Editing Gauss–Markov theorem (section)

===Linearity===

The dependent variable is assumed to be a linear function of the variables specified in the model. The specification must be linear in its parameters. This does not mean that there must be a linear relationship between the independent and dependent variables. The independent variables can take non-linear forms as long as the parameters are linear.  The equation <math> y = \beta_{0} + \beta_{1} x^2, </math> qualifies as linear while <math> y = \beta_{0} + \beta_{1}^2 x</math> can be transformed to be linear by replacing <math>\beta_{1}^2</math> by another parameter, say <math>\gamma</math>. An equation with a parameter dependent on an independent variable does not qualify as linear, for example <math>y = \beta_{0} + \beta_{1}(x) \cdot x</math>, where <math>\beta_{1}(x)</math> is a function of <math>x</math>.

[[Data transformation (statistics)|Data transformations]] are often used to convert an equation into a linear form. For example, the [[Cobb–Douglas production function|Cobb–Douglas function]]—often used in economics—is nonlinear:
:<math>Y = A L^\alpha K^{1 - \alpha} e^\varepsilon </math>

But it can be expressed in linear form by taking the [[natural logarithm]] of both sides:<ref>{{cite book |first=A. A. |last=Walters |title=An Introduction to Econometrics |location=New York |publisher=W. W. Norton |year=1970 |isbn=0-393-09931-8 |page=275 }}</ref>

: <math>\ln Y=\ln A + \alpha \ln L + (1 - \alpha) \ln K + \varepsilon = \beta_0 + \beta_1 \ln L + \beta_2 \ln K + \varepsilon</math>

This assumption also covers specification issues: assuming that the proper functional form has been selected and there are no [[Omitted-variable bias|omitted variables]].

One should be aware, however, that the parameters that minimize the residuals of the transformed equation do not necessarily minimize the residuals of the original equation.