HAC Overview Slides

HAC Overview Slides - NBER Summer Institute Minicourse...

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Lecture 9 1, July 21, 2008 NBER Summer Institute Minicourse – What’s New in Econometrics: Time Series Lecture 9 July 16, 2008 Heteroskedasticity and Autocorrelation Consistent Standard Errors
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Lecture 9 2, July 21, 2008 Outline 1. What are HAC SE’s and why are they needed? 2. Parametric and Nonparametric Estimators 3. Some Estimation Issues ( psd , lag choice, etc. ) 4. Inconsistent Estimators
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Lecture 9 3, July 21, 2008 1. What are HAC SE’s and why are they needed? Linear Regression: y t = x t β + e t 11 ˆ '' ' ' TT tt xx x y xe ββ −− == ⎛⎞ + ⎜⎟ ⎝⎠ ∑∑ 1 1 ˆ () ' ' X X T Tx x x e S A T T −= = . S XX p Σ XX = E ( x t x t ), A T (0, ) d N →Ω , where Ω = lim T →∞ Var 1 1 ' T t T = Thus l 1 ˆ , a NV T , where ˆˆ X XX X VS S HAC problem : ˆ Ω = ???
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Lecture 9 4, July 21, 2008 (Same problem arises in IV regression, GMM, … ) Notation: Ω = lim T →∞ Var 1 1 ' T tt t xe T = ⎛⎞ ⎜⎟ ⎝⎠ Let w t = x t e t (assume this is a scalar for convenience) Feasible estimator of Ω must use ˆˆ t wx e = , The estimator is { } () ˆ ˆ t w Ω , but most of our discussion uses { } ( ) ˆ t w Ω for convenience.
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Lecture 9 5, July 21, 2008 An expression for Ω : Suppose w t is covariance stationary with autocovariances γ j . (Also, for notational convenience, assume w t is a scalar). Then 12 1 01 1 2 2 1 1 11 () ( . . . ) 1 { (1 ) ( ) ( 2 ) ( ) 1 ( ) } 1 T tT t TT jj j jT j var w var w w w T T T T j T γγ = −− =− + = Ω= = + + + = + + + + +... × + =− + ∑∑ If the autocovariances are “1-summable” so that j j | |< ∞ then Ω = 1 1 T tj var w T == Recall spectrum of w at frequency ω is S ( ) = 1 2 ij j j e π =−∞ , so that Ω = 2 × S (0). Ω is called the “Long-run” variance of w .
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Lecture 9 6, July 21, 2008 II. Estimators (a) Parametric Estimators, w t ~ ARMA φ (L) w t = θ (L) ε t , S ( ω ) = 2 1( ) ( ) 2( ) ( ) ii ee θθ σ πφ Ω = 2 π × S (0) = 2 2 12 22 (1 ) (1) ) q p εε φφ −− = " " 2 2 2 ˆˆ ˆ ) ˆ ˆ ˆ ) q p Ω= " "
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Lecture 9 7, July 21, 2008 Jargon: VAR-HAC is a version of this. Suppose w t is a vector and the estimated VAR using w t is 11 ˆˆ ˆ ... tt p t p t ww w ε −− + +Φ + , where 1 1 ˆ ' T t T εε = Σ= Then ˆ ˆ ( ... ) ( ... ) ' pp II Ω= −Φ − −Φ Σ
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Lecture 9 8, July 21, 2008 (b) Nonparametric Estimators j j γ =−∞ Ω= ˆ ˆ m j j jm K =− with 1 1 ˆˆ T j tt j j tj ww T =+ == ( j 0)
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