# Updating regression solutions for removing a regressor without the original dependent variable

Note: This question is analagous to the question I asked here except instead of adding a column, I am removing it.

I am interested in a linear regression on the model;

$$Y= X\beta + \epsilon$$

And I have computed the OLS estimator $$\beta$$, $$\hat{\beta}$$;

$$\hat{\beta}=(X'X)^{-1}X'Y$$

I realize now though that my design is contained an extra column I did not mean to include. For simplicity say the first column $$x_1$$ should not have been included. I used the matrix $$X=[x_1, \tilde{X}]$$ when I should have used $$\tilde{X}$$.

I no longer have access to my $$Y$$ data - Is there a way for me to update $$\hat{\beta}$$ to be based on $$\tilde{X}$$ rather than $$X$$ when I don't have access to $$Y$$ anymore? i.e. can I update the OLS solution when I remove a column from the design matrix?

• Do you have the errors from the original regression? Those would allow you to reconstruct the original Y data. – James Phillips May 7 '19 at 16:18

The simplest case would be where your design matrix $$X$$ contained the intercept column and exactly one additional column... that just happens to be identical to the response, $$X=(1|Y)$$. In this case, your OLS estimate will be $$\hat{\beta}=(0,1)$$.
If you now remove the $$Y$$ column from $$X$$, you are left with the intercept column. Updating the model is equivalent to finding the mean of the $$Y$$, because that is the new OLS estimate for $$\beta_0$$. But you don't have access to $$Y$$ any more. Finding the mean of unknown data, if all we know is that the data correlates perfectly with itself, is impossible.