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← Revision 29 as of 2025-09-03 02:08:40 ⇥
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Comment: Apparently mutlivariate regression ~= multiple regression
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| ## page was renamed from Econometrics/LinearRegression | |
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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}} | === Single Regression === |
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| Take the generic equation form of a line: | In the case of a single predictor, the OLS regression is: |
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| {{attachment:b01.svg}} | {{attachment:estimate.svg}} |
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| Insert the first point into this form. | 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:b02.svg}} | The derivation can be seen [[Statistics/OrdinaryLeastSquares/Single|here]]. |
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| This can be trivially rewritten to solve for ''a'' in terms of ''b'': | |
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| {{attachment:b03.svg}} | |
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| Insert the second point into the original form. | |
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| {{attachment:b04.svg}} | === Multiple Regression === |
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| Now additionally insert the solution for ''a'' in terms of ''b''. | In the case of multiple predictors, the regression is fit like: |
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| {{attachment:b05.svg}} | {{attachment:mestimate.svg}} |
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| Expand all terms to produce: | But conventionally, this OLS system is solved using [[LinearAlgebra|linear algebra]] as: |
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| {{attachment:b06.svg}} | {{attachment:matrix.svg}} |
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| This can now be eliminated into: | Note that using a ''b'' here is [[Statistics/EconometricsNotation#Models|intentional]]. |
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| {{attachment:b07.svg}} | The derivation can be seen [[Statistics/OrdinaryLeastSquares/Multiple|here]]. |
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| Giving a solution for ''b'': | ---- |
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| {{attachment:b08.svg}} | |
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| This solution is trivially rewritten as: | |
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| {{attachment:b09.svg}} | == Estimated Coefficients == |
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| Expand the formula for correlation as: | 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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| {{attachment:b10.svg}} | 1. Linearity 2. Exogeneity, i.e. predictors are independent of the outcome and the error term 3. Random sampling 4. No perfect [[LinearAlgebra/Basis|multicolinearity]] 5. Homoskedasticity, i.e. error terms are constant across observations |
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| This can now be eliminated into: {{attachment:b11.svg}} Finally, ''b'' can be eloquently written as: {{attachment:b12.svg}} Giving a generic formula for the regression line: {{attachment:b13.svg}} |
#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.