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:

1. Linearity

2. Exogeneity

3. Random sampling

4. No perfect multicolinearity

5. 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.

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