Variance of a rational forecast In Chapter 24 of Richard Thaler's book Misbehaving, he writes: 


  An important property of rational forecasts---as a stock price is supposed to be---is that the predictions cannot vary more than the thing being forecast. 


He then goes on to explain this using an example of weather forecast. If the temperature in a region typically varies between 85°F and 95°F, then it would be wrong to predict 50°F one day and 115°F the next day (although this would be approximately right on average). 

What exactly is underlying mathematical principle for when he says "the predictions can't vary more than the thing being forecast"? Do forecasts have to get both the mean and the variance right to qualify as rational?
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}