Draft:RE-SVAR
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A Rational Expectations augmented Structural Vector Autoregression (RE-SVAR) model integrates forward-looking behavior directly into the traditional Structural Vector Autoregression (SVAR) framework. Unlike standard SVAR models, which rely on backward-looking assumptions, RE-SVAR incorporates rational expectations (RE) explicitly.
The RE-SVAR model is expressed as:
where:
xt: A vector of variables (e.g., inflation, output, monetary policy indicators), A0: The structural coefficient matrix, Ai: Lagged coefficient matrices, ϵt: A vector of structural shocks. Forward-Looking Structural Equations Monetary Policy Rule:
![{\displaystyle i_{t}=\phi _{\pi }\mathbb {E} _{t}[\pi _{t+h_{\pi }}]+\phi _{y}\mathbb {E} _{t}[y_{t+h_{y}}]+A_{MP}(L)x_{t-1}+\varepsilon _{MP,t},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb9a93135d6207816da7542461340a6745db0d10)
where:
φπ, φy: Policy response coefficients, Et: Represents the rational expectations operator at time t, hπ, hy: Forward-looking horizons for inflation and output. Output Equation:
![{\displaystyle y_{t}=\mathbb {E} _{t}[y_{t+1}]-\alpha _{1}(i_{t}-\mathbb {E} _{t}[\pi _{t+1}])+A_{IS}(L)x_{t-1}+\varepsilon _{IS,t},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b34973e40c8160ddbaa5580e0f25af5226da304f)
where α1 measures the sensitivity of output to interest rates and inflation expectations.
Inflation Equation:
![{\displaystyle \pi _{t}=\alpha _{2}\mathbb {E} t[\pi {t+1}]+\alpha _{3}y_{t}+A_{AS}(L)x_{t-1}+\varepsilon _{AS,t},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cd1b82f35cf8aa8c406257cdcbaa51a860253d7)
where:
α2, α3: Capture forward-looking and contemporaneous impacts on inflation. Identification of Structural Shocks Structural shocks are identified using an expectational difference method. For monetary policy shocks (ϵMP,t), the structural shock is derived as:
where:
ei,t: The reduced-form residual for the interest rate equation, Sπ, Sy: Selection vectors for inflation and output, Ψhπ, Ψhy: Companion-form matrices raised to the relevant horizons. This methodology extends to identifying shocks for output (ϵIS,t) and inflation (ϵAS,t) by substituting the appropriate structural equations.
The RE-SVAR approach provides a robust framework for analyzing forward-looking behavior and generating theoretically consistent impulse response functions (IRFs) to structural shocks.