Journal of Empirical Finance 19 (2012) 595–609

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Journal of Empirical Financejournal homepage: www.elsevier.com/locate/jempfin

Nonparametric estimation of scalar diffusion models of interest rates using
asymmetric kernels☆
Nikolay Gospodinov a,⁎, Masayuki Hirukawa b, 1ab

Department of Economics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G 1M8Faculty of Economics, Setsunan University, 17-8 Ikeda Nakamachi, Neyagawa, Osaka, 572-8508, Japana r t i c l e

i n f oArticle history:
Received 21 September 2011
Received in revised form 29 February 2012
Accepted 2 April 2012
Available online 11 April 2012
JEL classification:
C13
C14
C22
E43
G13

a b s t r a c t
This paper proposes an asymmetric kernel-based method for nonparametric estimation of scalardiffusion models of spot interest rates. We derive the asymptotic theory for the asymmetric kernel
estimators of the drift and diffusion functions for general and positive recurrent processes andillustrate the advantages of the Gamma kernel for bias correction and efficiency gains. The finitesample properties and the practical relevance of the proposed nonparametric estimators for bondand option pricing are evaluated using actual and simulated data for U.S. interest rates.
© 2012 Elsevier B.V. All rights reserved.

Keywords:
Nonparametric regression
Gamma kernel
Diffusion estimation
Spot interest rate
Derivative pricing

1. Introduction
Nonparametric methods have attracted growing attention in econometrics due to their flexibility for handling possible
nonlinearities in conditional moment function estimation. This paper focuses on nonparametric estimation of time-homogeneous
drift and diffusion functions in continuous-time models that are used to describe the underlying dynamics of spot interest rates.
More specifically, we improve the nonparametric estimators of drift and diffusion functions of Stanton (1997) by means of anonstandard smoothing technique. The main source for this improvement comes from asymmetric kernel functions which areemployed in place of standard symmetric kernels.
Kernel smoothing has been widely applied for estimating continuous-time diffusion processes; examples of recent contributions
include Florens-Zmirou (1993), Aït-Sahalia (1996a,b), Jiang and Knight (1997), Stanton (1997), Chapman and Pearson (2000), Bandi☆ We would like to thank the Co-Editor (Franz Palm), three anonymous referees, Evan Anderson, Zongwu Cai, Bruce Hansen, Arthur Lewbel, Peter Phillips,
Jeroen Rombouts, Ximing Wu, and participants at the 2007 MEG, 2008 CESG and 2008 SEA meetings and seminars at DePaul, Loyola, Purdue, and Queen's
University for helpful comments. The ﬁrst author gratefully acknowledges ﬁnancial support from FQRSC, IFM2 and SSHRC. The second author gratefully
acknowledges ﬁnancial support from Japan Society of the Promotion of Science (grant number 23530259).
⁎ Corresponding author. Tel.: + 1 514 848 2424x3935; fax: + 1 514 848 4536.
E-mail addresses: nikolay.gospodinov@concordia.ca (N. Gospodinov), hirukawa@econ.setsunan.ac.jp (M. Hirukawa).
URL's: http://alcor.concordia.ca/~gospodin/ (N. Gospodinov), http://www.setsunan.ac.jp/~hirukawa/ (M. Hirukawa).
1Tel.: + 81 72 839 8095; fax: + 81 72 839 8138.
0927-5398/$ – see front matter © 2012 Elsevier B.V. All rights reserved.
doi:10.1016/j.jempﬁn.2012.04.001

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Fig. 1. Shapes of the Gamma kernel function with a fixed smoothing parameter (0.02) at various design points.

(2002), Bandi and Phillips (2003), Fan and Zhang (2003), Nicolau (2003), and Arapis and Gao (2006). An interesting situation that
often arises in economics and finance is when the conditioning variables are nonnegative (i.e., they have a natural boundary at theorigin) such as nominal interest rates, volatility, etc. In this case, the Nadaraya–Watson (“NW”) regression estimator based on astandard symmetric kernel is not appropriate without a boundary correction for the region near the origin. This has generated a large
literature on correction methods for boundary effects in nonparametric regression estimation which includes Gasser and Müller
(1979), Rice (1984), Müller (1991), Fan and Gijbels (1992), Fan (1993), among others.
As a viable alternative to these boundary correction methods, Chen (2000) and Scaillet (2004) advocate the use of asymmetrickernel functions 2 that have the same support as the marginal density of the conditioning variables. In addition to eliminatingboundary effects, asymmetric kernels have a number of important advantages in the analysis of economic and financial data.
First, the shape of the asymmetric kernels varies according to the position at which smoothing is made; in other words, theamount of smoothing changes in an adaptive manner. Fig. 1 plots the shapes of a Gamma kernel function for five different design
points (x = 0.00, 0.02, 0.04, 0.08 and 0.16) at which the smoothing is performed. 3 It is worth noting that for all plotted functions,the smoothing parameter value of the Gamma kernel is fixed. In contrast, the amount of smoothing by symmetric kernels with asingle bandwidth parameter is fixed everywhere. Since the distribution of interest rates is empirically characterized by a modenear the boundary and the sparse tail region, a single bandwidth does not suffice in this situation; while it is appropriate toemploy a short bandwidth in the region near the boundary, a longer bandwidth is required to capture the shape of the tail. As aresult, when the data indicate that the distribution contains both dense and sparse regions, it may prove useful to turn to variablebandwidth methods (e.g. Abramson, 1982; Fan and Gijbels, 1992). The adaptive smoothing property of asymmetric kernels bears
some similarities to the variable bandwidth methods, but unlike these methods, the adaptive smoothing for asymmetric kernels isachieved by a single smoothing parameter which makes them much more appealing in empirical work.
Second, asymmetric kernels achieve the optimal rate of convergence (in mean integrated squared error sense) within the classof nonnegative kernel estimators. Third, unlike the case with symmetric kernels, the variances of asymmetric kernel estimators
tend to decrease as the design point moves away from the boundary. This property is particularly advantageous for the analysis ofinterest rate data since the support of the density has a sparse region for high interest rate values. In sum, we can view
asymmetric kernels as a combination of a boundary correction device and a “variable bandwidth” method.

2
Strictly speaking, asymmetric kernel functions should be referred to as kernel-type weighting functions. In a slightly different context, Gouriéroux andMonfort (2006) and Jones and Henderson (2007) argue that unlike the case of symmetric kernels, the roles of the data point X and the design point x in
asymmetric kernels are not exchangeable, which leads to a lack of normalization in density estimation using these kernels. Nevertheless, we follow the adopted
convention in the literature for these kernel-type functions. Also, note that the lack of normalization is not an issue in regression estimation.
3
These design points are chosen from the support of the distribution of interest rates.

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Although asymmetric kernels are relatively new in the literature, several papers report favorable evidence from applying themto empirical models in economics and finance. A non-exhaustive list includes: (i) estimation of recovery rate distributions on
defaulted bonds (Renault and Scaillet, 2004), (ii) income distribution estimation (Bouezmarni and Scaillet, 2005; Hagmann and
Scaillet, 2007); (iii) actuarial loss distribution estimation (Gustafsson et al., 2009; Hagmann and Scaillet, 2007); (iv) hazard
estimation (Bouezmarni and Rombouts, 2008); (v) regression discontinuity design (Fé, 2010); and (vi) realized integrated
volatility estimation (Kristensen, 2010).
Incorporating asymmetric kernel smoothing into the nonparametric drift and diffusion estimators of Stanton (1997) is
expected to shed additional light on the nonparametric estimation of spot rate diffusion models. The Monte Carlo simulations in
Chapman and Pearson (2000) indicate that when the true drift is linear in the level of the spot interest rate, there are two biases in
Stanton's (1997) symmetric kernel estimate of the drift function; namely, a bias near the origin and a pronounced downward bias
in the region of high interest rates where the data are sparse. While the existing boundary correction methods may provide a
remedy for improving the performance of the drift estimate near the boundary, they are of little help for removing the bias that
occurs for high values of the spot rate. In contrast, given the appealing properties of asymmetric kernels listed above, asymmetrickernel smoothing is expected to reduce substantially both of these biases in the estimate of the drift function.
The remainder of the paper is organized as follows. Section 2 develops the asymmetric kernel estimators of the drift and diffusion
functions and establishes their asymptotic properties. It also discusses the smoothing parameter selection and the implementation ofa bootstrap procedure for inference. In Section 3, the proposed estimators are implemented to study the dynamics of the U.S. risk-freerate. It also reports the results from a Monte Carlo simulation experiment that examines the finite-sample performance of thesymmetric and asymmetric kernel estimators of diffusion models and their practical relevance for bond and option pricing. Section 4summarizes the main results of the paper. Assumptions and proofs are provided in the Appendix.
2. Estimation of scalar diffusion models via asymmetric kernel smoothing
2.1. NW estimators using the Gamma kernel
The estimation of scalar diffusion models plays an important role in the analysis of term structure of interest rates andderivative pricing. The general form of the underlying continuous-time process for the spot rate Xt is represented by the stochastic
differential equation (“SDE”)
dX t ¼ μ ðX t Þdt þ σ ðX t ÞdW t ;ð1Þ

where Wt is a standard Brownian motion, μ(Xt) is the drift function and σ(Xt) is the diffusion (or instantaneous volatility)
function. We assume that the initial condition X0 is fixed, i.e. X0 = x0 ∈ (0, ∞). 4
Stanton (1997) develops a nonparametric approach to estimating the dynamics of spot interest rate driven by the scalardiffusion process (1). More specifically, Stanton (1997) uses the infinitesimal generator and a Taylor series expansion to express
the first-order approximations to μ(X) and σ 2(X) as
E X tþΔ −X t jX t Þ ¼ μ ðX t ÞΔ þ oðΔÞ;

ð2Þ

h
2
2
E X tþΔ −X t jX t  ¼ σ ðX t ÞΔ þ oðΔÞ;

ð3Þ

where Δ is a discrete, arbitrarily small, time step and o(Δ) denotes a remainder term which goes to 0 as Δ → 0. 5 Each of these
conditional expectations at the design point x can be estimated by running a nonparametric regression. The approach by Stanton(1997) is convenient because it enables us to estimate μ(⋅) and σ2(⋅) separately, whereas the approaches by Aït-Sahalia (1996a) andJiang and Knight (1997) require their sequential estimation; see Arapis and Gao (2006) for a comparison of these three methods.
Since the conditioning variable Xt is nonnegative, it is not appropriate to employ symmetric kernels for estimating μ(X) andσ 2(X) without a boundary correction. The key building block for nonparametric drift and diffusion estimation that we propose inthis paper is the NW kernel regression estimator based on the Gamma kernel function (Chen, 2000) 6
x=b

K Gðx=bþ1;bÞ ðuÞ ¼uexpð−u=bÞ

x=bþ1

b

Γ ðx=b þ 1Þ

1fu ≥ 0g;

ð4Þ

where Γ(α) = ∫ 0∞ y α − 1 exp(− y)dy, α > 0 is the Gamma function and b is the smoothing parameter. Since the density of a Gamma
distribution has support [0, ∞), the Gamma kernel function does not generate a boundary bias.
4This particular initialization is used in Florens-Zmirou (1993), among others. Alternative initializations are also possible.
Stanton (1997) also provides higher-order approximations to μ(X) and σ2(X) and argues that they are more accurate. However, Fan and Zhang (2003) showthat higher-order approximations tend to reduce biases at the expense of inﬂating variances nearly exponentially, and thus we focus only on ﬁrst-orderapproximations.
6
Two other asymmetric kernels, namely, the Inverse Gaussian and Reciprocal Inverse Gaussian kernels (Scaillet, 2004), can be also applied in this context, but
we focus exclusively on the Gamma kernel due to its better ﬁnite-sample performance (see Gospodinov and Hirukawa, 2007).
5

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Suppose that we observe a discrete sample {XiΔ}in= 1 at n equally spaced time points from the short-rate diffusion process{Xt : 0 ≤ t ≤ T} satisfying the SDE (1). Here Δ is the step size between observations and T = nΔ is the time span of the sample. Givena design point x > 0, we define the Gamma-NW estimator of the drift μ(X) and diffusion function σ 2(X) asn−1
1 ∑i¼1 X ðiþ1ÞΔ −X iΔ K Gðx=bþ1;bÞ ðX iΔ Þμ^ b ðxÞ ¼
Δ∑n−1
i¼1 K Gðx=bþ1;bÞ ðX iΔ Þ

ð5Þ

and
1
2
σ^ b ðxÞ ¼


2
n−1
∑i¼1
X ðiþ1ÞΔ −X iΔ K Gðx=bþ1;bÞ ðX iΔ ÞΔ

∑n−1
i¼1 K Gðx=bþ1;bÞ ðX iΔ Þ:ð6Þ

2.2. Asymptotic properties of Gamma-NW estimators
In this section, we present some large sample results for the Gamma-NW estimator in the diffusion model (1). Following Bandiand Phillips (2003; abbreviated as “BP” hereafter), we explore the in-fill and long span asymptotics such that n → ∞, T → ∞ (longspan), and Δ = T/n → 0 (in-fill). It is worth mentioning that the asymptotic results in BP are based on the assumption of symmetric,
nonnegative kernel functions; see Assumption 2 in BP. None of the asymmetric kernels proposed by Chen (2000) and Scaillet (2004)
can be expressed in the form of K(X − x ; b) for a data point X, design point x, and smoothing parameter b. Therefore, establishing thelimiting theory for the Gamma estimator does not directly follow from BP and is new to the literature.
The first results are based on the assumption that the short-rate process Xt is recurrent. Let S(x) denote the natural scale
function of Xt which is defined, for some generic constant c ∈ (0, ∞), as 
2μ ðuÞ
xySðxÞ ¼ ∫c exp ∫c − 2
du dy;σ ðuÞð7Þ

and s(x) be the speed function of Xt given bysðxÞ ¼

2:
σ 2 ðxÞS′ ðxÞð8Þ

Because the range of Xt is (0, ∞), Xt becomes recurrent if and only if limx → 0S(x) = − ∞ and limx → ∞S(x) = ∞ (Assumption 1(iii)in Appendix A). The full set of regularity conditions is provided in Appendix A.
In the subsequent analysis, the concept of local time plays a key role. The chronological local time of the diffusion process (1) isdefined as
L X ðT; xÞ ¼

1
1 T
2
lim ∫ 1½x;xþÞ ðX s Þσ ðX s Þds
σ 2 ðxÞ →0  0ð9Þfor every x > 0, where 1A(⋅) denotes the indicator function on the set A. L X ðT; xÞ is a normalized measure of the time spent by Xt in
a:s:the vicinity of a generic point x. Moreover, when Xt is strictly stationary, L X ðT; xÞ=T → f ðxÞ, where f(x) is the time-invariant
marginal density of Xt; see Bosq (1998, Theorem 6.3), for instance. In contrast, for a general recurrent diffusion, L X ðT; xÞ diverges toinfinity at a rate no faster than T. Using the Gamma kernel, L X ðT; xÞ for a fixed T can be (strongly) consistently estimated byXL^X ðT; x; bÞ ¼ Δ
K Gðx=bþ1;bÞ ðX iΔ Þ:
n

ð10Þ

i¼1

To describe the different asymptotic properties of the estimator depending on the position of the design point x, we denote by“interior x” and “boundary x” a design point x that satisfies x/b → ∞ and x/b → κ for some κ > 0 as n, T → ∞, respectively. Also, let “⇒”denote weak convergence to a random sequence. Theorem 1 states the asymptotic properties of the Gamma-NW estimator for
general recurrent diffusions.
Theorem 1. (general recurrent case)(i) (Drift estimator)
5=2
Suppose that Assumptions 1 and 3 in Appendix A hold. In addition, if b LX ðT; xÞ ¼ Oa:s: ð1Þ, then!
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃno
σ 2 ðxÞ
R
1=2 ^^b L X ðT; x; bÞ μ b ðxÞ−μ ðxÞ−Bμ ðxÞb ⇒N 0; pﬃﬃﬃ 1=2
2 πx

ð11Þ

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599

for interior x, and!
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
Γ ð2κ þ 1Þσ 2 ðxÞbL^X ðT; x; bÞ μ^ b ðxÞ−μ ðxÞ ⇒N 0; 2κþ1 2
Γ ðκ þ 1Þ
2

ð12Þ

for boundary x, where BμR(x) = μ′(x){1 + xs′(x)/s(x)} + (x/2)μ″(x).
(ii) (Diffusion estimator)
5=2
Suppose that Assumptions 1 and 3 in Appendix A hold. In addition, if b LX ðT; xÞ=Δ ¼ Oa:s: ð1Þ, thensﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!
4o
b1=2 L^X ðT; x; bÞn 2
σ ðxÞ
2
R
σ^ b ðxÞ−σ ðxÞ−Bσ 2 ðxÞb ⇒N 0; pﬃﬃﬃ 1=2Δ
πxfor interior x, andsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!
o ðT; x; bÞn 2bL^Γ ð2κ þ 1Þσ 4 ðxÞ
2Xσ^ b ðxÞ−σ ðxÞ ⇒N 0; 2κ 2Δ
2 Γ ðκ þ 1Þ

ð13Þ

ð14Þ

for boundary x, where BσR2 (x) = {σ 2(x)}′{1 + xs′(x)/s(x)} + (x/2){σ 2(x)}′′.
Proof. See Appendix B.
Theorem 1 establishes the limiting behavior of the drift and diffusion function estimators when a single smoothing parameter b isused for both interior and boundary regions. For each of the results, the rate on b balances orders of magnitude in squared bias andvariance for interior x. This rate undersmooths the curve for boundary x, and as a consequence, the O(b) leading bias term becomes
asymptotically negligible. The theorem also suggests that the diffusion estimator has a faster convergence rate than the drift
estimator, regardless of the position of the design point x. This result is consistent with Remark 8 in BP. However, the chronologicallocal time is random, and thus convergence rates of the two estimators are path-dependent (BP, Remark 4).
The existing literature often assumes stationarity or ergodicity of the process Xt. To derive the distributional theory of the GammaNW estimator in this case, we impose the additional condition that ∫ 0∞ s(x)dxb ∞ (Assumption 2 in Appendix A). Using this assumption,
it can be demonstrated that L^X ðT; x; bÞ=T a:s:
→ f ðxÞ and that s′(x)/s(x)= f′(x)/f(x). Accordingly, we have the following corollary.
Corollary 1. (positive recurrent case).
(i) (Drift estimator)
Suppose that Assumptions 1, 2 and 3 in Appendix A hold. In addition, if b = O(T − 2/5), then!
2
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃn
o
σ ðxÞ
S
1=2 ^
μ b ðxÞ−μ ðxÞ−Bμ ðxÞb ⇒N 0; pﬃﬃﬃ 1=2Tb
2 π x f ðxÞ

ð15Þ

for interior x, and!
pﬃﬃﬃﬃﬃﬃ
Γ ð2κ þ 1Þσ 2 ðxÞ^
Tb μ b ðxÞ−μ ðxÞ ⇒N 0; 2κþ1 2
Γ ðκ þ 1Þf ðxÞ2

ð16Þ

for boundary x, where BμS(x) = μ′(x){1 + xf′(x)/f(x)} + (x/2)μ′′(x).
(ii) (Diffusion estimator)
Suppose that Assumptions 1, 2 and 3 in Appendix A hold. In addition, if b = O(n − 2/5), then!
opﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃn
σ 4 ðxÞ2S
1=2 ^ 2σ b ðxÞ−σ ðxÞ−Bσ 2 ðxÞb ⇒N 0; pﬃﬃﬃ 1=2nb
πx f ðxÞ

ð17Þ

for interior x, and!
opﬃﬃﬃﬃﬃﬃn 2
Γ ð2κ þ 1Þσ 4 ðxÞ
2nb σ^ b ðxÞ−σ ðxÞ ⇒N 0; 2κ 2
2 Γ ðκ þ 1Þf ðxÞ
for boundary x, where BσS 2(x) = {σ 2(x)}′{1 + xf′(x)/f(x)} + (x/2){σ 2(x)}′ ′.

ð18Þ

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In Corollary 1, the leading bias terms in μ^ b ðxÞ and σ^ 2b ðxÞ for interior x, namely, BμS(x)b and BσS 2(x)b, are from nonparametricestimation of the conditional expectation using the Gamma kernel; on the other hand, the bias terms for boundary x become
asymptotically negligible due to undersmoothing as in Theorem 1. While the bias terms are of order O(b), it follows from Δ = T/n → 0
that μ^ b ðxÞ has a slower convergence rate of order O(T − 1/2b− 1/4) for interior x and O(T − 1/2b− 1/2) for boundary x. Therefore, if thesample size is not sufficiently large, it is much harder to estimate the drift accurately, especially for design points in the region of highvalues where the data are sparse. However, the property of asymmetric kernel smoothing that the variance of the estimator decreases
with x, is expected to provide an effective remedy.
Furthermore, Corollary 1 and the trimmingargument
squared error (“MISE”) of each in Chen (2000) yield the mean integrated

2
−1=2
2
−1=2
estimator given by MISE μ^ b ðxÞ ¼ O b þ T −1 band MISE σ^ 2b ðxÞ ¼ O b þ n−1 b. Then, the MISE-optimal smoothing
parameters are b*μ = O(T − 2/5) and b*σ 2 = O(n − 2/5) for μ^ b ðxÞ and σ^ 2b ðxÞ, respectively. This result suggests that a longer smoothing
parameter is required to estimate the drift, as documented in Chapman and Pearson (2000, p. 367). As a consequence, the optimal
MISEs are of order MISE μ^ b ðxÞ ¼ O T −4=5 and MISE σ^ 2b ðxÞ ¼ O n−4=5 .
2.3. Selection of smoothing parameter
The practical implementation of the proposed nonparametric estimator requires a choice of a smoothing parameter.7 Indeed, as
argued in Chapman and Pearson (2000), the performance of the NW estimator depends crucially on the choice of a bandwidthparameter, as well as the persistence of the data. In this respect, asymmetric kernels are not an exception, even if they possess theproperty of adaptive smoothing with a single smoothing parameter. While the result in the previous section provides some guidance inthis direction, the expression for the optimal smoothing parameter in mean squared error (“MSE”) or MISE sense involves unknown
quantities and a “plug-in rule” is difficult to obtain. Note also that the MSE-optimal smoothing parameter depends explicitly on thedesign point and, in principle, they should take different values at each x. Hagmann and Scaillet (2007), however, argue for a single,
global smoothing parameter since the dependence on the design point x may deteriorate the adaptability of asymmetric kernels.
In this paper, we adopt a cross-validation (“CV”) approach to choosing a global smoothing parameter for nonparametric curve
estimation based on asymmetric kernels. Since the data are dependent, the leave-one-out CV is not appropriate. Instead, we workwith the h-block CV version of Györfi et al. (1989) and Burman et al. (1994), where h data points on both sides of the s-th observation
are removed from T observations {Xt}tT= 1 and the unknown function of interest m(Xs) is estimated from the remaining T − (2h + 1)
observations. The idea behind this method is that for ergodic processes, the blocks of length h are asymptotically independent
although the block size may need to shrink (at certain rate) relative to the total sample size in order to ensure the consistency of theprocedure. Hall et al. (1995) establish the asymptotic properties of the h-block CV method.
^ −ði−hÞΔ:ðiþhÞΔ ðX iΔ Þ denote the estimate of the drift μ(XiΔ) or the
In our context with equally spaced observations {XiΔ}in= 1, let m
diffusion σ2(XiΔ) from n − (2h + 1) observations {XΔ, X2Δ, …, X(i − h − 1)Δ, X(i + h + 1)Δ, …, XnΔ(=XT)}. Then, the smoothing parametercan be selected by minimizing the least squares cross-validation functionCVðbÞ ¼ arg min
b∈B

n
−h
X

n
o2
^ −ði−hÞΔ:ðiþhÞΔ ðX iΔ Þ ψðX iΔ Þ;
Y iΔ −m

ð19Þ

i¼hþ1


2
where YiΔ = (X(i + 1)Δ − XiΔ)/Δ and X ðiþ1ÞΔ −X iΔ =Δ when estimating the drift and diffusion functions, respectively, B is apredetermined range of b, and ψ(⋅) is a weighting function that has compact support and is bounded by 1. In our numerical work,
we set ψ(⋅) ≡ 1. This procedure could be further extended to the modified (asymptotically optimal) h-block CV method proposedby Racine (2000).
Since formal results on the optimal selection of the block size h are not available in the literature, we propose an automatic,
data-dependent choice of h that takes into account the persistence of the data. While Burman et al. (1994) and Racine (2000)
adopt an ad hoc fixed fraction rule (h = n 1/4), this choice of h does not account for different degrees of persistence in theunderlying process. Given some similarities between the choice of h and the selection of a bandwidth parameter in theheteroskedasticity and autocorrelation consistent (“HAC”) variance estimation, we follow Andrews (1991) and modify the plugin-rule for h to h = (γn) 1/4, where γ ¼4ρ2and ρ b 1 is the AR coefficient from an AR(1) regression for fX iΔ gni¼1 . 8 When γ = 0ð1−ρÞ2 ð1þρÞ2(or equivalently ρ = 0), this procedure naturally reduces to the leave-one-out CV for serially uncorrelated data.
2.4. Bootstrap procedure
Inference on the conditional expectations μ(Xt) and σ 2(Xt) can be conducted using the asymptotic distributions derived in
Theorem 1 or Corollary 1 above. This involves explicit estimation of unknown functions of the data that enter the asymptotic
distributions. A more practical approach, that typically provides better approximations to the finite-sample distributions of the7
Bandwidth selection for kernel estimators of continuous-time diffusion models is an on-going research program. Some recent contributions tailored tocontinuous-time Markov processes include Bandi et al. (2010) and Kanaya and Kristensen (2011).
8
In practice, ρ is replaced by its least squares estimate.

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601

statistics of interest, employs simulation-based or bootstrap methods. 9 In particular, we propose a parametric bootstrap
procedure based on the discretized version of model (1)pﬃﬃﬃﬃ
X ðiþ1ÞΔ −X iΔ ¼ μ ðX iΔ ÞΔ þ σ ðX iΔ Þ ΔεiΔ ;

ð20Þ

iid

where ε iΔ ∼ Nð0; 1Þ. Our bootstrap method also deals explicitly with the discretization bias induced by the discrete-time version of theunderlying model. This is achieved by directly sampling from the continuous-time model with a step size set to △/M for some M ≥ 1(in our empirical application, M = 100) while keeping the design points the same across the bootstrap samples.10 More specifically,using the estimated drift and diffusion functions μ^ b ðxÞ and σ^ b ðxÞ, 11 we set x0* = XiΔ and compute recursively  Δ  þ σ^ b xk−1
xk ¼ xk−1 þ μ^ b xk−1
M

rﬃﬃﬃﬃﬃ
Δ 
ε ;
M k

ð21Þ

* and repeat this procedure for each design point to construct the bootstrap
for k = 1,… M, where εk ∼ Nð0; 1Þ. We then set X(i* + 1)Δ = xMnon** is used to re-estimate μ(X) and σ(X) at the same design points.
sample X ðiþ1ÞΔ. Finally, each bootstrap series X(i + 1)Δ − XiΔ

iid

i¼1

Let μb* (x) and σb*(x) denote the corresponding bootstrap estimates. Repeating this procedure a large number of times provides anapproximation to the distributions of μ b ðxÞ−μ^ b ðxÞ and σ b ðxÞ−σ^ b ðxÞ at each design point. Then, pointwise confidence bands can be
h


 i


 
constructed as μ^ b ðxÞ−qμ 1− α2 ; μ^ b ðxÞ−qμ α2 and σ^ b ðxÞ−qσ 1− α2 ; σ^ b ðxÞ−qσ α2 , where qμ*(α) and qσ* (α) denote the αquantiles of the bootstrap distributions of μ b ðxÞ−μ^ b ðxÞ and σ b ðxÞ−σ^ b ðxÞ. This method can also be used for bias correction of theoriginal estimates μ^ b ðxÞ and σ^ b ðxÞ which could further improve the accuracy of the approximation in a double bootstrap procedure.
3. Empirical and simulation analysis
3.1. Dynamics of U.S. risk-free rate
One outstanding question in the analysis of the term structure of U.S. interest rates is concerned with the presence of possible
nonlinearities in the underlying dynamics of the spot rate. Despite the large literature on this issue (Aït-Sahalia, 1996a,b; Stanton,
1997; Chapman and Pearson, 2000; Bandi, 2002; Durham, 2003; Fan and Zhang, 2003; Jones, 2003; Arapis and Gao, 2006; among
others), there is still no consensus on the presence of statistically significant nonlinearities in short-term interest rates and theireconomic importance for pricing bonds and interest rate derivative products. For instance, Chapman and Pearson (2000) argue thatthe nonlinearity in the drift of the spot rate at high values of interest rates documented by Stanton (1997) could be spurious due to thepoor finite-sample properties of Stanton's (1997) estimator. Fan and Zhang (2003) conclude that there is little evidence against
linearity in the short rate drift function. Similarly, Bandi (2002), Durham (2003) and Jones (2003) find no empirical support for
nonlinear mean reversion in short-term rates. In contrast, Arapis and Gao (2006) report that their specification testing strongly rejects
the linearity of the short rate drift at both daily and monthly frequency.
Here, we revisit this issue by estimating the drift function using our proposed Gamma kernel estimator. The data used for
estimation are 564 monthly observations for the annualized risk-free rate from the Fama files in the CRSP U.S. Government Bond Files
covering the period January 1952–December 1998. As it is typically the case for interest rate data, the U.S. risk-free rate exhibits highpersistence and strong conditional heteroskedasticity. Given the sensitivity of the nonparametric estimates of the drift function to the
bandwidth (Chapman and Pearson, 2000), this high persistence should be properly incorporated into the selection procedure for asmoothing parameter.
Since our interest lies in identifying possible nonlinearities in the drift function of the spot rate, it proves instructive toestimate the CIR (Cox et al., 1985) model
1=2

dX t ¼ κ ðθ−X t Þdt þ σ X t dW t ;ð22Þ

which specifies the drift as a linear function in X of the form μ(Xt) = κ(θ − Xt), where κ denotes the speed of mean reversion and θ
is the long-run mean of X. Given the nature of the observed data for the spot rate, the unknown parameters (κ, θ, σ) are estimatedfrom the discretized version of the model with Δ = 1/12 which induces a discretization bias. To correct for the discretization bias,
we follow Broze et al. (1995) and estimate the parameters of the CIR model by indirect inference using the same auxiliary modeland discretization step as suggested by Broze et al. (1995). The resulting estimates for the U.S. risk-free rate are κ^ ¼ 0:2804,
θ^ ¼ 0:0541 and σ^ ¼ 0:0876.
Fig. 2 presents the nonparametric estimates of the drift function for the risk-free rate from the Gamma NW and Gaussian NW
regressions along with the estimated parametric drift function from the CIR model. The smoothing parameter for the Gamma
9
Bootstrap methods for inference in nonparametric regression models have been developed by Härdle and Marron (1991) and Franke et al. (2002), among
others.
10
We would like to thank an anonymous referee for suggesting this to us.
11
The diffusion estimates are obtained by imposing the restriction σ(0) = 0 as in Stanton (1997) which substantially reduces the bias documented in Fan andZhang (2003) and ensures the nonnegativity of the simulated interest rates as △ → 0.

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N. Gospodinov, M. Hirukawa / Journal of Empirical Finance 19 (2012) 595–609

kernel estimator is chosen by h-block CV described in Section 2.3 while the smoothing parameter for the Gaussian kernel is set
equal to four times the bandwidth for iid data as in Stanton (1997). The differences in the nonparametric estimates reveal thatpart of the accelerated mean reversion at higher interest rates, reported in the previous literature, may be due to larger biases ofthe symmetric kernel-based estimators although the Gamma kernel estimator also suggests that the mean reversion becomesstronger for interest rates larger than 11–12%. To assess the statistical significance of these findings, we also report 95% confidence
bands for the Gamma kernel estimates obtained by the bootstrap procedure outlined in Section 2.4 with 999 bootstrap
replications. Despite the mild nonlinearity of the nonparametric estimates, the parametric estimate of the drift function from the
CIR model falls within the 95% confidence bands (with the exception of very low interest rates) and it appears that the null
hypothesis of a linear drift cannot be rejected for this particular sample. Also, the nonparametric estimate based on the Gaussian
kernel lies inside the 95% Gamma kernel-based confidence bands except for very large interest rates where the bias due tosparseness of the data becomes more pronounced. The Gamma kernel estimate of the diffusion function and the associated 95%
bootstrap confidence bands are plotted in Fig. 3. To gain further insights into the finite-sample properties of the nonparametricestimators and the economic importance of these seemingly small statistical differences for bond and option pricing, we conduct
a simulation experiment whose design closely mimics our empirical application.

3.2. Monte Carlo experiment
The data for the simulation experiment is generated from the CIR model (22) with true parameter values that are set equal tothe estimates from U.S. data in the previous section, (κ, θ, σ) = (0.2804, 0.0541, 0.0876), and a time step between two consecutive
observation equal to Δ = 1/12 corresponding to monthly data. The CIR model is convenient because the transition and marginaldensities are known and the bond and call option prices are available in closed form (Cox et al., 1985). 5000 sample paths for the
spot interest rate of 600 observations (n = 600, T = 50) are constructed recursively by drawing random numbers from thetransition non-central chi-square density and using the values for κ, θ, σ and Δ (see Chapman and Pearson, 2000, for moredetails).
The expressions for the price of a zero-coupon discount bond and a call option on a zero-coupon discount bond have an analyticalform and are given in Cox et al. (1985). We follow Jiang (1998) and Phillips and Yu (2005) and compute the prices of a three-yearzero-coupon discount bond and a one-year European call option on a three-year discount bond with a face value of $100 and an
exercise price of $87 with an initial interest rate of 7% by simulating spot rate data from the estimated diffusion process. The simulatedbond and derivative prices are then compared to the analytical prices based on the true values of the parameters.

Fig. 2. Estimates of the drift function μ(·) of the risk-free rate with the NW estimator based on the Gamma kernel (along with 95% bootstrap confidence bands),
NW estimator based on the Gaussian kernel and indirect inference estimator of the CIR model. The smoothing parameter for the Gamma kernel estimator ischosen by h-block cross-validation with h = (γn)1/4, where γ is defined in Section 2.3. The smoothing parameter for the Gaussian kernel estimator is set equal tofour times the bandwidth parameter for iid data (Stanton, 1997).

N. Gospodinov, M. Hirukawa / Journal of Empirical Finance 19 (2012) 595–609

603

Fig. 3. Estimate of the diffusion function σ(·) of the risk-free rate with the NW estimator based on the Gamma kernel (along with 95% bootstrap confidence
bands). The smoothing parameter for the Gamma kernel estimator is chosen by h-block cross-validation with h = (γn)1/4, where γ is defined in Section 2.3.

More specifically, the price of a zero-coupon bond with face value P0 and maturity (τ − t) is computed as
h

i
τ
τ
P t ¼ P 0 Et exp −∫t X~ u du ;

ð23Þ

 
 
where the risk-neutral process X~ t evolves as dX~ t ¼ μ^ X~ t dt þ σ^ X~ t dW t , and μ^ ð⋅Þ and σ^ ð⋅Þ denote the nonparametric estimatesof the drift and diffusion functions, respectively. 12 The expectation is evaluated by Monte Carlo simulation using a discretizedversion of the dynamics of the spot rate.
The price of a call option with maturity (m–t) on a zero-coupon bond with maturity (τ − t), face value P0 and exercise price K iscomputed as
h


 τ
i
m
m
C t ¼ Et exp −∫t X~ u du max P m −K; 0
h



h

i
i
τ ~
~
¼ Et exp −∫m−K; 0 ;t X u du max P 0 Em exp −∫m X u dv

ð24Þ

where m b τ and sample paths for X~ t are simulated from the nonparametrically estimated discretized model of spot rate.
We consider the NW estimators based on Gaussian and Gamma kernels with smoothing parameters that are selected by h-block
CV. The average Monte Carlo estimates and 95% Monte Carlo confidence bands for the drift function are plotted in Figs. 4 and 5. While
the Gamma NW kernel estimator is practically unbiased, the Gaussian NW kernel exhibits a downward bias for interest rates higher
than 9–10%. Furthermore, the asymmetric kernel estimator exhibits smaller variability both near the boundary and in the tail region
where the data are sparse. 13
The economic significance of the improved estimation of diffusion models of spot rate is evaluated by comparing bond andoption pricing errors based on symmetric and asymmetric kernel estimators 14 for the CIR model. The results are presented inTable 1. The bond and derivative prices based on the asymmetric kernel estimator are less biased than its Gaussian counterpart
which could translate in potentially substantial financial benefits and investment profits for large portfolios. Another importantfeature of the results is that the Gamma-based bond and option prices enjoy much smaller variability and tighter confidence
12

For simplicity, the market price of risk is assumed to be equal to zero since its computation requires another interest rate process of different maturity.
The estimates of the diffusion function σ(⋅) are not reported to conserve space. Overall, the diffusion function estimates based on the Gamma and Gaussian
kernels are similar due to the zero restriction of the diffusion function at the origin (see footnote 11). When this restriction is removed, the Gamma kernel
estimator tends to perform better as a result of its intrinsic boundary correction property.
14
Kristensen (2008) demonstrates that when the process Xt is stationary, derivative prices obtained by plug-in nonparametric estimates of the drift anddiffusion functions have the same rate of convergence as parametric estimators.
13

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N. Gospodinov, M. Hirukawa / Journal of Empirical Finance 19 (2012) 595–609

Fig. 4. Average Monte Carlo drift (Gamma and Gaussian NW) estimates from CIR model with (κ, θ, σ) = (0.2804, 0.0541, 0.0876). The smoothing parameters are
selected by h-block cross validation with h = (γn)1/4, where γ is defined in Section 2.3.

intervals than the symmetric kernel-based option prices. Finally, while the confidence intervals for the Gamma kernel-based callprices appear symmetric around the median estimate, the confidence intervals for the Gaussian kernel-based call prices tend to behighly asymmetric (with long right tail) which illustrates the economic impact of the boundary problems of symmetric kernels.

Fig. 5. 95% Monte Carlo confidence intervals of the drift (Gamma and Gaussian NW) estimates from CIR model with (κ, θ, σ) = (0.2804, 0.0541, 0.0876) andsmoothing parameters selected by h-block cross validation with h = (γn)1/4, where γ is defined in Section 2.3.

N. Gospodinov, M. Hirukawa / Journal of Empirical Finance 19 (2012) 595–609

605

Table 1Monte Carlo statistics of bond and option prices based on nonparametric (Gaussian NW and Gamma NW) estimators in the CIR model with κ = 0.2804, θ = 0.0541
and σ = 0.0876.

True price
Gaussian NW estimatorMedian estimate
Standard deviation
95% confidence interval
Gamma NW estimatorMedian estimate
Standard deviation
95% confidence interval

Bond priceCall option price82.4251.889

82.3591.322[80.420, 85.573]

1.656
0.515
[1.014, 3.026]

82.4471.115[80.665, 85.058]

1.704
0.347
[1.133, 2.463]

Notes: The statistics in the table are computed from 5000 samples generated from the CIR model with △ = 1/12 and n = 600. The prices of a three-year zerocoupon discount bond and a one-year European call option on a three-year bond with face value of $100, strike price of $87 and initial interest rate of 7% arecomputed by Monte Carlo simulation.

4. Conclusion
This paper proposes an asymmetric kernel estimator for estimating the underlying scalar diffusion model of spot interest rate. The
asymptotic properties of the Gamma kernel estimator of the drift and diffusion functions are established for both interior andboundary design points. We show that the Gamma kernel estimator possesses some appealing properties such as lack of boundary
bias and adaptability in the amount of smoothing. The paper adopts a block cross-validation method for dependent data in choosing
the smoothing parameter. The finite-sample performance of the asymmetric kernel estimator and its practical relevance for bond andoption pricing are evaluated using simulated data of spot interest rate. The empirical analysis of the U.S. risk-free rate reveals theexistence of some mild (and statistically insignificant) nonlinearity in the drift function although the estimated increased mean
reversion at higher interest rate levels is substantially smaller than originally reported by Stanton (1997).
Appendix A. Assumptions
This appendix provides a set of assumptions for Theorem 1 and Corollary 1 that are similar to those in Bandi (2002) and BP. Our
first assumption is basically the same as Assumption 1 in BP, which ensures that the SDE (1) has a unique strong solution Xt and that Xt
is recurrent. Assumption 2, together with Assumption 1, implies that the process Xt is positive recurrent (or ergodic) and ensures theexistence of a time-invariant distribution P0 with density f ðxÞ ¼ sðxÞ=∫∞0 sðxÞdx. Moreover, if X0 has distribution P 0, Xt becomes strictly
stationary. Assumption 3 controls the rates of convergence or divergence of the sequences used in the asymptotic results for generaland positive recurrent cases.
Assumption 1.
(i) μ(⋅) and σ(⋅) are time-homogeneous, B-measurable functions on (0, ∞), where B is the σ-field generated by Borel sets on(0, ∞). Both functions are at least twice continuously differentiable. Hence, they satisfy local Lipschitz and growth
conditions. Thus, for every compact subset J of the range (0, ∞), there exist constants C1J and C2J such that, for all x, y ∈ J,
|μ(x) − μ(y)| + |σ(x) − σ(y)| ≤ C1J |x − y| and |μ(x)| + |σ(x)| ≤ C2J {1 + |x|}.
(ii) σ 2(⋅) > 0 on (0, ∞).
(iii) The natural scale function S(x) satisfies limx → 0S(x) = − ∞ and limx → ∞S(x) = ∞.
Assumption 2. The speed function s(x) satisfies ∫∞0 sðxÞdxb∞.
Assumption 3. As n, T → ∞, Δ(=T/n) → 0, b(=bn, T) → 0 such thatLX ðT; xÞbsﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 ﬃ
1
Δlog
¼ oa:s: ð1Þ:Δ

ðA1ÞAppendix B. Preliminary lemma and Proof of Theorem 1
The Proof of Theorem 1 closely follows the proof of Theorem 3 in BP by adapting it to the case of the Gamma kernel. We
provide only the proof for the drift estimator μ^ b ðxÞ, because the one for the diffusion estimator σ^ 2b ðxÞ follows similar arguments. To
establish the results in Theorem 1, we need the following lemma.

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N. Gospodinov, M. Hirukawa / Journal of Empirical Finance 19 (2012) 595–609

Lemma 1. Under Assumptions 1 and 3 in Appendix A,
Δ

n
−1
X

T

K Gðx=bþ1;bÞ ðX iΔ Þμ ðX iΔ Þ ¼ ∫0 K Gðx=bþ1;bÞ ðX s Þμ ðX s Þds þ oa:s: ð1Þ:

ðB1Þ

i¼1

Proof of Lemma 1. Consider that


−1
 nX



T
K Gðx=bþ1;bÞ ðX iΔ Þμ ðX iΔ Þ−∫0 K Gðx=bþ1;bÞ ðX s Þμ ðX s Þds
Δ


i¼1


−1
nX

n
o


ðiþ1ÞΔ∫iΔ
K Gðx=bþ1;bÞ ðX iΔ Þ−K Gðx=bþ1;bÞ ðX s Þ μ ðX iΔ Þds
≤


i¼0


−1
nX
 


 

ðiþ1ÞΔ∫iΔ
K Gðx=bþ1;bÞ ðX s Þfμ ðX s Þ−μ ðX iΔ Þgds þ ΔK Gðx=bþ1;bÞ ðX 0 Þμ ðX 0 Þ
þ



ðB2Þ

i¼0

≡ A1 ðxÞ þ A2 ðxÞ þ Oa:s: ðΔÞ;where the bound of the third term is obtained from X0 = x0, Assumption 1 and boundedness of the kernel. Observe that
A1 ðxÞ ≤

n
−1
X


 

K Gðx=bþ1;bÞ X~ is jX s −X iΔ jjμ ðX iΔ Þjdsðiþ1ÞΔ  ′

∫iΔ

ðB3Þ

i¼0

for X~ is on the line segment connecting Xs and XiΔ. Since
kn;T ≡ max

supi≤n iΔ≤s≤ðiþ1ÞΔ

jX s −X iΔ j ¼ Oa:s:sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 ﬃ!
1
Δ log
Δ

ðB4Þ

(see p. 267 in BP), X~ is ¼ X s þ oa:s: ð1Þ uniformly over i ≤ n, and thus
A1 ≤ kn;T

n
−1
X




K Gðx=bþ1;bÞ ðX s þ oa:s: ð1ÞÞjμ ðX s þ oa:s: ð1ÞÞjds

ðiþ1ÞΔ  ′

∫iΔ

i¼0




T  ′
¼ kn;T ∫0 K Gðx=bþ1;bÞ ðX s þ oa:s: ð1ÞÞjμ ðX s þ oa:s: ð1ÞÞjds
¼

ðB5Þ


kn;T ∞  ′
∫0 bK Gðx=bþ1;bÞ ðu þ oa:s: ð1ÞÞjμ ðuÞjLX ðT; uÞdu:b

Note that′

K Gðx=bþ1;bÞ ðuÞ ¼

  x=b
1 u expð−u=bÞ
−:b bx=bþ1 Γ ðx=b þ 1Þ
b bx=bþ1 Γ ðx=b þ 1Þx ux=b−1 expð−u=bÞðB6Þ

Then, by the properties of the Gamma function,
x=b−1
x=b


u
expð−u=bÞexpð−u=bÞ∞ u∫∞0 bK ′ Gðx=bþ1;bÞ ðuÞdu ¼ x∫∞0 x=bþ1
du þ ∫0 x=bþ1
duΓ ðx=b þ 1Þ
Γ ðx=b þ 1Þ
b
b
x=b

¼x

b

Γ ðx=bÞ

bx=bþ1 Γ ðx=b þ 1ÞðB7Þ

þ 1 ¼ 1 þ 1 ¼ 2:

Combining this result with the continuity of μ(⋅) and LX ðT; ⋅Þ and Assumption 3, we can conclude that A1 is bounded by
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ


 
 
Oa:s: L X ðbT;xÞ Δlog Δ1 ¼ oa:s: ð1Þ. Similarly, A2(x) is bounded by Oa:s: L X ðbT;xÞ Δlog Δ1 ¼ oa:s: ð1Þ. ■N. Gospodinov, M. Hirukawa / Journal of Empirical Finance 19 (2012) 595–609

607

Proof of Theorem 1. Observe thatn−1

μ b ðxÞ−μ ðxÞ ¼

Δ∑i¼1 K Gðx=bþ1;bÞ ðX iΔ Þfμ ðX iΔ Þ−μ ðxÞgΔ∑n−1
i¼1 K Gðx=bþ1;bÞ ðX iΔ ÞX ðiþ1ÞΔ −X iΔ−μ ðX iΔ ÞΔ∑n−1
i¼1 K Gðx=bþ1;bÞ ðX iΔ ÞΔþn−1
K Gðx=bþ1;bÞ ðX iΔ Þ
Δ∑i¼1

ðB8Þ

≡ Bμ ðxÞ þ V μ ðxÞ:Using Lemma 1 above and Lemma 6 in BP, we haveT

∫0 K Gðx=bþ1;bÞ ðX s Þfμ ðX s Þ−μ ðxÞgds þ oa:s: ð1Þ

Bμ ðxÞ ¼∫T0 K Gðx=bþ1;bÞ ðX s Þds þ oa:s: ð1Þ

¼∫∞0 K Gðx=bþ1;bÞ ðuÞfμ ðuÞ−μ ðxÞgsðuÞds þ oa:s: ð1Þ
∫∞0 K Gðx=bþ1;bÞ ðuÞsðuÞds þ oa:s: ð1ÞðB9Þþ ðs:o:Þ;where (s. o.) denotes the term that is of smaller order than the first term. Ignoring smaller order terms, taking a second-order
Taylor expansion for μ(u) around u = x, and using E(θx − x) = b and Eðθx −xÞ2 ¼ bðx þ 2bÞ for θx ∼Gammaðx=b þ 1; bÞ, we obtain thefollowing approximation of the first term:∞∫0 K Gðx=bþ1;bÞ ðuÞfμ ðuÞ−μ ðxÞgsðuÞds
∫∞0 K Gðx=bþ1;bÞ ðuÞsðuÞds

(

)
#
′
xs ðxÞx ″þ μ ðxÞ b þ oðbÞ:¼ μ ðxÞ 1 þ
sðxÞ
2
"

′

ðB10Þ

On the other hand, Vμ(x) can be rewritten as

V μ ðxÞ ¼

1 ðiþ1ÞΔ∫fμ ðX s Þ−μ ðX iΔ ÞgdsΔ iΔ
n−1
Δ∑i¼1 K Gðx=bþ1;bÞ ðX iΔ ÞΔ∑n−1
i¼1 K Gðx=bþ1;bÞ ðX iΔ Þþ

1 ðiþ1ÞΔ∫σ ðX s ÞdWs
Δ iΔ
Δ∑n−1
K
ð
X
Þ
Gðx=bþ1;bÞ
iΔ
i¼1

n−1
Δ∑i¼1
K Gðx=bþ1;bÞ ðX iΔ Þ

ðB11Þ

≡ V 1 ðxÞ þ V 2 ðxÞ;qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
 Δlog Δ1 ¼ oa:s: ð1Þ following the same argument as in the Proof of Lemma 1 above. The leadingwhere V1(x) is bounded by Oa:s:variance term of the numerator of V2(x) can be determined by
Δ

n
−1
X

1 ðiþ1ÞΔ 2∫iΔ
σ ðX s Þds þ oa:s: ð1Þ
Δ
i¼0
n
o
T
2
2
¼ ∫0 K Gðx=bþ1;bÞ ðX s þ oa:s: ð1ÞÞ σ ðX s Þ þ oa:s: ð1Þ ds þ oa:s: ð1Þ:
2

K Gðx=bþ1;bÞ ðX iΔ ÞðB12Þ

Ignoring smaller order terms, and defining Ab(x) = b − 1Γ(2x/b + 1)/{2 2x/b + 1Γ 2(x/b + 1)}, we have
∫T0 K 2Gðx=bþ1;bÞ ðX s Þσ 2 ðX s ÞdsT2¼ Ab ðxÞ∫0 K Gð2x=bþ1;b=2Þ ðX s Þσ ðX s Þds

ðB13Þ¼ Ab ðxÞ∫T0 K Gð2x=bþ1;b=2Þ ðuÞσ 2 ðuÞLX ðT; uÞdu;where (see Chen, 2000)
8
>
>
>
<

−1=2


b
−1=2
pﬃﬃﬃ 1=2 þ o b
2 πx
Ab ðxÞ ¼>
b−1 Γ ð2κ þ 1Þ
−1
>
>
þo b
: 2κþ1 2Γ ðκ þ 1Þ
2

for interior x
ðB14Þ
for boundary x:

608

N. Gospodinov, M. Hirukawa / Journal of Empirical Finance 19 (2012) 595–609

Hence, for interior x,
1=2

b

2
a:s: σ ðxÞT2
2∫ K Gðx=bþ1;bÞ ðX s Þσ ðX s Þds → pﬃﬃﬃ 1=2 LX ðT; xÞ:
0
2 πx

ðB15Þ

It follows from the arguments in the proof of Theorem 3 in BP that!
σ 2 ðxÞ
V 2 ðxÞ ⇒ MN 0; pﬃﬃﬃ 1=2 :
2 π x L X ðT; xÞ

ðB16Þ

!
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
σ 2 ðxÞ
1=2 
b L X ðT; xÞV 2 ðxÞ ⇒ N 0; pﬃﬃﬃ 1=2 :
2 πx

ðB17Þ

1=4

b

Therefore,

5=2 The result for interior x can be established by combining (B8), (B10) and (B17) under the assumption that bthen replacing LX ðT; xÞ by L^X ðT; x; bÞ. The result for boundary x can be obtained similarly. ■L X ðT; xÞ ¼ Oa:s: ð1Þ andReferences
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