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
- Exogeneity, i.e. predictors are independent of the outcome and the error term
- Random sampling
No perfect multicolinearity
- Homoskedasticity, i.e. error terms are constant across observations
#5 mostly comes into the estimation of standard errors, and there are alternative estimators that are robust to heteroskedasticity.