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The proof can be seen [[Econometrics/OrdinaryLeastSquares/UnivariateProof|here]]. The derivation can be seen [[Statistics/OrdinaryLeastSquares/Univariate|here]].
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More conventionally, this is estimated with [[LinearAlgebra|linear algebra]] as:

{{attachment:matrix.svg}}

The derivation can be seen [[Statistics/OrdinaryLeastSquares/Multivariate|here]].
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If these assumptions can be made: The '''Gauss-Markov theorem''' demonstrates that (with some assumptions) the OLS estimations are the '''best linear unbiased estimators''' ('''BLUE''') for the regression coefficients. The assumptions are:
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 2. [[Econometrics/Exogeneity|Exogeneity]]  2. Exogeneity
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 4. No perfect multicolinearity
 5. [[Econometrics/Homoskedasticity|Homoskedasticity]]		  4. No perfect [[LinearAlgebra/Basis|multicolinearity]]
 5. Homoskedasticity
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Then OLS is the best linear unbiased estimator ('''BLUE''') for regression coefficients.

The variances for each coefficient are:

{{attachment:homo1.svg}}

Note that the standard deviation of the population's parameter is unknown, so it's estimated like:

{{attachment:homo2.svg}}

If the homoskedasticity assumption does not hold, then the estimators for each coefficient are actually:

{{attachment:hetero1.svg}}

Wherein, for example, ''r,,1j,,'' is the residual from regressing ''x,,1,,'' onto ''x,,2,,'', ... ''x,,k,,''.

The variances for each coefficient can be estimated with the Eicker-White formula:

{{attachment:hetero2.svg}}

See [[https://www.youtube.com/@kuminoff|Nicolai Kuminoff's]] video lectures for the derivation of the robust estimators.		 #5 mostly comes into the estimation of [[Statistics/StandardErrors|standard errors]], and there are alternative estimators that are robust to heteroskedasticity.

Ordinary Least Squares

Ordinary Least Squares (OLS) is a linear regression method. It minimizes root mean square errors.

Contents

  1. Ordinary Least Squares
    1. Univariate
    2. Multivariate
    3. Estimated Coefficients

Univariate

Given one independent variable and one dependent (outcome) variable, the OLS model is specified as:

model.svg

It is estimated as:

estimate.svg

This model describes (1) the mean observation and (2) the marginal changes to the outcome per unit changes in the independent variable.

The derivation can be seen here.

Multivariate

Given k independent variables, the OLS model is specified as:

mmodel.svg

It is estimated as:

mestimate.svg

More conventionally, this is estimated with linear algebra as:

matrix.svg

The derivation can be seen here.

Estimated Coefficients

The Gauss-Markov theorem demonstrates that (with some assumptions) the OLS estimations are the best linear unbiased estimators (BLUE) for the regression coefficients. The assumptions are:

  1. Linearity
  2. Exogeneity
  3. Random sampling

  4. No perfect multicolinearity

  5. Homoskedasticity

#5 mostly comes into the estimation of standard errors, and there are alternative estimators that are robust to heteroskedasticity.

CategoryRicottone

Statistics/OrdinaryLeastSquares (last edited 2025-09-03 02:08:40 by DominicRicottone)