1. 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
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
it is computationally intractable
(now and for the foreseeable future)

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. 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

5. 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)

6. 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

7. 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.

8. 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

9. 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

10. 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

11. 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.

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

27. 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

28. 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
Oslo Conference on Monetary Policy and Uncertainty
June 9, 2006