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

by Richard Hardy   Last Updated October 19, 2019 12:19 PM - source

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.

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