The point of confusion may be in thinking that a predictable price process is synonymous with a mean-reverting process while using the definitions in these papers, it's actually the opposite! In the context of these papers, a random walk would be 100% predictable: the unpredictable component of a random walk (i.e. the period specific shock which has finite variation), comprises 0% of the process's total variation (which is infinite).
Some broader points of caution
Be cautious when an author gives an English language word a mathematical definition that may not perfectly align with the word's common use in English or a particular field.
Also, predictable is not the same as exploitable. You can use information on the relative strength of teams to predict Vegas, sports-betting lines. That's different though than whether the sports betting lines are exploitable!
Back to this case...
Consider an AR(1):
$$ x_t = b x_{t-1} + \epsilon_t$$
- $b=1$ has no mean reversion in the sense that shocks are entirely persistent. It is also most predictable in the sense that as $b \rightarrow 1$, the fraction of the process's total variation that is forecastable also goes to 100 percent.
- $b=0$ has entirely transitory shocks and in that sense is most mean reverting. It's always expected to move back to the unconditional mean! It is also the least predictable in the sense that 0 percent of the processes variation can be forecast.
With an AR(1) structure, shocks decay in a simple, exponential fashion. With higher order lags (which these papers don't do), you can instead get more complicated behavior such as cycles and, at some points, step ahead forecasts further away from the mean.
These papers assume prices are a stationary process (rather than containing a unit root) and that the price process takes a simple, 1 lag autoregressive structure. (I'll sidestep a whole discussion as to if and when that's useful or realistic.)
Box and Tiao Decomposition
Let $\{z_t\}$ be a stationary process.
Define $\hat{z}_{t-1}$ as the expectation of $z_t$ based upon $t-1$ info:
$$ \hat{z}_{t-1} = \mathbb{E}[ z_t \mid z_{t-1}, z_{t-2}, \ldots ]$$
Box and Tiao then decompose total variation $\sigma^2_z$ into a predictable component (the step ahead forecast) $\sigma^2_\hat{z}$ and an unpredictable component $\sigma^2_\epsilon$:
$$ \underbrace{\mathbb{E} \left[ z_t^2 \right]}_{\sigma^2_z} = \underbrace{\mathbb{E}\left[ \hat{z}_{t-1}^2\right]}_{\sigma^2_{\hat{z}}} + \underbrace{\mathbb{E}[\epsilon^2_t]}_{\sigma^2_\epsilon}$$
They then define the predictability ratio $\lambda = \sigma^2_\hat{z} / \sigma^2_z$. If $\lambda = 0$, none of the total variation comes from variation in the step ahead forecast. If $\lambda \approx 1$, then nearly all of the total variation comes from the step ahead forecast.
Simple AR(1) case (what's effectively in these papers)
Let's assume we have a simple, mean zero AR(1):
$$ x_t = b x_{t-1} + \epsilon_t$$
Assume $-1 < b < 1$ so the process is stationary. The unconditional variance is $\sigma^2_x = \frac{1}{ 1 - b^2}\sigma^2_\epsilon$. The Box Tiao decomposition is:
$$ \underbrace{\frac{1}{ 1 - b^2}\sigma^2_e}_{\sigma^2_x} = \underbrace{\frac{b^2}{1 - b^2}\sigma^2_\epsilon}_{\sigma^2_\hat{x}} + \sigma^2_\epsilon $$
The predictability ratio is:
$$ \lambda = b^2 $$
Discussion
Mean reversion isn't a precisely defined term.
Finding mean reverting portfolios using canonical correlation analysis means minimizing predictability...
$b=0$ has high mean reversion in either the sense: (1) the step ahead forecast is always the unconditional mean or (2) shocks are entirely transitory. $b = 0$ leads to $\lambda = 0$, minimum predictability.
... while searching for portfolios with strong momentum can also be done
using canonical correlation analysis, by maximizing predictability.
When $b$ is close to 1, the process is close to a random walk. The author seems to be calling this momentum. I find that usage of "momentum" rather problematic.
The traditional way to identify the optimal sparse mean-reverting portfolio is to find a portfolio vector subject to maximizing its predictability.
Be aware that in the context of an AR(1) in prices, maximizing predictability implies finding a price process that decays towards its unconditional mean price as slowly as possible. A random walk in prices would have the most predictability (in prices).
The intuition behind this portfolio predictability is that the greater this ratio, the more $s_{t−1}$ dominates the noise, and therefore the more predictable $s_{t}$ becomes. Therefore, we will use this measure as a proxy for the portfolio’s mean reversion parameter $\lambda$ in (1). Maximizing this expression will yield the following optimization problem for finding the best portfolio vector $x_{opt}$.
As I discussed earlier $\lambda = b^2$ in the AR(1) context.
References
Box, G.E.P. and G.C. Tiao, "A canonical analysis of multiple time series," 1977, Biometrika
