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

Question

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?

Answer 1

I am afraid you are out of luck.

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.

The argument is analogous in higher dimensions.