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| '''Ordinary Least Squares''' ('''OLS''') is a linear regression method. It minimizes root mean square errors. | '''Ordinary Least Squares''' ('''OLS''') is a linear regression method, and is effectively synonymous with the '''linear regression model'''. |
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| == Univariate == | == Description == |
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| The regression line passes through two points: | A linear model is expressed as either {{attachment:model.svg}} (univariate) or {{attachment:mmodel.svg}} (multivariate with ''k'' terms). Either way, a crucial assumption is that the expected value of the error term is 0, such that the [[Statistics/Moments|first moment]] is ''E[y,,i,,|x,,i,,] = α + βx,,i,,''. |
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| {{attachment:regression1.svg}} | |
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| and | |
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| {{attachment:regression2.svg}} | === Univariate === |
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| These points, with the generic equation for a line, can [[Econometrics/OrdinaryLeastSquares/UnivariateProof|prove]] that the slope of the regression line is equal to: | In the univariate case, the OLS regression is: |
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| {{attachment:b12.svg}} | {{attachment:estimate.svg}} |
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| The generic formula for the regression line is: | This formulation leaves the components explicit: the y-intercept term is the mean outcome at ''x=0'', and the slope term is marginal change to the outcome per a unit change in ''x''. |
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| {{attachment:b13.svg}} | The derivation can be seen [[Statistics/OrdinaryLeastSquares/Univariate|here]]. === Multivariate === In the multivariate case, the regression is fit like: {{attachment:mestimate.svg}} But conventionally, multivariate OLS is solved using [[LinearAlgebra|linear algebra]] as: {{attachment:matrix.svg}} Note that using a ''b'' here is [[Statistics/EconometricsNotation#Models|intentional]]. The derivation can be seen [[Statistics/OrdinaryLeastSquares/Multivariate|here]]. |
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| == Multivariate == | == Estimated Coefficients == |
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| ---- == Linear Model == The linear model can be expressed as: {{attachment:model1.svg}} 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, i.e. predictors are independent of the outcome and the error term |
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| 4. No perfect multicolinearity 5. [[Econometrics/Homoskedasticity|Homoskedasticity]] |
4. No perfect [[LinearAlgebra/Basis|multicolinearity]] 5. Homoskedasticity, i.e. error terms are constant across observations |
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| Then OLS is the best linear unbiased estimator ('''BLUE''') for these coefficients. Using the computation above, the coefficients are estimated to produce: {{attachment:model2.svg}} 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, and is effectively synonymous with the linear regression model.
Description
A linear model is expressed as either 
(univariate) or 
(multivariate with k terms). Either way, a crucial assumption is that the expected value of the error term is 0, such that the first moment is E[yi|xi] = α + βxi.
Univariate
In the univariate case, the OLS regression is:
This formulation leaves the components explicit: the y-intercept term is the mean outcome at x=0, and the slope term is marginal change to the outcome per a unit change in x.
The derivation can be seen here.
Multivariate
In the multivariate case, the regression is fit like:
But conventionally, multivariate OLS is solved using linear algebra as:
Note that using a b here is intentional.
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:
- Linearity
- Exogeneity, i.e. predictors are independent of the outcome and the error term
- Random sampling
No perfect multicolinearity
- Homoskedasticity, i.e. error terms are constant across observations
#5 mostly comes into the estimation of standard errors, and there are alternative estimators that are robust to heteroskedasticity.