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← Revision 26 as of 2025-05-17 03:48:23 ⇥
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| The regression line passes through two points: | Given one independent variable and one dependent (outcome) variable, the OLS model is specified as: |
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| {{attachment:regression1.svg}} | {{attachment:model.svg}} |
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| and | It is estimated as: |
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| {{attachment:regression2.svg}} | {{attachment:estimate.svg}} |
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| Take the generic equation form of a line: | This model describes (1) the mean observation and (2) the marginal changes to the outcome per unit changes in the independent variable. |
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| {{attachment:b01.svg}} | The derivation can be seen [[Statistics/OrdinaryLeastSquares/Univariate|here]]. |
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| Insert the first point into this form. | ---- |
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| {{attachment:b02.svg}} | |
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| This can be trivially rewritten to solve for ''a'' in terms of ''b'': | |
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| {{attachment:b03.svg}} | == Multivariate == |
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| Insert the second point into the original form. | Given ''k'' independent variables, the OLS model is specified as: |
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| {{attachment:b04.svg}} | {{attachment:mmodel.svg}} |
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| Now additionally insert the solution for ''a'' in terms of ''b''. | It is estimated as: |
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| {{attachment:b05.svg}} | {{attachment:mestimate.svg}} |
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| Expand all terms to produce: | More conventionally, this is estimated with [[LinearAlgebra|linear algebra]] as: |
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| {{attachment:b06.svg}} | {{attachment:matrix.svg}} |
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| This can now be eliminated into: | The derivation can be seen [[Statistics/OrdinaryLeastSquares/Multivariate|here]]. |
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| {{attachment:b07.svg}} | ---- |
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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: | == Estimated Coefficients == |
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| {{attachment:b09.svg}} | 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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| Expand the formula for correlation as: | 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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| {{attachment:b10.svg}} 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. It minimizes root mean square errors.
Univariate
Given one independent variable and one dependent (outcome) variable, the OLS model is specified as:
It is estimated as:
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:
It is estimated as:
More conventionally, this is estimated with linear algebra as:
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.