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The linearity condition states that $\mathbb{E}[y_i]=(\vec{x}_i)^{T}\vec{\beta}$ for all $i$. Now, if we have fixed regressors, $\{\vec{x}_1,\vec{x}_2,\cdots\}$, our linearity condition only says for those $\vec{x}_i$, there is a linear relationship. So it does not assume linearity when we have some $\vec{x}_j$ that is not in the set of fixed regressors.

However, if the regressors are random as well, the linearity condition says for all $\vec{x}$ there are, linearity holds. Therefore, if we have an experimental design and had fixed regressors, we can only infer $\vec{x}$ that has happened in the fixed regressor during the experiment and cannot infer otherwise. However, with stochastic regressors, we can infer any $\vec{x}$. Is that right?

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  • $\begingroup$ Without an error term in the specification, this is a mathematical question, not a statistical one. $\endgroup$
    – Alecos Papadopoulos
    Commented May 11, 2016 at 2:14
  • $\begingroup$ @AlecosPapadopoulos Sorry. I forgot to attach the expected sign. Please see the edit. $\endgroup$
    – Kun
    Commented May 11, 2016 at 3:26

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No, I would say that the meaning/interpretation of linearity have nothing to do with the question of fixed or random regressors. You say

So it does not assume linearity when we have some $\vec{x}_j$ that is not in the set of fixed regressors.

But no, the linearity assumption is that the expectation is a linear function (well, really an affine function), and as a function the definition must hold for observed as well as non-observed values of the regressor.

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