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

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

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