= Five-point symmetric quadratic trend smoother
{title2=$(-3,12,17,12,-3)/35$}
For observations $X_t=T_t+\varepsilon_t$ 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.
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