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Keywords:
autoregressive process; estimating parameters; multidimensional process; nonlinear process; nonnegative process
autoregressive process; estimating parameters; multidimensional process; nonlinear process; nonnegative process
Summary:
Let $\mathbb{e}_t=(e_{t1},\dots ,e_{tp})^{\prime }$ be a $p$-dimensional nonnegative strict white noise with finite second moments. Let $h_{ij}(x)$ be nondecreasing functions from $[0,\infty )$ onto $[0,\infty )$ such that $h_{ij}(x)\le x$ for $i,j=1,\dots ,p$. Let $\mathbb{U}=(u_{ij})$ be a $p\times p$ matrix with nonnegative elements having all its roots inside the unit circle. Define a process $\mathbb{X}_t=(X_{t1},\dots ,X_{tp})^{\prime }$ by \[ X_{tj}=u_{j1}h_{1j}(X_{t-1,1})+\dots +u_{jp}h_{pj}(X_{t-1,p})+ e_{tj} \] for $j=1,\dots ,p$. A method for estimating $\mathbb{U}$ from a realization $\mathbb{X}_1,\dots ,\mathbb{X}_n$ is proposed. It is proved that the estimators are strongly consistent.
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