ols_derivation

ols_derivation - 1 Derivation of OLS estimators 1.1 Simple...

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1 Derivation of OLS estimators 1.1 Simple Regression: Two Variable Model The Ordinary Least Squares (OLS) technique involves finding parameter estimates by minimizing the sum of square errors, or, what is the same thing, minimizing the sum of square residuals (SSR) or ∑ n i =1 ( Y i- ˆ Y i ) 2 , where ˆ Y i = ˆ β 1 + ˆ β 2 X i is the fitted value of Y i corresponding to a particular observation X i . We minimize the SSR by taking the partial derivatives with respect to ˆ β 1 and ˆ β 2 , setting each equal to 0, and solving the resulting pair of simultaneous equations. δ δ ˆ β 1 n X i =1 ( Y i- ˆ β 1- ˆ β 2 X i ) 2 =- 2 n X i =1 ( Y i- ˆ β 1- ˆ β 2 X i ) (1) δ δ ˆ β 2 n X i =1 ( Y i- ˆ β 1- ˆ β 2 X i ) 2 =- 2 n X i =1 X i ( Y i- ˆ β 1- ˆ β 2 X i ) (2) Equating these derivatives to zero and dividing by- 2 we get n X i =1 ( Y i- ˆ β 1- ˆ β 2 X i ) = 0 (3) n X i =1 X i ( Y i- ˆ β 1- ˆ β 2 X i ) = 0 (4) Finally, rewriting eqns. 3 and 4 we obtain a pair of simultaneous equations (known as the normal equations ): n X i =1 Y i = n ˆ β 1 + ˆ β 2 n X i =1 X i (5) n X i =1 X i Y i = ˆ β 1 n X i =1 X i + ˆ...
View Full Document

This note was uploaded on 03/14/2010 for the course ECON econmetric taught by Professor Yy during the Spring '10 term at Seoul National.

Page1 / 3

ols_derivation - 1 Derivation of OLS estimators 1.1 Simple...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online