Bayesian and Adaptive Optimal Policy under Model Uncertainty Discussion of: Lars Svensson and Noah Williams 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 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 it is computationally intractable (now and for the foreseeable future)

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

Cogley- Colacito- Sargent Beck- Wieland Cogley- Colacito- Sargent Svensson- Williams I Svensson- Williams II Forward-looking variables No Partially (1970s style) Yes Not Yet Regime change Easy to incorporate Uncertainty about state of economy Parameter uncertainty Not really Model uncertainty Full Bayesian updating For a {0,1} indicator Realistic number of variables

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) Svensson-Williams II: “Bayesian Optimal Policy” Svensson-Williams I: “Distribution Forecast Targeting” 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)

Markov-Jumping Linear Quadratic Model LQ model with multiple regimes j є {1,2,…,n} Exogenous probability of regime change each period 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

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.

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

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

A Computationally Viable Alternative to S-W Cogley-Colacito- Sargent Svensson- Williams I Svensson- Williams II Forward-looking variables Partially (1970s style) Yes Not Yet Regime change Easy to incorporate Uncertainty about state of economy No Parameter uncertainty Not really Local uncertainty Model uncertainty Full Bayesian updating For a {0,1} indicator Realistic number of variables

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Summary The optimal policy problem is well understood in theory, but it 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

Bayesian and Adaptive Optimal Policy under Model Uncertainty Discussion of: Lars Svensson and Noah Williams 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