Let $u(n)$ be the input and $v(n)$ the output of a single-input-single-output system described by the Auto-Regressive-Moving-Average equation $$v(n)=\sum_{k=0}^{m_{0}}b_ku(n-k)+\sum_{k=1}^{n_{0}}a_{k}v(n-k).$$ Assume that $v(n)$ is known for all $n$ and $u(n)=0$ for some interval $n=t_{0},t_{0}+1,\dots,N$.
Then you can determine the coefficients $a_k$ by solving the equation $$\left(\begin{matrix}v(t_{0})\\v(t_{0}+1)\\\vdots\\v(N)\end{matrix}\right)=\left(\begin{matrix}v(t_{0}-1) & \dots & v(t_{0}-n_{0})\\ v(t_{0}) & \dots &v(t_{0}-n_{0}+1)\\ \dots & \dots & \dots\\ v(N-1)& \dots & v(N-n_{0})\end{matrix}\right) \left(\begin{matrix}a_1\\a_2\\\dots\\a_{n_{0}}\end{matrix}\right)$$
But how can you determine the coefficients $b_k$ and the values of $u(n)$ which are not equal to zero?