Ii use mean square ˆ var n n requires to be well

Info icon This preview shows pages 19–26. Sign up to view the full content.

View Full Document Right Arrow Icon
ii Use Mean Square ˆ Var[ | ]= ? n n Requires  to be   "well behaved" n Either converge to a constant matrix or diverge. Heteroscedasticity case: 1 1 n n Autocorrelatio - = σ ÷ ÷ = ϖ -1 -1 -1 X' X Ω X 0 β X' X Ω X' X Ω x x ' n n 2 i j i 1 j 1 ij n case: 1 1 . n  terms. Convergence is unclear. n n = = = ϖ ∑ ∑ -1 X' X Ω x x '
Image of page 19

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

View Full Document Right Arrow Icon
Part 14: Generalized Regression Asymptotic Normality ™  19/44 - - - - - - - = ÷ ÷ 1 1 1 1 1 1 ' 1 ˆ n( ) n ' n n Converge to normal with a stable variance O(1)? '  a constant matrix? n 1 '  a mean to which we can apply the n                    central limit theorem? Het X X Ω X β β Ω ε X X Ω X Ω ε - = ε ε = σ ÷ ÷ ÷ ÷ ϖ ϖ ϖ ϖ ϖ n 1 2 i i i i i 1 i i i i i i eroscedasticity case? 1 1 '   .  Var ,  is just data. n n   Apply Lindeberg-Feller.  (Or assuming  /  is a draw from a common   distribution with mean and fixed va x x X = Ω ε x riance - some recent treatments.) Autocorrelation case?
Image of page 20
Part 14: Generalized Regression Asymptotic Normality (Cont.) ™  20/44 n n 1 i j i i i 1 j 1 1 For the autocorrelation case 1 1 '   n n Does the double sum converge?  Uncertain.  Requires elements of   to become small as the distance between i and j increases. (Has to resem - = = - ε ε ∑ ∑ ij X =  x x Ω ε Ω Ω ble the heteroscedasticity case.)
Image of page 21

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

View Full Document Right Arrow Icon
Part 14: Generalized Regression Test Statistics (Assuming Known Ω) p With known Ω , apply all familiar results to the transformed model: p With normality, t and F statistics apply to least squares based on Py and PX p With asymptotic normality, use Wald statistics and the chi-squared distribution, still based on the transformed model. ™  21/44
Image of page 22
Part 14: Generalized Regression Generalized (Weighted) Least Squares Heteroscedasticity Case ™  22/44 1 2 2 2 n 1 -1/2 2 n 1 n n 1 i i i 1 i 1 i i i i i 2 0 ... 0 0 ... 0 Var[ ] 0 0 ... 0 0 0 ... 1/ 0 ... 0 0 1/ ... 0                 0 0 ... 0 0 0 ... 1/ 1 1 ˆ ( ) ( ) y ˆ y ˆ - - = = ϖ ϖ ε = σ Ω = σ ϖ ϖ ϖ = ϖ = = ÷ ÷ ϖ ϖ - ϖ σ = -1 -1 i i X' X X' y x x x β Ω Ω x β 2 n i 1 n K = ÷ ÷ -
Image of page 23

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

View Full Document Right Arrow Icon
Part 14: Generalized Regression Autocorrelation t = t-1 + ut (‘First order autocorrelation.’ How does this come about?) Assume -1 <  < 1. Why? ut = ‘nonautocorrelated white noise’ t = t-1 + ut (the autoregressive form) = (t-2 + ut-1) + ut = ... (continue to substitute) = ut + ut-1 + 2ut-2 + 3ut-3 + ... = (the moving average form) ™  23/44
Image of page 24
Part 14: Generalized Regression Autocorrelation ™  24/44 2 t t t 1 t 1 i t i i= 0 2 2i 2 u u 2 i=0 t t 1 t t 1 t 2 t t 1 t t 1 t Var[ ] Var[u u u ...]           =  Var u           = 1 An easier way: Since Var[ ] Var[ ] and  u Var[ ] Var[ ] Var[u ] 2 Cov[ ,u ]            = - - - - - - - ε = + ρ + ρ +
Image of page 25

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

View Full Document Right Arrow Icon
Image of page 26
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