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Econometrics-I-7 - 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 7. Estimating the Variance of the Least Squares Estimator
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Context     The true variance of  b  is  σ 2 E [(X X) -1 ]     We  consider how to use the sample data to  estimate this matrix.  The ultimate objectives  are to form interval estimates for regression  slopes and to test hypotheses about them.   Both require estimates of the variability of the  distribution.  We then examine a factor which  affects how "large" this variance is,  multicollinearity.
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Estimating σ 2 Using the residuals instead of the disturbances:   The natural estimator:   e e / n  as a sample  surrogate for  ε′ ε / n Imperfect observation of  ε    =  e i   + ( β  - b ) x i   Downward bias of  e e / n .  We obtain the result  E[ e e ]  =  (n-K) σ 2
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Expectation of e’e 1 1   ( ' ) '   [ ( ' ) ']   ( ) ( '( - - = = - = - = = + = β + = = e y - Xb y X X X X y I X X X X y My M X MX M M e'e M M 'M'M 'MM 'M β ε ε ε ε29 ε 29 = ε ε = ε ε = ε ε
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Method 1: E[ ] E[ trace ( ) ] scalar =  its trace E[ trace ( ) ] permute in trace [ trace E ( ) ] linear operators e'e| X 'M | X 'M | X M '| X M '| X = [ε ε ] = ε ε = εε = εε = 2 2 2 [ trace  E ( ) ] conditioned on X [ trace   ] model assumption [trace  ] scalar multiplication and   matrix trace [  -  ( ) ] σ σ σ -1 '| X I I I X X'X X'  εε = = = 2 2 2 2 { trace [ ] - trace[ ( ) ]} { n - trace[( ) ]}  permute in trace { n - trace[ ]} { n - K} σ σ σ σ -1 -1 I X X'X X'  X'X X'X  I = = = =
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Estimating σ 2 The unbiased estimator is s 2  =  e e /( n - K ).   “Degrees of freedom.” Therefore, the  unbiased  estimator is  s 2  =  e e /(n-K) =   ε′ M ε /(n-K).
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Method 2: Some Matrix Algebra 2 E[ ]  trace  What is the trace of   is idempotent, so its  trace equals its rank.  Its rank equals the number of nonzero characeristic roots. Characteric Roots : Signature of a Matrix = Spectral  σ e'e| X M M = Decomposition                  = Eigen (own) value Decomposition   =   ' where        = a matrix of columns such that  ' =  '  =       = a diagonal matrix of the characteristic roots           element A C C C CC C C I Λ Λ s of   may be zero Λ
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Decomposing M 2 2 2 2 Useful Result:  If   =   ' is the spectral decomposition, then  '  (just multiply)
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