- Open Access
Local polynomial estimations of time-varying coefficients for local stationary diffusion models
Journal of Inequalities and Applications volume 2014, Article number: 17 (2014)
This paper is dedicated to the study of local polynomial estimations of time-varying coefficients for a local stationary diffusion model. Based on local polynomial fitting, the estimations of drift parametric functions are obtained by using local weighted least squares method. By applying the forward Kolmogorov equation, the estimation of the diffusion coefficient is proposed. The consistency, asymptotic normality and uniform convergence of the estimations that we proposed are established.
The theory of local stationary process is introduced by Dahlhaus [1, 2]. Intuitively speaking, a process is locally stationary if the process behaves like a stationary diffusion process in a neighborhood of a chosen time point. In recent years, various efforts have been made to explicitly express the local stationary models. For example, Koo and Linton  discuss semi-parametric estimations of a class of locally stationary diffusion processes with locally linear drift. Vogt  considers nonparametric regression for a locally stationary time series model. To the best of our knowledge, few literature works have researched local polynomial estimations in locally stationary time-inhomogeneous diffusion models. This motivates us to consider the local estimations of the time-varying coefficients for locally stationary diffusion models.
In this paper, we study local polynomial estimations of time-varying coefficients for local stationary diffusion models. Firstly, we propose the local estimations of drift time-varying parameters by using the local weighted least squares method. The techniques that we employ here are based on local polynomial fitting (see Fan and Gijbels ). Secondly, by using the forward Kolmogorov equation, we obtain the estimation of the diffusion coefficient which is an unknown function of both time and state. Finally, we establish the consistency, asymptotic normality and uniform convergence of our proposed estimations.
The remainder of this paper is organized as follows. In Section 2, we introduce our diffusion models and give the definition of the local stationarity. In Section 3, the local polynomial estimations of the drift parameters are proposed, and the consistency, asymptotic normality and uniform convergence rate of the estimations for the drift functions are established. In Section 4, we give the estimation of diffusion coefficient and verify its asymptotic properties. Proofs of the main results are given in Section 5. In Section 6, the conclusions of this paper are given.
2 Methods and local stationary
Suppose that the diffusion process satisfies
where is a known Borel function, and is a standard Brownian motion defined on the filtered probability space , where is a σ-algebra, is a filtration. The stochastic process defined as the solution to model (1) has a domain , where . Here, is a given random variable taking values in R, which is independent of with . and are unknown drift parameters, while is an unknown diffusion coefficient.
Now we discuss the local stationary of model (1). We will approximate model (1) by a family of stationary processes indexed by . The stochastic process satisfies
where is a given random variable with for each . We shall require the following assumptions which ensure that model (2) has a family of stationary solutions (see Koo and Linton ).
(A1) The functions , , and are twice boundedly continuously differentiable on I.
(A2) for each .
(A3) For all , there exists such that .
(A4) The scale function defined as , where for , satisfies the following: for a fixed number , we have and .
(A5) The speed measure .
Under assumptions (A1)-(A5), for each time point , the process is strictly stationary and weekly dependent with a stationary density denoted by , please refer to Karatzas and Shreve .
Definition 1 The process is locally stationary if for each rescaled time point , there exists an associated process represented by (2) and satisfying assumptions (A1)-(A5) such that
where is a process of positive variables satisfying for some and independent of u, t and T. denotes an arbitrary norm on R.
From Definition 1, we have
Since equation (3) implies that in the neighborhood of a time point u, we can replace by for our asymptotic analysis, it allows us to approximate by a family of as .
Suppose that assumptions (A1)-(A5) hold, and , in model (1) satisfy
Then the diffusion process is locally stationary (see Theorem 1 of Koo and Linton ). Condition (4) is sufficient but need not be necessary.
3 Local polynomial estimations of drift parameters
In this section, we study the local polynomial estimations of time-varying drift parameters in model (1). Let the data be sampled at discrete time points, and suppose that the time points are equally spaced. Denote , and . The distance between two sampling points, Δ, will not shrink but remains fixed. According to the independent increment property of Brownian motion , the is independent and normally distributed with mean zero and variance Δ. Thus the discretized version of model (1) can be expressed as
Based on the findings in Stanton  and Fan , the first-order discretized approximation error to the continuous-time model is extremely small, as long as data are sampled monthly or more frequently. Their findings simplify the estimation procedure.
Similarly, the discretized version of model (2) can be expressed as
Based on equation (6), we discuss the local estimations of time-varying parameters and . By using p-polynomial fitting for and (see Fan and Gijbels ), we can obtain the local estimations of time-varying parameters and by minimizing the following objective function,
where is an unknown parameter vector, K is a real-valued kernel function, h is a bandwidth and . Let be the minimizer of locally weighted function (7). Then
where , is a diagonal matrix, and , where .
Let and . In matrixes and , we replace and by and , respectively. Then we can obtain the matrixes based on sample data, denoting X and Y, respectively. Let
Then the local polynomial estimations of and are
respectively. Due to equation (3), and share the same asymptotic properties, and so do and , and , respectively.
Now we establish the asymptotic properties of the local polynomial estimations of the drift parameters as the following. We first give regularity conditions for model (1). Denote
This can be interpreted as an estimation of .
(C1) The diffusion process is a locally stationary process represented by (2).
(C2) For all and , , and are twice continuously differentiable with and .
(C3) The kernel function is a bounded symmetric around zero Lipschitz continuous function with
(C4) Bandwidths h, and satisfy: as , and ; and ; .
(C5) For , the density of exists with .
(C6) For some positive η, .
The following Theorem 1 and Theorem 2 show the weak consistency and asymptotic normality of our proposed local estimations, respectively.
Theorem 1 Suppose that assumptions (C1)-(C6) hold. Let u be an interior point over . Then, as , we have
where means convergence in probability.
Theorem 2 Suppose that assumptions (C1)-(C6) hold. Let u be an interior point over . Then, as , we have
where means convergence in distribution, and , where , is a vector whose elements are the first two elements of vector , where
, where , , denotes the upper-left sub-matrix.
In order to discuss the uniform convergence of our proposed estimations, we further give the following two assumptions.
(C7) The diffusion process is strongly mixing with its mixing coefficients such that for ,
where converges exponentially fast to zero as .
(C8) The density of and joint densities of are uniformly bounded.
Assumptions (C7) and (C8) allow us to make use of the asymptotic independence property for heterogeneous data.
Theorem 3 Suppose that assumptions (C1)-(C8) hold and . Then, for a compact set I and any sequence such that and , we have
The proofs of Theorem 1, Theorem 2 and Theorem 3 are given in Section 5.
4 The estimation of the diffusion coefficient
In this section, we study the estimation of the diffusion coefficient. From the forward Kolmogorov equation (see (5.1.6) in Karatzas and Shreve ), we have
where is the density of . Since we already have estimations of , and , we can use these to estimate as follows,
The estimation in (14) is computed by numerical one-dimensional integration.
Theorem 4 Suppose that assumptions (C1)-(C6) hold. Let u be an interior point over . Then, as , we have
where , .
Theorem 5 Suppose that assumptions (C1)-(C8) hold and . Then, for a compact set I and any sequence such that and , we have
Theorem 4 shows the consistence and asymptotic normality of the estimation of the diffusion coefficient, and Theorem 5 gives the uniform convergence rate of our estimation.
5 Proofs of main results
By using equation (3) and is strongly mixing, in the neighborhood of u, we have
Let , where . Then
where , , . For , we have
Therefore, , which implies . This completes the proof of Theorem 1. □
Proof of Theorem 2 We start with the asymptotic normality of . Continuing from equation (18), we have
We have proved that in the proof of Theorem 1. We discuss in the following and we have
Note that is a martingale difference sequence. Then
where . By using and having the same asymptotic property, we have
This completes the proof of Theorem 2. □
Proof of Theorem 3 Let be the density of . Denote
where M is some positive number. Under assumptions (C1)-(C8), conditions for Theorem 1 in Kristensen  are met and application of Theorem 1 can be used.
Let us start with . Let
Note that , and are local demeaned values. According to Theorem 1 in Kristensen , we have
Also, due to standard nonparametric estimation manipulation, we obtain
By using the Taylor expansion, we have
Due to the same argument as in Kristensen , the uniform convergence rate of the above equation is determined by and . Therefore,
As for , the method in Hansen  applies with the previous result for and the rate of uniform convergence of is determined by that of . Therefore,
This completes the proof of Theorem 3. □
Proof of Theorem 4 We first discuss the consistency of our estimation. Recall that
Let . According to Koo and Linton , we have that is bounded away from zero and . Thus
which implies that . On the other hand, it can be easily shown that given the consistency of the marginal density and the drift shown before.
We can prove the asymptotic normality of by using a similar method of Theorem 3 in Koo and Linton . Here, we do not give the details. This completes the proof of Theorem 4. □
Proof of Theorem 5 Notice that the uniform convergence of the proposed estimation for the diffusion coefficient is determined by that of the estimation of the density function. Therefore, due to Lemma 6 in Koo and Linton , the proof of Theorem 5 is completed. □
In this paper, the local polynomial estimations of time-varying drift parametric functions are proposed. By applying the forward Kolmogorov equation, the estimation of the diffusion coefficient is obtained. The consistency, asymptotic normality and uniform convergence of the proposed estimations are proved.
Dahlhaus R: On the Kullback-Leibler information divergence of locally stationary processes. Stoch. Process. Appl. 1996, 62: 139-168. 10.1016/0304-4149(95)00090-9
Dahlhaus R: Fitting time series models to nonstationary processes. Ann. Stat. 1997, 25: 1-37.
Koo B, Linton O: Semiparametric estimation of locally stationary diffusion models. J. Econom. 2012, 170: 210-233. 10.1016/j.jeconom.2012.05.003
Vogt M: Nonparametric regression for locally stationary time series. Ann. Stat. 2012, 40: 2601-2633. 10.1214/12-AOS1043
Fan J, Gijbels I: Local Polynomial Modelling and Its Applications. Chapman & Hall, London; 1996.
Karatzas I, Shreve S: Brownian Motion and Stochastic Calculus. Springer, New York; 2000.
Stanton R: A nonparametric model of term structure dynamics and the market price of interest rate risk. J. Finance 1997, 52: 1973-2002. 10.1111/j.1540-6261.1997.tb02748.x
Fan J, Zhang C: A re-examination of Stanton’s diffusion estimations with applications to financial model validation. J. Am. Stat. Assoc. 2003, 98: 118-134. 10.1198/016214503388619157
Dahlhaus R, Subba Rao S: Statistical inference for time varying arch processes. Ann. Stat. 2006, 34: 1075-1114. 10.1214/009053606000000227
Fryzlewicz P, Sapatinas T, Subba Rao S: Normalised least squares estimation in time-varying ARCH models. Ann. Stat. 2008, 36: 742-786. 10.1214/07-AOS510
Kristensen D: Uniform convergence rates of kernel estimations with heterogeneous dependent data. Econom. Theory 2009, 25: 1433-1445. 10.1017/S0266466609090744
Hansen B: Uniform convergence rates for kernel estimations with dependent data. Econom. Theory 2008, 24: 726-748.
This work is supported by the National Natural Science Foundation of China (No. 11171221) and Shanghai Leading Academic Discipline Project (No. XTKX2012). The authors would like to thank the anonymous referees for several useful interesting comments and suggestions about the paper.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript, read and approved the final manuscript.
About this article
Cite this article
Wang, J., Xiao, Q. Local polynomial estimations of time-varying coefficients for local stationary diffusion models. J Inequal Appl 2014, 17 (2014). https://doi.org/10.1186/1029-242X-2014-17
- local stationary model
- local polynomial fitting
- asymptotic normality
- convergence rate