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

Info icon This preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon
Part 7: Estimating the Variance of  b Econometrics I Professor William Greene Stern School of Business Department of Economics
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Part 7: Estimating the Variance of  b Econometrics I Part 7 – Estimating  the Variance of b ™    1/35
Image of page 2
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
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 4
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
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 6
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
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 8
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?
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern