## Bootstrap of kernel smoothing in nonlinear time series.(English)Zbl 1006.62038

From the paper: Nonlinear modelling of time series is a promising approach in applied time series analysis. We consider nonparametric models of nonlinear autoregression. Motivated by econometric applications, we allow for heteroscedastic errors: $X_t=m(X_{t-1}, \dots,X_{t-p}) +\sigma(X_{t-1}, \dots, X_{t-q}) \varepsilon_t,\quad t=0,1,2, \dots \tag{1}$ Here the $$(\varepsilon_t)$$ are independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1. Furthermore, $$m$$ and $$\sigma$$ are unknown smooth functions. For the sake of simplicity, we consider only the case $$p=q=1$$. In this particular case, (1) can be interpreted as a discrete version of the general diffusion process with arbitrary (nonlinear) trend $$m$$ and volatility function $$\sigma$$, $\text{d} S_t=m(S_t)+ \sigma(S_t)\text{d}W_t,$ where $$W_t$$ is a standard Wiener process. The class of processes (1) also contains as a special case the qualitative threshold ARCH (QTARCH) processes. These processes were proposed as models for financial time series. Estimation of $$m$$ and $$\sigma$$ can be done by kernel smoothing of Nadaraya-Watson type. For the estimation of $$\sigma$$ we consider two estimates: $\widehat m_h(x)=\bigl(\widehat p_h(x)\bigr)^{-1}(T-1)^{-1}\sum^{T-1}_{t=1} K_h(x-X_t)X_{t+1},$
$\widehat\sigma^2_{1,h'}(x)=\bigl(\widehat p_{h'}(x)\bigr)^{-1}(T-1)^{-1}\sum^{T-1}_{t=1} K_{h'}(x-X_t) X^2_{t+1}-\widehat m^2_{h'}(x),$
$\widehat\sigma^2_{2, h'}(x)=\bigl(\widehat p_{h'}(x)\bigr)^{-1}(T-1)^{-1}\sum^{T-1}_{t=1}K_{h'}(x-X_t)\widehat r^2_{t+1}.$ Here $$K_h(\cdot)$$ denotes $$h^{-1}(\cdot/h)$$ for a kernel $$K$$. The residuals $$X_{t+1}-\widehat m_h(X_t)$$ are denoted by $$\widehat r_{t+1}$$. In the definition of $$\widehat \sigma^2_{2,h'} (x)$$ the residuals $$\widehat r_{t+1}$$ could be replaced by $$X_{t+1}- \widehat m_{h'} (X_t)$$ without changing the asymptotic first-order properties of $$\widehat \sigma^2_{2,h'} (x)$$. The estimate $$\widehat p_h$$ is a kernel estimate of the univariate stationary density $$p$$ of the time series $$\{X_t\}$$: $\widehat p_h(x) =(T-1)^{-1} \sum^{T-1}_{t=1} K_h(x-X_t).$ We show that the bootstrap can be used for estimating the distribution of kernel smoothers. This can be done by mimicking the stochastic nature of the whole process in the bootstrap resample or by generating a simple regression model. Consistency of these bootstrap procedures will be shown.

### MSC:

 62G08 Nonparametric regression and quantile regression 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62P05 Applications of statistics to actuarial sciences and financial mathematics 62G09 Nonparametric statistical resampling methods 62G07 Density estimation