Econometrics-I-6

? ? ? ? ε ε ε x x i o 2 2 | e var = ? ? ? ? = σ

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Unformatted text preview: ÷ ÷ ÷ ÷ ε ε ε X X I O { } 2 2 | E Var = . ... ÷ ÷ ÷ ÷ = σ + σ ÷ ÷ X I I ˜˜˜™™ ™ 21/34 Part 6: Finite Sample Properties of LS Variance of the Least Squares Estimator-------- = β ε + ε ε ε = =-- εε σ β β + β β β 1 1 1 1 1 1 1 2 1 ( ) = ( ) ( ) ( ) E[ | ]= ( ) [ | ] as [ | ] Var[ | ] E[( )( ) '| ] ( ) [ '| ] ( ) ( ) ( ) b X'X X'y X'X X' X + = X'X X' b X X'X X'E X = E X b X b b X = X'X X'E X X X'X = X'X X' I X X'X ----- σ σ σ 2 1 1 2 1 1 2 1 ( ) ( ) ( ) ( ) ( ) = X'X X'I X X'X = X'X X'X X'X = X'X ˜˜˜™ ™ 22/34 Part 6: Finite Sample Properties of LS Variance of the Least Squares Estimator---- = = σ σ σ β β 1 2 1 2 1 2 1 ( ) E[ | ] = Var[ | ] ( ) Var[ ] = E{Var[ | ] } + Var{E[ | ]} = E[( ) ] + Var{ } = E[( ) ] + We will ultimately ne b X'X X'y b X b X X'X b b X b X X'X X'X- 1 ed to estimate E[( ) ]. We will use the only information we have, , itself. X'X X ˜˜˜™ ™ 23/34 Part 6: Finite Sample Properties of LS Specification Errors-1 Omitting relevant variables: Suppose the correct model is y = X 1 1 + X 2 2 + . I.e., two sets of variables. Compute least squares omitting X 2. Some easily proved results: Var[ b 1] is smaller than Var[ b 1.2]. (The latter is the northwest submatrix of the full covariance matrix. The proof uses the residual maker (again!). I.e., you get a smaller variance when you omit X 2. (One interpretation: Omitting X 2 amounts to using extra information ( 2 = ). Even if the information is wrong (see the next result), it reduces the variance. (This is an important result.) ˜˜˜™ ™ 24/34 Part 6: Finite Sample Properties of LS Omitted Variables (No free lunch) E[ b 1] = 1 + ( X1X1 )- 1X1X2 2 1. So, b 1 is biased .(!!!) The bias can be huge. Can reverse the sign of a price coefficient in a “demand equation.” b 1 may be more “precise.” Precision = Mean squared error = variance + squared bias....
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ε ε ε X X I O 2 2 | E Var = = σ σ ...

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