A linear model is given by $y=X\beta + \epsilon$. Why do we often instead write it as $E(Y) = X\beta$?


2 Answers 2


Because this generalizes nicely to other types of regression models. That way we can express them all in a similar way (possibly with a link function - in your linear regression case the link function is just the identity function). This makes it easier to think of them in a similar way (and also to have similar programming interfaces to them).


  • logistic regression can be expressed as $\text{logit}(E(Y_i)) := \text{logit}(\pi_i) = \boldsymbol{X}\boldsymbol{\beta}$, with $Y_i \sim \text{Bernoulli}(\pi_i)$,
  • Poisson regression as $\log(E(Y_i)) := \text{log}(\mu_i) = \boldsymbol{X}\boldsymbol{\beta}$, with $Y_i \sim \text{Poisson}(\mu_i)$

and so on.

  • 1
    $\begingroup$ There are exceptions too. For example, quantile regressions are about conditional quantiles rather than conditional means. $\endgroup$
    – Nick Cox
    Oct 13, 2020 at 21:22

Well, if $E(XB + e) = XB$ we are showing that $E(e) = 0$.


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