Forecasting 2001- VAR Analysis

Forecasting 2001- VAR Analysis - Forecasting Financial...

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Copyright © 2000 – 2006 Investment Analytics Forecasting Financial Markets – VAR Analysis Slide: 1 Forecasting Financial Markets VAR Analysis Copyright © 2000-2006 Investment Analytics
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Copyright © 2000 – 2006 Investment Analytics Forecasting Financial Markets – VAR Analysis Slide: 2 Vector Autoregression ¾ Structural, first order VAR process ± y t = b 10 –b 12 z t + γ 11 y t -1 + γ 12 z t -1 + ε yt ± z t = b 20 21 y t + γ 21 y t -1 + γ 22 z t -1 + ε zt { ε yt } and { ε zt } are uncorrelated white noise processes ± y t and z t have contemporaneous effect on one another
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Copyright © 2000 – 2006 Investment Analytics Forecasting Financial Markets – VAR Analysis Slide: 3 VAR in Matrix Form ¾ Restate in matrix form: ± Bx t = Γ 0 + Γ 1 x t-1 + ε t ± x t = B -1 Γ 0 + B -1 Γ 1 x t-1 + B -1 ε t ± x t = A 0 + A 1 x t-1 + e t + + = zt yt t t t t z y b b z y b b ε γ 1 1 22 21 12 11 20 10 21 12 1 1
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Copyright © 2000 – 2006 Investment Analytics Forecasting Financial Markets – VAR Analysis Slide: 4 VAR in Standard Form ¾ Rewrite as two simultaneous equations: ± y t = a 10 + a 11 y t -1 + a 12 z t -1 + e 1t ± z t = a 20 + a 21 y t -1 + a 22 z t -1 + e 2t ¾ Error processes {e it } ± Have zero mean, constant variances and are individually serially uncorrelated ± May be correlated with one another
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Copyright © 2000 – 2006 Investment Analytics Forecasting Financial Markets – VAR Analysis Slide: 5 VAR Stationarity ¾ Substitution ± x t = A 0 + A 1 x t-1 + e t ± x t = A 0 + A 1 ( A 0 + A 1 x t-2 + e t-1 ) + e t ± x t = ( I + A 1 ) A 0 + A 1 2 x t-2 + A 1 e t-1 + e t I is 2 x 2 identity matrix ¾ After n iterations: 1 1 1 0 1 0 1 1 ) ... ( + = + + + + + = n t n i t n i i n t x A e A A A A I x
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Copyright © 2000 – 2006 Investment Analytics Forecasting Financial Markets – VAR Analysis Slide: 6 VAR Stability & Stationarity ¾ Stability condition ± A 1 n 0 as n i t i i t e A x = + = 0 1 µ 21 12 22 11 10 21 11 20 20 12 22 10 ) 1 )( 1 ( / ] ) 1 ( [ / ] ) 1 ( [ a a a a a a a a z a a a a y z y = + = + = =
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Copyright © 2000 – 2006 Investment Analytics Forecasting Financial Markets – VAR Analysis Slide: 7 Stationarity Conditions ¾ Mean is constant: E( x t ) = µ ¾ Variance is finite and time-invariant: ¾ Σ is variance covariance matrix of series {y t }and {z t } = = + + + + = = = 2 2 21 12 2 1 1 2 1 6 1 4 1 2 1 2 0 1 2 ) ( ...) ( ) ( σ µ A I A A A I e A E x E i i t i t
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Copyright © 2000 – 2006 Investment Analytics Forecasting Financial Markets – VAR Analysis Slide: 8 VAR Model with Lag Operator 2 21 12 22 11 1 1 21 2 11 10 21 11 20 2 21 12 22 11 1 2 12 1 22 20 12 22 10 22 2 21 20 2 22 21 20 1 12 11 10 ) 1 )( 1 ( ) 1 ( ) 1 ( ) 1 )( 1 ( ) 1 ( ) 1 ( ) 1 /( ) ( L a a L a L a e a e L a a a a a z L a a L a L a e a e L a a a a a y L a e Ly a a L Lz e Lz a Ly a a z e Lz a Ly a a y t t t t t t t t t t t t t t t t t + + + = + + + = + + = + + + = + + + =
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Copyright © 2000 – 2006 Investment Analytics Forecasting Financial Markets – VAR Analysis Slide: 9
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This document was uploaded on 11/04/2011 for the course ECON 421 at CUNY York.

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Forecasting 2001- VAR Analysis - Forecasting Financial...

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