# Number of significant linear predictors if predictors are not independent?

I would like to determine which of a set of candidate predictors $\{x_1, x_2,\ldots, x_n\}$ are significantly relevant to the linear prediction of $y$.

Typically, one can compare a full model

$$\hat y = a_0 + a_1x_1 + \cdots + a_nx_n$$

to a reduced model

$$\hat y = a_0 + a_1x_1 + \cdots + a_{i-1}x_{i-1} + a_{i+1}x_{i+1} + \cdots + a_nx_n$$

(where the reduced model does not include $x_i$ as one of the predictors) using an F-test, which is based on the sum of the squared residuals for each model. Then, if the p-value is below a certain threshold one can say that $x_i$ is significant.

In my case, however, there are definitely correlations among the predictors, and there may be unknown dependencies among them.

Will this affect the interpretation of the F-test p-value, and if so, are there anyways to account for these dependencies in calculating predictor significance?

This doesn't seem like that uncommon of an obstacle in statistics and may well have been answered before, so directions to relevant resources would also be very much appreciated.