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

Note: This question is analagous to the question I asked here except instead of a removing column, I am adding 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 missing the first column $$x_1$$ I should actually be using matrix $$\tilde{X}=[x_1, X]$$.

Unfortunately though 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 add a column to the design matrix?

• I assume you also don't have the residuals $\epsilon$ from your original model (which would allow you to recover the $Y$)? – Stephan Kolassa May 7 '19 at 13:22

Consider the simplest possible case. Assume $$X$$ contains no predictors at all, only the intercept column. Then the initial model would contain just a single coefficient estimate $$\hat{\beta}_0$$, which would be estimated by the mean of $$Y$$. So your question in this case becomes:
Given the mean $$\hat{\beta}_0$$ of my data, but not the data $$Y$$ themselves, can I estimate a regression slope $$\hat{\beta}_1$$ for $$Y$$ on a predictor $$x$$?
And of course, it is obvious that this cannot be done. This slope $$\beta_1$$ crucially depends on the original data $$Y$$, and by changing $$Y$$ (without affecting the mean), we can change $$\beta_1$$ to be any value we choose.