Econometrics-I-6

# Ordinary least squares regression lhsg mean 22609444

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---------------------------------------------------------------------- Ordinary least squares regression ............ LHS=G Mean = 226.09444 Standard deviation = 50.59182 Number of observs. = 36 Model size Parameters = 3 Degrees of freedom = 33 Residuals Sum of squares = 1472.79834 Standard error of e = 6.68059 Fit R-squared = .98356 Adjusted R-squared = .98256 Model test F[ 2, 33] (prob) = 987.1(.0000) --------+------------------------------------------------------------- Variable| Coefficient Standard Error t-ratio P[|T|>t] Mean of X --------+------------------------------------------------------------- Constant| -79.7535*** 8.67255 -9.196 .0000 Y| .03692*** .00132 28.022 .0000 9232.86 PG| -15.1224*** 1.88034 -8.042 .0000 2.31661 --------+------------------------------------------------------------- ™    18/34

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Part 6: Finite Sample Properties of LS The Extra Variable Formula A Second Crucial Result About Specification: y = X 1 1 + X 2 2 + but 2 really is 0 . Two sets of variables. One is superfluous. What if the regression is computed with it anyway? The Extra Variable Formula: (This is a VIR!) E[ b 1.2| 2 = 0 ] = 1 The long regression estimator in a short regression is unbiased.) Extra variables in a model do not induce biases . Why not just include them? ™    19/34
Part 6: Finite Sample Properties of LS Variance of b p Assumption about disturbances: p i has zero mean and is uncorrelated with every other j p Var[i| X ] = 2. The variance of i does not depend on any data in the sample. 2 1 2 2 2 2 n 0 ... 0 0 ... 0 Var | ... 0 0 0 0 0 ... ε σ ÷ ε σ ÷ = = σ ÷ ÷ ε σ X I O ™    20/34

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Part 6: Finite Sample Properties of LS 2 1 2 2 2 2 n 1 1 1 2 2 2 n n n 0 ... 0 0 ... 0 Var | ... 0 0 0 0 0 ... Var E Var | Var E ... ... ... ε σ ÷ ε σ ÷ = = σ ÷ ÷ ε σ ε ε ε ÷ ÷ ÷ ε ε ε ÷ ÷ ÷ = + ÷ ÷ ÷ ÷ ε ε ε X X I O { } 2 2 | 0 0 E Var = . ... 0 ÷ ÷ ÷ ÷ = 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 0 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

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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
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