Conducting the Pearson's Correlation test

[ashley.brown1]

My model is as follows:

Y=beta_0+beta_1*X_1+beta_2*X_2+beta_3*X_3+beta_4*X_4.

I also have two other variables X_5 and X_6 which I have not included in this model since running the Wald test told me there was no need to include them.

I want to determine if, by omitting X_5 and/or X_6, does my model suffer from omitted variable bias. In order to do this, I have run least squared regressions with Y against X_5 and Y against X_6 to see if there is any correlation (by looking at the R-Squared value). I have also done the same for X_1 against X_5, X_2 against X_5, etc. However, I also want to test whether these R-Squared values are significant or not (H_0: rho=0 and H_1: rho not equal to 0). I am finding it difficult to conduct this test using the Pearson's correlation test or the Spearmans Rank Correlation Coefficient. When I attempt to use the 'pancov' or 'corr' functions I am given an error.

Does anyone have any idea how to conduct these tests with the model I have?

Any information will be appreciated. Thank you.