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  • $\begingroup$ What kind of regression? $\endgroup$
    – Steven Gubkin
    Commented Oct 15, 2023 at 14:33
  • $\begingroup$ Let us make it simple, linear regression. $\endgroup$
    – Kernel
    Commented Oct 15, 2023 at 15:36
  • 1
    $\begingroup$ Your notation is unclear: could you please explain what you mean by "coefficients"? Are you trying to tell us that the ordinary regressions all have identical estimated slopes of $r,$ perhaps? Are you treating $x$ as the regressor or as the response in these regressions? Why would variance have anything to do with additivity of the responses? $\endgroup$
    – whuber ♦
    Commented Oct 15, 2023 at 15:57
  • $\begingroup$ Coefficients are the slopes. I updated my notions. X is the predictor. Regression is a function in variance and covariance. $\endgroup$
    – Kernel
    Commented Oct 15, 2023 at 16:33
  • 1
    $\begingroup$ One particular formula for slopes involves sums of squares. That's not the same as "a function of variance and covariance." You might find it more fruitful to think in terms of the kind of regression you are doing. For instance, with ordinary least squares regression, the slopes are chosen to minimize the sum of squares of residuals (no variances involved!). $\endgroup$
    – whuber ♦
    Commented Oct 15, 2023 at 16:51