Editing Gauss–Markov theorem (section)

===Strict exogeneity===
For all <math>n</math> observations, the expectation—conditional on the regressors—of the error term is zero:<ref>{{cite book |first=Fumio |last=Hayashi |author-link=Fumio Hayashi |title=Econometrics |publisher=Princeton University Press |year=2000 |isbn=0-691-01018-8 |page=7 |url=https://books.google.com/books?id=QyIW8WUIyzcC&pg=PA7 }}</ref>
:<math>\operatorname{E}[\,\varepsilon_{i}\mid \mathbf{X} ] = \operatorname{E}[\,\varepsilon_{i}\mid \mathbf{x}_{1}, \dots, \mathbf{x}_{n} ] = 0.</math>
where <math>\mathbf{x}_i = \begin{bmatrix} x_{i1} & x_{i2} & \cdots & x_{ik} \end{bmatrix}^{\operatorname{T}}</math> is the data vector of regressors for the ''i''th observation, and consequently <math>\mathbf{X} = \begin{bmatrix} \mathbf{x}_{1}^{\operatorname{T}} & \mathbf{x}_{2}^{\operatorname{T}} & \cdots & \mathbf{x}_{n}^{\operatorname{T}} \end{bmatrix}^{\operatorname{T}}</math> is the data matrix or design matrix.

Geometrically, this assumption implies that <math>\mathbf{x}_{i}</math> and <math>\varepsilon_{i}</math> are [[Orthogonality|orthogonal]] to each other, so that their [[Dot product|inner product]] (i.e., their cross moment) is zero.
:<math>\operatorname{E}[\,\mathbf{x}_{j} \cdot \varepsilon_{i}\,] = \begin{bmatrix} \operatorname{E}[\,{x}_{j1} \cdot \varepsilon_{i}\,] \\ \operatorname{E}[\,{x}_{j2} \cdot \varepsilon_{i}\,] \\ \vdots \\ \operatorname{E}[\,{x}_{jk} \cdot \varepsilon_{i}\,] \end{bmatrix} = \mathbf{0} \quad \text{for all } i, j \in n</math>
This assumption is violated if the explanatory variables are [[Errors-in-variables models|measured with error]], or are [[Endogeneity (econometrics)|endogenous]].<ref>{{cite book |first=John |last=Johnston |author-link=John Johnston (econometrician) |title=Econometric Methods |location=New York |publisher=McGraw-Hill |edition=Second |year=1972 |isbn=0-07-032679-7 |pages=[https://archive.org/details/econometricmetho0000john_t7q9/page/267 267–291] |url=https://archive.org/details/econometricmetho0000john_t7q9/page/267 }}</ref> Endogeneity can be the result of [[wikt:simultaneity|simultaneity]], where causality flows back and forth between both the dependent and independent variable. [[Instrumental variable]] techniques are commonly used to address this problem.