Incorporating the statistics of our old regression fit into our new regression

Let's assume we fit a linear regression to our data $$(X_i,Y_i)_{old}$$, we lose the old data points but we still have access to the coefficients and the statistics of our linear regression. New data $$(X_i,Y_i)_{new}$$ comes in. We want to fit a linear regression model again. What is the best method to incorporate the statistics from fit for $$(X_i,Y_i)_{old}$$ into our new regression?

• Are you open to using a bayesian method with the priors specified by the old coefficients and standard errors? Jun 30, 2021 at 15:09
• If you have the statistics from the old regression for $n, \sum x_i, \sum x_i^2, \sum y_i, \sum y_i^2, \sum x_i y_i$ (or their equivalents of number, means, variances and covariance) then you can incorporate them into the new regression Jun 30, 2021 at 15:10
• @DavidLukeThiessen If it works why not. Jun 30, 2021 at 15:11
• The simple linear regression coefficients are related to the means, variances and covariance and these statistics allow you to calculate these for the combined regression Jun 30, 2021 at 15:14
• Specific solutions along the lines indicated by @Henry appear here on CV: see stats.stackexchange.com/questions/6920 for instance. Also look for threads about updating means and covariances (or moments generally) with new data -- stats.stackexchange.com/questions/51622 will do -- and apply them to regression, using threads that describe how to do regression using only those moments -- such as stats.stackexchange.com/questions/107597.
– whuber
Jun 30, 2021 at 15:53