Gauss–Markov theorem#Strict exogeneity

Suppose we are given two random variable vectors, $X\; \backslash text\{,\; \}\; Y\; \backslash in\; \backslash mathbb\{R\}^k$ and that we want to find the best linear estimator of $Y$ given $X$, using the best linear estimator

$\backslash hat\; Y\; =\; \backslash alpha\; X\; +\; \backslash mu$

Where the parameters $\backslash alpha$ and $\backslash mu$ are both real numbers.

Such an estimator $\backslash hat\; Y$ would have the same mean and standard deviation as $Y$, that is, $\backslash mu\; \_\{\backslash hat\; Y\}\; =\; \backslash mu\; \_\{Y\}\; ,\; \backslash sigma\; \_\{\backslash hat\; Y\}\; =\; \backslash sigma\; \_\{Y\}$.

Therefore, if the vector $X$ has respective mean and standard deviation $\backslash mu\; \_x\; ,\; \backslash sigma\; \_x$, the best linear estimator would be

$\backslash hat\; Y\; =\; \backslash sigma\; \_y\; \backslash frac\{(X\; -\; \backslash mu\; \_x\; )\}\{\; \backslash sigma\; \_x\; \}\; +\; \backslash mu\; \_y$

since $\backslash hat\; Y$ has the same mean and standard deviation as $Y$.

Suppose we have, in matrix notation, the linear relationship

:$y\; =\; X\; \backslash beta\; +\; \backslash varepsilon,\backslash quad\; (y,\backslash varepsilon\; \backslash in\; \backslash mathbb\{R\}^n,\; \backslash beta\; \backslash in\; \backslash mathbb\{R\}^K\; \backslash text\{\; and\; \}\; X\backslash in\backslash mathbb\{R\}^\{n\backslash times\; K\})$

expanding to,

:$y\_i=\backslash sum\_\{j=1\}^\{K\}\backslash beta\_j\; X\_\{ij\}+\backslash varepsilon\_i\; \backslash quad\; \backslash forall\; i=1,2,\backslash ldots,n$

where $\backslash beta\_j$ are non-random but unobservable parameters, $X\_\{ij\}$ are non-random and observable (called the "explanatory variables"), $\backslash varepsilon\_i$ are random, and so $y\_i$ are random. The random variables $\backslash varepsilon\_i$ are called the "disturbance", "noise" or simply "error" (will be contrasted with "residual" later in the article; see errors and residuals in statistics). Note that to include a constant in the model above, one can choose to introduce the constant as a variable $\backslash beta\_\{K+1\}$ with a newly introduced last column of X being unity i.e., $X\_\{i(K+1)\}\; =\; 1$ for all $i$. Note that though $y\_i,$ as sample responses, are observable, the following statements and arguments including assumptions, proofs and the others assume under the only condition of knowing $X\_\{ij\},$ but not $y\_i.$

The Gauss–Markov assumptions concern the set of error random variables, $\backslash varepsilon\_i$:

• They have mean zero: $\backslash operatorname\{E\}[\backslash varepsilon\_i]=0.$
• They are homoscedastic, that is all have the same finite variance: for all $i$ and
• Distinct error terms are uncorrelated: $\backslash text\{Cov\}(\backslash varepsilon\_i,\backslash varepsilon\_j)\; =\; 0,\; \backslash forall\; i\; \backslash neq\; j.$

A linear estimator of $\backslash beta\_j$ is a linear combination

:$\backslash widehat\backslash beta\_j\; =\; c\_\{1j\}y\_1+\backslash cdots+c\_\{nj\}y\_n$

in which the coefficients $c\_\{ij\}$ are not allowed to depend on the underlying coefficients $\backslash beta\_j$, since those are not observable, but are allowed to depend on the values $X\_\{ij\}$, since these data are observable. (The dependence of the coefficients on each $X\_\{ij\}$ is typically nonlinear; the estimator is linear in each $y\_i$ and hence in each random $\backslash varepsilon,$ which is why this is "linear" regression.) The estimator is said to be unbiased if and only if

:$\backslash operatorname\{E\}\backslash left\; [\backslash widehat\backslash beta\_j\; \backslash right\; ]=\backslash beta\_j$

regardless of the values of $X\_\{ij\}$. Now, let $\backslash sum\_\{j=1\}^K\backslash lambda\_j\backslash beta\_j$ be some linear combination of the coefficients. Then the mean squared error of the corresponding estimation is

:$\backslash operatorname\{E\}\; \backslash left\; [\backslash left\; (\backslash sum\_\{j=1\}^K\backslash lambda\_j\; \backslash left(\backslash widehat\backslash beta\_j-\backslash beta\_j\; \backslash right\; )\; \backslash right)^2\backslash right\; ],$

in other words, it is the expectation of the square of the weighted sum (across parameters) of the differences between the estimators and the corresponding parameters to be estimated. (Since we are considering the case in which all the parameter estimates are unbiased, this mean squared error is the same as the variance of the linear combination.) The best linear unbiased estimator (BLUE) of the vector $\backslash beta$ of parameters $\backslash beta\_j$ is one with the smallest mean squared error for every vector $\backslash lambda$ of linear combination parameters. This is equivalent to the condition that

:$\backslash operatorname\{Var\}\backslash left(\backslash widetilde\backslash beta\backslash right)-\; \backslash operatorname\{Var\}\; \backslash left(\; \backslash widehat\; \backslash beta\; \backslash right)$

is a positive semi-definite matrix for every other linear unbiased estimator $\backslash widetilde\backslash beta$.

The ordinary least squares estimator (OLS) is the function

:$\backslash widehat\backslash beta=(X^\backslash operatorname\{T\}X)^\{-1\}X^\backslash operatorname\{T\}y$

of $y$ and $X$ (where $X^\backslash operatorname\{T\}$ denotes the transpose of $X$) that minimizes the sum of squares of residuals (misprediction amounts):

:$\backslash sum\_\{i=1\}^n\; \backslash left(y\_i-\backslash widehat\{y\}\_i\backslash right)^2=\backslash sum\_\{i=1\}^n\; \backslash left(y\_i-\backslash sum\_\{j=1\}^K\; \backslash widehat\backslash beta\_j\; X\_\{ij\}\backslash right)^2.$

The theorem now states that the OLS estimator is a best linear unbiased estimator (BLUE).

The main idea of the proof is that the least-squares estimator is uncorrelated with every linear unbiased estimator of zero, i.e., with every linear combination $a\_1y\_1+\backslash cdots+a\_ny\_n$ whose coefficients do not depend upon the unobservable $\backslash beta$ but whose expected value is always zero.

Proof that the OLS indeed minimizes the sum of squares of residuals may proceed as follows with a calculation of the Hessian matrix and showing that it is positive definite.

The MSE function we want to minimize is

$$f(\backslash beta\_0,\backslash beta\_1,\backslash dots,\backslash beta\_p)\; =\; \backslash sum\_\{i=1\}^n\; (y\_i-\backslash beta\_0-\backslash beta\_1x\_\{i1\}-\backslash dots-\backslash beta\_px\_\{ip\})^2$$

for a multiple regression model with p variables. The first derivative is

$$\backslash begin\{aligned\}$$

\frac{d}{d\boldsymbol{\beta}}f &= -2X^\operatorname{T} \left(\mathbf{y}-X\boldsymbol{\beta}\right)\\

&=-2\begin{bmatrix}

\sum_{i=1}^{n} (y_i - \dots - \beta_px_{ip})\\

\sum_{i=1}^nx_{i1} (y_i-\dots-\beta_px_{ip})\\

\vdots\\

\sum_{i=1}^nx_{ip} (y_i-\dots-\beta_px_{ip})

\end{bmatrix}\\

&= \mathbf{0}_{p+1},

\end{aligned}

where $X^\backslash operatorname\{T\}$ is the design matrix

$$X=\backslash begin\{bmatrix\}$$

1 & x_{11} & \cdots & x_{1p}\\

1 & x_{21} & \cdots & x_{2p}\\

&&\vdots\\

1 & x_{n1} & \cdots & x_{np}

\end{bmatrix}\in \R^{n\times(p+1)}; \qquad n\geq p+1

The Hessian matrix of second derivatives is

$$\backslash mathcal\{H\}\; =\; 2\backslash begin\{bmatrix\}$$

n & \sum_{i=1}^n x_{i1} & \cdots & \sum_{i=1}^n x_{ip} \\

\sum_{i=1}^n x_{i1}& \sum_{i=1}^n x_{i1}^2 & \cdots & \sum_{i=1}^nx_{i1}x_{ip}\\

\vdots & \vdots &\ddots & \vdots \\

\sum_{i=1}^n x_{ip} & \sum_{i=1}^n x_{ip}x_{i1}& \cdots & \sum_{i=1}^n x_{ip}^2

\end{bmatrix} = 2X^\operatorname{T}X

Assuming the columns of $X$ are linearly independent so that $X^\backslash operatorname\{T\}\; X$ is invertible, let , then

$$k\_1\backslash mathbf\{v\_1\}\; +\; \backslash dots\; +\; k\_\{p+1\}\; \backslash mathbf\{v\}\_\{p+1\}\; =\; \backslash mathbf\; 0\backslash iff\; k\_1=\; \backslash dots\; =k\_\{p+1\}=0$$

Now let $\backslash mathbf\{k\}\; =\; (k\_1,\backslash dots,k\_\{p+1\})^T\; \backslash in\; \backslash R^\{(p+1)\backslash times\; 1\}$ be an eigenvector of $\backslash mathcal\{H\}$.

$$\backslash mathbf\{k\}\; \backslash ne\; \backslash mathbf\{0\}\; \backslash implies\; \backslash left(k\_1\backslash mathbf\{v\_1\}+\backslash dots+k\_\{p+1\}\backslash mathbf\{v\}\_\{p+1\}\backslash right)^2\; >\; 0$$

In terms of vector multiplication, this means

\begin{bmatrix}\mathbf{v_1} \\ \vdots \\ \mathbf{v}_{p+1}\end{bmatrix}

\begin{bmatrix}\mathbf{v_1} & \cdots & \mathbf{v}_{p+1}\end{bmatrix}

\begin{bmatrix}k_1 \\ \vdots\\ k_{p+1}\end{bmatrix}

= \mathbf{k}^\operatorname{T}\mathcal{H}\mathbf{k} = \lambda \mathbf{k}^\operatorname{T}\mathbf{k}>0

where $\backslash lambda$ is the eigenvalue corresponding to $\backslash mathbf\{k\}$. Moreover,

$$\backslash mathbf\{k\}^\backslash operatorname\{T\}\backslash mathbf\{k\}\; =\; \backslash sum\_\{i=1\}^\{p+1\}k\_i^2\; >\; 0\; \backslash implies\; \backslash lambda\; >\; 0$$

Finally, as eigenvector $\backslash mathbf\{k\}$ was arbitrary, it means all eigenvalues of $\backslash mathcal\{H\}$ are positive, therefore $\backslash mathcal\{H\}$ is positive definite. Thus,

$$\backslash boldsymbol\{\backslash beta\}\; =\; \backslash left(X^\backslash operatorname\{T\}X\backslash right)^\{-1\}X^\backslash operatorname\{T\}Y$$

is indeed a global minimum.

Or, just see that for all vectors $\backslash mathbf\{v\},\; \backslash mathbf\{v\}^\backslash operatorname\{T\}\; X^\backslash operatorname\{T\}\; X\; \backslash mathbf\{v\}\; =\; \backslash |\backslash mathbf\{X\}\backslash mathbf\{v\}\backslash |^2\; \backslash ge\; 0$. So the Hessian is positive definite if full rank.

Let $\backslash tilde\backslash beta\; =\; Cy$ be another linear estimator of $\backslash beta$ with $C\; =\; (X^\backslash operatorname\{T\}X)^\{-1\}X^\backslash operatorname\{T\}\; +\; D$ where $D$ is a $K\; \backslash times\; n$ non-zero matrix. As we're restricting to unbiased estimators, minimum mean squared error implies minimum variance. The goal is therefore to show that such an estimator has a variance no smaller than that of $\backslash widehat\backslash beta,$ the OLS estimator. We calculate:

:$$

\begin{align}

\operatorname{E} \left[ \tilde\beta \right] &= \operatorname{E}[Cy] \\

&= \operatorname{E} \left [\left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right )(X\beta + \varepsilon) \right ]\\

&= \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right )X\beta + \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right ) \operatorname{E}[\varepsilon] \\

&= \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right )X\beta && \operatorname{E}[\varepsilon] =0 \\

&= (X^\operatorname{T}X)^{-1}X^\operatorname{T}X\beta + DX\beta \\

&= (I_K + DX)\beta. \\

\end{align}

Therefore, since $\backslash beta$ is unobservable, $\backslash tilde\backslash beta$ is unbiased if and only if $DX\; =\; 0$. Then:

:$$

\begin{align}

\operatorname{Var}\left(\tilde\beta\right) &= \operatorname{Var}(Cy) \\

&= C \text{ Var}(y)C^\operatorname{T} \\

&= \sigma^2 CC^\operatorname{T} \\

&= \sigma^2 \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D \right ) \left (X(X^\operatorname{T}X)^{-1} + D^\operatorname{T} \right ) \\

&= \sigma^2 \left ((X^\operatorname{T}X)^{-1}X^\operatorname{T}X(X^\operatorname{T}X)^{-1} + (X^\operatorname{T}X)^{-1}X^\operatorname{T}D^\operatorname{T} + DX(X^\operatorname{T}X)^{-1} + DD^\operatorname{T} \right) \\

&= \sigma^2(X^\operatorname{T}X)^{-1} + \sigma^2(X^\operatorname{T}X)^{-1} (DX)^\operatorname{T} + \sigma^2 DX (X^\operatorname{T}X)^{-1} + \sigma^2DD^\operatorname{T} \\

&= \sigma^2(X^\operatorname{T}X)^{-1}+ \sigma^2DD^\operatorname{T} && DX =0 \\

&= \operatorname{Var}\left(\widehat\beta\right) + \sigma^2DD^\operatorname{T} && \sigma^2(X^\operatorname{T}X)^{-1} = \operatorname{Var}\left(\widehat\beta\right)

\end{align}

Since $DD^\backslash operatorname\{T\}$ is a positive semidefinite matrix, $\backslash operatorname\{Var\}\backslash left(\; \backslash tilde\; \backslash beta\; \backslash right)$ exceeds $\backslash operatorname\{Var\}\backslash left(\backslash widehat\backslash beta\backslash right)$ by a positive semidefinite matrix.

As it has been stated before, the condition of $\backslash operatorname\{Var\}\; \backslash left(\; \backslash tilde\; \backslash beta\; \backslash right)-\; \backslash operatorname\{Var\}\; \backslash left(\backslash widehat\backslash beta\backslash right)$ is a positive semidefinite matrix is equivalent to the property that the best linear unbiased estimator of $\backslash ell^\backslash operatorname\{T\}\backslash beta$ is $\backslash ell^\backslash operatorname\{T\}\backslash widehat\backslash beta$ (best in the sense that it has minimum variance). To see this, let $\backslash ell^\backslash operatorname\{T\}\backslash tilde\backslash beta$ another linear unbiased estimator of $\backslash ell^\backslash operatorname\{T\}\backslash beta$.

:$$

\begin{align}

\operatorname{Var}\left(\ell^\operatorname{T}\tilde\beta\right) &= \ell^\operatorname{T} \operatorname{Var} \left(\tilde\beta\right) \ell \\

&=\sigma^2 \ell^\operatorname{T} (X^\operatorname{T}X)^{-1}\ell+\ell^\operatorname{T}DD^\operatorname{T}\ell \\

&= \operatorname{Var}\left(\ell^\operatorname{T}\widehat\beta\right)+(D^\operatorname{T}\ell)^\operatorname{T}(D^\operatorname{T}\ell) && \sigma^2 \ell^\operatorname{T} (X^\operatorname{T}X)^{-1}\ell = \operatorname{Var}\left(\ell^\operatorname{T}\widehat\beta\right) \\

&= \operatorname{Var}\left(\ell^\operatorname{T}\widehat\beta\right) +\|D^\operatorname{T}\ell\|\\

& \geq \operatorname{Var}\left(\ell^\operatorname{T}\widehat\beta\right)

\end{align}

Moreover, equality holds if and only if $D^\backslash operatorname\{T\}\backslash ell=0$. We calculate

:$$

\begin{align}

\ell^\operatorname{T}\tilde\beta &= \ell^\operatorname{T} \left (((X^\operatorname{T}X)^{-1}X^\operatorname{T} + D) Y \right ) && \text{ from above}\\

&= \ell^\operatorname{T}(X^\operatorname{T}X)^{-1}X^\operatorname{T}Y + \ell^\operatorname{T}DY \\

&= \ell^\operatorname{T}\widehat\beta +(D^\operatorname{T}\ell)^\operatorname{T} Y \\

&=\ell^\operatorname{T}\widehat\beta && D^\operatorname{T}\ell = 0

\end{align}

This proves that the equality holds if and only if $\backslash ell^\backslash operatorname\{T\}\backslash tilde\backslash beta=\backslash ell^\backslash operatorname\{T\}\backslash widehat\backslash beta$ which gives the uniqueness of the OLS estimator as a BLUE.

The generalized least squares (GLS), developed by Aitken,{{cite journal |first=A. C. |last=Aitken |title=On Least Squares and Linear Combinations of Observations |journal=Proceedings of the Royal Society of Edinburgh |year=1935 |volume=55 |pages=42–48 |doi=10.1017/S0370164600014346 }} extends the Gauss–Markov theorem to the case where the error vector has a non-scalar covariance matrix.{{cite book |first=David S. |last=Huang |title=Regression and Econometric Methods |location=New York |publisher=John Wiley & Sons |year=1970 |isbn=0-471-41754-8 |pages=[https://archive.org/details/regressioneconom0000huan/page/127 127]–147 |url=https://archive.org/details/regressioneconom0000huan |url-access=registration }} The Aitken estimator is also a BLUE.

In most treatments of OLS, the regressors (parameters of interest) in the design matrix $\backslash mathbf\{X\}$ are assumed to be fixed in repeated samples. This assumption is considered inappropriate for a predominantly nonexperimental science like econometrics.{{cite book |first=Fumio |last=Hayashi |author-link=Fumio Hayashi |title=Econometrics |publisher=Princeton University Press |year=2000 |isbn=0-691-01018-8 |page=13 |url=https://books.google.com/books?id=QyIW8WUIyzcC&pg=PA13 }} Instead, the assumptions of the Gauss–Markov theorem are stated conditional on $\backslash mathbf\{X\}$.

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 $y\; =\; \backslash beta\_\{0\}\; +\; \backslash beta\_\{1\}\; x^2,$ qualifies as linear while $y\; =\; \backslash beta\_\{0\}\; +\; \backslash beta\_\{1\}^2\; x$ can be transformed to be linear by replacing $\backslash beta\_\{1\}^2$ by another parameter, say $\backslash gamma$. An equation with a parameter dependent on an independent variable does not qualify as linear, for example $y\; =\; \backslash beta\_\{0\}\; +\; \backslash beta\_\{1\}(x)\; \backslash cdot\; x$, where $\backslash beta\_\{1\}(x)$ is a function of $x$.

Data transformations are often used to convert an equation into a linear form. For example, the Cobb–Douglas function—often used in economics—is nonlinear:

:$Y\; =\; A\; L^\backslash alpha\; K^\{1\; -\; \backslash alpha\}\; e^\backslash varepsilon$

But it can be expressed in linear form by taking the natural logarithm of both sides:{{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 }}

: $\backslash ln\; Y=\backslash ln\; A\; +\; \backslash alpha\; \backslash ln\; L\; +\; (1\; -\; \backslash alpha)\; \backslash ln\; K\; +\; \backslash varepsilon\; =\; \backslash beta\_0\; +\; \backslash beta\_1\; \backslash ln\; L\; +\; \backslash beta\_2\; \backslash ln\; K\; +\; \backslash varepsilon$

This assumption also covers specification issues: assuming that the proper functional form has been selected and there are no 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.

For all $n$ observations, the expectation—conditional on the regressors—of the error term is zero:{{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 }}

:$\backslash operatorname\{E\}[\backslash ,\backslash varepsilon\_\{i\}\backslash mid\; \backslash mathbf\{X\}\; ]\; =\; \backslash operatorname\{E\}[\backslash ,\backslash varepsilon\_\{i\}\backslash mid\; \backslash mathbf\{x\}\_\{1\},\; \backslash dots,\; \backslash mathbf\{x\}\_\{n\}\; ]\; =\; 0.$

where is the data vector of regressors for the ith observation, and consequently is the data matrix or design matrix.

Geometrically, this assumption implies that $\backslash mathbf\{x\}\_\{i\}$ and $\backslash varepsilon\_\{i\}$ are orthogonal to each other, so that their inner product (i.e., their cross moment) is zero.

:$\backslash operatorname\{E\}[\backslash ,\backslash mathbf\{x\}\_\{j\}\; \backslash cdot\; \backslash varepsilon\_\{i\}\backslash ,]\; =\; \backslash begin\{bmatrix\}\; \backslash operatorname\{E\}[\backslash ,\{x\}\_\{j1\}\; \backslash cdot\; \backslash varepsilon\_\{i\}\backslash ,]\; \backslash \backslash \; \backslash operatorname\{E\}[\backslash ,\{x\}\_\{j2\}\; \backslash cdot\; \backslash varepsilon\_\{i\}\backslash ,]\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash operatorname\{E\}[\backslash ,\{x\}\_\{jk\}\; \backslash cdot\; \backslash varepsilon\_\{i\}\backslash ,]\; \backslash end\{bmatrix\}\; =\; \backslash mathbf\{0\}\; \backslash quad\; \backslash text\{for\; all\; \}\; i,\; j\; \backslash in\; n$

This assumption is violated if the explanatory variables are measured with error, or are endogenous.{{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 }} Endogeneity can be the result of simultaneity, where causality flows back and forth between both the dependent and independent variable. Instrumental variable techniques are commonly used to address this problem.

The sample data matrix $\backslash mathbf\{X\}$ must have full column rank.

:$\backslash operatorname\{rank\}(\backslash mathbf\{X\})\; =\; k$

Otherwise $\backslash mathbf\{X\}^\backslash operatorname\{T\}\; \backslash mathbf\{X\}$ is not invertible and the OLS estimator cannot be computed.

A violation of this assumption is perfect multicollinearity, i.e. some explanatory variables are linearly dependent. One scenario in which this will occur is called "dummy variable trap," when a base dummy variable is not omitted resulting in perfect correlation between the dummy variables and the constant term.{{cite book |first=Jeffrey |last=Wooldridge |author-link=Jeffrey Wooldridge |title=Introductory Econometrics |publisher=South-Western |edition=Fifth international |year=2012 |isbn=978-1-111-53439-4 |page=[https://archive.org/details/introductoryecon00wool_406/page/n247 220] |url=https://archive.org/details/introductoryecon00wool_406|url-access=limited }}

Multicollinearity (as long as it is not "perfect") can be present resulting in a less efficient, but still unbiased estimate. The estimates will be less precise and highly sensitive to particular sets of data.{{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/159 159–168] |url=https://archive.org/details/econometricmetho0000john_t7q9/page/159 }} Multicollinearity can be detected from condition number or the variance inflation factor, among other tests.

The outer product of the error vector must be spherical.

:$\backslash operatorname\{E\}[\backslash ,\backslash boldsymbol\{\backslash varepsilon\}\; \backslash boldsymbol\{\backslash varepsilon\}^\{\backslash operatorname\{T\}\}\; \backslash mid\; \backslash mathbf\{X\}\; ]$

= \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

This implies the error term has uniform variance (homoscedasticity) and no serial correlation.{{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 }} If this assumption is violated, OLS is still unbiased, but inefficient. The term "spherical errors" will describe the multivariate normal distribution: if $\backslash operatorname\{Var\}[\backslash ,\backslash boldsymbol\{\backslash varepsilon\}\backslash mid\; \backslash mathbf\{X\}\; ]\; =\; \backslash sigma^\{2\}\; \backslash mathbf\{I\}$ in the multivariate normal density, then the equation $f(\backslash varepsilon)=c$ is the formula for a ball centered at μ with radius σ in n-dimensional space.{{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 }}

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.

• Independent and identically distributed random variables
• Linear regression
• Measurement uncertainty

• Best linear unbiased prediction (BLUP)
• Minimum-variance unbiased estimator (MVUE)

{{Reflist|30em}}

• {{cite book |first=James |last=Davidson |chapter=Statistical Analysis of the Regression Model |title=Econometric Theory |location=Oxford |publisher=Blackwell |year=2000 |pages=17–36 |isbn=0-631-17837-6 }}
• {{cite book |first=Arthur |last=Goldberger |chapter=Classical Regression |title=A Course in Econometrics |url=https://archive.org/details/courseeconometri00gold_524 |url-access=limited |location=Cambridge |publisher=Harvard University Press |year=1991 |pages=[https://archive.org/details/courseeconometri00gold_524/page/n92 160]–169 |isbn=0-674-17544-1 }}
• {{cite book |first=Henri |last=Theil |author-link=Henri Theil |chapter=Least Squares and the Standard Linear Model |title=Principles of Econometrics |url=https://archive.org/details/principlesofecon0000thei |url-access=registration |location=New York |publisher=John Wiley & Sons |year=1971 |pages=[https://archive.org/details/principlesofecon0000thei/page/101 101]–162 |isbn=0-471-85845-5 }}

• [http://jeff560.tripod.com/g.html Earliest Known Uses of Some of the Words of Mathematics: G] (brief history and explanation of the name)
• [http://www.xycoon.com/ols1.htm Proof of the Gauss Markov theorem for multiple linear regression] (makes use of matrix algebra)
• [https://web.archive.org/web/20040213071852/http://emlab.berkeley.edu/GMTheorem/index.html A Proof of the Gauss Markov theorem using geometry]

{{Least squares and regression analysis|state=expanded}}

{{DEFAULTSORT:Gauss-Markov theorem}}

Category:Theorems in statistics