Why do we write models in the form of E(Y)?

Question

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

Answer 1

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).

E.g.

• 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.

Answer 2

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