Stochastic Time Variation Slides

# Stochastic Time Variation Slides - NBER Summer Institute...

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Lecture 6 - 1, July 21, 2008 NBER Summer Institute Minicourse – What’s New in Econometrics: Time Series Lecture 6: July 15, 2008 Specification and estimation of models with stochastic time variation

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Lecture 6 - 2, July 21, 2008 Outline 1. Break Models (Break Dates) 2. Markov Switching Models 3. Martingale TVP a. MLEs and alternatives b. Data Augmentation (EM) c. TVPs as nuisance parameters
Lecture 6 - 3, July 21, 2008 1. Break Models Inference about Break Dates (Bai (1997), Hansen (2001)) Example (a special case of the linear regression): y t = β t + ε t for for t t t τ δτ = + > π = /T = Break “Fraction” o = true break date o = true break fraction ˆ = Least squares estimator of , ˆ = ˆ / T

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Lecture 6 - 4, July 21, 2008 Some results that are useful for inference: Bai(1997) shows ˆ π o ~ O p ( T 1 δ 2 ), so that T 2 ( ˆ o ) ~ O p (1) 2 ( ˆ τ o ) ~ O p (1) Thus, ˆ is consistent for o , but ˆ is not consistent for o . The speed at which ˆ converges to o depends on . In general, the distribution of ˆ and ˆ depends on the distribution of the errrors ε t . This is true even when T is large. Thus, robust inference is problematic. There are approximations that can be used when is appropriately small.
Lecture 6 - 5, July 21, 2008 An asymptotic approximation: Recall T δ 2 ( ˆ π o ) ~ O p (1), so assume T 2 →∞ . Also, is small, so assume 0. (More formally, = T which approaches zero as T grows large.). (Example, both of these are satisfied if T = aT 0.49 ) The challenge is to compute a convenient expression ˆ . The trick is to use empirical process methods like those used for the FCLT. The main ideas can be understood in a situation in which β and are known, so that estimating τ (equivalently ) is the only problem. The least squares objective function is SSR( ) = 22 11 () ( ) T tt yy βδ == + −+ ∑∑ and the trick is to study the behavior of this function as T gets large.

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Lecture 6 - 6, July 21, 2008 Suppose τ > o . Then we can write 22 11 2 1 2 111 [( ) ] [] 1 () ( ) ( ) ( ) ( ( ) ) 2 ()2 o o o o o o o T tt T t t T t t T to t T T T SSR y y yy y T ττ ππ π τβ β δ βδ εε ε ετ επ == + + = + + = + + + =− + + + + =+ + + −+ + ∑∑ Where the last expression substitures = /T . The first term does depend on ; ignore it when thinking about the function that is being maximizing.
Lecture 6 - 7, July 21, 2008 Thus, we can think about choosing τ or π to minimize [( ) ] 22 [] 1 11 () 2 ( ) 2 o o o T ot o T tt T ππ ττ δ ε + =+ = −+ = + ∑∑ Let υ = ( o ) T 2 / 2 σ . Then minimizing SSR over is the same as minimizing g T over , where g T ( ) = + 2( / ) 2 [( / ) ] 1 1 (/ ) o T t δσ + = and the division by is for later convenience.

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Lecture 6 - 8, July 21, 2008 Recall δ 0, so 2 , so that 2 [( / ) ] [] 1 1 (/ ) ( / ) ( ) o d T t W ε δσ υ επ σε σ + = . (For analogy with the standard formula, think of ( / ) 2 = sample size , so that ( / ) = 1 sample size ). Thus g T ( ) = + 2( / ) 2 [( / ) ] 1 1 ) o T t π + = () 2 () d gW υυ →= + .
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Stochastic Time Variation Slides - NBER Summer Institute...

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