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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. [[Statistics/Exogeneity|Exogeneity]] 3. Random sampling 4. No perfect [[LinearAlgebra/Basis|multicolinearity]] 5. [[Statistics/Homoskedasticity|Homoskedasticity]] |
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| {{attachment:b10.svg}} | The variances for each coefficient are: |
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| This can now be eliminated into: | {{attachment:homo1.svg}} |
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| {{attachment:b11.svg}} | Note that the standard deviation of the population's parameter is unknown, so it's estimated like: |
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| Finally, ''b'' can be eloquently written as: | {{attachment:homo2.svg}} |
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| {{attachment:b12.svg}} | If the homoskedasticity assumption does not hold, then the estimators for each coefficient are actually: |
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| Giving a generic formula for the regression line: | {{attachment:hetero1.svg}} |
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| {{attachment:b13.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. |
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
- Random sampling
No perfect multicolinearity
The variances for each coefficient are:
Note that the standard deviation of the population's parameter is unknown, so it's estimated like:
If the homoskedasticity assumption does not hold, then the estimators for each coefficient are actually:
Wherein, for example, r1j is the residual from regressing x1 onto x2, ... xk.
The variances for each coefficient can be estimated with the Eicker-White formula:
See Nicolai Kuminoff's video lectures for the derivation of the robust estimators.