Efficient estimation in heteroscedastic varying coefficient models

Wei, Chuanhua; Wan, Lijie Article Efficient estimation in heteroscedastic varying coefficient models Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Wei, Chuanhua; Wan, Lijie (2015) : Efficient estimation in heteroscedastic varying coefficient models, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 3, Iss. 3, pp. 525-531, https://doi.org/10.3390/econometrics3030525 This Version is available at: https://hdl.handle.net/10419/171837 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ Econometrics 2015,3, 525-531; doi:10.3390/econometrics3030525 OPEN ACCESS econometrics ISSN 2225-1146 www.mdpi.com/journal/econometrics Article Efficient Estimation in Heteroscedastic Varying Coefficient Models Chuanhua Wei 1,* and Lijie Wan 2 1Department of Statistics, Minzu University of China, Beijing 100081 , China 2Department of Statistics, University of Kentucky, Lexington, KY 40508, USA; E-Mail: [email protected] *Author to whom correspondence should be addressed; E-Mail:[email protected]; Tel: +86-10-6893-3910 (ext. 8062). Academic Editor: Kerry Patterson Received: 16 January 2015 / Accepted: 2 July 2015 / Published: 15 July 2015 Abstract: This paper considers statistical inference for the heteroscedastic varying coefficient model. We propose an efficient estimator for coefficient functions that is more efficient than the conventional local-linear estimator. We establish asymptotic normality for the proposed estimator and conduct some simulation to illustrate the performance of the proposed method. Keywords: heteroscedasticity; local linear; varying coefficient models JEL classifications: C13; C14 1. Introduction Recently, the varying coefficient model has attracted much attention among econometricians and statisticians. One attractive feature of this model is its ability to capture the nonlinearity of the data without suffering from the “curse of dimensionality”. In general, it is of the form Yi=XT iα(Ui) + εi, i = 1,2,··· , n (1.1) where Y, isare responses; Xi= (Xi1, Xi2,··· , Xip)Tand Uiare associated covariates; α(·) = (α1(·),α1(·),··· ,αp(·))Tis a p-dimensional vector of unknown functions; ε, isare independent and identically distributed random errors with E(εi|Xi, Ui) = 0 and Var(εi|Xi, Ui) = σ2(Xi, Ui). Econometrics 2015,3526 Due to its flexibility, the varying coefficient model has been studied in many different contexts and has been successfully applied to nonlinear time series analysis, longitudinal and functional data analysis, panel data analysis, spatial data analysis, and time-varying models in finance. See, for example, the work of Cai et al. [1], Cai [2], Cai and Li [3], Cai et al. [4], Fan and Zhang [5], Fan et al. [6], Fotheringham et al. [7], Hoover et al. [8], Li et al. [9] and Xiao [10], among others. In the above models, the varying coefficient model is generally estimated by the local-linear approach. Usually, the errors are assumed to be i.i.d. to start. However, in applications, heteroscedasticity is often found in residuals from both cross-sectional and time series modelling. In the context of the linear regression model, it is well known that if the errors are heteroscedastic, then the generalized least-squares (GLS) estimator is more efficient than ordinary least-squares (OLS) estimator. To the best of our knowledge, there has been no work on the problem of designing an efficient estimation method for varying coefficient models with heteroscedastic errors. In this paper, we propose an efficient estimator for varying coefficients based on the local linear approach. The paper is structured as follows. We introduce an efficient estimator in Section 2, and their asymptotic properties are given in Section 3. We report the results of some Monte Carlo simulations in Section 4. 2. Efficient Estimation Without considering heteroscedasticity, we apply a local linear regression technique to estimate the varying coefficient functions. For each given u, the local linear estimator ˆ α(u)of α(u)is the part corresponding to aof the minimizer of n X i=1 Yi−XT ia−XT i(Ui−u)b2Kh(Ui−u0),(2.1) where Kis a kernel function, his a bandwidth and Kh(·) = K(·/h)/h. Then we have ˆ α(u) = [ˆ α1(u),··· ,ˆ αp(u)]T= (Ip0p){DT uWδ uDu}−1DT uWδ uY.(2.2) where X=       XT 1 XT 2 . . . XT n       ,Y=       Y1 Y2 . . . Yn       ,Du=       XT 1 U1−u hXT 1 XT 2 U2−u hXT 2 . . .. . . XT nUn−u hXT n       , and Wu= diag{Kh(U1−u), Kh(U2−u),··· , Kh(Un−u)}. The estimator ˆ α(u)ignores the information contained in the variance matrix and it is inefficient. To overcome this, we propose a class of efficient estimators in the following. Denote σi=pσ2(xi, ui), for the moment where we assume that σiis known. Multiply both sides of model (1.1) by 1/σi, we have the following homoscedastic varying coefficient model Y∗ i=ZT iα(ui) + ei, i = 1,2,··· , n, (2.3) where Y∗ i=yi/σi,Zi=Xi/σiand ei=εi/σiwith E(ei|xi, ui)=0,Var(ei|xi, ui) = 1. Econometrics 2015,3527 Applying the local linear approach to model (2.3), the efficient estimator of α(u)is given as follows ˜ α(u)=(Ip0p){BT uWuBu}−1BT uWuY∗.(2.4) where Y∗= (Y∗ 1, Y ∗ 2,··· , Y ∗ n), Bu=       ZT 1ZT 1 U1−u h ZT 2ZT 2 U2−u h . . .. . . ZT nZT nUn−u h       . 3. Asymptotic Property First, we make the following assumptions. Let µi=R∞ 0tiK(t)dt, νi=R∞ 0tiK2(t)dt. Assumption 1. The errors εi(i= 1,2,··· , n)are independent and 0<E(ε2 i)<∞and Var(ε2 i)>0. Assumption 2. The random variable Uhas a bounded support Π. Its density function f(·)is Lipschitz continuous and bounded away from 0on its support. Assumption 3. The p×pmatrixes E[XXT|U]and E[XXT/σ2(X, U)|U]are non-singular for each U∈Π. Assumption 4. There is an s > 2such that EkXk2s<∞and for some k < 2−s−1such that n2k−1h→ ∞as n→ ∞. Assumption 5. {αj(·), j = 1,··· , p}have continuous second derivatives in U∈Π. Assumption 6. The function K(·)is a symmetric density function with compact support and the bandwidth hsatisfies nh8→0and nh2/(log n)2→ ∞as n→ ∞. For the estimator ˆ α(u), Cai et al. [1] proved the following result: Theorem 1 Under the assumptions 1–6, the estimator ˆ α(u)is asymptotically normal, namely, √nh ˆ α(u)−α(u)−1 2h2µ2α00(u)→N(0, ν0Ψ/f(u)), where Ψ=Γ(U)−1E[XXTσ2(X, U)|U]Γ(U)−1,Γ(U) = E[XXT|U]. For the estimator ˜ α(u), we obtain the following result by the Theorem 1 directly. Theorem 2 Under the assumptions 1–6, the estimator ˜ α(u)is asymptotically normal, namely, √nh ˜ α(u)−α(u)−1 2h2µ2α00(u)→N(0, ν0Φ−1/f(u)), Econometrics 2015,3528 where Φ= E(ZZT|U) = E(XXT/σ2(X, U)|U). Denote ¯ X= ( ¯ X1,··· ,¯ Xn)T,¯ Xi=pKh(Ui−u)Xi,Σ= diag {σ2(X1, U1),··· , σ2(Xn, Un)}. By the proof of Theorem 1 in Cai et al. [1], we have 1 n n X i=1 Kh(Ui−u)XiXT i=¯ XT¯ Xp −→ E(XXT|U)f(u), 1 n n X i=1 Kh(Ui−u)XiXT iσ2(X, U) = ¯ XTΣ¯ Xp −→ E(XXTσ2(X, U)|U)f(u), and 1 n n X i=1 Kh(Ui−u)XiXT i/σ2(X, U) = ¯ XTΣ−1¯ Xp −→ E(XXT/σ2(X, U)|U)f(u). Since (¯ XT¯ X)−1¯ XTΣ¯ X(¯ XT¯ X)−1≥(¯ XTΣ−1¯ X)−1, then we have Ψ≥Φ−1. This implies that ˜ α(u)is asymptotically more efficient than ˆ α(u)in terms of asymptotic covariance matrix. Remark 1. Since ˜ α(u)depends on the unknown parameters σ2(X, U), it is infeasible. To provide a feasible efficient estimator of α(u), we need to estimate σ2(X, U)consistently. It is not difficult to show that the resultant feasible estimator has the asymptotic property as ˜ α(u). Remark 2. To obtain the consistent estimator of the variance function σ2(z, u), it is important to model σ2(z, u). Several kinds of variance function have been proposed. Discussion on the parametric variance function can be found in Carroll and Ruppert [11]. Muller and Stadtmuller [12], Chiou and Muller [13] and Ruppert et al. [14] studied nonparametric variance estimation. Muller and Zhao [15] proposed a general semiparametric variance function model in a fixed design regression setting. Keilegom and Wang [16] considered a general class of mean-variance regression models, in which both the mean function and the variance function were semiparametrically modeled. Zhu et al. [17] consider a single-index structure to study heteroscedasticity in a single-index regression model with high-dimensional predictors. 4. Simulation Studies In this section we compare the behavior of the conventional estimator ˆ α(u)with that of the new estimator ˜ α(u), given in (2.2) and (2.4), respectively, when the sample size is finite. The data are generated from the following varying coefficient model yi=xiα(ui) + σ(xi, ui)εi, i = 1,2,··· , n, (4.1) where xi∼N(0,1), ui=i/n,α(ui) = ui+ sin(2πui). Firstly, we consider the following four known variance functions: (A) : σ(xi, ui) = exi; (B) : σ(xi, ui) = eui; Econometrics 2015,3529 (C) : σ(xi, ui) = 1 + xi; (D) : σ(xi, ui) = 1 + ui. Secondly, we consider the case that the variance function is unknown. For simplicity, the variance function is assumed to have the following parametric structure, (E) : E(ε2 i|xi, ui) = σ2(xi, ui) = exp(γ0+γ1ui), with γ0= 1, γ1= 2. Obviously, we can build the following linear regression model ln ε2 i=γ0+γ1ui+ξi,(4.2) with Eξi= 0. In practice, εiis not available, but it may be estimated by ˆεi=yi−xT iˆ α(ui), where ˆ α(ui)are the local linear estimates of model (4.1) without considering the heteroscedasticity structure. Applying the least squares approach to liner model (4.2) with εiwas replaced by ˆεi, we can obtain the estimators of γ0, γ1, denoted by ˆγ0and ˆγ1respectively. Accordingly, we get the estimator of σ2(xi,zi, ui), as ˆσ2(xi,zi, ui) = exp(ˆγ0+ ˆγ1ui). To study the effect of the distribution of the error for our method, we take the following three different types of the error distribution, (1) εi∼N(0,0.52), (2) εi∼U(−√3/2,√3/2), (3) εi∼1 8χ2 8−1. The Gauss kernel function and h=n−1/5are used in our simulation studies. We compare the proposed efficient estimator ˜ α(u)with that of the ordinary local linear estimator ˆ α(u)by using the estimated mean average squared error (MASE), MASE{ˆ α(·)}=1 1000 ∗n N X l=1 n X i=1 [ˆ αl j(ui)−αj(ui)]2, where ˆ αl(ui), l = 1,2,··· , N, are the estimate of the coefficient α(ui)in N= 1000 replications. The simulation results are presented in Table 1, and for all the scenarios we studied, the proposed efficient estimators outperform the ordinary local linear estimators. Table 1. Mean average squared error (MASE) index for the estimators of varying coefficients. Variance Sample N(0,0.52)U(−√3/2,√3/2) 1 8χ2 8−1 Function nˆ α(·)˜ α(·)ˆ α(·)˜ α(·)ˆ α(·)˜ α(·) A 30 0.4873 0.1354 0.5125 0.1277 0.5109 0.1374 50 0.4063 0.0805 0.3820 0.0804 0.3686 0.0805 80 0.3146 0.0539 0.3054 0.0523 0.2910 0.0525 B 30 0.2018 0.1936 0.1872 0.1791 0.1835 0.1752 50 0.1129 0.1077 0.1174 0.1126 0.1114 0.1072 80 0.0728 0.0696 0.0734 0.0706 0.0733 0.0706 C 30 0.1469 0.1302 0.1502 0.1269 0.1571 0.1304 50 0.1024 0.0920 0.1039 0.0893 0.1015 0.0884 80 0.0670 0.0679 0.0731 0.0660 0.0734 0.0645 D 30 0.1446 0.1422 0.1390 0.1368 0.1487 0.1474 50 0.0841 0.0831 0.0844 0.0832 0.0819 0.0807 80 0.0612 0.0606 0.0556 0.0549 0.0595 0.0589 E 30 0.3950 0.3936 0.4227 0.4133 0.3996 0.3913 50 0.3094 0.3039 0.3137 0.3068 0.3077 0.3015 80 0.2549 0.2508 0.2513 0.2508 0.2544 0.2520 Econometrics 2015,3530 5. Conclusions In this paper, we focus on the estimation problem of the varying coefficient model with heteroscedastic errors. Based on the local linear method, we develop a simple approach to estimate the nonparametric coefficient functions by taking the estimated error heteroscedasticity into account. The resulting estimators are shown to have smaller asymptotic variances than the conventional local-linear estimators. The asymptotic normality of the proposed estimator is established. Furthermore, some simulation experiments are performed to evaluate the finite sample behaviors of the proposed estimators. Acknowledgments We are grateful to two anonymous referees for helpful comments on this paper. Chuanhua Wei’s research was supported by the National Natural Science Foundation of China (No.11301565), and Beijing Higher Education Young Elite Teacher Project(No.YETP1316). Author Contributions The authors contributed equally to this work. Conflicts of Interest The authors declare no conflict of interest. References 1. Cai, Z.W.; Fan, J.Q.; Yao, Q.W. 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