Econometrics-I-7

# Econometrics-I-7 - Econometrics I Professor William Greene...

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Part 7: Estimating the Variance of  b Econometrics I Professor William Greene Stern School of Business Department of Economics

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Part 7: Estimating the Variance of  b Econometrics I Part 7 – Estimating  the Variance of b ™    1/35
Part 7: Estimating the Variance of  b Context The true variance of b|X is 2 (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. ™    2/35

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Part 7: Estimating the Variance of  b Estimating 2 Using the residuals instead of the disturbances: The natural estimator: ee /n as a sample surrogate for / n Imperfect observation of i = ei + (  - b ) x i Downward bias of ee /n. We obtain the result E[ ee|X ] = (n-K)2 ™    3/35
Part 7: Estimating the Variance of  b Expectation of ee - - = = - = - = = + = β + = = β ε ε ε ε 29 ε 29 = ε ε = ε ε = ε ε 1 1       ( ' ) '       [ ( ' ) ']       ( ) ( '( 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 ™    4/35

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Part 7: Estimating the Variance of  b Method 1: E[ ] 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 = εε = εε = σ σ σ 2 2 2 [ trace  E ( ) ] conditioned on X [ trace   ] model assumption [trace  ] scalar multiplication and   matrix trace [  -  ( ) ] -1 '| X I I I X X'X X'  εε = = = σ σ σ σ 2 2 2 2 { trace [ ] - trace[ ( ) ]} { n - trace[( ) ]}  permute in trace { n - trace[ ]} { n - K} Notice that E[ | ] is not a -1 -1 I e X X'X X'  X'X X'X  e I X = = = =  function of  . X ™    5/35
Part 7: Estimating the Variance of  b Estimating σ2 The unbiased estimator is s2 = ee /(n-K). “Degrees of freedom correction” Therefore, the unbiased estimator of 2 is s2 = ee /(n-K) ™    6/35

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Part 7: Estimating the Variance of  b 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 ™    7/35
Part 7: Estimating the Variance of  b Decomposing M 2 2 2 2 Useful Result:  If   =   ' is the spectral decomposition, then  '  (just multiply)  =   ,  so  .  All of the characteristic roots of   are 1 or 0.  How many of each?

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