Discussion of:Discussion of:
Eric T. SwansonFederal Reserve Bank of San Francisco
http://www.ericswanson.pro/
Bayesian and Adaptive Optimal Policy under Model Uncertainty
Lars Svensson and Noah Williams
Oslo Conference on Monetary Policy and UncertaintyJune 9, 2006
The Optimal Policy ProblemThe Optimal Policy Problem
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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, butit is computationally intractable (now and for the foreseeable future)
Sampling of LiteratureSampling 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
Beck-Wieland
Cogley-Colacito-Sargent
Svensson-Williams I
Svensson-Williams II
Forward-looking variables
NoPartially
(1970s style)Yes Not Yet
Regime change NoEasy to
incorporateYes Yes
Uncertainty about state of economy
No Yes No Yes
Parameter uncertainty
Yes Not really Not really Not really
Modeluncertainty No Yes Yes Yes
Full Bayesian updating
YesFor a {0,1} indicator
NoFor a {0,1} indicator
Realistic number of variables
No No Yes No
1. Extend Markov-Jumping-Linear-Quadratic (MJLQ) model from engineering literature to forward-looking LQ models
2. Non-optimal/quasi-optimal policy analysisa. Discuss computation of optimal simple rules in the MJLQ
frameworkb. Discuss making “distribution forecast plots”
3. Turn to question of optimal policy in the MJLQ frameworka. “No Learning” policyb. “Anticipated Utility” policy (learning, but no experimenting)c. Full Bayesian updating (learning and experimenting)
Outline of Svensson-Williams I & IIOutline of Svensson-Williams I & II
1. Extend Markov-Jumping-Linear-Quadratic (MJLQ) model from engineering literature to forward-looking LQ models
2. Non-optimal/quasi-optimal policy analysisa. Discuss computation of optimal simple rules in the MJLQ
frameworkb. Discuss making “distribution forecast plots”
3. Turn to question of optimal policy in the MJLQ frameworka. “No Learning” policyb. “Anticipated Utility” policy (learning, but no experimenting)c. Full Bayesian updating (learning and experimenting)
Svensson-Williams II:“Bayesian Optimal Policy”
Svensson-Williams I:“Distribution Forecast Targeting”
Markov-Jumping Linear Quadratic ModelMarkov-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
“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
Svensson-WilliamsSvensson-Williams
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 DifficultiesComputational 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 DifficultiesComputational 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-WA Computationally Viable Alternative to S-WCogley-Colacito-
SargentSvensson-Williams I
Svensson-Williams II
Forward-looking variables
Partially (1970s style)
Yes Not Yet Yes
Regime changeEasy to
incorporateYes Yes Yes
Uncertainty about state of economy
Yes No Yes Yes
Parameter uncertainty
Not really Not really Not reallyLocal
uncertainty
Model uncertainty Yes Yes Yes No
Full Bayesian updating
For a {0,1} indicator
NoFor a {0,1} indicator
Yes
Realistic number of variables
No Yes No Yes
Swanson (2006JEDC, 2006WP)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
1. Is this framework general enough?2. Does this framework reduce the dimensionality of the problem?3. Does this framework make the problem computationally tractable?
Yes.Yes.Yes.
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
Full Bayesian Updating of u*, U.S. 1997-2001Full Bayesian Updating of u*, U.S. 1997-2001
SummarySummary
• 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 are1. general enough2. tractable3. fit the data very well
Discussion of:Discussion of:
Eric T. SwansonFederal Reserve Bank of San Francisco
http://www.ericswanson.pro/
Bayesian and Adaptive Optimal Policy under Model Uncertainty
Lars Svensson and Noah Williams
Oslo Conference on Monetary Policy and UncertaintyJune 9, 2006
