Local polynomial estimations of time-varying coefficients for local stationary diffusion models
© Wang and Xiao; licensee Springer. 2014
Received: 20 October 2013
Accepted: 3 December 2013
Published: 14 January 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
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.
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 .
where is a process of positive variables satisfying for some and independent of u, t and T. denotes an arbitrary norm on R.
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 .
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
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.
where , is a diagonal matrix, and , where .
respectively. Due to equation (3), and share the same asymptotic properties, and so do and , and , respectively.
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 .
(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.
where means convergence in probability.
, 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.
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.
The proofs of Theorem 1, Theorem 2 and Theorem 3 are given in Section 5.
4 The estimation of the diffusion coefficient
The estimation in (14) is computed by numerical one-dimensional integration.
where , .
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
Therefore, , which implies . This completes the proof of Theorem 1. □
This completes the proof of Theorem 2. □
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.
This completes the proof of Theorem 3. □
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.
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.
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