The effect upon the ordinary least squares estimator

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The effect upon the ordinary least-squares estimator can be seen by exam- ining the partitioned form of the formula ˆ β = ( X X ) 1 X y . Here, we have (11) X X = X 1 X 2 [ X 1 X 2 ] = X 1 X 1 X 1 X 2 X 2 X 1 X 2 X 2 = X 1 X 1 0 0 X 2 X 2 , where the final equality follows from the condition of orthogonality. The inverse of the partitioned form of X X in the case of X 1 X 2 = 0 is (12) ( X X ) 1 = X 1 X 1 0 0 X 2 X 2 1 = ( X 1 X 1 ) 1 0 0 ( X 2 X 2 ) 1 . We also have (13) X y = X 1 X 2 y = X 1 y X 2 y . On combining these elements, we find that (14) ˆ β 1 ˆ β 2 = ( X 1 X 1 ) 1 0 0 ( X 2 X 2 ) 1 X 1 y X 2 y = ( X 1 X 1 ) 1 X 1 y ( X 2 X 2 ) 1 X 2 y . In this special case, the coefficients of the regression of y on X = [ X 1 , X 2 ] can be obtained from the separate regressions of y on X 1 and y on X 2 . It should be understood that this result does not hold true in general. The general formulae for ˆ β 1 and ˆ β 2 are those which we have given already under (4) and (9): (15) ˆ β 1 = ( X 1 X 1 ) 1 X 1 ( y X 2 ˆ β 2 ) , ˆ β 2 = X 2 ( I P 1 ) X 2 1 X 2 ( I P 1 ) y, P 1 = X 1 ( X 1 X 1 ) 1 X 1 . 2
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THE PARTITIONED REGRESSION MODEL It can be confirmed easily that these formulae do specialise to those under (14) in the case of X 1 X 2 = 0. The purpose of including X 2 in the regression equation when, in fact, interest is confined to the parameters of β 1 is to avoid falsely attributing the explanatory power of the variables of X 2 to those of X 1 . Let us investigate the effects of erroneously excluding X 2 from the regres- sion. In that case, the estimate will be (16) ˜ β 1 = ( X 1 X 1 ) 1 X 1 y = ( X 1 X 1 ) 1 X 1 ( X 1 β 1 + X 2 β 2 + ε ) = β 1 + ( X 1 X 1 ) 1 X 1 X 2 β 2 + ( X 1 X 1 ) 1 X 1 ε.
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