Very preliminary
Comments welcome
Robust.tex
Robust control made simple
Lars E.O. Svensson
First draft: March 2000. This draft: April 12, 2000
1. Introduction
Some of the recent literature on robust control in economics (for instance, Giannoni [3], Hansen
and Sargent [4], Onatski [5], Onatski and Stock [6], Sargent [9]) is somewhat technical and
di¢cult to see through. This note attempts to use a simple example of optimal monetary policy
to convey the gist of robust control.
Let m 2 M denote a particular model of the transmission mechanism of monetary policy,
where M denotes a feasible set of models. Let f 2 F denote a particular monetary policy, where
F is the feasible set of policies. Let V be an (expected) loss function, so V (f; m) denotes the
(expected) loss of policy f in model m.
A Bayesian approach to optimal policy under model uncertainty, as in Brainard [2], assigns
a prior probability measure, ©, on the feasible set of models M. The expected loss for given
policy is then
EM V (f; m) ´
Z
V (f; m)d©(m):
m2M
The optimal policy f ¤ minimizes the expected loss,
f ¤ = arg min EM V (f; m):
f 2F
The optimal policy will be a function of M, © and V , f ¤ (M; ©; V ).
A robust-control approach to optimal policy under model uncertainty, as in Hansen and
Sargent [4] and Onatski and Stock [6], does not assign any prior probability measure on the
feasible set of models. Instead, it focuses on the maximum loss for any given policy f ,1
max V (f ; m).
m2M
The optimal policy f^ is then the policy that minimizes the maximum loss,
f^ = arg min max V (f; m):
f 2F m2M
1
If maxm2M V (f ; m) does not exist, it is replaced by the least upper bound, supm2M V (f ; m).
Under robust control, the worst possible model for a given policy, m(f),
^
de…ned by
m(f)
^
= arg max V (f; m);
m2M
is crucial, and the optimal policy will ful…ll
f^ = arg min V (f; m(f
^ )):
f 2F
^
The optimal policy will be a function of M and V , f(M;
V ):
I use a particular simple model to illustrate these concepts.
2. Optimal policy with known model
Consider a simple backward-looking Phillips curve,
¼ t+1 = °¼t + ®xt + "t+1 ;
where ¼t and xt is in‡ation and the output gap, respectively, in period t, the coe¢cients ® and
° are positive, and "t is an iid cost-push shock with E["t ] = 0 and Var["t ] = ¾2" .
Let the period loss function be
Lt = ¼ 2t + ¸x2t ;
where ¸ ¸ 0 is the relative weight on output-gap stability, and let the social loss function be
E[Lt ] = E[¼2t ] + ¸E[x2t ] = Var[¼t ] + ¸Var[xt ];
where the last equality follows if we restrict policy to result in E[¼t ] = E[xt ] = 0.
For simplicity we consider the output gap as the central bank’s control variable and note
that in‡ation is an endogenous predetermined variable. We …rst want to …nd the optimal policy
when ® and ° are known. The optimal policy will be a feedback on the only predetermined
variable, ¼t ;
xt = ¡f ¼t ;
where the response coe¢cient f remains to be determined. In‡ation is then given by
¼t+1 = (° ¡ ®f)¼t + "t :
We realize that the optimal policy will ful…ll 0 · f · ®° .
2
(2.1)
The variances of in‡ation and the output gap ful…ll
Var[xt ] = f 2 Var[¼t ];
1
¾2 ;
Var[¼t ] =
1 ¡ (° ¡ ®f)2 "
(2.2)
(2.3)
and the social loss will equal
E[Lt ] =
1 + ¸f 2
¾2 ´ V (f; ®; °; ¸):
1 ¡ (° ¡ ®f )2 "
(2.4)
The optimal policy f = f ¤ (®; °; ¸) is then given by
f ¤ (®; °; ¸) ´ arg min V (f; ®; °; ¸):
f
It will ful…ll the …rst-order condition
Vf (f ¤ ; ®; °; ¸) ´ 2
¸f ¤ [1 ¡ (° ¡ ®f ¤ )2 ] ¡ ®(1 + ¸f ¤2 )(° ¡ ®f ¤ )
=0
[1 ¡ (° ¡ ®f ¤ )2 ]2
(2.5)
and the second-order condition
Vf f (f ¤ ; ®; °) > 0:
It is easy to show (see appendix) that the derivatives of the optimal policy with respect to
®, ° and ¸ ful…ll
f®¤ (®; °; ¸) < 0;
f°¤ (®; °; ¸) > 0;
f¸¤ (®; °; ¸) < 0:
Furthermore, we have
f ¤ (®; °; 0) =
°
and f ¤ (®; °; 1) ´ lim f ¤ (®; °; ¸) = 0:
¸!1
®
3. Optimal policy under robust control
Let us now introduce some model uncertainty. Assume that we know the supports of the
coe¢cients ® and °, but not their (constant) true values. Let the supports be given by ® 2
[®1 ; ®2 ] and ° 2 [° 1 ; ° 2 ], where 0 < ®1 < ®2 and 0 < ° 1 < ° 2 . Furthermore, I restrict the
bounds of the supports to ful…ll
®2 ° 2 ¡ ®1 ° 1 < ®1 + ®2 :
3
(3.1)
As explained below, this condition is su¢cient for the existence of a policy that results in bounded
in‡ation and output-gap variance for all possible models. The condition can be written
° 2 < 1 + (1 + ° 1 )
®1
®2
and implies that ° 2 must not be too large and is ful…lled for any ° 2 · 1.
Thus, the models can be indexed by m = (®; °) and the feasible set of models M is given by
M ´ fm = (®; °)j®1 · ® · ®2 ; ° 1 · ° · ° 2 g:
Let the feasible set of policies F be given by (2.1) with f ¸ 0, that is, F = ffjf ¸ 0}.
Selecting the optimal policy under robust control means committing initially to a policy f^ 2 F
so as to minimize the social loss for the worst possible model; the latter being the one that
results in maximum social loss for a given policy. That is,
f^ = arg min max V (f ; ®; °; ¸):
f 2F (®;°)2M
The restriction to f ¸ 0 will not be binding, as we shall see below.
3.1. The worst possible model
From (2.4) follows that for any given f, the worst possible model is the one that maximizes
j° ¡ ®fj. Let m(f)
^
denote the worst possible model for any given f 2 F , that is,
m(f)
^
´ arg max j° ¡ ®fj.
(®;°)2F
We have
8
< ° ¡ ®f if ° ¡ ®f ¸ 0;
j° ¡ ®f j =
: ®f ¡ ° if ° ¡ ®f < 0:
Because f ¸ 0, we realize that the maximum of j° ¡ ®fj for a given f is the maximum of
° 2 ¡ ®1 f and ®2 f ¡ ° 1 ,
max j° ¡ ®fj = max(° 2 ¡ ®1 f; ®2 f ¡ ° 1 ):
(®;°)
Thus, m(f)
^
is either (®1 ; ° 2 ) or (®2 ; ° 1 ), depending on whether ° 2 ¡ ®1 f ? ®2 f ¡ ° 1 , that is,
whether f 7 f¹, where
° + °2
> 0:
f¹ ´ 1
®1 + ®2
4
(3.2)
It follows that m(f)
^
is given by
8
< (®1 ; ° ) if f · f;
¹
2
m(f)
^
=
: (® ; ° ) if f ¸ f:
¹
2 1
We note that the worst possible model is on the boundary of feasible set of models, and hence
depends crucially on the assumed feasible set of models.
Now we can see why I imposed the condition (3.1); this condition insures that
¹ < 1;
max j° ¡ ®fj
(®;°)2M
which is su¢cient for the existence of a policy that results in …nite in‡ation and output-gap
variance for all feasible models.
3.2. Strict in‡ation targeting
The simplest case is strict in‡ation targeting, ¸ = 0, in which case
V (f; ®; °; 0) ´
1
¾2:
1 ¡ (° ¡ ®f )2 "
The optimal policy under strict in‡ation targeting and robust control is then given by
f^ = arg min max j° ¡ ®fj = arg min max(° 2 ¡ ®1 f; ®2 f ¡ ° 1 ):
f 2F (®;°)2M
f 2F
In order to minimize max(° 2 ¡ ®1 f; ®2 f ¡ ° 1 ), it is best to select f = f^ where f^ ful…lls
° 2 ¡ ®1 f^ = ®2 f^ ¡ ° 1 ;
that is,
¹
f^ = f;
where f¹ is given by (3.2).
3.3. Flexible in‡ation targeting
Let us now consider the case of ‡exible in‡ation targeting, ¸ > 0, when V (f ; ®; °; ¸) is given by
(2.4). Since the term ¸f 2 in the numerator adds to the social loss, we realize that, for su¢ciently
¹ and to trade o¤ some increase in the denominator for
large ¸, it is optimal to lower f below f,
¹ we then have m(f
a reduction in the numerator. For f < f,
^ ) = (®1 ; ° 2 ), hence we have
f^ = arg min max V (f ; ®; °; ¸) = arg min V (f ; ®1 ; ° 2 ; ¸) = f ¤ (®1 ; ° 2 ; ¸);
f 2F (®;°)2M
f 2F
5
where f ¤ (®1 ; ° 2 ; ¸) denotes the optimal policy if the true model is known to be (®1 ; ° 2 ):
^
Thus, the optimal policy for an arbitrary ¸, f(¸),
is given by
f^(¸) = min[f ¤ (®1 ; ° 2 ; ¸); f¹]:
Note that for small ¸, the optimal policy if (®1 ; ° 2 ) were the true model will imply f ¤ (®1 ; ° 2 ; ¸) >
¹ Indeed, we know that f ¤ (®1 ; ° ; 0) =
f.
2
°2
®1
> f¹ =
° 1 +° 2
®1 +®2 .
This is because, if (®1 ; ° 2 ) were the
true model, one would not need to consider that the worst possible model shifts to (®2 ; ° 1 ) when
f rises above f¹.
Put di¤erently, there is a level ¹̧ > 0 for the weight on output-gap stabilization for which
f ¤ (®1 ; ° 2 ; ¹̧ ) = f¹:
Then we have
8
< f¹ · f ¤ (®1 ; ° ; ¸) if ¸ ·
2
^ =
f(¸)
: f ¤ (® ; ° ; ¸) < f¹ if ¸ >
1 2
¹̧ ;
¹̧ :
Since f ¤ (®1 ; ° 2 ; ¸) ¸ 0 and f¹ > 0, this also con…rms that the restriction f^ ¸ 0 imposed above
is not binding.
4. Discussion
Robust control consists of selecting the policy that results in the best outcome for the worst
possible model. In this example, the worst possible model ends up being on the boundary of
the assumed feasible set of models. Thus, the assumptions about the feasible set of models is
crucial for the outcome. Robust control has been proposed as a non-Bayesian alternative that
utilizes less prior assumptions about the model. If the worst possible model somehow ended up
in the interior of the feasible set of models, one could perhaps argue that the outcome is less
sensitive to the assumptions about the feasible set of models. When the worst possible model is
on the boundary of the feasible set of models, this is no longer so. Prior assumptions about the
feasible set of models become crucial to the outcome.
Furthermore, if a Bayesian prior probability measure were to be assigned to the feasible set
of models, one might …nd that the probability for the models on the boundary are exceedingly
small. Thus, highly unlikely models can come to dominate the outcome of robust control.
For weights on output-gap stabilization below a critical value (¸ · ¹̧ ), optimal policy is given
by f¹ in (3.2), which equals the “middle” °,
° 1 +° 2
2 ,
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divided by “middle” ®,
®1 +®2
2 .
Thus, policy
is “intermediate” in this sense, neither particularly “aggressive” nor particularly “cautious”
(although still depending on the bounds of the supports). For weights above that critical value
(¸ > ¹̧ ), policy is given by the optimal policy when the economy has a low ® and a high °.
The worst possible model is one where the inherent serial correlation in in‡ation is the largest
and the e¤ect of the output gap on in‡ation is the smallest. Since the response coe¢cient is
decreasing in ® and increasing in °, this means that the response coe¢cient is the largest for
the di¤erent models (when the model is known). In this sense, the policy under robust control
ends up being aggressive.
Finally, under robust control, the commitment to the reaction function is made initially
and forever, given only an assumption about the feasible set of models. Clearly, it would be
better to observe the economy for some time and learn more about the model before making a
commitment to a given policy. This leads into optimal learning, as in Wieland [15]–[17].
A. Properties of the optimal policy with known model
Di¤erentiating the …rst-order condition (2.5), it directly follows that the optimal policy f ¤ (®; °; ¸)
ful…lls
Vf ® (f ¤ ; ®; °; ¸)
2¸f ¤2 (° ¡ ®f ¤ ) + ®2 (1 + ¸f ¤2 )f ¤
=
¡
2
< 0;
Vf f (f ¤ ; ®; °; ¸)
[1 ¡ (° ¡ ®f ¤ )2 ]2 Vf f (f ¤ ; ®; °; ¸)
Vf ° (f ¤ ; ®; °; ¸)
¡2¸f ¤ (° ¡ ®f ¤ ) ¡ ®(1 + ¸f ¤2 )
=
¡
2
> 0;
f°¤ (®; °; ¸) ´ ¡
Vf f (f ¤ ; ®; °; ¸)
[1 ¡ (° ¡ ®f ¤ )2 ]2 Vf f (f ¤ ; ®; °; ¸)
¤ [1¡(°¡®f ¤ )2 ]¡®f ¤2 (°¡®f ¤ )
Vf ¸ (f ¤ ; ®; °; ¸)
= ¡ 2 f [1¡(°¡®f
f¸¤ (®; °; ¸) ´ ¡
¤ )2 ]2 V (f ¤ ;®;°;¸)
ff
Vf f (f ¤ ; ®; °; ¸)
®(°¡®f ¤ )=¸
= ¡ 2 [1¡(°¡®f ¤ )2 ]2 Vf f (f ¤ ;®;°;¸) < 0;
f®¤ (®; °; ¸) ´ ¡
where the last equality uses the …rst-order condition.
References
[1] Ball, Laurence (1999), “Policy Rules for Open Economies,” in Taylor [14].
[2] Brainard, William (1967), “Uncertainty and the E¤ectiveness of Policy,” American Economic Review 57, Papers and Proceedings, 411–425.
[3] Giannoni, Marc P. (1999), “Does Model Uncertainty Justify Caution? Robust Optimal
Monetary Policy in a Forward-Looking Model,” Working Paper.
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[4] Hansen, Lars Peter, and Thomas J. Sargent (1998), “Alternative Representation of Discounted Robust Linear Quadratic Control,” Working Paper.
[5] Onatski, Alexei (1998), “On the Robustness of Policy Rules to Di¤erent Kinds of Model
Uncertainties,” Working Paper.
[6] Onatski, Alexei, and James H. Stock (2000), “Robust Monetary Policy Under Model Uncertainty in a Small Model of the U.S. Economy,” NBER Working Paper No. 7490.
[7] Rudebusch, Glenn D. (1999), “Is the Fed Too Timid? Monetary Policy in an Uncertain
Word,” Working Paper, FRBSF.
[8] Rudebusch, Glenn D., and Lars E.O. Svensson (1999), “Policy Rules for In‡ation Targeting,” in Taylor [14].
[9] Sargent, Thomas J. (1999), “Comment on Ball (1999),” in Taylor [14].
[10] Söderström, Ulf (1999), “Monetary Policy with Uncertain Parameters,” Working Paper.
[11] Stock, James H. (1999), “Comment on Rudebusch and Svensson (1999),” in Taylor [14].
[12] Svensson, Lars E.O. (1997), “In‡ation Forecast Targeting: Implementing and Monitoring
In‡ation Targets,” European Economic Review 41, 1111–1146.
[13] Svensson, Lars E.O. (1999), “In‡ation Targeting: Some Extensions,” Scandinavian Journal
of Economics 101, 337–361.
[14] Taylor, John B., ed. (1999), Monetary Policy Rules, Chicago University Press.
[15] Wieland, V. (1996a) “Monetary Policy, Parameter Uncertainty and Optimal Learning,”
Working Paper, Federal Reserve Board.
[16] Wieland, Volker (1996b), “Learning by Doing and the Value of Optimal Experimentation,”
FEDS Working Paper No. 96–5.
[17] Wieland, V., 1998, “Monetary Policy and Uncertainty about the Natural Unemployment
Rate,” FEDS Working Paper No. 22.
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