Econometrics-I-14

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

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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 '

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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?
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.)

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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
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 = ÷ ÷ -

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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
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 +

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