Three Essays on Robustness and Asymmetries inCentral Bank Forecasting
Taro Ikeda

Kurume University, Faculty of Economics
1635 Mii, Kurume, Fukuoka, Japan

Kobe University, Graduate School of Economics
2-1 Rokko-dai, Nada, Kobe, Hyogo, Japan(E-mail : young tubo80@msn.com)

Table of contents
1. Robust control and asymmetries in central bank forecasting

p.1-18

2. Monetary policy delegation with robust control and forecasting asymmetries

p.19-28

3. Robustness for asymmetric forecasting in the presence of heterogeneity with a New Keynesian model

p.29-41

Robust control and asymmetries in central bankforecasting
Taro Ikeda ∗†August 17, 2012

Abstract
This paper introduces asymmetric central bank forecasting into the standard
New Keynesian model within the context of robust control theory. Asymmetric
forecasting expresses policymakers’ reservations about economic forecasts, and thedegree of their reservations is reﬂected as an asymmetric preference whose existence
warrants laying aside the assumption that policymakers’ base decisions primarily
on rational expectations.
This study concludes that monetary policy becomes more aggressive because ofpolicymakers’ reservations about forecasts stemming from asymmetry, and preference for policies robust enough to overcome unanticipated situations. In addition,
adopted policies will likely amplify economic ﬂuctuations and signiﬁcantly reducesocial welfare.
Keywords: robust control, asymmetric forecasting, bounded rationality
JEL classification: E50, E52, E58

1

Introduction

The importance of considering deviations from rational expectations is receiving growing
recognition in monetary policy analysis. The customary view is that policymakers, following rational expectations, forecast the economy and implement an optimal plan with
conﬁdence in their constructed model. Unfortunately, the customary view implies thatoutcomes are vulnerable to uncertainties that policymakers failed to consider.
Hansen and Sargent (2008) posit a robust control approach to address the fact that therational expectations model overlooks uncertainties, and therefore adopted policies need∗†

Kurume University, Faculty of Economics: young tubo80@msn.com
Kobe University, Graduate School of Economics

1

to be more robust. In their approach, monetary policymakers have reservations about
their model and organize an optimal plan for the uncertain economic environment.
To further incorporate deviations from assumptions about rational expectations, Branch(2011), Capistrán and Timmermann (2009), and others empirically and theoretically emphasize asymmetric forecasting. Expectations based on asymmetric preferences introduce
additional bias in policymakers’ rational expectations, and bias is ampliﬁed by forecasting
asymmetry. Asymmetric forecasting suitably describes the realistic context of economic
forecasting―namely, policymakers’ actual forecasts are subjective and imperfect; however, they simultaneously seem to include their best rational expectations.
This paper inserts a robust control algorithm and asymmetric forecasting into thestandard New Keynesian model. We focus on the diﬀerence between the standard modeland our model, which incorporates uncertainty stemming from asymmetric forecasting.
This paper proceeds as follows. Section 2 explains robust control and asymmetricpreference and develops the model. Section 3 presents numerical results for asymmetricpreferences and policymakers’ preference for robustness. Section 4 concludes the paper.

2

The model

2.1

Basic settings

Branch (2011), Capistrán and Timmermann (2009), Surico (2007), and Riboni and RugeMurcia (2010) suggest that central bank forecasting may become asymmetrical. In theirviews, asymmetric preferences generate positive and negative deviations from rational
expectations about inﬂation and output gaps. The resulting subjective forecast for any
variable is speciﬁed as follows:Êt zt+1 = Et zt+1 + φz σz,t ,(1)

where φz is an asymmetric preference in subjective forecasting and σz,t is a conditionalvariance for the variable. According to (1), asymmetric forecasting Êt zt+1 adds bias φz σz,tto the rational expectation level, Et zt+1 . In other words, the existence of asymmetricpreferences introduces a deviation from rational expectations into policymakers’ forecasts.
Asymmetric preferences embodied in forecasts reﬂect the central bank’s reservations about its forecast (i.e., bankers question whether their expectations are rationallygrounded). As a result, policymakers wish to include a margin of error in their evaluationof the economy. Since real policy and policymakers lack a determinant reference for the
economy and form expectations about factors outside of the economic model, it is essen2

tial to consider forecasting asymmetry. In our approach, policymakers’ insecurity about
their forecast leads them to adopt policies that introduce uncertainty into the economy.
In addition, the central bank implements policies by considering the worst case thatmight arise because of uncertainties stemming from subjective forecasting. For instance,
the policymakers ﬁrst consider the situation in which uncertainty increases welfare loss
and then adopt a policy likely to oﬀset it. Hansen and Sargent (2008) establish this minmax algorithm and call the attempted response robust control. Adopting this robust
control algorithm, policymakers are concerned about model misspeciﬁcations generated
by ill-considered uncertainties and wish to fortify their model against them.
Hansen and Sargent (2008) ascribe uncertainty to a (ﬁctitious) malevolent agent who
wants to distort the economy. As the leader in a Stackelberg game, the monetary policymaker takes the strategy of the malevolent agent as given and plans his optimal strategy.
The strategy involves creating an approximating model that speciﬁes a robust decision
rule which remains in force even if a non-modeled uncertainty materializes. Applications of the robust control theory for monetary policy analysis appear in Leitemo andSöderström (2008), Tillmann (2009), Walsh (2004), and elsewhere.
The distinction of this paper is that it combines asymmetric forecasting with a robust
control algorithm and incorporates them into a standard New Keynesian model. We
assume that monetary policymakers recognize that a bias is induced by their reduced
conﬁdence in their forecast, and they wish to pursue the policy that produces robust
results even if the bias disturbs the economy. Our contribution is to consider the scenariothat emerges because monetary policymakers and policy operate without the conﬁdence
that conditions have been fully evaluated.

2.2

Robust control and asymmetric forecastingThis section combines asymmetric forecasting and robust control and incorporates theminto the canonical New Keynesian model. Monetary policy obeys the min-max strategy
of attempting to minimize welfare loss after uncertainty maximizes it.
We assume the following explicit speciﬁcation of the New Keynesian model for the
economic system:
πt = β Êt πt+1 + κyt + et ,( )1
yt = Êt yt+1 −(it − Êt πt+1 ) + ut ,
σ

(2)
(3)

Equation (2) corresponds to the New Keynesian Phillips curve based on proﬁt maximization by ﬁrms in Calvo’s sticky-price setting, and Equation (3) is the dynamic IS curve3induced by consumption, a Euler equation that captures the agent’s intertemporal optimization. Note that expectation Êt is the asymmetric forecast for the variables speciﬁed
as follows:Êt πt+1 = Et πt+1 + φπ σπ,t ,(4)Êt yt+1 = Et yt+1 + φy σy,t ,(5)where φπ and φy are the central bank’s asymmetric preferences, which recognize thetendency for forecasting bias. Inserting asymmetric forecasting into the system (2 and3) yields the following relationships:
πt = βEt πt+1 + κyt + βφπ σπ,t + et ,( )1(it − Et πt+1 )yt = Et yt+1 −
σ
( )
φπ
+
σπ,t + φy σy,t + ut ,
σ

(6)

(7)

Equations (6) and (7) are the distorted Phillips curve and the IS equation with asymmetric forecasting biases, respectively. In addition, the budget constraint for the malevolent
agent is
E0

∞
∑

2
2
β t+1 (σπ,t+1+ σy,t+1) ≤ η0 ,(8)

t=0where η0 is the supremum of the budget which represents the model misspeciﬁcations
that concern policymakers.
Taking the model structure (6-8) as given, discretionary monetary policy encounters
the problem of minimizing the welfare loss function. To this end, set the Lagrangian as
follows:
( )
( )
1
θ
2
2
2
2
(πt + λyt ) −(σπ,t+ σy,t)Lt =22−µπt (πt − βEt πt+1 − κyt − βφπ σπ,t − et )
(
( )
( )
)
1
φπ
y
−µt yt − Et yt+1 +(it − Et πt+1 ) −σπ,t − φy σy,t − ut .
(9)σσ
After rearranging for the ﬁrst-order condition, we obtainκy t = − πt ,λβφπσπ,t =
πt ,
θ

(10)
(11)

4

The condition does not restrict the IS curve because the interest rate can adjust fully
to the demand shock with elasticity σ. Pursuant to this condition, the bias for inﬂation
forecasting σπ,t increases with inﬂation asymmetry φπ and decreases with the inverse of
the preference for robustness θ.
To derive worst-case inﬂation, insert conditions (10) and (11) into the distorted
Phillips curve (6). We then obtain
πt = βAEt πt+1 + Aet ,

(12)

where
A≡

λθ
λθ + κ2 θ − λβ 2 φ2π.

Let us guess the solution of (12) with the AR(1) cost-push shock. The solution for
worst-case inﬂation is guessed as πt = a1 et , and its expectation Et πt+1 = a1 ρet to obtain:
πtW =

A
et .
1 − βρA(13)

Next, determine the worst-case output gap through the ﬁrst-order conditionytW = −κAet .
λ(1 − βρA)

(14)Insert (13) and (14) into the distorted IS curve and rearrange with respect to it , whichyields the worst-case interest rate asiW
t = Bet + σut ,

(15)

where
B≡

σκθ(1 − ρ)A + λ(θρ + βφ2π )A.
λθ(1 − βρA)

ρ = 0 in (13-15) corresponds to solutions with a white noise cost-push shock.
The worst-case model deﬁnes the economy’s movement when policymaker’s pessimism
proves to be justiﬁed. However, in reality, a useful policy to accommodate uncertainty is
to consider an approximating model in which the worst-case distortion does not appear
when the policy has a preference for robustness. To derive the approximating model, we
cannot utilize ﬁrst-order conditions as we did for deriving the worst-case model. Instead,the worst-case inﬂation and output gap are inserted into the IS curve without distortion5

to obtain the approximate output gap. Namely
( )
1
W
A
W
yt = Et yt+1 −(iW
t − Et πt+1 ) + ut
σ(λ − σκ)ρA − (1 − βρA)λB=
et .
σλ(1 − βρA)

(16)

Similarly, inﬂation is approximated with the pure New Keynesian Phillips curve asπtA =λ − κ2 Aet .
λ(1 − βρA)

(17)

Along with our models, response coeﬃcients for the shock are substantially complicatedto investigate the qualitative property about asymmetric preference and preference forrobustness, so that we conﬁrm the numerical implications for them in Section 3.

2.3

System speciﬁcation

The system speciﬁcation (6 and 7) solves using an iterative algorithm with Bellman’sequation. We brieﬂy explore the solution for our system based on Giordani and Söderlind(2004).
Most important, asymmetries φπ and φy are included as a loading for control variables,
and uncertainties σπ,t and σy,t are entirely disguised by the shocks. Keeping these facts
in mind, we situate backward- and forward-looking variables in the following matrix:


 

 

1 0 0 0
ut+1
ρy 0
0 0
ut
0


 

 

 0 1 0 0   et+1   0 ρπ 0 0   et   0 


=

+
 it
 0 0 1 1/σ   E y
 

 

1 0 

  t t+1   −1 0  yt   1/σ 
0 0 0 βEt πt+1
0 −1 −κ 1
πt
0




φy φπ /σ ()  1 0 ()

 0 βφπ  σy,t+1
 0 1  εut+1


+
+
.(18)
 0
 0 0  εe0  σπ,t+1


t+1
0
0
0 0
More succinctly,
Ψt+1 = AΨt + B ∗ u∗t + CΓt+1 .

(19)

Further, re-rewrite the system with a partitioned matrix for the backward- and forward-

6

looking variables as
(
) (
)(
)
(
ξt+1
A11 A12
ξt
=
+ B
zt+1
A21 A22
zt

)
Φ

(

it
Ωt+1

)

(
+C

ϵt+1
02×1

)
, (20)

where
(
ξt ≡

ut
et

)

(
zt ≡

,

yt
πt

)

(
,

Ωt+1 ≡

σy,t+1σπ,t+1)(
ϵt+1 ≡

,

εut+1
εet+1

)
,

and



φy φπ /σ 0 βφπ 

.
Φ≡
0 
 0

0
0
Note that forecasting asymmetries are included in loading matrix Φ. From the system,
we construct the Bellman equation as follows:()
Jt = rt + βEt Ψ′t+1 Vt+1 Ψt+1 + υt+1 ,

(21)

where
(

rt ≡ ξtzt)(

Q11 Q12Q21 Q22

)(

ξt
zt

)

(
+ 2 ξt

zt

)

(

U1
U2

)
u∗t + u∗t ′ Ru∗t .

(22)After guessing the solution for the expectation term as Et zt+1 = Ct+1 Et ξt , the following
relationship between backward- and forward-looking variables can be derived (for detailderivation, see Söderlind (2003)):zt = Dt ξt + Gt (Φ)u∗t ,(23)

ξt+1 = Ãt ξt + B̃t (Φ)u∗t + εt+1 ,

(24)

where
Dt = (A22 − Gt+1 A12 )−1 (Ct+1 A11 − A21 ),Gt (Φ) = (A22 − Ct+1 A21 )−1 (Ct+1 B1∗ (Φ) − B2∗ ),Ã1 = A11 + A12 Dt ,B̃t (Φ) = B1∗ (Φ) + A12 Gt (Φ).

7where B1∗ (Φ) is the upper side of partition matrix and Ct+1 is the solution for the guess.
Insert (23 and 24) into (22) and optimize the Bellman equation to obtain the following
policy function of the system:u∗t = −Ft (Φ)ξt .

(25)

Then forward-looking variables (23) can be expressed as
zt = (Dt − Gt (Φ)Ft (Φ))ξt .

(26)

From Equations (25) and (26), it is clear that the forecasting asymmetries directoryaﬀects the optimal policy plan and the dynamics of the system.1 From (25), it also isevident that policy decision Ft (Φ) aﬀects both its instrument and the worst-case shocks.

1

Based on these relationships, the Bellman equation is iterated until Ft (Φ) and Ct+1 converge toﬁxed values.

8

3

Numerical results

3.1

Response coeﬃcients for a cost-push shock
Inflation

Interest rate

Output gap

1.25

−1.1

1.8

−1.15

1.7

1.2
−1.2

1.6

1.15
−1.25
1.5
1.1

−1.3
1.4
−1.35

1.05
1.3

−1.4
1

1.2

−1.45
0.95
2

0
ϕπ

−2

0

200
θ

400

−1.5
2

0
ϕπ

−2

0

200
θ

400

1.1
2

0
ϕπ

−2

0

200

400

θ

Figure 1: Response coeﬃcients for a cost-push shock in the worst-case model

Table 1 speciﬁes the calibrated parameters used in the analysis. These parameters
are basic values taken mainly from Clarida, Galin, and Gertler (2000). The remaining
parameters, which they excluded, are trade-oﬀ parameters for the central bank’s targeted
output gap λ and the degree of inertia in the cost-push shock ρπ .2 We set 0.25 for the
output trade-oﬀ according to McCallum and Nelson (2000), Walsh (2003), and Tillmann(2009) among others, and 0.35 for the shock inertia as in Giannoni and Woodford (2003).
Taking these parameters as given, and based on relationships articulated in the worstcase model (12-14), we depict coeﬃcients of the variables for the cost-push shock in
Figure 1. Inﬂation and interest rates respond positively to the shock while output reacts
negatively. Note that these tendencies become more vigorous when the absolute valueof the inﬂation asymmetry increases. In contrast, the response is progressively muted aspolicymakers’ preference for robustness diminishes (θ increasing).
Alongside the worst-case model, Figure 2 shows the reaction coeﬃcients in the approximating model. They diﬀer from previous results in that inﬂation displays a decreasing
2

The inertia for demand shock, ρy , is also reported, although the parameter is less important because
the demand shock aﬀects only the interest rate adjustment, which is always equal to elasticity σ.

9Inflation

Output gap

0.9875

−1.1

0.987

−1.2

0.9865−1.3
0.986
0.9855

−1.4

0.985

−1.5

0.9845
−1.6
0.984
−1.7

0.9835
0.983
2

0
ϕπ

−2

0

200

−1.8
2

400

0
ϕπ

θ

−2

0

200

400

θ

Figure 2: Response coeﬃcients for a cost-push shock in the approximating model

responsiveness to inﬂation asymmetry, although sensitivity is extremely low. Accordingly,
the monetary policy response in the approximating model seems governed by output gap
adjustments.3

3This result is not consistent with the impulse response functions in later sections. The contradiction
might arise because the guessed solution is only among many possible solutions.

10

3.2

Detection Error Probabilities
0.5
0.45
0.4

p(θ−1)

0.35
0.3
0.25
0.2
ϕπ =0.5
ϕπ =1.0
ϕπ =1.5
ϕπ =2.0

0.15
0.1−0.025−0.02−0.015−0.01−0.0050θ−1Figure 3: Detection error probabilities with various φπ

The detection error probability determines the desired lower bound for robustness
parameter θ. If planners have extremely low preferences for robustness, the min-max
theory of robustly attempting to minimize welfare loss breaks down. At the breakdown
point, the objective function for the inner maximization problem of the multiplier game
becomes convex, and therefore the solution goes to inﬁnity (Hansen and Sargent, 2008,
Chapter 6; Giordani and Söderlind, 2004).
To avoid that misfortune, we adopt the following error detection probability:p(θ−1 ) = Prob(ln(LA /LW ) < 0|Approx.) + Prob(ln(LA /LW ) > 0|W orst),(27)

where Prob(ln(LA /LW ) < 0|Approx.) is the probability that the worst-case model is chosen when the data are generated from the approximating model, and Prob(ln(LA /LW ) >0|W orst) is vice versa. Hansen and Sargent (2008, Chapter 9 and 14), ﬁnd it reasonable
to select a robustness parameter between 0.10 or 0.20 for the probability points of thedetection error.
Figure 3 presents the simulated error detection probability. The ﬁgure suggests theprobability is 0.10 when the preference for robustness approaches 40. Accordingly, we
set 50 as the preference to preserve theoretical desirability.
11

3.3

Impulse response functions
Inflation

Output1.4
rational
approx.
worst

1.2

Interest rate

0

1.4

−0.2

1.2

−0.4
1

1
−0.6

0.8

0.8
−0.8

0.6

0.6
−1

0.4

0.4
−1.2

0.2

0
0

0.2

−1.4

2

4

6

8

−1.6
0

2

4

6

8

0
0

2

4

6

8

Figure 4: Impulse responses with φπ = 2.0 and θ = 50

Figure 4 presents impulse response functions for the rational expectations model, approximating model, and worst-case model. The asymmetry and robustness parameters
are set at 2.0 and 50. From the ﬁgure, asymmetry in inﬂation forecasting enhances theaggressiveness of monetary policy given policymakers’ preferences for robust control. In
fact, the asymmetry drives a larger surge in inﬂation, contraction in output, and risein interest rates. These responses imply that doubts about forecasts and fear of uncertainty inspire excessive aversion to forecasting uncertainty among central bankers;
thereby, making the economy more volatile and the interest rate response more vigorous. A similar interpretation of the robustness parameter can be seen in Leitemo andSöderström (2008).
To conﬁrm the phenomenon, Figure 5 plots impulse responses in the approximating
model with various values of φπ . From this, according to increases in φπ , deviations fromrational expectations increase and bankers’ aversion to forecast uncertainty grows.
On the other hand, Figure 6 shows that the vigor of impulse responses to inﬂation
asymmetry are fully suppressed if policymakers eschew robustness in rectifying model
misspeciﬁcations (corresponding to a larger θ).

12

Inflation
1.4

Outputϕπ = 0.0
ϕπ = 1.5
ϕπ = 2.0

1.2

Interest rate

0

1.4

−0.2

1.2

−0.4
1

1
−0.6

0.8

0.8
−0.8

0.6

0.6
−1

0.4

0.4
−1.2

0.2

0
0

0.2

−1.4

2

4

6

8

−1.6
0

2

4

6

8

0
0

2

4

6

8

Figure 5: Impulse responses of the approximating model for various φπ when θ = 50

13

Inflation

Output1
rational
approx.
worst

0.9

Interest rate

0

1.4

−0.2

1.2

−0.4

1

−0.6

0.8

−0.8

0.6

−1

0.4

−1.2

0.2

0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0

2

4

6

8

−1.4
0

2

4

6

8

0
0

2

4

6

8

Figure 6: Impulse responses with φπ = 2.0 and θ = 5000

These results are consistent with Giannoni (2002), Giordani and Söderlind (2004),
Hansen and Sargent (2001), and Leitemo and Söderström (2008). Policymakers’ heightened preference for robustness enhances their aversion to uncertainty and results in moreaggressive monetary policy.

3.4

Welfare loss

Welfare losses calculated by values of φπ and θ are reported in Figure 7. In the ﬁgure,
together with lower conﬁdence in forecasting and heightened preference for robustness,
welfare loss is monotonically increasing. In contrast, if the central bank is conﬁdent
that its forecasts embody rational expectations or if it disregards deviations from therational expectations model, the welfare loss tends to be identical with the level of rational
expectations.
Table 2 reports the numerical results of the welfare loss. Recovery of the welfare loss
for rational expectations level (φπ = 0.0) is achieved when (inverse of willingness for)robustness θ equals 1700 in the line of φπ = 2.0. This result suggests that discarding
robustness is required to approach the loss for the rational expectations model when
conﬁdence in forecasting is relatively low.
All told, if policymakers deal assertively with uncertainty stemming from asymmetric

14

260
240
220
200
180
160
0.02
140
2

0.015
1.5

0.01
1

0.005

0.5
0
ϕπ

0

θ−1

Figure 7: Welfare losses with various φπ and θ

preferences, the economy becomes more volatile and the anticipated welfare losses are
greater.

4

Conclusion

This paper has examined robust control for uncertainty generated by asymmetric forecasting of monetary policy. Beginning with the basic New Keynesian model, we assume
the central bank lacks perfect conﬁdence in economic forecasts and prefers a robust response to counteract uncertainty. The degree of the central bank’s conﬁdence or lackthereof is represented by an asymmetric preference in forecasting.
The simulation results suggest that an increase in inﬂation asymmetry prompts amore vigorous policy reaction to a cost-push shock because the asymmetry provokes
central bankers’ aversion to uncertainty. Moreover, welfare loss increases as asymmetry
increases. Overall, these tendencies are exacerbated when policymakers lack conﬁdence
in forecasts and prefer robustness.

References[1] Branch, William. (2011).

Nowcasting and the Taylor Rule.
15

mimeograph.

[2] Capistrán, Carlos, and Allan Timmermann. (2009). Disagreement and Biases inInﬂation Expectations. Journal of Money, Credit and Banking, 41, 365-396.
[3] Clarida, Richard, Jordi Gali, and Mark Gertler. (2000). Monetary Policy Rules
And Macroeconomic Stability: Evidence And Some Theory. The Quarterly Journalof Economics, 115, 147-180.
[4] Giannoni, Marc. (2002). Does Model Uncertainty Justify Caution? Robust Optimal
Monetary Policy In A Forward-Looking Model. Macroeconomic Dynamics, 6, 111144, February.
[5] Giannoni, Mark, and Michael Woodford. (2003). How Forward-Looking is Optimal
Monetary Policy? Journal of Money, Credit and Banking, 35, 1425-1469.
[6] Giordani, Paolo, and Paul Söderlind. (2004). Solution of macromodels with HansenSargent robust policies: some extensions. Journal of Economic Dynamics andControl, 28, 2367-397.
[7] Hansen, Lars, and Thomas Sargent. (2001). Acknowledging Misspeciﬁcation in
Macroeconomic Theory. Review of Economic Dynamics, 4, 519-535.
[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] 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.
[11] Söderlind, Paul. (2003).
UNISG). mimeograph.

Lecture Notes for Monetary Policy (PhD course at[12] Surico, Paolo. (2007). The Fed’s monetary policy rule and U.S. inﬂation: Thecase of asymmetric preferences. Journal of Economic Dynamics and Control, 31,
305-324.
[13] Tillmann, Peter. (2009).
nomica, 76, 486-504.

Robust Monetary Policy with the Cost Channel.

Eco-[14] Walsh, Carl. (2003). Speed Limit Policies: The Output Gap and Optimal Monetary
Policy. American Economic Review, 93, 265-278.
16

[15] Walsh, Carl. (2004).
Equivalence Result.

Robustly Optimal Instrument Rules and Robust Control: An
Journal of Money, Credit and Banking, 36, 1105-13.

17βTable 1: Calibration
σ
κ
λ
ρy

ρπ

0.99

1.00

0.35

0.30

0.25

0.35

Table 2: Welfare losses for the approximating modelParameters θ = 50 θ = 200 θ = 1700 θ = 5000
φπ = 0.0
149.17 149.17
149.17149.17φπ = 0.5
151.36 149.70
149.23149.19φπ = 1.0
158.92 151.36
149.42149.25φπ = 2.0
246.10 158.92
150.18149.5118

Monetary policy delegation with robust control andforecasting asymmetries
Taro Ikeda ∗†August 17, 2012

Abstract
This paper considers problems in delegating monetary policy to a central banker
that prefers a robust optimal policy when its economic model suﬀers uncertainty
stemming from asymmetric forecasting. It distinguishes the outcomes of delegating
to conservative versus activist monetary policymakers under conditions of asymmetric forecasting. Results suggest the economy ﬂuctuates less under guidance by
a conservative central banker than under an activist banker. Accordingly, social
welfare approaches the level attained under rational expectations if policy is delegated to a conservative banker.
Keywords: delegation problem for monetary policy, robust control, asymmetric
forecasting
JEL classification: E52, E58

1

Introduction

The recent ﬁnancial crisis illuminates numerous issues in conducting monetary policy
during periods of uncertainty. In examining diﬃculties imposed by the crisis, numerous researchers have investigated how monetary policymakers tackle events that are notexplicitly anticipated in their constructed models―in a word, uncertainty.
Hansen and Sargent (2008) incorporated uncertainty into the agent decision problem
by adapting robust control techniques. In their algorithm, the agent assumes that theworst case induced by uncertainty is a given, and then it chooses the course that maximizes social welfare. Uncertainty aﬀects the economic dynamics through the min-max
decision rule. Robust control notably has been applied to monetary policy analysis by
Leitemo and Söderström (2008), Tillmann (2009a, b), and Walsh (2004) among others.
∗†

Kurume University, Faculty of Economics: young tubo80@msn.com
Kobe University, Graduate School of Economics

19Branch (2011), Capistrán and Timmermann (2009), and Surico (2007) stressed asymmetric forecasting in considering deviations from rational expectations (RE). Ikeda (2012)combined a robust control algorithm and forecasting asymmetry and built them into thestandard New Keynesian system. He found that forecasting asymmetry promotes aggressive monetary policy excessively and reduces social welfare.
Following Ikeda (2012), this paper examines the monetary policy delegation problem.
Examining policy tradeoﬀs between stabilizing inﬂation and output, we discuss how thecentral banker’s characterization as aggressive or conservative aﬀects the New Keynesian
economy.
The paper proceeds as follows. Section 2 brieﬂy introduces the model. Section 3
reports numerical results of the economy’s impulse responses and welfare gains among
the models considered. Section 4 concludes the study.

2

Summary of the model

The New Keynesian Phillips curve and the dynamic IS curve based on asymmetric forecasting are
πt = β Êt πt+1 + κyt + et , and( )1yt = Êt yt+1 −(it − Êt πt+1 ) + ut ,
σ

(1)
(2)

where Êt denotes the asymmetric forecast. The conventional speciﬁcation for asymmetric
forecasting in period t + 1 takes the form ofÊt πt+1 = Et πt+1 + φπ σπ,t(3)Êt yt+1 = Et yt+1 + φy σy,t(4)

The asymmetries φπ and φy reﬂect the degree of reservations for the central bank in theforecast. If the asymmetry is large, the bank regards its forecast as less accurate and theforecast actually becomes to include a margin toward a rational expectation.
Inserting asymmetric forecasting (3-4) into the model, the standard New Keynesiansystem can be rewritten as:
πt = βEt πt+1 + κyt + βφπ σπ,t + et ,
( )
( )
1
φπ
yt = Et yt+1 −(it − Et πt+1 ) +σπ,t + φy σy,t + ut ,
σ
σ

20

(5)
(6)

Equations (5-6) denote the distorted Phillips curve and the distorted IS curve, respectively. Cost-push shock et and demand shock ut follow the AR (1) process:et = ρπ et−1 + επtwith 0 ≤ ρπ < 1,επt ∼i.i.d.N (0, 1)(7)

ut = ρy ut−1 + εyt

with 0 ≤ ρy < 1,

εyt ∼i.i.d.N (0, 1)(8)

The matrix for this system of backward-looking and forward-looking variables can be
expressed as follows:1
 

 



0
ρy 0
0 0
ut
ut+1
1 0 0 0
 

 



 0 1 0 0   et+1   0 ρπ 0 0   et   0 
+
i
=






 0 0 1 σ −1   E y  −1 0
−1  t1 0   yt   σ   t t+1  
0 −1 −κ 1πt
0Et πt+1
0 0 0 β
φy σ −1 φπ)  1 0 (
)

(
 0
 σy,t+1
 0 1  εut+1
βφ
π


+
+
(9)
 0



e
0  σπ,t+1
 0 0  εt+1
0
0
0 0
Stacking these as follows,Ψt+1 = AΨt + B ∗ u∗t + CΓt+1 .

(10)

where


1

 0
B ∗ u∗t = 
 0

0

0
1
0
0

−1 



0 0
0 φy σ −1 φπ
 

it

0 0 
0
βφπ 

  0

 σy,t+1 


−1 
−1
1 σ   σ00 σπ,t+1
0 β
0
0
0

(11)

Then, the monetary policy objective is speciﬁed as a quadratic form of
E0∞∑

β t (πt2 + λyt2 ),

(12)

t=0

where λ is a tradeoﬀ parameter between inﬂation and the output gap. In this sense, thecentral banker that confronts a small λ emphasizes stabilizing inﬂation (conservatism),
whereas the banker confronting a large λ addresses output stabilization (activism).
In the context of optimization, the min-max decision rule adds a penalty for uncer-

1

See Giordani and Söderlind (2004) for manipulations of the matrix.

21

tainty to the standard loss (12). Therefore, the problem is formulated as
min max E0

∞
∑

(
β

t

)
2
2
πt2 + λyt2 − θ(σπ,t+1+ σy,t+1
)

(13)

t=0

s.t. Ψt+1 = AΨt + B ∗ u∗t + CΓt+1 .
The explicit ﬁrst-order conditions of the problem with a pen and pencil method areκyt = − πt andλβφπσπ,t =
πt .
θ

(14)
(15)

These conditions imply that the asymmetry in output forecasting has no eﬀect, sinceconditions related to the IS equation do not aﬀect the solution.
In pursuit of a numerical solution, formulating and solving Bellman equations yields
the policy functionu∗t = −Ft (Φ)ξt ,(16)

where ξt is the bundle of shocks and Φ denotes a loading matrix including asymmetricparameters. Also, the bundle of forward-looking variables zt is determined as
zt = (Dt − Gt (Φ)Ft (Φ))ξt .

(17)

According to (16-17), the dynamics of monetary policy and the economy are directly
aﬀected by loading matrix Φ, which consists of forecasting asymmetries. With thesesolutions, we can derive the approximating model under which as a practical matter
uncertainty does not appear (corresponding to the system of σπ,t+1 = σy,t+1 = 0 with theworst-case solutions for inﬂation and output).
The next section presents numerical results centered on the tradeoﬀ represented by
diﬀering values of λ. Conditioned on the uncertainty in asymmetric forecasting, howdoes the economic dynamics depend on the central banker’s characteristics, and when is
it desirable to delegate policymaking to a conservative or an activist central banker?

22

3

Numerical results

3.1

Impulse responses

We calibrate the intertemporal discount rate β equal to 0.99, the demand elasticity forinterest rate σ at 1.00, and the slope of the Phillips curve κ at 0.30, as in Clarida, Gali,
and Gertler (2000). Also, inertia for the cost-push shock is 0.35 from Giannoni andWoodford (2003). The (inverse of) the central bank’s preference for robustness θ is set at
50 according to a detection error probability. See Hansen and Sargent (2008) for criteria.
Figure 1 shows the impulse responses under a conservative central banker (λ = 0.05)
and Figure 2 under an activist banker (λ = 0.20). As per Figure 1, under a conservative
central banker intent on stabilizing inﬂation, the impulse responses of the approximating
model are close to those of the RE model. As a result, monetary policy adjusts thenominal interest rate moderately.2 On the other hand, as in Figure 2, the activism of thebanker causes vigorous movements of economy and its instrument toward a cost-pushshock. The level of contraction in output is smaller than under the conservative case,but, the surge in inﬂation justiﬁes the excess tightening of monetary policy. Also, notethat these responses do not depend on the sign of asymmetry φπ , as asymmetry is always
squared in solutions within a linear-quadratic framework.
As Figure 3 shows, the monetary policy response varies according to the values oftradeoﬀ parameter λ. Figure 3 conﬁrms the evidence in Figures 1 and 2. Policy responses
for a cost-push shock in the approximating model are slightly attenuated compared toresponses in the RE model for lower values of λ. For larger values of λ, in contrast,
the interest rate is gradually hiked higher than is called for in the RE model. When λreaches 0.20, the interest rate is raised about 1.2% in the RE model, but 1.3% in theapproximating model (with φπ = 2.0). The monetary policy stance to a cost-push shock
depends on the inﬂation asymmetry as well as the tradeoﬀ between stabilizing inﬂationand the output gap.

2

The attenuated policy response in the approximating model implies the Brainard principle in which
cautiousness toward uncertainty weakens a policy’s response to it.

23

Inflation

Output0.45

Interest rate
1.8

0
RE
Approx.

0.4

1.6
−0.5

0.35

1.4

0.3

1.2
−1

0.25

1

0.2

0.8
−1.5

0.15

0.6

0.1

0.4

−2

0.05
0
0

0.2

2

4

6

8

−2.5
0

2

4

6

8

0
0

2

4

6

8

6

8

Figure 1: Impulse Responses with φπ = 2.0 and λ = 0.05
Inflation

Output1.4

Interest rate

0

1.4

−0.2

1.2

RE
Approx.
1.2

−0.4
1

1
−0.6

0.8

0.8
−0.8

0.6

0.6
−1

0.4

0.4
−1.2

0.2

0
0

0.2

−1.4

2

4

6

8

−1.6
0

2

4

6

8

0
0

2

4

Figure 2: Impulse Responses with φπ = 2.0 and λ = 0.20

24

RE model

Approximating model

1.8

1.8

1.6

1.6

1.4

1.4

1.2

1.2

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0
0

0
0

tim

n

0.1
λ

2

o
iz

n

0.05

0.2

or

o
iz

4

0.15

eh

or

eh

tim

2

4

0.05

0.1
λ

0.15

0.2

Figure 3: Interest Rate Responses with Various λ when φπ = 2.0 (approximating model)

3.2

Welfare gains under delegation

Figure 4 depicts welfare gains between approximating and RE models for values of tradeoﬀ parameter. The left panel shows the welfare gains (losses) of the approximating modelin the conservative case compared to the RE model. The middle panel compares theactivist case to the RE case. The right panel compares the approximating model of conservatism to that of activism. The left and middle panels suggest that the central bank’s
preference for robustness and the existence of asymmetric preferences reduce social welfare compared to the RE model, but the degree of welfare loss is substantially less undera conservative banker. For instance, delegating policy to a conservative central banker
can signiﬁcantly suppress deviation from the RE model. Also, convergence to the RE
model can be achieved when inﬂation asymmetry and preference for robustness approachzero. This tendency makes sense, for the RE model can be recovered with asymmetry
and the (inverse of) robustness parameter being zero.
The right panel of Figure 4 supports the implications above. The conservative central banker enhances welfare, and this tendency is notable when inﬂation asymmetry
and preference for robustness are increased. If the central bank has reservations about
its economic forecast and a preference for robustness when its model displays forecasting uncertainty, welfare gains are substantially larger under a conservative than underan activist banker. This implication resembles Tillman (2009a), who also reports that
25

−WRE
WApprox.
λ=0.05λ=0.05−WRE
WApprox.
λ=0.20λ=0.200−WApprox.
WApprox.
λ=0.20λ=0.05

0

130

−10

120

−20

110

−30

100

−40

90

−0.2
−0.4
−0.6
−0.8
−1
−1.2
0.02

−1.4
2

0.02

−50
2

2

0.01

0.01

0
ϕπ

0.02

80
0.01

0
−2 0

θ−1

ϕπ

0
−2 0

θ−1

ϕπ

−2 0

θ−1

Figure 4: Comparison of Welfare Gains between Approximating and RE Models for
Values of λApprox.
Note: Social welfare is calculated as a minus of social loss. Wλ=0.05
denotes welfare whenRE
λ = 0.05 in the approximating model and Wλ=0.05
in the RE case. Wλ=0.20 corresponds to thecase of λ = 0.20.

26

conservative central banking could curtail welfare losses under uncertain circumstances.

4

Conclusion

We have analyzed the problem of delegating monetary policy when the central bankhas a preference for robust optimal policy under conditions of asymmetric forecasting.
Numerical results suggest that social welfare toward a cost-push shock is high and close
to the outcome achieved under rational expectations if monetary policy is delegated tothe conservative central banker.

References[1] Branch, William. (2011).

Nowcasting and the Taylor Rule.

mimeograph.

[2] Capistrán, Carlos, and Allan Timmermann. (2009). Disagreement and Biases inInﬂation Expectations. Journal of Money, Credit and Banking, 41, 365-396.
[3] Clarida, Richard, Jordi Gali, and Mark Gertler. (2000). Monetary Policy Rules
And Macroeconomic Stability: Evidence And Some Theory. The Quarterly Journalof Economics, 115, 147-180.
[4] Giannoni, Marc, and Michael Woodford. (2003) How Forward-Looking is Optimal
Monetary Policy? Journal of Money, Credit and Banking, 35, 1425-1469.
[5] Giordani, Paolo, and Paul Söderlind. (2004). Solution of macromodels with HansenSargent robust policies: some extensions. Journal of Economic Dynamics andControl, 28, 2367-397.
[6] Hansen, Lars, and Thomas Sargent. (2008). Robustness. Princeton University Press.
[7] Ikeda, Taro. (2012).
mimeograph.

Robust control and asymmetries in central bank forecasting.

[8] Leitemo, Kai, and Ulf Söderström. (2008). Robust monetary policy in a small open
economy. Journal of Economic Dynamics and Control, 32, 3218-3252.
[9] Surico, Paolo. (2007). The Fed’s monetary policy rule and U.S. inﬂation: Thecase of asymmetric preferences. Journal of Economic Dynamics and Control, 31,
305-324.

27

[10] Tillmann, Peter. (2009a). Does Model Uncertainty Justify Conservatism? Robustness and the Delegation of Monetary Policy, The B.E. Journal of Macroeconomics,
9 (1).
[11] Tillmann, Peter. (2009b).
nomica, 76, 486-504.
[12] Walsh, Carl. (2004).
Equivalence Result.

Robust Monetary Policy with the Cost Channel.

Eco-

Robustly Optimal Instrument Rules and Robust Control: An
Journal of Money, Credit and Banking, 36, 1105-13.

28

Robustness for asymmetric forecasting in the presenceof heterogeneity with a New Keynesian model
Taro Ikeda ∗†August 17, 2012

Abstract
This paper investigates three types of heterogeneity stemming from asymmetricpreferences (Type I), uncertainty about an economic model (Type II), and both
asymmetric preferences and model uncertainty (Type III) in asymmetric forecastingby a central bank and private agents. Results suggest that asymmetric preferences
amplify volatility of economic system, similar to that in homogeneous forecasts.
However, if the central bank’s and private agents’ asymmetric preferences sharethe same directionality under Type III heterogeneity, social welfare under Type III
is higher than that under Type I heterogeneity. In contrast, Type I heterogeneity
is desirable in diﬀerent signs of asymmetric preferences.
Keywords: robust control, asymmetric forecasting, heterogeneous expectations
JEL classification: E50, E52, E58

1

Introduction

Rational expectations have dominated macroeconomic analysis since 1970s, and monetary policy analysis is no exception. However, the central assumption of rational expectations that agents always possess complete information about underlying economic
structures has been sharply criticized, and homogeneity in formulating expectations has
not been supported by empirical studies. In response, there exist two major reﬁnements
to the expectation theory, the learning and the robust control. In the learning approach,∗

Kurume University, Faculty of Economics, 1635 Mii, Kurume, Fukuoka, Japan(E-mail :young tubo80@msn.com)†
Kobe University, Graduate School of Economics, 2-1 Rokko-dai, Nada, Kobe, Hyogo, Japan29

economic agents learn to recognize economic laws of motion, and eventually, reach conclusions consistent with a rational expectations solution (Evans and Honkapohja, 2001).1
On the other hand, the robust control assumes that the agents have willingness to adjust misspeciﬁcations between data-generated and approximating models (Hansen andSargent, 2008). Agents adopt a max―min optimization principle to address uncertainty
arising from model misspeciﬁcations. In a sense, robust controls incorporate agents’beliefs about the reliability of an economic model, an issue excluded from rational expectations theory.
Time is the major diﬀerence between the learning and robust control. Learning consumes time, whereas agents can acquire a decision rule based on robust controls quickly
with limited data. In addition, the agents in learning pursue the rational expectations
solution as a end point for a decision when learnability condition is satisﬁed, whereas therobust control approach regards it as one of many possible solutions within an entropy
ball deduced by an algorithm.
In addition to these two reﬁnements, asymmetric forecasting is also an important
departure from the conventional rational expectations theory. Branch (2011), Capistrán
and Timmermann (2009), and Surico (2007) empirically support the existence of asymmetric preferences in monetary policy. Agents with asymmetric preferences introduce
asymmetries into expectations and breed asymmetric biases into expectations, either ofwhich could be optimistic or pessimistic, depending on the particular asymmetric preference.
Understanding heterogeneity in expectations is important for injecting reality into
macroeconomics and monetary policy analysis. The robust control approach is taking
hold in modern macroeconomics, including monetary economics, because it imparts a
degree of realism to decision making that escapes the conventional rational expectations
theory. Therefore, this paper considers heterogeneous asymmetry in forecasting under arobust control algorithm to cast the analysis in a New Keynesian framework. In this analysis, the central bank and public agents proﬀer diﬀering economic forecasts arising fromtheir asymmetric preferences and uncertainties. Further, we consider that social planners wish to defend policy outcomes against uncertainties stemming from asymmetric
forecasting. Heterogeneity arises from diﬀerences in forecasters’ asymmetric preferences
and their respective uncertainties about an economic model. We consider three types ofheterogeneity: heterogeneity arising from asymmetric preferences (Type I), uncertainty
about an economic model (Type II), and both asymmetric preferences and model uncer1

Giannitsarou (2003), Honkapohja and Mitra (2006), and Branch and McGough (2009) analyzed
heterogeneous expectations under learning and reported that system stability conditions could diﬀer
through heterogeneity.

30

tainty (Type III).
This paper proceeds as follows. Section 2 analyzes these three types of heterogeneity.
Section 3 conﬁrms theoretical results with numerical simulations to compare welfare
outcomes. Section 4 concludes this paper.

2

Model settings

This section introduces three types of forecasting heterogeneity with asymmetry. We address them through a robust control algorithm and deduce their respective social welfare
gains presented in the models. Analysis reveals that asymmetric preference is the mostimportant factor because it disciplines diﬀerences in results by the types of heterogeneity.

2.1

Type I: heterogeneity from asymmetric preferences

First, we consider Type I heterogeneity in which asymmetric forecasting is induced solely
by diﬀerences in the central bank’s and private agents’ asymmetric preferences.
The central bank’s asymmetric forecast for any variable zt can be expressed as
ÊtCB zt+1 = Et zt+1 + 2φCBz σz,t(1)

where φCBis its asymmetric preferences and σz,t is uncertainty inferred by asymmetricz
forecasting.
Similarly, private agents’ asymmetric forecasting is expressed as
ÊtP zt+1 = Et zt+1 + 2φPz σz,t ,(2)

where φPz denotes their asymmetric preferences. Then, social expectations can be expressed as an average of the bank’s and private agents’ forecasts as
1+ φPz )σz,t .
Êt zt+1 = (ÊtCB zt+1 + ÊtP zt+1 ) = Et zt+1 + (φCB
z
2

(3)

Accordingly, the basic New Keynesian economy with asymmetric forecasting yields
the following relations:P
πt = βEt πt+1 + κyt + β(φCB
π + φπ )σπ,t + et ,( )1
yt = Et yt+1 −(it − Et πt+1 )σ)( CB
φπ + φPπ+ φPy )σy,t + ut ,σπ,t + (φCB
+
y
σ

31

(4)

(5)

Equation (4) is the New Keynesian Phillips curve (NKPC) distorted by asymmetric
forecasting. Equation (5) is the distorted dynamic IS (DIS) curve. The NKPC implies
intertemporal optimization by monopolistic ﬁrms facing Calvo-type price rigidities. The
DIS is derived from the private sector Euler equation.
Taking these distorted structures as given, social planners wish to fortify outcomesagainst uncertainty by establishing decision rules. To this end, we set the Lagrangian as
( )
( )
θ
1
2
2
2
2
)
+ σy,tLt =(πt + λyt ) −
(σπ,t
2
2
(
)
P
−µπt πt − βEt πt+1 − κyt − β(φCB
π + φπ )σπ,t − et
(
( )
1
y
−µt yt − Et yt+1 +(it − Et πt+1 )
σ
( CB
)
)
φπ + φPπCBP−
σπ,t − (φy + φy )σy,t − ut .
σ
The optimization conditions of this problem areκy t = − πt ,
and
λ
P
β(φCB
π + φπ )σπ,t =
πt .
θ

(6)
(7)

Substituting conditions (6) and (7) into the distorted systems and positing a cost-pushshock that follows a ﬁrst-order autoregressive process yields the worst-case solutions as
πtW =

Â

et ,

(8)

et ,

(9)

iW
t = B̂et + σut ,(10)ytW = −

1 − βρÂκÂ

λ(1 − βρÂ)

where
Â ≡

λθ
.
P 2
λθ + κ2 θ − λβ 2 (φCB
π + φπ )

and
B̂ ≡

P 2
σκθ(1 − ρ)Â + λ(θρ + β(φCB
π + φπ ) )Âλθ(1 − βρÂ).

Inserting worst-case solutions into non-distorted systems yield the following approxi-

32mating model:ytA =(λ − σκ)ρÂ − (1 − βρÂ)λB̂
σλ(1 − βρÂ)

et .

(11)

Similarly, inﬂation is approximated with the pure New Keynesian Phillips curve asπtA =λ − κ2 Â
λ(1 − βρÂ)

et .

(12)

With these solutions, we can restore homogeneity to asymmetric forecasting if (φCBπ +P 2
φPπ )2 = φ2π in Â and B̂. We can do so because the expression (φCB
π + φπ ) is the sole difference between worst-case solutions under heterogeneous and homogeneous forecastingmodels.

2.2

Type II: heterogeneity from uncertainty

Next, we consider Type II heterogeneity that arises solely from uncertainty about themodel. Assuming the central bank’s asymmetric forecast is
CB
ÊtCB zt+1 = Et zt+1 + 2φz σz,t
,

(13)

CB
where σz,tis the uncertainty considered by the policy. The private agents’ asymmetric
forecast is
P
ÊtP zt+1 = Et zt+1 + 2φz σz,t
,

(14)

P
where σz,tdenotes the uncertainty that they must resolve. Note that the asymmetricpreference, φz , is identical for policymakers and private agents. When Type II uncertainty
prevails, social expectations are formed as the average of Equations (13) and (14), as
shown in the following equation:

1
P
CB
).
+ σz,tÊt zt+1 = (ÊtCB zt+1 + ÊtP zt+1 ) = Et zt+1 + φz (σz,t2(15)

Employing the expectation formulation of (15) yields the following economic structures:PCB) + et ,+ σπ,tπt = βEt πt+1 + κyt + βφπ (σπ,t

33

(16)

and
yt

( )
1
= Et yt+1 −(it − Et πt+1 )
σ
( )
φπ
CB
P
CB
P
(σπ,t
+
+ σπ,t) + (σy,t+ σy,t) + ut ,σ(17)Social planners who seek robustness set the Lagrangian as
( )
( )
)
θ ( CB 2
1
2
2
P 2
CB 2
P 2
(πt + λyt ) −(σπ,t ) + (σπ,t) + (σy,t) + (σy,t
)
Lt =
2
2
(
)
CB
P
−µπt πt − βEt πt+1 − κyt − βφπ (σπ,t+ σπ,t) − et(( )1y(it − Et πt+1 )−µt yt − Et yt+1 +
σ
)
( )
φπ
CB
P
CB
P
(σπ,t + σπ,t ) − φy (σy,t + σy,t ) − ut .
−σFirst-order conditions of the problem areκy t = − πt ,
λ

(18)

CB
σπ,t
=

βφπ
πt ,
θ

(19)

P
σπ,t
=

βφπ
πt ,
θ

(20)

and

Conditions (19) and (20) suggest that policymaker’s and private agents’ uncertainties
are equivalently determined. Substituting these conditions into the distorted NKPC andDIS yields
πtW =

Ā
et ,
1 − βρĀ

ytW = −

(21)

κĀ
et ,
λ(1 − βρĀ)

(22)

iW
t = B̄et + σut .

(23)

34

whereĀ ≡λθλθ + κ2 θ − 2λβ 2 φ2π

,

and
B̄ ≡

σκθ(1 − ρ)Ā + λ(θρ + 2βφ2π )Ā.
λθ(1 − βρĀ)Solutions of the approximating model areytA =(λ − σκ)ρĀ − (1 − βρĀ)λB̄
et .
σλ(1 − βρĀ)

(24)

πtA =

λ − κ2 Āet .
λ(1 − βρĀ)

(25)

and

These solutions are exactly the same as those for the homogeneous model. Therefore,
Type II heterogeneity falls from consideration.

2.3

Type III: heterogeneity from both asymmetric preferences
and uncertainty

Now, we consider heterogeneity that arises from the central bank’s and private agents’
asymmetric preferences and model uncertainty. When Type III heterogeneity prevails,
the expectation formulation of central bank is speciﬁed as
CB
ÊtCB zt+1 = Et zt+1 + 2φCBz σz,t ,(26)

and that of private agents is
P
ÊtCB zt+1 = Et zt+1 + 2φPz σz,t,(27)

Therefore, the averaged social expectation is given byCB
P P
Êt zt+1 = Et zt+1 + φCBz σz,t + φz σz,t .

(28)

35

From these, the distorted New Keynesian system becomesCB
P P
πt = βEt πt+1 + κyt + βφCBπ σπ,t + βφπ σπ,t + et ,( )1(it − Et πt+1 )yt = Et yt+1 −σ( CB )
( P)
φπ
φπ
CB
P
CB
P P
+
σπ,t +σπ,t+ φCB
y σy,t + φy σy,t + ut ,
σ
σ

(29)

(30)

For Type III heterogeneity, we set the Lagrangian as
( )
( )
θ
1
2
2
2
2
)
+ σy,t(πt + λyt ) −
(σπ,t
Lt =
2
2
(
)
CB
P P
−µπt πt − βEt πt+1 − κyt − βφCBπ σπ,t − βφπ σπ,t − et
(
( )
1
y
−µt yt − Et yt+1 +(it − Et πt+1 )σ( CB )
( P)
)
φπ
φπ
CB
P
CB CB
P P
−
σπ,t −
σπ,t − φy σy,t − φy σy,t − ut .
σσThis optimization yields the following ﬁrst-order conditions:κ
y t = − πt ,
λ

(31)

CB
σπ,t
=

βφCB
π
πt ,
θ

(32)

P
σπ,t
=

βφPπ
πt ,
θ

(33)

and

Worst-case solutions can be derived from these conditions as:
πtW =

Ã
et ,
1 − βρÃ

ytW = −

(34)

κÃ
et ,
λ(1 − βρÃ)

(35)

and
iW
t = B̃et + σut ,

(36)

36

where
Ã ≡

λθ
),
(
2 + (φP )2
(φCB
)
π
π

λθ + κ2 θ − λβ 2and()( CB 2)P 2σκθ(1 − ρ)Ã + λ θρ + β (φπ ) + (φπ )
Ã
B̃ ≡

.

λθ(1 − βρÃ)In addition, the approximating model’s solutions areytA =(λ − σκ)ρÃ − (1 − βρÃ)λB̃
et .
σλ(1 − βρÃ)

(37)

πtA =

λ − κ2 Ãet .
λ(1 − βρÃ)

(38)

and

Altogether, solutions exhibit closely similar forms under all three types of heterogeneity. The only diﬀerences are in the denominator of A and the numerator of B as
2
P 2
(φCB
+ φPπ )2 in Type I and (φCBππ ) + (φπ ) in Type III. The corresponding parts areintegrated as φ2π in the homogeneous model.

2.4

Comparison of Types I and III

The diﬀerence between Type I and Type III heterogeneity is how their solutions incorporate asymmetries. To consider the diﬀerence, we deﬁne the following convolution ofasymmetries in solutions for each type:P 2
ΨI ≡ (φCB
π + φπ ) ,

(39)

2
P 2
ΨIII ≡ (φCBπ ) + (φπ ) .

(40)

and

ΨI and ΨIII generate the entire diﬀerence of consequences between Type I and TypeIII heterogeneity, since the social welfare is monotonously decreasing with them (Ikeda,2012). Therefore, when ΨI equals ΨIII , social welfare is identical under Types I and III.

37

This situation is conditioned asPΨI − ΨIII = 2φCBπ φπ = 0.

(41)

To achieve condition (41), at least one party (the central bank or private agents) mustbe fully rational.
Social welfare under Type I heterogeneity exceeds Type III whenPΨI − ΨIII = 2φCBπ φπ < 0.

(42)

In this condition, the directionality of agents’ asymmetric preferences diﬀers from that of
the central bank, suggesting that the belief for uncertainty is entirely adverse among thepolicymaker and private agents. The central bank’s and private agents’ beliefs oﬀset eachother, weakening the transmission of uncertainty into the economy through asymmetric
forecasting (Equation (3)). As a result, the ﬂuctuation stemming from model uncertainty
is reduced.
In addition, when the conditionPΨI − ΨIII = 2φCBπ φπ > 0(43)

holds, Type III heterogeneity imparts greater social welfare because uniformly directional
asymmetries amplify uncertainty under Type I heterogeneity.
In summary, asymmetric preferences interact when asymmetry is heterogeneous anduncertainty is homogeneous, whereas they are independent when asymmetry and uncertainty are simultaneously heterogeneous. Diﬀerences in eﬀects on social welfare depend
on whether the interaction of asymmetric preferences between agents is active.

3

Numerical comparison

This section numerically conﬁrms the results obtained in Section 2. To this end, we
calculate and compare welfare loss under Type I and III heterogeneity. Our strategy
is simply to replace the solution of the standard-homogeneous model2 , φ2π , with ΨI andΨIII in calculating losses under Types I and III. For instance, under Type I heterogeneity,
ΨI can be substituted in the homogeneous model as
P 2
φ2π = (φCB
π + φπ ) ,2

For the diﬀerence in solutions between Type I and III models and the standard model, see theconstruction of Ā and B̃ in the Type II model or Ikeda (2012).

38

since this is the only diﬀerence from the standard-homogeneous model. As a result, thesimulation can be implemented following the relation
φπ =

√
P 2
(φCB
π + φπ ) .

(44)

We replace the asymmetric parameter according to condition (44) in the standard modeland calculate social loss.
In addition, the relation between Type III and the standard model is2P 2
φ2π = (φCBπ ) + (φπ ) .

Accordingly,
φπ =

√
2
P 2
(φCB
π ) + (φπ ) .

(45)

After calculating losses under both types of heterogeneity, we manipulate the welfare
gain between them as() ()
W elf aregain = − var(π) + λvar(y) | case I + var(π) + λvar(y) | case III .(46)In condition (46), we deﬁne social welfare as negative social loss. For simulation, parameters are calibrated in the manner shown in Ikeda (2012). The discount rate β is 0.99,
the elasticity of interest rate σ is 1.00, the slope of the NKPC κ is 0.30, the objective
trade-oﬀ parameter λ is 0.25, and the inertia of the cost-push shock is 0.35. In addition,the (inverse of) willingness for robustness θ is set at 50 according to the detection error
probabilities (Hansen and Sargent, 2008, chapter 9).
Figure 1 displays the simulated welfare gains. Its numerical results also support thoseobtained in Section 2. In the region where asymmetries bear diﬀerent signs, social welfare
under Type I exceeds that under Type III (the gain becomes positive). In the region
where their signs are identical, welfare is higher under Type III as the interaction termof Type I asymmetries become positive. This vigorously ampliﬁes economic ﬂuctuations,and therefore, gains in the region are negative. These results sustain the analysis in
Section 2.

4

Conclusion

This paper analyzed the heterogeneity of asymmetric forecasting with a robust control
algorithm. We considered heterogeneity stemming from asymmetric preferences (Type I),
uncertainty about the economic model (Type II), and both asymmetric preferences and
39

40
20
0
−20
−40
−60

1
0.5

−80
1

0.8

0
0.6

0.4

0.2

0

−0.2

−0.5
−0.4

−0.6

−0.8

−1

−1

ϕP
π

B
ϕC
π

Figure 1: Welfare gains between Type I and Type III heterogeneity

model uncertainty (Type III). Signiﬁcant diﬀerences from the homogeneous model appear
only in Types I and III. Our results suggest that social welfare under Type I heterogeneity
exceeds that under Type III when the direction of heterogeneous asymmetric preferences
diﬀers between the central bank and private agents. That is, the interaction of themturns negative in Type I, reducing the size of a transmission pass for uncertainty to themodel. This in turn reduces the economic variance under Type I heterogeneity, whereas
the outcome under Type III heterogeneity escape the interaction eﬀect of asymmetricpreferences.

References[1] Branch, William. (2011).

Nowcasting and the Taylor Rule.

mimeograph.

[2] Capistrán, Carlos, and Allan Timmermann. (2009). Disagreement and Biases inInﬂation Expectations. Journal of Money, Credit and Banking, 41, 365-396.
[3] Branch, William, and Bruce McGough. (2009). A New Keynesian model withheterogeneous expectations, Journal of Economic Dynamics and Control, 33(5),1036-1051.

40

[4] Evans, George, and Seppo Honkapohja. (2001). Learning and Expectations in
Macroeconomics. Princeton University Press.
[5] Giannitsarou, Chryssi. (2003).
namics, 6(4), 885-906.

Heterogeneous Learning,” Review of Economic Dy-[6] Hansen, Lars, and Thomas Sargent. (2008). Robustness. Princeton University Press.
[7] Honkapohja, Seppo, and Kaushik Mitra. (2006). Learning Stability in Economies
with Heterogeneous Agents, Review of Economic Dynamics, 9(2), 284-309.
[8] Ikeda, Taro. (2012). Robust control and asymmetries in central bank forecasting,mimeograph.
[9] Surico, Paolo. (2007). The Fed’s monetary policy rule and U.S. inﬂation: Thecase of asymmetric preferences. Journal of Economic Dynamics and Control, 31,
305-324.

41