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 proof can be seen here.

Multivariate

Linear Model

The linear model can be expressed as:

If these assumptions can be made:

1. Linearity

2. Exogeneity

3. Random sampling
4. No perfect multicolinearity

5. Homoskedasticity

Then OLS is the best linear unbiased estimator (BLUE) for these coefficients.

Using the computation above, the coefficients are estimated to produce:

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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