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x3p9d - EXERCISE 3.9 3.9 e Demonstrate that is equal to...

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3.9 Demonstrate that is equal to zero in multiple regression analysis. ( Note : The proof is a generalization of the proof for the simple regression model, given in Section 1.7.) 1 EXERCISE 3.9 e

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2 i ki k i i u X X Y + + + + = β ... 2 2 1 ki k i i X b X b b Y + + + = ... ˆ 2 2 1 We will consider the general model with k – 1 explanatory variables. The true relationship and the fitted equation are shown. The first subscript on the x variables identifies the variable and the second the observation. EXERCISE 3.9
3 ki k i i i i i X b X b b Y Y Y e - - - - = - = ... ˆ 2 2 1 i ki k i i u X X Y + + + + = β ... 2 2 1 ki k i i X b X b b Y + + + = ... ˆ 2 2 1 The residual in observation i is the difference between the actual value of Y and the fitted value. EXERCISE 3.9

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4 ki k i i i i i X b X b b Y Y Y e - - - - = - = ... ˆ 2 2 1 - - - - = ki k i i i X b X b nb Y e ... 2 2 1 i ki k i i u X X Y + + + + = β ... 2 2 1 ki k i i X b X b b Y + + + = ... ˆ 2 2 1 We sum the residuals over the n observations. EXERCISE 3.9
5 ki k i i i i i X b X b b Y Y Y e - - - - = - = ...

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x3p9d - EXERCISE 3.9 3.9 e Demonstrate that is equal to...

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