PARTIMOD - 3. THE PARTITIONED REGRESSION MODEL Consider...

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3. THE PARTITIONED REGRESSION MODEL Consider taking a regression equation in the form of (1) y =[ X 1 X 2 ] · β 1 β 2 ¸ + ε = X 1 β 1 + X 2 β 2 + ε. Here, [ X 1 ,X 2 ]= X and [ β 0 1 0 2 ] 0 = β are obtained by partitioning the matrix X and vector β of the equation y = + ε in a conformable manner. The normal equations X 0 = X 0 y can be partitioned likewise. Writing the equations without the surrounding matrix braces gives X 0 1 X 1 β 1 + X 0 1 X 2 β 2 = X 0 1 y, (2) X 0 2 X 1 β 1 + X 0 2 X 2 β 2 = X 0 2 y. (3) From (2), we get the equation X 0 1 X 1 β 1 = X 0 1 ( y X 2 β 2 ), which gives an expres- sion for the leading subvector of ˆ β : (4) ˆ β 1 =( X 0 1 X 1 ) 1 X 0 1 ( y X 2 ˆ β 2 ) . To obtain an expression for ˆ β 2 , we must eliminate β 1 from equation (3). For this purpose, we multiply equation (2) by X 0 2 X 1 ( X 0 1 X 1 ) 1 to give (5) X 0 2 X 1 β 1 + X 0 2 X 1 ( X 0 1 X 1 ) 1 X 0 1 X 2 β 2 = X 0 2 X 1 ( X 0 1 X 1 ) 1 X 0 1 When the latter is taken from equation (3), we get (6) n X 0 2 X 2 X 0 2 X 1 ( X 0 1 X 1 ) 1 X 0 1 X 2 o β 2 = X 0 2 y X 0 2 X 1 ( X 0 1 X 1 ) 1 X 0 1 On de±ning (7) P 1 = X 1 ( X 0 1 X 1 ) 1 X 0 1 , can we rewrite (6) as (8) n X 0 2 ( I P 1 ) X 2 o β 2 = X 0 2 ( I P 1 ) whence (9) ˆ β 2 = n X 0 2 ( I P 1 ) X 2 o 1 X 0 2 ( I P 1 ) 1
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TOPICS IN ECONOMETRICS Now let us investigate the efect that conditions oF orthogonality amongst the regressors have upon the ordinary least-squares estimates oF the regression parameters. Consider a partitioned regression model, which can be written as (10) y =[ X 1 ,X 2 ] · β 1 β 2 ¸ + ε = X 1 β 1 + X 2 β 2 + ε. It can be assumed that the variables in this equation are in deviation Form. Imagine that the columns oF X 1 are orthogonal to the columns oF X 2 such that X 0 1 X 2 = 0. This is the same as assuming that the empirical correlation between variables in X 1 and variables in X 2 is zero.
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PARTIMOD - 3. THE PARTITIONED REGRESSION MODEL Consider...

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