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Ordinary Least Squares

Ordinary Least Squares (OLS) is a linear regression method. It minimizes root mean square errors.

Contents

Ordinary Least Squares

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
Random sampling
No perfect multicolinearity
Homoskedasticity

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, r_1j is the residual from regressing x₁ onto x₂, ... x_k.

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.

CategoryRicottone

Statistics/OrdinaryLeastSquares (last edited 2025-09-03 02:08:40 by DominicRicottone)

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  Comment: Killing Econometrics page
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+## page was renamed from Econometrics/OrdinaryLeastSquares
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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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+Line 16:
-{{attachment:regression1.svg}}
+{{attachment:model.svg}}
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-and
+It is estimated as:
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+Line 20:
-{{attachment:regression2.svg}}
+{{attachment:estimate.svg}}
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+Line 22:
-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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+Line 24:
-{{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 [[Econometrics/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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+Line 34:
-{{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 [[Econometrics/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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-. Exogeneity
+. [[Econometrics/Exogeneity|Exogeneity]]
 3. Random sampling
 4. No perfect [[LinearAlgebra/Basis|multicolinearity]]
 5. [[Econometrics/Homoskedasticity|Homoskedasticity]]
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+Line 60:
-{{attachment:model2.svg}}
+The variances for each coefficient are:
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-.#3 Random sampling
 4. No perfect multicolinearity
 5. Heteroskedasticity
+{{attachment:homo1.svg}}
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+Line 64:
-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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+Line 66:
-Using the computation above, the coefficients are estimated to produce:
+{{attachment:homo2.svg}}
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+Line 68:
-{{attachment:model3.svg}}
+If the homoskedasticity assumption does not hold, then the estimators for each coefficient are actually:
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+Line 70:
-The variance for each coefficient is estimated as:
+{{attachment:hetero1.svg}}
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+Line 72:
-{{attachment:model4.svg}}
+Wherein, for example, ''r,,1j,,'' is the residual from regressing ''x,,1,,'' onto ''x,,2,,'', ... ''x,,k,,''.
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+Line 74:
-Where R^2^ is calculated as:
+The variances for each coefficient can be estimated with the Eicker-White formula:
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+Line 76:
-{{attachment:model5.svg}}
+{{attachment:hetero2.svg}}
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+Line 78:
-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.

Diff for "Statistics/OrdinaryLeastSquares"

Ordinary Least Squares

Univariate

Multivariate

Estimated Coefficients