Definition of OLS / Ordinary Least Squares

Definition: Definition of OLS / Ordinary Least Squares: OLS stands for Ordinary Least Squares, the standard linear regression procedure. One estimates a parameter from data and applying the linear model

y = Xb + e

where y is the dependent variable or vector, X is a matrix of independent variables, b is a vector of parameters to be estimated, and e is a vector of errors with mean zero that make the equations equal.

The estimator of b is: (X'X)-1X'y

A common derivation of this estimator from the model equation (1) is:

y = Xb + e

Multiply through by X'. X'y = X'Xb + X'e

Now take expectations. Since the e's are assumed to be uncorrelated to the X's the last term is zero, so that term drops. So now:

E[X'Xb] = E[X'y]

Now multiply through by (X'X)-1

E[(X'X)-1X'Xb] = E[(X'X)-1X'y]

E = E[(X'X)-1X'y]

Since the X's and y's are data the estimate of b can be calculated.(Econterms)

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Econterms. "Definition of OLS / Ordinary Least Squares." ThoughtCo, Jun. 16, 2014, thoughtco.com/ols-or-ordinary-least-squares-definition-1146178. Econterms. (2014, June 16). Definition of OLS / Ordinary Least Squares. Retrieved from https://www.thoughtco.com/ols-or-ordinary-least-squares-definition-1146178 Econterms. "Definition of OLS / Ordinary Least Squares." ThoughtCo. https://www.thoughtco.com/ols-or-ordinary-least-squares-definition-1146178 (accessed November 22, 2017).