Editing Gauss–Markov theorem (section)

===Spherical errors===
The [[outer product]] of the error vector must be spherical.
:<math>\operatorname{E}[\,\boldsymbol{\varepsilon} \boldsymbol{\varepsilon}^{\operatorname{T}} \mid \mathbf{X} ]
= \operatorname{Var}[\,\boldsymbol{\varepsilon} \mid \mathbf{X} ]
= \begin{bmatrix}
\sigma^{2} & 0 & \cdots & 0 \\
0 & \sigma^{2} & \cdots & 0 \\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \cdots & \sigma^{2}
\end{bmatrix}
= \sigma^{2} \mathbf{I}
\quad \text{with } \sigma^{2} > 0</math>
This implies the error term has uniform variance ([[homoscedasticity]]) and no [[serial correlation]].<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=10 |url=https://books.google.com/books?id=QyIW8WUIyzcC&pg=PA10 }}</ref> If this assumption is violated, OLS is still unbiased, but [[Efficiency (statistics)|inefficient]]. The term "spherical errors" will describe the [[multivariate normal distribution]]: if <math>\operatorname{Var}[\,\boldsymbol{\varepsilon}\mid \mathbf{X} ] = \sigma^{2} \mathbf{I}</math> in the multivariate normal density, then the equation <math>f(\varepsilon)=c</math> is the formula for a [[Ball (mathematics)|ball]] centered at μ with radius σ in n-dimensional space.<ref>{{cite book |first=Ramu |last=Ramanathan |chapter=Nonspherical Disturbances |title=Statistical Methods in Econometrics |url=https://archive.org/details/statisticalmetho00rama |url-access=limited |publisher=Academic Press |year=1993 |isbn=0-12-576830-3 |pages=[https://archive.org/details/statisticalmetho00rama/page/n339 330]–351 }}</ref>

[[Heteroskedasticity]] occurs when the amount of error is correlated with an independent variable. For example, in a regression on food expenditure and income, the error is correlated with income.  Low income people generally spend a similar amount on food, while high income people may spend a very large amount or as little as low income people spend. Heteroskedastic can also be caused by changes in measurement practices. For example, as statistical offices improve their data, measurement error decreases, so the error term declines over time.

This assumption is violated when there is [[autocorrelation]].  Autocorrelation can be visualized on a data plot when a given observation is more likely to lie above a fitted line if adjacent observations also lie above the fitted regression line. Autocorrelation is common in time series data where a data series may experience "inertia." If a dependent variable takes a while to fully absorb a shock. Spatial autocorrelation can also occur geographic areas are likely to have similar errors. Autocorrelation may be the result of misspecification such as choosing the wrong functional form. In these cases, correcting the specification is one possible way to deal with autocorrelation.

When the spherical errors assumption may be violated, the generalized least squares estimator can be shown to be BLUE.<ref name="Huang1970" />