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briefnts11_regrs

# briefnts11_regrs - Brief Notes#11 Linear Regression(a...

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Brief Notes #11 Linear Regression (a) Simple Linear Regression Model: Y = β 0 + β 1 g(X) + ε , with ε ~ (0, σ 2 ) Y = β 0 + β 1 X + ε Data: (X i , Y i ), with i = 1, … , n Y i = β 0 + β 1 X i + ε i x y x y x i y i i y ˆ ) y ˆ y ( i i ) y y ˆ ( i ) y y ( i X ˆ ˆ Y ˆ 1 0 β + β = Least Squares Estimation of ( β 0 , β 1 ) ˆ 2 ˆ Find ( β ˆ , β ˆ ) such that (Y Y ) = min , where Y i β = ˆ 0 β + ˆ 1 X . 0 1 i i i i

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Solution S β ˆ 0 = Y β ˆ 1 X = Y S XY X , β ˆ 1 = S XY , XX S XX 1 where X = X i , Y = 1 Y i n i n i S XX = (X X) 2 , S XY = (X Y )( X Y) . i i i i i Properties of β ˆ ˆ 0 for ε i ~ iid N(0, σ 2 ) β 1 ⎡⎛ 1 X 2 ⎜− X ⎞⎤ β ˆ 0 ⎜⎡β 0 , σ 2 n + S XX S XX ˆ N ~ β 1 X 1 β 1 ⎜− ⎣ ⎝ S XX S XX ˆ
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briefnts11_regrs - Brief Notes#11 Linear Regression(a...

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