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By linearity, I do not mean the linearity of the estimators in the OLS regression equation, what I am trying to ask is why are the beta coefficients a linear function of y(i)'s. Why is linearity a desirable property for estimators?

Let me give some context: Y(i) = b0 + b1X(i) + e(i) is the equation of a simple linear regression model. It is evident that b0 and b1 are linear in the above equation, i.e., neither raised to power more than 1 nor appearing as a reciprocal in the equations.

When one is proving the BLUE properties of beta coefficients, one needs to prove that both the beta coefficients are a linear function of y(i)'s. My question, therefore, is why is this type of linearity desirable?

badavadapav
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  • I think your question is answered here: https://stats.stackexchange.com/questions/276661/why-dont-we-consider-nonlinear-estimators-for-the-parameters-of-linear-regressi – Estacionario Oct 30 '19 at 07:14
  • one reason is that solutions are in closed form and don't require numerical methods to be obtained. once you get into numerical methods, things get a lot more complicated as far as A) knowing that you've reached a global minimum, B) checking that resulting hessian is positive definite and on and on and on. – mlofton Oct 30 '19 at 08:39
  • I would venture to say that linearity of estimators is hardly desirable at all. Unfortunately, too much has been made of the concept. – BigBendRegion Feb 28 '20 at 00:25

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