Page 109 - Basic Statistics

104

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Vector Y is a linear function of Y, with coefficients H = [ X ( X’X ) X’ ]. The
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matrix H called "hat matrix" is a matrix which is determined by the matrix X.

This matrix is a very important role in the regression analysis. The matrix H is
symmetric and idempotent matrix, ie H’ = H dan H H = H.

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e = Y - Y .
= Y - [ X ( X’X ) X’] Y
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= Y - H Y

= [I - H] Y (G.7)

The vector e is a linear function of Y, with coefficients [I - H]. Teh matrix [I - H]
is also symmetric and idempotent matrix.

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5.2.4 DISTRIBUTION OF STATISTICS β , Y , AND e

If the model is correct, then the expected value of Y is X. Since statistics,
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β , Y , dan e is a linear function of Y, and Y is a random vector is known, then
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β , Y , dan e are also random vectors. The properties of each statistic as a linear
function of Y can be expressed as follows.

a. The Expectation

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E ( β ) = E( ( X’X ) X’Y )
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=[( X’X ) X’ ] E ( Y )
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= [( X’X ) X’] (X)
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= [( X’X ) ( X’X)]
= 

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This show that β is unbiased estimate of , if the chosen model is correct. If the
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chosen model is not correct, then the ( X’X ) ( X’X)  I and E ( β ) does not
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simplify to as in equation above.

~~* CHAPTER 5 THE MULTIPLE LINEAR REGRESSION MODEL *~~