CENTRE FOR DYNAMIC MACROECONOMIC ANALYSISWORKING PAPER SERIES

CDMA07/19

Robust Learning Stability with Operational
Monetary Policy Rules*
Seppo Honkapohja†
Bank of Finland,
University of Cambridge

George Evans
University of Oregon andUniversity of St Andrews

OCTOBER 2007
REVISED JANUARY 2008
ABSTRACT
We consider “robust stability” of a rational expectations equilibrium,
which we define as stability under discounted (constant gain) least-squares
learning, for a range of gain parameters. We find that for operational forms ofpolicy rules, i.e. rules that do not depend on contemporaneous values ofendogenous aggregate variables, many interest-rate rules do not exhibit robust
stability. We consider a variety of interest-rate rules, including instrument rules,
optimal reaction functions under discretion or commitment, and rules thatapproximate optimal policy under commitment. For some reaction functions
we allow for an interest-rate stabilization motive in the policy objective. The
expectations-based rules proposed in Evans and Honkapohja (2003, 2006)deliver robust learning stability. In contrast, many proposed alternatives
become unstable under learning even at small values of the gain parameter.
Keywords: Commitment, interest-rate setting, adaptive learning, stability,
determinacy.
JEL Classifications: E52, E31, D84.

* Paper prepared for the 11th Annual Conference of the Central Bank of Chile, “Monetary Policy
Under Uncertainty and Learning,” November 15-16, 2007. We thank our discussant Claes Berg, Carl
Walsh and the conference participants for valuable comments. Support from National Science
Foundation Grant No. SES-0617859 and ESRC grant RES-000-23-1152 is gratefully acknowledged.

CASTLECLIFFE, SCHOOL OF ECONOMICS & FINANCE, UNIVERSITY OF ST ANDREWS, KY16 9AL
TEL: +44 (0)1334 462445 FAX: +44 (0)1334 462444 EMAIL: cdma@st-andrews.ac.uk
www.st-andrews.ac.uk/cdma

1

Introduction

Recently, the conduct of monetary policy in terms of interest rate rules has
been examined from the viewpoint of imperfect knowledge and learning by
economic agents. In this literature stability of rational expectations equilibrium (REE) is taken as a key desideratum for good monetary policy design.1
Most of this literature postulates that agents use least squares or related
learning algorithms to carry out real-time estimations of the parameters of
their forecast functions as new data becomes available. Moreover, it is usually assumed that the learning algorithms have a decreasing gain; in the mostcommon case the gain is the inverse of the sample size so that all data points
have equal weights. Use of such a decreasing-gain algorithm makes it possible
for learning to converge exactly at the REE in environments without structural change. Convergence requires that REE satisfies a stability condition,
known as E-stability.
Decreasing-gain algorithms do not, however, perform well when occasional unobservable structural changes take place. So-called constant-gain
algorithms are a natural alternative for estimating parameters in a way thatis alert to possible structural changes. If agents use a constant-gain algorithm, then parameter estimates of the forecast functions do not fully converge to the REE values. Instead, they remain random, even asymptotically.
However, for small values of the gain parameter the estimates remain for
most of the time in a small neighborhood of the REE, provided that the
REE is E-stable.2 Recently, constant-gain algorithms have been employedin empirical work, e.g. see Milani (2005), Milani (2007a), Orphanides andWilliams (2005b), Orphanides and Williams (2005a) and Branch and Evans(2006).
It should be emphasized that the connection between convergence ofconstant-gain learning and E-stability noted above is a limiting result forgain parameters suﬃciently small. For finite values of the gain parameter, the stability condition for constant-gain learning is more stringent thanE-stability. In this paper we examine the stability implications of various
interest rate rules when agents use constant-gain learning rules with plausi1

For surveys see Evans and Honkapohja (2003a), Bullard (2006) and Evans andHonkapohja (2007).
2
See Chapters 3 and 7 of Evans and Honkapohja (2001) for the basic theoretical resultson constant-gain learning. See also Evans, Honkapohja, and Williams (2006) for references
on recent papers on constant-gain learning.

2

ble positive values of the gain. We will say that an interest rate rules yields
robust learning stability of the economy if stability under constant-gain learning obtains for all values of the gain parameter in the range suggested by theempirical literature.3
In this study we focus on interest rate rules that are operational in thesense discussed by McCallum (1999). He argues that monetary policy cannot
be conditioned on current values of endogenous aggregate variables. Thus,
the rules we consider assume that policy responds to expectations of contemporaneous (or future) values of inflation and output but not on their actual
values in the current period.
We consider robust learning stability for a variety of operational interest
rate rules that have been suggested in the recent literature. These include
Taylor rules and optimal reaction functions under discretion and commitment when central bank policy aims for interest-rate stabilization in addition
to the usual motives for flexible inflation targeting. The reaction function
may be expectations-based in the spirit of Evans and Honkapohja (2003b)
and Evans and Honkapohja (2006), or of the Taylor-type form suggested by
Duﬀy and Xiao (2007). We also analyze two interest rate rules that approximate optimal policy under commitment and were suggested by Svensson andWoodford (2005) and McCallum and Nelson (2004). Our results show that
expectations-based rules deliver robust learning stability, whereas the proposed alternatives often become unstable under learning even at quite smallvalues of the constant gain parameter.

2

Constant Gain Steady-State Learning

2.1

Theoretical Results

In this paper we employ multivariate linear models. In this simplest casein which the shocks are white noise and there are no lagged endogenous
variables, the REE takes the form of a stochastic steady state. We now
3

There are numerous concepts of robustness that are relevant to policymaking reflecting, e.g., uncertainty about the structure of the economy, and a desire by both private
agents and policymakers to guard against the risk of large losses. We do not mean to
downplay the importance of such factors, but we here abstract from them in order tofocus on the importance of setting policy in such a way as to ensure stability in the faceof constant-gain learning.

3

briefly review the basics of steady state learning in linear models.4
The steady state can be computed by postulating that agents’ beliefs,
called the “perceived law of motion” (PLM), take the form
yt = a + et ,for a vector yt , where et ∼ iid(0, σ 2 ). Using the model, one then computes the“actual law of motion” (ALM), which describes the temporary equilibriumin the current period, given the PLM. We write the ALM using a linear
operator T as
yt = α + T a + et ,where the matrix T depends on the structural parameters of the model.
Examples of the T map will be given below. An REE is a fixed point ā ofthe T map, i.e.
ā = α + T ā
We assume that I − T is non-singular, so that there is a unique solutionᾱ = (I − T )−1 α. For convenience, and without loss of generality, we now
assume that the model has been written in deviation from the mean form,
so that α = 0. Thus in our analysis the REE corresponds to ā = 0. Under
learning agents attempt to learn the value of ā, and hence in deviation frommean form we are examining whether agents’ estimates of the mean converge
to a = 0.
Steady-state learning under decreasing gain is given by the recursive algorithm
at = at−1 + γ t (yt − at−1 ),(1)where the gain γ t is a sequence of small decreasing numbers such as γ t = 1/t.
Assuming that yt = T at−1 + et , i.e. that expectations are formed using theestimate at−1 based on data through time t − 1, the convergence conditionof algorithm (1) is given by the conditions for local asymptotic stability of āunder an “associated diﬀerential equation”da= T a − a,dτ
4

See Chapters 8 and 10 of Evans and Honkapohja (2001) for a detailed discussion ofadaptive learning in linear models.

4

known as the E-stability diﬀerential equation. Here τ denotes notional or
virtual time. It is easily seen that the E-stability condition holds if and only
if all eigenvalues of the matrix T have real parts less than one.5
Under constant-gain learning, the estimate at of a is updated accordingtoat = at−1 + γ(yt − at−1 ),(2)where 0 < γ ≤ 1 is the constant gain parameter. The only diﬀerence to (1)is constancy of the gain sequence. We now haveat = at−1 + γ (T at−1 + et − at−1 ) , orat = (γT + (1 − γ)I) at−1 + γet .
This converges to a stationary stochastic process around the REE value ā = 0(in deviation from mean form) provided all roots of the matrix γT + (1 − γ)I
lie inside the unit circle.
It is evident that stability under constant-gain learning depends on thevalue of γ, and we have the following result.
Proposition 1 For a given 0 < γ ≤ 1, the stability condition is that the
eigenvalues of T lie inside a circle of radius 1/γ and origin at (1 − 1/γ, 0).
This condition is therefore stricter for larger values of γ.
Proof. The stability condition is that the roots of γ(T + γ −1 (1 − γ)I)
lie inside the unit circle centered at the origin. Equivalently, the roots of(T + γ −1 (1 − γ)I) must lie inside a circle of radius 1/γ centered at the origin.
Since the roots of T + γ −1 (1 − γ)I are the same as the roots of T plus
γ −1 (1 − γ), this is equivalent to the condition given.
Note that the right edge of the circle is at (1, 0) in the complex plane andthat as γ → 0 we obtain the standard (decreasing gain) E-stability condition
that the real parts of all roots of T are less than than one. Looking at theother extreme γ = 1 gives the following:
Corollary 2 We have stability for all 0 < γ ≤ 1 if and only if all eigenvaluesof T lie inside the unit circle.
5
Throughout, we rule out boundary cases in which the real part of some eigenvalue of
the T -map is one.

5

We remark that stability for all constant gains 0 < γ ≤ 1 is equivalent
to a condition known as iterative E-stability, sometimes called “IE-stability.”
Iterative E-stability is said to hold when T j → 0 as j → ∞.6
Note that when the stability condition holds, the parameter at converges
to a stationary stochastic process that we can fully describe. This in turninduces a stationary stochastic process for yt = T at−1 + et .

2.2

Application to Taylor Rules

Consider the standard forward-looking New Keynesian (NK) model,
xt = −ϕ(it − π et+1 ) + xet+1 + gt
π t = λxt + βπ et+1 + ut .

(3)
(4)

For convenience we here assume that (gt , ut )0 are iid, so that the preceding
technical results can be applied. Later we will consider cases with AR(1)shocks. We use xet+1 and π et+1 to denote expectations of πt+1 and xt+1 . Below
we will be precise about the information sets available to agents when forming
expectations and throughout the paper we will be exploring the implicationsof alternative assumptions.
Bullard and Mitra (2002) consider Taylor rules of various forms, including
the “contemporaneous data” ruleit = χπ π t + χx xt ,(5)and the “contemporaneous expectations” ruleit = χπ π et + χx xet .

(6)

In this section, when analyzing the contemporaneous expectations rule wefollow Bullard and Mitra (2002) in assuming that all expectations are basedon information at time t − 1, i.e. πet = Êt−1 π t , xet = Êt−1 xt , π et+1 = Êt−1 π t+1
and xet+1 = Êt−1 xt+1 . Since we have iid shocks, forecasts are based purely on
the estimated intercept.
Bullard and Mitra (2002) show that the determinacy and E-stability conditions are the same and are identical for both interest rate rules, and givenbyλ(χπ − 1) + (1 − β)χx > 0.
(7)
6

In many models, iterative E-stability is known to be a necessary condition for thestability of “eductive” learning, e.g. see Evans and Guesnerie (1993).

6

They considered this finding important because of the argument by McCallum (1999) that it is not plausible that interest-rate rules can be conditioned
on contemporaneous observations of endogenous aggregate variables like inflation and output, whereas they could plausibly be conditioned on central
bank forecasts or “nowcasts” Êt−1 π t , Êt−1 xt .
We reconsider this issue from the vantage point of constant-gain learning.
For the interest-rate rule (6) the model takes the formeyt = M0 yte + M1 yt+1
+ P vt ,(8)

where yt0 = (xt , π t ) and vt = (gt , ut ), and where¶¶µµ−χx ϕ −χπ ϕ
1
ϕ
,
(9)
and M1 =M0 =λ β + ϕλ−χx ϕλ −χπ ϕλµ¶
1 0and P =.
λ 1
Since our shocks are iid the PLM is simply yt = a + et , and the corresponding ALM is yt = (M0 +M1 )a+et , where et = P vt . The usual E-stability
condition is that the eigenvalues of M0 + M1 have real parts less than one,which leads to the condition (7). Applying Corollary 2, for convergence ofconstant-gain learning for all gains 0 < γ ≤ 1 we need that both eigenvaluesof M0 + M1 lie inside the unit circle.
We investigate stability of constant-gain learning numerically, using the
Woodford calibration of ϕ−1 = 0.157, λ = 0.024, β = 0.99. Setting χπ = 1.5,
eigenvalues with real parts less than −1 arise for χx > 0.31 and eigenvalues
with real parts less than −9 arise for χx > 1.57. This implies that when χπ =1.5 and χx > 1.57 the equilibrium is unstable under learning for constantgains γ ≥ 0.10. This is perhaps not a significant practical concern since
Taylor’s recommended parameters are χπ = 1.5 and (based on the quarterly
calibration of Woodford) χx = (0.5)/4 = 0.125. However, it does show apreviously unrecognized danger that arises under constant-gain learning if the
Taylor rule has too strong a response to Êt−1 xt , and this finding foreshadows
instability problems that arise in more sophisticated rules discussed below.
Finally, we remark that the potential for instability under constant-gain
learning arises specifically because of the necessity to use forecasts Êt−1 yt .
For the current data Taylor rule (5) it can be shown that the condition (7)guarantees stability under learning for all constant gains 0 < γ ≤ 1.7
7

The model now takes the form yt = M1 Êt yt+1 + P vt and the required condition is thesame as the determinacy condition.

7

3

Optimal Discretionary Monetary Policy

We now consider optimal policy under constant-gain learning, starting in thisSection with optimal discretionary policy. We focus on homogeneous learningby private agents and the policy-maker. We initially restrict attention to thecase of iid exogenous shocks, so that steady-state learning is appropriate.
However, we also analyze the more general case, where the observable shocks
follow AR(1) processes.
Consider the loss function
E0

∞
X
£
¤
(π t − π∗ )2 + αx (xt − x∗ )2 + αi (it − i∗ )2 ,

(10)

t=0

where π ∗ , x∗ and i∗ represent target values. For simplicity, we set π ∗ = x∗ =0. The weights αx , αi > 0 represent relative weights given by policy-makers
to squared deviations of xt and it from their targets, compared to squared
deviations of π t from its target.
The first-order condition (FOC) for discretionary optimal policy is
λπ t + αx xt − αi ϕ−1 (it − i∗ ) = 0.

(11)

We first consider a Taylor-type rule proposed by Duﬀy and Xiao (2007)
and then discuss the expectations-based rule recommended by Evans andHonkapohja (2003b).

3.1

Taylor-type Optimal Rules

Duﬀy and Xiao (2007) propose using the equation (11) directly to obtain a
Taylor-type rule that implements optimal discretionary policy. Solving the
FOC for it yields the rule
it =

ϕλ
ϕαx
πt +
xt ,
αi
αi

where at this point we drop the term i∗ since for brevity we are suppressing
all intercepts. As discussed by Duﬀy and Xiao (2007), this is formally acontemporaneous-data Taylor rule. They show that for calibrated values ofstructural parameters and policy weights this leads to a determinate andE-stable equilibrium.
8Because it is problematical that the Central Bank can observe contemporaneous output and inflation,8 we instead examine the rule
it =

ϕλ
ϕαx
Êt−1 π t +Êt−1 xt ,
αi
αi

(12)

where the information set for the “nowcasts” π et = Êt−1 π t , xet = Êt−1 xt is pastendogenous variables and exogenous variables. This again leads to a modelof the form (8) with coeﬃcients (9), where χπ = ϕλ/αi and χx = ϕαx /αi .
We assume that private agents and Central Banks estimate the same PLM,
Since we are here assuming steady-state learning we also have π et = Êt−1 π t+1
and xet = Êt−1 xt+1 .
We first note that for αi suﬃciently large the model under this Taylortype rule will suﬀer from indeterminacy. This follows from the Bullard-Mitra
result that the determinacy condition is (7), from which the critical value of
αi can be deduced. The condition for determinacy is
αi < ᾱi ≡ ϕλ + (1 − β)λ−1 ϕαx .

(13)

If the central bank’s desire to stabilize the interest rate is too strong, i.e.
condition (13) is not met, then the central bank fails to adjust the interest
rate suﬃciently to ensure that the generalized Taylor principle (7) is satisfied.
To assess this point numerically, we use the calibrated parameter values ofTable 6.1 of Woodford (2003), with αx = 0.048, ϕ = 1/0.157, λ = 0.024,
β = 0.99 and get approximately ᾱi = 0.28. Woodford’s calibrated values of
αi are 0.077 or 0.233, where the latter value is from Woodford (1999). Thus
the condition for determinacy does hold for these calibrations.
We next consider stability under learning. For the PLM yt = a + et weagain get the ALM yt = (M0 + M1 )a + et , and
¶
µ
−1
2
2
α
ϕ
ϕ
−
α
λϕ
1 − α−1xiiT ≡ M0 + M1 =.
−1 2 22λ − α−1i λϕ αx β + λϕ − αi λ ϕIt can be shown that2
det(T ) = β(1 − α−1i αx ϕ ).
8

An alternative would be to assume that agents and the policymaker sees the contemporaneous value of the exogenous shocks but not the contemporaneous values of xt andπt . This would not alter our results.

9For stability under all values 0 < γ < 1 we need¯¯¯β(1 − α−1 αx ϕ2 )¯ < 1,i

and it is clear that for given β, αx , ϕ this condition will not be satisfied for
αi > 0 suﬃciently small. Hence
Proposition 3 Let α̂i = β(1 + β)−1 αx ϕ2 . For 0 < αi ≤ α̂i there exists
0 < γ̂(β, ϕ, αi , αx ) < 1 such that the optimal discretionary Taylor-type rule(12) renders the REE unstable under learning for γ̂ < γ ≤ 1.
Thus, in addition to the indeterminacy problem for “large values” of αi ,the Taylor-type optimal rule suﬀers from a more serious problem of instability
under constant-gain learning for “small values” of αi . The source of thisdiﬃculty is the interaction of strong policy responses seen in equation (12)and a large gain parameter. This combination leads to cyclical overshootingof inflation and output gap. This is particularly evident as αi tends to zero
since in this case, e.g., a positive change in inflation expectations Êt−1 π t leads
to large increase in it , which in turn leads to large negative changes in xt andπt via equations (3) and (4). The severity of this problem depends on thevalue of γ̂ in Proposition 3. Ideally, stability would hold for all 0 < γ ≤ 1,
but if γ̂ is high the problem might not be a major concern.
We investigate the magnitude of γ̂ numerically by computing the eigenvalues of γT + (1 − γ)I. As an example, for the Woodford calibration
β = 0.99, ϕ = 1/0.157, λ = 0.024, we find that with αx = 0.048 and
αi = 0.077, the critical value γ̂ ≈ 0.04. Since estimates in the macro literature suggest gains in the range 0.02 to 0.06, this indicates that optimal
Taylor-type rules may not be stable under learning.9 The source of the problem is that with low αi the implied weights on Êt−1 π t and especially Êt−1 xt
are very high. Under constant-gain learning this can lead to instability unlessthe gain parameter is very low. We will demonstrate later that this problem
is avoided by using a suitable expectations-based optimal rule.
We next consider the case in which the exogenous shocks are AR(1)processes. In this setting various information assumptions have been usedin the literature. Perhaps the most common assumption is that agents see
9

Milani (2007b) considers a setting in which agents switch between decreasing gain
and constant gain estimators, depending on recent average mean-square errors. In theconstant-gain regime the estimated gains are even higher, around 0.07 to 0.08.

10

current and lagged exogenous variables and lagged, but not current, endogenous variables. Expectations under this assumption are denoted as Êt π t ,Êt xt , Êt π t+1 and Êt xt+1 . An alternative would replace these by Êt−1 π t ,Êt−1 xt , Êt−1 π t+1 and Êt−1 xt+1 , indicating that agents only see lagged information.10 Whether agents see current or only lagged exogenous shocks is notparticularly crucial and does not aﬀect our main results. Consequently, we
follow the most common assumption that expectations are specified as Êt π t ,Êt xt , Êt πt+1 and Êt xt+1 .11 In contrast, as we have already seen, whether
agents and policy-makers are able to see current endogenous variables is an
important issue for stability under learning. This is why we use the term operationality to indicate an interest rate rule that does not depend on current
endogenous variables.
We now assume that the exogenous shocks gt and ut follow AR(1) processes,
i.e.
gt = μgt−1 + g̃t and ut = ρut−1 + ũtwhere 0 < |μ| , |ρ| < 1 and g̃t ∼ iid(0, σ 2g ), ũt ∼ iid(0, σ 2u ) are independent
white noise processes. We write this in vector form as
vt = F vt−1 + ṽt .
Under the current assumptions, the PLM of the agents is
yt = a + cvt ,and the forecasts are now Êt yt = a + cvt and Êt yt+1 = a + cF vt . Using thegeneral model (8), the ALM isyt = (M0 + M1 )a + (M0 c + M1 cF + P )vt ,and the E-stability conditions are that all eigenvalues of the matrices M0 +M1
and I ⊗ M0 + F 0 ⊗ M1 have real parts less than one. Here ⊗ denotes the
Kronecker product of two matrices.12
10

A third alternative, which is occasionally used in the literature, allows agents to see thecontemporaneous values of endogenous variables. However, this assumption runs against
the requirement of operationality that we want to emphasize here.
11
The standard assumption under RE is that agents have contemporaneous information.
Our information assumption takes account of the operationality critique, but nonethelessallows for the possibility of convergence under learning to the REE.
12
In the case of lagged information the PLM is specified as yt = a + cvt−1 + η t and the
ALM is then yt = (M0 + M1 )a + (M0 c + M1 cF + P F )vt−1 + P ṽt .

11

deviation of x from RE
deviation of π from RE

1
0
-1
-2
0

100

200

300

400

500
time

600

700

800

900

1000

100

200

300

400

500
time

600

700

800

900

1000

0.02
0
-0.02
-0.04
0

Figure 1: Stability of optimal Taylor-type rule with γ = 0.02.
To examine stability under constant-gain learning, we simulate the model
under constant-gain recursive least squares (RLS) estimation of the PLM
parameters a and c.13 Under constant-gain least squares agents discount old
data geometrically at the rate 1 − γ. Let at , ct denote the estimates basedon data through t − 1. For the recursive formulation of (constant-gain) leastsquares see the Appendix. Given these estimates, expectations are formede
as yte = Êt yt = at + ct vt and yt+1
= Êt yt+1 = at + ct F vt and the temporary
equilibrium is then given by (8) with these expectations.
We use the previous values for the structural parameters and also set
μ = ρ = 0.8. Simulations of the system indicate instability under constantgain RLS learning for gain parameters at or in excess of 0.024. Thus, withregressors that include exogenous AR(1) observables instability arises at evenlower gain values than in the case of steady state learning. Figures 1 and
2 illustrate the evolution of parameters over time under constant-gain RLS
learning with the Taylor-type rule (12) in stable and unstable cases.

3.2

Expectations-Based Optimal Rules

Assume now that at time t the exogenous shocks gt , ut and private-sector
expectations Êt π t+1 and Êt xt+1 are observed by the Central Bank. The
expectations-based (EB) rule is constructed so that it exactly implements
13

The RLS formulae are given in the Appendix.

12

x 10

deviation of x from RE

1

8

0.5
0
-0.5
-1
-1.5

0

deviation of π from RE

4

x 10

10

20

30

40

50
time

60

70

80

90

100

10

20

30

40

50
time

60

70

80

90

100

6

2
0
-2
-4

0

Figure 2: Instability of optimal Taylor-type rule with γ = 0.04.
(11), the FOC under discretion, even outside an REE for given expectations,as suggested by Evans and Honkapohja (2003b). To obtain the rule, we
combine (3), (4) and (11), and solve for it in terms of the exogenous shocks
and the expectations.
The resulting EB-rule isit =(αx + λ2 )ϕβλϕ + (αx + λ2 )ϕ2x+ÊÊt πt+1
t t+1
αi + (αx + λ2 )ϕ2αi + (αx + λ2 )ϕ2(αx + λ2 )ϕ
λϕ
+
gt +
ut .
2
2
αi + (αx + λ )ϕαi + (αx + λ2 )ϕ2This leads to a reduced form
yt = M Êt yt+1 + P vt .

(14)

Determinacy of the REE corresponding to optimal discretionary monetary
policy requires that M has both eigenvalues inside the unit circle.14 We again
have the condition αi < ᾱi , where ᾱi is given by (13).
For stability under learning, first consider the case where the exogenous
shocks vt are iid and agents use steady state learning under constant gain.
14

Equivalently we need |tr(M )| < 1 + det(M ) and |det(M )| < 1.

13

For this reduced form the PLM yt = a + et gives the ALM yt = Ma + et(where et = P vt ), as discussed in Section 2.1. Thus T = M and there is avery close connection between determinacy and stability under learning. We
have:
Proposition 4 Assume αi < ᾱi and the shocks are iid. Then the EB-rule,
which implements the FOC, yields a reduced form that is stable under steadystate learning for all constant-gain rules 0 < γ ≤ 1.
Provided αi < ᾱi , so that determinate optimal policy is possible, the EBoptimal rule will successfully implement the optimal REE: under decreasing
gain learning there will be convergence to the REE, and under small constant gain it will converge to a stochastic process near the optimal REE.
Furthermore, for all constant gains 0 < γ ≤ 1 there will be convergence to astationary process centered at the optimal REE.
Second, we examine numerically the case of AR(1) shocks with (constantgain) RLS learning. For the Woodford calibration β = 0.99, ϕ = 1/0.157, λ =0.024, αx = 0.048 and αi = 0.077 (and ρ = μ = 0.8) we find that learning converges for gain values at or below γ = 0.925. In other words, theexpectations-based optimal discretionary rule is quite robustly stable underlearning. We also make a technical remark that when the agents have to
run genuine regressions, as in the current case, then the IE-stability condition does not imply convergence of constant-gain learning for all 0 < γ ≤ 1.
However, we see that stability does hold even for γ quite close to one.

4

Optimal Policy with Commitment

For brevity, in the remainder of the paper we assume that αi = 0, i.e. theCentral Bank does not have an interest rate stabilization objective.15 Given
the model (3)-(4) and the loss function (10) with αi = 0, it is well-known
that optimal monetary policy under commitment (in a timeless perspective)
is characterized by the condition16
λπ t = −αx (xt − xt−1 ),
15

(15)

See Duﬀy and Xiao (2007) for the extension to the case where the Central Bank also
has an interest-rate stabilization motive.
16
See e.g. Clarida, Gali, and Gertler (1999) and Woodford (1999). For the exposition,
we follow Evans and Honkapohja (2006).

14

which is often called the optimal targeting rule. It can be shown that the
optimal rational expectations equilibrium of interest has the form
xt = bx xt−1 + cx ut
π t = bπ xt−1 + cπ ut ,where we choose the unique 0 < bx < 1 that solves the equation βb2x −(1+β +λ2 /αx )bx +1 = 0 and bπ = (αx /λ)(1−bx ), cx = −[λ+βbπ +(1−βρ)(αx /λ)]−1
and cπ = −(αx /λ)cx .
Diﬀerent optimal reaction functions that implement the optimal targeting
rule (15) have been proposed in the literature. Under rational expectations
one obtains the fundamentals-based reaction function
it = ψx xt−1 + ψg gt + ψu ut ,

(16)

where
ψx = bx [ϕ−1 (bx − 1) + bπ ]
ψg = ϕ−1
ψu = [bπ + ϕ−1 (bx + ρ − 1)]cx + cπ ρ.
Evans and Honkapohja (2006) show that the reaction function (16) often
leads to indeterminacy and always leads to expectational instability. They
propose instead the expectations-based reaction function
it = δL xt−1 + δπ Êt π t+1 + δ x Êt xt+1 + δ g gt + δ u ut ,(17)

where the coeﬃcients are17
δL =

−αx
λβ
λ
−1
.
2 , δπ = 1 +2 , δx = δg = ϕ , δu =ϕ(αx + λ )ϕ(αx + λ )ϕ(αx + λ2 )Under the interest-rate reaction rule (17) the reduced form model is of theform
yt = M1 Êt yt+1 + Nyt−1 + P vt ,with yt0 = (xt , π t ) and vt0 = (gt , ut ). The corresponding REE takes the form
yt = b̄yt−1 + c̄vt . Evans and Honkapohja (2006) show that the optimal
17

In the discretionary case with αi = 0 the same coeﬃcients would obtain, except that
δ L = 0.

15

expectations-based reaction function (17) delivers a determinate and E-stable
optimal REE for all values of the parameters. It is therefore clearly preferredto the fundamentals-based rule (16).
In connection with constant gain learning we have the following partial
result:18
Proposition 5 The EB-rule under commitment (17) yields a reduced form
for which the eigenvalues of the T −map are inside the unit circle for all
values of the structural parameters.
This result is partial in the sense that the eigenvalues condition is no longer
suﬃcient for stability of constant-gain learning for all 0 < γ ≤ 1. This isbecause in the model the regressors include exogenous and lagged endogenous
variables.
We now examine numerically the performance of constant-gain RLS learning under the expectations-based optimal rule with commitment. Using
Woodford’s parameter values (but with αi = 0), we find that constant-gain
RLS learning converges for values of the gain parameter below γ̂ ≈ 0.25.
The inclusion of a lagged variable among the regressors appears to have asignificant eﬀect on stability of learning for large gains. However, the rule isstill robust for all plausible values of the gain parameter.
As noted above, the Duﬀy and Xiao (2007) formulation under commitment breaks down when αi = 0 (as it does in the discretionary case). One
might investigate numerically the performance of the Duﬀy-Xiao rule underconstant-gain RLS for calibrated values of αi . Based on the results in thediscretionary case, we are not optimistic about robust learning stability of
the Duﬀy-Xiao rule with commitment.

5

Alternative Rules for Optimal Policy underCommitment

5.1

Svensson-Woodford Rule

Given that the fundamentals-based optimal rules (without interest rate stabilization) lead to problems of indeterminacy and learning instability, Svensson
and Woodford (2005) suggest a modification to such a rule. In this rule the18

See the Appendix for a proof.

16

fundamentals-based rule (16) is complemented with a term that is based on
the commitment optimality condition. We again assume that contemporaneous data are not available to the policy-maker, so that current values ofinflation πt and the output gap xt are replaced by their nowcasts Êt π t andÊt xt . This results in the interest rate ruleit = ψx xt−1 + ψg gt + ψu ut + θ[Êt π t +αx(Êt xt − xt−1 )],λ(18)

where θ > 0.
The full model is now given (3), (4) and (18). By substituting (18) into(3) this model can be reduced to a bivariate model of the form
yt = M0 Êt yt + M1 Êt yt+1 + Nyt−1 + P vt ,(19)

where the information set in the forecasts and nowcasts includes current
values of the exogenous shocks but not of the endogenous variables. It is also
assumed for convenience that vt = F vt−1 + ṽt is a known, stationary process.
The coeﬃcient matrices are
µ
µ
¶
¶
1
ϕ
−ϕαx θλ−1 −ϕθ, M1 =M0 =λ β + λϕ
−ϕαx θ
−ϕθλ
¶
µ
¶
µ
0
−ϕψu
−ϕψx + ϕαx θλ−1 0
.
N =
,P =
0 1 − λϕψ u−λϕψ x + ϕαx θ 0
The PLM has the form
yt = a + byt−1 + cvt
and the T −mapping is
T (a, b, c) = ((M0 + M1 (I + b))a, M1 b2 + M0 b + N, M0 c + M1 (bc + cF ) + P ).
The usual E-stability conditions are stated in terms of the eigenvalues of thederivative matrices
DTa = M0 + M1 (I + b̄)DTb = b̄0 ⊗ M1 + I ⊗ M1 b̄ + I ⊗ M0
DTc = F 0 ⊗ M1 + I ⊗ M1 b̄ + I ⊗ M0 ,where ⊗ is the Kronecker product and b̄ is the RE value of b.
17

We compute numerically the E-stability eigenvalues for the Woodford
calibration with αx = 0.048 and θ = 1.19 For this case the eigenvalues of DTa
are −9.570 and 0.99, while the eigenvalues of DTb are −10.605, −9.672, 0.878
and −0.0118. However, θ = 1 is very close to the lower bound on θ needed
for E-stability (since one root of DTa is almost one), and the eigenvalues are
sensitive to the value of θ. For example, for θ = 1.5 the eigenvalues of DTa
are −15.975 and 0.949, while the eigenvalues of DTb are −17.059, −16.082,
0.842 and −0.0110. It is seen that large negative eigenvalues appear.
The calculation of the E-stability eigenvalues suggests that the interest
rate rule (18) can be subject to instability if learning is based on constantgain. We now examine numerically the performance of the rule (18) underdiﬀerent values of the constant gain using the Woodford calibrated values ofthe model parameters and θ = 1.5. Numerical simulations show that under
the interest rate rule (18) constant-gain RLS learning becomes unstable forvalues of γ at 0.019 or higher.
We also examine numerically the sensitivity of the stability upper boundon γ for diﬀerent values of αx , i.e. the degree of flexibility of inflation targeting. Table 1 gives the approximate highest value γ̂ of the gain for which
stability under constant-gain learning obtains.
αxγ̂

0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.1
0.185 0.06 0.035 0.02 0.018 0.014 0.009 0.007

Table 1: Critical values of γ for stability, Svensson-Woodford rule
It is seen from Table 1 that robust learning stability of the Svensson-Woodford
hybrid rule is very sensitive to the degree of flexibility in inflation targeting.
Robust stability obtains only when the Central Bank is an “inflation hawk”.

5.2

McCallum-Nelson Rule

McCallum and Nelson (2004) propose a diﬀerent rule that approximates optimal interest-rate policy in the timeless-perspective sense. They suggest that19

We remark that the eigenvalues of the same model but with contemporaneous data
available would not deliver large negative eigenvalues in the E-stability calculation for thisparameterization.

18

the interest rate be raised above inflation whenever the timeless-perspective
optimality condition is above zero.
Their rule performs well if yt is observable, but as McCallum and Nelson(2004) themselves point out, such a rule would be subject to the operationality problem that we have encountered several times: it presupposes that
contemporaneous data on inflation and the output gap is available. One way
to overcome this problem is to replace unknown contemporaneous data by
nowcasts of the variables. In this case the interest-rate rule becomes
αx
it = Êt π t + θ[Êt π t + (Êt xt − xt−1 )].
(20)
λUnder RE this rule approximates optimal policy under (timeless perspective)
commitment, provided θ > 0 is large.
The model is then given by equations (3), (4) and (20). The model can be
reduced to a bivariate model of the form (19), where the coeﬃcient matrices
are
¶
µ
µ
¶
1
ϕ
−ϕαx λ−1 −ϕ(1 + θ)M0 =, M1 =
λ β + λϕ
−ϕαx−λϕ(1 + θ)
µ
¶
µ
¶
−1
1 0
ϕαx λ
0
N =
,P =
.
λ 1
ϕαx
0

Using the same parameter values as above in the case of the SvenssonWoodford hybrid rule, with αx = 0.048, we obtain that for θ = 1 the eigenvalues of DTa are −9.719 and 0.869, while the eigenvalues of DTb are −10.780,−9.833, 0.750 and −0.213. For θ = 1.5 the eigenvalues of DTa are −9.997
and 0.841, while the eigenvalues of DTb are −11.087, −10.138, 0.701 and−0.213.
The results are very sensitive to αx . For αx = 0.1, we obtain that forθ = 1 the eigenvalues of DTa are −22.954 and 0.912, while the eigenvaluesof DTb are −24.042, −23.033, 0.835 and −0.143.
It can be seen that the problem of large negative eigenvalues appears withthis rule, so that the potential of instability under constant-gain learning
exists. Using the Woodford calibration (including αx = 0.048) and choosingθ = 1.5, we find that constant-gain RLS learning becomes unstable for valuesof the gain at or above 0.029.
We again examine numerically the sensitivity of the stability upper boundon γ for diﬀerent values of αx , i.e. the degree of flexibility of inflation targeting. Table 2 gives the approximate highest value γ̂ of the gain for which
stability under constant-gain learning obtains.
19

αx
γ̂

0.01 0.02 0.03 0.04 0.05 0.06 0.08 0.1
0.395 0.107 0.058 0.037 0.026 0.021 0.015 0.01

Table 2: Critical values of γ for stability, McCallum-Nelson ruleComparing Tables 1 and 2, it is seen that the stability performance of the
McCallum-Nelson rule (20) is somewhat better than that of the hybrid rule(18) for the same parameter values. However, it is still the case that the
McCallum-Nelson rule is not robust for many plausible values of the gain
parameter.
We remark that McCallum and Nelson (2004) suggest that a preferable
alternative to (20) is to use forward expectations in place of nowcasts, since
this delivers superior results under rational expectations. In this case, themodel has no lagged endogenous variables, i.e. N = 0 in (19). This casehas been analyzed numerically in Evans and Honkapohja (2003a) and Evansand Honkapohja (2006). In this formulation, large negative eigenvalues nolonger arise. However, we found that determinacy and E-stability require asmall value of the parameter θ, while for small values of θ the welfare losses
for optimal policy can be significant.

6

Conclusions

A lot of recent applied research on learning and monetary policy has emphasized discounted (constant-gain) least-squares learning by private agents. We
have examined the stability performance of various operational interest-rate
rules under constant-gain learning for diﬀerent values of the gain parameter.
Since estimates of the gain parameter tend to be in the range of 0.02 to 0.06
for quarterly macro data, ideally there should convergence of learning forgain parameters up to 0.1. Based on this criterion, we have found that many
proposed interest-rate rules are not robustly stable under learning in thissense. An exception to this finding is the class of expectations-based optimal
rules in which the interest rate feeds directly back on private expectations in
an appropriate way.

20

A

Appendix

Constant-gain RLS Algorithm
Suppose the economy is described in terms of a multivariate linear model,
which includes possible dependence on lagged endogenous variables.
Under least-squares learning agents have the PLM
yt = a + byt−1 + cvt + et ,(21)

where a, b and c denote parameters to be estimated. Here yt is a p × 1 vectorof endogenous variables. vt is k × 1 vector of observable exogenous variables,and et is a vector of white noise shocks. If the model does not have lagged
endogenous variables, then the term byt−1 is omitted.
At time t agents compute their forecasts using (21) with the estimatedvalues (at , bt , ct ) based on data up to period t − 1. Constant-gain RLS takes
the formξ t = ξ t−1 + γRt−1 Zt−1 (yt−1 − ξ 0t−1 Zt−1 )0 ,0
− Rt−1 )Rt = Rt−1 + γ(Zt−1 Zt−10
, vt0 ) and 1 > γ > 0. The algorithm starts
where ξ 0t = (at , bt , ct ), Zt0 = (1, yt−1
at t = 1 with a complement of initial conditions. We remark that the only
diﬀerence from standard RLS is that the latter assumes a decreasing gain
γ t = 1/t.20

Proof of Proposition 5
We now sketch a proof of Proposition 5. We examine the formulas given
in equations (A7)-(A9) on p. 36 of Evans and Honkapohja (2006). Two of
the eigenvalues of DTb are 0, while the remaining eigenvalues are those of
the matrix
!
Ã
Kb =

−λβbπ
αx +λ2αx βbπαx +λ220−λβbx
αx +λ2αx βbxαx +λ2

The formal analysis of recursive least squares (RLS) learning in linear multivariate
models is developed e.g. in Evans and Honkapohja (1998) and Chapter 10 of Evans andHonkapohja (2001).

21

x −1)The eigenvalues of Kb are 0 and −1 < αxαβ(2b< 1. Likewise, two of2
x +λthe eigenvalues of DTc are 0 while the other two eigenvalues of those of thematrix
!
Ã

Kc =

−λβbπ
αx +λ2αx βbπαx +λ2−λβρ
αx +λ2αx βρ
αx +λ2,x −1+ρ), which is inside the unit circle
and the eigenvalues of Kc are 0 and αx β(bαx +λ2
unless ρ is negative and large in magnitude. Finally,
!
Ã

DTa =

−λβbπ
αx +λ2αx βbπαx +λ2−λβ
αx +λ2αx β
αx +λ2βbx
and its eigenvalues are 0 and 0 < ααxx+λ2 < 1.

22

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Debate. University of Chicago Press, Chicago, USA.
Branch, W., and G. W. Evans (2006): “A Simple Recursive Forecasting
Model,” Economic Letters, 91, 158—166.
Bullard, J. (2006): “The Learnability Criterion and Monetary Policy,”Federal Reserve Bank of St. Louis Review, 88, 203—217.
Bullard, J., and K. Mitra (2002): “Learning About Monetary Policy
Rules,” Journal of Monetary Economics, 49, 1105—1129.
Clarida, R., J. Gali, and M. Gertler (1999): “The Science of Monetary
Policy: A New Keynesian Perspective,” Journal of Economic Literature,
37, 1661—1707.
Duffy, J., and W. Xiao (2007): “The Value of Interest Rate Stabilization Policies When Agents are Learning,” Journal of Money, Credit andBanking, forthcoming.
Evans, G. W., and R. Guesnerie (1993): “Rationalizability, Strong Rationality, and Expectational Stability,” Games and Economic Behaviour,
5, 632—646.
Evans, G. W., and S. Honkapohja (1998): “Economic Dynamics with
Learning: New Stability Results,” Review of Economic Studies, 65, 23—44.
(2001): Learning and Expectations in Macroeconomics. Princeton
University Press, Princeton, New Jersey.
(2003a): “Adaptive Learning and Monetary Policy Design,” Journalof Money, Credit and Banking, 35, 1045—1072.
(2003b): “Expectations and the Stability Problem for Optimal Monetary Policies,” Review of Economic Studies, 70, 807—824.
(2006): “Monetary Policy, Expectations and Commitment,” Scandinavian Journal of Economics, 108, 15—38.

23

(2007): “Expectations, Learning and Monetary Policy:
Overview of Recent Research,” mimeo.

An

Evans, G. W., S. Honkapohja, and N. Williams (2006): “Generalized
Stochastic Gradient Learning,” mimeo.
McCallum, B. T. (1999): “Issues in the Design of Monetary Policy Rules,”in Taylor and Woodford (1999), chap. 23, pp. 1483—1530.
McCallum, B. T., and E. Nelson (2004): “Timeless Perspective vs. Discretionary Monetary Policy in Forward-Looking Models,” Federal Reserve
Bank of St.Louis Review, 86, 43—56.
Milani, F. (2005): “Adaptive Learning and Inflation Persistence,” Working
paper, University of California, Irvine.
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(2007b): “Learning and Time-varying Macroeconomic Volatility,”
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Svensson, L. E. O., and M. Woodford (2005): “Implementing Optimal
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24

www.st-and.ac.uk/cdma
ABOUT THE CDMAThe Centre for Dynamic Macroeconomic Analysis was established by a direct grant fromthe University of St Andrews in 2003. The Centre funds PhD students and facilitates a programmeof research centred on macroeconomic theory and policy. The Centre has research interests in areas
such as: characterising the key stylised facts of the business cycle; constructing theoretical models
that can match these business cycles; using theoretical models to understand the normative andpositive aspects of the macroeconomic policymakers' stabilisation problem, in both open andclosed economies; understanding the conduct of monetary/macroeconomic policy in the UK andother countries; analyzing the impact of globalization and policy reform on the macroeconomy;
and analyzing the impact of financial factors on the long-run growth of the UK economy, fromboth an historical and a theoretical perspective. The Centre also has interests in developing
numerical techniques for analyzing dynamic stochastic general equilibrium models. Its affiliatedmembers are Faculty members at St Andrews and elsewhere with interests in the broad area ofdynamic macroeconomics. Its international Advisory Board comprises a group of leading
macroeconomists and, ex officio, the University's Principal.
Affiliated Members of the School
Dr Fabio Aricò.
Dr Arnab Bhattacharjee.
Dr Tatiana Damjanovic.
Dr Vladislav Damjanovic.
Prof George Evans.
Dr Gonzalo Forgue-Puccio.
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Dr Peter Macmillan.
Prof Rod McCrorie.
Prof Kaushik Mitra.
Prof Charles Nolan (Director).
Dr Geetha Selvaretnam.
Dr Ozge Senay.
Dr Gary Shea.
Prof Alan Sutherland.
Dr Kannika Thampanishvong.
Dr Christoph Thoenissen.
Dr Alex Trew.

Prof Joe Pearlman, London Metropolitan
University.
Prof Neil Rankin, Warwick University.
Prof Lucio Sarno, Warwick University.
Prof Eric Schaling, Rand Afrikaans University.
Prof Peter N. Smith, York University.
Dr Frank Smets, European Central Bank.
Prof Robert Sollis, Newcastle University.
Prof Peter Tinsley, Birkbeck College, London.
Dr Mark Weder, University of Adelaide.
Research Associates
Mr Nikola Bokan.
Mr Farid Boumediene.
Mr Johannes Geissler.
Mr Michal Horvath.
Ms Elisa Newby.
Mr Ansgar Rannenberg.
Mr Qi Sun.

Senior Research Fellow

Advisory Board

Prof Andrew Hughes Hallett, Professor of
Economics, Vanderbilt University.

Prof Sumru Altug, Koç University.
Prof V V Chari, Minnesota University.
Prof John Driffill, Birkbeck College London.
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Dr Brian Lang, Principal of St Andrews University.
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Institute of Economic and Social Research.
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