hw3 - solution b = ( X X )-1 X Y yields the values of b and...

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Stat 704, Fall 2011: Homework 3 Due Tuesday Sept. 20 Regression through the origin, again : Let Y i = x i τ + ± i , where E ( ± i ) = 0 and var( ± i ) = σ 2 . (a) Write the model as Y = X τ + ± , defining each matrix/vector. (b) Show that ˆ τ = ( X 0 X ) - 1 X 0 Y = n i =1 x i Y i n i =1 x 2 i . A 1 × 1 matrix is just a number. (c) Show var(ˆ τ ) = σ 2 ( X 0 X ) - 1 = σ 2 n i =1 x 2 i . 5.1 5.5, 5.13, 5.15 5.17 Hint : If cov( Y ) = Σ then cov( AY ) = AΣA 0 . Extra credit The normal equations for simple linear regression can be written in matrix terms ± n x i x i x 2 i ²± b 0 b 1 ² = ± ∑ Y i x i Y i ² Let X = 1 x i 1 x 2 . . . . . . 1 x n , b = ± b 0 b 1 ² , Y = Y 1 Y 2 . . . Y n . (a) Verify that the normal equations are equivalent to [ X 0 X ] b = X 0 Y and that the
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Unformatted text preview: solution b = ( X X )-1 X Y yields the values of b and b 1 discussed in class and in the book. (b) According to results from linear models, the variance of b is the upper left element of 2 ( X X )-1 , the variance of b 1 is the lower right, and the covariance between b and b 1 is given by the o-diagonal. Verify that 2 ( X X )-1 gives the correct values (i.e. in the text or notes) for the two variance terms. (c) Are b and b 1 independent? Why or why not? 1...
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