Bank of Canada
Banque du Canada
Working Paper 2002-30 / Document de travail 2002-30
Inflation Expectations and Learning
about Monetary Policy
by
David Andolfatto, Scott Hendry, and Kevin Moran
ISSN 1192-5434
Printed in Canada on recycled paper
Bank of Canada Working Paper 2002-30
October 2002
Inflation Expectations and Learning
about Monetary Policy
by
David Andolfatto,1 Scott Hendry,2 and Kevin Moran2
1Department of Economics
Simon Fraser University
Burnaby, British Columbia, Canada V5A 1S6
dandolfa@sfu.ca
2Monetary and Financial Analysis Department
Bank of Canada
Ottawa, Ontario, Canada K1A 0G9
shendry@bankofcanada.ca
kmoran@bankofcanada.ca
The views expressed in this paper are those of the authors.
No responsibility for them should be attributed to the Bank of Canada.
iii
Contents
Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Abstract/Résumé . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.
Empirical Evidence of Inflation Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.
The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1
3.2
3.3
3.4
4.
Monetary Policy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.1
4.2
4.3
5.
Preferences and technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
Parameters of the interest-rate-targeting rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Shifts and shocks to monetary policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Monte Carlo Simulation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.1
6.2
6.3
6.4
7.
The monetary policy rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Monetary policy shocks and monetary policy shifts. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Incomplete information and learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Calibration and Solution of the Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.1
5.2
5.3
6.
Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Financial intermediaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Impulse responses following a regime shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
The experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Results for the benchmark case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Appendix A: Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
iv
Acknowledgements
The authors wish to acknowledge the expert research assistance of Veronika Dolar. We thank
Walter Engert, Paul Gomme, Andrew Levin, Césaire Meh, and seminar participants at the Bank of
Canada, the 2002 CEA meeting, and the 2002 North American Summer Meeting of the
Econometric Society for useful comments and discussions.
v
Abstract
Various measures indicate that inflation expectations evolve sluggishly relative to actual inflation.
In addition, they often fail conventional tests of unbiasedness. These observations are sometimes
interpreted as evidence against rational expectations.
The authors embed, within a standard monetary dynamic stochastic general-equilibrium model,
an information friction and a learning mechanism regarding the interest-rate-targeting rule that
monetary policy authorities follow. The learning mechanism enables optimizing economic agents
to distinguish between transitory shocks to the policy rule and occasional shifts in the inflation
target of monetary policy authorities.
The model’s simulated data are consistent with the empirical evidence. When the information
friction is activated, simulated inflation expectations fail conventional unbiasedness tests much
more frequently than in the complete-information case when this friction is shut down. These
results suggest that an important size distortion may occur when conventional tests of
unbiasedness are applied to relatively small samples dominated by a few significant shifts in
monetary policy and sluggish learning about those shifts.
JEL classification: E47, E52, E58
Bank classification: Business fluctuations and cycles; Economic models
Résumé
Divers indicateurs donnent à penser que les attentes d’inflation évoluent plus lentement que
l’inflation observée. En outre, dans bien des cas, ils se révèlent entachés de biais lorsqu’on les
soumet aux tests usuels d’absence de biais. Le rejet de ces tests est parfois interprété comme une
réfutation de l’hypothèse de rationalité des attentes.
Les auteurs intègrent, dans un modèle monétaire stochastique dynamique d’équilibre général
standard, un élément de friction relatif à l’acquisition de l’information et un mécanisme
d’apprentissage concernant la règle de taux d’intérêt qu’appliquent les autorités monétaires. Le
mécanisme d’apprentissage permet aux agents économiques ayant un comportement
d’optimisation de distinguer les chocs temporaires que subit la règle de politique monétaire des
modifications apportées à l’occasion à la cible d’inflation poursuivie.
Les données simulées dans le modèle sont conformes aux observations empiriques. Lorsque
l’élément de friction est pris en compte, les attentes d’inflation simulées comportent un biais
vi
d’après les tests standard, et ce, beaucoup plus souvent que dans le cas d’une information
complète, sans friction. Ces résultats indiquent qu’une forte distorsion de niveau est possible
lorsque les tests standard d’absence de biais sont appliqués à des échantillons relativement
restreints caractérisés par seulement quelques changements d’orientation importants de la
politique monétaire et que les agents mettent beaucoup de temps à reconnaître ces derniers.
Classification JEL : E47, E52, E58
Classification de la Banque : Cycles et fluctuations économiques; Modèles économiques
1
1.
Introduction
Various measures indicate that inflation expectations evolve sluggishly relative to
actual inflation. Expectations tend to underpredict inflation during periods of rising
inflation and overpredict it during periods of diminishing inflation.1 Related to this
sluggishness phenomenon is the stylized fact, documented by an extensive empirical
literature, that measured inflation expectations often reject the hypotheses of unbiasedness and efficiency.2 These results have sometimes been interpreted as evidence
against rational behaviour on the part of economic agents.
This paper assesses whether an information friction over, and a learning mechanism about, the interest-rate-targeting rule followed by monetary policy authorities, once embedded in a standard monetary dynamic stochastic general-equilibrium
(DSGE) model, can lead simulated data to quantitatively replicate the empirical evidence against the unbiasedness of inflation expectations.
The following information friction is introduced. We assume that the interestrate-targeting rule followed by monetary policy authorities is affected by transitory
shocks but also, occasionally, by persistent shifts in the inflation target that anchors
the rule. We interpret the transitory shocks in the standard way, as instances of
monetary policy authorities wishing to deviate from their rule for a short period; for
example, to react to financial shocks. We assume that the occasional shifts in the
inflation target reflect changes in economic thinking about the optimal inflation rate,
or the appointment of a new central bank head with different preferences for inflation
outcomes. Importantly, we also assume that these transitory shocks and persistent
shifts cannot be separately observed (nor credibly revealed). Consequently, market
participants must solve a signal-extraction problem to distinguish between the two
components, giving rise to a learning rule that shares some features with adaptiveexpectations processes.3
Next, we calibrate the parameters of this signal-extraction problem and embed
it within the limited-participation environment developed by Christiano and Gust
(1999). We then repeatedly simulate the model and perform unbiasedness tests on
the artificial data equivalent to those performed on measured inflation expectations.
Our simulations identify substantially different outcomes in the unbiasedness
tests when the information friction is active compared with the complete-information
case when it is shut down. Specifically, the fraction of rejections when complete information is assumed never deviates significantly from the level suggested by the
1
For example, Dotsey and DeVaro (1995) uncover economic agents’ expectations about U.S.
inflation—using commodity futures data—over the disinflationary episode of 1980Q1–1983Q3, and
find that expected inflation exceeded actual inflation in all but three periods for the eight-month
forecasts and in each period for the one-year forecasts. DeLong (1997) reports that, during the
U.S. inflationary episode of the 1970s, a consensus, private sector inflation forecast underestimated
the actual inflation rate in every year and that, remarkably, in each and every year inflation was
actually expected to fall (Figure 6.9, 267).
2
See Thomas (1999), Roberts (1997), and Croushore (1997), and the references they cite.
3
Muth (1960) demonstrates that the optimal learning rule in such a signal-extraction problem
resembles adaptive-expectation processes.
2
size of the tests. In contrast, when the information friction is activated, the tests
reject the null hypothesis of unbiasedness much more frequently—between two to
fives times—although our model embodies the “rational expectations” solution concept by construction. Interestingly, these differences are much attenuated when the
sample size of each simulation is increased significantly.
Given these results, we propose the following interpretation of the empirical
rejections of the unbiasedness hypothesis. The process by which economic agents
form inflation expectations may be fundamentally sound, but a few significant shifts
in monetary policy, coupled with relatively sluggish learning about those shifts, can
lead to significant size distortions of the tests. Furthermore, while this effect may
be sufficient to trigger excessive rejections of the null hypothesis in small samples,
it should disappear as the sample size grows.
Environments with information frictions and learning effects similar to the one we
describe have been used previously, notably to rationalize the persistent responses
of real variables following monetary policy shocks.4 Our paper makes a twofold
contribution to this literature.
First, we locate the signal-extraction problem within an interest-rate-targeting
rule, rather than a monetary-growth process. This feature, which our model shares
with that of Erceg and Levin (2001), allows the learning literature to connect with
the now-standard view of the proper modelling of monetary policy.
Second, we evaluate the incomplete information and learning framework not by
its capacity to generate persistence in the dynamics of real variables, as Erceg and
Levin (2001) do, but by its ability to replicate the dynamic relationship that exists
between realized and expected inflation. More precisely, we specify parameter values
for the underlying components of the monetary policy process and verify whether
the rejection of the unbiasedness hypothesis emerges as an implication of these parameter values. Conversely, Erceg and Levin (2001) choose parameter values to
match the relationship between realized and expected inflation and concentrate on
the implication of their chosen specification for real variables. Our analysis complements theirs and broadens the scrutiny of the empirical relevance of incomplete
information and learning effects.
Our strategy is similar to that of Kozicki and Tinsley (2001a), who argue that the
frequent empirical rejections of the term structure’s expectation hypothesis could
be the result of economic agents learning only gradually about shifts in the Federal
Reserve’s objectives. Kozicki and Tinsley embed a learning mechanism similar to
ours in a simple macroeconomic environment and assess whether the expectation
hypothesis is rejected by the simulated data. In earlier contributions, Lewis (1988,
1989) uses similar intuition to verify whether sluggish learning can generate the
4
Recent contributions include Andolfatto and Gomme (1999), Moran (1999), Andolfatto,
Hendry, and Moran (2000), Erceg and Levin (2001), who analyze closed economies, and Sill and
Wrase (1999), who study an open-economy environment. In an early contribution using a different
modelling technology, but appealing to very similar ideas, Brunner, Cukierman, and Meltzer (1980)
analyze the properties of a stochastic IS–LM model in which agents cannot distinguish between
permanent and transitory shocks to real and nominal variables.
3
“forward discount” puzzle observed in foreign exchange market data.
This paper is organized as follows. Section 2 describes the stylized fact that
measured inflation expectations fail simple unbiasedness tests. Section 3 describes
the model used in our simulation, essentially the one developed by Christiano and
Gust (1999). Section 4 details our view of monetary policy as an interest-ratetargeting rule affected by two types of disturbances: transitory shocks to the rule,
and occasional, persistent shifts in the inflation target of monetary policy authorities.
Section 4 also describes the mechanics of the Kalman filter, used by economic agents
to solve the signal-extraction problem and to distinguish one component of monetary
policy disturbances from another. Section 5 explains the calibration strategy we
utilize. Section 6 describes our Monte Carlo simulations and our results. Section 7
concludes.
2.
Empirical Evidence of Inflation Expectations
Survey data are one of the tools commonly used to identify economic agents’ inflation
expectations.5 To illustrate a typical path for such data, Figure 1 depicts the (mean)
forecast for one-year-ahead inflation (as measured by the Livingston survey) as well
as the inflation rate that eventually prevailed.6 The sluggishness described earlier
is clear: in times of generally rising inflation, such as the 1970s, expected inflation
tends to underpredict realized inflation. In contrast, in times of falling inflation,
such as the 1980s and 1990s, the forecasts appear to overpredict inflation.
Several studies examine the statistical properties of such inflation expectations,
with the objective of testing for departures from rationality. Such departures, usually identified as rejections of unbiasedness and efficiency, appear to be a common
conclusion of this literature.7 The unbiasedness tests are typically conducted by
testing H0 : a0 = 0; a1 = 1 using the following simple regression equation:
πt = a0 + a1 πte + ǫt ,
(1)
where πt is the net, annualized rate of inflation from period t-k to period t and πte
is the expectation of πt formed at time t-k.
We identify the rejection of unbiasedness, defined using (1), as the stylized fact
that the model should replicate. For illustrative purposes, we reproduce below one
5
Other methods include uncovering inflation expectations from futures market data (as in Dotsey and DeVaro 1995) or comparing yields on inflation-indexed and non-indexed treasuries (see
Shen and Corning 2001).
6
The Livingston survey was started by J.A. Livingston, a business journalist in the Philadelphia
area, and is now maintained by the Federal Reserve Bank of Philadelphia. Croushore (1997)
describes the history of the survey and its current structure. Other survey data on inflation
expectations include those from the survey of households conducted by the Institute for Social
Research at the University of Michigan, and the more recently established Survey of Professional
Forecasters. Thomas (1999) describes the three surveys. The Conference Board of Canada also
has produced, since 1988, survey data on (Canadian) inflation expectations.
7
Thomas (1999) conducts unbiasedness and efficiency tests on the three sources of survey data.
Croushore (1997) reviews the tests conducted on the Livingston data over the years.
4
of Thomas’ (1999) regressions. Run with data from the Livingston survey, it shows
the following estimated equation:
πt = 0.134 + 0.88Et−2 [πt ],
(0.41)
(0.08)
(2)
where the sample used is 1980Q3 to 1997Q4. The data are of semi-annual frequency and the expectations have a one-year-ahead horizon (two periods).8 These
estimation results lead to a rejection of H0 .9
The rejection of H0 can be overturned in some large samples, where the positive
forecasting errors of the 1970s appear to cancel the mainly negative ones of the
1980s. This suggests that the rejections of the unbiasedness hypothesis could simply
be owing to a small sample problem. As stated in section 1 and described in section
6, this is precisely what our results imply.
Interestingly, once it is defined with a quasi-difference specification, the unbiasedness hypothesis continues to be rejected in large samples.10 Future research
might investigate whether this facet of the relationship between realized and expected inflation could be replicated by our incomplete information and learning
environment.
8
This frequency is not standard across all sources of inflation-expectations data. In the model
we developed, a period corresponds to one quarter and we report simulation results obtained with
one-quarter-ahead and four-quarters-ahead expectations.
9
A correction for serial correlation in the residuals must be introduced when constructing the
test statistic. Thomas (1999) reports the results of estimating (1) on other samples and with
alternative measures of inflation expectations. On balance, the evidence points to rejections of
the null hypothesis, especially when the sample being considered is small. A similar regression
run with the Canadian data on inflation expectations, over the sample running from 1988Q1 to
2001Q1, yields the following estimate:
πt = 0.29 + 0.77Et−4 [πt ],
(0.29)
(0.11)
with, again, an easy rejection of H0 .
10
Consider estimating the following regression:
πt+k − πt = a0 + a1 (Et [πt+k ] − πt ) + ut ,
and testing H0 : a0 = 0, a1 = 1. Under the null hypothesis, both this regression and (1)
are identical. Nevertheless, Dolar and Moran (2002) report that the evidence against the null
hypothesis is much more robust using this regression. Furthermore, the estimates of a1 arising
from the regression are almost always between zero and one. This specification is very similar to
the one often used to document the forward discount puzzle:
et+1 − et = b0 + b1 (ft − et ) + ut ,
with et the spot exchange rate at time t and ft the forward exchange rate. Researchers have
often proposed learning effects as one potential explanation for the frequent empirical rejections of
H0 : b0 = 0, b1 = 1 obtained from this regression. See Froot and Thaler (1990) and Taylor (1995)
for a discussion.
5
3.
The Model
The model we used is very similar to the one developed by Christiano and Gust
(1999). We therefore provide only an overview and refer interested readers to the
original paper. The main nominal rigidity appearing in the model is the assumption
of limited participation, one of the standard ways of introducing monetary nonneutralities in a DSGE model. In contrast, Erceg and Levin (2001) use nominal
price and wage stickiness to achieve this non-neutrality. We could redo our analysis
with these nominal rigidities, but the robustness of our results, described in section
6.4, suggests that using nominal price or wage rigidity would not alter our main
conclusions.
3.1
Households
The model economy comprises a continuum of identical, infinitely lived households.
At the start of every period, a household’s wealth consists of kt units of capital,
Mtc units of liquid financial assets, and Mtd units of illiquid assets (deposited at a
financial intermediary).11 During the course of the period, households rent their
capital to firms, allocate their time between work and leisure, choose desired levels
of consumption and investment, and choose how to allocate their financial assets
into the cash and deposits they will carry over to the next period.
The purchase of consumption and investment goods must be carried out with
liquid assets. Available liquid assets consist of beginning-of-period balances (Mtc )
and wage payments. This assumption leads to the following liquidity constraint:
Pt ct + Pt (kt+1 − (1 − δ) kt ) ≤ Mtc + Wt nt ,
(3)
where ct is per-household consumption, (kt+1 − (1 − δ)kt ) is investment, Pt is the
nominal price of goods, Wt is the nominal wage, and nt is labour supply.
At the end of the period, households receive their capital rental income and
return on deposits. These revenues, combined with any liquid assets remaining
from their goods purchases, sum up to their end-of-period financial wealth, which
is allocated between next period’s liquid and illiquid assets. The following budget
constraint arises:
c
d
Mt+1
≤ rkt kt + Rtd Mtd + (Mtc + Wt nt − Pt ct − Pt (kt+1 − (1 − δ)kt )),
+ Mt+1
(4)
where rkt is the rental rate on capital and Rtd the return on illiquid assets.
Households choose a plan for consumption, investment, labour supply, and financial asset allocation to maximize their lifetime utility. Hence, they solve the
11
In what follows, lower-case variables kt and nt express the levels of capital and work supplied
by the households; upper-case variables Kt and Nt represent the quantities of these variables
demanded by firms. Moreover, Mtd and Mtc express households’ liquid asset holdings, while Mt
denotes the total supply of such assets.
6
following problem:
max
c
d
[ct+k ,nt+k ,kt+k+1 ,Mt+k+1
,Mt+k+1
]|∞
k=0
Et
∞
X
η t+k U(ct+k , nt+k + ACt+k )
(5)
k=0
where U(., .) is the period utility function and η the time discount, and where the
maximization is done with respect to (3),(4), and initial levels kt , Mtc , and Mtd .
The term nt+k + ACt+k represents the time costs of market activities, in terms
of leisure foregone. The term ACt represents the costs households must incur to
adjust their liquid asset portfolios. The functional form selected for these costs is
as follows:12
c
(6)
/Mtc − µ)2 ,
ACt = τ (Mt+1
with µ the steady-state growth rate of the total supply of liquid assets.
3.2
Firms
Firms combine labour and capital inputs to produce the economy’s output. They
have access to the following constant-returns-to-scale production function:
Yt = At Ktθ Nt1−θ ,
where At denotes a transitory productivity shock that evolves according to the
following process:
At = (1 − ρA )A + ρA · At−1 + νtA , νtA ∼ N(0, σA2 ).
(7)
Firms rent capital and hire labour to maximize per-period profits. Since firms
pay their capital rental expenditures directly from revenues, the first-order condition
for the choice of capital is the familiar one:
rkt = θAt (Kt /Nt )θ−1 .
(8)
In contrast, it is assumed that a given fraction (denoted 1−Jt ) of the firms’ wage
costs must be paid in advance. To do so, firms must borrow the necessary funds
from financial intermediaries at the rate Rtl . This assumption leads to the following
first-order condition for labour demand:13
((1 − Jt )Rtl + Jt )Wt /Pt = (1 − θ)At (Kt /Nt )θ .
(9)
The evolution of Jt (which we call a money-demand shock) is exogenous and
obeys the following:
Jt = ρJ Jt−1 + νtJ , νtJ ∼ N(0, σJ2 ).
(10)
12
Expressing these costs in terms of leisure rather than goods is not important for the results.
Because the production function features constant returns to scale, these efficiency conditions
also hold for the aggregate values of capital and labour demand in the economy. Hereafter, Kt and
Nt represent those aggregate quantities.
13
7
3.3
Financial intermediaries
Financial intermediaries accept deposits from households and lend the receipts to
firms. Furthermore, they are the recipients of any liquid assets that the central bank
injects into the economy to support its monetary policy rule.14 The revenues of the
intermediaries are therefore the total amount lent multiplied by the lending rate,
while their expenses are the total deposits received multiplied by the deposit rate.
Profits are thus
Rtl Bt − Rtd (Mtd + Xt ),
where Bt is total lending and Xt represents injections into the economy of liquid
assets by the central bank. The assumption of perfect competition in the financial
sector ensures that, in equilibrium, Rtl = Rtd ≡ Rt .
3.4
Equilibrium
An equilibrium for this artificial economy consists of a vector of allocations (ct+k ,
c
d
nt+k , kt+k+1 , Mt+k+1
, Mt+k+1
, Nt+k , Kt+k , Bt+k )|∞
k=0 , of prices (Pt+k , Wt+k , Rt+k ,
∞
rkt+k )|k=0 , of exogenous variables (At+k , Jt+k |∞
),
and of starting values (kt , Mtc ,
k=0
Mtd ). These allocations, prices, exogenous variables, and starting values are such
that households maximize lifetime utility, as stated by (5); firms and financial intermediaries maximize profits; and the following market-clearing equilibrium conditions
are met:
ct + Kt+1 − (1 − δ)Kt = Yt ;
Mtc + Mtd = Mt ;
Mtd + Xt = Bt = Wt Nt ;
Nt = nt ;
Kt = k t .
4.
Monetary Policy
4.1
The monetary policy rule
Monetary policy authorities target the nominal interest rate. This targeting is made
precise by assuming that the desired nominal rate, denoted i∗t , is the following function of macroeconomic conditions:
i∗t = r ss + πtT + α(πt − πtT ) + βyt ,
(11)
where r ss is the steady-state value of the real interest rate, πtT is the inflation target
of the monetary policy authorities at time t, and yt represents the output gap.
14
They cannot profit from these injections, however, because it is assumed that the injections
are deposited in the households’ accounts.
8
Note that r ss + πtT represents the steady-state value of the nominal rate; (11) thus
signifies that monetary authorities will increase rates relative to steady-state when
price pressures threaten to push inflation over the current target, or when the output
gap is positive.
It is often conjectured that, instead of rapidly moving the nominal rate to reach
the targeted level, monetary policy authorities implement gradual changes in rates
that only eventually converge to that level. Such a smoothing motive can be represented mathematically by assuming that the actual rate implemented by the central
bank will be the following weighted sum of the targeted rate and the preceding
period’s rate:
it = (1 − ρ)i∗t + ρit−1 ,
(12)
where the coefficient ρ governs the extent of smoothing exercised by the monetary
policy authorities.
Monetary policy authorities regularly deviate from their rule. These deviations
(described in section 4.2) are called monetary policy shocks and are denoted by the
variable ut . Combining equations (11) and (12), as well as introducing the ut shocks,
leads to the following characterization of monetary policy:
it = (1 − ρ)[r ss + πtT + α(πt − πtT ) + βyt ] + ρit−1 + ut .
(13)
The instrument by which monetary authorities implement the rule (13) remains
the growth rate of money supply. The significance of this rule is that the central
bank manipulates this growth rate, (µt = MMt+1
), such that the observed relationship
t
between nominal rates, inflation, and output that emerges obeys (13).
4.2
Monetary policy shocks and monetary policy shifts
We assume that monetary policy, as expressed by the interest rate rule in (13),
is subject to two types of disturbances. The first consists of the monetary policy
shocks referred to in section 4.1 (the variable ut ). We interpret these disturbances
as the reaction of monetary authorities to economic factors, such as financial stability concerns, not articulated in the rule (13). Alternatively, these shocks could
be understood to be errors stemming from the imperfect control exercised by monetary policy authorities over the growth rate of the money supply (µt ). Under
either interpretation, however, we envision that these shocks have little persistence.
Accordingly, we assume that their evolution is governed by the following process:
ut+1 = φ1 ut + et+1 ,
(14)
with 0 ≤ φ1 << 1 and et+1 ∼ N(0, σe2 ).
The second disturbance that monetary policy is subjet to is as follows. We assume that, while remaining constant for extended periods of time, the monetary
policy authorities’ inflation target, πtT , is nevertheless subject to occasional, persistent shifts. We see two possible interpretations of these shifts. First, they could
9
correspond with changes in economic thinking that lead monetary policy authorities to modify their views about the proper rate of inflation to pursue. DeLong
(1997), for example, argues that the Great Inflation of the 1970s, and its eventual
termination by the Federal Reserve at the beginning of the 1980s, was a result of
shifting views about the shape of the Philips curve and, more generally, about the
nature of the constraints under which monetary policy is conducted. Alternatively,
a change in the inflation target could reflect the appointment of a new central bank
head, whose preferences for inflation outcomes differ from their predecessor’s. Under
either interpretation, we envision that these shifts will have a significant duration,
in the order of, say, five to ten years.
Mathematically, we express these shifts in the inflation target by the variable
zt ≡ πtT − π0T , so that zt constitutes the deviation of the current target of authorities
(πtT ) from its long-term unconditional mean (π0T ). We assume that the following
process, a mixture of a Bernoulli trial and a normal random variable, expresses how
zt evolves over time:
zt
with probability φ2 ,
zt+1 =
(15)
gt+1 with probability 1 − φ2 , gt+1 ∼ N(0, σg2 ),
with 0 << φ2 < 1. In some ways, the process for zt shares some similarities with
a random walk specification. Specifically, with a high value of φ2 the conditional
expectation of zt+1 is close to zt . In contrast with a random walk, however, the
process is not affected by innovations every period and is, ultimately, stationary.
On the other hand, the process differs from a standard autoregressive process in
that the decay of a given impulse will be sudden and complete, rather than gradual.
We believe that this characterization of the regime shifts accords well with recent
episodes of monetary history and with our suggested interpretations of these shifts.15
In section 4.3, we refer to some model simulations as representing complete
information. By this we mean that economic agents can observe the exact decomposition of monetary policy disturbances between their zt and ut parts. In such a
case, although uncertainty remains (it arises from the innovations et+1 and gt+1 ),
agents have sufficient information to compute the correct conditional expectations
concerning future monetary policy.
4.3
Incomplete information and learning
To credibly communicate shifts in the inflation target might be difficult for monetary policy authorities. For example, although a new central bank head with a
strong aversion to inflation might indicate this aversion in public announcements,
economic agents may be uncertain as to what these announcements mean for the
quantitative inflation targets. As a result, they might treat the announcements with
skepticism and modify their beliefs about the monetary policy authorities’ inflation
target only after observing several periods of lower inflation. Announcements of
15
It is left for future research to determine how much difference it would make, in practice, to
model the shifts as arising from a random-walk process with very low innovation variance.
10
explicit, quantitative changes in the inflation target might suffer, at least initially,
from similar credibility problems.16 Alternatively, central banks sometimes do not
make explicit announcements about their inflation target, but let economic agents
decipher as best they can announcements of a more general nature.
To capture the spirit of this information problem, we assume that the zt shifts
are unobservable to economic agents. They observe only a mixture of the zt shifts
and the ut shocks. Agents thus face a signal-extraction problem that is solved using
the Kalman filter.
Recalling (13), assume that, at time t, the long-run inflation target is changed
from its unconditional mean of π0T to πtT . Assume also, for notational purposes, that
the response to the output gap—the coefficient β—is zero. The rule is thus:
it = (1 − ρ)[r ss + πtT + α(πt − πtT )] + ρit−1 + ut .
(16)
Rewrite (16) by adding and subtracting π0T two times:
it = (1 − ρ)[r ss + πtT + (π0T − π0T ) + α(πt − πtT + π0T − π0T )] + ρit−1 + ut ,
(17)
or, rearranging terms,
it = (1 − ρ)[r ss + π0T + α(πt − π0T )] + ρit−1 + (1 − ρ)(1 − α)(πtT − π0T )) + ut . (18)
|
{z
}
ǫ∗t
Equation (18) illustrates that, from the viewpoint of an economic agent whose
initial belief about the monetary policy authorities’ inflation target was π0T , the
observed shock to the policy rule (ǫ∗t ) is a combination of a persistent shift (1 −
ρ)(1 − α)(πtT − π0T ) and the transitory disturbance to rule ut . The signal-extraction
problem that economic agents face thus entails separating ǫ∗t into its persistent and
transitory components. Then, given their knowledge of the rule and its parameters
(α and ρ), agents can back-out an estimate of πtT − π0T , the shift in the inflation
target.17
As stated earlier, the signal-extraction problem is solved using the Kalman filter.
The evolution of ǫ∗t , the observed shock to the monetary policy rule in (18), can be
16
Even after such announcements are made and credibility is largely established, substantial
uncertainty over the weight attributed by the central bank to inflation outcomes within a targeted
range might still remain. Ruge-Murcia (2001), for example, argues that, contrary to stated weights,
the inflation outcomes of the 1990s in Canada are consistent with asymmetric preferences of the
Bank of Canada over its official target range.
17
A different type of learning could also be modelled. Agents could be considered to have
imperfect knowledge about the coefficients of the rule (α, β, and ρ) and to learn about these shifts
by repeated observations of the interest rate changes engineered by monetary policy authorities.
Empirical estimations of Taylor-type monetary policy rules have identified structural shifts in the
parameters of such rules occurring around 1980. See Clarida, Galı́, and Gertler (2000), for example.
We plan to pursue the implications of this type of imperfect information in future work.
11
expressed within the following system:
zt+1
Nt+1
zt
φ2 0
;
+
·
=
0 φ1
et+1
ut
ut+1
zt
∗
(1 − ρ)(1 − α) 1 ·
;
ǫt =
ut
where Nt+1 is defined as follows:
(1 − φ2 ) zt , with probability φ2 ;
Nt+1 =
gt+1 − φ2 zt , with probability 1 − φ2 .
(19)
(20)
(21)
Under the definition of Nt , Et [Nt+1 ] = 0. The fact that Et [et+1 ] = 0 was already
assumed in equation (14).
Equations (19) and (20) define a state-space system (e.g., Hamilton 1994, chapter
13), where (19) is the state equation and (20) the observation equation. When
applied to such a system, the Kalman filter delivers forecasts of the two unobserved
states (zt and ut ), conditional on all observed values of ǫ∗t . We assume that economic
agents know the value of all the parameters of the problem, so that the forecasts
arising from the filter are feasible.
The projections underlying the Kalman filter are updated sequentially and represent the best linear forecasts of the unobserved variables based on available information. Furthermore, if the variables in the dynamic system are normal, the forecasts
arising from the filter are optimal.18 We denote the forecasts of the two unobserved
variables, given the information available at time t, as zbt|t and u
bt|t . Equations (19)
and (20) can then be used to compute expected future deviations of the interest rate
from the benchmark rule, as follows:
zbt+1 |t
zbt |t
φ2 0
;
(22)
·
=
0 φ1
u
bt |t
u
bt+1 |t
zbt+1 |t
∗
(1 − ρ)(1 − α) 1 ·
.
(23)
Et [ǫt+1 ] =
u
bt+1 |t
Additional details on our implementation of the Kalman filter (which requires
nothing more than to establish a formal correspondence between our notation and
that by Hamilton) are provided in Appendix A.
The information friction that we assume is stronger, in some sense, than others
often used in the literature—notably by Andolfatto and Gomme (1999), who assume
that the “regime” part of monetary policy can take only a finite number of values
(usually two). Such a restriction simplifies the learning problem of economic agents
18
Examination of (15) shows that, conditional on the value of zt , zt+1 is not normally distributed.
But when one considers that the only source of variation in zt arises from a normal variable, it
must be that, in an unconditional sense, zt is distributed normally. Considering the high values of
φ2 used in our calibration, however, this unconditional, normal behaviour will appear only after a
very large number of data have been observed.
12
and usually produces quick transition of the beliefs following regime shifts. We consider, however, that these “two-point” learning problems understate the severity of
the information friction over monetary policy faced by real-world economic agents.19
5.
Calibration and Solution of the Model
Three distinct areas of the model require calibration: the model itself (preferences,
technology, etc.), the parameters of the interest rate rule (13), and the parameters
governing the evolution of the shocks and shifts in the rule. The model period
corresponds to one quarter.
5.1
Preferences and technology
The first part of the calibration exercise is straightforward, as we adopt most of the
choices made in Christiano and Gust (1999). For example, the utility function is
specified to be:
(nt + ACt )1+ψ1
U(ct , nt + ACt ) = log[ct − ψ0
].
1 + ψ1
Under this specification of utility, no intertemporal smoothing motive is present in
the labour supply; the only factor affecting the decision of households is the real
wage, with an elasticity of 1/ψ1 .20 We choose ψ1 so that the elasticity is 2.5. The
parameter ψ0 is mainly a scale parameter and we fix its value to 2.15, which implies
a steady-state value of around 1.0 for employment.
The parameter τ expresses the severity of the portfolio adjustment costs. We fix
its value to 10.0, which, in a version of the model that expresses monetary policy
as an exogenous process for money growth, generates sizable persistence following a
monetary policy shock.
Other parameters governing preferences and technology appear in most models,
and standard values for their calibration are established: we thus fix η to 0.99, θ to
0.36, and δ to 0.025.
We conduct two types of experiments regarding our assumption about the technology (At ) and money-demand (Jt ) shocks. We first envision a world where disturbances to the monetary policy rule (ut and zt ) are the only source of volatility. In
such a world, we fix technology and money demand at their long-run mean, so that
At ≡ 1 and Jt ≡ 0, ∀t.
In addition, we want to add these two extra sources of volatility into the model.
We thus reintroduce the technology shocks by using the familiar values of 0.95 for ρA
and 0.005 for σA . Because no similar values are established for the money-demand
shock, we follow Christiano and Gust and apply the technology shock values to the
process for Jt : we thus have ρJ = 0.95 and σJ = 0.005.
19
20
See Kozicki and Tinsley (2001b, 165) for a similar argument.
See Greenwood, Hercowitz, and Huffman (1988) for further details.
13
The model is solved using the first-order approximation method and algorithms
given in King and Watson (1998). Details of the solution method are available from
the authors upon request.
5.2
Parameters of the interest-rate-targeting rule
According to the rule in (13), current interest rates are determined by the deviation
of inflation from its current target (with a coefficient α), by the output gap (β), and
by its own lagged values (ρ).
The evidence about the correct values for these coefficients is not precise, particularly because empirical studies of interest-rate-targeting rules (Taylor 1993, Clarida,
Galı́, and Gertler 2000, Nelson 2000) often use specifications of (13) that, although
similar in spirit to the one used here, differ in the details of the timing assumptions and definitions used. Furthermore, some values of the triple (α, β, ρ) lead to
non-uniqueness (or non-existence) of stable equilibria in the model.21
We thus use such empirical evidence to suggest a range of reasonable values for
the parameters and conduct a sensitivity analysis of our results to different values
within that range. For example, many empirical studies report evidence that the
behaviour of monetary policy authorities is consistent with significant smoothing of
interest rate changes. We thus use a range of [0, 0.5] for the parameter ρ. To ensure
the uniqueness of equilibria, we must fix the coefficient describing the response to
inflation, α, to a relatively high value. We thus explore values in the range [2.0, 4.0]
for that parameter. The same requirement of uniqueness suggests relatively low
values for the response to the output gap, β. We thus use a range of [0, 0.25]. Our
benchmark specification sets α = 2.0, ρ = 0.25, and β = 0.25.22
5.3
Shifts and shocks to monetary policy
We now describe the calibration of the processes governing the evolution of the
shocks (the ut variables) and the shifts (the zt variables) in monetary policy.
Recall that φ2 and σg govern the dynamics of the zt variable. These parameters
respectively express the expected duration of a particular regime and the standard
deviation of the distribution from which the value of a regime shift, when one occurs,
is drawn. φ1 and σe denote the autocorrelation and innovation variance of the ut
shocks.
The interpretations suggested above for the shifts in the variable zt —changes in
economic thinking or appointments of new central bank heads—suggest that these
21
See Christiano and Gust (1999) for a detailed examination of the ranges of values for which
non-uniqueness obtains.
22
We define the output gap as the deviation of current output from its steady-state value. This
is an incorrect definition, particularly in the presence of technology shocks that modify potential
output significantly. A better measure of the output gap results when potential output is defined
as the level of output that would obtain in a version of the model where all nominal frictions have
been removed.
14
shifts occur only infrequently, perhaps once every five or ten years. Transposed to
the quarterly frequency we use, this corresponds to one shift, on average, every 20
to 40 periods. Such an average duration between shifts corresponds to values of φ2
between 0.95 and 0.975. We use the slightly wider range of [0.95, 0.99] for φ2 , with
0.975 as the benchmark value.
Calibrating the standard deviation of the innovation in regime shifts, σg , is less
straightforward. In our benchmark specification, we set it to 0.005, which implies
that when a one-standard-deviation shift does occur, it corresponds to a change
of 2 per cent, on an annualized basis, in the inflation target of monetary policy
authorities. We also explore the consequences of lower (0.0025) and higher (0.01)
values for this parameter.
One interpretation of the Romer and Romer (1989, 1994) dates is that they represent changes in the inflation target of the Federal Reserve, and therefore occurrences
of zt shifts.23 Because seven such dates are identified over a 40-year sample, this
would correspond to an expected duration of five to six years (or 20 to 25 quarters)
for these shifts, placing the duration parameter within the range we use.24
To calibrate the transitory shocks, ut , we simply use a range of [0, 0.2] for
the autocorrelation parameter, φ1 , with 0.1 as the benchmark value. We set the
benchmark value of the variance of the innovations to these shocks, σe , to 0.005, in
a symmetric way with the variance of the regime shifts, and experiment with lower
values. Because ut is equivalent to the interest rate shock in the monetary policy
rule, a one-standard-deviation value of 0.005 corresponds to a 2 per cent innovation
in the (annualized) rate. Considering that central banks usually change interest
rates by much lower increments, a value of 0.005 for σe is probably an upper bound.
Table 1 summarizes the calibration values that we use.
Figure 2 illustrates the impact of the information friction (in our benchmark
calibration) following a negative, one-standard-deviation shift in zt . Again, this
shift corresponds to a decrease in the inflation target from 5 per cent to 3 per cent.
The true zt , along with agents’ best estimate of that variable, appears in the top
panel of the figure.
Following the shift, economic agents assign some weight to the possibility that
the observed disturbance to monetary policy was a regime shift, and thus the top
panel of Figure 2 shows that the agents’ best estimate of zt starts to decline towards
the true value. Agents also assign some weight, however, to the possibility that
the observed disturbance was a transitory shock, ut . Thus the middle panel of the
graph shows that agents’ best estimate of ut rises after the initial period. Eventually,
because the shift is persistent, agents doubt more and more that it might have come
23
In their papers, Romer and Romer analyze the minutes of FOMC deliberations, and identify
dates at which the Federal Reserve Board decided to cause a recession to stop inflationary pressures.
See Hoover and Perez (1994) and Leeper (1997) for a discussion of Romer and Romer’s methodology
and results.
24
Other results with which we could match our calibration of the zt shifts are those in Owyang
and Ramey (2001), where the authors identify shifts in the preferences of monetary policy authorities over inflation, within an empirical expression of the classic Barro and Gordon (1983)
model.
15
from the largely transitory ut , and they become convinced that it must have come
from a zt shift. Accordingly, the agents’ best estimate of zt and ut , respectively,
converges towards the true value and to zero.
The bottom panel of Figure 2 shows the progress of the agents’ estimate of the
monetary policy authorities’ annualized inflation target (implied by their estimates
of zt : recall the definition of ǫ∗t in equation (18)). The panel shows that beliefs
smoothly converge towards the true value of 3 per cent.
6.
Monte Carlo Simulation of the Model
6.1
Impulse responses following a regime shift
To develop intuition about the Monte Carlo results that follow, Figure 3 shows the
impulse responses of the artificial economy following a shift in the monetary policy
authorities’ inflation target. The shift is identical to the one illustrated in Figure 2:
the inflation target is lowered, at time t = 5, from 5 per cent on an annualized basis
to 3 per cent.
In Panel A of Figure 3, the solid lines represent the case where agents have
complete information about the shift. In contrast, the dashed lines represent the
case where information friction is active and the learning mechanism governs the
formation of expectations.
The solid lines indicate that, following the implementation of the monetary policy
shift, a very short downturn affects the economy: consumption, output, and employment shrink for only one or two periods. Very rapidly, the positive, long-term effects
of the decrease in inflation begin to take hold and all real variables increase, passing
their initial levels and converging towards a higher steady-state. The dashed lines
indicate that, in the incomplete-information case, this process takes several periods
to firmly establish itself, during which all real aggregates are lower than they were
in the full-information case. This occurs because economic agents assign a positive
probability to the interest rate shock being a transitory hike in interest rates, with
straightforward negative effects on the real economy.
Panel B of Figure 3 illustrates the situation from a slightly different angle. The
solid lines depict realized inflation, and the dashed lines expected inflation. The left
graph in Panel B shows the complete-information case: apart from the initial surprise
in the first period of the shock, economic agents have the correct inflation expectations. The right graph in Panel B depicts the incomplete-information case: inflation
expectations lag actual inflation for several periods before converging. This feature
replicates, in a qualitative fashion, the behaviour of inflation expectations during
the 1980s, as shown in Figure 1. In that figure, during a period of generally decreasing inflation, expectations—as measured by the Livingston survey—overpredicted
actual numbers for several quarters.
The gap between realized and expected inflation in the right graph of Panel B,
Figure 3, is a direct result of the learning behaviour described in section 4. Initially,
16
agents assign some weight to the possibility that the observed monetary disturbance
was a transitory disturbance to the rule. They therefore do not expect it to last
and they think that inflation might return to the initial, higher level of 5 per cent.
Over time, agents become convinced that a shift has indeed occurred and their
expectations converge to values that are closer to the actual ones.
In the simulations performed using the model, transitory shocks occur simultaneously with the shifts in the inflation target; therefore, a picture of the artificial
economy’s responses will not be very informative. The main picture given by the
graphs in Figure 3, however, remains: when a shift occurs, agents are likely to underestimate them for some time, and inflation expectations are likely to erroneously
predict actual inflation for some time. It remains to be seen whether this effect is
strong enough to generate empirical rejections of the unbiasedness hypothesis.
6.2
The experiment
We treat our model as if it was the true data generating process (DGP) of economic
variables, and assess what an empirical researcher, given outcomes from this DGP,
would conclude about the unbiasedness of inflation expectations. To this end, Monte
Carlo simulations of the model economy are performed 1000 times.
In each of these simulations, a random realization of 80 periods is generated for
both unobserved disturbances to the interest rate rule (the ut and zt variables).25
Economic agents’ estimates of these shocks are computed and the model is solved
according to this information. Two alternative measures of inflation, one-quarterahead inflation (πt ≡ Pt+1 /Pt ) and four-quarters-ahead inflation (πt ≡ Pt+4 /Pt ),
are stored, along with the corresponding expectations of these quantities (πte ≡
Et [Pt+1 /Pt ] and πte ≡ Et [Pt+4 /Pt ]).
Next, for each of these simulations, we perform the unbiasedness test described
in section 2. Recall that this involves estimating the regression
πt = a0 + a1 πte + εt ,
(24)
and testing the null hypothesis H0 : a0 = 0; a1 = 1. For each simulation, we record
the estimates ab0 and ab1 , as well as the appropriate test statistic about H0 .26
Figures 4 to 7 show the results of these simulations using the benchmark calibration. In each of those figures, the top panel is a histogram that depicts the estimates
of a0 across the 1000 replications. It also depicts the median of the estimates. The
middle panel depicts the estimates of a1 , again identifying the median. The bottom
25
The empirical rejections of the unbiasedness hypothesis described in section 2 are typically
obtained with data samples of limited length.
26
Under the null hypothesis and for one-quarter-ahead expectations, the expectation errors (the
residuals in (24)) should not be serially correlated and we therefore use a simple F -statistic to test
H0 . In the case of four-quarters-ahead expectations, the expectation errors would be correlated up
to three lags even under H0 , because of the overlap between the horizon of the expectations and
the frequency of the data. We thus use the Newey-West procedure to correct for serial correlation
when computing the standard errors of the estimates. The test statistic is distributed as a χ2 .
17
panel illustrates the results of the 1000 tests of H0 , showing a histogram of the test
statistic along with its median and the 5 per cent and 1 per cent critical values
associated with the test. We also indicate the fraction of the simulations for which
the test statistic rejects H0 at a significance level better than 5 per cent. When
the null hypothesis is true and the test is correctly specified, this fraction should
be close to 5 per cent, the size of the test. On the other hand, one can interpret
results where this fraction is significantly higher than 5 per cent to suggest that the
learning effects reduce the capability of the test to properly identify unbiasedness.
6.3
Results for the benchmark case
Figures 4 and 5 illustrate the cases of one-quarter-ahead and four-quarters-ahead
expectations, respectively, when information is complete. Figures 6 and 7 illustrate the cases of one-quarter-ahead and four-quarters-ahead expectations for the
incomplete-information case, in which the information friction that we emphasized
is activated.
The top panel of Figure 4 is a histogram that depicts the estimates of a0 across
the 1000 replications. It shows that the median estimate of a0 is very close to zero.
Similarly, the middle panel, which depicts the estimates of a1 , shows a median very
close to the hypothesized value of 1. The bottom panel confirms that these deviations
from the respective values of 0 for α0 and 1 for α1 were not often significant: the test
statistic for the hypothesis has a median around 0.70, when the 5 per cent rejection
region starts above 3. In fact, only 5.3 per cent of the simulations lead the test
statistic to reject the null at better than the 5 per cent significance level. It appears
that, in the case of one-quarter-ahead expectations with complete information, the
unbiasedness test performs just as it should.
Figure 5 depicts the case when inflation expectations are measured as fourquarters-ahead expectations, with the information still complete. While the estimates of a0 remain close to zero, the middle panel of the figure shows that the
median estimate of a1 is now around 0.96. The correction for serial correlation,
however, makes rejections harder to achieve, so that the test statistic rejects the
null hypothesis in only about 9 per cent of the cases, not drastically away from the
5 per cent size of the test.
Overall, the complete-information results in Figures 4 and 5 suggest that, in
such an economy, the simple tests of unbiasedness that are often performed in the
empirical literature behave much as they should.
Let us now examine the cases for which the information friction is activated.
Figure 6 shows that, for the one-quarter-ahead expectations, while the estimate of
the constant parameter is again not drastically different from zero, the distribution
of the slope estimates is significantly skewed away from the hypothesized value of 1,
yielding a median value of 0.82. The bottom panel of Figure 6 illustrates that this
skewness is reflected in the number of times H0 is rejected: more than 20 per cent
of the cases feature a rejection of the null hypothesis, even though, by construction,
our solution embodies the “rational expectations” hypothesis.
18
Figure 7 shows that, for the case of four-quarters-ahead expectations with incomplete information, results are similar to those in Figure 6: the slope estimates are
distributed significantly away from the hypothesized value of 1 and imply rejections
of H0 that are about five times more frequent than the normal rate of 5 per cent.
These benchmark results suggest that the joint hypothesis of the model, the
learning mechanism, and the calibration of the problem introduce significant size
distortions in unbiasedness tests of inflation expectations. These distortions arise
because the relatively small samples with which these tests are performed are dominated by a few significant shifts in monetary policy that surprise agents and lead
them, at least initially, to be confused about the true intentions of monetary policy
authorities. Section 6.4 analyzes the extent to which the qualitative nature of the
results expressed in Figures 4 to 7 are sensitive to the calibration of the model.
6.4
Sensitivity analysis
To analyze the sensitivity of the results to modifications in the calibration, we redo
the above analysis for several alternative specifications. Table 2 reports the results.
Column one indicates the kind of departure from the benchmark calibration that
is under study. Columns two and three indicate the frequency with which the unbiasedness hypothesis is rejected when one-quarter-ahead expectations are used, in
the complete-information and incomplete-information cases, respectively. Columns
four and five report the corresponding results when the four-quarters-ahead expectations are utilized. To facilitate the comparison, the results from the benchmark
cases are repeated at the beginning of the table.
Table 2 gives the general impression that the complete-information cases generate
rejections of H0 as often, roughly, as the size of the tests implies. Particularly in
column one, the fraction of rejections seldom departs significantly from the level
(5 per cent) suggested by the size of the test. Although the numbers in column
three do depart more significantly from 5 per cent, the departures are never excessive.
In contrast, the incomplete-information cases feature rejections of H0 that are far
more frequent. Although the precise numbers change from one case to the next, we
observe rejections of H0 two to five times more often when the information friction
is activated. Interestingly, the fraction of rejections does not seem to depend upon
whether one-quarter-ahead or four-quarters-ahead expectations are used.
For specific cases, in the first three departures from the benchmark case, eliminating the response of monetary policy authorities to the output gap or modifying
the extent to which interest rate changes are smoothed-in does not change the results
markedly. However, increasing the aggressiveness of the monetary policy authorities’ response to deviations of inflation from the target (an increase of α from 2.0 to
either 3.0 or 4.0) does modify the results substantially. The unbiasedness hypothesis is then rejected around 35 per cent of the time when the information friction is
active, while the corresponding numbers for the complete-information case increase
only slightly. The frequency of rejections increases because a high value for the coefficient α acts like a multiplier on the monetary policy shift. This is best illustrated
19
by recalling the definition of ǫ∗t in (18): a high value of α implies, for a given shift
(πt − π0T ), a stronger increase in interest rates.27
We also experiment with modifications to the processes governing the evolution
of the two components to monetary policy. We modify the expected duration of a
given shift in zt , first from 0.975 to 0.99, then back to 0.95. Increasing the duration
opens the gap between the complete- and incomplete-information cases somewhat,
compared with the benchmark case. Decreasing the duration closes that gap. Modifications to the persistence of the transitory shocks, the next departures from benchmark that we consider, do not modify the results markedly. Next, we experiment
with changes in the variances of the shocks and shifts. These experiments show
that increasing the variance in the shifts zt , relative to the variance of the shocks
ut , increases the gap between the complete- and incomplete-information cases. Considering our previous assessment that the benchmark value for the variance in the
shocks ut (the parameter σe ) is an upper bound, this result points to significant and
continued differences between the complete- and incomplete-information cases.
The last series of modifications to the benchmark calibration, in Panel C of Table 2, bring interesting observations. First, increasing the number of repetitions for
the benchmark case (to 2500 from 1000) brings only small changes to the results.
This experiment shows that 1000 repetitions is enough to get a good sense of the
population distribution of the test statistics. Second, Panel C shows that modifying
the level of the costs of adjustments in the portfolio of economic agents or including money-demand shocks as specified in (9) and (10) does not change the results
significantly.
On the other hand, adding technology shocks, as specified in (7), attenuates
the difference between the complete- and incomplete-information cases. Such a result actually validates our approach; because the technology shocks are perfectly
observed by both economic agents and monetary policy authorities, one does not
expect that the introduction of those shocks would imply a distorted relationship
between realized and expected inflation. The rejections of the unbiasedness hypothesis that we identify thus truly arise from the limited information about monetary
policy shocks.28
The last experiment shows that using 1000 repetitions of much longer samples
(with 500 periods in each sample) drastically reduces the rejections, particularly
for the incomplete-information case. This last experiment strongly suggests that
27
While it may seem that stronger shifts would make learning easier, the high duration of a given
shift implies that economic agents will not, at first, identify even sharp spikes in interest rates as
arising from shifts in the inflation target. This intuition is also at play when the departure from
benchmark analyzed is an increase in the standard deviation of the shifts themselves.
28
Of course, the belief that macroeconomic volatility in the last 40 years was solely the result of
technology shocks would imply, in the environment we describe, that the empirical rejections of the
unbiasedness hypothesis could not have come from learning about monetary policy. We believe,
however, that ample evidence exists of very significant monetary policy shocks having affected the
macroeconomic outcomes of all major economies in the last 40 years. In an environment with
learning about the parameters of the monetary policy rule, technology shocks could potentially
affect learning about monetary policy to a greater degree.
20
the rejections of the unbiasedness hypothesis may be the result of size distortions
caused by a few significant shifts in monetary policy in the small samples typically
used to study inflation expectations.
7.
Conclusion
Figure 1, which graphs the realized and expected consumer price index inflation
(as measured by the Livingston survey data), suggests that a few significant shifts
in monetary policy during the 1970s and at the beginning of the 1980s surprised
economic agents and, for a while, left them unsure of the true intentions of monetary
policy authorities. Apart from the periods immediately following the shifts, the
expectations of economic agents do not appear to be completely out of line with
realized inflation.
We have shown that these empirical features can be represented by modelling the
shifts in a standard monetary DSGE model, and by assuming that agents must learn
about the shifts over time. In the case of complete information about the shifts, the
unbiasedness hypothesis is rejected with very low frequency, in keeping with the size
of the tests. In contrast, when the information friction is active, the unbiasedness
hypothesis is rejected much more often—between two and five times—than the size
of the tests would imply, even though our model embeds the rational expectations
solution concept by construction. Furthermore, the likelihood of rejection tends to
be eliminated when the sample size increases, even though the information friction
remains.
We acknowledge that the Kalman filter may not be optimal in small samples,
for which the (asymptotic) normal behaviour of the two unobserved components of
monetary policy has not established itself. It would be interesting to verify the extent
to which economic agents could improve on the Kalman filter estimates by using
non-linear filters. Furthermore, we must caution that we identify a single inflation
expectation as the average of survey participants’ responses. The dispersion of the
survey’s participants around that average is neither analyzed nor modelled.
Overall, our results support the view that learning effects with regard to monetary policy, in addition to creating persistence in the responses of most macroaggregates following monetary policy shocks, imply dynamics in the expectations of
agents that replicate well some of the empirical evidence about measured inflation
expectations.
This view, and thus the importance of incomplete information for modelling
macroeconomic activity, could be reinforced by verifying that the incomplete information and learning framework replicates other facets of the relationship between
realized and (measured) expected inflation. Notably, our framework could be used
to determine whether the learning effects replicate the additional results against
unbiasedness discussed in section 2, or the evidence that inflation expectations are
not efficient predictors of realized inflation. Alternatively, the framework could be
used to determine whether the learning effects could lead simulated data to match
21
the dynamic correlation patterns linking realized and expected inflation.
The evidence of shifts in the parameters of the interest rate rule (see Clarida,
Galı́, and Gertler 2000) opens interesting avenues for future research. Straightforward modifications to our framework could be used to determine whether such shifts,
and least-square learning about the shifts on the part of economic agents, are responsible for the evidence against unbiasedness and efficiency in measured inflation
expectations.
22
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25
Table 1. Parameter Calibration
Parameter
Symbol
Benchmark Value
Range Examined
Preferences and Technology
Elasticity of labour supply
Scaling of labour supply
Portfolio adjustment costs
Discount factor
Capital share in production
Capital depreciation rate
ψ1
ψ0
τ
η
θ
δ
0.4
2.15
15
0.99
0.36
0.025
-
Interest-Rate-Targeting Rule
Response to inflation
Response to output gap
Smoothing of interest rates
α
β
ρ
2.0
0.25
0.25
[2.0, 4.0]
[0, 0.25]
[0, 0.5]
Shifts and Shocks to Monetary Policy
Duration of shifts
Standard deviation of shifts
Persistence in shocks
Standard deviation of shocks
φ2
σg
φ1
σe
0.975
0.005
0.1
0.005
[0.95, 0.99]
[0.0025 0.01]
[0, 0.2]
[0.0025, 0.005]
26
Table 2. Sensitivity Analysis of the Results: Frequency of Rejections
Specification Examined
Benchmark case
One Quarter Ahead
Complete Incomplete
5.3%
20.5%
Four Quarters Ahead
Complete Incomplete
8.9%
26.3%
Panel A: Modifications to the Monetary Policy Rule
No response to output gapa
No smoothing of interest ratesb
Increased smoothing of interest ratesc
More aggressive response to inflationd
Most aggressive response to inflatione
5.5%
5.5%
5.1%
5.7%
7.2%
19.9%
22.2%
18.0%
34.7%
37.7%
8.5%
9.4%
7.8%
10.6%
12.7%
26.2%
29.5%
24.3%
34.7%
34.6%
Panel B: Modifications to the Calibration of Monetary Shocks and Shifts
Very high duration of regime shiftsf
Lower duration of regime shiftsg
No persistence in transitory shocksh
Higher persistence of transitory shocksi
Higher variance of regime shiftsj
Lower variance of regime shiftsk
Lower variance of transitory shocksl
5.2%
5.6%
4.7%
5.8%
5.6%
4.8%
5.6%
22.4%
15.4%
19.8%
19.4%
37.9%
8.1%
33.9%
6.7%
10.9%
8.7%
8.4%
15.1%
7.3%
15.2%
35.3%
20.6%
27.8%
25.5%
34.6%
19.4%
34.8%
5.8%
5.3%
5.4%
5.4%
5.0%
21.8%
19.6%
20.1%
6.9%
10.8%
9.6%
9.0%
8.8%
9.2%
2.7%
28.6%
26.5%
26.1%
10.1%
4.9%
Panel C: Other Modifications
Increased number of repetitionsm
Higher portfolio adjustment costsn
Inclusion of money-demand shocks
Inclusion of technology shocks
Increased length of simulated time serieso
a
β = 0.0
ρ = 0.0
c
ρ = 0.5
d
α = 3.0
e
α = 4.0
f
φ2 = 0.99
g
φ2 = 0.95
h
φ1 = 0.0
i
φ1 = 0.2
j
σg = 0.01
k
σg = 0.0025
l
σe = 0.0025
m
2500 repetitions using an 80-period sample and benchmark calibration
n
τ = 15.0
o
1000 repetitions using a 500-period sample
b
27
Figure 1: Realized Inflation versus Measured Expectation (Livingston
Survey)
15.0
Actual Inf.
Expected Inf.
12.5
Inflation rate
10.0
7.5
5.0
2.5
0.0
1960
1966
1972
1978
1984
1990
1996
28
Figure 2: The Learning Mechanism: The Benchmark Calibration
zt Shift: Actual Value and Best Estimate
−3
1
x 10
0
Estimate
Actual Value
−1
−2
−3
−4
−5
−6
0
5
15
20
25
20
25
20
25
ut Shock: Actual Value and Best Estimate
−3
2.5
10
x 10
2
1.5
1
0.5
0
−0.5
0
5
10
15
Inflation Target: Actual Value and Best Estimate
5.5
5
4.5
4
3.5
3
2.5
0
5
10
15
Time (Shift occurs at t = 5)
29
Figure 3: Complete- and Incomplete-Information Responses to a Shift
in the Inflation Target
Panel A: Comparison of Complete− and Incomplete−Information Responses
−3
5
x 10
5
4
0
−5
10
Complete Information
3
Incomplete Information
0
5
10
15
20
−3 Consumption, dev. fr. ss.
x 10
2
25
0
5
10
15
20
25
Net, annualized, inflation in %
5
4.5
5
4
0
−5
3.5
0
−3
10
x 10
5
10
15
20
Work effort, dev. fr. ss.
3
25
10
5
9
0
8
−5
0
0
5
10
15
20
25
Net, annualized, money growth in %
5
10
15
20
Output, dev. fr. ss.
7
25
0.015
0
5
10
15
20
25
Net ann. nominal interest rate, in %
6
5.5
0.01
5
0.005
0
4.5
0
5
10
15
20
Investment, dev. fr. ss.
4
25
0
5
10
15
20
Ex−ante real rate (ann. in %)
25
Panel B: Comparison of Realized and Expected Inflation
5
4
5
Realized
Expected
4.5
4
3
2
0
5
10
15
20
25
Realized vs. Expected: Complete information
3.5
3
0
5
10
15
20
25
Realized vs. Expected: Incomplete information
30
Figure 4: Results of Monte Carlo Simulations: One-Quarter-Ahead
Expectations, Complete Information
Estimate of a
0
400
↓ median (= −1.6005e−05)
Frequency
300
200
100
0
−0.025
−0.02
−0.015
−0.01
−0.005
0
0.005
0.01
2.5
3
3.5
Estimate of a1
400
↓ median (= 1.0009)
Frequency
300
200
100
0
0
0.5
1
1.5
2
F−statistic testing H0: a0 = 0 and a1 = 1
300
Frequency
250
→5.3% of the distribution
↓ median (= 0.69202)
200
150
100
50
0
0
1
2
3
↑ 5%
4
5
6
↑ 1% significance
7
8
9
31
Figure 5: Results of Monte Carlo Simulations: Four-Quarters-Ahead
Expectations, Complete Information
Estimate of a
0
300
Frequency
250
↓ median (= 0.0014978)
200
150
100
50
0
−0.06
−0.04
−0.02
0
0.02
0.04
0.06
0.08
Estimate of a1
300
↓ median (= 0.96086)
Frequency
250
200
150
100
50
0
0
0.5
1
1.5
2
2.5
2
χ −statistic testing H : a = 0 and a = 1 (HAC−robust)
0
0
1
1000
↓ median (= 1.4247)
Frequency
800
600
→8.9% of the distribution
400
200
0
0
10
20
30
↑ 1% significance level
↑ 5%
40
50
60
70
80
90
100
32
Figure 6: Results of Monte Carlo Simulations: One-Quarter-Ahead
Expectations, Incomplete Information
Estimate of a0
↓ median (= 0.0020535)
250
Frequency
200
150
100
50
0
−0.005
0
0.005
0.01
0.015
0.02
0.025
0.03
Estimate of a1
200
↓ median (= 0.82256)
Frequency
150
100
50
0
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
F−statistic testing H0: a0 = 0 and a1 = 1
250
→20.5% of the distribution
Frequency
200
↓ median (= 1.6124)
150
100
50
0
0
2
4
↑ 5%
6
8
↑ 1% significance
10
12
14
16
33
Figure 7: Results of Monte Carlo Simulations: Four-Quarters-Ahead
Expectations, Incomplete Information
Estimate of a0
160
↓ median (= 0.010079)
140
Frequency
120
100
80
60
40
20
0
−0.02
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Estimate of a
1
120
↓ median (= 0.78364)
Frequency
100
80
60
40
20
0
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
χ2−statistic testing H : a = 0 and a = 1 (HAC−robust)
0
0
1
350
300
↓ median (= 3.2466)
Frequency
250
200
→26.3% of the distribution
150
100
50
0
0
5
10
15
20
↑ 5%
↑ 1% significance level
25
30
35
40
45
50
34
Appendix A: Kalman Filter
Recall equations (19), (20), describing the evolution of the observed monetary policy
deviations from the benchmark rule:
zt+1
Nt+1
zt
φ2 0
;
(A.1)
+
·
=
0 φ1
et+1
ut
ut+1
zt
∗
(1 − ρ)(1 − α) 1 ·
;
(A.2)
ǫt =
ut
with Nt+1 defined as follows:
(1 − φ2 ) zt with probability φ2 ;
Nt+1 =
gt+1 − φ2 zt with probability 1 − φ2 ;
(A.3)
and where, again, it was assumed that et+1 ∼ N(0, σe2 ) and gt+1 ∼ N(0, σg2 ).
Compare this system with the one described in Hamilton’s (1994, chapter 13)
discussion of state-space models and the Kalman filter:
yt =
ξt+1 =
E(vt vt′ ) =
E(wt wt′ ) =
A′ · xt + H′ · ξt + wt ;
F · ξt + vt+1 ;
Q;
R.
(A.4)
The equivalence between the two systems is established by defining yt = ǫ∗t ,
xt = 0, ξt = [zt ut ]′ , wt = 0, vt = [Nt et ]′ , as well as the following matrices:
2
φ2 0
(1 − ρ)(1 − α)
σN 0
;F=
; R = 0.
;Q=
A = 0; H =
0 σe2
0 φ1
1
2
Note that one can show that σN
= (1 − φ2 )(1 + φ2 )σg2 .
Denote the mean squared error of the one-step-ahead forecasts of the unobserved
states, conditional on time-t information, as Pt+1 |t .29 Conditional on starting values
ξˆ1 |0 and P1 |0 30 , the following recursive structure that describes the evolution of ξˆt+1 |t
and Pt+1 |t emerges:
Kt = FPt |t−1 H(H′Pt |t−1 H)−1 ;
ξbt+1 |t = Fξˆt |t−1 + Kt (yt − H′ ξ̂ t |t−1 );
′
′
Pt+1 |t = (F − Kt H )Pt |t−1 (F −
′
HKt )
(A.5)
+ Q.
(A.6)
(A.7)
The intuition behind this updating sequence is that, at each step, agents will
use their observed forecasting errors (yt − H′ · ξbt |t−1 ) and their knowledge of the
parametric form of the system to update their best estimates of the unobserved
29
30
So that Pt+1 |t = Et [(ξt+1 − ξbt+1 |t )(ξt+1 − ξbt+1 |t )′ ].
We use the unconditional expectations.
35
states, ξt . The mechanics of this updating take the form of linear projection and are
detailed in Hamilton.
Under the assumption of normality of both processes (for zt and ut ), the sequence (ξbt+1 |t )|Tt=1 represents the optimal one-step-ahead forecasts of the unobserved
states.31
31
Even without the assumption of normality, the sequence of filtered estimates remains the best
linear forecasts of the unobserved states conditional on time-t information.
Bank of Canada Working Papers
Documents de travail de la Banque du Canada
Working papers are generally published in the language of the author, with an abstract in both official
languages. Les documents de travail sont publiés généralement dans la langue utilisée par les auteurs; ils sont
cependant précédés d’un résumé bilingue.
2002
2002-29
Exponentials, Polynomials, and Fourier Series:
More Yield Curve Modelling at the Bank of Canada
2002-28
Filtering for Current Analysis
2002-27
Habit Formation and the Persistence
of Monetary Shocks
2002-26
2002-25
2002-24
D.J. Bolder and S. Gusba
S. van Norden
H. Bouakez, E. Cardia, and F.J. Ruge-Murcia
Nominal Rigidity, Desired Markup Variations, and
Real Exchange Rate Persistence
Nominal Rigidities and Monetary Policy in Canada
Since 1981
A. Dib
Financial Structure and Economic Growth: A NonTechnical Survey
V. Dolar and C. Meh
2002-23
How to Improve Inflation Targeting at the Bank of Canada
2002-22
The Usefulness of Consumer Confidence Indexes in the
United States
2002-21
N. Rowe
B. Desroches and M-A. Gosselin
Entrepreneurial Risk, Credit Constraints, and the Corporate
Income Tax: A Quantitative Exploration
2002-20
Evaluating the Quarterly Projection Model: A Preliminary
Investigation
2002-19
Estimates of the Sticky-Information Phillips Curve
for the United States, Canada, and the United Kingdom
2002-18
H. Bouakez
Estimated DGE Models and Forecasting Accuracy:
A Preliminary Investigation with Canadian Data
2002-17
Does Exchange Rate Policy Matter for Growth?
2002-16
A Market Microstructure Analysis of Foreign Exchange
Intervention in Canada
2002-15
Corporate Bond Spreads and the Business Cycle
2002-14
Entrepreneurship, Inequality, and Taxation
C.A. Meh
R. Amano, K. McPhail, H. Pioro,
and A. Rennison
H. Khan and Z. Zhu
K. Moran and V. Dolar
J. Bailliu, R. Lafrance, and J.-F. Perrault
C. D’Souza
Z. Zhang
C.A. Meh
Copies and a complete list of working papers are available from:
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