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Comment: Rewrite of coefficients section
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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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| Given one independent variable and one dependent (outcome) variable, the OLS model is specified as: | 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:model.svg}} | |
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| It is estimated as: | === Single Regression === In the case of a single predictor, the OLS regression is: |
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| This model describes (1) the mean observation and (2) the marginal changes to the outcome per unit changes in the independent variable. | 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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| The derivation can be seen [[Econometrics/OrdinaryLeastSquares/Univariate|here]]. ---- |
The derivation can be seen [[Statistics/OrdinaryLeastSquares/Single|here]]. |
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| == Multivariate == | |
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| Given ''k'' independent variables, the OLS model is specified as: | === Multiple Regression === |
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| {{attachment:mmodel.svg}} It is estimated as: |
In the case of multiple predictors, the regression is fit like: |
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| More conventionally, this is estimated with [[LinearAlgebra|linear algebra]] as: | But conventionally, this OLS system is solved using [[LinearAlgebra|linear algebra]] as: |
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| The derivation can be seen [[Econometrics/OrdinaryLeastSquares/Multivariate|here]]. | Note that using a ''b'' here is [[Statistics/EconometricsNotation#Models|intentional]]. The derivation can be seen [[Statistics/OrdinaryLeastSquares/Multiple|here]]. |
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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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| 5. [[Econometrics/Homoskedasticity|Homoskedasticity]] | 5. Homoskedasticity, i.e. error terms are constant across observations |
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| 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.
Contents
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.
Single Regression
In the case of a single predictor, 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.
Multiple Regression
In the case of multiple predictors, the regression is fit like:
But conventionally, this OLS system 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.