On p. 113 of Cryer's Time Series Analysis, in the context of ACFs and partial ACFs, I read
Consider predicting $ Y_t $ based on a linear function of the intervening variables $ Y_{t-1}, Y_{t-2}, \ldots, Y_{t-k+1} $, say,
$$ \beta_1 Y_{t-1} + \beta_2 Y_{t-2} + \cdots + \beta_{k-1} Y_{t-k+1},$$
with the $ \beta $'s chosen to minimize the mean square error of prediction. If we assume that the $ \beta $'s have been so chosen and then think backward in time, it follows from stationarity that the best “predictor” of $ Y_{t-k} $ based on the same $ Y_{t-1}, Y_{t-2},\ldots, Y_{t-k+1} $ will be
$$ \beta_1 Y_{t-k+1} + \beta_2 Y_{t-k+2} + \cdots + \beta_{k-1}Y_{t-1}. $$
This statement "reverses" the coefficients, in a way suggests that stationarity is symmetric. This seems counter-intuitive to me; could anyone help verify this result?
For example, one simple case of this claim is that if $c$ is the optimal coefficient for $Y_{t} = cY_{t-1}$, then $Y_{t-1} = cY_{t}$.
