|
Size: 1653
Comment: Renamed attachments
|
← Revision 24 as of 2025-01-10 14:33:38 ⇥
Size: 2156
Comment: Killing Econometrics page
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 13: | Line 13: |
| The regression line passes through two points: | Given one independent variable and one dependent (outcome) variable, the OLS model is specified as: |
| Line 15: | Line 15: |
| {{attachment:regression1.svg}} | {{attachment:model.svg}} |
| Line 17: | Line 17: |
| and | It is estimated as: |
| Line 19: | Line 19: |
| {{attachment:regression2.svg}} | {{attachment:estimate.svg}} |
| Line 21: | Line 21: |
| These points, with the generic equation for a line, can [[Econometrics/OrdinaryLeastSquares/UnivariateProof|prove]] that the slope of the regression line is equal to: | This model describes (1) the mean observation and (2) the marginal changes to the outcome per unit changes in the independent variable. |
| Line 23: | Line 23: |
| {{attachment:b12.svg}} The generic formula for the regression line is: {{attachment:b13.svg}} |
The derivation can be seen [[Statistics/OrdinaryLeastSquares/Univariate|here]]. |
| Line 33: | Line 29: |
| == Linear Model == | == Multivariate == |
| Line 35: | Line 31: |
| The linear model can be expressed as: | Given ''k'' independent variables, the OLS model is specified as: |
| Line 37: | Line 33: |
| {{attachment:model1.svg}} | {{attachment:mmodel.svg}} |
| Line 39: | Line 35: |
| 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: |
| Line 42: | Line 54: |
| 2. [[Econometrics/Exogeneity|Exogeneity]] | 2. [[Statistics/Exogeneity|Exogeneity]] |
| Line 44: | Line 56: |
| 4. No perfect multicolinearity 5. [[Econometrics/Homoskedasticity|Homoskedasticity]] Then OLS is the best linear unbiased estimator ('''BLUE''') for these coefficients. Using the computation above, the coefficients are estimated to produce: {{attachment:model2.svg}} |
4. No perfect [[LinearAlgebra/Basis|multicolinearity]] 5. [[Statistics/Homoskedasticity|Homoskedasticity]] |
| Line 65: | Line 71: |
| It follows that the variances for each coefficient are: | 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: |
| Line 69: | Line 77: |
| These variances can be estimated with the Eicker-White formula: {{attachment:hetero3.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.