The Nonparametric Empirical Likelihood Goodness-of-fit Test for Integrated Diffusion Processes

Liu Zaiming Tang Mingtian? Wang Yunyan

(1. School of Mathematics and Statistics,Central South University,Changsha 410075,China；2. School of Science,Jiangxi University of Science and Technology,Ganzhou 341000,China)

Abstract This paper is devoted to the nonparametric goodness-of-fit test for integrated diffusion processes. Firstly,a nonparametric test is constructed for testing whether the drift function of a integrated diffusion process is of a known parametric form with unknown parameters. Secondly,a test statistic for goodness-of-fit test is obtained by applying the empirical likelihood technique. And finally,the asymptotic distribution of the test statistic is established,and then the proposed test method is applied to an example to verify its effectiveness.

Key words Integrated diffusion process Empirical likelihood Goodness-of-fit test Nadaraya-Watson estimator

1 Introduction

In the past few decades diffusion processes have been proved to be immensely useful, not only in finance and economics science,but also in other fields such as biology,medicine,physics and engineering.It?’s diffusions,which are defined by the following stochastic differential equation are commonly used in finance to model asset prices, interest or exchange rates. Here{Bt,t ≥0}is a Brownian motion,μis a locally bounded predictable drift function andσis a cadlag volatility process.During the past decades, specification of model (1.1) has attracted a lot of attention in both theoretical studies and practical applications. For example,Ait-Sahalia[1]considered two approaches for testing the parametric specification. The first one was based on aL2-distance between a kernel stationary density estimator and the parametric stationary density, and the second one was based on certain discrepancy measure regarding the transitional distribution of the process derived from the Kolmogorov forward and backward equations. Chen,H?rdle and Kleinow[2]used the empirical likelihood technique to construct a test procedure for the goodness-of-fit of a diffusion model. Chen, Gao and Tang [3] proposed a test for model specification of a parametric diffusion process based on a kernel estimation of the transitional density of the process. Kutoyants[4]considered the goodness-of-fit testing problem for ergodic diffusion processes. Chen,Hu and Song[5]developed a nonparametric specification test for the volatility functions of diffusion processes.

Despite the fact that integrated diffusion models have been proved to be extremely useful,the problem of goodness-of-fit tests for integrated diffusion processes has still been a new issue in recent years.Goodness-of-fit tests play important roles in theoretical and applied statistics because they bridge the mathematical models and the real data,and the study for them in a general framework has a long history.Theχ2test is the best-known parametric goodness-of-fit test,while one of the most popular nonparametric tests is the Kolmogorov-Smirnov test.

In this paper,we develop a goodness-of-fit test for the drift function in an integrated diffusion process with given high-frequency observations. The proposed goodness-of-fit test is based on the empirical likelihood method,which is a nonparametric technique of constructing confidence intervals introduced by Owen[16,17]. Here we use the empirical likelihood technique to construct the test statistic because it has two attractive features. Just as Chen,H?rdle and Kleinow[2]pointed out that the method had two attractive features (for further details, see Chen, H?rdle and Kleinow [2]). As for the updated comprehensive overview of the empirical likelihood one can refer to Owen [18] and a general framework of empirical likelihood based on estimating function was discussed in Qin and Lawless[19].

The remainder of the paper is organized as follows. Section 2 introduces the integrated diffusion process,outlines the hypothesis and the kernel smoothing of drift function. The assumptions and our main results are provided in Section 3. Section 4 gives an example to investigate the empirical likelihood test procedure. Some useful lemmas and mathematical proofs of the lemmas and our main results are presented in Section 5.

2 Integrated diffusion models and hypothesis testing

Consider an integrated diffusion process defined by the following second-order stochastic differential equation:where{Bt,t ≥0}is a standard 1-dimensional Brownian motion, and(Xt)t≥0is a scalar process. The functionsμ(·)andσ2(·)are the so-called drift and diffusion term respectively.

The aim of this paper is to test a parametric model for the drift functionμ(·)against a nonparametric alternative,that is,consider the following null hypothesis:

against a series of nonparametric alternatives:

whereθis an unknown parameter within a parametric space Θ,Cnis a nonnegative sequence tending to zero asn →∞and Δn(x)is a sequence of bounded functions. Letp(x)be the density function of the processXt,andI={x ∈R|p(x)≥β}for someβ> 0 be a compact set. For simplicity,we assume thatI=[0,1].

Since Δ tends to zero,the values ofXiΔ,X(i-1)Δand ?XiΔare near with each other,hence our estimation can be based on the sample{?XiΔ,i=1,2,···}.

The estimation of the drift function in the second-order stochastic differential equation(2.1)depends on the following equation:

as time increment Δ→0,whereFt=σ(Xs,s ≤t). Readers can refer to Nicolau[10]or Lemma 5.1 in section 5 for more details about(2.2). So the Nadaraya-Watson kernel estimator of the drift functionμ(x)is as follows:

3 Goodness-of-fit statistic and main results

3.1 The empirical likelihood goodness-of-fit test statistic

To extend the empirical likelihood ratio statistic to a global measure of goodness-of-fit test,we choosekn-equally spaced grid pointst1,t2,··· ,tknon[0,1], wheret1= 0,tkn= 1 andti ≤tjfor 1≤i

Remark 3.2 Assumption 3.1 assures thatXis stationary [20], and by the Kolmogorov forward equation,the stationary densityp(x)ofXis

(ii)The mixing coefficientα(k)satisfies:α(k)≤aρkfor somea>0 andρ ∈(0,1).

Remark 3.3 Assumption 3.2(i)is similar to Assumption 4 in Nicolau[10]which assures that the processxisα-mixing (see, e.g., Chen, Hansen and Carrasco [22], Definition 2.2) and Assumption 3.2 means that the process is geometricα-mixing. Since the measurable functions of mixing processes are alsoα-mixing,so{?XiΔ}isα-mixing,and the mixing size of{?XiΔ}is the same as that of{XiΔ}. Under Assumptions 3.1-3.2,we know from Nicolau[10]that{?XiΔ,i=0,1,···}is a stationary process.

Assumption 3.7 Δn(x)is uniformly bounded with respect toxandn, andCn=n-1/2h-1/4is the order of the difference betweenH0andH1.

Remark 3.5 Assumption 3.6 and Assumption 3.7 are common in nonparametric goodness-of-fit tests.

whereK(2)is the convolution ofK.

4 Applications

where{Bt,t ≥0}is a standard Brownian motion. The functionsμ(·)andσ2(·)are the drift and diffusion term respectively.

This process is ergodic, and its stationary distribution is the normal distribution with expectation 0 and varianceσ2/(2μ), provided thatμ> 0. Recently, this process was used to model the ice-core data from Greenland. The observations were of the integrated process,for details see Ditlevsen,Ditlevsen and Andersen[24].

Figure 1 EPICA Dronning Maud Land Ice Core 10-51 KYrBP δ18O Data

Figure 2 The P-values of the ice-core data

5 Lemmas and proofs

Lemma 5.1 ([10]) LetZbe ad-dimensional diffusion process given by the stochastic integral equation

whereμ(z) = [μi(z)]d×1is ad×1 vector,σ(z) = [σij(z)]d×dis ad×ddiagonal matrix, andBtis ad×1 vector of independent Brownian motions. Assume thatμandσhave continuous partial derivatives of order 2s. Letf(z)be a continuous function defined on Rdwith values in Rdand with continuous partial derivative of order 2s+2. Then

where the former one holds by Lemma 5.3 and the latter one holds by Lemma 5.1 and the same lines of arguments as in Theorem 2 of Nicolau[10].

The proof of Corollary 3.1 follows from the proof of Theorem 3.1.

6 Conclusions

In this paper, we used the empirical likelihood technique to construct a test procedure for goodness-of-fit test for an integrated diffusion process which is commonly used for modeling in the field of finance. The asymptotic null distributions of the proposed test statistic were developed and the ice-core data was used to investigate the empirical likelihood test procedure.

We considered a nonparametric test for testing whether the drift function is of a known parametric form indexed by a vector of unknown parameters. In fact,for the parameter identification of a diffusion function in the integrated diffusion,test statistics based on empirical likelihood can be obtained similarly,but they are more complicated. On the other hand,the Nadaraya-Watson estimator of a drift function was used to construct the test statistic. Considering the good deviation property of a local linear estimator,the Nadaraya-Watson estimator of the drift function can be replaced by the local linear estimator.