MLR 1 - 1 Introduction n i X X X Y i i p p i i i 1 2 2 1 1...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 3/18/2009 1 Introduction n i X X X Y i i p p i i i ,..., 1 , ... , , 2 2 , 1 1 = + + + + + = ε β β β β ` ε i are random error terms assumed to be identically and independently distributed Normal with mean 0 and variance ` “Linear” means that the equation is linear in the parameters β , β 1 , …, β p ` The X j i are assumed to be fixed and known. 2 ε σ j,i ◦ They can be continuous, discrete, or categorical ` Y is random and continuous ` Observations are independent of each other 3/18/2009 2 Y i is Normally Distributed with ` Mean = ` Variance = = = variance of ε 2 i Y σ 2 ε σ n i X X X i p p i i i ,..., 1 , ... , , 2 2 , 1 1 = + + + + = β β β β μ ` Typically use least squares to obtain parameter estimates ` The equations are more complicated but the approaches are identical – tests are based on the t-statistic and CI s are developed as usual. 3/18/2009 3 X X β 1 ε X Y β ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = − − − − n n np n p n n p p n n p X X X X X X X X Y Y Y Y ε ε ε ε β β β 1 2 1 1 , 1 1 , 1 2 21 1 11 1 2 1 1 , 1 1 1 1 , , M L M L O M M L L M M ε X β Y + = ⎦ ⎣ ⎦ ⎣ ˆ ˆ ˆ ˆ 2 2 1 1 2 1 2 1 1 ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ − − ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ ⎥ ⎥ ⎤ ⎢ ⎢ ⎡ Y Y Y Y e e Y Y b b HY Y X X X X Xb Y e Y b 1 ' ' ˆ ˆ ˆ , ˆ ˆ ˆ , 1 1 1 1 1 = = = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ − − = ⎥ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎢ ⎣ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ = − − − − − Y Y Y Y e e Y Y b n n n n n n n n p M M M M Y H I Y Y X b Y Y Y X X X b HY Y X X X X Xb Y 1 ) ( ' ' ' ' ' ) ' ( ) ( − = − = = = = = − SSE 3/18/2009 4 Parameter estimators, , are Normally i β ˆ distributed with ` Means = ` Variance = or i i β μ β = ˆ ) , ,..., ( 2 1 2 ˆ ε β σ σ p X X f i = equivalently 1 2 ˆ ) ' ( − = X X Σ β ε σ ` The predicted values, X X X...
View Full Document

This note was uploaded on 07/22/2011 for the course STA 4702 taught by Professor Staff during the Spring '08 term at University of Florida.

Page1 / 12

MLR 1 - 1 Introduction n i X X X Y i i p p i i i 1 2 2 1 1...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online