Distribution is much easier to think about this

This preview shows page 24 - 33 out of 33 pages.

distribution is Much easier to think about this variance under an additional assumption, so Assume Var( u|x 1 , x 2 ,…, x k ) = σ 2 (Homoskedasticity)
Image of page 24

Subscribe to view the full document.

Fall 2008 under Econometrics Prof. Keunkwan Ryu 25 Variance of OLS (cont) Let x stand for ( x 1 , x 2 ,…x k ) Assuming that Var( u | x ) = σ 2 also implies that Var( y | x ) = σ 2 The 4 assumptions for unbiasedness, plus this homoskedasticity assumption are known as the Gauss-Markov assumptions
Image of page 25
Fall 2008 under Econometrics Prof. Keunkwan Ryu 26 Variance of OLS (cont) ( 29 ( 29 ( 29 s ' other all on regressing from the is and where , 1 ˆ s Assumption Markov - Gauss Given the 2 2 2 2 2 x x R R x x SST R SST Var j j j ij j j j j - = - = σ β
Image of page 26

Subscribe to view the full document.

Fall 2008 under Econometrics Prof. Keunkwan Ryu 27 Components of OLS Variances The error variance: a larger σ 2 implies a larger variance for the OLS estimators The total sample variation: a larger SST j implies a smaller variance for the estimators Linear relationships among the independent variables: a larger R j 2 implies a larger variance for the estimators
Image of page 27
Fall 2008 under Econometrics Prof. Keunkwan Ryu 28
Image of page 28

Subscribe to view the full document.

Fall 2008 under Econometrics Prof. Keunkwan Ryu 29 Misspecified Models ( 29 ( 29 ( 29 same the re ' then they ed, uncorrelat are and unless ˆ ~ Thus, ~ that so , ~ ~ ~ model ed misspecifi again the Consider 2 1 1 1 1 2 1 1 1 0 x x Var Var SST Var x y β β σ β β β < = + =
Image of page 29
Fall 2008 under Econometrics Prof. Keunkwan Ryu 30 Misspecified Models (cont) While the variance of the estimator is smaller for the misspecified model, unless β 2 = 0 the misspecified model is biased As the sample size grows, the variance of each estimator shrinks to zero, making the variance difference less important
Image of page 30

Subscribe to view the full document.

Fall 2008 under Econometrics Prof. Keunkwan Ryu 31 Estimating the Error Variance We don’t know what the error variance, σ 2 , is, because we don’t observe the errors, u i What we observe are the residuals, û i We can use the residuals to form an estimate of the error variance
Image of page 31
Fall 2008 under Econometrics Prof. Keunkwan Ryu 32 Error Variance Estimate (cont) ( 29 ( 29 ( 29 ( 29 [ ] 2 1 2 2 2 1 ˆ ˆ thus, 1 ˆ ˆ j j j i R SST se df SSR k n u - = - - = σ β σ df = n – ( k + 1), or df = n k – 1 df (i.e. degrees of freedom) is the (number of observations) – (number of estimated parameters)
Image of page 32

Subscribe to view the full document.

Fall 2008 under Econometrics Prof. Keunkwan Ryu 33 3.5The Gauss-Markov Theorem Given our 5 Gauss-Markov Assumptions it can be shown that OLS is “BLUE” Best Linear Unbiased Estimator Thus, if the assumptions hold, use OLS
Image of page 33

{[ snackBarMessage ]}

Get FREE access by uploading your study materials

Upload your study materials now and get free access to over 25 million documents.

Upload now for FREE access Or pay now for instant access
Christopher Reinemann
"Before using Course Hero my grade was at 78%. By the end of the semester my grade was at 90%. I could not have done it without all the class material I found."
— Christopher R., University of Rhode Island '15, Course Hero Intern

Ask a question for free

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