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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}} Insert the first point into this form. {{attachment:b02.svg}} This can be trivially rewritten to solve for ''a'' in terms of ''b'': {{attachment:b03.svg}} Insert the second point into the original form. {{attachment:b04.svg}} Now additionally insert the solution for ''a'' in terms of ''b''. {{attachment:b05.svg}} Expand all terms to produce: {{attachment:b06.svg}} This can now be eliminated into: {{attachment:b07.svg}} Giving a solution for ''b'': {{attachment:b08.svg}} This solution is trivially rewritten as: {{attachment:b09.svg}} Expand the formula for correlation as: {{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}} |
The derivation can be seen [[Statistics/OrdinaryLeastSquares/Univariate|here]]. |
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| == Linear Model == | == Multivariate == |
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| The linear model can be expressed as: | Given ''k'' independent variables, the OLS model is specified as: |
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| {{attachment:model1.svg}} | {{attachment:mmodel.svg}} |
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| If these assumptions can be made: | It is estimated as: {{attachment:mestimate.svg}} More conventionally, this is estimated with [[LinearAlgebra|linear algebra]] as: {{attachment:matrix.svg}} The derivation can be seen [[Statistics/OrdinaryLeastSquares/Multivariate|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: |
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| 2. Exogeneity | 2. [[Statistics/Exogeneity|Exogeneity]] 3. Random sampling 4. No perfect [[LinearAlgebra/Basis|multicolinearity]] 5. [[Statistics/Homoskedasticity|Homoskedasticity]] |
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| {{attachment:model2.svg}} | The variances for each coefficient are: |
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| 3.#3 Random sampling 4. No perfect multicolinearity 5. Heteroskedasticity |
{{attachment:homo1.svg}} |
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| Then OLS is the best linear unbiased estimator ('''BLUE''') for these coefficients. | Note that the standard deviation of the population's parameter is unknown, so it's estimated like: |
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| Using the computation above, the coefficients are estimated to produce: | {{attachment:homo2.svg}} |
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| {{attachment:model3.svg}} | If the homoskedasticity assumption does not hold, then the estimators for each coefficient are actually: |
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| The variance for each coefficient is estimated as: | {{attachment:hetero1.svg}} |
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| {{attachment:model4.svg}} | Wherein, for example, ''r,,1j,,'' is the residual from regressing ''x,,1,,'' onto ''x,,2,,'', ... ''x,,k,,''. |
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| Where R^2^ is calculated as: | The variances for each coefficient can be estimated with the Eicker-White formula: |
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| {{attachment:model5.svg}} | {{attachment:hetero2.svg}} |
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| Note also that the standard deviation of the population's parameter is unknown, so it's estimated like: {{attachment:model6.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.