Made conditional on the potential future paths of

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made conditional on the potential future paths of specified variables in the model.      In addition to data description and forecasting, the VAR model is also used for structural inference and policy analysis. In structural analysis, certain assumptions about the causal structure of the data under investigation are imposed, and the resulting causal impacts of unexpected shocks or innovations to specified variables on the variables in the model are summarized. These causal impacts are usually summarized with impulse response functions and forecast error variance decompositions. Eric Zivot: http://faculty.washington.edu/ezivot/econ584/notes/varModels.pdf ™  40/45
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Part 15: Generalized Regression Applications VAR ™  41/45 1 11 1 12 2 13 3 1 1 2 21 1 22 2 23 3 2 2 3 31 1 32 2 33 3 3 3 ( ) ( 1) ( 1) ( 1) ( ) ( ) ( ) ( 1) ( 1) ( 1) ( ) ( ) ( ) ( 1) ( 1) ( 1) ( ) ( ) (In Zivot's examples, 1. Exchange rates 2. y(t y t y t y t y t x t t y t y t y t y t x t t y t y t y t y t x t t = γ - + γ - + γ - + δ + ε = γ - + γ - + γ - + δ + ε = γ - + γ - + γ - + δ + ε )=stock returns, interest rates, indexes of industrial production, rate of inflation
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Part 15: Generalized Regression Applications ™  42/45 12 2 13 3 2 1 11 1 1 2 1 1 1 VAR Formulation (t) = (t-1) + x(t) + (t) SUR with identical regressors. Granger Causality: Non ( 1) ( 1) ( zero off diagonal elements in ( ) ( 1) ( ) ( ) ( ) 1) y t y t y t y t y t x t t y t γ - γ - γ Γ Γ = γ - + + + δ + ε - = + y y δ ε 22 2 2 2 3 33 3 23 3 31 1 3 3 3 1 2 2 2 12 ( 1) ( ) ( ) ( ) ( 1) ( ) ( ) Hypothesis: does not Granger cause : ( 1) ( 1) ( 1 0 ) = y t x t t y t y t x t t y t y t y y t y γ - + + δ + ε = + + γ - + δ + ε γ γ - γ - γ -
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Part 15: Generalized Regression Applications ™  43/45 2 2 Impulse Response (t) = (t-1) + x(t) + (t) By backward substitution or using the lag operator (text, 943) (t) x(t) x(t-1) x(t-2) +... (ad infinitum) + (t) (t-1) (t-2) Γ = + Γ + Γ + Γ + Γ y y y δ ε δ δ δ ε ε ε 2 1 + ... [ must converge to as P increases. Roots inside unit circle.] Consider a one time shock (impulse) in the system, = in period t Consider the effect of the impulse on y ( ), s=t, t+1,... Ef P s Γ λ ∆ε 0 2 2 12 2 12 fect in period t is 0. is not in the y1 equation. affects y2 in period t, which affects y1 in period t+1. Effect is In period t+2, the effect from 2 periods back is ( ) ... and so on ε ∆ε γ ×λ Γ ×λ .
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Part 15: Generalized Regression Applications Zivot’s Data ™  44/45
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Part 15: Generalized Regression Applications Impulse Responses   45/45
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