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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...
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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.
- Spring '10