# 02_09_2016 - Stat 151A Notes Lecture 7 Jeff Weng Why OLS is...

• Notes
• 2
• 100% (1) 1 out of 1 people found this document helpful

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

Stat 151A Notes Lecture 7 (02/9/16) Jeff Weng Why OLS is BLUE (Proof + Interpretation) Gauss-Markov theorem states that in a linear regression model in which errors have expec- tation zero and are uncorrelated and have equal variance, the best linear unbiased estimator (BLUE) of the coefficients is given by OLS (best means lowest variance of estimate). Assumptions: E ( i ) = 0 V ar ( i ) = σ 2 < Cov ( i , j ) = 0 , i 6 = j Model: y = + , ( y, R n , β R k , X R n × k ) X is not random, but is observable. β is not random and not observable. is random, y is random. BLUE stands for Best Linear Unbiased Estimator. Show that OLS is BLUE: E ( ˆ β ) = β which holds for OLS. ˆ β = ( X T X ) - 1 X T y, ˆ β is not linear in x, but in y. Also y is random variable so ˆ β and thus ˆ β is linear in Y. Let ˜ B = Cy, C = ( X T X ) - 1 X T + D, where D is k × n non-zero matrix. E [ Cy ] = E [((( X T X ) - 1 X T + D )( + )] = (( X T X ) - 1 X T + D ) + (( X ( X T ) - 1 X T + D ) E [ ] = β + DXβ = 0 V ar [ ˜ B ] = V ar [ Cy ] = CV ar [ y ] C T = σ 2 CC T = σ 2 (( X T X ) - 1 + D )( X ( X T X ) - 1 + D T ) = σ 2 (( X T X ) - 1 X T X ( X T X ) - 1 ) + ( X T X ) - 1 X T D T + DX ( X

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

This is the end of the preview. Sign up to access the rest of the document.
• Spring '14
• BEAN,DM

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern