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Unformatted text preview: ˜™™™ ™ 7/34 Part 6: Finite Sample Properties of LS Least Squares = = = = + = + = + ε + ε + ε ∑ ∑ ∑ β ε β ε =β ε β β β 1 1 1 1 n 1 i i i 1 1 n 1 i i i 1 n 1 i i i 1 i i i ( ) = ( ) ( ) = The true parameter plus sampling error. Also ( ) = ( ) y ( ) ( ) = ( ) = b X'X X'y X'X X'(X + ) = X'X X' b b X'X X'y X'X x X'X X' X'X x X'X x v = ∑ n 1 = The true parameter plus a linear function of the disturbances. b ˜™™™ ™ 8/34 Part 6: Finite Sample Properties of LS Deriving the Properties b = a parameter vector + a linear combination of the disturbances, each times a vector. Therefore, b is a vector of random variables. We analyze it as such. The assumption of nonstochastic regressors. How it is used at this point. We do the analysis conditional on an X , then show that results do not depend on the particular X in hand, so the result must be general – i.e., independent of X . ˜˜™™™ ™ 9/34 Part 6: Finite Sample Properties of LS Properties of the LS Estimator: b is unbiased Expected value and the property of unbiasedness. E[ bX ] = E[ + ( XX )1XX ] = + ( XX )1X E[ X ] = + E[ b ] = E X {E[ b  X ]} = E[ b ]. (The law of iterated expectations.) ˜˜™™™ ™ 10/34 Part 6: Finite Sample Properties of LS Sampling Experiment ˜˜™™™ ™ 11/34 Part 6: Finite Sample Properties of LS Means of Repetitions bx ˜˜™™™ ™ 12/34 Part 6: Finite Sample Properties of LS Partitioned Regression A Crucial Result About Specification: y = X 1 1 + X 2 2 + Two sets of variables. What if the regression is computed without the second set of variables? What is the expectation of the "short" regression estimator? E[ b 1( y = X 1 1 + X 2 2 + )] b 1 = ( X1X1 )1 X1y ˜˜™™™ ™ 13/34 Part 6: Finite Sample Properties of LS The Left Out Variable Formula “Short” regression means we regress y on X 1 when y = X 1 1 + X 2 2 + and 2 is not (This is a VVIR!) b 1 = ( X1X1 )1 X1y = ( X1X1 )1 X1 ( X 1 1 + X 2 2 + ) = ( X1X1 )1 X1X 1 1 + ( X1X1 )1 X1 X 2 2 + ( X1X1 )1 X1 ) E[ b 1] = 1 + ( X1X1 )1 X1X 2 2 ˜˜™™ ™ 14/34 Part 6: Finite Sample Properties of LS Application The (truly) short regression estimator is biased. Application: Quantity = 1Price + 2Income + If you regress Quantity on Price and leave out Income. What do you get? ˜˜™™ ™ 15/34 Part 6: Finite Sample Properties of LS Application: Left out Variable Leave out Income. What do you get?...
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 Fall '10
 H.Bierens
 Econometrics, Least Squares, Standard Deviation, Variance, Mean squared error, Bias of an estimator

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