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Discussion of:

Lars Svensson and Noah Williams. Discussion of:. Bayesian and Adaptive Optimal Policy under Model Uncertainty. Eric T. Swanson Federal Reserve Bank of San Francisco http://www.ericswanson.pro/. Oslo Conference on Monetary Policy and Uncertainty June 9, 2006. The Optimal Policy Problem.

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Discussion of:

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  1. Lars Svensson and Noah Williams Discussion of: Bayesian and Adaptive Optimal Policy under Model Uncertainty Eric T. Swanson Federal Reserve Bank of San Francisco http://www.ericswanson.pro/ Oslo Conference on Monetary Policy and Uncertainty June 9, 2006

  2. The Optimal Policy Problem subject to: Allow: • Forward-looking variables • Model nonlinearities • e.g., regime change • Uncertainty • about state of economy (e.g., output gap, NAIRU, prod. growth) • about parameters • about model • Realistic number of variables, lags The solution to the optimal policy problem is well understood in theory, but (now and for the foreseeable future) it is computationally intractable

  3. Sampling of Literature • Wieland (2000JEDC, 2000JME) – parameter uncertainty, experimentation • Levin-Wieland-JWilliams I & II (1999Taylor, 2003AER) – model uncertainty • Meyer-Swanson-Wieland (2001AER) – simple rules, pseudo-Bayesian updating • Swanson (2006JEDC, 2006WP) – full Bayesian updating, LQ w/regime change • Beck-Wieland (2002JEDC) – parameter uncertainty, experimentation • Levin-JWilliams (2003JME) – model uncertainty • Cogley-Sargent (2005RED) – model uncertainty, quasi-Bayesian updating • Cogley-Colacito-Sargent (2005WP) – full Bayesian updating • Küster-Wieland (2005WP) – model uncertainty • Zampolli (2004WP), Blake-Zampolli (2005WP) – Markov-switching LQ model • Svensson-NWilliams I & II (2005WP, 2006WP) – Markov-switching LQ model Note: the above excludes robust control, least-squares learning, LQ w/trivial filtering

  4. Extend Markov-Jumping-Linear-Quadratic (MJLQ) model from engineering literature to forward-looking LQ models • Non-optimal/quasi-optimal policy analysis • Discuss computation of optimal simple rules in the MJLQ framework • Discuss making “distribution forecast plots” • Turn to question of optimal policy in the MJLQ framework • “No Learning” policy • “Anticipated Utility” policy (learning, but no experimenting) • Full Bayesian updating (learning and experimenting) Svensson-Williams I: “Distribution Forecast Targeting” Svensson-Williams II: “Bayesian Optimal Policy” Outline of Svensson-Williams I & II • Extend Markov-Jumping-Linear-Quadratic (MJLQ) model from engineering literature to forward-looking LQ models • Non-optimal/quasi-optimal policy analysis • Discuss computation of optimal simple rules in the MJLQ framework • Discuss making “distribution forecast plots” • Turn to question of optimal policy in the MJLQ framework • “No Learning” policy • “Anticipated Utility” policy (learning, but no experimenting) • Full Bayesian updating (learning and experimenting)

  5. Markov-Jumping Linear Quadratic Model Case 1: The regime you are in is always observed/known: • then the optimal policy is essentially linear • there is separation of estimation and control • optimal policy problem is extremely tractable Case 2: The regime you are in is always unobserved/unknown: • then the framework is very general, appealing • but all of the above properties are destroyed • LQ model with multiple regimes j є {1,2,…,n} • Exogenous probability of regime change each period

  6. Svensson-Williams “Aside from dimensional and computational limitations, it is difficult to conceive of a situation for a policymaker that cannot be approximated in this framework” (Svensson-Williams I, p. 11) “Aside from dimensional and computational limitations • Obviously, we want a modeling framework that is general enough, but: • Do the methods of the paper reduce the dimensionality of the problem? • Do the methods of the paper make the problem computationally tractable? (i.e., do they reduce the dimensionality enough?) Yes. No.

  7. Computational Difficulties Svensson-Williams do reduce the dimensionality of the problem: • By restricting attention to a discrete set of regimes {1,…,n}, full Bayesian updating requires only n-1 additional state variables (p1,…,pn-1)t • Note: Cogley-Colacito-Sargent use the same trick Still, dynamic programming in a forward-looking model is computationally challenging, limited to a max of ≈4 state variables even using Fortran/C • Each predetermined variable is a state variable • Each forward-looking variable introduces an additional state variable because of commitment • Each regime beyond n=1 introduces an additional state variable Svensson-Williams can only solve the model for simplest possible case: • 1 predetermined variable, 0 forward-looking variables, 2 regimes • Note: Svensson-Williams are still working within Matlab • Cogley-Colacito-Sargent use Fortran, solve a similar model with 4 state variables

  8. Computational Difficulties Svensson-Williams, Sargent et al. hope to find “Anticipated Utility” policy (no experimentation) is a good approximation to Full Bayesian policy • “Anticipated Utility” policy is much easier to compute (though not trivial) • Cogley-Colacito-Sargent find “Anticipated Utility” works well for their simple model However: • Wieland (2000a,b), Beck-Wieland (2002) find experimentation is important for resolving parameter uncertainty • particularly if a parameter is not subject to natural experiments • Just because “Anticipated Utility” works well for one model does not imply it works well in general • we would need to solve any given model for the full Bayesian policy to know whether the approximation is acceptable • There may be better approximations than “Anticipated Utility” • e.g., perturbation methods probably provide a more fruitful avenue for developing tractable, accurate, rigorous approximations

  9. A Computationally Viable Alternative to S-W

  10. Swanson (2006JEDC, 2006WP) • Adapts forward-looking LQ framework to case of regime change in: • NAIRU u*, potential output y* • Rate of productivity growth g • Variances of shocks ε • Framework maintains separability of estimation and control • Even in models with forward-looking variables • Even when there is local parameter uncertainty • Due to separability, full Bayesian updating is computationally tractable • Allows application to models with realistic number of variables • Optimal policy matches behavior of Federal Reserve in 1990s • Evidence that framework is useful in practice as well as in principle • Is this framework general enough? • Does this framework reduce the dimensionality of the problem? • Does this framework make the problem computationally tractable? Yes. Yes. Yes.

  11. Full Bayesian Updating of u*, U.S. 1997-2001

  12. Full Bayesian Updating of u*, U.S. 1997-2001

  13. Full Bayesian Updating of u*, U.S. 1997-2001

  14. Full Bayesian Updating of u*, U.S. 1997-2001

  15. Full Bayesian Updating of u*, U.S. 1997-2001

  16. Full Bayesian Updating of u*, U.S. 1997-2001

  17. Full Bayesian Updating of u*, U.S. 1997-2001

  18. Full Bayesian Updating of u*, U.S. 1997-2001

  19. Full Bayesian Updating of u*, U.S. 1997-2001

  20. Full Bayesian Updating of u*, U.S. 1997-2001

  21. Full Bayesian Updating of u*, U.S. 1997-2001

  22. Full Bayesian Updating of u*, U.S. 1997-2001

  23. Full Bayesian Updating of u*, U.S. 1997-2001

  24. Full Bayesian Updating of u*, U.S. 1997-2001

  25. Full Bayesian Updating of u*, U.S. 1997-2001

  26. Summary • The optimal policy problem is well understood in theory, butit is computationally intractable • Svensson-Williams propose using MJLQ framework to reduce dimensionality of the problem • MJLQ framework can be very general • but when it is general, it is also computationally intractable • MJLQ framework with “Anticipated Utility” may provide a tractable approximation in the future • but there are some reasons to be skeptical • other approximation methods may be more promising • In the meantime, consider alternatives that are • general enough • tractable • fit the data very well

  27. Lars Svensson and Noah Williams Discussion of: Bayesian and Adaptive Optimal Policy under Model Uncertainty Eric T. Swanson Federal Reserve Bank of San Francisco http://www.ericswanson.pro/ Oslo Conference on Monetary Policy and Uncertainty June 9, 2006

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