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Solution techniques

Solution techniques. Martin Ellison University of Warwick and CEPR Bank of England, December 2005. State-space form. Generalised state-space form. Many techniques available to solve this class of models We use industry standard: Blanchard-Kahn. Alternative state-space form.

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Solution techniques

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  1. Solution techniques Martin Ellison University of Warwick and CEPR Bank of England, December 2005

  2. State-space form Generalised state-space form Many techniques available to solve this class of models We use industry standard: Blanchard-Kahn

  3. Alternative state-space form

  4. Partitioning of model backward-looking variablespredetermined variables forward-looking variables control variables

  5. Jordan decomposition of A eigenvectors diagonal matrix ofeigenvalues

  6. Blanchard-Kahn condition The solution of the rational expectations model is unique if the number of unstable eigenvectors of the system is exactly equal to the number of forward-looking (control) variables. i.e., number of eigenvalues in Λgreater than 1 in magnitude must be equal to number of forward-looking variables

  7. Too many stable roots multiple solutions equilibrium path not unique need alternative techniques

  8. Too many unstable roots no solution all paths are explosive transversality conditions violated

  9. Blanchard-Kahn satisfied one solution equilibrium path is unique system has saddle path stability

  10. Rearrangement of Jordan form

  11. stable unstable Partition of model

  12. Transformed problem

  13. Decoupled equations stable unstable Decoupled equations can be solved separately

  14. Solution strategy Solve unstable transformed equation Solve stable transformed equation Translate back into original problem

  15. As , only stable solution is Solution of unstable equation Solve unstable equation forward to time t+j Forward-looking (control) variables are function of backward-looking (predetermined) variables

  16. As , no problems with instability Solution of stable equation Solve stable equation forward to time t+j

  17. Solution of stable equation Future backward-looking (predetermined) variables are function of current backward-looking (predetermined) variables

  18. Full solution All variables are function of backward-looking (predetermined) variables: recursive structure

  19. Baseline DSGE model State space form To make model more interesting, assume policy shocks vt follow an AR(1) process

  20. New state-space form One backward-looking variable Two forward-looking variables

  21. Blanchard-Khan conditions Require one stable root and two unstable roots Partition model according to

  22. Next steps Exercise to check Blanchard-Kahn conditions numerically in MATLAB Numerical solution of model Simulation techniques

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