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

min
is ( I s )st
subject to:
Et   s t L( X s )
s t
Et F ( X t 1 , X t , X t 1 , it ,  t )  0
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
BeckWieland
CogleyColacitoSargent
SvenssonWilliams I
SvenssonWilliams II
Forward-looking
variables
No
Partially
(1970s style)
Yes
Not Yet
Regime change
No
Easy to
incorporate
Yes
Yes
Uncertainty about
state of economy
No
Yes
No
Yes
Parameter
uncertainty
Yes
Not really
Not really
Not really
Model
uncertainty
No
Yes
Yes
Yes
Full Bayesian
updating
Yes
For a {0,1}
indicator
No
For a {0,1}
indicator
Realistic number
of variables
No
No
Yes
No
Outline 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 analysis
a. Discuss computation of optimal simple rules in the MJLQ
framework
b. Discuss making “distribution forecast plots”
3. Turn to question of optimal policy in the MJLQ framework
a. “No Learning” policy
b. “Anticipated Utility” policy (learning, but no experimenting)
c. Full Bayesian updating (learning and experimenting)
Svensson-Williams I:
“Distribution Forecast Targeting”
Svensson-Williams II:
“Bayesian Optimal Policy”
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
limitations, it
is difficult to conceive of a situation for a policymaker
that cannot be approximated in this framework”
(Svensson-Williams I, p. 11)
Obviously, we want a modeling framework that is general enough, but:
• Do the methods of the paper reduce the dimensionality of the problem?
Yes.
• Do the methods of the paper make the problem computationally
tractable? (i.e., do they reduce the dimensionality enough?)
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-ColacitoSargent
SvenssonWilliams I
SvenssonWilliams II
Forward-looking
variables
Partially (1970s
style)
Yes
Not Yet
Yes
Regime change
Easy to
incorporate
Yes
Yes
Yes
Yes
No
Yes
Yes
Not really
Not really
Not really
Local
uncertainty
Yes
Yes
Yes
No
For a {0,1}
indicator
No
For a {0,1}
indicator
Yes
No
Yes
No
Yes
Uncertainty about
state of economy
Parameter
uncertainty
Model uncertainty
Full Bayesian
updating
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
1. Is this framework general enough?
Yes.
2. Does this framework reduce the dimensionality of the problem?
Yes.
3. Does this framework make the problem computationally tractable? 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
1. general enough
2. tractable
3. fit the data very well
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