Gauss–Markov theorem

Gauss–Markov theorem Not to be confused with Gauss–Markov process. "BLUE" redirects here. For queue management algorithm, see Blue (queue management algorithm). Part of a series on Regression analysis Models Linear regressionSimple regressionPolynomial regressionGeneral linear model Generalized linear modelDiscrete choiceBinomial regressionBinary regressionLogistic regressionMultinomial logistic regressionMixed logitProbitMultinomial probitOrdered logitOrdered probitPoisson Multilevel modelFixed effectsRandom effectsLinear mixed-effects modelNonlinear mixed-effects model Nonlinear regressionNonparametricSemiparametricRobustQuantileIsotonicPrincipal componentsLeast angleLocalSegmented Errors-in-variables Estimation Least squaresLinearNon-linear OrdinaryWeightedGeneralized PartialTotalNon-negativeRidge regressionRegularized Least absolute deviationsIteratively reweightedBayesianBayesian multivariateLeast-squares spectral analysis Background Regression validationMean and predicted responseErrors and residualsGoodness of fitStudentized residualGauss–Markov theorem  Mathematics portal vte In statistics, the Gauss–Markov theorem (or simply Gauss theorem for some authors)[1] states that the ordinary least squares (OLS) estimator has the lowest sampling variance within the class of linear unbiased estimators, if the errors in the linear regression model are uncorrelated, have equal variances and expectation value of zero.[2] The errors do not need to be normal, nor do they need to be independent and identically distributed (only uncorrelated with mean zero and homoscedastic with finite variance). The requirement that the estimator be unbiased cannot be dropped, since biased estimators exist with lower variance. See, for example, the James–Stein estimator (which also drops linearity), ridge regression, or simply any degenerate estimator.

The theorem was named after Carl Friedrich Gauss and Andrey Markov, although Gauss' work significantly predates Markov's.[3] But while Gauss derived the result under the assumption of independence and normality, Markov reduced the assumptions to the form stated above.[4] A further generalization to non-spherical errors was given by Alexander Aitken.[5] Contents 1 Statement 1.1 Remark 2 Proof 3 Remarks on the proof 4 Generalized least squares estimator 5 Gauss–Markov theorem as stated in econometrics 5.1 Linearity 5.2 Strict exogeneity 5.3 Full rank 5.4 Spherical errors 6 See also 6.1 Other unbiased statistics 7 References 8 Further reading 9 External links Statement Suppose we have in matrix notation, {displaystyle {underline {y}}=X{underline {beta }}+{underline {varepsilon }},quad ({underline {y}},{underline {varepsilon }}in mathbb {R} ^{n},{underline {beta }}in mathbb {R} ^{K}{text{ and }}Xin mathbb {R} ^{ntimes K})} expanding to, {displaystyle y_{i}=sum _{j=1}^{K}beta _{j}X_{ij}+varepsilon _{i}quad forall i=1,2,ldots ,n} where {displaystyle beta _{j}} are non-random but unobservable parameters, {displaystyle X_{ij}} are non-random and observable (called the "explanatory variables"), {displaystyle varepsilon _{i}} are random, and so {displaystyle y_{i}} are random. The random variables {displaystyle 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 {displaystyle beta _{K+1}} with a newly introduced last column of X being unity i.e., {displaystyle X_{i(K+1)}=1} for all {displaystyle i} . Note that though {displaystyle 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 {displaystyle X_{ij},} but not {displaystyle y_{i}.} The Gauss–Markov assumptions concern the set of error random variables, {displaystyle varepsilon _{i}} : They have mean zero: {displaystyle operatorname {E} [varepsilon _{i}]=0.} They are homoscedastic, that is all have the same finite variance: {displaystyle operatorname {Var} (varepsilon _{i})=sigma ^{2}0} In terms of vector multiplication, this means {displaystyle {begin{bmatrix}k_{1}&cdots &k_{p+1}end{bmatrix}}{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} ^{mathsf {T}}{mathcal {H}}mathbf {k} =lambda mathbf {k} ^{mathsf {T}}mathbf {k} >0} where {displaystyle lambda } is the eigenvalue corresponding to {displaystyle mathbf {k} } . Moreover, {displaystyle mathbf {k} ^{mathsf {T}}mathbf {k} =sum _{i=1}^{p+1}k_{i}^{2}>0implies lambda >0} Finally, as eigenvector {displaystyle mathbf {k} } was arbitrary, it means all eigenvalues of {displaystyle {mathcal {H}}} are positive, therefore {displaystyle {mathcal {H}}} is positive definite. Thus, {displaystyle {boldsymbol {beta }}=left(X^{mathsf {T}}Xright)^{-1}X^{mathsf {T}}Y} is indeed a global minimum.

Proof Let {displaystyle {tilde {beta }}=Cy} be another linear estimator of {displaystyle beta } with {displaystyle C=(X'X)^{-1}X'+D} where {displaystyle D} is a {displaystyle Ktimes 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 {displaystyle {widehat {beta }},} the OLS estimator. We calculate: {displaystyle {begin{aligned}operatorname {E} left[{tilde {beta }}right]&=operatorname {E} [Cy]\&=operatorname {E} left[left((X'X)^{-1}X'+Dright)(Xbeta +varepsilon )right]\&=left((X'X)^{-1}X'+Dright)Xbeta +left((X'X)^{-1}X'+Dright)operatorname {E} [varepsilon ]\&=left((X'X)^{-1}X'+Dright)Xbeta &&operatorname {E} [varepsilon ]=0\&=(X'X)^{-1}X'Xbeta +DXbeta \&=(I_{K}+DX)beta .\end{aligned}}} Therefore, since {displaystyle beta } is unobservable, {displaystyle {tilde {beta }}} is unbiased if and only if {displaystyle DX=0} . Then: {displaystyle {begin{aligned}operatorname {Var} left({tilde {beta }}right)&=operatorname {Var} (Cy)\&=C{text{ Var}}(y)C'\&=sigma ^{2}CC'\&=sigma ^{2}left((X'X)^{-1}X'+Dright)left(X(X'X)^{-1}+D'right)\&=sigma ^{2}left((X'X)^{-1}X'X(X'X)^{-1}+(X'X)^{-1}X'D'+DX(X'X)^{-1}+DD'right)\&=sigma ^{2}(X'X)^{-1}+sigma ^{2}(X'X)^{-1}(DX)'+sigma ^{2}DX(X'X)^{-1}+sigma ^{2}DD'\&=sigma ^{2}(X'X)^{-1}+sigma ^{2}DD'&&DX=0\&=operatorname {Var} left({widehat {beta }}right)+sigma ^{2}DD'&&sigma ^{2}(X'X)^{-1}=operatorname {Var} left({widehat {beta }}right)end{aligned}}} Since DD' is a positive semidefinite matrix, {displaystyle operatorname {Var} left({tilde {beta }}right)} exceeds {displaystyle operatorname {Var} left({widehat {beta }}right)} by a positive semidefinite matrix.

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

{displaystyle {begin{aligned}operatorname {Var} left(ell ^{t}{tilde {beta }}right)&=ell ^{t}operatorname {Var} left({tilde {beta }}right)ell \&=sigma ^{2}ell ^{t}(X'X)^{-1}ell +ell ^{t}DD^{t}ell \&=operatorname {Var} left(ell ^{t}{widehat {beta }}right)+(D^{t}ell )^{t}(D^{t}ell )&&sigma ^{2}ell ^{t}(X'X)^{-1}ell =operatorname {Var} left(ell ^{t}{widehat {beta }}right)\&=operatorname {Var} left(ell ^{t}{widehat {beta }}right)+|D^{t}ell |\&geq operatorname {Var} left(ell ^{t}{widehat {beta }}right)end{aligned}}} Moreover, equality holds if and only if {displaystyle D^{t}ell =0} . We calculate {displaystyle {begin{aligned}ell ^{t}{tilde {beta }}&=ell ^{t}left(((X'X)^{-1}X'+D)Yright)&&{text{ from above}}\&=ell ^{t}(X'X)^{-1}X'Y+ell ^{t}DY\&=ell ^{t}{widehat {beta }}+(D^{t}ell )^{t}Y\&=ell ^{t}{widehat {beta }}&&D^{t}ell =0end{aligned}}} This proves that the equality holds if and only if {displaystyle ell ^{t}{tilde {beta }}=ell ^{t}{widehat {beta }}} which gives the uniqueness of the OLS estimator as a BLUE.

Generalized least squares estimator The generalized least squares (GLS), developed by Aitken,[5] extends the Gauss–Markov theorem to the case where the error vector has a non-scalar covariance matrix.[6] The Aitken estimator is also a BLUE.

Gauss–Markov theorem as stated in econometrics In most treatments of OLS, the regressors (parameters of interest) in the design matrix {displaystyle mathbf {X} } are assumed to be fixed in repeated samples. This assumption is considered inappropriate for a predominantly nonexperimental science like econometrics.[7] Instead, the assumptions of the Gauss–Markov theorem are stated conditional on {displaystyle mathbf {X} } .

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 {displaystyle y=beta _{0}+beta _{1}x^{2},} qualifies as linear while {displaystyle y=beta _{0}+beta _{1}^{2}x} can be transformed to be linear by replacing {displaystyle beta _{1}^{2}} by another parameter, say {displaystyle gamma } . An equation with a parameter dependent on an independent variable does not qualify as linear, for example {displaystyle y=beta _{0}+beta _{1}(x)cdot x} , where {displaystyle beta _{1}(x)} is a function of {displaystyle 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: {displaystyle Y=AL^{alpha }K^{1-alpha }e^{varepsilon }} But it can be expressed in linear form by taking the natural logarithm of both sides:[8] {displaystyle ln Y=ln A+alpha ln L+(1-alpha )ln K+varepsilon =beta _{0}+beta _{1}ln L+beta _{2}ln K+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.

Strict exogeneity For all {displaystyle n} observations, the expectation—conditional on the regressors—of the error term is zero:[9] {displaystyle operatorname {E} [,varepsilon _{i}mid mathbf {X} ]=operatorname {E} [,varepsilon _{i}mid mathbf {x} _{1},dots ,mathbf {x} _{n}]=0.} where {displaystyle mathbf {x} _{i}={begin{bmatrix}x_{i1}&x_{i2}&cdots &x_{ik}end{bmatrix}}^{mathsf {T}}} is the data vector of regressors for the ith observation, and consequently {displaystyle mathbf {X} ={begin{bmatrix}mathbf {x} _{1}^{mathsf {T}}&mathbf {x} _{2}^{mathsf {T}}&cdots &mathbf {x} _{n}^{mathsf {T}}end{bmatrix}}^{mathsf {T}}} is the data matrix or design matrix.

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

{displaystyle 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,jin n} This assumption is violated if the explanatory variables are stochastic, for instance when they are measured with error, or are endogenous.[10] 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.

Full rank The sample data matrix {displaystyle mathbf {X} } must have full column rank.

{displaystyle operatorname {rank} (mathbf {X} )=k} Otherwise {displaystyle mathbf {X} '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.[11] 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.[12] Multicollinearity can be detected from condition number or the variance inflation factor, among other tests.

Spherical errors The outer product of the error vector must be spherical.

{displaystyle operatorname {E} [,{boldsymbol {varepsilon }}{boldsymbol {varepsilon ^{mathsf {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} This implies the error term has uniform variance (homoscedasticity) and no serial dependence.[13] If this assumption is violated, OLS is still unbiased, but inefficient. The term "spherical errors" will describe the multivariate normal distribution: if {displaystyle operatorname {Var} [,{boldsymbol {varepsilon }}mid mathbf {X} ]=sigma ^{2}mathbf {I} } in the multivariate normal density, then the equation {displaystyle f(varepsilon )=c} is the formula for a ball centered at μ with radius σ in n-dimensional space.[14] 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.

In the presence of spherical errors, the generalized least squares estimator can be shown to be BLUE.[6] See also Independent and identically distributed random variables Linear regression Measurement uncertainty Other unbiased statistics Best linear unbiased prediction (BLUP) Minimum-variance unbiased estimator (MVUE) References ^ See chapter 7 of Johnson, R.A.; Wichern, D.W. (2002). Applied multivariate statistical analysis. Vol. 5. Prentice hall. ^ Theil, Henri (1971). "Best Linear Unbiased Estimation and Prediction". Principles of Econometrics. New York: John Wiley & Sons. pp. 119–124. ISBN 0-471-85845-5. ^ Plackett, R. L. (1949). "A Historical Note on the Method of Least Squares". Biometrika. 36 (3/4): 458–460. doi:10.2307/2332682. ^ David, F. N.; Neyman, J. (1938). "Extension of the Markoff theorem on least squares". Statistical Research Memoirs. 2: 105–116. OCLC 4025782. ^ Jump up to: a b Aitken, A. C. (1935). "On Least Squares and Linear Combinations of Observations". Proceedings of the Royal Society of Edinburgh. 55: 42–48. doi:10.1017/S0370164600014346. ^ Jump up to: a b Huang, David S. (1970). Regression and Econometric Methods. New York: John Wiley & Sons. pp. 127–147. ISBN 0-471-41754-8. ^ Hayashi, Fumio (2000). Econometrics. Princeton University Press. p. 13. ISBN 0-691-01018-8. ^ Walters, A. A. (1970). An Introduction to Econometrics. New York: W. W. Norton. p. 275. ISBN 0-393-09931-8. ^ Hayashi, Fumio (2000). Econometrics. Princeton University Press. p. 7. ISBN 0-691-01018-8. ^ Johnston, John (1972). Econometric Methods (Second ed.). New York: McGraw-Hill. pp. 267–291. ISBN 0-07-032679-7. ^ Wooldridge, Jeffrey (2012). Introductory Econometrics (Fifth international ed.). South-Western. p. 220. ISBN 978-1-111-53439-4. ^ Johnston, John (1972). Econometric Methods (Second ed.). New York: McGraw-Hill. pp. 159–168. ISBN 0-07-032679-7. ^ Hayashi, Fumio (2000). Econometrics. Princeton University Press. p. 10. ISBN 0-691-01018-8. ^ Ramanathan, Ramu (1993). "Nonspherical Disturbances". Statistical Methods in Econometrics. Academic Press. pp. 330–351. ISBN 0-12-576830-3. Further reading Davidson, James (2000). "Statistical Analysis of the Regression Model". Econometric Theory. Oxford: Blackwell. pp. 17–36. ISBN 0-631-17837-6. Goldberger, Arthur (1991). "Classical Regression". A Course in Econometrics. Cambridge: Harvard University Press. pp. 160–169. ISBN 0-674-17544-1. Theil, Henri (1971). "Least Squares and the Standard Linear Model". Principles of Econometrics. New York: John Wiley & Sons. pp. 101–162. ISBN 0-471-85845-5. External links Earliest Known Uses of Some of the Words of Mathematics: G (brief history and explanation of the name) Proof of the Gauss Markov theorem for multiple linear regression (makes use of matrix algebra) A Proof of the Gauss Markov theorem using geometry hide vte Least squares and regression analysis Computational statistics Least squaresLinear least squaresNon-linear least squaresIteratively reweighted least squares Correlation and dependence Pearson product-moment correlationRank correlation (Spearman's rhoKendall's tau)Partial correlationConfounding variable Regression analysis Ordinary least squaresPartial least squaresTotal least squaresRidge regression Regression as a statistical model Linear regression Simple linear regressionOrdinary least squaresGeneralized least squaresWeighted least squaresGeneral linear model Predictor structure Polynomial regressionGrowth curve (statistics)Segmented regressionLocal regression Non-standard Nonlinear regressionNonparametricSemiparametricRobustQuantileIsotonic Non-normal errors Generalized linear modelBinomialPoissonLogistic Decomposition of variance Analysis of varianceAnalysis of covarianceMultivariate AOV Model exploration Stepwise regressionModel selection Mallows's CpAICBICModel specificationRegression validation Background Mean and predicted responseGauss–Markov theoremErrors and residualsGoodness of fitStudentized residualMinimum mean-square errorFrisch–Waugh–Lovell theorem Design of experiments Response surface methodologyOptimal designBayesian design Numerical approximation Numerical analysisApproximation theoryNumerical integrationGaussian quadratureOrthogonal polynomialsChebyshev polynomialsChebyshev nodes Applications Curve fittingCalibration curveNumerical smoothing and differentiationSystem identificationMoving least squares Regression analysis categoryStatistics category Mathematics portalStatistics outlineStatistics topics Categories: Theorems in statistics

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