Variance of a rational forecast
A colleague of mine pointed me to Steven M. Sheffrin's Rational Expectations, where I found the explanation (on p.122): 


  [L]et $P_t^*$ be the variable to be forecast. Then the optimal forecast is given by 
  \begin{equation}P_t=E[P_t^*|I_{t-1}].\end{equation}
  Let $u_t$ denote the forecast error, which, by the orthogonality principle, must be uncorrelated with the forecast $P_t$. Thus $P_t^*=P_t+u_t$ and, taking the variance of both sides, 
  \begin{equation}
\mathrm{var}(P_t^*)=\mathrm{var}(P_t)+\mathrm{var}(u_t)
\end{equation}
  or 
  \begin{equation}
\mathrm{var}(P_t^*)\ge \mathrm{var}(P_t),\quad\text{since $\mathrm{var}(u_t)\ge0$}.
\end{equation}