15.4 Technical appendix

In this section we give the precise conditions under which the bound (15.4) holds. Define:

$\displaystyle V_I = \sigma^{-2} \sum_{t\in I} X_t X_t^{\top} \qquad W_I = V_I^{-1},$    

furthermore let $ w_{ij,I}$ denote the elements of $ W_I$. For some positive constants $ b>0, \, B>1, \, \rho<1, \, r \geq 1, \, \lambda>\sqrt{2}$ and for $ i = 1 \ldots p$ consider the random set were the following conditions are fulfilled:
$\displaystyle A_{i,I}$ $\displaystyle =$ \begin{displaymath}\left\{
\begin{array}{c}
b \leq w_{jj,I}^{-1} \leq bB \\ \\
...
...vert \leq \rho \quad \forall \,i=1,\ldots,p
\end{array}\right\}\end{displaymath}  

Let $ (Y_1,\,X_1)\ldots (Y_\tau\,,X_\tau)$ obey (15.1), where the regressors are possibly stochastic, then it holds holds for the estimate $ \widehat\theta_I$:

\begin{multline*}
\textrm{P}\left(\vert\widehat\theta_{i,I} -\theta_{i,\tau} \v...
...\lambda)^{p-1} \lambda \exp(-\lambda^2/2),\qquad i = 1,\ldots,p.
\end{multline*}

A proof of this statement can be found in Liptser and Spokoiny (1999). For a further generalization, where the hypothesis of local time homogeneity holds only approximatively, see Härdle et al. (2000).