IMES DISCUSSION PAPER SERIESJapanese and U.S. Inflation Dynamics in the 21st Century

Jeff Fuhrer
Discussion Paper No. 2017-E-5

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
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IMES Discussion Paper Series 2017-E-5
June 2017

Japanese and U.S. Inflation Dynamics in the 21st Century
Jeff Fuhrer*
Abstract
This paper examines the behavior of inflation in the U.S. and Japan over the past
twenty-five years. The paper estimates structural models of inflation dynamics forboth countries, relying on survey measures as proxies for expectations. The results
suggest some promising directions for inflation modeling in both countries. First,
the use of survey expectations as proxies for the expectations in conventionalmodels appears to be helpful, aiding in identification of the inflation process inboth countries. Second, methods for endogenizing such expectations are tractable
and replicable. They require the measurement of longer-term expectations, butsuch data are available for many key variables in many developed economies. 1
Third, models that incorporate such expectations identify a rationale for the
behavior of US and Japanese inflation:
a. Long-run expectations anchor the process, although long-run expectations
can be influenced over time by persistent deviations of inflation (or output) fromtheir long-run equilibria;
b. Short-run expectations are tied to their long-run counterparts, but they can
deviate quite persistently from long-run expectations, due to persistent deviationsof output from potential, and due to intrinsic persistence in the expectations;c. Inflation appears well-explained by short-run expectations and a traditional
output gap;
Fourth, the balance sheet actions in Japan appear to have boosted short-run
inflation expectations, compared to where they would have been without suchactions. The estimates in this paper suggest that this in turn has helped to raise
realized inflation by about one-half percentage point.
Keywords: Inflation dynamics; Intrinsic Persistence; Survey expectations
JEL classification: E31, E32, E52, D84*Federal Reserve Bank of Boston (E-mail: jeff.fuhrer@bos.frb.org)
The views expressed in this paper are those of the author and do not necessarily reflect theofficial views of the Bank of Japan, the Federal Reserve Bank of Boston or the Federal Reserve
System.
1

A separate but important issue is whose expectations are measured in the available surveys.
To be sure, a survey of important price-setters, rather than professional forecasters, would
likely be preferred. Such a dataset is not available for the U.S. or Japan at present, to myknowledge.

1. Introduction
After decades of inflation-fighting around the world, many central banks recently have enjoyedrelatively placid inflation behavior. To be sure, the world has faced its share of economic upheaval.
But the imprint of these upheavals on inflation has been remarkably small. Figure 1 shows the US
experience, highlighting pre-2000 and post-1999 experience, with recession shading indicated by thegray bars.
12.0

Figure 1
Core PCE inflation, 4-qtr.

Post-1999 period

10.0
8.0
6.0
4.0
2.0

1960:Q1
1962:Q1
1964:Q1
1966:Q1
1968:Q1
1970:Q1
1972:Q1
1974:Q1
1976:Q1
1978:Q1
1980:Q1
1982:Q1
1984:Q1
1986:Q1
1988:Q1
1990:Q1
1992:Q1
1994:Q1
1996:Q1
1998:Q1
2000:Q1
2002:Q1
2004:Q1
2006:Q1
2008:Q1
2010:Q1
2012:Q1
2014:Q1
2016:Q1

0.0

Remarkably for the U.S., the decline in inflation during the Great Recession is nearly a nonevent—consider the trajectory of inflation from 2009 to 2016 in figure 1. Inflation dropped about
one percentage point in the wake of the recession, but very quickly rebounded toward two percent,and has fluctuated between 1.4 and 2% since 2012. Many forecasters expected a more pronounced
decline in inflation, abstracting from the temporary effects of the large decline in oil prices, as figure
2 shows.

2

2.0

Figure 2: Core CPI actual and forecasts

1.8
1.6
1.4
1.2
1.0
0.8

Core CPI inflation, 4-qtr.

0.6

GSU, Feb. 2009

0.4

Global Insight, Jan. 2009

0.2

SPF, Feb. 2009

0.0
2009:Q1

2009:Q2

2009:Q3

2009:Q4

2010:Q1

2010:Q2

For those who still cling to the old-style accelerationist Phillips curve, the prediction for inflation
would have been even more pessimistic, as figure 3 shows.

4

Figure 3: Accelerationist Phillips curve inflation predictions
Flatter PC
Estimated PC
Toy PC

2
0
-2
-4
-6
-8

2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018

Such a forecast may seem extreme, but for a recession that featured approximately twenty pointyears of cumulative unemployment gaps (see figure 4), the relative stability of inflation is quite
remarkable. Even a modest Phillips curve slope of -0.1 would imply a two percentage point drop in3

inflation over this period (the red dashed line in the figure above). Instead, the decline was barely apercentage point, and quite short-lived at that. 2
Figure 4: Cumulative point-years of unemployment gaps, from beginning of recession
25

Great Recession

2007

20

Index, series value minus peak

15

1981
10
1973

Average, 1949-2007

5

2001

1990
1969
0

1957
1953

-5

1960
-10
0

5

10

15

20

25

30

35

Number of quarters after business cycle peak
Heavy black line represents average of recessions
Heavy red line represents current experience

In Japan, the story has been somewhat different but in many ways equally puzzling. As figure
5 indicates, despite significant and persistent negative output gaps, inflation fluctuated in a relatively
narrow range, averaging about -1% through much of the past twenty years, then rising to positive
territory in 2013-4, presumably related to the Bank of Japan’s announcements about an evenstronger commitment to bringing about positive inflation. The surge to two percent appears to havebeen temporary, and more recently inflation has been hovering around zero.
Note that this pattern of inflation and output gaps does not conform well to either anaccelerationist or a New-Keynesian model of inflation. In the former case, the long sequence ofnegative output gaps should have resulted in a series of declines in inflation over this period, whichevidently was not the case. In the latter case, absent other forces, inflation should have mirrored theoutput gap, moving toward its trend as the output gap closed. This dynamic is not obvious in the
figure either, although there is some tendency for inflation to turn positive in some cases with a lag
after the output gap nears or exceeds zero.

2 In comments delivered at the Federal Reserve Bank of Boston’s annual economic conference, I discussed thepredictions of NKPC-style models, which rely on the expected future path of real marginal costs. These models farealmost as poorly.

4

Figure 5

4
2
0
-2
-4
-6

Japanese core CPI inflation (4-qtr.)
Japanese output gap, Cabinet office
Japanese output gap (BOJ)-8

2016:Q4

2016:Q1

2015:Q2

2014:Q3

2013:Q4

2013:Q1

2012:Q2

2011:Q3

2010:Q4

2010:Q1

2009:Q2

2008:Q3

2007:Q4

2007:Q1

2006:Q2

2005:Q3

2004:Q4

2004:Q1

2003:Q2

2002:Q3

2001:Q4

2001:Q1

2000:Q2

1999:Q3

1998:Q4

1998:Q1-10

In the wake of this now extended period of stable inflation, despite historic fluctuations inreal activity for both Japan and the US, many have turned to a number of explanations. One
possibility is that the slope of the Phillips curve has diminished greatly in many countries, which at ahigh level comports well with the relative inactivity of inflation in the face of large output andemployment fluctuations. However, even small Phillips curve slopes would have produced larger
responses to the Great Recession in the US, as noted above. Others have proposed nonlinearities in
the Phillips curve—both regions of “inactivity” around a zero output gap, and asymmetric gap
effects as inflation nears zero.
Most have settled on an “anchored expectations” model of inflation, which will be discussed
in more detail below. 3 The next figure presents data for Japanese core inflation along with
Consensus Forecasts’s one- and 6-to-10-year forecasts for CPI inflation, in the solid blue and dashedblue lines respectively. The sample is restricted by the availability of the Consensus forecasts, which
begin semi-annually in 1990.

3

One exception to this “explanation” for inflation dynamics is the work of Gilchrist et al (2017), whichemphasizes the role that financial constraints play in determining the price response of firms to recessions.

5Figure 6

Japanese core inflation and inflation expectations
4Core inflation
1-year expectation
6-10-year expectation

3

2

1

0

-1

-2
1990

1995

2000

2005

2010

2015

2020

Year

This figure reveals several important facts about inflation and inflation expectations in Japan: (1)long-run expectations have remained quite stable in the face of prolonged economic weakness; (2)short-run expectations have often deviated quite noticeably, mostly below long-run expectations; (3)actual inflation tends to track the short-run expectations reasonably well; and (4) one might infer atendency for short-run expectations to revert towards long-run expectations, although the rate ofany such reversion is not rapid. A simple error-correction model linking short- and long-run
inflation expectations ( π teS , π teL ) for this data develops the following error-correction coefficients:
Table 1
Inflation expectations error-correction coefficients
1991-2016 (semiannual data)ekek
π t − π t −=α (π teS−1 − π teL−1 ) + b∆π teS−1 + c∆π teL−1 + d , k = [s,L]
1
Error-correction coefficient ( α )p-value
Short-run expectations
-0.46
0.005
Long-run expectations
0.036
0.772
Qualitatively similar observations hold for the US, and while the specifics of Japanese inflation
history clearly deviate from those of the US, some of the features noted above are common to boththe US and Japan. The sections that follow will draw on some of these empirical observations toform a model of inflation dynamics that links actual inflation to short- and long-run expectations, aswell as the output gap. 4
4 The results for the US data over the same sample period, using SPF inflation expectations on quarterly data, are as

follows. Note that because the Japanese data are observed semi-annually, these error-correction coefficients imply anerror-correction rate that is more than twice as fast for the U.S.
Error-correction coefficient ( α )p-value
Short-run expectations
-0.53
0.000
Long-run expectations-0.021
0.49

6

2. Anchored expectations models of inflation: Theoretical considerations.
If one assumes long-run expectations anchoring, then some care must be taken in how one
specifies long-run expectations. In the specification for long-run expectations displayed below,
inflation will be anchored regardless of what actions monetary policy takes. This seems unlikely to
be true. The model comprises an anchored-expectations Phillips curve commonly used in theFederal Reserve System, a trivial IS curve, and a policy rule that targets only the output gap.
l1π t −1 + (1 − l1 )π tLR + β ( yt − yt* )
π=
t

π tLR
= ρπ tLR−1 + (1 − ρ )π *yt = Eyt +1 − σ ( ft − Eπ t +1 − 2)

(2.1)

ft =2 + π * + a y ( yt − yt* )Long-run expectations are exogenous in the model—subject to shocks, but always returning to the
inflation goal, apart from any action by the central bank. In most standard models with rational
expectations, this is not going to work. Unless the policy rule obeys the Taylor principle (orsomething like it, depending on variations in the specification), the model will be unstable—inflation
will not settle to any particular rate from arbitrary initial conditions. 5
But this model works just fine with no central bank response to inflation. 6 It will always converge to
π t = π * , as long as the central bank closes the output gap.

5 In fact, the stability properties of the model are more complicated than this. For a given value of l , the range of value
1of ay that stabilizes the model ranges from 0.1 to infinity, given no emphasis on inflation. The greater the weight on thelong-run expectation, the lower value of ay that is required to stabilize the model. Inflation target can also stabilize themodel, but that is not the novel feature of the specification.
6 It behaves fine in the sense that standard solution techniques will produce the right “root count,” implying a modelthat is unique and stable, converging to the inflation goal from arbitrary initial conditions.

7

Figure 7
Stability region for l1 and ay in the reference model(No response to inflation gap)
3

2.5

a

y

2

1.5

1

0.5

0
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

l
1

Figure 8 shows the dynamic response of the model to a set of random shocks that push it away fromthe steady state when the inflation response is 0, l1=0.7, and ay=0.5.
Figure 8=0Simulation of anchored expectations model with a
Arbitrary initial conditions
9

8

ff
ygap
e

7

6

5

4

3

2

1

0
1985

1990

1995

2000

2005

Year

This stands standard monetary theory on its head: The central bank provides the real anchor, andthat automatically provides a nominal anchor.

8

This outcome contrasts sharply with the more conventional incorporation of long-run
expectations, in which they represent the trend for inflation, often represented in the following way:
LR
*
(2.2)
π t − π tLR= l1 ( Eπ t +1 − π tLR+1 ) + (1 − l1 )(π t −1 − π t −1 ) + β ( yt − yt )Thus inflation is modeled relative to its long-run trend, here represented by the long-run expectation
as is conventional. The stability properties in this case are quite different but well-known: The model
converges only if the central bank responds sufficiently to the inflation gap, as illustrated figure 9
below.

3

Figure 9
Stability region for aπ and ay in the trend inflation model

2.5

a

y

2

1.5

1

0.5

0
0.5

1

1.5

2

2.5

3

a

This exercise points to what’s missing in the anchored-expectations model: Short-term
expectations that explicitly link the evolution of inflation to expected monetary policy actions.
Alternatively, one can posit a mechanism whereby central bank actions (and expectations about
future actions) feed directly into π tLR . Depending on how this is done, that mechanism can also
solve the problem of upside-down monetary theory.

9One simple remedy is to make the long-run expectations explicitly dependent on thesequence of short-run expectations that they (presumably) embody. A version of this model thatalters equation (2.1) in this way is as follows:π=
l1π t −1 + (1 − l1 )π tLR + β ( yt − yt* )tLRπ=ω Eπ tLR+1 + (1 − ω )π t
t

(2.3)

This model displays exactly the same stability properties as the canonical trend-inflation model
above—insufficient responses to the inflation gap do not pin down the inflation rate, but following
the Taylor Principle works as expected. The model differs from standard theory in placing primaryemphasis on long-run expectations for short-run inflation dynamics. But because long-run
expectations will only converge to the target if the central bank is expected to move inflation toward
its target, it returns to the domain of sensible monetary policy predictions.
Another version of this model, in which the long-run expectations are modeled as a long moving
average of lagged (realized) inflation, also achieves conventional stability properties.
(2.4)
Π tLR =aΠ tLR−1 + (1 − a )π t −1 + c
The reason is that the dependence of inflation on long-run expectations implies dependence of
inflation on lagged inflation. Unless the central bank over time guides inflation back towards a
specific goal, long-run inflation expectations (and thus realized inflation) will not be anchored, andinflation will not be determinate.
It’s unlikely that central bank staff anywhere literally believe the simple anchored expectation
model of equation (2.1), and I know that Board staff have in place some mechanisms by which thelong-run expectation can become temporarily unanchored from the target, but (presumably, withappropriate policy action) will ultimately return to the target. What we know about such mechanisms
as an empirical matter is precious little. That makes reliance on the anchoring power risky, to myeye.
While this section proposed some “fixes” to ensure that the anchored-expectation model
behaves sensibly, that model is still divorced from theory. There is no standard model that suggeststhat inflation should depend only on long-run expectations. Most all incarnations embody a primaryrole for short-run expectations. Equation (2.2) is a simple version of such a model.
3. Some empirical estimates of the influence of short- and long-run inflation
expectations in the U.S.
Much of the rationale for a focus on anchored expectations models lies in their empirical appeal,
coupled with a loose connection to central banks’ desire that expectations not become un-anchored,as they likely were in the 1970s. Starting with Williams’ empirical note (2006), authors documented
that inflation since the late-1990s might be well-modeled by an empirical model in which inflation
was stationary around a (barely) time-varying long-run trend or (if not varying) constant, perhaps
with adjustments for the influence of the output gap and a few transitory relative prices, usuallyenergy and import prices. An extremely simple version of this model, employed by Williams is(3.1)
πt= c + εt .
10Williams shows that this model beats both a random walk model of inflation and a simple Phillips
curve specification. Perhaps that’s because expectations are anchored, and to a first approximation,
inflation fluctuates randomly around the central bank’s goal?
Figure 10 shows the in-sample fit and RMSE for a few simple models. The first is just the
sample mean of core PCE from 1998-2016. The second adds the long-run expectation (PTR), alongwith an intercept. The third and fourth add a lag of inflation and the change in the unemployment
rate. Once one knows the ex post average value of inflation, the long-run expectation adds virtuallynothing to explaining inflation. Put differently, PTR is essentially an intercept in these simple
forecasting equations.
Figure 10
Fit of various inflation models, 1998-2016
2

Model 1: ptr and constant, R
Model 2: ptr,
Model 3: ptr,

t-1

t-1

, constant, R

and Ugap

t-i= 0.00118
2= 0.0872, constant, R2= 0.175

3Core PCE
4-qtr. PCE
Mean, 98-16

2.5

Model 1Model 2Model 3

2

1.5

1
RMSEs
Mean : 0.52
Model 1: 0.52
Model 2: 0.49
Model 3: 0.46

0.5

0
1998

2000

2002

2004

2006

2008

2010

2012

2014

2016

2018

Year

The addition of lags and a gap measure improve the fit and RMSE a bit, but not
dramatically. A key point here is that apart from knowing the intercept term, it has been difficult to
explain fluctuations in inflation over the past twenty years. That is somewhat less true if one
smoothes the data (as in the four-quarter % change, the solid black line above), but it is still true.
The change in the unemployment rate picks up some of the fluctuations associated with therecession and its immediate aftermath. Lagged inflation picks up some of the very modest
autocorrelation in inflation over the period. But these are small helps.

11What about short-run expectations, for which there may be stronger theoretical rationale?
There have been many periods during which the inclusion of short-run survey expectations has been
essential to inflation forecasting equations (and Phillips curves). It is not the case that such short-run
expectations measures enter significantly in all periods, nor that the coefficients on short-run
expectations are stable across all sub-periods. But the same is true of long-run expectations. The
sign, magnitude and significance of long-run expectations all vary over the past 35 years in a varietyof empirical inflation models. Representative empirical results bearing on these points are discussed
in the next subsection.

Estimates of Phillips curves with short- and long-run expectations over recent samples:
The model employed in this section allows for the influence of• Lagged inflation;• Short-run (survey-based) expectations of inflation;• Long-run (survey-based) expectations of inflation. Here we use the variable “PTR” from the
FRB/US model;• An unemployment gap;
• Inflation terms that sum to one or not;• The inclusion of an intercept.
S=
π t a1π t −1 + a2π tS+1 + a3π LR+ b(U t − U t* ) + c + ε t(3.2)

The intercept is not theoretically required, but it serves as a robustness check on the role that longSS
run expectations π LR
play in the inflation model. That is, does inflation track fluctuations in π LR
, orS
or an intercept simply serve as the average inflation rate around which inflation
do either π LR
fluctuates in an atheoretic statistical representation of recent inflation, such as

π t =+c b( yt − yt* ) + ε t or just π t = c + ε t ?
Short-run expectations π tS+1 are also taken from the SPF; this memo uses the four-quarter
expectation in the core CPI, although the results will be similar for the one-quarter or other shortrun expectations measures. The unemployment gap is computed using the civilian unemployment
rate and the CBO’s estimate of the natural rate of unemployment.
The unit sum restriction brings the specification closer to a “structural” relationship; without
imposing this restriction, the estimate is more of a practical forecasting equation. For most models,an intercept would be an unnecessary addition, once one has a long-run settling point built into theequation. But as suggested above, in recent years it’s a open question whether inflation is bestmodeled as fluctuating around the long-run expectation, or fluctuating around a constant. Of course,
it’s a close call because the long-run expectation has been relatively constant in recent years. But one
would hope that the long-run expectation would dominate the intercept, if variation in the long-run
expectation means anything.
12The tables below summarize a few different samples and specifications for simple Phillips curves
that are versions of equation (3.2). The key insight from these estimates is that one can obtain quite
an array of results, depending on the sample one looks at, and also on the specification one employs.
Over the full sample (table 2), short-run survey-based expectations play a quantitatively importantrole in explaining inflation. The long-run expectations are considerably less reliable. Identification of
the unemployment gap is poor.
In samples that focus on the data since the turn of the century (table 3), identification ismuch weaker. In models that include both short- and long-run expectations, neither consistently
enters significantly, and the unemployment gap terms are rarely significant. In many cases, theinclusion of an intercept drives out the long-run expectation, which suggests that it is not variationin that variable, but its relative constancy, that serves as an intercept-anchor in this simple inflation
specification.
Table 2Estimated “Phillips curves”, 1982:Q1-2016:Q3
Various specifications, with restrictions (unit sum) or not, intercept or not, short- and long-run
expectations
Inflation variable = core PCE
Variable
(1)
(2)
(3)
(4)
(5)
(6)
Lagged π
0.04060.437***
0.04380.172*
PTR
0.182
0.331*
0.189
0.306*
0.189
0.194
0.811***
0.799***
0.777***
0.232
0.757***
0.390**
Short-run π e
Ugap
0.0446-0.123*0.0426-0.01910.04730.0461
Intercept
Yes
No
Yes
No
Yes
Yes
Unit sum
Yes
Yes
No
No
Yes
No
restriction?
Newey-West corrected standard errors* = .05 significance or better** = .01 significance or better*** = .001 significance or better
Table 3
Estimated “Phillips curves”, 2000-2016:Q3
Various specifications, with restrictions (unit sum) or not, intercept or not, short- and long-run
expectations
Inflation variable = core PCE
Variable
1
2
3
4
5
6
7
8
0.139
0.237
0.141
0.139
0.01330.247
0.02030.00951
Lagged π
0.861*** 0.762*** 0.768*** 1.33
0.307
0.929*** 0.152
0.853
PTR
e
0.68
-0.176
0.653
0.698
Short-run π-0.1*-.136*** -0.0997* -0.107*-0.0217-.147*** -0.0231-0.0272
Ugap-0.193*
None
None
-1.16
-0.375** None
None
-1.55
Intercept
Unit sum
Yes
Yes
No
No
Yes
Yes
No
No
restriction?
Newey-West corrected standard errors
13* = .05 significance or better** = .01 significance or better*** = .001 significance or better
While the PCE is the official inflation variable of the Fed, it is still of some interest to see
how robust the results above are to the moderately different core CPI measure. The table below
shows the full-sample estimates of Phillips curves using the core CPI measure. The key differences
are that (a) PTR is never significant, with an estimated coefficient quite close to zero; (b) the shortrun expectation is even more uniformly large and significant. The recent sample results (not show)
differ relatively little from those for the core PCE.
Table 4
Estimated “Phillips curves”, 1982:Q1-2016:Q3
Various specifications, with restrictions (unit sum) or not, intercept or not, short- and long-run
expectations
Inflation variable = core CPI
Variable
(1)
(2)
(3)
(4)
(5)
(6)
Lagged π
0.208*
0.208*
0.131
0.191*
PTR
0.07570.0693-0.0256
0.097
0.0683-0.0572e
0.717***
0.722***
1.03***
0.737***
0.932***
1.23***
Short-run π
Ugap-0.0856-0.0812-0.0678-0.109-0.101-0.0709
Intercept
Yes
No
Yes
No
Yes
Yes
Unit sum
Yes
Yes
No
No
Yes
No
restriction?
Newey-West corrected standard errors* = .05 significance or better** = .01 significance or better*** = .001 significance or better
Figures 11-13 provide rolling-sample estimates of variants of equation (3.2), with only longrun expectations, only short-run expectations, and a horse race (figure 13) that pits short-run against
long-run expectations. 7 The size and significance of all of these variables’ coefficients have varied
over the sample, for all of these specifications. In figure 13, one can see that short- and long-run
expectations have both hovered around 0.5, remaining reasonably significant. The significance of theunemployment gap (the green line) is much more consistent in this specification.

7 Note that the lagged inflation term is excluded here, as it rarely played a prominent role.

14

Figure 11

Board inflation model, rolling regression estimates, window = 40
Estimated coefficient sums
1.4
Lagged1.2e1Long-run
U-gap

0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
1970

1975

1980

1985

1990

1995

2000

2005

2010

1995

2000

2005

2010

Estimation start datep-values
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1970

1975

1980

1985

1990

Estimation start date

Figure 12
Substitute short-run for long-run expectations
Rolling regression estimates, window = 40 quarters
Estimated coefficient sums
1.2

1

0.8
Lagged

0.6
Short-run

e

U-gap

0.4

0.2

0

-0.2
1980

1985

1990

1995

2000

2005

2010

2000

2005

2010

Estimation start datep-values
1

0.8

0.6

0.4

0.2

0
1980

1985

1990

1995

Estimation start date15

Figure 13

Horse race between shortlong-run expectations,Windowand= 60
Rolling regression estimates,
window= 60 quarters
Estimated coefficients
Rolling restricted OLS estimates of Phillips curves with SPF expectations

1

0.5) 10-yr.

E(

) SPF

E(U) SPF

0-0.5
1983:Q1

E(

1985:Q3

1988:Q1

1990:Q3

1993:Q1

1995:Q3

1998:Q1

2000:Q3

2003:Q1

1995:Q3

1998:Q1

2000:Q3

2003:Q1

Estimation start datep-values
0.8

0.6

0.4

0.2

0
1983:Q1

1985:Q3

1988:Q1

1990:Q3

1993:Q1

Estimation start date

What to conclude from the empirical evidence?
I would be hard-pressed to argue that only short- or long-run expectations belong in an
empirical Phillips curve for U.S. data. My reading is that, especially in recent data (the past twenty
years or so), there is not enough variation to distinguish between the influence of short- versus longrun expectations. The fact that our favorite measure of long-run expectations has barely budged
since 1998 makes it difficult to know (a) how important that stability is, or (b) how a shift in thatmeasure (or properly-measured long-run expectations) might affect inflation. 8
In addition, the lack of a theoretical founding for a simple model that includes only long-run
expectations should factor in to our search for a preferred specification, at least to some degree.
More prosaically, Truman Bewley has surveyed hundreds of producers over the past 10 years, andthe number of times that the long-run inflation goal is mentioned (unprompted) as a factor in
pricing decisions is precisely zero9. If price-setters are not aware that the central bank’s inflation goal
8 As defined, the expected inflation rate on average over the next 10 years really should have moved more than it has. It

suggests that the SPF survey respondents think of this not as a ten-year average, but as a proxy for their estimate of thecentral bank’s inflation goal, or at least as a proxy for where inflation will be 10 years hence.
9 See especially point (5) in the Bewley presentation here: https://www.bostonfed.org/great-recovery2016/agenda/

16

(or long-run expectations, should they become detached from the inflation target) matters at all in
their pricing decisions, then the mechanism by which the goal ends up as a key determinant of
aggregate price dynamics is mysterious indeed.
4. A DSGE model with survey expectations
In a recently published paper (Fuhrer 2017), 10 I explore a framework in which both shortand long-run expectations matter, in which long-run expectations are well-anchored, but in which
the convergence of inflation to the central bank target is not mechanically guaranteed. The noveltyof the paper is that it employs survey measures for all expectations (both inflation and real side), andattempts to endogenize the expectations in a way that is generally consistent with the logic of DSGE
models, but importantly, it is not assumed that expectations are rational. It finds a role for both
short-run and long-run expectations, in a way that is described below.
1.
2.
3.

4.
5.

6.
7.
8.

The essence of the paper is as follows:
Inflation depends on short-run expectations and on a gap measure;
The short-run expectations can be iterated forward, and thus depend on longer-run
expectations and on short-run expectations of the gap measure;This iteration can of course go on forever, so I proxy for is by using long-run expectationsof inflation, which should contain a sequence of shorter-run expectations of the gap
measure, if the model is approximately correct. It is not necessary that expectations arerational in order to take this step.
This anchors short-run expectations to longer-run expectations, which can be assumed toultimately return to the central bank’s inflation target (or not).
A parallel mechanism is employed for output. Output depends on short-run output
expectations and a real interest rate. The same iteration arguments imply that short-run
output expectations depend on long-run output expectations (which embody a sequence ofshort-run real interest rate expectations).
This model is estimated on US data, and I find that it helps a lot with a number ofidentification issues.
I also find that the inertia that is intrinsic to expectations better explains macro inertia thaninflation indexation, or habits, or serially correlated shocks in DSGE models.
These results imply that short-run expectations for inflation and output—properly
measured, via surveys—are critical inputs to empirical macro models.

An example of the modeling strategy embodied in the paper may be found in the Phillips
curve/inflation expectations system below. The first equation posits an expectations-augmented
Phillips curve with survey expectations for the one-period-ahead inflation rate:

π t = βπ tS+1,t − π u (U t − U t* ) .
The survey expectations in turn are assumed to follow the law of iterated expectations, so that they
may be iterated forward in time. However, lacking an infinite forward sequence of survey

10

http://www.sciencedirect.com/science/article/pii/S0304393216301362/pdfft?md5=6cf7d546a2ea8ec9a185ec14e28f31
43&pid=1-s2.0-S0304393216301362-main.pdf

17expectations, we approximate this process by allowing the long-run (10-year) expectation for
inflation to proxy for the iterated-forward sequence of expected unemployment rates:
π tS+*1,t =AΠ SLR ,t − π u (U tS+1,t − U t*+1 ) .
Finally, one-period survey expectations are assumed to adjust gradually towards this sequence oflong-run expectations via an error-correction mechanism, indexed by parameter µ π :Sπ=µ π π tS+*1,t + (1 − µ π )(π tS,t −1 ) .
t +1,tThe “intrinsic” inertia in expectations is captured by the size of the parameter µ π , and its
significance is estimated on time series data, allowing for other sources of persistence that are now
standard in the DSGE literature (indexation of prices, habit formation, autocorrelated shocks).
Of course, a similar system must be employed to solve for the survey expectations of
unemployment U tS+i ,t . The paper provides more detail.
The paper develops some interesting results, namely:••••

Survey expectations, both short- and long-run, enter significantly in the model, with therestrictions indicated above;
The intrinsic inertia in expectations is strongly preferred by the data as a means of explaining
persistence in aggregate time series for inflation and unemployment, as opposed to othermethods that the DSGE literature has embraced in recent years;
To be specific, terms reflecting inflation indexation, habit persistence, and correlated
structural errors all but vanish from the model in terms of quantitative and statisticalsignificance;
In a nested test, the DSGE model with survey expectations strongly dominates the modelwith rational expectations. The test shows that the data place a small and statisticallyinsignificant weight on rational expectations.

This does not mean that long-run expectations do not matter. They do, inasmuch as they proxy
for a sequence of short-run expectations (which appear not to be the rational expectations impliedby the model) that are linked to inflation. The model described above allows long-run expectations
to deviate from the central bank’s goal, but does not provide any deep rationale for suchdeviations. 11
Evidence in micro data bearing on the behavior of aggregate survey expectations
In a separate unpublished paper, I show that the micro data for the Michigan, SPF and EuroSPF surveys exhibit behavior that is fully consistent with intrinsic inertia in expectations formation.
Remarkably consistent empirical results show that individual survey participants consistently movetheir own forecasts toward the lagged central tendency of forecasts, which can impart inertia to theaggregate expectations measure beyond that in the driving processes for inflation, output growth,interest rates or unemployment.
11 The model allows for shocks that have persistence effects on the long-run inflation expectation, but ultimately, thelong-run expectations will converge to the inflation goal. The model does not identify any deeper structural explanation
for such shocks.

18

The figure below provides a scatter plot of the individual observations on forecast revisions (thevertical axis) agains the discrepancy between an individual forecast and the last observed median
forecast. A significant negative correlation is evident in the figure; the table following documents therobustness of this negative correlation to the inclusion of many different controls.
Figure 14

Table 4
Response of forecast revisions to lagged discrepancies between individual forecasts and central
tendency measures
i , SPFMedianiii
π ti+, SPF
d [π ti+, SPF+1,t −1
1,t − π t=1,t −1 − π t +1,t −1 ] + aπ t −1 + cZ t + d i + e t

Variable

π ti,t −1 − π tMedian|t −1

π ti+1,t −1 − π tMedian+1|t −1

π ti−1
U ti+, SPF
1,t
U ti+, SPF
2,t∆Yt i,tInflation results-0.30(0.002)0.08(0.193)
-0.58
-0.68
-0.47
-0.56
(0.000) (0.000) (0.000) (0.000)
0.01(0.648)0.04(0.883)-0.10(0.690)0.06(0.000)19-0.56(0.000)-0.03(0.011)
0.33(0.178)-0.02(0.904)0.03(0.030)0.01(0.640)-0.05(0.019)0.01(0.465)i
0.04
Rt +1,t(0.703)All controls*
Yes
Adjusted R-squared
0.069
0.206
0.251
0.172
0.224
0.473
Observations
3272
3274
1591
1926
3029
2945
* “All controls” includes real-time estimates of lagged inflation, unemployment, Treasury bill rate,current period, t+1, t+2, t+3 forecasts of inflation, unemployment, Treasury bill rate, output growth.
Inflation revision on discrepancy from lagged central tendency:other forecast horizons∆Yt i+1,t

Revision from t-1 to t for forecast period
t+1
t+2
t+3
t+1

π

i
t +1,t −1−πMediant +1|t −1-0.58 (0.000)-0.56 (0.000)-0.54(0.000)π ti+ 2,t −1 − π tMedian+ 2|t −1
π ti+3,t −1 − π tMedian+ 3|t −1

-0.61
(0.000)π ti−1Other forecast controls
Observations

N
3274

N
3257

N
3180

t+2
-0.53
(0.000)0.01 (0.648)
Y
3029

0.03
(0.019)
Y
3017

t+3

-0.61
(0.000)0.05(0.000)Y
2960

Thus expectations inertia is a feature in the micro data for survey expectations. The result for the
SPF holds up quite strongly for the Michigan and the Euro-SPF surveys. It cannot be explained as astatistical artifact of forecasters who make rational forecasts about a persistent variable (this isdiscussed in detail in the paper).
While this work raises questions about how expectations are formed, and what role they play ininflation and output dynamics, the results in these papers appear to be strong and promising.
5. A survey-based DSGE model of inflation dynamics for Japan
As suggested above, although the specifics of Japan’s inflation experience differ from thosein the U.S., it may be that like the U.S., inflation is well-modeled as depending on both short- andlong-run survey expectations of inflation, along with a measure of a resource gap. This section
follows the methodology of the preceding sections, using data on Japanese inflation, the output gap,
the expected output gap, and short- and long-run inflation expectations. All of the expectations data
are taken from the Consensus Forecasts database, which began collecting such data semi-annually in
1990. [A look at anchoring and other key dynamics issues in Japanese data]
Simple Phillips curves with short- and long-run expectations
We begin by examining single-equation relationships among the key data.

20

Lagged inflation
Short-run expectations
Long-run expectations
Output gap
Lagged inflation
Short-run expectations
Output gap
Lagged inflation
6-10 year expectations
Output gap

Table 5
Dependent variable: core inflation
Coefficientp-value
0.288*
0.567**-0.0358
0.102
Dependent variable: core inflation
0.285*
0.558**
0.1
Dependent variable: one-year expectation
0.253**
0.557**
0.14**

0.020
0.007
0.904
0.124
0.018
0.004
0.117
0.002
0.006
0.002

Inflation depends significantly on short-run expectations, lagged inflation and (for the
expectations themselves) the output gap. Long-run expectations appear not to be directly importantfor core inflation, but quite important for anchoring short-run expectations. A system test of the
exclusion of long-run expectations from the inflation equation (but not from the short-run
expectations equation) cannot reject this restriction, developing a p-value of 0.99.
Table 6 presents simple regressions of short-run inflation expectations on lagged inflation,
long-run expectations, and an output gap measures, for zero- to five-year-ahead horizons, expandingon the bottom panel of Table 5. The results suggest a very consistent dependence on long-run
expectations (perhaps as an “anchor” for short-run expectations), a modest dependence on lagged
inflation (depending on the horizon), and modest dependence on the output gap for the first threehorizons in the table. Taken together, tables 5 and 6 provide initial evidence that suggests that asurvey-based model for inflation may be fruitful for Japanese data. Of course, a more complete
assessment of such a model is required to be more confident of this assessment. The next section
turns to this task.
Table 6
Dependence of short-run inflation expectations on lagged inflation, long-run expectations and theoutput gap
Horizon (year)
Coefficients 0
1
2
3
4
5
Lagged0.48***
0.25**
0.17*
0.091
0.15***
0.076*
inflation
6-10 year
0.3
0.55**
0.66***
0.86***
0.74***
0.78***
expectation
Output gap 0.17***
0.14**
0.097*
0.027-0.0045
0.026
Asterisks indicate significance at the 5, 1, and .1% level (*, ** and *** respectively)21

A survey-based model of Japanese inflation and output
In this section, I construct a DSGE-like model for Japanese data, drawing on the approach
used for US data in Fuhrer (2017). That paper employs survey expectations in a DSGE model, using
methods described therein to endogenize short-run survey expectations of key variables. The model
below is a simplified version of the model explored in that paper, and focuses primarily on inflationand inflation expectation dynamics. The description of the IS block of the model is more reducedform in nature, although it follows the outlines of the approach to expectations taken in the inflation
block.
•
•

•
•
•
•

The model comprises
A New Keynesian Phillips curve in which inflation depends on short-run expected inflation
as measured in the Consensus Forecasts survey of forecasters, as well as the current value of
the Cabinet Office’s estimate of the output gap;
A semi-structural forward-looking “IS” curve in which an estimate of the output gap
depends on a lag of the output gap, the short-run expectation of the output gap as measured
by the Consensus survey, and a short-term real interest rate defined as the difference
between the policy rate and the one-year inflation expectation from the same survey;
Equations that link the short-run expectations for inflation and the output gap to long-run
survey expectations of inflation and output, as well as survey-based expectations of output
and real rates, respectively;
In addition, we allow for intrinsic inertia in short-run inflation expectations, indexed by theparameter µπ as indicated in equation (5.1) below;
Long-run expectations are assumed to follow simple AR(1) processes that revert to sampleestimated long-run values.
The policy rate follows a simple, inertial policy rule that responds to deviations of inflationfrom its (for the moment) assumed goal of one percent. The rule is estimated via OLS on asample prior to the ZLB period (1980-1999), and its parameters are also held fixed for thesystem estimation.

Thus the model may be represented by the following system of equations:=
π t bπ tS+1 + π y ytCab

π tS+1,t =(1 − µπ )( bΠ SLR ,t + π y ytS+1,t ) + µπ π tS,t −1SytCab = yl ytCab−1 + y E yt +1,t − y r ( rrt − )rt ≡ rt − π tS+1,tS
S
ytS+1,t =(1 − µ y )( yLR y LR,t − y r ( Et rr
t +1 − )) + µ y yt ,t −1
S
S
y LR
,t = 0.68 y LR ,t −1

Π SLR ,t = 0.95Π SLR ,t −1 + 0.05cΠ
=
rt 0.92rt −1 + 0.08[2.5(π tS+1,t − 1) + r ]r ≡ r +π22

, (5.1)

in which π denotes inflation, y is the output gap, r is the central bank policy rate (from the IFS
database), Π SLR is the Consensus survey’s 6-10 year inflation forecast, rt S+1 is the Consensus survey
observation on the one-year forecast of the three-month Treasury yield, the superscript “S” denotes
a variable from the Consensus survey, and coefficients that are held fixed at their single-equation
OLS estimates are indicated in the model. Critically, we estimate the policy rule over the sample
prior to the extended period during which the policy rate was constrained at its effective lower
bound. The discount rate β is held constant at 0.98 in the estimation. 12
This leaves eight key parameters to estimate: π y , the slope of the Phillips curve, which iscommon to the Phillips curve and to the one-period-ahead inflation expectations equation; µπ , theparameter that indexes the intrinsic persistence in inflation expectations; yl , the coefficient on thelagged output gap in the IS curve; µ y , the parameter indexing the intrinsic persistence in the output
expectation equation; yE , the coefficient on the one-year-ahead expectation of the output gap in the
IS curve; yLR , the coefficient on long-run output expectations in the output expectations equation;yρ , the slope of the IS curve; and ρ , the equilibrium real interest rate, which is common to boththe IS curve and to the reduced-form representation of output expectations. 13
The Consensus Survey does not provide forecasts for the output gap. Instead, I use theCabinet Office’s estimate of the current output gap, combined with the Consensus Survey’s
forecasts for real GDP growth in ensuing years, along with an estimated trend growth rate, toconstruct t+1 and long-run output gap forecast estimates. 14
Several features are worthy of note from these estimates. First, the degree of “intrinsic
persistence” in inflation expectations is significant, with the coefficient on lagged inflation
expectations ( µπ ) estimated at 0.79 (equivalently, the error-correction coefficient suggests about 21
percent of the gap is closed per half-year). Second, the “slope” parameters in the Phillips and IS
curves ( π y , yρ ) are sizable and reasonably precisely estimated. Third, the estimated value of theequilibrium real interest rate, ρ is negative, and the 90% confidence interval lies entirely in thenegative domain. Fourth, as suggested by the results in the table, Figure xx shows that for all of theparameters, the data move the estimated values noticeably away from the priors and/or sharpen the
precision of the estimates relative to the priors, suggesting that the data contain useful identifying
information for this model. For example, the posterior distributions of the inflation error-correction
parameter µπ implies larger and more precisely estimated error-correction than its priordistribution, and the posterior distribution for the equilibrium real rate similarly is more negative
than its prior, evincing little overlap with its prior distribution.
12 Estimating β rather than imposing yields a value above 0.95, using priors that put significant weight on a value above

0.8. Freeing up the priors to allow for lower values, while running counter to the notion of the NKPC, yields lower
values of β of 0.6-0.7. We choose to retain the structural interpretation of these equations, and thus keep the higher
values that conform better to the notion of a discount rate.
13 The prior distributions for all of the parameters in this and subsequent estimations are displayed in the Appendix
table.
14 The output gap equals the current output gap plus the cumulative sum of real growth forecasts less the estimate of thetrend growth rate. Thus output gaps are constructed using the identity that links the output gap from period to period
and the discrepancy between real growth and potential or trend growth over the forecast horizon. The trend growth rate
is simply an HP-filtered version of the 6-10 year real GDP growth forecast from the Consensus survey.

23

Table 7
Bayesian estimates

0.14Standard
Deviation
0.051

tstatistic
2.8

5th
95th
percentile percentile
0.059
0.23

πy

Posterior
mode
0.14

µπ

0.78

0.77

0.77

0.057

14

0.67

0.86

yl

0.59

0.59

0.59

0.11

5.3

0.41

0.77

yE

0.33

0.32

0.32

0.083

3.9

0.19

0.46

yρ

0.3

0.3

0.3

0.092

3.3

0.15

0.45

ρ
µy

-1.5
0.64

-1.4
0.64

-1.4
0.64

0.32
0.058

-4.5
11

-1.9
0.54

-0.87
0.73

0.52
0.51
0.52
yLR
4 blocks, 65,000 replications per block

0.084

6.1

0.37

0.65

Parameter

MeanMedian0.14

24

Figure 15
Posterior parameter distributions
260000 replications

10

10
Posterior
Prior

5

5

0
-0.5

0
0

1

0.5

1.5

0

y

4

0.2

0.4

0.6

1

0.8

1.2

5
4

3

3
2
2
1

1

0
-0.5

0

1

0.5

0
-0.5

1.5

0

1

0.5

y

y

E

l

2

5
4

1.5

3
1
2
0.5

1

0

0
-0.5

0

1

0.5

-4

1.5

-2

4

2

0

6

y
10

5
4
3

5
2
1
0
0

0.2

0.4

0.6

0.8

1

0
-0.5

1.2

0

0.5

1

1.5

y
LR

y

By these diagnostics, the model performs quite well. An in-sample simulation of the model
verifies that the equations are capturing most of the significant fluctuations in these series over the
past thirty-five years. The simulation imposes the zero lower bound constraint for the nominal
policy rate. The figure shows the results of these in-sample simulations for inflation and inflation
expectations, along with the simulation when the error-correction parameter for inflation
expectations is set close to zero. That figure emphasizes the point made in Fuhrer (2017): The
intrinsic inertia in inflation expectations is important for explaining the dynamics of inflation andinflation expectations. In Japan, the lowest readings for inflation are best explained as the result of25

intrinsic persistence in short-run inflation expectations in response to chronically negative output
gaps.
Figure 16
Inflation

Inflation expectations, 1-year4

4ActualSimulated,

3

=0.05

3
Simulated, Bayesian estiamtes

2
2
1
1
0
0

-1

-2
1990

-1
1995

2000

2005

2010

2015

2020

1990

Year

1995

2000

2005

2010

2015

2020

Year

Note that at these estimates, the specification for inflation does not require either indexing or
autocorrelated structural shocks to match the data reasonably well. The first autocorrelation of theestimated shock processes is 0.089, suggesting little in the way of omitted and highly-correlated
structural shocks.
This could suggest that the need to include lagged inflation (via an indexation argument) is
obviated, although a more formal test is required to confirm this. Re-estimating the model above,
but adding lagged inflation to the Phillips curve equationπ=ωπ t −1 + ( b − ω )π tS+1 + π y ytCabtallows us to test the importance of lagged inflation, once the inertial survey expectations areincluded. We set the prior for ω to a gamma distribution with mean of 0.1 and standard deviationof 0.05, consistent with the lack of autocorrelation in the estimated structural shocks in the baselinemodel. The posterior is maximized at a value of ω of 0.10 with an estimated standard deviation of
0.063. The 90% confidence interval for ω extends from 0.032 to 0.24. Other parameter estimatesare qualitatively unchanged from those presented above. The data do not suggest moving theestimate from its prior. Put differently, there is no clear role for lagged inflation in this model once
survey expectations are incorporated.
Modeling long-run inflation expectations
The model described above includes long-run expectations, which are important as a longrun attractor for short-run expectations (and thus ultimately for realized inflation). But the model
simply posits an exogenous AR(1) process for long-run expectations, assuming that they revert
ultimately to the central bank’s inflation goal. This is not a fully satisfactory description of long-run
expectations, particularly not if they are central to determining the steady-state value for inflation. In
fact, the model has similar stability properties to the model discussed in section 2 above—because
long-run expectations always return to the central bank’s goal, the central bank need only return the
output gap to zero to stabilize inflation.
26

One way to model long-run expectations (see, e.g. Clark 2010) is to assume that theyultimately revert to the central bank inflation goal, but in the short-run are influenced by persistentfluctuations in short-run inflation that differ from the central bank target. A simple way to represent
this isK

Π tLR =aΠ tLR−1 + ∑ bk π t − k + c

(5.2)

k =0

An estimated version of this equation on Japanese data, using Consensus 6-10 year forecasts asproxies for the long-run expectations, yields (p-values in parentheses)
0.27 Π tLR−1 + 0.27 Σπ t − k + 0.91
(0.059)(0.007)(0.000)And the in-sample fit of this equation, displayed below, is quite good. The implied long-run value
for the long-run expectation is 1.25, with an asymptotic standard error of 0.072.
=
Π tLR

Figure 17
Fit of simple lagged inflation model
4
6-10-year expectation
Fit
Actual inflation

3

2

1

0

-1

-2
1990

1995

2000

2005

2010

2015

2020

Year

This model captures in a simple way the possibility that long-run expectations can be dragged awayfrom their anchor, if actual inflation deviates from target for long enough. If we write the model
more generally as a well-defined autoregression
Π tLR =aΠ tLR−1 + (1 − a )π t −1 + c ,then we can show generically that any coefficient 0<a<1 will yield more sensible stability results forthe model. To focus on stability properties associated with long-run inflation expectations, we
simplify the model above so that it comprises the following equations27

=
π t bπ tS+1 + π y ytCab

π tS+1,=µπ ( bΠ SLR ,t + π y ytS+1,t ) + (1 − µπ )π tS,t −1tΠ SLR ,t =aΠ SLR ,t −1 + (1 − a )π t −1,ytCab =bytCab−1 − y r ( rrt − )rt ≡ rt − π tS+1,trr
r=
rt −1 + (1 − )[r + aπ (π tS+1,t − 1) + a y ytCab ]twhere we simplify the IS curve, and where long-run inflation expectations are now modeled asabove, that is as a geometrically weighted average of lagged actual inflation.
The next figure displays the region of stability for this model as we vary the policy response
to inflation and output from 0 to 3, holding the lag coefficient a from at 0.25, conditioned on a valueof a y 0.5,====
µπ 0.25,
π y 0.12,yρ 0.4 , as estimated above. 15 Note that the simple version of the“Taylor principle” does not hold in this model, although clearly for modest values of a y , the central
bank must respond to inflation, unlike the simplest anchored-expectations model.
Stability region for [aFigure 18,a ] in the test model
y

3

2.5

a

y

2

1.5

1

0.5

0
0

0.5

1

1.5

2

2.5

3

a

But this is still a fairly mechanical representation of long-run expectations. An alternative
method that captures the spirit of standard models makes long-run expectations a function of the
15 The value of a has no effect on stability, as long as it is strictly greater than zero.

28

unemployment gap: One can think of this as the implication of a short-run expectations relationshipbetween inflation and expected unemployment. If inflation is high, the central bank raises the policyrate, which raises unemployment, which lowers current and expected (short- and long-run) inflation.
This stabilizes the model in exactly the conventional way.
We augment the theory-model above consistent with this notion:
Π SLR ,t =aΠ SLR ,t −1 + (1 − a )π + bytCab
where π is the inflation goal, which is assumed equal to one. This equation for long-run
expectations makes expectations a long moving average of the output gap. Below, we examine thestability properties for values of aπ , a y . For this model, we obtain the Taylor Principle exactly:
Figure 19

The next figure displays the least-squares fit of this equation for long-run inflation expectations overthe past 25 years on Japanese data:

29

Figure 20
Fit of simple output gap model for long-run expectations3

6-10-year expectation
Fit

2.5

2

1.5

1

0.5

0
1990

1995

2000

2005

2010

2015

2020

Year

We now estimate the full model with feedback from the output gap to long-run
expectations. The results are presented in table 8 and figure 21.
Table 8
Bayesian estimates
Standard
MedianDeviation
0.13
0.045

tstatistic
2.9

5th
95th
percentile percentile
0.06
0.21

πy

Posterior
mode
0.13

µπ

0.74

0.73

0.73

0.063

12

0.62

0.83

yl

0.59

0.59

0.59

0.1

5.9

0.42

0.76

yE

0.31

0.3

0.3

0.085

3.6

0.16

0.44

yρ

0.27
-1.3
0.6

0.27
-1.2
0.6

0.27
-1.2
0.6

0.092
0.42
0.064

3
-3.1
9.5

0.11
-1.9
0.49

0.42
-0.52
0.7

yLR
0.53
0.51
0.51
a
0.38
0.36
0.36
b
0.045
0.046
0.045
4 blocks, 65,000 replications per block

0.082
0.099
0.013

6.4
3.8
3.5

0.37
0.2
0.025

0.65
0.52
0.068

Parameter

ρ
µy

Mean
0.13

30

Figure 21
Posterior parameter distributions
260000 replications

10

10

4

Posterior

3

Prior

5

5

2
1

0
-0.5

0
0

1

0.5

1.5

0

1

0.5

1.5

0
-0.5

0

0.5

1

1.5

y
y

5

5

4

4

3

3

2

2

1

1

l

2
1.5
1

0
-0.5

0

0.5

1

0
-0.5

0.5
0
0

y

1

-5

1.5

5

0

y
E

10

0.5

5

5

4

4

3

3

2

2

1

1

5

0
0

1

0.5

1.5

0
-0.5

0

0.5

1.5

0
-0.5

0

0.5

1

1.5

a

y
LR

y

40

1

30
20
10
0
-0.5

0

0.5

1

b

As one can see from the estimated parameters and their distributions, the addition of this feedback
leaves the other estimated parameter values essentially as they were in the baseline. But now themodel has somewhat more desirable stability properties—more specifically, the model incorporates
an explicit link from monetary policy to long-run expectations, via its effects on the output gap, sothat only if policy devotes sufficient attention to inflation will the economy converge to the central
bank’s long-run inflation goal.
.
31

The zero lower bound and survey expectations
In rational expectations models, failure to incorporate the zero lower bound in estimation
can seriously bias parameter estimates. The logic is straightforward: If the policy rule is estimatedover a sample during which the policy rule is binding for part of the period, then the estimatedcoefficient can imply that the policy rule will take on negative values. While policy rates in someother countries have been set to negative values, rates in Japan over this period were held at or
above zero.
This section examines the extent to which the zero lower bound may confound estimates ofthe model presented thus far. Recall that the model was estimated imposing a policy rule that wasestimated on the sample prior to the effective lower bound period that started in 1999. But given
low inflation, that rule might imply a policy rate well below zero for much of the sample.
r=rr
rt −1 + (1 − )[r + aπ (π tS+1,t − π )]t
We look at the model’s implication for the policy rate under two policy rules. The first is theone imposed in the sections above. The second is a rule that is estimated over the full sample,
including the effective lower bound period. The results of that estimation are presented below. The
parameters of the model do not change much, with the exception of the equilibrium real interest
rate. While the 90 percent confidence interval for the real rate is still strictly negative, the modal
estimate is -0.88, versus -1.4 or lower in the previous estimates. Note that this estimate employs
information from both the policy rule and the real side of the model economy ( ρ enters in the
output and output expectations equations as well as the equilibrium level of the policy rate in thepolicy rule).
The policy rule parameters imply a moderate response to inflation ( aπ = 1.2 ), a significant
degree of interest rate smoothing ( ρ = 0.82 ), and an inflation target of about one-half percent (π = 0.58 ). All are precisely estimated.
Table 9Bayesian estimates
Standard
MedianDeviation
0.15
0.053

tstatistic
2.7

5th
95th
percentile percentile
0.059
0.23

πy

Posterior
mode
0.14

µπ

0.77

0.75

0.75

0.062

12

0.64

0.84

yl

0.64

0.63

0.63

0.099

6.5

0.46

0.79

yE

0.26

0.27

0.26

0.079

3.3

0.14

0.4

yρ

0.32
-0.95
0.6

0.32
-0.95
0.59

0.32
-0.97
0.59

0.089
0.32
0.062

3.6
-2.9
9.7

0.17
-1.5
0.49

0.47
-0.41
0.69

0.5
0.84
1.1
1.3

0.49
0.83
1.1
1.3

0.49
0.83
1
1.3

0.077
0.044
0.14
0.12

6.4
19
8
11

0.36
0.76
0.85
1.1

0.62
0.9
1.3
1.4

Parameter

ρ
µy
yLR

ρ
aπ
π

Mean
0.15

32

a
0.34
0.34
0.34
b
0.05
0.051
0.051
4 blocks, 65,000 replications per block

0.098
0.013

3.4
3.8

0.18
0.03

0.5
0.073

We then simulate the model over the estimation sample period, taking the long-run
expectation as exogenous, performing a period-by-period (static) simulation. The question iswhether given the model parameters and data, the model implies significantly negative policy rates
during the sample. If so, we will need to re-estimate the model to account for not taking the zero
lower bound into account.
As the first figure shows, both policy rules fit the actual policy rate quite well. And theperiods of implied sub-zero policy rates are relatively modest in magnitude. The dashed blue line
shows the effect of the inflation gap on the policy rate according to the rule, that is, the simulated
value of (1-ρ )a π (π tS+1,t -π ) . When the policy rate is approximately zero, this is the only term that
affects the policy rate, as the interest rate smoothing term r rt −1 is approximately zero.
Figure 22
Policy rates implied by model and policy rules
Model estimated over ZLB period
8
Policy rate
Simulated

6

Inflation term

4

2

0
1990

1995

2000

2005

2010

2015

2020

2010

2015

2020

Year
Model estimated pre-ZLB period
8

6

4

2

0
1990

1995

2000

2005

Year

33

In order to examine more closely these periods, the next figure displays the same simulations,focusing on the period from 2000 forward.
Figure 23
Policy rates implied by model and policy rules
Model estimated over ZLB period
0.8
Policy rate

0.6

Simulated
Inflation term

0.4
0.2
0
-0.2
-0.4
-0.6
2000

2002

2004

2006

2008

2010

2012

2014

2016

2018

2012

2014

2016

2018

Year
Model estimated pre-ZLB period
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6
2000

2002

2004

2006

2008

2010

Year

As the figure indicates, the implied policy rates dip slightly negative—from -0.1 to -0.3 percentage
points for either of the policy rules. These are within the range of negative rates that the Bank ofJapan subsequently and other central banks earlier put into place. In addition, these small deviations
below zero are unlikely to have significant effects on the estimates of other model parameterspresented above. Of course, interest rate smoothing helps in this regard, as the period-by-period
coefficient on the inflation gap is premultiplied by one minus the smoothing coefficient, which isabout 0.7 or 0.9 for the two policy rules. Based on these results, one might not expect the impositionof the zero lower bound to have much impact on the estimated parameters. Still, because theconstraint implied by the zero lower bound (or near-zero lower bound) played such a prominentrole in many developed economies in the post-Great Recession period, we will next consider amethod that imposes the zero lower bound in estimation.
Estimating the model with a zero lower bound constraint
Here we estimate the full model that includes feedback from the output gap into long-run
inflation expectations, and simultaneously estimated policy parameters, while imposing the zero34

lower bound. We do so in a way that is similar to the methods of Erceg and Lindé (2014) and Hirose
and Inoue (2014). Specifically, we impose the zero lower bound in the filter that maps parametersand data into estimated shocks for the model above. 16 In this way we impose the constraint on thepolicy rule*rrrt=
rt −1 + (1 − )[r + aπ (π tS+1,t − π )](5.3)
rt = max(rt* , 0)
The results for estimation of the “full” model are presented below. As the first table shows,the parameters for the model are little changed, apart from the policy rule parameters. As the second
table shows, imposing the zero lower bound and assuming a constant policy rule from 1990-2016
implies a larger inflation response aπ (1.4 versus 1.1 in the unconstrained model), a lower
equilibrium real rate ρ (-1.1 versus -0.95), but about the same degree of interest rate smoothing at
0.83 versus 0.84. All other parameters lie within a fraction of a standard deviation of theunconstrained parameter estimates.
Table 10
Bayesian estimates, 1990:H1-2016:H1
Standard
tMean
MedianDeviation statistic
0.16
0.16
0.056
2.8

5th
95th
percentile percentile
0.07
0.25

πy

Posterior
mode
0.16

µπ

0.77

0.76

0.76

0.059

13

0.66

0.85

yl

0.63

0.58

0.58

0.11

5.6

0.4

0.76

yE

0.27

0.31

0.31

0.088

3

0.17

0.46

yρ

0.32
-1.1
0.59

0.33
-1.3
0.59

0.33
-1.3
0.59

0.088
0.34
0.062

3.6
-3.3
9.7

0.18
-1.9
0.49

0.47
-0.82
0.69

0.48
0.48
0.48
ρ
0.83
0.83
0.83
aπ
1.4
1.4
1.4
1.3
1.2
1.2
π
a
0.35
0.35
0.35
b
0.049
0.049
0.049
4 blocks, 65,000 replications per block

0.083
0.034
0.28
0.11
0.099
0.014

5.8
25
5
11
3.6
3.6

0.35
0.77
0.98
1
0.18
0.027

0.62
0.88
1.9
1.4
0.5
0.072

Parameter

ρ
µy
yLR

16 Appendix B provides the details for computing the model solution and residuals under the zero lower boundconstraint. The filter employed in my software is a specialized version of the Kalman filter that is optimized for linear
rational expectations models. In this implementation, I check for each set of parameter values and each observation
whether the implied solution for the policy rate is less than zero. If it is, I impose that the policy rate be zero by resetting
the appropriate filter parameters, and then resolve for the set of residuals that are consistent with that constraint. Thismethod allows a simple alteration of the linear Bayesian software that is used to estimate and simulate the models in thispaper.

35

Table 11
Compare estimates with and without imposition of ZLB
Standard
Parameter
Posterior modet-statistic
Deviation
with
with
with
without
without
without
ZLB
ZLB
ZLB
πy
0.16
0.14
0.056
0.053
2.8
2.7

µπ

0.77

0.77

0.059

0.062

13

12

yl

0.63

0.64

0.11

0.099

5.6

6.5

yE

0.27

0.26

0.088

0.079

3

3.3

yρ

0.32
-1.1
0.59

0.32
-0.95
0.6

0.088
0.34
0.062

0.089
0.32
0.062

3.6
-3.3
9.7

3.6
-2.9
9.7

0.48
0.83
1.4
1.3
0.35
0.049

0.5
0.84
1.1
1.3
0.34
0.05

0.083
0.034
0.28
0.11
0.099
0.014

0.077
0.044
0.14
0.12
0.098
0.013

5.8
25
5
11
3.6
3.6

6.4
19
8
11
3.4
3.8

ρ
µy
yLR

ρ
aπ
π
a
b

36

Figure 24
Posterior parameter distributions
260000 replications

10

4

10
Posterior

3

Prior

5

5

2
1

0
-0.5

0
0

1

0.5

1.5

0

1

0.5

1.5

0
-0.5

0

1

0.5

1.5

y
y

5

5

4

4

3

3

2

2

1

1

l

2
1.5
1

0
-0.5

0

1

0.5

0.5

0
-0.5

0
0

y

1

-5

1.5

5

0

y
E

10

0.5

5

20

4

15

3
5

10
2
5

1
0
0

1

0.5

0
-0.5

0
0

0.5

1

1.5

0

0.5

0
-0.5

0

1

1.5

0.5

1

y
y

2

LR

4

1.5

3

1

2

0.5

1

5
4
3
2

0

1

0
0

1

2

3

4

-2

0

4

2

6

a

a

Adding an output response to the policy rule
While price stability is the primary goal of the Bank of Japan, there have no doubt beentimes when the BOJ has devoted some attention to output. While much of the period has featured
both output and inflation below their desired levels, there have been times when they have moved(at least temporarily) in opposite directions. This section estimates a policy rule, imposing the zero
lower bound, that allows a response to the one-year-ahead forecasts of both inflation and output, or:r=rr
rt −1 + (1 − )[r + aπ (π tS+1,t − π ) + a y ytS+1,t ]t37

The prior distribution for the additional parameter a y is shown in the table in Appendix A. The
results of the estimation are summarized in the following table.
Table 12
Bayesian estimates, 1990:H1-2016:H1
Zero lower bound imposed
PosteriorStandardt5th
95th
Parameter
MeanMedianmode
Deviation statistic percentile percentile
πy
0.14
0.15
0.15
0.054
2.6
0.066
0.24

µπ

0.78

0.76

0.77

0.056

14

0.67

0.85

yl

0.59

0.57

0.57

0.086

6.9

0.42

0.71

yE

0.3

0.34

0.34

0.09

3.4

0.19

0.49

yρ

0.31
-1.4
0.61

0.33
-1.5
0.61

0.33
-1.4
0.61

0.069
0.27
0.058

4.5
-5.3
11

0.22
-1.9
0.51

0.45
-1
0.7

0.51
0.75
1.2

0.5
0.74
1.2

0.5
0.74
1.2

0.082
0.059
0.22

6.1
13
5.3

0.37
0.64
0.81

0.65
0.83
1.5

1.3
1.2
1.2
0.65
0.66
0.68
π
a
0.34
0.38
0.38
b
0.05
0.046
0.046
4 blocks, 65,000 replications per block

0.11
0.16
0.095
0.012

12
4.1
3.6
4.2

1
0.33
0.23
0.027

1.4
0.88
0.54
0.066

ρ
µy
yLR

ρ
aπ

ay

The estimates suggest a significant role for both inflation and output gap forecasts in thedetermination of the policy rate. However, the inclusion of the output response does not alter theconclusions about inflation dynamics in Japan during this period. There is still a prominent role for
expectations inertia, and a strong link from short-run to long-run expectations.
Balance Sheet effects
Missing from the analysis so far is the effect of the Bank of Japan’s significant actions to
expand its balance sheet. Using the ratio of total BOJ assets to nominal GNP as a proxy for balance
sheet actions, I have estimated their impact on inflation and inflation expectations, augmenting themodel above with an additional linear term in the change in the share of BOJ assets to GNP. Using
this measure, the BOJ expanded its balance sheet beginning in the late 1990s, rising as a share of
GDP from about 10% in 1997 to just over 80% by 2016.
The direct effect on realized inflation appears to be tiny and statistically insignificant. 17 The
effect on inflation expectations, which in the model affects realized inflation one-for-one, appears to

17 This conclusion is the same for both single-equation OLS regressions, and for system estimates in which this effect isincluded into the model described up to this point.

38

be more significant. A least-squares regression of one-year inflation expectations on the key variablesin the model above, augmented to include the change in the GNP share of BOJ assets yields:A
π tS+1,t = 0.77Π SLR ,t + 0.18 ytS+1,t + 0.11∆[ t −1 ]Yt −1
All of the estimated coefficients have p-values of 0.001 or smaller. Thus the estimated effect ofbalance sheet expansion on inflation expectations is economically and statistically significant. We
include this effect into the full model by augmenting the inflation expectations equation as follows:A
π tS+1,t =(1 − µπ )( βΠ SLR ,t + π y ytS+1,t + π BS ∆[ t −1 ]) + µπ π tS,t −1 .
Yt −1This yields a Bayesian posterior estimate for the coefficient on the BOJ assets variable of 0.091, withan estimated standard deviation for the posterior distribution of 0.041; the 90% confidence interval
ranges from 0.055 to 0.19. Other estimated parameters are not much affected by the inclusion ofthis variable. The prior and estimated posterior distributions for π BS are displayed below. 18
Figure 25
20
15
10
5
0
-0.5

0

0.5

1

πBS
BS

Quantiative importance of balance sheet actions
With the estimated model, we can examine the quantitative impact of the BOJ’s balance
sheet actions on inflation over time. We compare a baseline simulation of the model that assumesA
the actual balance sheet trajectory (the realized t ratio) with an alternative in which the ratio isYtheld fixed at its level in the first half of 1997. Given the coefficient estimated above, the increase in
the balance sheet is estimated to have raised short-term inflation expectations by a bit more than
one-half percentage point over the period since 1997. This passes directly into higher inflation,according to the estimated model. 19
18 The priors for this parameter are somewhat informed by the point estimate and standard error from the OLS

estimates above.

19 This estimated effect is smaller than the coefficient

π BS times the increase in the balance sheet-to-GDP ratio, as it ismediated by the error-correction parameter (1 − µπ ) , and the effect on inflation expectations is also smoothed by theeffect of lagged inflation expectations on current inflation expectations.

39

Figure 26
Simulated inflation, contribution of balance sheet actions beginning in 1997:H2
Comparison to baseline holding A/Y = 0.1
0.8
Contribution to inflation expectations
Change in A/Y

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

-0.2
1990

1995

2000

2005

Year

2010

2015

2020

.

6. Conclusions
It would be a bit of an over-reach to claim that the preliminary exercises in this paper have
explained the inflation puzzles in the U.S. and Japan. However, the results presented herein suggest
some promising directions for inflation modeling in both countries:
1. The use of survey expectations as proxies for the expectations in conventional models
appears to be of empirical value. That is, surveys aid in identification of the inflation process
in both countries;
2. Methods for endogenizing such expectations, which are necessary once one moves awayfrom the rational expectations assumption, are tractable and replicable. They require themeasurement of longer-term expectations, but such data are available for many key variables
in many developed economies; 20
3. Models that incorporate such expectations identify the following rationale for the behaviorof US and Japanese inflation:
a. Long-run expectations anchor the process, although long-run expectations can be
influenced over time by persistent deviations of inflation (or output) from their longrun equilibria;
b. Short-run expectations are tied to their long-run counterparts, but they can deviate
quite persistently from long-run expectations, due to persistent deviations of output
from potential, and due to intrinsic persistence in the expectations;

20 A separate but important issue is whose expectations are measured in the available surveys. To be sure, a survey ofimportant price-setters, rather than professional forecasters, would likely be preferred. Such a dataset is not available for
the U.S. or Japan at present, to my knowledge.

40

c. In this regard, while both the U.S. and Japan evince a linkage (“anchoring”) fromshort-run to long-run expectations, the size and duration of deviations between
short- and long-run expectations is larger in Japan than in the U.S.
d. Inflation appears well-explained by short-run expectations and a traditional output
gap;
4. The balance sheet actions in Japan appear to have boosted short-run inflation expectations,compared to where they would have been without such actions. Higher expectations in turnhave helped to raise realized inflation.
References
Bewley, Truman, 2016. Remarks at the Federal Reserve Bank of Boston Annual Economic
Conference, October. https://www.bostonfed.org/-/media/Documents/economic/conf/great-recovery2016/bewley-presentation.pdf

Clark, Todd, and Troy Davig, 2008. “An Empirical Assessment of the Relationships Among
Inflation and Short- and Long-Term Expectations,” Research Working Paper 08-05, Federal
Reserve Bank of Kansas City, November .
Erceg, Christopher and Jesper Lindé, 2014. “Is There a Fiscal Free Lunch in a Liquidity Trap.”
Journal of European Economic Association, 12(1), 73–107.
Fuhrer, Jeffrey 2017. “Expectations as a source of macroeconomic persistence: Evidence fromsurvey expectations in a dynamic macro model.” Journal of Monetary Economics, 86, pp. 22–35.
Fuhrer, Jeffrey, 2017. “Expectations as a Source of Macroeconomic Persistence: Evidence fromMicro-Survey Data,” FRB Boston Working paper, 2017.
Gilchrist, Simon, Schoenle, Raphael, Sim, Jae, and Egon Zakrajšek, 2017. “Inflation dynamics during
the financial crisis,” American Economic Review, 107(3): 785–823.
Hirose, Yasuo and Atsushi Inoue. “The Zero Lower Bound and Parameter Bias in an Estimated
DSGE Model,” Bank of Japan Discussion Paper No. 2014-E-9.
Williams, John 2006. “The Phillips Curve in an Era of Well-Anchored Inflation Expectations,”
Working Paper, FRB San Francisco.
Appendix A
Prior Distributions

Lower

Upper

0.2

Prior
Standard
Deviation
0.1

0.001

1.2

Beta

0.5

0.1

0.001

0.99

Beta

0.3

0.1

0.001

0.99

Parameter Distribution

Prior
Mean

πy

Gamma

µπ
yl

41

Support

yE

Beta

0.5

0.2

0.001

0.99

yρ

Gamma
Normal
Beta

0.5
0.1
0.5

0.2
0.5
0.1

0.001
-5
0.001

1.5
5
1

Beta
Beta
Gamma

0.5
0.5
1.5

0.1
0.1
0.3

0.01
0.01
0.01

0.99
1
3

Gamma
Normal
Beta
Gamma

0.5
1
0.2
0.1

0.3
0.7
0.1
0.05

0.01
-1
0.001
0.001

1.5
5
0.99
0.7

Diffuse

-

-

-

-

ρ
µy
yLR

ρ

aπ
ay

π
a
b
Shock
variances

Appendix B
Imposing the zero lower bound in estimation
Each of the models in this paper may be cast in the formθ0

,∑t H x + ∑ H E ( x ) =i t +i
i t
t +i
1
i=
−
i=

t

(6.1)

where τ and θ are positive integers, xt is a vector of variables, and the H i are conformable nsquare coefficient matrices, where n is the number of endogenous variables in the model. The
coefficient matrices H i are completely determined by a set of underlying structural parameters Θ .
The expectation operator Et (·) denotes mathematical expectation conditioned on the process
history through period t (or t-1),(6.2)=
Et ( xt +i ) E ( xt +i | xt , xt −1 , …).
The model may include expectational and accounting identities; the former are important because
they define variables that can only be observed within the context of the model. Because t is white
noise, Et (t + k ) = 0, ∀k > 0. Leading equation (6.1) by one or more periods and taking expectations
conditioned on period-t information yields a deterministic forward-looking equation in expectations,
θ

(x
) 0,
∑t H E =i= −i

k > 0.

t + k +it

(6.3)

We use the AIM procedure detailed above to solve equation (6.3) for expectations of the future interms of expectations of the present and the past. For a given set of initial conditions,{Et ( xt + k +i ) :k > 0,i =−t , …, −1} , if equation (6.3) has a unique solution that grows no faster than agiven upper bound, that procedure computes the vector autoregressive representation of thesolution path,=Et ( xt + k )−1∑ B E (xi = −titt + k +i),k > 0.

In the models we consider here, the roots of equation (6.4) lie on or inside the unit circle.
42

(6.4)

Using the fact that Et ( xt − k ) = xt − k for k ≥ 0 , equation (6.4) is used to derive expectations of the
future in terms of the realization of the present and the past. These expectations are then substituted
into equation (6.1)to derive a representation of the model that we call the observable structure,0∑t S x =  .

i= −i t +it(6.5)

Equation (6.5) is a structural representation of the model because it is driven by the structural
disturbance, t ; the coefficient matrix S0 contains the contemporaneous relationships among theelements of xt . It is an observable representation of the model because it does not contain
unobservable expectations.
Computing the Model Shocks
Consider concatenating the n by n coefficient matrices Si , ordered left to right from i = −τ to 0. We
denote this n by n × (τ + 1) matrix SCOF. Define the vector stack of the endogenous variables attime t as =X t [ xt −t , …, xt ]′ . Thus equation (6.5) may be rewritten(6.6)
SCOF X t = t
In computing the residuals, it will be useful to partition SCOF as follows. Denote stochastic
equations by the subscript s, identity equations by the subscript i, and denote data variables with thesubscript d and “not-data”' variables (such as the unobserved ex ante long real rate) with the subscript
n. Arbitrarily ordering the observable structure so that stochastic equations appear in the top rows
and data variables in the left columns of each block, we can write equation (6.6) as X t −1 S s ,d S s ,n  
 S s ,t −1
 =  t 
(6.7)
S

Si ,d Si ,n   X d ,t   0 
 i ,t −1 X n ,t 
S s ,t −1 denotes the coefficient block of SCOF for the lagged variables that enter the stochastic

equations; Si ,t −1 is the corresponding block for identity equations. The right-hand-most n by nblock of equations, representing the coefficients on contemporaneous variables, is further
partitioned vertically into its data and not-data components. For each observation t, we use thisconcatenated, partitioned version of the observable structure to solve for the residuals t . First,
solve for the period-t not-data variables as
X n ,t =− Si−,n1[ Si ,t −1 X t −1 + Si ,d X d ,t ](6.8)
Now substitute the solution for X n ,t into the top rows of equation (6.7) to solve for t :

t = S s ,t −1 X t −1 + S s ,d X d ,t − S s ,n Si−,n1[ Si ,t −1 X t −1 + Si ,d X d ,t ](6.9)
Equation (6.9) corresponds to the filter equation (it is a specific filter equation for the problem
outlined in this appendix) used in standard implementations of likelihood-based and Bayesianestimation. The residuals for each time period t= 1, …, T are computed.
One may also define the non-stochastic version of the system in (6.7) as

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 X −1 
=0 , S − S x0 S n0  
(6.10)
 X n ,t 
 X x ,t 
where the partition distinguishes between exogenous and endogenous variables, and the superscripts“-” and “0” denote all lagged observations and the contemporaneous observation, respectively. Thispartitioning is useful for solving for any of the endogenous variables in terms of lagged data andexogenous variables. 21 If the model (at current parameter settings and lagged data) implies that theconstrained variable should be less than zero, then the zero lower bound constraint would beviolated.
When one of the variables in the vector X is constrained to be strictly non-negative, as in thecase of a policy rate that faces a zero lower bound (or an effective lower bound), we modify thecomputation of the residuals so as to impose the zero lower bound as follows. First, solve for the
residuals in period t as above. Now compute the solution values for the data variables in X using
equation (6.10) asXˆ t = -P_(S0x x x +S- x - ),P_ ≡ ( S n0 S n0 ') -1 S n0 'which follows directly from the partition in equation (6.10). When the zero-bound-constrained
element of Xˆ t is less than zero, we impose the zero lower bound on this variable by inserting theequation X tk, zlb = 0 into the H matrix of equation (6.1) in place of the original equation for thatvariable in equation (6.7). 22 We then re-solve for SCOF in (6.6); call this new matrix SCOFZLB , withpartitions exactly as in (6.7). We then re-solve for the residuals conditional on this constraint, using
equation (6.9) with the appropriate partitions of the SCOFZLB matrix. This generates a new set ofresiduals that are consistent with imposing the zero lower bound on the variable of interest.
Using this method, we compute the log-likelihood and the posterior for the model toestimate parameters that maximize the log posterior. The simulated posterior distributions impose
the zero lower bound in precisely the same way.

21 Here, we take only the constant variable to be exogenous.
22 That is, we set the k,t n + k th element of H s ,d = 1 , and the other elements of the kth row of H s ,d = 0 .
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