Measuring Predictability

A friend writes the following.  (I have edited very slightly for clarity.)

Based on forecasts you've seen, what would you say is a "reasonable" ratio of the standard deviation of the forecast error to the standard deviation of a covariance-stationary series being forecast? ... It would be great if you can tell me "I'd consider x reasonable and y too high." 

The problem is that the premise underlying the question (namely, that there is such a "reasonable" value of the ratio $r$ of innovation variance to unconditional  variance) is false.  That is, there's no small value $c$ of $r$ such that $r<c$ means that we've done a good forecasting job.  Equivalently, there's no large value $c'$ of the predictive $R^2~ (R^2 = 1 - r^2)$ such that $R^2 > c'$ means that we've done a good forecasting job.  Instead, "good" $c$ or $c'$ values depend critically on the dynamic nature of the series being forecast.  Consider, for example, a covariance-stationary AR(1) process, $y_t = \phi y_{t-1} + \varepsilon_t$, where $\varepsilon_t \sim iid (0, \sigma^2)$. The innovation variance is $\sigma^2$ and the unconditional variance is $\sigma^2 / (1 - \phi^2)$, so the lower bound on  $r$ (and hence the upper bound on  $R^2$) depends entirely on $\phi$ and can be anywhere in the unit interval! This is an important lesson: "predictability" can (and does) differ greatly across economic series. For more than you ever wanted to know, see Diebold and Kilian (2001), "Measuring Predictability: Theory and Macroeconomic Applications".