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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.

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