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For the given model $y = x\beta + \varepsilon$, the four assumptions would be

  1. The model is linear in $x$ and $\beta$, also additive in $\varepsilon$
  2. The conditional mean of the error terms are zero (i.e. $E[\varepsilon|x] = 0)$
  3. $Var[\varepsilon|x] = \sigma^2I_n$
  4. Exogeneity of $x$ (i.e. either $x$ is fixed or independent of $\varepsilon$)

My questions is that in the case of the following situations which assumptions have been violated.

Case 1. $E[x'\varepsilon] \neq 0$ [I am not sure whether this implies the second or fourth assumption has been violated]

Case 2. $Cov(x_i,\varepsilon_i) \neq 0$ [I suppose the third one has been violated but could it imply the violation of other assumptions?]

Any help will be greatly appreciated.

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  • $\begingroup$ Just a quick point. These are some assumptions that are appropriate for certain things. There are not the assumptions, as there is no one unique set of assumptions that covers all uses of the model. $\endgroup$ – Matthew Drury Jun 5 '18 at 23:58
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I encourage you to rewrite Case 1 out in matrix form with just 1 variable and i=3 observations. This would give you that the expected value of the inner product of the ith variable and the ith residual are 0. Hopefully this makes it clear that Case 1 violates assumptions 2 and 4 (since assumptions 2 and 4 are the same).

Case 2 violates assumptions 2, 3, and 4. To see this, you can rewrite the covariance as the product of the standard deviation of the variables, the standard deviation of the residuals, and the correlation of the variables and the residuals. If the covariance listed in Case 2 is non-zero, then all three of these factors are non-zero. This means that the correlation of the variables and the residuals is non-zero (aka endogeneity [violating assumption 4, and therefore 2]).

As for assumption 3, if there is non-zero covariance between your explanatory variables and the error terms, then the variance of the error terms is obviously non-constant.

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    $\begingroup$ I also want to add that these are the econometrics assumptions for classical linear regression, as specified by the Gauss-Markov theorem. Feel free to look that up, as it should give you more detail. Beware, statisticians outside of econometrics will likely give you different assumptions (assumptions 2 and 3 are usually combined, and also require the normality of the residuals). Also, exogeneity is not really discussed in the sciences, as experiments are easy (or at least easier) to design in the sciences. $\endgroup$ – Octavio Urista Jun 6 '18 at 15:26
  • $\begingroup$ Yup. There are also uses of regression that necessitate MUCH weaker assumptions. If you're only trying to make good predictions, a lot of the classical assumptions are pretty irrelevent. $\endgroup$ – Matthew Drury Jun 6 '18 at 15:30

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