Testing for Regime Switching Talk - Transparencies

Testing for Regime Switching Talk - Transparencies -...

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Testing for Regime Switching joint work with Drew Carter and Ben Hansen Douglas G. Steigerwald UC Santa Barbara August 2010 D. Steigerwald (UCSB) Regime Switching August 2010 1 / 37
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Roadmap Null Hypothesis (Ghosh and Sen 1985) QLR (Cho and White 2007) 1 QLR Test Consistency 1 QMLE inconsistent for autoregressions 2 su¢ cient condition for QMLE consistency Asymptotic Null Distribution - Gaussian Processes (CW) Model Dependent Covariance 1 Construct Covariance of Gaussian Processes 2 Parameter Space Dependence 3 Subsample D. Steigerwald (UCSB) Regime Switching August 2010 2 / 37
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Regime Structure data generating process dependent ( β mixing) stationary unobserved state (indicates regimes) S t 2 f 0 , 1 g Markov process P ( S t = 1 j S t 1 = 0 ) = p ± 0 P ( S t = 0 j S t 1 = 1 ) = p ± 1 Example f ( Y t , θ ± 0 ) = c exp h 1 2 ( Y t θ ± 0 ) 2 i if S t = 0 f ( Y t , θ ± 1 ) = c exp h 1 2 ( Y t θ ± 1 ) 2 i if S t = 1 D. Steigerwald (UCSB) Regime Switching August 2010 3 / 37
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Null Hypothesis Ghosh and Sen 1985 Under H 0 : f ( Y t , θ ) equivalently represented by 3 curves p & 0 6 = 0 p 1 6 = 0 θ 0 = θ 1 = θ boundary of parameter space p & 0 = 0 ( P ( S t = 0 j S t ± 1 = 0 ) = 1) θ & 0 = θ (curve 2) p 1 = 0 ( P ( S t = 1 j S t ± 1 = 1 ) = 1) θ 1 = θ (curve 3) D. Steigerwald (UCSB) Regime Switching August 2010 4 / 37
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Quasi-Log-Likelihood Cho and White If p & 0 = 0 (or p 1 = 0), then asymptotic di¢ culties Use quasi-log-likelihood L n ( π , θ 0 , θ 1 ) = 1 n n t = 1 l t ( π , θ 0 , θ 1 ) where l t ( π , θ 0 , θ 1 ) = log [( 1 ± π ) f ( Y t , θ 0 ) + π f ( Y t , θ 1 )] stationary probability π = P ( S t = 1 ) ignores serial correlation in f S t g D. Steigerwald (UCSB) Regime Switching August 2010 5 / 37
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Consistency Consistency of QMLE under H 1 ) consistency of QLR test Cho and White (Theorem 1.b): QMLE is consistent under H 1 for class of processes that includes Y t = θ i + α Y t 1 + U t for S t = i issue: QMLE ignores dependence in S t dependence in S t is confounded with dependence in Y t QMLE inconsistent for α D. Steigerwald (UCSB) Regime Switching August 2010 6 / 37
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Example: AR(1) M ( π , α , θ 0 , θ 1 ) = E [ L n ( π , α , θ 0 , θ 1 )] f ( Y t , θ 0 ) = c exp h 1 2 ( Y t α Y t 1 θ 0 ) 2 i f ( Y t , θ 1 ) = c exp h 1 2 ( Y t α Y t 1 θ 1 ) 2 i ∂α M ( π ± , α ± , θ ± 0 , θ ± 1 ) = C π ± , θ ± 1 θ ± 0 ² ( θ ± 1 θ ± 0 ) ² Cov ( Y t 1 , S t ) = C π ± , θ ± 1 θ ± 0 ² ( θ ± 1 θ ± 0 ) 2 ² ( π ± p ± 0 ) π ± ( 1 π ± ) π ± α ± ( π ± p ± 0 ) ± derivative vanishes if θ ± 1 = θ ± 0 , π ± = 0, π ± = 1 ( H 0 ) π ± = p ± 0 ) P ( S t = j ) = P ( S t = j j S t 1 ) ( S t iid, QMLE is MLE) D. Steigerwald (UCSB) Regime Switching August 2010 7 / 37
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Consistency QMLE vs. QLRT QMLE inconsistent for autoregression Y t = θ i + α Y t 1 + U t U t ± iidN ( 0 , 1 ) if S t = i moving average Y t = θ i + U t U t = ε t + δε t 1
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This note was uploaded on 12/26/2011 for the course ECON 245b taught by Professor Staff during the Fall '08 term at UCSB.

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Testing for Regime Switching Talk - Transparencies -...

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