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On semiparametric EV models with serially correlated errors in both regression models and measured covariates. (English) Zbl 1142.62043
The response $$y_i$$ depends on the covariates $$\xi_k(t)$$ and $$z_{ki}$$ through the following linear regression model: $y_i=\beta_1\xi_1(t_i)+\dots+\beta_p\xi_p(t_i)+\alpha_1z_{1i}+\dots+a_qz_{qi} +\varepsilon_i.$ The variables $$y$$, $$z$$ and $$t$$ are fully observed but $$\xi(t)$$ is measured with errors: the observations are $$x_{ki}=\xi_k(t_i)+u_{ki}$$. The errors $$\varepsilon_i$$ are modelled by an autoregression model and $$u_{ki}$$ are modelled by a vector autoregression. The authors describe a weighted estimating-equations based estimator for $$\alpha$$ and $$\beta$$ in which kernel smoothing is used to approximate $$\xi_k(t)$$ and difference-based algorithms are applied for the estimation of the covariance structure of the errors. It is shown that the obtained estimates are asymptotically normal in $$\sqrt{n}$$-asymptotics. Results of simulations and an application to Microsoft stock data are presented.

##### MSC:
 62J05 Linear regression; mixed models 62G08 Nonparametric regression and quantile regression 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation
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