an class="math inline">\(u_i\sim N(0, w^2)\)</span> and all <span class="math inline">\(u_i\)</span> are independent and inpendent of <span class="math inline">\(e\)</span>. Then <span class="math inline">\((e,u)\)</span> can be expressed in terms of <span class="math inline">\((x,y)\)</span> as <span class="math display">\[
(e,u) = \left[\begin{matrix}e_{0}\\e_{1}\\e_{2}\\e_{3}\\u_{1}\\u_{2}\\u_{3}\end{matrix}\right] = \left[\begin{matrix}1 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0\\- a &amp; 1 &amp;