Gauss–Markov theorem
Gauss–Markov theorem Not to be confused with Gauss–Markov process. "BLUE" redirige ici. 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. Voir, par exemple, 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] Contenu 1 Déclaration 1.1 Remarque 2 Preuve 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 Voir également 6.1 Other unbiased statistics 7 Références 8 Lectures complémentaires 9 External links Statement Suppose we have in matrix notation, {style d'affichage {souligner {y}}=X{souligner {bêta }}+{souligner {varepsilon }},quad ({souligner {y}},{souligner {varepsilon }}en mathbb {R} ^{n},{souligner {bêta }}en mathbb {R} ^{K}{texte{ et }}Xin mathbb {R} ^{ntimes K})} expanding to, {style d'affichage y_{je}=somme _{j=1}^{K}bêta _{j}X_{ij}+varepsilon _{je}quad forall i=1,2,ldots ,n} où {style d'affichage bêta _{j}} are non-random but unobservable parameters, {style d'affichage X_{ij}} are non-random and observable (called the "explanatory variables"), {displaystyle varepsilon _{je}} are random, et donc {style d'affichage y_{je}} are random. The random variables {displaystyle varepsilon _{je}} are called the "disturbance", "bruit" 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 {style d'affichage bêta _{K+1}} with a newly introduced last column of X being unity i.e., {style d'affichage X_{je(K+1)}=1} pour tous {style d'affichage i} . Note that though {style d'affichage y_{je},} as sample responses, are observable, the following statements and arguments including assumptions, proofs and the others assume under the only condition of knowing {style d'affichage X_{ij},} but not {style d'affichage y_{je}.} The Gauss–Markov assumptions concern the set of error random variables, {displaystyle varepsilon _{je}} : They have mean zero: {nom de l'opérateur de style d'affichage {E} [varepsilon _{je}]=0.} They are homoscedastic, that is all have the same finite variance: {nom de l'opérateur de style d'affichage {A été} (varepsilon _{je})=sigma ^{2}
Proof Let {style d'affichage {tilde {bêta }}=Cy} be another linear estimator of {style d'affichage bêta } avec {displaystyle C=(X'X)^{-1}X'+D} où {displaystyle D} est un {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 {style d'affichage {chapeau large {bêta }},} the OLS estimator. We calculate: {style d'affichage {commencer{aligné}nom de l'opérateur {E} la gauche[{tilde {bêta }}droit]&=operatorname {E} [Cy]\&=operatorname {E} la gauche[la gauche((X'X)^{-1}X'+Dright)(Xbeta +varepsilon )droit]\&=left((X'X)^{-1}X'+Dright)Xbeta +left((X'X)^{-1}X'+Dright)nom de l'opérateur {E} [varepsilon ]\&=left((X'X)^{-1}X'+Dright)Xbeta &&operatorname {E} [varepsilon ]=0\&=(X'X)^{-1}X'Xbeta +DXbeta \&=(JE_{K}+DX)beta .\end{aligné}}} Par conséquent, puisque {style d'affichage bêta } is unobservable, {style d'affichage {tilde {bêta }}} is unbiased if and only if {displaystyle DX=0} . Alors: {style d'affichage {commencer{aligné}nom de l'opérateur {A été} la gauche({tilde {bêta }}droit)&=operatorname {A été} (Cy)\&=C{texte{ A été}}(y)C'\&=sigma ^{2}CC'\&=sigma ^{2}la gauche((X'X)^{-1}X'+Dright)la gauche(X(X'X)^{-1}+D'right)\&=sigma ^{2}la gauche((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 {A été} la gauche({chapeau large {bêta }}droit)+sigma ^{2}DD'&&sigma ^{2}(X'X)^{-1}=nomopérateur {A été} la gauche({chapeau large {bêta }}droit)fin{aligné}}} Since DD' is a positive semidefinite matrix, {nom de l'opérateur de style d'affichage {A été} la gauche({tilde {bêta }}droit)} exceeds {nom de l'opérateur de style d'affichage {A été} la gauche({chapeau large {bêta }}droit)} by a positive semidefinite matrix.
Remarks on the proof As it has been stated before, the condition of {nom de l'opérateur de style d'affichage {A été} la gauche({tilde {bêta }}droit)-nom de l'opérateur {A été} la gauche({chapeau large {bêta }}droit)} is a positive semidefinite matrix is equivalent to the property that the best linear unbiased estimator of {displaystyle ell ^{t}bêta } est {displaystyle ell ^{t}{chapeau large {bêta }}} (best in the sense that it has minimum variance). Pour voir ça, laisser {displaystyle ell ^{t}{tilde {bêta }}} another linear unbiased estimator of {displaystyle ell ^{t}bêta } .
{style d'affichage {commencer{aligné}nom de l'opérateur {A été} la gauche(euh ^{t}{tilde {bêta }}droit)&=ell ^{t}nom de l'opérateur {A été} la gauche({tilde {bêta }}droit)ell \&=sigma ^{2}euh ^{t}(X'X)^{-1}ell +ell ^{t}DD^{t}ell \&=operatorname {A été} la gauche(euh ^{t}{chapeau large {bêta }}droit)+(D^{t}aune )^{t}(D^{t}aune )&&sigma ^{2}euh ^{t}(X'X)^{-1}ell =operatorname {A été} la gauche(euh ^{t}{chapeau large {bêta }}droit)\&=operatorname {A été} la gauche(euh ^{t}{chapeau large {bêta }}droit)+|D^{t}aune |\&geq operatorname {A été} la gauche(euh ^{t}{chapeau large {bêta }}droit)fin{aligné}}} En outre, equality holds if and only if {displaystyle D^{t}euh =0} . We calculate {style d'affichage {commencer{aligné}euh ^{t}{tilde {bêta }}&=ell ^{t}la gauche(((X'X)^{-1}X'+D)Yright)&&{texte{ from above}}\&=ell ^{t}(X'X)^{-1}X'Y+ell ^{t}DY\&=ell ^{t}{chapeau large {bêta }}+(D^{t}aune )^{t}Y\&=ell ^{t}{chapeau large {bêta }}&&D^{t}ell =0end{aligné}}} This proves that the equality holds if and only if {displaystyle ell ^{t}{tilde {bêta }}=ell ^{t}{chapeau large {bêta }}} 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 {style d'affichage mathbf {X} } are assumed to be fixed in repeated samples. This assumption is considered inappropriate for a predominantly nonexperimental science like econometrics.[7] À la place, the assumptions of the Gauss–Markov theorem are stated conditional on {style d'affichage 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}+bêta _{1}x^{2},} qualifies as linear while {displaystyle y=beta _{0}+bêta _{1}^{2}X} can be transformed to be linear by replacing {style d'affichage bêta _{1}^{2}} by another parameter, dire {gamma de style d'affichage } . An equation with a parameter dependent on an independent variable does not qualify as linear, par exemple {displaystyle y=beta _{0}+bêta _{1}(X)cdot x} , où {style d'affichage bêta _{1}(X)} is a function of {style d'affichage x} .
Data transformations are often used to convert an equation into a linear form. Par exemple, 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}+bêta _{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, toutefois, 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] {nom de l'opérateur de style d'affichage {E} [,varepsilon _{je}mid mathbf {X} ]=nomopérateur {E} [,varepsilon _{je}mid mathbf {X} _{1},des points ,mathbf {X} _{n}]=0.} où {style d'affichage mathbf {X} _{je}={commencer{bmatrice}X_{i1}&x_{i2}&cdots &x_{ik}fin{bmatrice}}^{mathématiques {J}}} is the data vector of regressors for the ith observation, and consequently {style d'affichage mathbf {X} ={commencer{bmatrice}mathbf {X} _{1}^{mathématiques {J}}&mathbf {X} _{2}^{mathématiques {J}}&cdots &mathbf {X} _{n}^{mathématiques {J}}fin{bmatrice}}^{mathématiques {J}}} is the data matrix or design matrix.
Géométriquement, this assumption implies that {style d'affichage mathbf {X} _{je}} et {displaystyle varepsilon _{je}} are orthogonal to each other, so that their inner product (c'est à dire., their cross moment) est zéro.
{nom de l'opérateur de style d'affichage {E} [,mathbf {X} _{j}cdot varepsilon _{je},]={commencer{bmatrice}nom de l'opérateur {E} [,{X}_{j1}cdot varepsilon _{je},]\nom de l'opérateur {E} [,{X}_{j2}cdot varepsilon _{je},]\vdots \operatorname {E} [,{X}_{jk}cdot varepsilon _{je},]fin{bmatrice}}= mathbf {0} quad {texte{pour tous }}je,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 {style d'affichage mathbf {X} } must have full column rank.
{nom de l'opérateur de style d'affichage {rang} (mathbf {X} )=k} Autrement {style d'affichage mathbf {X} 'mathbf {X} } is not invertible and the OLS estimator cannot be computed.
A violation of this assumption is perfect multicollinearity, c'est à dire. 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.
{nom de l'opérateur de style d'affichage {E} [,{symbole gras {varepsilon }}{symbole gras {varepsilon ^{mathématiques {J}}}}mid mathbf {X} ]=nomopérateur {A été} [,{symbole gras {varepsilon }}mid mathbf {X} ]={commencer{bmatrice}sigma ^{2}&0&cdots &0\0&sigma ^{2}&cdots &0\vdots &vdots &ddots &vdots \0&0&cdots &sigma ^{2}fin{bmatrice}}=sigma ^{2}mathbf {je} quad {texte{avec }}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: si {nom de l'opérateur de style d'affichage {A été} [,{symbole gras {varepsilon }}mid mathbf {X} ]=sigma ^{2}mathbf {je} } in the multivariate normal density, alors l'équation {style d'affichage 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. Par exemple, 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. Par exemple, 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. Volume. 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. est ce que je: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. ^ Sauter à: a b Aitken, UN. C. (1935). "On Least Squares and Linear Combinations of Observations". Actes de la Royal Society of Edinburgh. 55: 42–48. est ce que je:10.1017/S0370164600014346. ^ Sauter à: 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. Presse de l'Université de Princeton. p. 13. ISBN 0-691-01018-8. ^ Walters, UN. UN. (1970). An Introduction to Econometrics. New York: O. O. Norton. p. 275. ISBN 0-393-09931-8. ^ Hayashi, Fumio (2000). Econometrics. Presse de l'Université de Princeton. p. 7. ISBN 0-691-01018-8. ^ Johnston, John (1972). Econometric Methods (Deuxième éd.). 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 (Deuxième éd.). New York: McGraw Hill. pp. 159–168. ISBN 0-07-032679-7. ^ Hayashi, Fumio (2000). Econometrics. Presse de l'Université de Princeton. p. 10. ISBN 0-691-01018-8. ^ Ramanathan, Ramu (1993). "Nonspherical Disturbances". Statistical Methods in Econometrics. Presse académique. 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: Presse universitaire de Harvard. 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 (statistiques)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: Théorèmes en statistiques
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