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Spurious nonparametric regression in econometrics

Spurious nonparametric regression in econometrics -...

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Spurious nonparametric regression in econometrics * Li Songchen 1 , Chen Haiyan 2 1 College of Mathematics and Computational Science, Shenzhen University, Guangdong Shenzhen, 518060, China 2 School of Management, Tianjin University, Tianjin, 300072,China Abstract This paper develops the asymptotic theory for the Nadaraya-Watson kernel estimator and local polynomial estimator when two independently integrated processes are used in a nonlinear regression. It is shown that the Nadaraya- Watson kernel estimator and the local polynomial estimator do not possess limiting distributions in this context but actually diverge at rate 1/ 2 n as the sample size n → ∞ , and this is slower than that of parameters in linear regression. In spite of the difference in the rate of divergence between the parametric and nonparametric cases, they all can induce spurious regression. Keywords : Nadaraya-Watson kernel estimation; Local polynomial estimation; Spurious regression; Integrated processes; Local time; Quadratic variation. 1. Introduction The spurious regression was first studied by Granger and Newbold (1974) using simulation, and the simulation results are supported by Phillips’ (1986) theoretical analysis. Phillips proves that the usual t test statistic in a spurious regression does not have a limiting distribution but diverges as the sample size approaches infinity. He also shows that 2 R has a nondegenerate limiting distribution while the DW statistic converges in probability to zero. These results has been generalized by Marmol (1998), Tsay and Chung (2000) to cases with fractionally integrated processes. Most studies of the spurious regression concentrate on linear parametric regression and the asymptotic distributions of parameters and related statistics. By contrast, spurious regression in nonparametric cases and the asymptotic distributions for Nadaraya-Watson (NW, henceforth) kernel estimator and local polynomial estimator of nonlinear relationship of independently integrated processes are presently undeveloped. The present paper relates to some existing work in nonlinear transformations of integrated time series and an asymptotic theory of inference for nonlinear regression were developed in Park and Phillips (1999, 2001). de Jong (2004), de Jong and Wang (2005), Potscher (2004), and Berkes and Horvath (2006) extended these results for nonlinear transformations to cover a wider class of functions. Bandi and Phillips (2003) developed an asymptotic theory of function estimation and inference in possibly nonstationary difussions. * This project was supported by the National Natural Science Foundation of China (No: 70471050). Email: [email protected] 1
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Some results to the study of kernel density estimation, nonparametric regression and nonparametric autoregression in the context of integrated processes or near-integrated processes were developed in Phillips and Park (1998), Bandi (2004). Karlsen and Tjostheim (2001), Guerre (2004), Karlsen, Myklebust and Tjostheim (2007) studied nonparametric estimation for recurrent
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Spurious nonparametric regression in econometrics -...

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