Five-point symmetric quadratic trend smoother ($(-3,12,17,12,-3)/35$)

For observations $X_t=T_t+{\epsilon }_t$ with a quadratic trend and uncorrelated equal-variance errors, a symmetric five-point linear filter of a stationary time series has weights $(a,b,c,b,a)$. Reproducing every quadratic requires $2a+2b+c=1$ and $8a+2b=0$. Substitution gives $b=-4a$, $c=1+6a$ and error variance ${\sigma }^2(70a^2+12a+1)$. Its unique minimum occurs at $a=-3/35$, producing the displayed weights and variance $17{\sigma }^2/35$. Negative weights are allowed: this is a linear weighted smoother, not a convex average. Applying a linear filter to a deterministic trend plus stationary errors does not assert that the complete observation process is stationary.