Independence of error in Linear Regression I was reading the book ``Genetics and Analysis of Quantitative Traits'', by Lynch and Walsh. In chapter $5$ of the book while discussing the concept of heritability, they seem to be claiming the following:
In a linear regression (where $y_i$ is observation $i$, $A$ is the known design matrix, $x$ is slopes we're estimating) minimising $||y - Ax||_2^2$, the error term, $y_i-a_i^{T}x$ is uncorrelated with $y_j-a_j^{T}x$, if $i \neq j$ and $a_i \neq a_j$. To me it is not clear why this is the case. Am I missing something here?
 A: Independence of the errors is a regression assumption; it doesn't follow from anything else.
But rather than quote the list there, I'll quote Andrew Gelman. Gelman gives the following list on his blog (itself, quoting Gelman&Hill):


*

*Validity. Most importantly, the data you are analyzing should map to the research question you are trying to answer. This sounds obvious but is often overlooked or ignored because it can be inconvenient. . . .


*Additivity and linearity. The most important mathematical assumption of the regression model is that its deterministic component is a linear function of the separate predictors . . .


*Independence of errors. . . .


*Equal variance of errors. . . .


*Normality of errors. . . .
Further assumptions are necessary if a regression coefficient is to be given a causal interpretation . .

The first isn't really a statistical assumption; if we include nonstatistical assumptions I think one could argue for some others, and I'd push them all into "further assumptions", but the remaining assumptions can be found in most lists; many people would add one or two additional assumptions that Gelman doesn't present. I largely agree with his ordering of the assumptions there.
[The normality is really only of any importance if we're performing inference that relies on it -- and in large samples often not even then; the estimates are efficient if normality holds, but they're usually reasonable if the distributions aren't such that all linear estimators are fairly bad.]
