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Projection pursuit autoregression in time series. (English) Zbl 0940.62083
Suppose that \(y_t\) is an observed stationary time series defined by \( y_t=G(x_t)+\varepsilon_t \), where \(\varepsilon_t\) are i.i.d. random variables, \(x_t\in R^p\) are the lag variables for \(y_t\) and possibly other explanatory variables \(G(x)=E(y_t |x_t=x)\). For any \(p\)-dimensional unit vector \(\vartheta\) denote \( \phi_\vartheta(\vartheta^Tx)=E(y_t |\vartheta^Tx_t=\vartheta^T x) \). Then the first projection pursuit component of \(G\) within a bounded region \(\mathcal A\) is \(\vartheta_1\), which minimizes \[ S(\vartheta)=E((G(x_1)-\phi_\vartheta(\vartheta^T x))^2I\{x\in {\mathcal A}\}). \] One can define the second, the third,…, components analogously. The authors concentrate on estimation of \(\vartheta_1\). An estimator \(\hat\vartheta_1\) is constructed using a kernel regression estimator for \(\phi_\vartheta\) and a residual sum of squares \(\hat S_n(\vartheta)\) instead of \(S\). It is demonstrated that \(\hat\vartheta_1-\vartheta_1=O(n^{-1/(2r+1)})\), where \(n\) is the number of observations and \(G\) has continuous derivatives of \((r+1)\)-th order. Results of simulations and a real data example are presented.

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62H12 Estimation in multivariate analysis
62H99 Multivariate analysis
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