TESTING SERIAL CORRELATION IN SEMIPARAMETRIC VARYING COEFFICIENT PARTIALLY LINEAR ERRORS-IN-VARIABLES MODEL

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1 Jrl Syst Sci & Complexity (2009) 22: TESTIG SERIAL CORRELATIO I SEMIPARAMETRIC VARYIG COEFFICIET PARTIALLY LIEAR ERRORS-I-VARIABLES MODEL Xuemei HU Feng LIU Zhizhong WAG Received: 19 September 2006 / Revised: 25 August 2008 c 2009 Springer Science + Business Media, LLC Abstract The authors propose a V,p test statistic for testing finite-order serial correlation in a semiparametric varying coefficient partially linear errors-in-variables model. The test statistic is shown to have asymptotic normal distribution under the null hypothesis of no serial correlation. Some Monte Carlo experiments are conducted to examine the finite sample performance of the proposed V,p test statistic. Simulation results confirm that the proposed test performs satisfactorily in estimated size and power. Key words Asymptotic normality, local linear regression, measurement error, modified profile least squares estimation, partial linear model, testing serial correlation, varying coefficient model. 1 Introduction Consider the following semiparametric varying coefficient partially linear errors-in-variables model: { Y = X β + Z α(t) + ε, (1.1) ξ = X + η, where Y is the response, X = (X 1, X 2,, X p ) and Z = (Z 1, Z 2,, Z q ) are regressors, β = (β 1, β 2,, β p ) is a vector of p-dimensional unknown parameters, and α( ) = (α 1 ( ), α 2 ( ),, α q ( )) is a vector of unknown smooth functions. Due to the curse of dimensionality, we assume, for simplicity, that T is univariate. The superscript denotes the transpose of a vector or matrix. The error ε is assumed to be covariance stationary with zero mean and variance σ 2, with common cumulative distribution function (cdf) F ε, where σ 2 > 0 is an unknown parameter. The covariate X is measured with additive error, and Z and T are error free. The Xuemei HU Mathematics and Statistics College, Chongqing Technology and Business University, Chongqing , China. huxuem@163.com Feng LIU School of Mathematics and Physics, Chongqing Institute of Technology, Chongqing , China. Zhizhong WAG School of Mathematics and Computing Technology, Central South University, Changsha , China. This research is supported by the ational atural Science Foundation of China under Grant os and ; the Science and Technology Project of Chongqing Education Committee under Grant o. KJ080609; the Doctor s Start-up Research Fund under Grant o ; and the Youth Science Research Fund of Chongqing Technology and Business University under Grant o

2 484 HU XUEMEI liu FEG WAG ZHIZHOG measurement error η is independent of (X, Z, T, ε). In order to identify model (1.1), we assume that Σ η, the covariance matrix of η, is known. Obviously, model (1.1) includes many usual parametric, semi-parametric, and nonparametric regression models. For example, when α( ) = a (where a is a constant vector), model (1.1) reduces to the usual linear errors-in-variables model. Cui and Chen [1] proposed a constrained empirical likelihood confidence region for a parameter β 0 in the linear errors-in-variables model. When q = 1 and z = 1, model (1.1) becomes the partially linear errors-in-variables model. Liang et al. [2 3] and Cui et al. [4 5], among others, have studied the estimation and application of the model. When the covariate X is measured exactly and β = 0, model (1.1) reduces to varying coefficient regression model that has been widely studied as Hastie and Tibshirani [6], Xia and Li [7], Fan and Zhang [8], Chiang et al. [9], Huang et al. [10], among others. When β 0, model (1.1) becomes a semiparametric varying coefficient partially linear model. Fan and Huang [11] proposed the profile least squares estimation of β and local linear estimation of α( ) and made profile likelihood inference. When the covariate X is measured with error, You and Chen [12] obtained a modified profile least squares estimator for β and a local linear estimator for the varying coefficient functions α( ). Testing for serial correlation has long been a standard practice in applied econometric analysis because the presence of serial correlation invalidates the classical formula for the variancecovariance matrix of ordinary least squares estimators and leads to the inconsistency of these estimators if the regressors contain lagged dependent variables. Especially, strong serial correlation is often an indication of omitting important explanatory variables or functional form misspecification. Therefore, ignoring serial correlation when it is significant can lead to misleading inference. Chi and Reinsel [13 14] tested serial correlation in an AR(1) random effect model, while Kyriazidou [15] and Godfrey [16] tested serial correlation in multivariate regression models and dynamic regression models, respectively. When models are only partially specified, Li and Hsiao [17] first proposed two test statistics: S n and V n,p for testing serial correlation in a semiparametric partially linear panel data model that could allow lagged dependent variables as explanatory variables. The first is for testing a first-order serial correlation, and the second one is for testing higher-order serial correlation. Then, Li and Stengos [18] suggested two test statistics: S T and V T,p, to test finite-order serial correlation in a semiparametric times series model. Recently, Gong et al. [19] proposed one similar test statistic to a linear errors-in-variables model for testing serial correlation. In this paper, we propose a V,p test statistic for testing finiteorder serial correlation in semiparametric varying coefficient partially linear errors-in-variables model (1.1). The proposed V,p test is valid not only for first-order serial correlation but also for higher-order serial correlation. In addition, it is distributed-free, i.e., V,p test is valid not only for a symmetry distribution, but also for an asymmetry distribution (e.g., the tail heavy t distribution). The paper is organized as follows. In Section 2, we set up the regression model, estimation and assumptions. Section 3 obtains Theorem 3.1 and proposes the V,p serial correlation test. Monte Carlo studies of size and power of the V,p test are provided in Section 4. Section 5 discusses the proof of Theorem Regression Model, Estimation, and Assumptions Suppose that we have a random sample {Y i, ξ i = (ξ i1, ξ i2,,ξ ip ), Z i = (Z i1, Z i2,, Z iq ), T i } n from model (1.1), i.e., { Yi = Xi β + Z i α(t i) + ε i, (2.1) ξ i = X i + η i,

3 TESTIG SERIAL CORRELATIO 485 where X i cannot be observed and η i are i.i.d. random vector with mean zero and covariance matrix Σ η. The null hypothesis to be tested is that the errors ε i are serially uncorrelated, and the interested alternative hypothesis is a p-th order autoregression, denoted by AR(p) and written as ε i = a 1 ε i a p ε i p + u i, u i i.i.d. (0, σ 2 ), in which a i satisfies the stationary condition that the roots of equation a(u) = 1 a 1 u a 2 u 2 a p u p = 0 lie outside the unit circle, or a p-th order moving average, denoted by MA(p) and written as ε i = u i + a 1 u i a p u i p, u i i.i.d. (0, σ 2 ). Let a = (a 1, a 2,, a p ) be the error coefficient vector. If a = 0, serial correlation in model (2.1) does not exist; otherwise, serial correlation exists. Especially, when a 1 0, a 2 = = a p = 0, first-order serial correlation exists in model (2.1). Thus, our aim is to test whether a = 0 or not, i.e., H 0 : a = 0 H 1 : a 0 (2.2) Denote γ = (γ 1, γ 2,,γ p ), γ k = Eε i ε i+k, k = 1, 2,,p; i = 1, 2,,; = n p. (2.3) According to time series analysis theory, we easily show that the test problem (2.2) is equivalent to the following test problem: H 0 : γ = 0 H 1 : γ 0. Let v i = Y i ξi β Z i α(t i) = ε i ηi β. Then, under the null hypothesis of no serial correlation, v i are i.i.d. random variables with mean 0 and finite variance σ 2 + β Σ η β. Denote γ = ( γ 1, γ 2,, γ p ), γ k = Ev i v i+k, k = 1, 2,,p; i = 1, 2,,; = n p. (2.4) The fact that ε i is independent of the i.i.d. random vector η i suggests that γ k = E(ε i η i β)(ε i+k η i+k β) = Eε iε i+k = γ k holds. Thus, (2.2) is equivalent to the following hypotheses: H 0 : γ = 0 H 1 : γ 0, (2.5) i.e., testing whether ε i is serially uncorrelated or not is translated into testing whether v i is serially uncorrelated or not. Since v i = Y i ξ i β Z i α(t i), i = 1, 2,,n, include the unknown β and unknown α(t). Thus, to test whether v i is serially uncorrelated, we first need to estimate β and α(t). Following You and Chen [12], we will adopt local linear regression technique to estimate α(t), and adopt the modified profile least squares approach to estimate β. For any given β, model (2.1) can be written as Y i p X ij β j = j=1 q Z ij α j (T i ) + ε i, i = 1, 2,,n, (2.6) j=1 where X ij and Z ij are the j-th element of X i and Z i, respectively. This is a varying coefficient model. Hence, we apply a local linear regression technique to estimate the varying coefficient functions {α j ( ), j = 1, 2,,q}. For each given point T 0, approximate α j (T) locally by a linear function α j (T) α j (T 0 ) + α j (T 0)(T T 0 ) a j + b j (T T 0 ), j = 1, 2,,q, for T in

4 486 HU XUEMEI liu FEG WAG ZHIZHOG a small neighborhood of T 0. This leads to the following weighted local least-squares problem: find {(a j, b j ), j = 1, 2,,q} to minimize n [( Y i p ) X ij β j j=1 q ] 2 {a j + b j (T i T 0 )}Z ij K h (T i T 0 ), (2.7) j=1 for a given kernel function K( ) and bandwidth h, where K h ( ) = K( /h)/h. Denote X1 X 11 X 12 X 1p Z X2 1 Z 11 Z 12 Z 1q X =. = X 21 X 22 X 2p......, Z = Z 2. = Z 21 Z 22 Z 2q......, Xn X n1 X n2 X np Zn Z n1 Z n2 Z nq Z1 Z2 D T =. Z n T 1 T 0 h Z1 T 2 T 0 h Z2. Tn T0 h Z n, Y = (Y 1, Y 2,,Y n ) W T = diag(k h (T 1 T 0 ), K h (T 2 T 0 ),, K h (T n T 0 )). Let ã 1 (T), ã 2 (T),,ã q (T), b 1 (T), b 2 (T),, b q (T) denote the estimator of α 1 (T), α 2 (T),, α q (T), b 1 (T), b 2 (T),,b q (T), respectively. Then, the solution to the least squares problem (2.7) can be expressed as ( α 1 (T), α 2 (T),, α q (T), h b 1 (T), h b 2 (T),, h b q (T)) = (D TW T D T ) 1 D TW T (Y Xβ). Hence, ã(t) = ( α 1 (T), α 2 (T),, α q (T)) = (I q q 0 q q )(D T W TD T ) 1 D T W T(Y Xβ). Substituting ã(t) into model (2.6), we obtain Y i Ŷi = (X i X i ) β + ε i ε i, i = 1, 2,, n, (2.8) where Ŷ = (Ŷ1, Ŷ2,, Ŷn) = SY, X = ( X1, X2,, Xn ) = SX, ε = ( ε 1, ε 2,, ε n ) = Sε with S = (S1,S 2,,Sn), where Si = (Z i 0)(D T i W Ti D Ti ) 1 DT i W Ti. If we take ε i ε i as the residual errors, then, (2.8) is a version of the ordinary linear regression model. Applying the least squares to model (2.8) results in the profile least squares estimator β n = { n (X i X i )(X i X i ) } 1 n (X i X i )(Y i Ŷi) of β. However, X i is measured with additive errors. To overcome inconsistency caused by the measurement error, we use the following modified profile least squares estimator: { n } 1 β n = (ξ i ξ i )(ξ i ξ n i ) nσ η (ξ i ξ i )(Y i Ŷi), (2.9) where ξ = ( ξ 1, ξ2,, ξn ) = Sξ and ξ = (ξ 1,ξ 2,, ξ n). Moreover, the fact that E(Z i T i) α(t i ) = E(Y i X i β T i) = E(Y i ξ i β T i) suggests to estimate α(t) by α(t) = ( α 1 (T), α 2 (T),, α q (T)) = (I q 0 q q )(D T W TD T ) 1 D T W T(Y ξ β n ). (2.10) To derive the main result, we make the following assumptions, which are also assumed by You and Chen [12].

5 TESTIG SERIAL CORRELATIO 487 Assumption 2.1 The random variable T has a bounded support Q, and its density function f( ) is Lipschitz continuous and bounded away from 0 on its support. Assumption 2.2 The q q matrix E(ZZ T) is nonsingular for each T Q, and E(XX T), E(ZZ T) and E(XZ T) are all Lipschitz continuous. Assumption 2.3 There is an s > 2 such that E X 2s <, E Z 2s <, E ε 2s <, and E η 2s <, and for some δ < 2 s 1, there is n 2δ 1 h as n Assumption 2.4 {α j ( ), j = 1, 2,,q} have the continuous second derivative in T Q. Assumption 2.5 The function K( ) is a symmetric density function with compact support, and the bandwidth h satisfies nh 8 0 and nh 2 /(log n) 2 as n. Following You and Chen [12], the above technical conditions guarantee that the n consistency of β n, also guarantee that the estimator â(t) attains the optimal strong convergence rate of the usual nonparametric regression. 3 V,p Test In this section, we suggest a test statistic for testing the absence of serial correlation. Under the null hypothesis of no serial correlation, γ = ( γ 1, γ 2,, γ p ) = 0. Therefore, we shall use γ as the basis to construct our test statistic. Let U i = (U i1, U i2,,u ip ), U ik = υ i υ i+k, i = 1, 2,,p; i = 1, 2,,; = n p; (3.1) Û i = (Ûi1, Ûi2,,Ûip), Û ik = υ i υ i+k, i = 1, 2,,p; i = 1, 2,,; = n p; (3.2) where υ i = Y i ξ i β n Z i α(t i) = υ i + ξ i (β β n ) + Z i (α(t i) α(t i )), i = 1, 2,,n, are the estimated residuals of the semiparametric varying-coefficient partially linear errors-invariables model (2.1). Following Lemma 5.1, it can be shown that α(t) α(t) = O p (c n ), β n β = O p (n 1/2 ). ote that E ξ 2 and E Z 2 are finite. Using the Cauchy-Schwartz inequality, it follows that υ i = υ i + o p (1). Hence, we propose the statistic V,p = 1 Û i (3.3) for testing finite-order serial correlation in model (2.1). The following theorem gives the asymptotic distribution of V,p. Theorem 3.1 Suppose that Assumptions hold. Then, under the null hypothesis, V,p L (0, σ 2 0 I p) and σ 2 0 V,p V,p L x 2 p as, where L denotes convergence in distribution, σ 0 = σ 2 + β Σ η β, and I p is a p-order unit matrix. In order to apply the asymptotic distribution for making statistical inferences, we need to estimate σ 0. Based on β n and α(t), we have obtained the estimated residuals υ i = Y i ξi β n Zi α(t i), i = 1, 2,, n. Thus, under the null hypothesis, an estimator of the error variance σ 0 n (Y i ξ i β n Z i α(t i)) 2. It is easy to show that σ 2 0 V,p V,p L x 2 p is given by σ 0 = 1 n as. Remark 1 In practice, the V,1 test is analogous to the Durbin-Watson [20 21] (hereafter DW) test. However, the DW test only uses AR(1) with a normal error distribution as the alternative model, while V,1 test can use AR(1) or MA(1) with a symmetric or asymmetric

6 488 HU XUEMEI liu FEG WAG ZHIZHOG error distribution (e.g., the heavy tail distribution) as the alternative model. The V,p, p > 1 test is similar to the LM tests of Breusch [22] and Godfrey [23]. One main difference is that the V,p test use residuals from a semiparametric varying-coefficient partially linear errors-invariables model, while LM tests use residuals from a parametric model. Also, since we do not derive our test from the maximum likelihood principle, we do not need the normality assumption for the error term ε i. Thus, the proposed V,p test is distributed-free. Remark 2 Gong et al. [19] proposed one similar test statistics for testing serial correlation in a linear EV (i.e., errors-in-variables) model and derived one similar results as Theorem 3.1. In fact, the test statistic is just the special case of our V,p test statistic. 4 Simulation Results In this section, we report some simulation results to examine the finite sample performance of V,p test. The data are generated from model Y i = X i β + Z i α(t i ) + ε i, ξ i = X i + η i, i = 1, 2,, n, where X i (1, 2), Z i (0, 2), T i U(0, 1), β = 1, α(t i ) = sin(2πt i ), η i (0, 1), and {ε i } are generated by from the following four different processes: 1) ε i = ρε i 1 + u i, ρ < 1, (AR(1)); 2) ε i = u i + ρu i 1, ρ < 1, (MA(1)); 3) ε i = a 1 ε i 1 + a 2 ε i 2 + u i, (AR(2)); 4) ε i = a 1 u i 1 + a 2 u i 2 + u i, (MA(2)), where u i satisfies a) u i (0, 1), b) u i U( 3, 3), or c) u i Student t(3). Samples of size are n=50, 100, 150, and 200, respectively. The kernel is the Gaussian kernel K h ( ) = 1 h 2π exp( ( )2 /2h 2 ), and the bandwidth is selected by h = n 1 5 Tsd, where T sd is the standard error of T. The number of simulated realizations is We compute the V,p test s probabilities of rejecting the null hypothesis at 5 % nominal significance levels. Tables 1 6 correspond to a one-order alternative hypothesis and report the V,1 test s estimated sizes and powers. Tables 7 12 correspond to two-order alternative hypothesis, and give the V,2 test s estimated sizes and powers. Table 1 V,1 test s estimated size and power, AR(1) with u i (0, 1) ρ n (size) Table 2 V,1 test s estimated size and power, AR(1) with u i U( 3, 3) ρ n (size)

7 TESTIG SERIAL CORRELATIO 489 Table 3 V,1 test s estimated size and power, AR(1) with u i Student t(3) ρ n (size) Table 4 V,1 test s estimated size and power, MA(1) with u i (0, 1) ρ n (size) Table 5 V,1 test s estimated size and power, MA(1) with u i U( 3, 3) ρ n (size) Table 6 V,1 test s estimated size and power, MA(1) with u i Student t(3) ρ n (size) Table 7 V,2 test s estimated size and power, AR(2) with u i (0, 1) a = (a 1, a 2) n (0, 0)(size) (0, 0.5) (0.3, 0.4) (0.4, 0.6) ( 0.2, 0.6)

8 490 HU XUEMEI liu FEG WAG ZHIZHOG Table 8 V,2 test s estimated size and power, AR(2) with u i U( 3, 3) a = (a 1, a 2) n (0, 0)(size) (0, 0.5) (0.3, 0.4) (0.4, 0.6) ( 0.2, 0.6) Table 9 V,2 test s estimated size and power, AR(2) with u i Student t(3) a = (a 1, a 2) n (0, 0)(size) (0, 0.5) (0.3, 0.4) (0.4, 0.6) ( 0.2, 0.6) Table 10 V,2 test s estimated size and power, MA(2) with u i (0, 1) a = (a 1, a 2) n (0, 0)(size) (0, 0.5) (0.3, 0.4) (0.4, 0.6) ( 0.2, 0.6) Table 11 V,2 test s estimated size and power, MA(2) with u i U( 3, 3) a = (a 1, a 2) n (0, 0)(size) (0, 0.5) (0.3, 0.4) (0.4, 0.6) ( 0.2, 0.6) Table 12 V,2 test s estimated size and power, MA(2) with u i Student t(3) a = (a 1, a 2) n (0, 0)(size) (0, 0.5) (0.3, 0.4) (0.4, 0.6) ( 0.2, 0.6)

9 TESTIG SERIAL CORRELATIO 491 From these tables, we can see that under the null hypothesis and different error distributions, the estimated sizes are quite good and close to their nominal levels Under the alternative hypothesis and different error distributions, test s powers are also quite good. Higher-order cases are also similar. 5 Proof of Theorem 3.1 ote that = n p, hereafter we do not distinguish O p (n) (or o p (n)) from O p () (or o p ()). Lemma 5.1 Suppose that Assumptions hold. Then, under the null hypothesis: 1) The modified profile least squares estimator β n of β is asymptotically normal, namely, as n, n( βn β) L (0,Σ1 1 Σ 2Σ1 1 ), where Σ 1 and Σ 2 are similar to those of You and Chen [12]. 2) For any given β, the local linear estimator α( ) of α( ) attains the following strong convergence rate, max sup α j (T) α j (T) = O(c n ), 1 j q T Q where c n = h 2 + (log n/nh) 1 2. The lemma comes from You and Chen [12]. Lemma 5.2 Let G i be i.i.d. r.v.s. with EG i = 0 and EG 2 i <. Then, for any permutation (j 1, j 2,, j n ) of (1, 2,,n), k max G ji = O p (n 1 2 log n). 1 k n The lemma comes from Gao [24]. Proof of Theorem 3.1 Let e i = ξ i (β β n ) + Z i (α(t i) α(t i )). Then, for any integer k, 1 k p, where 1 Û ik = 1 v i v i+k = 1 (v i + e i )(v i+k + e i+k ) a.s., = 1 v i v i+k + 1 e i e i+k + 1 e i v i+k + 1 v i e i+k = 1 U ik + I 1 + I 2 + I 3, I 1 = 1 [ξi (β β n )ξi+k(β β n )]

10 492 HU XUEMEI liu FEG WAG ZHIZHOG + 1 [Zi (α(t i) α(t i ))Zi+k (α(t i+k) α(t i+k ))] + 1 [ξi (β β n )Zi+k(α(T i+k ) α(t i+k ))] + 1 [Zi (α(t i ) α(t i ))ξi+k(β β n )] = I 11 + I 12 + I 13 + I 14. It is easy to see that I 11 = (β β n ) [ 1 ξ iξ i+k ](β β n ) holds. Applying the ergodic theorem to every element of the matrix 1 ξ iξ i+k, we find that 1 ξ iξ i+k converges to a bounded matrix. Hence, I 11 = o p (1). ext combining Assumption 2.3, Assumption 2.5, and Lemma 5.1, we have I 12 = 1 [ q [ = 1 q j 1=1 q j 1=1 j 2=1 q max sup 1 j 1 q j 1=1 j T Q 2=1 1 Z i,j1 Z i+k,j2 = O(c 2 n)o p (1) = o p (1), ][ q Z i,j1 (α j1 (T i ) α j1 (T i )) j 2=1 Z i+k,j2 (α j2 (T i+k ) α j2 (T i+k ))] q Z i,j1 Z i+k,j2 (α j1 (T i ) α j1 (T i ))(α j2 (T i+k ) α j2 (T i+k ))] α j1 (T) α j1 (T) max sup α j2 (T) α j2 (T) 1 j 2 q T Q I 13 1 { }1 { [ξi (β β n )] 2 2 } 1 [Zi+k (α(t i+k) α(t i+k ))] [ ]1 [ ξ i 2 β β n 2 2 ] 1 Z i+k 2 α(t i+k ) α(t i+k )) 2 2 [ β β n 2 1 ξ i 2 ] 1 2 = O p ( 1 2 )Op (1)O(c n )O p (1) = o p (1). Similar to I 13, I 14 = o p (1). Thus, I 1 = o p (1). [ sup α(t) α(t) 2 1 T Q Z i+k 2 ] 1 2 I 2 = 1 [ξi (β β n )v i+k ] + 1 [Zi (α(t i ) α(t i ))v i+k ] = I 21 + I 22. Obviously, I 21 (β β n ) 1 ξ iv i+k, where the k-dependent random vector ξ i v i+k = (X i + η i )(ε i+k η i+k β) satisfies Eξ iv i+k = 0 and E ξ i v i+k 2 <. Thus, according to

11 TESTIG SERIAL CORRELATIO 493 the ergodic theorem, it follows that 1 ξ iv i+k = o p (1). Therefore, I 21 = o p (1). Let Z i v i+k = G i. Then, by Abel s Inequality and Lemma 5.2, I 22 = 1 (α(t i ) α(t i )) Z i v i+k 1 sup α(t) α(t) max T Q 1 k = 1 n O(c n )O p (n 1 2 log n) = op (1). Thus, I 2 = o p (1). Similar to I 2, I 3 = o p (1). Hence, Obviously, 1 Û ik = 1 U ik + o p (1). 1 Û i = 1 U i + o p (1). G ji ote that U i = (U i1, U i2,,u ip ) is a p-dependent random vector, and for any i j, Cov(U i, U j ) = (EU ik U jl ) 1 k,l p = 0. Then, for any non-zero p-dimensional real vector b, b U i is a p-dependent random variable, and for any i j, Cov(b U i, b U j ) = 0. Thus, combining the central limit theorem for a p-dependent random sequence of Brockwell and Davis [25], it holds that 1 b U i L (0, b bσ0). 2 By applying Crámer-Wold device, we can prove that the p-dimension random vector U i = (U i1, U i2,, U ip ) follows the multivariate normal distribution (0, σ0i 2 p ) iff any linear combination b U i of U i follows the normal distribution (0, b bσ0 2 ). Therefore, ( ) V,p = Û i1, Û i2,, Û ip ( ) = U i1, U i2,, U ip + o p (1) L (0, σ0i 2 p ). k References [1] H. J. Cui and S. X. Chen, Empirical likelihood confidence regions for parameter in the errors-invariable models, Journal of Multivariate Analysis, 2003, 84(1): [2] H. Liang, W. Härdle, and R. J. Carroll, Estimation in a semiparametric partially linear errors-invariables model, The Annals of Statistics, 1999, 27(5): [3] H. Liang, Asymptotic normality of parametric part in partially linear models with measurement error in the nonparametric part, Journal of Statistical Planning and Inference, 2000, 86(1): [4] H. J. Cui and R. C. Li, On parameter estimation for semi-linear error-in-variable models, Journal of Multivariate Analysis, 1998, 64(1): 1 24.

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