About residual of linear regression What assumptions does one make about the residuals in linear regression? Say if we didn't have an intercept, would the sum of the residuals be guaranteed to equal zero? if we used a different loss function (e.g., absolute loss), would the sum be guaranteed to equal zero?
 A: What assumptions does one make about the residuals in linear regression?
For estimates to be unbiased and consistent, residual must


*

*have mean zero, i.e. $E[\varepsilon_i|X]=0$ (conditional mean zero assumption)

*must be independent and identically distributed; i.e. $cov(\varepsilon_i,\varepsilon_j)=0$ (i.i.d. assumption).


For the estimates being efficient, one also requires


*Constant variance; i.e. $Var(\varepsilon_i|X)=\sigma_\varepsilon$ (homoscedasticity) 

*Residuals are normally distributed; i.e. $\varepsilon_i$~$N(0,\sigma_\varepsilon)$ (normality assumption).
Generally, assumptions 1-3 are the important ones with assumption 4 being asymptotically fulfilled for large samples (say more than 50).
Say if we didn't have an intercept, would the sum of the residuals be guaranteed to equal zero?
Yes, see Comments below.
If we used a different loss function (e.g., absolute loss), would the sum be guaranteed to equal zero?
It of course depends on the loss function. But if you don't want to violate the conditional mean zero assumption, residuals should better add up to 0.
