Use of expression "statistically significantly predicts" based on in-sample analysis

Question

Suppose one estimates a linear time series model $$ y_t=\beta_0+\beta_1 x_{t-1}+\varepsilon_t $$ and finds that $\hat\beta_1>0$ and the $p$-value associated with $\hat\beta_1$ is lower than the chosen significance level. Can one say, without any caveats, that

$x$ statistically significantly predicts $y$?

Similarly, can one say

$x$ positively predicts $y$?

My main concern is that the claim about prediction is based on in-sample analysis without stating the implicit assumptions that are required to make a conclusion about out-of-sample results from in-sample results. Another, minor concern is that the claim is based on a significance test of $\beta_1$ rather than a measure of change in prediction errors of $y$ when $x_1$ is added to the model. However, the latter is probably not a problem as the significance of $\beta_1$ probably implies the prediction errors of $y$ will be decreased by use of $x_1$ in the model.

Answer 1

Yes, and Yes. Note that you predict coefficients "only in-sample". To say something about statistical significance, you need a distribution for coefficients. But this is estimated using the in-sample only. You can make a OOS prediction, but technically speaking you cannot say anything about out-of-sample statistical significance.

In your minor concern, I guess you are essentially comparing "$\beta_{1}$" and "$\beta_{1} * x_{1}$". In Statistics it often the relation of concern, not prediction. So as long as $x_{1}$ significantly affects $y$, how much it changes $y$ is of less concern.