Econometrics-I-6 - Applied Econometrics William Greene...

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Applied Econometrics William Greene Department of Economics Stern School of Business
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Applied Econometrics 6. Finite Sample Properties of the Least Squares Estimator
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Terms of Art Estimates and estimators Properties of an estimator - the sampling  distribution “Finite sample” properties as opposed to  “asymptotic” or “large sample” properties
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The Statistical Context of Least Squares Estimation The sample of data from the population The stochastic specification of the regression  model Endowment of the stochastic properties of the  model upon the least squares estimator 
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Least Squares - - - - - = - = - = = = + = + ε + ε + ε 1 1 1 n 1 1 i i i 1 n 1 i i i 1 n 1 i i i 1 n i i i 1 ( )    = ( ) ( ) Also ( ) = ( ) y    ( )    = ( )    =    (Influence functions) b X'X X'y X'X X'(X + ) = X'X X' b X'X X'y X'X x = X'X x X'X x v β ε β ε β β β
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Deriving the Properties So,  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  in  hand, so the result must be general – i.e.,  independent of  X
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Properties of the LS Estimator Expected value and the property of unbiasedness.  E[ b|X ] =  β  = E[ b ].   Prove this result. A Crucial Result About Specification:      y   =   X 1 β 1  +  X 2 β 2  +  ε 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?       b 1   =  ( X 1 X 1 ) -1 X 1 y
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The Left Out Variable Formula (This is a VVIR!)       E[ b 1 ]  =   β 1   +  ( X 1 X 1 ) -1 X 1 X 2 β 2 The (truly) short regression estimator is biased. Application:       Quantity  =   β 1 Price  +   β 2 Income  +   ε If you regress Quantity on Price and leave out  Income.  What do you get?  (Application below)
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The Extra Variable Formula A Second Crucial Result About Specification:  =   X 1 β 1  +  X 2 β 2  +  ε   but  β 2  really is 0. Two sets of variables.  One is superfluous.  What if the 
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This note was uploaded on 06/20/2011 for the course ECON 803 taught by Professor Pp during the Spring '11 term at Thammasat University.

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Econometrics-I-6 - Applied Econometrics William Greene...

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