Local Polynomial Estimator Of The Regression Function In Autoregressive Models With Errors In Variables

In this paper, we study the problem of the nonparametric estimation of the autoregression function in autoregressive models with errors-in-variables. With the application of the deconvolution kernel, we give the local polynomial estimator of the autoregression function , we also prove the deconvolution kernel estimator proposed by Comte(2004) is a special case of the polynomial estimator. Under some specific conditions , the properties of consistence and asymptotic normality of the local polynomial estimator can be obtained here. The performance of the method are studied by some simulation experiments.The well-known Black and Scholes'(1973) model assumed the following dynamics for an asset price S_t at time t:where B_t is a standard Brownian motion. The solution of this equation leads to the following model for h_t = In(S_t/S_t-1), the returns associated with the asset price S_t at time t:where is a sequences of i.i.d Gaussian variables.It appears from financial data that the volatility , Var(e) = 0% < +00, fc denotes the density of e.[Assumption 2] et = A(ln(^t)2 -u),u = E(\n(^)2), {^} i-i-d., ^t ^Only the data of {Yt} can be observed , fe is known, Comte(2004) gave the deconvolution kernel estimator of the autoregression function m(x) in the model above.We consider the estimator of m{x) based on the observationswhere Kn(x) and /n(x) are given by:1 ? Yt - xwhere K(-) is a kernel, hn is a bandwidth parameter and g* denotes the Fourier transform:p*(x) = / eiteg(t)dt.For the data of {Xt} can not be observed , the estimator of m(x) just can be constructed on the data of {It} and Yt = Xf\-et.ln light of the method that Comte(2004) gave the deconvolution kernel estimator , and considering the special function of the deconvolution kernel , we give the local polynomial estimator m^x) of m(x) constructed on the data of {Yt} in section 2, and we also discuss the deconvolution kernel estimator rhn(x) is a special case of the local polynomial estimator rh{pn(x);In section 3, we exhibit the properties of consistence and asymptotic normality of the local polynomial estimator under some specific conditions;In section 4 , we deal with a test of the method through simulation study;We present some elements of proofs in section 5.We give the local polynomial estimator of the autoregression function in the autoregressive model with errors-in-variables:[Assumption 3] {Xt} is a stationary sequence , fx() is bounded away from 0 on the interval [a,b](a < b), and fx(-) admits bounded derivatives until order I — 1.[Assumption 4] The conditional moment E{Xf\Xt-i = i) is continuous on [a, 6] (a < 6), moreover EX?1 < +oo.[Assumption 5] m(x) admits continuous derivatives until order /-Ion the interval [a, b] (a < b).[Assumption 6] K{-) is a kernel ,K(u) = K(—u),moreover, K(-) is a kernel of I — 1 order, i.e.:jK(y)dy=l, jyl-lK(y)dy^0 , f yiK(y)dy = 0, i = 1,2,. ..,1 - 2.For instance, the Gaussian kernel K(x) = e^{^}/2/y/2ir is a kernel of order 2.[Assumption 7] f*(t) = ï¿¡(eite) ^ 0, t G JS1.If al the conditions of the assumptions above are fulfil ed, and Yb.Yi, ??-,^n are observations , fe is known, we consider the local polynomial estimator of m(x) in the autoregressive model :) = F(0)TCn(x),where Cn(x) is defined asCn(x) = arg minC6fl't=lwhereSee Stefanski and Carroll(1991),if i^n() is a real function, then Kn(u) = iiTn(—u), and #(?) is a even bounded function that integrates to 1 although it is not nonnegative. Here, Kn(-) is taken as the weight function, we define Kn((itn) as Tnax{Kn(fhn),0} , andc =QF(u) =Mtn -t—i 3*KBased on the definition of the estimator ,we can obtain the properties of consistence and asymptotic normality of the local polynomial estimator under some specific conditions.We assume the following:= l,E(rn) = E(t&) = 0,m4 = E{(rj( - I)2} < +ao.Erj? <+oo.The density p(-) of 7/1 exists and satisfies inf p(x) > 0 for any compact HcR1.(A2) There exists 0 < cx < 1-E\rn\ such that |m(a;)| < Ci(l + |x|). There exists C2 > 0, such that |m(i) - m(y)| < &i\x - y\, x,y G R1.(A3) Eef < +00.Lemma 1. Under assumptions (Al) - (A3) the Markov process {Yt} defined above is geometrical y ergodic ,i.e. it is ergodic, with stationary probability measure tt() such that , for almost every y,for some 0 < p < 1. Here Pn(B\y) = P{Yn € B\Y0 = y}, for a Borel subset B C R1, and | ? | tv is the total variation distance. (A4) The function m(x) is (/ — 1) times continuously differentiable and there exists one-sided derivatives m± (x), at the point x 6 Rl-(A5) The density fi(-) of the stationary distribution ir(-) exists , is bounded, continuous and strictly positive in a neighbourhood of the point x.(A6) As we defined above ,the deconvolution kernel Kn : R1 —+ i?4" U {0} , uniformly bounded, moreover, max{max{|uj;tx € suppKn}} < +00. There exists k > 0, maxH-ftTnUoo < k. (A7) hn = Pn-W+1\ where (3 > 0.(A8) The distribution of Yo is definite,which is the stationary distribution tt() in Lemma 1 .Define the fol owing matrices= fF(u)FT(u)Kn(u)du,according to the definition of Kn(-) and condition (A6),we know for any positive interger n, A, and $n are positive definite (Tsybakov'10!).Set Dn = A¹^nA^1.Letu > 0, u<0.^-J F(u)ulKn(uWl\x,u)du.Denotem(x) m'(*)/?nC{x) =we have the theorems below:Theorem 1. Assume (A1)-(A8). Then {Cn(x) - C(x)}TF(0) -^ 0, asn —*â–  +00.Theorem 2. Assume(Al)-(A8).Then(x) - C(x)) - &;(*)] -^ iV(0, /,),n - +oo,where...