N

a%LX ðt; sÞ ds

: ð4Þ

Eq. (4) has a standard meaning: when applied to interest rates it gives the conditionalrisk over the period ½0; t� of an interest rate level of a; given that interest rates are atleast as big as a: Again, a complete asymptotic theory for the nonparametricestimates of spatial hazard rates assists our investigation.

The notions of spatial density and spatial hazard rate assume importanceparticularly when the underlying process is nonstationary as they furnish thepossibility of characterizing some of the features of the data, i.e., those related to thelocation of the process. In effect, in the presence of nonstationarity, conventionaldescriptive statistics fail to provide reliable information given the tendency of thedata to drift away from a particular point. Spatial densities and their functionals,such as the above-mentioned spatial hazard rates, can then be regarded as newdescriptive tools for series that are nonstationary1 or whose stationarity cannot beguaranteed, as in this paper. Based on estimated spatial densities and spatial hazardrates, we discuss some of the features of the specific data set at hand. We study theannualized seven-day Eurodollar rate (June 1, 1973 through February 25, 1995) fromBank of America. The data were previously used in A.ıt-Sahalia (1996a, b).

1 This observation was first made by Peter C.B. Phillips in the context of nonstationary discrete-time

series during the Irving Fisher Conference at Yale University, May 1998. Here, we extend his idea to

possibly nonstationary continuous-time processes.


Finally, we carefully discuss the sense in which the information embodied in thespatial density of the interest rate process can be used to implement a flexible andrigorous approach to the nonparametric estimation of the two functions driving theinterest rate dynamics in continuous-time, viz. the drift and the diffusion function.As in many recent papers (A.ıt-Sahalia, 1996a, b; Jiang, 1998; Stanton, 1997, forexample), the main source of the rejection of traditional linear mean-revertingstructures in the constant elasticity of variance class is the specification of the driftfunction. Our estimated drift is virtually zero between 3% and about 15% (A.ıt-Sahalia, 1996b). It mean-reverts in a nonlinear fashion only at the upper edge of therange of the sample process. We emphasize the importance of the martingalebehavior of the spot-rate series over most of its range in disputing linear mean-reverting models. As for the marked nonlinearity of the drift at the upper edge of thesample process, its empirical relevance is clouded by the availability of fewobservations in this range. This idea can be phrased in a more rigorous fashion inour framework. We will show that to draw precise inference on the drift of theprocess at a point (i.e., to achieve statistically consistent estimates), we need torequire the estimated local time of the process at that point to be large. Differentlyput, we need to require the time spent by the sample process in the spatial vicinity ofthat point to be large. Since the sample process barely visits interest rate levels at theupper edge of its range, we cannot draw firm conclusions about the behavior of thedrift at high interest rate levels, where nonlinearities arise. The problem of the lack ofsufficient observations at high rates has been noted earlier by A.ıt-Sahalia (1996a).Nonetheless, we believe this issue should suggest a more cautious interpretation ofthe economic and statistical content of the estimated nonlinearities in the literature.Between 3% and 16% the nonparametric diffusion function exhibits a marked CEVshape.

The paper is organized as follows. Section 2 introduces the model and illustratesthe difference between the methodology employed in this study and somerepresentative existing procedures. Section 3 discusses the estimation techniqueadopted here and the sense in which spatial arguments assist in developing a generalapproach to the functional analysis of the dynamics of the short-term interest rateprocess in continuous-time. In Section 4 we present the data and implement themethod. Section 5 concludes. Technical details, comments on methodology, andproofs are provided in Appendices A and B.

2. The model

We assume a Markov, possibly nonlinear, continuous data-generating process forthe short-term interest rate series. Its evolution over time is described by thestochastic differential equation

drt ¼ mðrtÞ dt þ sðrtÞ dBt; ð5Þ

where fBt; tX0g is a standard Brownian motion. The functions mð:Þ : D ¼ ð%r; %rÞ-R

and s2ð:Þ : D ¼ ð%r; %rÞ-Rþ represent the conditional expected rate of change of the


process (drift) and the conditional rate of change of volatility (diffusion) forinfinitesimal time changes, respectively. Traditional parametric forms for drift anddiffusion in the finance literature on the spot interest rate process are in Table 1 ofA.ıt-Sahalia (1996a), for example. BP (1998) discuss conditions on the two functionswhich guarantee the existence and uniqueness of a strong and recurrent solution toEq. (5).

A thorough description of the existing literature on the estimation of mð:Þ and s2ð:Þis virtually an impossible task. Jiang and Knight (1997) provide a partial survey ofrecent methods. A useful, somewhat unusual, discussion for the purpose of thepresent paper hinges on the following observation. There are two standard ways toperform asymptotics in the context of diffusion estimation (and prove consistency ofthe proposed estimators for the unknown functions of interest). Either the timehorizon is assumed to lengthen to infinity (viz. long span asymptotics) orobservations are sampled at higher and higher frequency (viz. infill asymptotics).Some existing work relies on the former without necessitating the latter. Thiscategory includes papers by A.ıt-Sahalia (1996a, b, 2002), Conley et al. (1997),Gallant and Tauchen (1996), Gouri!eroux et al. (1993) and Hansen and Scheinkman(1995), among others. A second category, including work by Florens-Zmirou (1993)and Jiang and Knight (1997) to name just a few, exploits the local information in thepath of the process over a fixed time span of observations. It readily appears that thediscrete nature of the data renders procedures that allow for fixed time distancesbetween observations preferable. Nonetheless, fully functional estimation of driftand diffusion function does not appear feasible in the presence of potentialnonstationarities and equally spaced data points unless the data are sampled moreand more frequently over time. In the light of the discussion in the introduction, wechoose to hinge on infill asymptotics to achieve complete generality as far as thestatistical properties of the spot interest rate process are concerned. However, thestringency of the infill assumption adopted here has been subjected to thoroughinvestigation. The recent Monte Carlo evidence in Bandi and Nguyen (1999) andJiang and Knight (1999) suggests that even daily frequencies are valid approxima-tions to frequent sampling for estimators relying on increasingly frequentobservations. We use daily data in what follows.

Along with increasingly frequent sampling, robust (to deviations from stationar-ity) nonparametric identification of the infinitesimal first moment requires anenlarging time span of observations (Merton, 1980). Below, we specify acommonsense estimator (see, e.g., Florens-Zmirou (1993), in the diffusion functioncase) for the drift term that is constructed as a straight sample analogue to thetheoretical function. The following result proves that consistency cannot be achievedover a fixed time span and provides the rate of divergence of the proposed estimator.We assume that we observe rt at ft ¼ t1; t2;y; tng in the time interval ½0; %T�; where %T

is a positive constant. We also assume equispaced data. In consequence, frt ¼rDn

; r2Dn; r3Dn

;y; rnDng are n observations at ft1 ¼ Dn; t2 ¼ 2Dn; t3 ¼ 3Dn;y; tn ¼

nDng; where Dn ¼ %T=n: The asymptotic result is obtained as the sample frequencyincreases for a given ending time %T: Note that 1A below denotes the indicator kernelof the set A: The use of a continuous kernel would not affect the result.


Theorem 2.1. Given a bandwidth hnð-0Þ such that nhn-N and nh4n-0; the estimator

#mðnÞðrÞ defined as

#mðnÞðrÞ :¼1

Dn

Pn1i¼1 1fjriDnrjohng½rðiþ1ÞDn

riDn�Pn

i¼1 1fjriDnrjohngð6Þ

diverges at a rate given by the square root of hn as n-N:

In what follows we propose alternative estimators of the two functions of interestbased on the necessity of reducing the local variability induced by discreteobservations and enhancing the availability of information through the use of alonger time span. In other words, we perform both infill and long span asymptotics.We retain the sample analog structure and do not impose cross-restrictions on the twofunctions based on the existence of a time-invariant marginal density as in A.ıt-Sahalia(1996a, b) or Jiang (1998), for example, in order to avoid invoking strong requirementson the distribution of the underlying process. A complete description of the asymptotictheory is in BP (1998). We refer the interested reader to that paper for details.

3. The econometric approach

We assume that observations on the short-term interest rate process frt; tX0g arerecorded as in the previous section but in the interval ½0;T �; where T is notnecessarily fixed at %T: We suggest Eqs. (7)–(10) below to estimate s2ðrÞ; mðrÞ; %Lrð %T; rÞ;and %Hrð %T; rÞ; respectively:

#s2ðn;TÞðrÞ :¼

Pni¼1 K

riDn;Tr

hn;T

� �1

mn;T ðiDn;T ÞDn;T

Pmn;T ðiDn;T Þ1j¼0 ½rtðiDn;T ÞjþDn;T rtðiDn;T Þj �

2

Pni¼1 K

riDn;Tr

hn;T

� � ;ð7Þ

#mðn;TÞðrÞ :¼

Pni¼1 K

riDn;Tr

hn;T

� �1

mn;T ðiDn;T ÞDn;T

Pmn;T ðiDn;T Þ1j¼0 ½rtðiDn;T ÞjþDn;T rtðiDn;T Þj �Pn

i¼1 KriDn;T

r

hn;T

� � ; ð8Þ

#%Lrð %T; rÞ :¼Dn; %T

hn; %T

Xn

i¼1

KriDn; %T

r

hn; %T

� �; ð9Þ

and

#%Hrð %T; rÞ :¼#%Lrð %T; rÞR

N

r#%Lrð %T; sÞ ds

: ð10Þ

The terms ftðiDn;T Þjg in Eqs. (7) and (8) denote a sequence of random times definedas

tðiDn;T Þ0 ¼ infftX0 : jrt riDn;T jpen;Tg ð11Þ


and

tðiDn;T Þjþ1 ¼ infftXtðiDn;T Þj þ Dn;T : jrt riDn;T jpen;Tg; ð12Þ

for all i: The number mn;T ðiDn;T Þpn counts the stopping times associated with thevalue riDn;T and is defined as

mn;T ðiDn;T Þ ¼Xn

j¼1

1fjrjDn;TriDn;T

jpen;Tg 8i; ð13Þ

where 1A denotes, as earlier, the indicator of A: The quantity en;T is a bandwidth-like(spatial) parameter depending on the time span (T) and on the sample size (n). Thefunction Kð:Þ that appears in Eqs. (7)–(10) is a smooth kernel whose properties aredescribed in Appendix A, Assumption A1.

The estimators in Eqs. (7) and (8) are straight sample analogues to thetheoretical functions s2ðrÞ and mðrÞ: They can be interpreted as the resultof a two-step procedure. First, sample analogues to the values that drift anddiffusion take on at the sample points are defined. Second, estimated driftand diffusion values at the sample points are averaged using weights basedon smooth kernels to recover the theoretical functions at levels that are notvisited by the sample process. Eqs. (7) and (8) can be rewritten as weightedaverages with weights based on indicator functions convoluted with smooth kernels(BP, 1998). Up to a constant of proportionality in the limiting variances, the use ofmore general kernels than discontinuous indicators would not affect the asymptoticresults (BP, 1998). Furthermore, Eqs. (7) and (8) can be regarded as the product of ageneral approach to the functional estimation of diffusions that encompassesspecifications based on single smoothing. The relative merits of double-smoothingover its simple counterpart in finite samples are discussed in Bandi and Nguyen(1999).

BP (1998) show that Eqs. (7) and (8) are consistent for the true functionswith probability one and normally distributed in the limit under appropriateconditions on the relevant smoothing sequences. In particular, as n-N; T-N;Dn;T ¼ T=n-0; hn;T-0 (with n;T-N) so that ð %LrðT ; rÞ=hn;T ÞðDn;T logð1=Dn;T ÞÞ

1=2

¼ oa:s:ð1Þ and en;T-0 (with n;T-N) so that ð %LrðT ; rÞ=en;T ÞðDn;T logð1=Dn;T ÞÞ1=2 ¼

oa:s:ð1Þ; the estimator #s2ðn;TÞðrÞ is such that

#s2ðn;TÞðrÞ-

a:s:s2ðrÞ: ð14Þ

Provided ðe5n;T=Dn;T Þ %LrðT ; rÞ ¼ oa:s:ð1Þ and hn;T ¼ oðen;T Þ; thenffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

#%LrðT ; rÞen;T

Dn;T

sð #s2

ðn;TÞðrÞ s2ðrÞÞ ) Nð0; 2s4ðrÞÞ: ð15Þ

Furthermore, if n-N; T-N; Dn;T ¼ T=n-0; hn;T-0 (as n;T-N) so thatð %LrðT ; rÞ=hn;T ÞðDn;T logð1=Dn;T ÞÞ

1=2 ¼ oa:s:ð1Þ and en;T-0 (as n;T-N) so that


ð %LrðT ; rÞ=en;T ÞðDn;T logð1=Dn;T ÞÞ1=2 ¼ oa:s:ð1Þ with en;T %LrðT ; rÞ-a:s:

N; then

#mðn;TÞðrÞ-a:s:

mðrÞ ð16Þ

and ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#%LrðT ; rÞen;T

qð #mðn;TÞðrÞ mðrÞÞ ) Nð0; ð1=2Þs2ðrÞÞ ð17Þ

given e5n;T

%LrðT ; rÞ ¼ oa:s:ð1Þ and hn;T ¼ oðen;T Þ:The normalizations in the weak convergence results in Eqs. (15) and (17) above are

random because of the presence of the local time factor #%LrðT ; rÞ: As T diverges toinfinity, the chronological local time %LrðT ; rÞ of the process rt diverges to infinity aswell by recurrence. Hence, the rates of convergence depend on the asymptoticdivergence features of the chronological local time %LrðT ; rÞ: Such rates can becharacterized in closed-form if the underlying process is Brownian motion orstationary. In the former case the convergence rates are

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffien;T T1=2=Dn;T

pandffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

en;T T1=2p

for diffusion and drift, respectively. In the latter, they areffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffien;T T=Dn;T

pand

ffiffiffiffiffiffiffiffiffiffiffien;T T

p(BP, 1998).

The limit theory assumes that T and n go to infinity. If we had at our disposalincreasingly frequent data over a fixed time span ðT ¼ %TÞ; we could still estimate thediffusion function consistently. T going to infinity is a technical device introduced toexploit the recurrence of the process in the estimation procedure. Recurrence iscrucial in the estimation of the drift function. In the case of diffusion estimation wedo not need to require infinite passage times to identify the true function (seeFlorens-Zmirou, 1993; Geman, 1979, for example). For a fixed T (¼ %T), we canrewrite Eq. (15) as

ffiffiffiffiffiffiffiffiffiffinen; %T

p#s2ðn; %TÞðrÞ s2ðrÞ

� �) MN 0; 2

s4ðrÞ%Lrð %T; rÞ= %T

� �; ð18Þ

where MN signifies mixed normal distribution. The conditions on the diffusionbandwidths hn; %T and en; %T approximately reduce to en; %Tpnk1 with k1Að1=4; 1=2Þ;hn; %Tpnk2 with k2Að0; 1=2Þ and hn; %T ¼ oðen; %TÞ (BP, 1998).

The drift term cannot be identified over a fixed-time interval, no matter howfrequently the data is sampled (see Theorem 2.1) unless cross-restrictions based onthe existence of a time-invariant probability density are imposed (see, e.g., Jiang andKnight, 1997). Here, we are lengthening the sample span ðT-NÞ as the frequencyof observations increases ðT=n-0Þ: We do so to gain information on the theoreticalfunction through repeated visits to each spatial set (which are guaranteed by theassumption of recurrence). Since local arguments as in the case of diffusionestimation can not be utilized, there are three main consequences. First, contrary todiffusion estimation, the stochastic properties of the underlying process play a vitalrole in drift estimation (Bandi and Nguyen, 1999). Second, the rate of convergence of

the diffusion estimator ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#%LrðT ; rÞen;T=Dn;T

qÞ is faster than the rate of convergence of

the drift estimator ðffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi#%LrðT ; rÞen;T

qÞ: Third, it is generally not possible to express in

closed-form (i.e., as a simple function of the number of observations) the main


admissibility condition that the drift window width edriftn;T needs to satisfy

ð %LrðT ; rÞedriftn;T -a:s:

N; that is). Furthermore, the drift optimal spatial bandwidthsequence edrift

n;T generally converges to zero at a slower pace than the correspondingsequence in the diffusion case ðediff

n;T Þ: Notice that, should T be fixed, then the localtime factor, %LrðT ; rÞ; would be Opð1Þ rather than tend to infinity almost surely and

the drift estimator would diverge at a speed equal toffiffiffiffiffiffiffiffiffiedrift

n;T

q; as also revealed by

Theorem 2.1 in this paper.Empirically, it is possible to identify the drift function in a situation where the

diffusion term is treated as a nuisance parameter, provided the data is sufficientlyrecurrent. Asymptotically, recurrence makes the chronological local time processdiverge to infinity. Divergence to infinity of the local time factor is a necessarycondition for the theoretical consistency of the drift estimator since%LrðT ; rÞedrift

n;T -a:s:N if %LrðT ; rÞ-a:s:

N and edriftn;T converges to zero slowly enough.

In our case, we require the data set to be characterized by repeated observations inthe neighborhood of each interest rate level. We can phrase this condition differentlyand say that we require the data set to be affected by few outliers. Apparently, this isa standard requirement for reliable statistical inference. Then, the issue is how toassess the recurrence properties of the specific data set at hand. A possible rule ofthumb in our framework is to estimate the spatial density of the sample process ateach point and verify its magnitude. A large spatial density at an interest rate levelimplies repeated passages in the vicinity of that level and, possibly, satisfactoryinference.

In what follows we will provide traditional descriptive statistics, such as samplemeans, standard deviations and so forth, whose interpretation is straightforwardshould stationarity hold. Furthermore, we will thoroughly study the local time andspatial hazard process to describe the locational features of the short-term interestrate series and detect areas where inference might be imprecise due to the lack ofsufficient data points. A complete asymptotic theory will assist our analysis. Florens-Zmirou (1993) and BP (1998) show that if hn; %T-0 as n-N for a fixed time span T

(¼ %T) in such a way that ð1=hn; %TÞðDn; %T logð1=Dn; %TÞÞ1=2 ¼ oð1Þ; then

#%Lrð %T; rÞ-a:s:

%Lrð %T; rÞ: ð19Þ

An application of Slutsky theorem allows us to prove that

#%Hrð %T; rÞ-a:s:

%Hrð %T; rÞ ð20Þ

under the same conditions on the smoothing sequence that guarantee the consistencyof #%Lrð %T; rÞ: We now discuss the limit distributions of #%Lrð %T; rÞ and #%Hrð %T; rÞ and givetheir rates of convergence in Theorems 3.1 and 3.2 below.

Theorem 3.1. If hn; %T-0 (as n-N for a fixed T ¼ %T) in such a way that

1

h3=2

n; %T

ðDn; %T logð1=Dn; %TÞÞ1=2 ¼ oð1Þ; ð21Þ


then

1ffiffiffiffiffiffiffiffihn; %T

p ð #%Lrð %T; rÞ %Lrð %T; rÞÞ ) MN 0; 8k1

s2ðrÞ%Lrð %T; rÞ

� �; ð22Þ

where k ¼RN

0

RN

0 minðs; qÞKðsÞKðqÞ ds dq:

Theorem 3.2. If hn; %T-0 (as n-N for a fixed T ¼ %T) in such a way that

1

h3=2

n; %T

ðDn; %T logð1=Dn; %TÞÞ1=2 ¼ oð1Þ; ð23Þ

then


p ð #%Hrð %T; rÞ %Hrð %T; rÞÞ ) MN 0;8kð %Hrð %T; rÞÞ

2

s2ðrÞ %Lrð %T; rÞ

� �; ð24Þ

where k ¼RN

0

RN

0 minðs; qÞKðsÞKðqÞ ds dq:

These results enable us to construct asymptotic confidence intervals which closelyresemble conventional intervals for probability densities and standard hazardfunctions obtained from kernel estimates (Phillips, 1998). The asymptotic 95%confidence intervals for %Lrð %T; rÞ and %Hrð %T; rÞ are given by

#%Lrð %T; rÞ71:96 8khn; %T

#s2ðrÞ#%Lrð %T; rÞ

� �1=2

ð25Þ

and

#%Hrð %T; rÞ71:96 8khn; %T

#s2ðrÞ

ð #%Hrð %T; rÞÞ2

#%Lrð %T; rÞ

!1=2

; ð26Þ

respectively. The scale factor 8k accounts for the time dependence in theobservations as the Brownian covariance kernel appears in the definition of k: Itis noted that the limit processes %Lrð %T; rÞ and %Hrð %T; rÞ are random. Additionally,spatial densities and spatial hazard rates have a time dimension, as opposed to theirstandard counterparts. Their time dimension will be explored in our subsequentdescriptive analysis. We now turn to the implementation of the method.

4. Implementation

The proposed methodology requires a long time series of high frequencyobservations. Nevertheless, due to the risk of spurious microstructure contamina-tions, the proper use of such series is an issue which goes beyond the scope of thiswork. We compromise by using a less ideal but still suitable series, namely the seven-day Eurodollar deposit rate, bid–ask midpoint, from Bank of America.

The data were previously used in A.ıt-Sahalia (1996a, b). As A.ıt-Sahalia (1996a)points out, choosing a seven-day rate is ‘‘ya compromise between: (i) literally


taking an instantaneous rate and (ii) avoiding some of the spurious microstructureeffects associated with overnight ratesy’’. Our description of the data is based onA.ıt-Sahalia’s work and we refer the reader to his papers for a complete discussion.Here it suffices to say that the data points are daily observations from June 1, 1973 toFebruary 25, 1995. The total number of observations is 5,505. The rates quoted wereoriginally bond-equivalent yields. They were transformed to continuously com-pounded yields to maturity (annualized rate). Weekends and holidays do not receivespecial treatment but weekend effects (e.g., French and Roll, 1986) do not seem to bea major concern for money-market instruments. Monday is taken as the first dayafter Friday. A time-series plot of the data is contained in Fig. 1. Fig. 2 provides atime-series plot of the first differences. Table 1 contains a summary of thecharacteristics of the data whereas Table 2 gives some standard descriptive statistics.

We stressed before the necessity of devising estimation procedures robust todeviations from strong distributional assumptions. In effect, it is hard to claim thatthe data in question is, without any doubt, stationary. We perform conventional

0.24

0.16

0.08

0.00

Spo

t rat

e

1974 1978 1982 1986 1990 1994 1998

Year

Fig. 1. The short-term interest rate series in levels. The data used is the Bank of America seven-day

Eurodollar deposit spot rate midpoint bid–ask. The sample period is June 1, 1973 through February 25,

1995 (5,505 annualized daily observations).

Table 1

Summary of the features of the short-term interest rate data used in this study.

Source Bank of America seven-day Eurodollar (midpoint bid–ask)

Frequency Daily

Sample period 1 June, 1973 through 25 February, 1995

Sample size 5,505 observations

Type Continuously compounded yields to maturity (annualized rate)


augmented Dickey-Fuller (ADF) tests both with a constant term and a trend, andwith only a constant term in the deterministic part of the fitted regression.

The order of the time polynomial is set equal to one whereas the number of laggedfirst differences is set equal to five. The test systematically fails to reject the null at allconventional levels (Tables 3 and 4). In the literature, even slight rejections (but thisis not the case here) are interpreted as strong evidence in favor of stationarity due tothe low power of the test (for example, A.ıt-Sahalia, 1996a, b; Jiang and Knight,1997). Different choices of the number of lagged first differences do not affect theresults. For the sake of comparison, we worked with up to 30 lags. This is the

0.06

0.02

_0.02

_0.061974 1978 1982 1986 1990 1994 1998

Year

Firs

t diff

eren

ces

spot

rat

e

Fig. 2. The short-term interest rate series in first differences. The data used is the Bank of America seven-

day Eurodollar deposit spot rate midpoint bid–ask. The sample period is June 1, 1973 through February

25, 1995 (5,505 annualized daily observations).

Table 2

Descriptive statistics for the seven-day Eurodollar deposit spot rate bid–ask midpoint (5,505 annualized

daily observations from June 1, 1973 to February 25, 1995). We report the mean, the standard deviation

and a set of daily autocorrelations (rj denotes the autocorrelation at lag j) for the series in levels and first

differences. Monthly autocorrelations for the same data set are reported in A.ıt-Sahalia (1996a, b).

Spot interest rate First differences

Mean 0.0836 0.0000035

Standard deviation 0.0359 0.004070

r1 0.9936 0.2710

r2 0.9908 0.0347

r3 0.9883 0.0377

r4 0.9863 0.0297

r5 0.9839 0.1789

r10 0.9779 0.0173


number of lags used in A.ıt-Sahalia (1996a). In the constant/trend case the test fails toreject at all conventional levels. In the constant case, the first rejection occurs with 30lags, at the 10% level. This outcome is consistent with the result reported in A.ıt-Sahalia (1996a).

We verify the previous outcome by implementing a different testing procedurebased on Phillips’ (1987) ZðaÞ and ZðtÞ statistics. Z tests generally have more powerthan the ADF test. As far as size is concerned, the ADF test is generally less subjectto distortions, especially in the presence of MA(1) errors with parameter close to one(see Phillips and Perron, 1988). Again, we consider both the constant/time trend caseand the constant case. The number of autocovariance terms to compute the spectrumat frequency zero is set equal to five. The automated optimal bandwidth case is alsoevaluated as it usually delivers quite different results. In all cases the test fails to

Table 3

Nonstationarity tests with a constant and a trend in the fitted regressions

We show the results of two nonstationarity tests applied to the seven-day Eurodollar deposit spot rate bid–

ask midpoint examined in this study (5,505 annualized daily observations from June 1, 1973 to February

25, 1995). We implement the Augmented Dickey-Fuller test (ADF) and the Z tests in Phillips (1987). The

ADF t-statistic is the ratio #a=s#a from the estimated model

Drt ¼ a þ bt þ art1 þX5

i¼1

jiDrti þ ut;

where s#a is the standard error of the parameter estimate #a: We derive the statistics Zð#aÞ and ZðtÞ from the

estimated model

rt ¼ a þ bt þ art1 þ et:

In particular,

Zð#aÞ ¼ nð#a 1Þ #l

ðn2Pn

t¼1%r2

t1Þ

with%rt ¼ rt #a #bt and

ZðtÞ ¼#s#$

tð#aÞ #l

#$ðn2Pn

t¼1%r2

t1Þ1=2

;

where tð#aÞ is equal to ð#a 1Þ=s#a; #l is a consistent (kernel) estimate of l ¼P

N

h¼1 Eðe0ehÞ; #$2 is a consistent

(kernel) estimate ofP

N

h¼NEðe0ehÞ; and #s2 is a consistent estimate of Eðe2

t Þ: In the Zð#aÞ and ZðtÞ tests the

number of autocovariance terms to compute the spectrum at frequency zero is set equal to five. We reject

the null hypothesis of a unit root if the statistics #a=s#a; Zð#aÞ; and ZðtÞ are smaller than the critical values or

equal to them. Critical values and corresponding values for the relevant statistics are reported in the table.

The second column contains estimates of the autoregressive parameter, #f say.

#f Test statistic 1% value 5% value 10% value

ADF test 0.9963 2.3447 3.9978 3.4318 3.1617

Zð#aÞ test 0.9923 19.4572 28.9388 21.2162 17.9117

ZðtÞ test 0.9923 3.1383 3.9978 3.4318 3.1617

Automatic window width

Zð#aÞ test 0.9923 21.0867 28.9388 21.2162 17.9117

ZðtÞ test 0.9923 3.2655 3.9978 3.4318 3.1617


reject at the 1% critical level. Results are mixed at the 5% level and generally infavor of stationarity at the 10% level, with the sole exception being the ZðtÞ testwhen both a constant and a trend are included and the window width is not chosenin an automatic fashion. The results are not qualitatively altered by differentspecifications.

Since the series displays time intervals of fairly regular behavior and relatively lowvolatility (1973–1980 and 1982–1995), we apply the same tests to subperiods in orderto assess the influence of higher volatility periods on the test results. A pattern seemsto emerge: higher volatility periods inject stationarity in the data. Even though thenull is rarely rejected, the test values appear to be closer to the corresponding criticalvalues in the presence of more volatile data. For instance, over the period 1973–1980the null of nonstationarity is never rejected and the statistic values are very safely

Table 4

Nonstationarity tests with a constant in the fitted regressions

We show the results of two nonstationarity tests applied to the seven-day Eurodollar deposit spot rate bid–

ask midpoint examined in this study (5,505 annualized daily observations from June 1, 1973 to February

25, 1995). We implement the Augmented Dickey-Fuller test (ADF) and the Z tests in Phillips (1987). The

ADF t-statistic is the ratio #a=s#a from the estimated model

Drt ¼ a þ art1 þX5

i¼1

jiDrti þ ut;

where s#a is the standard error of the parameter estimate #a: We derive the statistics Zð#aÞ and ZðtÞ from the

estimated model

rt ¼ a þ art1 þ et:

In particular,

Zð#aÞ ¼ nð#a 1Þ #l

ðn2Pn

t¼1%r2

t1Þ

with%rt ¼ rt #a and

ZðtÞ ¼#s#$

tð#aÞ #l

#$ðn2Pn

t¼1%r2

t1Þ1=2

;

where tð#aÞ is equal to ð#a 1Þ=s#a; #l is a consistent (kernel) estimate of l ¼P

N

h¼1 Eðe0ehÞ; #$2 is a consistent

(kernel) estimate ofP

N

h¼NEðe0ehÞ; and #s2 is a consistent estimate of Eðe2

t Þ: In the Zð#aÞ and ZðtÞ tests the

number of autocovariance terms to compute the spectrum at frequency zero is set equal to five. We reject

the null hypothesis of a unit root if the statistics #a=s#a; Zð#aÞ; and ZðtÞ are smaller than the critical values or

equal to them. Critical values and corresponding values for the relevant statistics are reported in the table.

The second column contains estimates of the autoregressive parameter, #f say.

#f Test statistic 1% value 5% value 10% value

ADF test 0.997 2.0918 3.4583 2.8710 2.5936

Zð#aÞ test 0.9935 16.2600 19.8270 13.7251 11.0755

ZðtÞ test 0.9935 2.8608 3.4583 2.8710 2.5936

Automatic window width

Zð#aÞ test 0.9935 17.6300 19.8270 13.7251 11.0755

ZðtÞ test 0.9935 2.9781 3.4583 2.8710 2.5936


located in the acceptance region. When we add the more volatile data between 1980and 1982, the overall picture remains quite unequivocally nonstationary, but ourstatistics appear to deliver values closer to the rejection thresholds. The same appliesto the data in the period 1982–1995 versus the longer period 1980–1995.

For the time being, it is safe to stress that standard testing procedures do not offerunambiguous support to the stationarity assumption and justify the use ofinvestigation methods robust to deviations from it.

4.1. A look at the spatial characteristics of the data

Some authors have recently modelled the spot interest rate process as a randomlyshifting process with no fixed mean (see, for example, Das, 1994; Naik and Lee,1993). A.ıt-Sahalia (1996b) proposes a simpler modelling alternative to time-inhomogeneity which is based on the nonlinearity of the drift and diffusionfunctions. The idea is that sufficiently general specifications in the stationary time-homogeneous class can determine multimodal densities resembling regime shifts.Here, we suggest an alternative way to study data that display irregular behavior. Welook at the time spent by the sample process in each point of its range and examinehow this evolves over time. Our underlying process maintains the simplicity of time-homogeneous specifications but, at the same time, is general enough to allow fornonstationarities.

We consider three different time horizons: 1973–1980, 1973–1982, and 1973–1995.Appendix A provides details on the implementation. Our interest in the change thatoccurred between 1980 and 1982 is motivated by the corresponding atypicalbehavior of the series. As briefly mentioned earlier, this period is characterized byhigh volatility (see Fig. 2) and high interest rates (see Fig. 1). These features makeinference more difficult. Below, we clarify the nature of these difficulties.

We start with the period 1973–1980. The spatial density of the process appears tobe bimodal (Fig. 3(a)). The modes show up at around 6% and 11%. When we addthe data from 1980 to 1982 (Fig. 3(b)), we recognize persistence in the previousfeatures and the emerging of two additional modes associated with higher interestrate levels around 15% and 19%. As we move to considering the whole data set, weexpect to find evidence of a prolonged passage of the series below the 4% line butstill in its vicinity. This is confirmed by the height of the estimated spatial density atcorresponding values (Fig. 3(c)).

Given the features of the estimation procedure, in a finite sample we expect to beable to identify well the true functions of interest at points that are visited often.After a quick look at the graph of the estimated local time in the full period 1973–1995 (Fig. 3(c)) we anticipate that problems will arise for interest rates in the 20–24%range, as the time spent by the sample process in this range is fairly small.

In Fig. 4 we report our results for the spatial hazard rate process. In the period1973–1980 we recognize a nonmonotonic increase in the interest rate risk (Fig. 4(a)).Two peaks can be detected, around 6% and 11%. They correspond to the modes ofthe sojourn density in the same period. When we include the observations from 1980to 1982 (Fig. 4(b)), the already identified peaks survive and two new ones emerge


roughly at 16% and 20%. Notice, also, that the confidence bands are broad atinterest rates above the 20% threshold, implying less reliable inference than at lowerrates. It is worth recalling that the same information is contained in the estimatedspatial density. In effect, the empirical process appears not to spend much time in the20–24% range. This justifies the uncertainty embodied in the wide confidence bands.A similar feature will characterize the estimates of the two functions of interest,whose asymptotic confidence bands will be broad around the upper bound of therange of the sample process. Similar considerations apply to the full period 1973–1995 (Fig. 4(c)).

A simple accounting exercise further clarifies the information contained in theestimated spatial densities. Between 16% and 18% we have at our disposal 80observations. The number increases to 114 between 18% and 20%, but it is onlyequal to 28 between 20% and 22%. The number of observations in the 22–24%range is three. Most of the data is concentrated in the 6–8% and 8–10% ranges:1,197 and 1,516, respectively. In what follows we inspect the drift and diffusionfunctions in the range up to 22%. In fact, the dimension of the empirical local time

180

160

140

120

100

80

60

40

20

00.04 0.08 0.12 0.16

Spot rate Spot rate

Loca

l tim

e es

timat

es

Loca

l tim

e es

timat

es

Loca

l tim

e es

timat

es0

20

40

60

80

100

120

140

0.04 0.08 0.12 0.16 0.20 0.24 0.28

Spot rate

0.04 0.08 0.12 0.16 0.20 0.24 0.280

40

80

120

160

200

240

280

320

(c)(b)(a)

Fig. 3. Functional estimates of the local time process of the short-term interest rate series examined in this

study, i.e., %Lrðt; aÞ ¼ ð1=s2ðaÞÞlime-0 ð1=eÞR t

0 1½a;aþe½ðrsÞs2ðrsÞ ds: The data used is the Bank of America

seven-day Eurodollar deposit spot rate midpoint bid–ask. The sample period is June 1, 1973 through

February 25, 1995 (5,505 annualized daily observations). The straight lines are the pointwise

nonparametric estimates of the local time process over the periods 1973–1980, 1973–1982, and 1973–

1995, respectively. The dashed lines are the corresponding 95% asymptotic confidence bands.


of the process for interest rate values above the 22% threshold would make inferencevery unreliable.

4.2. Nonparametric results

Moving to the estimation (see Appendix A for details) of the curves of interest,Figs. 5(a) and (b) plot the nonparametric estimates, with their 95% asymptoticconfidence bands.2

240

200

160

120

80

40

00.04 0.08 0.12 0.16

Spot rate

Loca

l haz

ard

estim

ates

Loca

l haz

ard

estim

ates

200

180

160

140

120

100

80

60

40

20

0Lo

cal h

azar

d es

timat

es

200

180

160

140

120

100

80

60

40

20

00.04 0.08 0.12 0.16 0.20 0.24 0.28

Spot rate

0.04 0.08 0.12 0.16 0.20 0.24 0.28

Spot rate(a) (b) (c)

Fig. 4. Functional estimates of the spatial (local) hazard process of the short-term interest rate series

examined in this study, i.e., %Hrðt; aÞ ¼ %Lrðt; aÞ=RN

a%Lrðt; sÞ ds; where %Lrðt; aÞ ¼ ð1=s2ðaÞÞlime-0 ð1=eÞR t

0 1½a;aþe½ðrsÞs2ðrsÞ ds: The data used is the Bank of America seven-day Eurodollar deposit spot rate

midpoint bid–ask. The sample period is June 1, 1973 through February 25, 1995 (5,505 annualized daily

observations). The straight lines are the pointwise nonparametric estimates of the spatial hazard process

over the periods 1973–1980, 1973–1982, and 1973–1995, respectively. The dashed lines are the

corresponding 95% asymptotic confidence bands.

2 The Monte Carlo evidence in Bandi and Nguyen (1999) demonstrates that the asymptotic distributions

used to derive the confidence bands are satisfactory approximations to the finite sample distributions.

Interestingly, this is particularly true in the drift case. Bandi and Nguyen’s results rely on simulated

processes that have been, or could be, employed as descriptions of the short-term interest rate process in

continuous-time finance. Nonstationary and stationary processes with different levels of persistence are

simulated. The number of observations is set equal to 5,000 to replicate almost 20 years of daily data. This

figure is consistent with the magnitude of the data set used in this paper and in most recent work on the

analysis of the short-term interest rate process such as A.ıt-Sahalia (1996a, b), Jiang (1998), and Stanton

(1997).


The drift term appears to be nonlinear as opposed to most conventionalparametric specifications such as those in Table 1 in A.ıt-Sahalia (1996a), forexample, and in the next subsection. Nevertheless, the process mean-reverts stronglyonly when it approaches the upper bound of its range. Up to values close to 15%, theshort-term rate virtually behaves as a martingale, since the drift is statisticallysignificant but economically negligible. In A.ıt-Sahalia (1996b) the drift reverts atboth ends of the theoretical domain due to the adopted parametric specification but

0.06

0.05

0.04

0.03

0.02

0.01

0.000.00

Diff

usio

n es

timat

esD

rift e

stim

ates

0.04 0.08 0.12 0.16 0.20 0.24

Spot rate

0.00 0.04 0.08 0.12 0.16 0.20 0.24

Spot rate

_1.6

_1.2

_0.8

_0.4

_0.0

0.4

0.8

(a)

(b)

Fig. 5. Functional estimates of the diffusion and drift function of the short-term interest rate series

examined in this study, s2ð:Þ and mð:Þ that is. The data used is the Bank of America seven-day Eurodollar

deposit spot rate midpoint bid–ask from June 1, 1973 through February 25, 1995 (5,505 annualized daily

observations). The straight lines are the pointwise nonparametric estimates. The dashed lines are the

corresponding 95% asymptotic confidence bands. The dotted line is the zero line.


is very close to zero between 3% and 24%, that is over the sample domain. A.ıt-Sahalia (1996a) assumes a linear mean-reverting drift from the start. Stanton (1997)and Jiang (1998) work with a different series but their drift dynamics resemble thegeneral features of our findings. They both use daily values of the secondary marketyields on three-month US Treasury Bills. The time horizon is January 1965–July1995 in Stanton (1997) and January 1962–January 1996 in Jiang (1998). Similar tothe results in A.ıt-Sahalia (1996b) are the drift estimates in Jiang and Knight (1997).Jiang and Knight (1997) use daily data of the Canadian three-month treasury billrate from January 2, 1982 to January 31, 1995. A common feature of this literature isto imply the unpredictability of the short rate over most of its range since the processappears to evolve over time as a martingale. Mean-reversion comes into play only atthe extremities of the sample range.

The nonlinearity of the drift can account for atypical dynamics. In effect, duringthe 1980–1982 period a change in parameters seems to occur (see Fig. 1). A.ıt-Sahalia(1996b) reports that ‘‘y the mean a of the process with drift mðr; a;bÞ ¼ bða rÞestimated over 1980 to 1982 is significantly higher than the mean estimated on therest of the sampley ’’. Our estimated nonparametric drift displays nonlinearities forhigh interest rate values, corresponding to the same period. In general, as A.ıt-Sahalia(1996b) points out, misspecified linear models for the drift can hide nonlinearities aschanges in parameters.

We now turn to the estimates of the diffusion function. The diffusion term exhibitsquite conventional dynamics (CEV diffusions, for example) up to interest rate valuesaround 16%. Volatility tapers off between 18% and 20% and then rises again. Acomparable result is contained in Jiang (1998). Up to 16%, his nonparametricdiffusion mimics the behavior of a parametric CEV diffusion. Above 16%, hisestimates suggest less volatility than implied by a monotonically increasingparametric function in the CEV class. The diffusion function in Jiang and Knight(1997) displays a smile with lower volatility associated with interest rates at the lowerend of the range and in the middle. In A.ıt-Sahalia (1996b) the instantaneousconditional volatility of the process has U-shaped dynamics. It is equal to about1.7% at zero. It decreases to about zero at 11% and then rises steadily. Its domain ofvariation over the sample range is between 1.25% (at 4%) and about 4.3% (at 24%).The estimated diffusion in A.ıt-Sahalia (1996a) is increasing over the range of thesample process but this increase is nonmonotonic. The function has an absolute peakat about 17%. The diffusion function in Stanton (1997) is monotonically increasing.

In the next subsection we will see that a CEV diffusion function with an exponentequal to three, i.e. s2ðrÞ ¼ kr3 where k is a positive constant, appears to match ournonparametric model reasonably well between 3% and about 16%. When pairedwith a drift which is zero over the same range, such a diffusion function determinesnonstationary dynamics. In effect, the model

drt ¼ kr3=2t dBt; ð27Þ

which appears in Cox (1975) and Cox et al. (1980) (CIR, henceforth) impliesnonstationary behavior for the spot interest rate series over ð0;NÞ: Nonetheless, theprocess in Eq. (27) stops evolving as soon as a realization of the Brownian motion


lowers it sufficiently. In other words, Eq. (27) can be a reasonable representation ofthe (late) dynamics of the series, but more reversion to the center of the interest raterange should be built into the model to guarantee recurrence.

To conclude, here we are not taking a stance on stationarity nor, as just pointedout, are we suggesting that Eq. (27) represents a valid description of the data over itsentire admissible range. We simply notice that using a methodology that is robust todeviations from the existence of a time-invariant marginal density, we obtain shapesfor the two functions of interest that give support to the necessity of being cautiousabout the stationarity of the series in question. This point is coherent with the resultsof our preliminary unit-root tests. We now discuss the significance of the estimatednonlinearities at high rates.

4.3. A parametric comparison

A quick look at the overall shape of the two functions suggests that someconventional parametric models might not be severely misspecified. For instance,between 3% and about 16% the dynamics of the diffusion function can be wellreplicated by a conventional CEV model for the instantaneous variance. Thisobservation is important. By not specifying a particular parametric structure,functional methods avoid misspecifications, but do so at the expense of a greaterestimation error than their parametric counterparts.

Here, we undertake a simple exercise to assess whether credible traditionalparametric models lead to reliable inference in the presence of well-behavedfunctions like the ones we have just estimated. The estimation method we use is aconventional GMM (Hansen, 1982). Chan et al. (1992) provide a well-knownapplication to the study of the spot interest rate process. Notice that we do notattempt to estimate consistently parametric continuous-time models by use ofdiscretely sampled data. We only intend to compare the outcome of a naive3 but verycommonplace technique to our previous findings.

We assume, for simplicity, a continuous-time model as in Chan et al. (1992), that iswe parametrize mð:Þ and sð:Þ as

mðrÞ ¼ a0 þ a1r ð28Þ

and

sðrÞ ¼ g0rg1 ð29Þ

3 Discretizations are approximations. The relation between parameters in the continuous-time format

and in the discrete-time analogue is not straightforward (see, for instance, Drost and Werker, 1996;

Nelson, 1990). Recall, also, the temporal aggregation problem in Breeden et al. (1989), Grossman et al.

(1987), and Longstaff (1989). Nevertheless, the error introduced by discretizing is of second-order

importance if changes are measured over very short periods of time (Campbell, 1986; Chan et al., 1992, for

instance). This point provides some justification for using the procedure with daily data (Chan et al. (1992)

use monthly observations).


with g0 > 0: Some conventional models of the short-term riskless rate can be nestedin Eqs. (28) and (29) with appropriate parameter restrictions (see Chan et al., 1992).For instance, a1 ¼ 0 and g1 ¼ 0 deliver the Merton model (Merton, 1973). Provideda1o0 and g1 ¼ 0; Eqs. (28) and (29) give the Ornstein-Uhlenbeck process in Vasicek(1977), whereas a1o0 and g1 ¼ 1=2 characterize the process introduced by Cox et al.(1985). The constant elasticity of variance (CEV) specification proposed by Cox(1975) and Cox and Ross (1976) requires a0 ¼ 0: In Brennan and Schwartz (1980), g1

is equal to one. We consider the discrete-time econometric specification

rtþ1 rt ¼ a0 þ a1rt þ etþ1; ð30Þ

E½etþ1� ¼ 0; E½e2tþ1� ¼ g2

0r2g1t : ð31Þ

We follow Chan et al. (1992) in defining the relevant moment conditions. The resultsof this exercise are reported in Table 5. A graphical comparison between ourfunctional estimates and their parametric counterparts is contained in Figs. 6(a) and(b).

We start with the drift function. All the parameter estimates are significant at the5% level. In Chan et al. (1992) the drift parameter estimates are statisticallyinsignificant, implying that a linear mean-reverting structure fits our data set betterthan the data set examined in Chan et al. (1992).

Obviously, nonlinearities cannot be captured by a linear parametric structure.Nevertheless, nonlinear dynamics do not play a substantial role up to the upperextremity of the range of the sample process. As pointed out earlier, between 3% andabout 15% our nonparametric drift is measured precisely in a tight neighborhood ofzero. Still, the parametric specification displays mild mean-reversion. Wherenonlinearities arise, the unrestricted parametric model seems to mimic sufficientlywell the behavior of the functional estimates with the exception of interest rate levelsabove 20%. Moreover, the parametric curve lies within our nonparametric 95%bands. These are important points. A vanishing nonparametric drift up to about15% implies that the interest rate process behaves as a martingale over a region of itsrange. Further, the tight nonparametric confidence bands in the same region and theshape of the asymptotic bands of the parametric drift function suggest that thedifference between our nonparametric specification and the parametric model isstatistically significant (see Fig. 7).

Less clear-cut is the behavior of the drift function at higher interest rates, that isfrom 15% to 20%, and around the upper bound of the range of the sample process,that is above 20%. Nonlinearities come into play in a region where the available datais fairly thin. Here, the linear parametric model can hardly be rejected on purelystatistical grounds. Nevertheless, as demonstrated by the large nonparametricconfidence bands and by the relatively large parametric bands, the overalluncertainty in this region suggests caution in interpreting our results.

The parametric model is more satisfactory for estimating the diffusion function.The specification is nonlinear. The CEV structure only fails to capture dynamicssuch as those detected for interest rate levels around the upper bound of the sampleprocess. Between 3% and 16%, the nonparametric and the parametric curves


overlap almost perfectly. Not surprisingly, the parametric exponent in the varianceterm is estimated very precisely. This is a conventional result. In Chan et al. (1992) g1

is about 1.5 and almost two standard errors above one. Further, all models which

Table 5

We estimate a linear model for the drift in the constant elasticity of variance class as in Chan et al. (1992)

using the seven-day Eurodollar deposit spot rate bid–ask midpoint (5,505 annualized daily observations

from June 1, 1973 to February 25, 1995). The interest rate model we examine is

drt ¼ ða0 þ a1rtÞ dt þ ðg0rg1t Þ dBt;

where Bt is a standard Brownian motion. We consider the discrete-time econometric specification

rtþ1 rt ¼ a0 þ a1rt þ etþ1;

E½etþ1� ¼ 0; E½e2tþ1� ¼ g2

0r2g1t

and estimate the model by using GMM (Hansen, 1982) as in Chan et al. (1992). We follow Chan et al.

(1992). In particular, define a vector y with elements a0; a1; g20 and g1: Given etþ1 ¼ rtþ1 rt a0 a1rt; let

the vector ftðyÞ be

ftðyÞ ¼

etþ1

etþ1rt

e2tþ1 g2

0r2g1t

ðe2tþ1 g2

0r2g1t Þrt

266664

377775:

If the restrictions implied by the discrete-time model hold, then EðftðyÞÞ ¼ 0: This observation provides us

with a set of four moment conditions that we employ to estimate the parameters of the model. As in Chan

et al. (1992), we define the optimal weighting matrix as

#WnðyÞ ¼1

n

Xn1

t¼1

ftðyÞftðyÞ0 !1

and estimate the asymptotic covariance matrix by

1

n

1

n

Xn1

t¼1

@ftð#ynÞ@y0

!0

ð #Wnð#ynÞÞ1

n

Xn1

t¼1

@ftð#ynÞ@y0

! !1

:

The standard errors of the parameter estimates are in parentheses. We impose various restrictions on the

parameters (bold figures) to obtain standard parametric models for the short-term interest rate process.

The results of Hansen’s (1982) test of overidentifying restrictions are reported in the last column. The

corresponding p-values are in squared brackets.

a0 a1 g20 g1 w2

Model

Unrestricted CEV 0.1320 1.5918 4.1295 1.49

(0.0494) (0.6981) (2.0264) (0.1095)

CIR 0.0776 0.8549 0.0323 0.5 38.37

(0.0488) (0.6901) (0.0023) [5.83e 0.10]

Vasicek 0.0621 0.6614 0.0021 0 59

(0.0488) (0.6903) (0.0001) [1.5e 0.14]

Restricted CEV 0 0.2375 3.3289 1.44 6.98

(0.1340) (1.7105) (0.113) [0.0082]

Brennan and Schwartz (1980) 0.1035 1.1919 0.4136 1 14.27

(0.0489) (0.6911) (0.0279) [0.000158]


make g1o1 are rejected. Our estimated g1 takes on a similar value. Even though amore complex specification is needed to fit the diffusion curve around the upperbound of the interest rate range, the estimated parametric diffusion lies almosteverywhere within our nonparametric asymptotic bands, or very close to them.

0.06

0.05

0.04

0.03

0.02

0.01

0.00

Diff

usio

n es

timat

esD

rift e

stim

ates

0.00 0.04 0.08 0.12 0.16 0.20 0.24

Spot rate

0.00 0.04 0.08 0.12 0.16 0.20 0.24

Spot rate

0.8

0.4

_0.0

_0.4

_0.8

_1.2

_1.6

(a)

(b)

Fig. 6. Comparison between the functional estimates of the drift and diffusion function of the short-term

interest rate series analyzed in this study and a parametric linear model for the drift (i.e., mðrtÞ ¼ a0 þ a1rt)

in the CEV class (i.e., s2ðrtÞ ¼ g20r

2g1t ) as in Chan et al. (1992). The data used is the Bank of America seven-

day Eurodollar deposit spot rate midpoint bid–ask from June 1, 1973 through February 25, 1995 (5,505

annualized daily observations). The straight lines are the pointwise nonparametric estimates. The dashed

lines are the corresponding 95% asymptotic confidence bands. The dotted lines represent the parametric

estimates. We estimate the parametric specification by GMM (Hansen, 1982) as in Chan et al. (1992).


Complications arise when investigating the plausibility of a nonlinear mean-reverting drift at high interest rate levels. First, even assuming stationarity, the driftfunction is not constrained to display a specific shape at high rates provided theelasticity of the CEV instantaneous variance is sufficient to balance the driftdynamics and determine reversion to the center of the stationary distribution of theprocess. This point is made in Conley et al. (1997). They show that what matters formean-reversion is a pull measure defined as the ratio between the drift and two timesthe diffusion function. Stationarity can be volatility-induced. In consequence, shouldthe series be stationary, then the uncertainty related to the lack of sufficientobservations at high rates would add up to the absence of a strong theoreticalmotivation for drift-induced mean-reversion. This would make conclusions on thedynamics of the drift at high rates quite arbitrary. The potential nonstationarity ofthe series complicates matters even further. These observations are somewhatstronger than one of the conclusions put forward in concurrent work by Jones(1998). In a Bayesian framework, he shows that the use of uninformative Jeffreyspriors does not result in statistical evidence for a nonlinear drift unless stationarity isimposed.

0.8

0.4

_0.0

_0.4

_0.8

_1.2

_1.60.00 0.04 0.08 0.12 0.16 0.20 0.24

Spot rate

Drif

t est

imat

es

Fig. 7. Comparison between the nonparametric drift of the short-term interest rate series analyzed in this

study and a linear specification for the drift (i.e., mðrtÞ ¼ a0 þ a1rt) as in Chan et al. (1992). The data used is

the Bank of America seven-day Eurodollar deposit spot rate midpoint bid–ask from June 1, 1973 through

February 25, 1995 (5,505 annualized daily observations). The straight line is the nonparametric drift. The

dashed lines are its 95% asymptotic confidence bands. The dotted line represents the parametric curve.

The dashed–dotted lines are its 95% asymptotic confidence bands. The parametric model is estimated by

GMM (Hansen, 1982) as in Chan et al. (1992).


Second, at the edges of the sample range the drift is more easily biased in smallsamples. As for the upper bound, we know with near certainty that the maximumvalue achieved by the interest rate process in our sample is almost surely lower thanthe theoretical maximum value. Hence, the drift term is almost surely too low at theupper edge of the distribution of the data. The contrary is true for interest rate valuesat the lower bound. Still, the size of the bias depends on the volatility of the sampleprocess. The volatility is very low at the lower edge of the sample but, as discussedearlier, quite high at the upper edge. Therefore, a (downward) bias is more likely tooccur at high interest rates,4 thus strengthening our concerns related to the thin dataavailable.5

Finally, as pointed out earlier, the estimated parametric drift curve lies within ourasymptotic nonparametric confidence bands for high interest rate values. Hence, thesimple unrestricted parametric model proposed here cannot be statistically rejectedfor values in the vicinity of the upper bound of the sample process.

To summarize, small sample biases do not affect our functional estimationprocedure for the drift in the range between 3% and 15%. The spot rate behaves as amartingale up to about 15%. At higher values it mean-reverts nonlinearly but astandard parametric linear mean-reverting model for the drift cannot be rejected inthis region. Our nonparametric results appear statistically unreliable, even thougheconomically sensible, for interest rate values between 20% and 22%. Thisobservation can be applied to most papers in the literature since nonlinearitiesusually play a role in scarcely populated regions of the spot interest rate domain.

5. Conclusion

Using new functional techniques, we provide further statistical evidence ofmartingale behavior for the spot interest rate process over the restricted rangebetween 3% and about 15% (A.ıt-Sahalia, 1996b). This suggests that the short-terminterest rate process displays less predictability than implied by a linear mean-reverting structure for the drift. As for the nonlinearity of the drift at values between16% and 22%, we cannot interpret it as a purely economically driven phenomenon,due to the possibility of a small sample bias in this range. This is particularly true inthe range between 20% and 22%. In effect, the time spent by the empirical process atvalues about its upper edge is very small, thus making conclusions based onstatistical inference potentially arbitrary. In consequence, we suggest caution whendrawing conclusions about the shape of the drift around the upper extreme of the

4 I thank Chris Sims for pointing this out to me.5 In independent and parallel work, Chapman and Pearson (2000) reach a similar conclusion using a

weighted least-squares estimation procedure applied to the data set in this paper and in Stanton (1997).

They also show that the estimation methods proposed by A.ıt-Sahalia (1996b) and Stanton (1997) suggest

nonlinearities of the type reported in the corresponding papers even when applied to sample paths

simulated from a linear mean-reverting square root process. They conclude that the nonlinearity of the

short-term interest rate drift is not a ‘‘stylized fact’’.


empirical range of the process. A parametric CEV structure mimics rather well thebehavior of the diffusion function over most of its sample domain.

In a recent paper, Pritsker (1998) points out that methods based on the estimationof the marginal density of the interest rate process (A.ıt-Sahalia, 1996a, b; Siddique,1994, for example) fail to account for the effects of time-dependence in finitesamples. The method suggested in this work relies on a more general notion ofdensity and the temporal dependence in the trajectory of the short-term interest rateseries of interest plays a role.

It is noted that our functional drift and diffusion functions can be used to testalternative parametric models of the spot interest rate process based on a testingmethodology that matches parametric specifications to their nonparametriccounterparts. Due to the larger identifying information and the generality of spatialmethods, this procedure is likely to have better size properties and more power thantesting methods based on density-matching (Pritsker, 1998).

Extensions along this line are being conducted and will be reported in laterwork.

Appendix A. Econometric estimation

Assumption A1. The kernel Kð:Þ is a Lipschitz continuous, symmetric,and nonnegative function whose derivative K0 is absolutely integrable and forwhichZ

N

N

KðsÞ ds ¼ 1;

ZN

N

K2ðsÞ dsoN; sups

KðsÞoN; ðA:1Þ

and ZN

N

s2KðsÞ dsoN: ðA:2Þ

Assumption A2. The smoothing parameter hn; %T¼1 and the spatial smoothingparameter en; %T¼1 are set as

hn; %T¼1 ¼ch

logðnÞ#srn

kh ; ðA:3Þ

en; %T¼1 ¼ce

logðnÞ#srn

ke ; ðA:4Þ

where #sr is the estimated unconditional standard deviation of the series over theperiod of interest, n is the number of observations, kh; ke are positive exponentsdepending on the limit theory of the specific estimator, and ch; ce are constants ofproportionality. For practical purposes, %T is set equal to 1:

To estimate the sojourn times and the spatial hazard rates, we use a secondorder Epanechnikov kernel as it simplifies calculations for the hazard


functions, i.e.,

KðxÞ ¼ 3=4ð1 x2Þ1f1;1g: ðA:5Þ

We set kltimeh ¼ 1=4: The value for the constant ðcltime

h ; that is) is chosen to minimizethe integrated mean-squared error of the local time estimator for the three timeperiods that we consider. The results are robust to changes of the bandwidth aroundits optimal value. In the periods 1973–1980, 1973–1982, and 1973–1995, thenumerical values of the smoothing parameters are 1.394%, 2.05%, and 2.09%,respectively.

As standard in the literature on the nonparametric estimation of diffusions, weemploy a Gaussian kernel to estimate the drift and the diffusion function. The use ofan Epanechnikov or an exponential kernel would not change qualitatively theresults.

For coherence with kltimeh and c

ltimeð197321995Þh ; the pairs kdrift

h ; cdrifth and kdiff

h ; cdiffh are

chosen equal to 1=4 and to the value that minimizes the integrated mean-squarederror of the local time estimator in the presence of a Gaussian kernel, respectively.The numerical value of hdrift

n; %T¼1and hdiff

n; %T¼1is 1.426%. The exponent of the leading

(spatial) bandwidth kdiffe is set equal to 1=4 to achieve a close-to-optimal rate of

convergence over a fixed time span %T; undersmooth slightly and eliminate theinfluence of the bias term from the limiting distribution. Given cdiff

h ; kdiffh and kdiff

e ; weselect cdiff

e to minimize the integrated mean-squared error of the diffusion estimatorunder the constraint hdiff

n; %T¼1oediff

n; %T¼1(see Section 3). The numerical value of the leading

bandwidth ediffn; %T¼1

is equal to 1.5% and coherent with values already found in theliterature on the estimation of the infinitesimal second moment of continuous-timeprocesses of the diffusion type for the spot interest rate (A.ıt-Sahalia, 1996a).

It is noted that the admissible (and optimal) condition that the drift bandwidthought to satisfy cannot be expressed as a simple function of the number of datapoints ðnÞ: Furthermore, standard automated methods to select the optimalbandwidth might lead to misleading conclusions in the drift case (Bandi andNguyen, 1999). Nonetheless, the discussion in Section 3 (and the simulation resultsin Bandi and Nguyen, 1999) indicate that the optimal leading bandwidth for the driftis generally larger than the corresponding value for diffusion estimation.Furthermore, the magnitude of the optimal smoothing parameter for the drift isdirectly related to the persistence of the process. Table 2 shows that high persistenceis an unquestionable feature of the time series investigated here.

It turns out that bandwidths for the drift that are set larger than ediffn; %T¼1

ð¼ 1:5%Þdeliver almost indistinguishable results in the range between 3% and about 18%,that is where firm statistical conclusions are drawn in the paper. At higher rateslarger smoothing parameters have a slight tendency to flatten the estimated curves,thus confirming our doubts about the estimated nonlinearities in the literature andreinforcing the conclusions put forward in this study. We decided to be conser-vative and report a drift curve estimated using a smoothing parameter that is setlarger than the optimal diffusion bandwidth but relatively close to it (i.e.,edrift

n; %T¼1¼ 1:7%). The results are very robust to alternative, credible bandwidth

choices for the drift.


Appendix B. Proofs

Proof of Theorem 2.1. We assume the same diffusion model as in Florens-Zmirou(1993) and the same sampling scheme as in Section 2. Let %T ¼ 1 for simplicity.Consider the bounded and continuous function mðrÞ: We wish to assess theasymptotic properties offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP½nt�1

i¼1 1fjriDnrjohng

n

s P½nt�1i¼1 1fjriDnrjohngn½ðrðiþ1ÞDn

riDnÞ mðrÞ=n�P½nt�1

i¼1 1fjriDnrjohng

!

¼1ffiffiffiffiffiffiffi2hn

p P½nt�1i¼1 1fjriDnrjohng½ðrðiþ1ÞDn

riDnÞ mðrÞ=n�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

#%Lrðt; rÞq ; ðB:1Þ

where tAð0; 1�; ½nt� denotes the largest integer that is less than or equal to nt

and

#%Lrðt; rÞ :¼

P½nt�1i¼1 1fjriDnrjohng

2nhn

: ðB:2Þ

Define

MðnÞðtÞ ¼1ffiffiffiffiffiffiffi2hn

p X½nt�1

i¼1

1fjriDnrjohng½ðrðiþ1ÞDn riDn

Þ mðrÞ=n� ðB:3Þ

and

miþ1 ¼1ffiffiffiffiffiffiffi2hn

p 1fjriDnrjohng½ðrðiþ1ÞDn riDn

Þ mðrÞ=n�: ðB:4Þ

Hence,

MðnÞðtÞ ¼X½nt�1

i¼1

miþ1: ðB:5Þ

We represent by Ii=n the conditional expectation with respect to the filtration Iri=n ¼

sðrs; spi=nÞ: We know that Iri=nDIB

i=n; where IBi=n is the filtration generated by the

Brownian motion Bs with spi=n: If, under appropriate conditions on hn; thefollowing four expressions hold, namely

X½nt�1

i¼1

Ii=nðmiþ1Þ-p

0; ðB:6Þ

X½nt�1

i¼1

Ii=nðm2iþ1Þ-

pWðrÞ %Lrðt; rÞ; ðB:7Þ


X½nt�1

i¼1

Ii=njmiþ1j3 -

p0; ðB:8Þ

X½nt�1

i¼1

Ii=nðmiþ1biþ1Þ-p

0; ðB:9Þ

where WðrÞ is some function of r; then the sequence of processes ðMðnÞðtÞ;BðnÞðtÞÞ; withMðnÞðtÞ defined as before and BðnÞðtÞ ¼

P½nt�1i¼1 biþ1 with biþ1 ¼ Bðði þ 1Þ=nÞ Bði=nÞ;

converges in distribution to the process ðUWðrÞ %Lrðt;rÞ;BðtÞÞ; where U and B areindependent Brownian motions. We start by verifying Eq. (B.6):

Ii=nðmiþ1ÞpjIi=nðmiþ1Þj ðB:10Þ


p 1fjriDnrjohngIi=n

Z ðiþ1ÞDn

iDn

mðrsÞ ds

��þZ ðiþ1ÞDn

iDn

sðrsÞ dBs mðrÞ=n

�� ðB:11Þ

pconst:1ffiffiffiffiffiffiffi2hn

p 1fjriDnrjohngIi=n supspðiþ1ÞDn

jrs riDnj

" #1

n

��

þ const:1ffiffiffiffiffiffiffi2hn

p 1fjriDnrjohnghn

n

��: ðB:12Þ

We apply the Burkholder-Davis-Gundy (BDG, hereafter) inequality (Revuz andYor, 1994, Exercise 4.13, Section 4) to the random quantity

Ii=n supspðiþ1ÞDn

jrs riDnj

" #ðB:13Þ

to obtain

Ii=nðmiþ1Þpconst:1ffiffiffiffiffiffiffi2hn

p 1fjriDnrjohng1

n

1ffiffiffin

p��

��þ const:

1ffiffiffiffiffiffiffi2hn

p 1fjriDnrjohnghn

n

��: ðB:14Þ

Summing up over i’s, we get

X½nt�1

i¼1

Ii=nðmiþ1Þp const:


2nhn

!ðnhnÞ

1=2 1

nðB:15Þ


þ const:


2nhn

!ðh3=2

n Þ

pconst: Opð1Þh

1=2n

n1=2þ h3=2

n

!-p

0: ðB:16Þ

The stochastic order term in Eq. (B.16) derives from the fact that

#%Lrðt; rÞ :¼


2nhn

¼ Opð1Þ ðB:17Þ

provided nh4n-0 (see Florens-Zmirou, 1993). We now verify Eq. (B.7). We apply

It #o’s formula and add and subtract ð1=2hnÞ1fjriDnrjohngs2ðrÞ=n to obtain

Ii=nðm2iþ1Þ ¼

1

2hn

1fjriDnrjohngIi=n

Z ðiþ1ÞDn

iDn

2ðrs riDnÞmðrsÞ ds

�

þZ ðiþ1ÞDn

iDn

2ðrs riDnÞsðrsÞ dBs þ

Z ðiþ1ÞDn

iDn

ðs2ðrsÞ s2ðrÞÞ ds

�

1

hn

1fjriDnrjohngIi=nðrðiþ1ÞDn riDn

ÞmðrÞ

n

þ1

2hn

1fjriDnrjohngm2ðrÞ

n2þ

1

2hn

1fjriDnrjohngs2ðrÞ

n: ðB:18Þ

We examine the first term. By virtue of the Cauchy-Schwartz (CS, henceforth)inequality and the BDG inequality, write

1

2hn

1fjriDnrjohng Ii=n

Z ðiþ1ÞDn

iDn

2ðrs riDnÞmðrsÞ ds þ

Z ðiþ1ÞDn

iDn

ðs2ðrsÞ s2ðrÞÞ ds

� �

p1

2hn

1fjriDnrjohng

"Ii=n sup

spðiþ1ÞDn

2ðrs riDnÞ

" #20@

1A

1=2

:

� Ii=n

Z ðiþ1ÞDn

iDn

jmðrsÞj ds

� �2 !1=2

þIi=n

Z ðiþ1ÞDn

iDn

ðs2ðrsÞ s2ðriDnÞÞ ds

� �


þZ ðiþ1ÞDn

iDn

ðs2ðriDnÞ s2ðrÞÞ ds

#ðB:19Þ

pconst:1ffiffiffi

np 1fjriDnrjohng

2nhn

þ const: hn

1fjriDnrjohng

2nhn

:ðB:20Þ

Now we sum up over i’s and bound the previous expression by const:ð1=ffiffiffin

pþ

hnÞOpð1Þ; which tends to zero in probability as n-N: The second term is

mðrÞn

1

hn


Þ

¼mðrÞ

n

1

hn

1fjriDnrjohngIi=n

Z ðiþ1ÞDn

iDn

mðrsÞ ds

� �ðB:21Þ

pconst:1

n

1

2nhn

1fjriDnrjohng: ðB:22Þ

Summing up over i’s again, we obtain

mðrÞn

1

hn

X½nt�1

i¼1


Þpconst:1

nOpð1Þ-

p0: ðB:23Þ

The third term is

1

2hn


n2pconst:

1

n

1

2nhn


Then,

1

2nhn

X½nt�1

i¼1


npconst:

1

nOpð1Þ-

p0: ðB:25Þ

The fourth term is

1

2hn

1fjriDnrjohngs2ðrÞ

n¼ s2ðrÞ

1

2nhn

1fjriDnrjohng

� �: ðB:26Þ

Hence,

s2ðrÞX½nt�1

i¼1

1

2nhn

1fjriDnrjohng -ps2ðrÞ %Lrðt; rÞ: ðB:27Þ


In consequence,

X½nt�1

i¼1

Ii=nðm2iþ1Þ-

ps2ðrÞ %Lrðt; rÞ: ðB:28Þ

We now turn to Eq. (B.8). By using previous results, it is easy to show that

X½nt�1

i¼1

Ii=njmiþ1j3 -

p0 ðB:29Þ

as nhn-N: Eq. (B.9) deserves more attention. Write

Ii=nðmiþ1biþ1Þp jIi=nðmiþ1biþ1Þj


p 1fjriDnrjohng

� jIi=n½ððrðiþ1ÞDn riDn

Þ mðrÞ=nÞðBðiþ1ÞDn BiDn

Þ�j ðB:30Þ

p1ffiffiffiffiffiffiffi2hn

p 1fjriDnrjohngjIi=n½ðrðiþ1ÞDn riDn

ÞðBðiþ1ÞDn BiDn

Þ�j

þ1ffiffiffiffiffiffiffi2hn

p 1fjriDnrjohngmðrÞ

n

��jIi=nðBðiþ1ÞDn

BiDnÞj: ðB:31Þ

Since IrtDIB

t ; the Brownian motion Bt is a martingale with respect to the filtrationIr

t generated by frs; sptg: Hence, Ii=nðBðiþ1ÞDn BiDn

Þ ¼ 0: A simple application ofthe CS inequality gives us

Ii=nðmiþ1biþ1Þp1ffiffiffiffiffiffiffi2hn

p 1fjriDnrjohngjðIi=nðrðiþ1ÞDn riDn

Þ2Þ1=2

� ðIi=nðBðiþ1ÞDn BiDn

Þ2Þ1=2j ðB:32Þ

p1ffiffiffiffiffiffiffiffiffiffi2nhn

p 1fjriDnrjohngðIi=nðrðiþ1ÞDn riDn

Þ2Þ1=2: ðB:33Þ

From previous results we know that the order of magnitude of Ii=nðrðiþ1ÞDn riDn

Þ2 is1=n; then

Ii=nðmiþ1biþ1Þpconst:1ffiffiffiffiffiffiffiffiffiffi2nhn

p 1fjriDnrjohng1ffiffiffi

np pconst:

h1=2n

2nhn


Hence,

X½nt�1

i¼1

Ii=nðmiþ1biþ1Þpconst: h1=2n

X½nt�1

i¼1

1fjriDnrjohng

2nhn

pconst: h1=2n Opð1Þ-

p0:

ðB:35Þ


This last inequality verifies Eq. (B.9). To conclude, if hn-0 as n-N in sucha way that nhn-N and nh4

n-0; then Eqs. (B.6)–(B.8) guarantee that MðnÞðtÞconverges to the local martingale MðtÞ with increasing process s2ðrÞ %Lrðt; rÞ:Such martingale is equivalent in law to the time-changed Brownian motionUs2ðrÞ %Lrðt;rÞ (Revuz and Yor, 1994, Theorem 1.6, Section 5). Furthermore, Eq. (B.9)implies that Us2ðrÞ %Lrðt;rÞ (i.e., the so-called Dambis, Dubins-Schwartz Brownianmotion of M) and Bt are independent (Revuz and Yor, 1994, Theorem 1.9,Section 5). Then,

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP½nt�i¼1 1fjriDnrjohng

n

sð #mðnÞðrÞ mðrÞÞ ) Nð0;s2ðrÞÞ: ðB:36Þ

But this last expression gives us

ffiffiffiffiffihn

pð #mðnÞðrÞÞ ) ð %Lrðt; rÞÞ

1=2Nð0; 1=2s2ðrÞÞ ðB:37Þ

and

#mðnÞðrÞ ¼ Op1ffiffiffiffiffihn

p !

: ðB:38Þ

This proves the stated result. &

Proof of Theorem 3.1. Here, and in the proof of Theorem 3.2 below, we assume thedynamic evolution of the short-term interest rate process is described by Eq. (5).Further, we assume Assumption 1 in BP (1998) is satisfied. We write the estimationerror decomposition of the local time estimator as follows:


p ð #%Lrð %T; rÞ %Lrð %T; rÞÞ

¼1ffiffiffiffiffiffiffiffihn; %T

p ZN

N

KðqÞ1

s2ðhn; %Tq þ rÞ

� �Lrð %T; hn; %Tq þ rÞ dq %Lrð %T; rÞ

� �

þ Oa:s:1

h3=2

n; %T

ðDn; %T logð1=Dn; %TÞÞ1=2

0@

1A: ðB:39Þ

We omit the stochastic order term since it is negligible in the limit under theassumptions made on the bandwidth parameter hn; %T: Then,


p ZN

N

KðqÞ1

s2ðhn; %Tq þ rÞ

� �Lrð %T; hn; %Tq þ rÞ dq %Lrð %T; rÞ

� �ðB:40Þ


¼Z

N

N

KðqÞ1

s2ðhn; %Tq þ rÞ

� �2

1


p ðLrð %T; hn; %Tq þ rÞ Lrð %T; rÞÞ dq

!

þ1ffiffiffiffiffiffiffiffihn; %T

p ZN

N

KðqÞs2ðrÞ s2ðhn; %Tq þ rÞs2ðhn; %Tq þ rÞs2ðrÞ

� �Lrð %T; rÞ dq

� �

¼ A þ B: ðB:41Þ

By the Lipschitz property of the diffusion function (see Assumption 1 in BP, 1998)the term B is negligible as n-N (and hn; %T-0). We now examine the term A usingLemma 4 in BP (1998). Write

ZN

0

KðqÞ1

s2ðrÞ þ oa:s:ð1Þ

� �2

1



!

þZ 0

N

KðqÞ1

s2ðrÞ þ oa:s:ð1Þ

� �2

1



!

) 2

ZN

0

KðqÞ1

s2ðrÞ

� �B"ðLrð %T; rÞ; qÞ dq

þ 2

Z 0

N

KðqÞ1

s2ðrÞ

� �B#ðLrð %T; rÞ;qÞ dq; ðB:42Þ

where B" and B# are independent standard Brownian sheets from thePapanicolaou, Stroock, and Varadhan theorem (see Revuz and Yor, 1994, Theorem2.6, Section 13). But,

2

ZN

0

KðqÞ1

s2ðrÞB"ðLrð %T; rÞ; qÞ dq

¼d 21

sðrÞ

� �2

ðLrð %T; rÞÞ1=2

ZN

0

KðqÞB"ð1; qÞ dq ðB:43Þ

¼d 21

sðrÞ

� �ð %Lrð %T; rÞÞ

1=2

ZN

0

KðqÞBðqÞ dq ðB:44Þ

¼d 21

sðrÞ


1=2N 0;

ZN

0

ZN

0

minðs; qÞKðqÞKðsÞ dq ds

� �ðB:45Þ


and, in consequence,


p ð #%Lrð %T; rÞ %Lrð %T; rÞÞ

)1

sðrÞ


1=2N 0; 8

ZN

0

ZN

0

minðs; qÞKðqÞKðsÞ dq ds

� �: ðB:46Þ

This proves the stated result. &

Proof of Theorem 3.2. Using the method of proof of Theorem 3.1 above, we writethe estimation error decomposition as


p ð #%Hrð %T; rÞ %Hrð %T; rÞÞ


p #%Lrð %T; rÞRN

r#%Lrð %T; sÞ ds

%Lrð %T; rÞR

N

r%Lrð %T; sÞ ds

!ðB:47Þ


p #%Lrð %T; rÞRN

r#%Lrð %T; sÞ ds

%Lrð %T; rÞR

N

r#%Lrð %T; sÞ ds

!þ Opð

ffiffiffiffiffiffiffiffihn; %T

qÞ: ðB:48Þ

The termRN

r#%Lrð %T; sÞ ds can be studied as follows:Z

N

r

#%Lrð %T; sÞ ds

¼Z

N

r

%T

nhn; %T

Xn

i¼1

Ks riDn; %T

hn; %T

� �ds ðB:49Þ

¼Z

N

r

1

hn; %T

Z %T

0

Ks ru

hn; %T

� �du

!ds þ oa:s:ð1Þ ðB:50Þ

¼Z

N

r

ZN

N

1

hn; %T

Ks u

hn; %T

� �1

s2ðuÞLrð %T; uÞ du

� �ds þ oa:s:ð1Þ ðB:51Þ

¼Z

N

r

ZN

N

KðcÞ1

s2ðs chn; %TÞLrð %T; s chn; %TÞ dc

� �ds þ oa:s:ð1Þ ðB:52Þ

-a:s:Z

N

r

%Lrð %T; sÞ ds: ðB:53Þ

The result in Eq. (B.53) derives from an application of the occupation time formula(see Revuz and Yor, 1994, Corollary 1.6, Section 6) and dominated convergencegiven the properties of the kernel from Assumption A1. Hence, by Theorem 3.1


provided 1=h3=2

n; %T

� �Dn; %T log 1=Dn; %T

� �� 1=2-0

� �; it follows that


p ð #%Hrð %T; rÞ %Hrð %T; rÞÞ


p #%Lrð %T; rÞ %Lrð %T; rÞRN

r%Lrð %T; sÞ ds þ oa:s:ð1Þ

!þ opð1Þ ðB:54Þ

) MN 0;1

sðrÞ

� �28k %Lrð %T; rÞ

ðRN

r%Lrð %T; sÞ dsÞ2

!ðB:55Þ

¼d MN 0;1

sðrÞ

� �28kð %Hrð %T; rÞÞ

2

%Lrð %T; rÞ

!; ðB:56Þ

where k ¼RN

0

RN

0 minðs; qÞKðsÞKðqÞ ds dq: This proves the stated result. &

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short-term interest rate dynamics: a spatial approach

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