Econometrics-I-6

Also y b xx xy xx xx xx x b b xx xy xx x xx x xx x xx

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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

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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[ b|X ] = E[ + ( XX ) -1X|X ] = + ( XX ) -1X E[ |X ] = + 0 E[ b ] = E X {E[ b | X ]} = E[ b ]. (The law of iterated expectations.) ™    10/34

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Part 6: Finite Sample Properties of LS Sampling Experiment ™    11/34
Part 6: Finite Sample Properties of LS Means of Repetitions b|x ™    12/34

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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 = ( X1X1 )-1 X1y ™    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 0 (This is a VVIR!) b 1 = ( X1X1 )-1 X1y = ( X1X1 )-1 X1 ( X 1 1 + X 2 2 + ) = ( X1X1 )-1 X1X 1 1 + ( X1X1 )-1 X1 X 2 2 + ( X1X1 )-1 X1 ) E[ b 1] = 1 + ( X1X1 )-1 X1X 2 2 ™    14/34

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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? In time series data, 1 < 0, 2 > 0 (usually) Cov[ Price , Income ] > 0 in time series data. So, the short regression will overestimate the price coefficient. It will be pulled toward and even past zero. Simple Regression of G on a constant and PG Price Coefficient should be negative. ÷ 1 1 2 Cov[Price,Income] E[b ] =β + β Var[Price] ™    16/34

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Part 6: Finite Sample Properties of LS Estimated ‘Demand’ Equation Shouldn’t the Price Coefficient be Negative? ™    17/34
Part 6: Finite Sample Properties of LS Multiple Regression of G on Y and PG. The Theory Works!

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