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Functional coefficient regression models for non-linear time series: a polynomial spline approach. (English) Zbl 1062.62184

This paper deals with functional coefficient regression model described as follows. $\left\{{Y}_{t},{X}_{t},{U}_{t}\right\}$ are jointly strictly stationary processes with ${X}_{t}=\left({X}_{t1},...,{X}_{td}\right)$ taking values in ${R}^{d}$ and ${U}_{t}$ in $R$. Let $E\left({Y}_{t}^{2}\right)<\infty$. The multivariate regression function is defined as $f\left(x,u\right)=E\left({Y}_{t}|{X}_{t}=x,{U}_{t}=u\right)$. The functional coefficient regression model requires that the regression function has the form $f\left(x,u\right)={\sum }_{j=1}^{d}{a}_{j}\left(u\right){x}_{j}$, where ${a}_{j}\left(·\right)$ are measurable functions from $R$ to $R$ and $x=\left({x}_{1},\cdots ,{x}_{d}\right)$.

The authors propose a global smoothing method based on polynomial splines for estimation of the functional coefficient regression models for nonlinear time series. Different coefficient functions are allowed to have different smoothing parameters. Consistency and the rate of convergence of the spline estimates are established. Methods for automatic selection of the threshold variable and significant variables are discussed. The authors propose a method to produce multi-step-ahead forecasts, including point forecasts, interval forecasts and density forecasts using the estimated model. Two real data examples, US GNP and Dutch guilder-US dollar exchange rate time series, are used to illustrate the proposed method.

##### MSC:
 62M10 Time series, auto-correlation, regression, etc. (statistics) 62G08 Nonparametric regression 62M20 Prediction; filtering (statistics)