Asymmetric forecasting and commitment policy in arobust control problem
Taro Ikeda ∗†April 5, 2013

Abstract
This paper provides a piece of results regarding asymmetric forecasting andcommitment monetary policy with a robust control algorithm. Previous studies
provide no clariﬁcation of the connection between asymmetric preference and robust commitment policy. Three results emerge from general equilibrium modelingwith asymmetric preference: (i) the condition for system stability implies an average inﬂation bias with respect to asymmetry (ii) the eﬀect of asymmetry can be
mitigated if policy makers relinquish a concern for robustness, and (iii) commitment policy may be superior to discretionary policy under widely used calibration
sets, regardless of asymmetry.
Keywords: asymmetric forecasting, commitment monetary policy, robust control
JEL classification: E50, E52, E58

1

Introduction

Recent macroeconomic scholarship has dramatically advanced in understanding economic
expectations. Research has examined not only standard rational expectations paradigms
but also more general forms of agent expectations. Amid the recent fruitful environment∗

Lecturer, Faculty of Economics, Kurume University, 1635 Mii, Kurume, Fukuoka, Japan(E-mail :young tubo80@msn.com)†
Research Fellow, Graduate School of Economics, Kobe University, 2-1 Rokko-dai, Nada, Kobe,
Hyogo, Japan1

for macroeconomics, this paper distinguishes how asymmetric forecasting (asymmetricpreference or asymmetry) aﬀects a robustly optimized commitment policy. To the bestof my knowledge, this study is the ﬁrst to implement a general equilibrium analysis ofasymmetric preference under state-contingent policy.
Asymmetric forecasting is one of the general forms of expectations and has been established by several authors, including distinguished contributions by Nobay and Peel(2003), Ruge-Murcia (2003), and Surico (2007). Asymmetric preference is useful in identifying a bias in macroeconomic expectations. The standard rational expectations requirethat agents assign a ﬁxed-symmetric weight in their assessment of risk. In practice, however, it is natural to suppose that agents assigns a diﬀerent weighting when assessing
future values (and risk) for inﬂation and/or output.
Previous studies of asymmetric preference have an advantage to formulate the agent’s
expectations realistically, but traditional methods involving asymmetry oﬀer no generalequilibrium analysis (system analysis). Prior to this paper, inquiries have lagged developments in mainstream macroeconomics by entirely depending on a Euler equationof the model and in being based on outcomes of a partial equilibrium analysis. This
study develops a theory of asymmetric preference via general equilibrium modeling torevise the macroeconomic study of asymmetric preference. It sheds light on relationships
between asymmetric preference and commitment monetary policy with a robust control
algorithm.
This paper proceeds as follows. Section 2 explains the developments of asymmetricpreference and motivations of this paper. Section 3 provides analytical and numericalresults of the model. Section 4 concludes.

2

Literature review and Motivation

Asymmetric preference conventionally has been used to identify distortions in the objectives of monetary authority and criticize their economic consequences in the context
2

of Barro and Gordon’s (1983) average inﬂation bias. Nobay and Peel (2003) introduced
a linex loss function into monetary policy analysis, in which the shape of the objectivefunction is disciplined by asymmetric preference. Existence of asymmetric preference
generates a bias in the forecasts of inﬂation and/or output. On the basis of the theory, Ruge-Murcia (2003) provided empirical evidence for asymmetry in the output (unemployment) of Federal Reserve. Surico (2007) tailored asymmetry a New Keynesian
theory and indicated the existence of expansionary-asymmetric preference for output inthe pre-Volcker period, which in turn triggered an average inﬂation bias.1 Riboni and
Ruge-Murcia (2010) considered the voting protocol of the monetary policy committee
and mentioned member-speciﬁc preferences as an asymmetry. The estimated interest
rate rule suggested that most monetary policies in developed countries do not follow a
consensus voting protocol. Again, all scholarly contributions are based on the conceptof a partial equilibrium; presently, no contribution exists in the context of a generalequilibrium perspective.
Recently, studies on asymmetric preference have been rapidly developed by econometric researchers―that is, the elicitation of agent expectations with general assumptions
rather than those used in the standard rational expectations approach. Among the vast
literature, studies closely related to this paper are Capistrán and Timmermann (2009)
and Branch (2013).2 Capistrán and Timmermann (2009) pursued a determinant formof inﬂation expectations using the Survey of Professional Forecasters and reported thatasymmetry exists in the professionals’ forecasts. They also observed a correlation between asymmetry and cross-sectional dispersion of forecasts and suggested irrationality
in expectations. Branch (2013) sought to explain the “too low for too long” phenomenonof federal funds rates by referring to the central bank’s use of a “nowcasting” Taylor

1

He reported that the asymmetry (and inﬂation bias) disappeared after the Volcker era.
See Pesaran and Weale (2006) and Manski (2004) for surveys of the ﬁelds. Engelberg et al. (2009)
also reported asymmetric tendencies in the historical evolution of market expectations with the Survey ofProfessional Forecasters, although they simultaneously expressed skepticism about whether asymmetricpreference exists.
2

3

rule with asymmetric preference in assessing current period expectations for inﬂationand output gap. He supported the existence of asymmetry and stressed the importanceof considering asymmetry in policy rules for understanding the recent behavior of U.S.
monetary policy.
The next advances in understanding asymmetry is likely to emerge from these recent
studies, so I draw heavily from Branch (2013), who modeled asymmetric forecasting Ê
for zt+1 in period t as
Êt zt+1 = Et zt+1 + φz σz,t ,(1)

where Et zt+1 is the level of standard rational expectations, φz denotes the asymmetricpreference of forecaster, and σz,t is uncertainty induced by asymmetry. If forecasters’
estimates are severely asymmetrical, the deviation from symmetric (rational) expectations is enlarged. The positive sign for asymmetry indicates that they prefer negative
deviations of zt+1 from its target level to positive deviations, and vice versa.3
Based on the relationship in (1), the New Keynesian Phillips curve with asymmetric
expectations takes the form
πt = βEt πt+1 + κxt + βφπ σπ,t + ut ,(2)

where φπ is asymmetry in inﬂation forecasting and σπ,t is its distortion. Note that
I assume that ﬁrms oﬀering asymmetric forecasts breed subsequent uncertainty in thebasic New Keynesian model, in which policymakers attempt to control for uncertainty
robustly. This assumption allows for specifying the policy objective as a quadratic form:
( ) ∑
∞
[
1
2
].
Losst =
Et
β t πt2 + λ(xt − x∗ )2 − θσπ,t
2
t=0

(3)

To complete this approach, I also beneﬁt from the robust control theory developed
3

See Nobay and Peel (2003) for details of intuition for asymmetric preference with a linex loss function.

4

by Hansen and Sargent (2008). They suppose that a robust policy suspects this modeland worry about misspeciﬁcations. To mitigate failures brought by model uncertainty,
policy makers engage in a mental zero-sum game: a robust policy minimizes social loss,
given the ﬁctitious malevolent agent’s strategy of maximizing uncertainty in the economy.
Applications of this algorithm in monetary policy include Giordani and Söderlind (2004)
for the most basic New Keynesian model and Leitemo and Söderström (2008) for a small
open economy. Numerical results suggest that the robust policy reacts more severely to a
cost-push shock than in the case of rational expectations because it becomes uncertainty
averse. Walsh (2004) analyzes robust control with a New Keynesian model and indicatesan “equivalent result” in the sense that concern for robustness does not aﬀect optimization
conditions with respect to inﬂation and output. In contrast to these ﬁndings, Tillmann(2009) insists the weakened response of robust policy when a cost channel exists.4
Per the policy objective in (3), parameter θ denotes the policy’s disregard for robustness. An inﬁnite value of θ implies that policy has no concern with robustness. The
robust policy in this study ﬁrst elicits the market’s expectations, and then optimizes thepolicy objective with model distortions speciﬁed as asymmetry-induced uncertainty inthe market’s expectations: policy makers worry about asymmetric distortions in agent
expectations and try to robustify the model against it. It is natural to assume this sequence in the robust control problem since the monetary authority elicits actual market
expectations from survey data before making a policy decision conditioned by economic
uncertainty. Moreover, Capistrán and Timmermann (2009) and Branch (2013) empirically establish asymmetric expectations in the market.5 Their ﬁndings support theapproach in this paper.

4

He interpreted the tendency as the “Brainard principle” of the robust policy.
Although Branch (2013) assumed policy has asymmetry, his assumption is essentially the same asthat taken in this paper. He elicited actual asymmetry from survey data of market participants. This
paper models the policy decision process in more detail with respect to the policy eliciting market
expectations.
5

5

3

Model

This section fully exposes the model and its properties. It reveals the model’s analyticalsolutions and presents numerical simulations based on them.

3.1

Analytical solution

This subsection distinguishes the relationship between commitment policy and asymmetry and then derives social losses under both commitment and discretionary policies.

3.1.1

Characterizing commitment policy with asymmetry

The robust policy solves the Lagrangian of the form:
( ) ∑
∞
[
]
1
2
Lt =
Et
β t πt2 + λ(xt − x∗ )2 − θσπ,t+ 2st (βEt πt+1 + κxt + βφπ σπ,t − πt ) ,2
t=0
where (β, κ, λ, θ, x∗ ) ∈ R5+ without the loss of accuracy. These parameters are generally
positive in optimal monetary policy analyses.
Optimal conditions for state-contingent policy are obtained as
πt = st − st−1 ,κ
xt = x∗ − st ,λβφπσπ,t =
st ,
θ

(4)
(5)
∀t ≥ t0 .

(6)

As usual, policymakers commit to an inertial plan to control the economy. History
dependency requires their policy to include both a purely forward-looking component
and a past state of the economy.
The distorted Phillips curve leads to the second-order diﬀerence equation of the La-

6grange multiplier st as
[

]
) 1
1(
1
L − 1 + β + g(φπ , θ) + L st = − (κx∗ + ut ),
β
β
β
2
2 2
θκ − λβ φπ
g≡
,
λθ
−1

(7)

and the following proposition arises.
proposition 1 The parameterization in standard rational expectations solutions are restored aslim g(φπ , θ) = lim g(φπ , θ) =

φπ →0

θ→∞

λ
.
κ2

This condition also implies no uncertainty in the model since limφπ →0 σπ,t = limθ→∞ σπ,t =0 in optimal condition. The rational expectations approach implies a policy that is
unconcerned with robustness, in which ﬁrms make symmetric forecasts, or both.
I factorize the diﬀerence equation as1(L−1 − µ1 )(L−1 − µ2 )Lst+1 = − (κx∗ + ut ),β
whereµ1 + µ2 =µ1 µ2 =

1+β+g
,
β

1
,
β

and obtain the solution for the smaller root asµ1 =(1 + β + g) −√(1 + β + g)2 − 4β
.
2β

(8)

The solution of the root (8) induces the following proposition.
proposition 2 Given (β, κ, λ, θ, x∗ ) ∈ R5+ , diﬀerence equation (7) has two real roots of7

which 0 < µ1 < 1 < µ2 if and only if
φπ ∈ Iφπ =  −



√

√
θκ2
λβ 2

,

θκ2
λβ 2

.

After solving the lag-polynomial, backward recursion produces∞∑
1st = κx∗ + µ1
µj1 ut−j ,
g
j=0

(9)

if the cost-push shock is i.i.d.6
The solution of the diﬀerence equation directly disciplines the price level dynamics,
so that the unconditional expectation of controlled price is1
Ept = κx∗ .
g(10)

In equation (10), average inﬂation bias arises if 1/g > 0 with κ ̸= 0. Accordingly, I
obtain following corollary.
corollary 2.1 Proposition 2 implies the average inﬂation bias w.r.t. φπ . Otherwise,
φπ ∈ IφCπ implies a deﬂationary-explosive price pass.
Corollary 2.1 stresses the harmfulness of the asymmetric preference for the outcome of
optimal policy. The asymmetry enhances the distortion in the Phillips curve and, atbest, worsens long-run inﬂation. The deﬂationary scenario will destroy the economy.

3.1.2

Loss evaluation

I derive the unconditional expectations for the worst-case loss of the form:( ) { ∑
}
∞
[ 2
]
1
t
∗ 2
2
ELosst =E Et
β πt + λ(xt − x ) − θσπ,t ,2t=06∀t > t0 .

(11)

I restrict the discussion within the i.i.d. shock case since evaluations of the expectation and loss
function are especially diﬃcult in the serial correlation case.

8

The Appendix shows that the unconditional loss under a commitment policy is calculated as
ELosscom
=
t

{
}
] µ21
1
1 2 ∗2 [2κ x + g + 2(1 − µ1 )σ ,2(1 − β) g
1 − µ21 u(12)

and the loss under a discretionary policy is
ELossdist =µ̃
2(1 − β)[(κx∗
1 − β µ̃

)2

]
+ σu2

,

(13)

where µ̃ ≡ 1/(1 + g). I compare these two through numerical simulation in the next
section, as they involve a complicated bundle of parameters.

9(a) Smaller root(c) Relative loss(b) Price level expectation

0.6

3.4

2.2
θ = 50θ = 100
θ = 200
θ = 500
θ = 1e10 (RE)

0.595
0.59

3.3

2.1

3.2

0.585
2
0.58

3.1

0.575

1.9
3

0.57
1.8
0.565

2.9

0.56
1.7

2.8

0.555
0.55

0

0.5

1

1.5

|ϕπ |

2

1.6

0

0.5

1

|ϕπ |

1.5

2

2.7

0

0.5

1

1.5

2

|ϕπ |

Figure 1: Stable root, Expected price level and Relative loss w.r.t. φπ and θ

3.2

Simulation

Figure 1 displays three results from the numerical simulation. Panel (a) indicates thesmaller root of (8). Panel (b) is the expectation for the average price level from (10).
com
Panel (b) indicates the relative unconditional loss calculated as ELossdis
int /ELosst

Section 2. In this simulation, I set κ = 0.30 as in Clarida, Galı́ and Gertler (2000),
λ = 0.25 as in McCallum and Nelson (2000) and β = 0.99 as usual. I examine φπ ∈[−2, 2] ⊂ ∩θ Iφπ (θ) with θ = {50, 100, 200, 500, 1e10} under the parameterizations.7
Panel (a) indicates that the stable roots increase with asymmetry, which expands theeﬀects of a cost-push shock (see (9)). In panel (b), an increase in asymmetry produces
increases in expectations for the price level. Under these parameterizations, both thestochastic and deterministic components of the economy increase with asymmetry. Panel(c) stresses the superiority of a commitment policy over a discretionary policy given theparameterizations. The greatest gain under commitment exceeds 300% relative to theBranch (2013) and Capistràn and Timmermann (2009) reported the interval of |φπ | from about 0
to 2.
7

10

discretionary policy. These tendencies are monotone under the calibration. Note that themonotonically increasing eﬀects of asymmetry can be suppressed with monetary policythat relinquishes concern for robustness (has larger θ).
Many researchers have emphasized the superiority of commitment policies over discretionary policies (e.g., Benigno and Woodford, 2005). I support the prevailing view ofmonetary policy analysis because the existence of asymmetry does not essentially aﬀect
the stylized fact in the simulation. The notable diﬀerences from asymmetry are its eﬀects
on the determinants of system stability and its role in creating the average inﬂation bias.

4

Conclusion

This study examined relationships between asymmetric forecasting and a robust commitment policy and established three results. First, the asymmetry providing stable roots
produces an average, long-run inﬂationary bias in commitment policy. Second, the eﬀectof asymmetry can be mitigated if policymakers disregard robustness. Third, a commitment policy is superior to discretionary policy even if asymmetry exists. This superiority
depends on the calibration in this study, but parameters were chosen from a widely used
set and are not speciﬁc to this study.

References[1] Barro, Robert and David Gordon. (1983). “Rules, discretion and reputation in amodel of monetary policy.” Journal of Monetary Economics, 12, 101-121.
[2] Benigno, Pierpaolo and Michael Woodford. (2005). “Inﬂation Stabilization And Welfare: The Case Of A Distorted Steady State.” Journal of the European Economic
Association, 3, 1185-1236.
[3] Branch, William. (2013). “Nowcasting and the Taylor Rule.” mimeograph.
11[4] Capistrán, Carlos and Allan Timmermann. (2009). “Disagreement and Biases inInﬂation Expectations.” Journal of Money, Credit and Banking, 41, 365-396.
[5] Clarida, Richard, Jordi Galı́ and Mark Gertler. (2000). “Monetary Policy Rules And
Macroeconomic Stability: Evidence And Some Theory.” The Quarterly Journal of
Economics, 115, 147-180.
[6] Engelberg, Joseph, Charles Manski and Jared Williams. (2009). “Comparing the
Point Predictions and Subjective Probability Distributions of Professional Forecasters.” Journal of Business and Economic Statistics, 27, 30-41.
[7] Giordani, Paolo and Paul Söderlind. (2004). “Solution of macromodels with HansenSargent robust policies: some extensions.” Journal of Economic Dynamics and Control, 28, 2367-397.
[8] Hansen, Lars and Thomas Sargent. (2008). Robustness. Princeton University Press.
[9] Leitemo, Kai and Ulf Söderström. (2008). “Robust monetary policy in a small open
economy.” Journal of Economic Dynamics and Control, 32, 3218-3252.
[10] Manski, Charles (2004). “Measuring Expectations.” Econometrica, 72, 1329-1376.
[11] McCallum, Bennett and Edward Nelson. (2000). “Timeless Perspectives vs. Discretionary Monetary Policy In Forward-Looking Models.” NBER Working Papers,
7915, National Bureau of Economic Research.
[12] Nobay, Robert and David Peel. (2003). “Optimal discretionary monetary policy ina model of asymmetric central bank preferences.” Economic Journal, 113, 657-665.
[13] Pesaran, Hashem and Martin Weale. (2006). “Survey Expectations.” Handbook of
Economic Forecasting, 1, 715-776.

12

[14] Riboni, Alessandro and Francisco Ruge-Murcia. (2010). “Monetary Policy by Committee: Consensus, Chairman Dominance, or Simple Majority?” The Quarterly
Journal of Economics, 125(1), 363-416.
[15] Ruge-Murcia, Francisco. (2003). “Does the Barro-Gordon model explain the behavior of US inﬂation? A reexamination of the empirical evidence.” Journal of Monetary
Economics, 50(6), 1375-1390.
[16] Surico, Paolo. (2007). “The Fed’s monetary policy rule and U.S. inﬂation: The caseof asymmetric preferences.” Journal of Economic Dynamics and Control, 31, 305324.
[17] Tillmann, Peter. (2009). “Robust Monetary Policy with the Cost Channel.” Economica, 76(303), 486-504.
[18] Walsh, Carl. (2004). “Robustly Optimal Instrument Rules and Robust Control: An
Equivalence Result.” Journal of Money, Credit and Banking, 36, 1105-13.

13

Appendix:

Calculating unconditional lossa.) commitment case
The Lagrange multiplier with an i.i.d. cost-push shock takes the AR(1) form ofst = µ1 st−1 +µ1κx∗ (1 + µ1 ) + µ1 ut .
1 − βµ1

Backward recursion yields∞∑
1 ∗st = κx + µ1
µj1 ut−jgj=0with µ1 + µ2 = (1/β)(1 + β + g) and µ1 µ2 = 1/β.
Accordingly, the squared sequence of multiplier
[
s2t =

][
]
1 ∗
1 ∗
2
2
κx + µ1 (ut + µ1 ut−1 + µ1 ut−2 + · · · )κx + µ1 (ut + µ1 ut−1 + µ1 ut−2 + · · · )ggtakes an unconditional expectation of the form(
Es2t =

1 ∗
κx
g

)2

µ21
+
σ2 .
2 u
1 − µ1Moreover, the cross product of st and st−1 is[

1 ∗st st−1 =κx + µ1 (ut + µ1 ut−1 + µ21 ut−2 + · · · )
g

][

]
1 ∗
2
κx + µ1 (ut−1 + µ1 ut−2 + µ1 ut−3 + · · · ) ,gand, therefore, its expectation is derived as(
Est st−1 =

1 ∗
κx
g

)2
+

µ21
µ1 σu2 .
1 − µ21

Inserting these expectations into the loss function in (8) in Sec. 3.1.2 yields
ELosscom
=
t

{
}
] µ21
1
1 2 ∗2 [2κ x + g + 2(1 − µ1 )σ .
2(1 − β) g
1 − µ21 u
14b.) discretionary case
Discretionary robust policy solves a one-period optimization problem subject to a distorted Phillips curve. The ﬁrst-order necessary conditions are
πt = q t ,κ
xt = x∗ − qt ,λβφπ
σπ,t =qt ,θ
where qt is the Lagrange multiplier associated with discretionary optimization.
By the distorted Phillips curve, the ﬁrst-order diﬀerence equation(1 − β µ̃L−1 )qt = µ̃(κx∗ + ut )is obtained, and it is solved asqt =µ̃κx∗ + µ̃ut .
1 − β µ̃The second moment of the solution is immediately obtained as(
Eqt2 =µ̃κx∗
1 − β µ̃)2
+ µ̃2 σu2 .

Consequently, the unconditional loss function under discretionary policy is evaluated as
ELossdist =µ̃
2(1 − β)[(κx∗
1 − β µ̃

)2

]
+ σu2

15

.