= Steady-state local-level Kalman gain
{title2=$\alpha_\infty=(\sqrt{c^2+4c}-c)/2$}
For the <state-space model> with both scalar coefficients equal to one, $\alpha_t=(P_{t-1}+W)/(P_{t-1}+W+V)$ and $P_t=\alpha_tV$. If the variance converges and $c=W/V$, its limiting gain satisfies $\alpha^2+c\alpha-c=0$. The nonnegative root lies below one for finite $c>0$; it tends to zero with $c$ and to one as $c$ grows. Thus noisier state evolution assigns more weight to the latest observation.
Back to article page
