Econometrics-I-7

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

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Unformatted text preview: Applied Econometrics William Greene Department of Economics Stern School of Business Applied Econometrics 7. Estimating the Variance of the Least Squares Estimator Context The true variance of b|X is σ 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. 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 i = e i + ( β - b ) ′ x i Downward bias of e ′ e / n . We obtain the result E[ e ′ e|X ] = (n-K) σ 2 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 ε29 = ε ε = ε ε = ε ε 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 M '| X M I M 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 = = = = Estimating σ 2 The unbiased estimator is s 2 = e ′ e /( n- K ). “Degrees of freedom correction” Therefore, the unbiased estimator is s 2 = e ′ e /(n-K) = ε′ M ε /(n-K). 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 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 CC I Λ Λ s of may be zero Λ 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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Econometrics-I-7 - Applied Econometrics William Greene...

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