Real-Time Estimate of the Equilibrium Real Interest Rate:
Evidence from Japan
Shingo Umino 1
Kobe University

1 Introduction
Most central banks in developed countries adopt a short-term nominal interest rate
as a policy rate to stabilize economic activity and financial systems: the Uncollateralized
Overnight Call Rate in Japan and Federal Funds Target Rate in the United States areexamples of such rates. Under such a consensus of monetary policy rates, when the central
bank sets a target rate in real-time monetary policy, the real equilibrium interest rate (ERR),
which is the real interest rate consistent with output equal to the potential level, can be
regarded a benchmark rate among monetary policy rates 2. When the actual real rate isequal to the ERR, the actual output is coincident with the potential output and prices
become stable. For central banks, if they raise their policy rates, then the actual real rate
exceeds the ERR and the economy contracts. However, if they decrease their policy rates,
then the actual real rate is below the ERR and the economy is stimulated upward. Therefore,
for central banks that can control the actual real rate, perceiving the real-time ERR issignificant for stabilization of economic activity and prices. Therefore, one can readily
imagine that the real-time ERR reflects the economy of the moment and that it varies overtime. Nevertheless, many studies which have conducted monetary policy analysis haveshown that the ERR is assumed to be as constant as the long-term equilibrium rate. In stark
contrast, some evidence presented herein shows that the ERR has a character of sufficient1
2

E-mail: shingo.umino@gmail.com
Some authors refer to the ERR as the natural rate of interest (NRI) in the same meaning. Wicksell (1896)

presented a pioneer study of both the ERR and NRI. Blinder (1998) differently names the ERR as the neutral
rate.

time-variation to the degree that it is difficult to estimate it accurately 3.
As previous works have discussed, it is difficult to measure the true ERR.
Nevertheless, many researchers continue to study estimation of the ERR because estimating
the ERR is believed to enable the sensing of economic conditions and support theappropriate conduct of monetary policy. For instance, the central bank might consider the
reliable real-time ERR as a benchmark rate to stabilize the economy when conducting
monetary policy in real time. However, as Orphanide and van Norden (2002) proposed
when they estimated the real-time output gap in the United States, the monetary authority
might confront two uncertainties from available data, such that the ERR is discredited inreal time. First, the data that are used in estimating the ERR are often revised. Therefore,
the estimated ERR using real-time data differs from the second ERR using revised data forthe same time. Secondly, even if the data are not revised, additional data become available
in a subsequent quarter. Therefore, the previously estimated ERR might vary according to
the acquisition of new data 4. These possibilities imply that re-estimation of the ERR
perturbs the central bank. Consequently, it cannot conduct monetary policy appropriately5.
Orphanide and van Norden (2002) reported that uncertainty in measuring the output gap iscaused by data revision and accumulation, especially real GDP data. Kamada (2009) also
examined the uncertainty of the Japan’s ERR caused by the revision of real GDP 6.
However, the uncertainty that occurs because of variation of the ERR fromacquisition of additional data over time is unavoidable. Nevertheless, it is necessary tocontrol the uncertainty by which the variation of the ERR is caused by the revision of data.
As is well known, from Faust et al. (2005), the feature of the Japan’s data revision, that is,
the reason why the preliminary GDP data differ from the true value is ’noise.’ And, when
one estimates the true value of GDP as the latent variable from the featured GDP data, thestudies by Mankiw et al. (1984) and Mankiw and Shapiro (1986) revealed that the filtering
approach is useful. Hence, the estimation model for the ERR as a latent variable is applied
to the Kalman filter. Our main goal in this paper is to obtain credible estimates of the ERR
3For the Japanese case, see Oda and Muranaga (2003) and Kamada (2009). For the United States, see

Laubach and Williams (2003), Clark and Kozicki (2004) and Trehan and Wu (2007).
4

The latter is generally regarded as the end-of-sample problem.

5

Of course, Orphanide and van Norden (2002) also present a third uncertainty: model uncertainty.

6

Real GDP data entail several estimates in Japan: the first preliminary (quick) estimate, the second

preliminary (quick) estimate, the semi-final estimate and the final estimate.

and other latent variables using a conventional method. Indeed, few methods exist tomeasure the ERR yet. Consequently, from some candidate models, it is not productive toselect the best estimation model that holds the smallest uncertainty. Instead, it is necessary
to incorporate into a conventional model those ideas and artifices that reduce uncertainty of
the ERR estimates and which improve the credibility of the ERR estimates. Therefore, thispaper specifically applies the method proposed by Laubach and Williams (2003) (LW)model 7.
The main objective is divided into two more parts. First, to ascertain the degree of
uncertainty by revising GDP data, the uncertainties generated by an original LW model areevaluated when the ERR is measured in real time. Using a different method from that
presented herein, Kamada (2009), based on the classification of four uncertainties by
Orphanides and van Norden (2002), reported that the uncertainty of the data revision is afraction of 1% in the estimated ERR. This might mean that this uncertainty causes a policy
misunderstanding under the nominal zero lower bound in Japan. However, in Kamada(2009), the greatest uncertainty was the end-of-sample problem. These results wereidentical to those presented by Orphanides and van Noden (2002), which evaluated theuncertainties in the US real-time output gap using various methods. Moreover, the result
reported by Kamada (2005) after examination of Japan’s real-time output gap resembled
that obtained by Orphanides and van Norden (2002).
Second, data used herein are reviewed based upon hints offered by Kamada (2005,2009) to reduce their associated uncertainty. The baseline LW model is modified if
necessary. Furthermore, the uncertainties of the modified LW model are evaluated using
Kamada’s conceptualization. In real-time estimation of the ERR, Kamada (2009) proposed
the use of the revision-free data instead of real GDP data. There has not been such a
proposition to reduce uncertainty into the ERR in previous studies. The usual GDP data
must estimate the latent variables as the potential output or the ERR. However, when theordinary GDP data is not used, the revision-free data are expected to hold the characteristicthat the data reflect the level of aggregate demand or the output gap. Fortunately, fullrevision-free data for Japan are readily accessible, for example the Short-Term Economic
Survey of Enterprises in Japan (TANKAN).
7

As methods to measure the ERR, a simple Hodrick–Prescott (HP) filter to estimate the trend of actual real

rate is also available. Meanwhile, the ERR can be shown from a dynamic stochastic general equilibrium(DSGE) model. See Neiss and Nelson (2003), Smets and Wouter (2004) and Edge et al. (2007).

Results of this study can be summarized as follows. As a report by Kamada (2009)and other reports of previous studies have described, the real-time estimate obtained fromthe baseline LW model includes considerable uncertainty because of the inconsistency
between vintage data and updated data for the real GDP. Moreover, the baseline modelshows that the components used to construct the ERR amplify the uncertainty. Therefore, toreduce this uncertainty attributable to the estimates, the baseline LW model was modified.
Then the usage of data that can be revised was reviewed. Consequently, the real-time ERR,
as estimated by the modified LW and the usage of a new data, can be shown to have lower
uncertainty than that of the baseline model. In general, only a few ERR estimation methods
exist. Nevertheless, without eliminating the idea of the baseline model, a modified
approach that reduces the uncertainties, which are an obstacle to policymakers, can bepresented.
The remainder of the paper is organized as follows. Section 2 presents a descriptionof the baseline Laubach–Williams model. Section 3 presents evaluation of the uncertainties
from the baseline LW model in real time and presents some issues for consideration.
Section 4 presents the revision-free data and the modified models, with comparison of theuncertainties from the modified model with those of the baseline model. Section 5
concludes this presentation.

2 ERR estimation model
2.1 Baseline LW model
The equilibrium real interest rate (ERR) is estimated using a model resembling thatof Laubach and Williams (2003). The dynamics of the output gap that are expressed as theIS equation are a backward-looking formulation.
yt − yt* = a y ,1 ( yt −1 − yt*−1 ) + a y ,2 ( yt − 2 − yt*− 2 )−[(r − r ) + (r − r )]+ ε ,
2

ar ,3

t −1

*
t −1

t −2

*
t −2

y ,t

where yt is the real logarithm of GDP and yt* is the potential output. Consequently, the(1)

output gap yt − yt* is defined as the difference between yt and yt* . rt is the real
uncollateralized call rate and rt* denotes the equilibrium real interest rate. Similarly, thereal interest rate gap is expressed as the difference between rt and rt* . Therefore, an IS
equation is constructed from the lags of the output gap, the lagged real interest rate and aserially uncorrelated shock.
Second, the AS equation based on the Phillips curve is presented as follows.
4b
1 − bπ ,1 − bπ ,2 8
π t = bπ ,1π t −1 + π ,2 ∑π t −i +π t −i
∑
3 i=2
4
i =5
+ bπ ,3 ( yt −1 − yt*−1 ) + bπ ,4π tl + bπ ,5π to−1 + ε π ,t .

(2)

The current inflation π t is explained by the first through eighth lagged inflation. The sumof all coefficients of the terms of inflation is assumed to be one. Moreover, two restrictions
are imposed on the inflation terms in the AS equation. First, the coefficients of the second
through fourth lagged variable are assumed to be equal. Similarly, the coefficients of thefifth through eighth lagged variable are assumed to be equal. Gordon (1998) and Laubach
and Williams (2003) are followed to set such restrictions. Two variables are set: core
import price inflation π l (excluding petroleum, computers and semiconductors) andlagged crude imported oil price inflation π o . Relative oil price shocks are measured using
them. Finally, in addition, AS equation consists of the lagged output gap and a serially
uncorrelated error.
As described by optimal growth theory and New Keynesian theory, ERR is usuallycorrelated with the growth rate trend. It also depends upon the household’s rate of time
preference. In this study, therefore, the form of the ERR is defined as follows.
rt* ≡ c ⋅ g t + zt .

(3)

In that equation, g t signifies the annualized trend growth rate of real output, and zt isdefinable as the factor that influence on household’s consumption demand. Therefore, zt
is regarded as the demand component, for which, if it holds high persistency, the process
shown below follows.
zt = zt −1 + ε z ,t(4)

Consequently, zt is set as a random walk process.
Finally, the data generating processes of the potential output and the trend growthrate are given as

yt* = yt*−1 + g t + ε * ,(5)g t = g t −1 + ε g ,t .

(6)

y ,t

Therein, g t is defined as the annualized trend growth rate of the potential output.
Furthermore, ε *y ,tand ε g ,t are assumed as serially uncorrelated and contemporaneously

uncorrelated error. Our formulation of the trend growth rate is given by Kuttner (1994) andis adjusted by many previous studies. In this form, the trend growth rate is assumed to vary
at small variance every quarter because it is difficult to assume that the trend growth isconstant.
The baseline LW model includes some unobservable variables. This model isdesignated as an unobserved-component (UC) model. A Kalman filter is useful to estimate
the UC model and to generate unobserved data. This paper presents the stat-space
representation of baseline LW model and applies Kalman filtering to this representation.

2.2 Estimation results
The estimate of ERR for Japan and the data used are shown as quarterly from
1980Q3 to 2010Q2 in Japan. The logarithmic real GDP (seasonal adjusted) is used as themeasure of output and consumer price index (excluding fresh food, seasonal adjusted) as
the measure of prices. Expected inflation is calculated from the forecast of the univariate
AR(3) model. The nominal interest rate is the quarterly average of the overnight
uncollateralized call rate, expressed in units of percent per year. The price index of non-oil
imports is not available in Japan. The imported price index is applied to π l and π o .
As Laubach and Williams (2003) described, the maximum likelihood estimates ofthe standard deviations of the innovations to the trend growth rate g t and the demand
components zt , σ g and σ z are likely to be biased toward zero. This phenomenon is the‘pile-up’ problem that Stock (1994) and Stock and Watson (1998) pointed out. To addressthis problem, this study uses the medium unbiased estimator suggested by Stock andWatson (1998). Consistent estimates of the ratios, λg ≡ σ g /σ y* and λz ≡ ar ,3 (σ z /σ y ) , areobtained in the first estimation. In the final estimation, these restrictions are imposed toobtain the maximum likelihood estimation.
Table 1 presents estimation results of the baseline model parameters for different
model specifications. In all cases, the coefficients relating the output gap to the real rate gap,
ar ,3 , are negative and statistically significant. The second and third columns report resultsin the models in which the baseline LW model is modified. Especially in the third column,
the coefficient ar ,3 is quite small compared to those reported by Kamada (2009) in Japanand Clark and Kozicki (2005) in the US, which were obtained assuming the same model.
However, when the baseline and a different specification model are estimated in a
sub-sample from 1999Q1, even then Japan’s nominal interest rate remains at around zero,
which confirms that each estimation result of model parameters is robust.

3

Evaluation of uncertainty

3.1 Definitions of uncertainty
The uncertainty caused by data revision and accumulation can affect the action ofa central bank. In this context of the output gap, Orphanides and van Norden (2005) define
output gap estimates of four kinds: real time, quasi-real, quasi-final and final output gaps.
In this case of the ERR, the ERR is estimated using only the vintage of data available foreach point in time. Consequently, the real-time ERR is obtained. Next, when the
counterfactual assumption is made that the ERR at period t is calculated using the fullsample data series 1 through T (> t ) , the quasi-real ERR is estimated. For the quasi-final
ERR, another counterfactual assumption is made that the ERR at period t is calculatedusing the estimated model parameters at period T and the final data series. Then, thequasi-final ERR is calculated. The final ERR is a series that is estimated from all data inperiod t . It is used in the usual analysis. Moreover, for UC models, the quasi-final ERR isa filtered estimate, whereas the final ERR is a smoothed estimate.

As discussed in some previous reports of studies, three uncertainties are defined
through each difference in the above four ERR estimates. First, the difference between thereal-time and quasi-real ERR estimates is attributable to the effect of data revision. This
study refers to the effect of the first uncertainty as Error 1. Second, the difference between
the quasi-real and quasi-final ERR estimates is attributable to the effect of model
parameters revision. This study specifically examines the effect of the second uncertainty
as Error 2. Finally, the difference between the quasi-final and final ERR estimates isgenerally attributable to the effect of the sample-end problem. This study refers to the effectof the third uncertainty as Error 3.

3.2 Results
Complete real-time GDP data for Japan were unavailable. A dataset such as the US
real-time dataset by the Federal Reserve Bank of Philadelphia is unfortunately not
published as historical data. Therefore, the first published real GDP data were used, that is
the first quick estimate, from the Nikkei (Nihon Keizai Shinbun). Table 2 below presents
those results:
1. Error 1 in most estimated state variables is larger than Error 2 or 3.
2. Error 1 in the potential output that is apparently influenced directly by therevision of GDP is noticeable.
3. Two components of which the ERR consists, ‘ g ’ and ‘ z ’, are apparently
sources of high uncertainty in the ERR.
4. Because of the data-generating processes of g and z , the uncertainty of the
ERR can be amplified to a greater degree than necessary.
Results (a) and (b) present that the real-time latent variables are affected by therevision of real GDP. From the first column of Table 1, the uncertainty of the ERR (Error
1) is 0.58%. Considering the situation of long-run low interest rate in the estimate period,
this value is large. Similarly, Error 1 in the potential output is larger than other errors.
However, Error 1 of the ERR is larger than that of the potential output. Moreover, the twocomponents, ‘ g ’ and ‘ z ’, that must be estimated first to calculate the ERR have respective
uncertainties that are greater than 1%. Comparison of the results obtained by Kamada

(2009) must be done in light of an important point: the results obtained in this study reflect
Japan’s long-run slump after the bubble burst, whereas Kamada (2009) includes the bubble
period. Furthermore, it is noteworthy that results reported by Kamada (2009) differ withthose of this study in terms of the use of real-time data.

3.3 Implications
According to the results described above, it is necessary to reduce the uncertainty of
ERR in real-time estimate as summarized in the following two points: (1) rethinking datathat have a possibility of revision and (2) reviewing the data generating processes of ‘ g ’
and ‘ z ’.
The best means to reduce uncertainty is to avoid using data that have a possibility ofrevision, but this way of measuring the ERR is not feasible. Data that have a possibility ofrevision, such as real GDP, are important to estimate latent economic indicators. Therefore,
if real GDP is not used, then data must be prepared that are not revised and which haveinformation about current economic activity such as real GDP. Kamada (2005, 2009)
reported such data in the context that it remedies the real-time output gap. Suchsupplementary data are taken from the Short-term Economic Survey of Enterprise in Japan,
TANKAN. The TANKAN aims at contributing to the appropriate implementation ofmonetary policy by capturing business trends of enterprises accurately. Kamada (2005,2009) also suggests the use of a Business Conditions diffusion index (DI) of 10 items in the
TANKAN. This DI indicates judgment of general business conditions of the responding
enterprise in light of individual profits.
However, this DI is not applied for this study. Instead, the weighted DI is proposed.
It is frequently seen on the minutes of monetary policy meetings because this DI is superior
to the business condition DI in that a DI that reflects the output gap is desired here. The
weighted DI actually consists of two DIs:
the weighted DI = (1 − α ) × Production Capacity DI + α × Employment
Condition DI,
In that equation, α is a labour income share 8. The Production Capacity DI is a8

It is assumed to be 0.64.

judgment of excessiveness, adequacy, or shortage of production capacity or businessequipment of the responding enterprise 9. The Employment Conditions DI is a judgment ofexcessiveness, adequacy, or shortage of the number of employees at the responding
enterprise 10.
When the revision-free data are used and the processes of both ‘ g ’ and ‘ z ’ arereconsidered, it is necessary to modify the baseline LW model. Herein, the following two
versions of the modified LW model are proposed:
Version 1 is to use both the weighted DI and ordinary GDP data, and toreconsider both ‘ g ’ and ‘ z ’ processes.
Version 2 is to use the weighted DI only and to reconsider both the ‘ g ’ and ‘ z ’processes.

4 Modifications and re-evaluation
First of all, let me consider the relation between the TANKAN and e noisef before
applying the TANKAN to the ERR estimation model. Publicly, the real-time data set has
not yet been published in Japan. So, in this paper, I gathered both the first preliminary andsecond preliminary GDP published in 2006Q1-2011Q2 (benchmark year 2000). I ensured
that the GDP data published first at some point takes around three years to be the stable
number. Therefore, if the GDP published three years later is the true one, the difference
between each revised GDP and the three-years-later GDP is regarded to be noise. Figure 1
illustrates the relation between noise and TANKAN. The figure displays a positive
correlation. Namely, I ensure that TANKAN is helpful to estimate the true output gap.
Based upon the proposition of the two types of baseline LW model in the previous
section, for Version 1, the two measurement equations of the baseline model are retainedwith no change and modify the transition ones only. This model uses both data that have a
possibility of revision (ordinary real GDP) and non-revised data (the weighted DI). As
discussed in the preceding section, because the published weighted DI already includes a9This DI excludes a shortage caused by temporary conditions such as a factory closure for regular repairs.

10

the Business Condition DI and the Employment Condition DI are available from 1980Q1. However, theProduction Capacity DI is available from 1990Q1. Consequently, the business condition DI 1980Q3 through
1989Q4 is used. Subsequently, the weighted DI is adopted.

gap form, it plays a role as a guide of the dynamics of the output gap, as
rt* = 4c( yt* − yt*−1 ),(7)( yt* − yt*−1 ) = fxt + ε * ,

(8)

y ,t

where variable xt corresponds to the weighted DI in the data series and ε *

y ,t

is a

serially uncorrelated error. This model is designated as the Version 1 model, which canestimate the potential output simultaneously.
However, when the weighted DI is fully reviewed as an alternative measure of theoutput gap, the weighted DI can be substituted for the output gap y − y * in Eq. (1). Thenthe baseline LW model transforms into a very simple model. The Phillips curve, Eq. (2),drops from the baseline LW model and the ERR is simply a random walk process, as in
specifications explained by Clark and Kozicki (2005).
xt = a x ,1 xt −1 + a x ,2 xt − 2 +[(r − r ) + (r − r )],
2

ar ,3

t −1

*
t −1

t −2

*
t −2

rt* = rt*−1 + ε * ,

(10)

r ,t

Therein, the variable xt which corresponds to the weighted DI in the data series is theoutput gap and ε *r ,tis a serially uncorrelated error. This model is designated as the

Version 2 model. Unlike the Version 1 model, the Version 2 model can not estimate thepotential output simultaneously.
Table 3 shows that most uncertainties in the ERR and output gap become smaller
than those in the baseline LW model 11. For the Version 1 model, the ERR’s uncertainty
caused by the data revision (Error 1) is decreased by 85% relative to that obtained using
the baseline model. In addition, uncertainty in the output gap drops to a much lower value.
The sample-end problem in both ERR and output gap (Error 3) gets low. From results
obtained for the Version 2 model, of course, because real GDP is not used, the uncertainty
by the data revision does not rise. Instead, the Error 3 increases by 2.6%.

11

The case of the United States could not be examined because no statistical survey exists similar to the

TANKAN in US.

(9)5 Conclusion
Although knowledge of the usefulness of the ERR is widely shared, the problem of
the real-time estimation that Orphanides and van Norden (2002) pointed out also arises in
estimating the ERR when the central bank attempts to capture the dynamics of the ERR inreal time using all available data in time. Results of this study showed that the real-time
estimate of the ERR by LW model includes several types of uncertainty that arise throughprocesses of data revision and accumulation. Especially, the effect of the data revision is
not negligible in the context of a zero bound interest rate, and the model specification
amplifies such effects. However, to reduce the uncertainties of the real-time ERR estimate,when the revision-free data (the weighted DI for TANKAN) were used as an indicator of
the output gap and the LW model was modified appropriately, most uncertainties in themodified LW models became lower than those of the baseline LW model.

Acknowledgements
I would like to thank Toshiki Jinushi, Yoichi Matsubayashi, Shigeto Kitano,
Teruyoshi Kobayashi, (Kobe University) and Masashi Saito (BOJ). I would like to thankSusanto Basu (Boston College) especially for his valuable comments. All errors are theresponsibility of the author.

References
Blinder, A. S. (1998). Central Banking in Theory and Practice, MIT.
Edge, R. M., Kiley, M. T. & Laforte, J.-P. (2008). Natural rate measures in an
estimated DSGE model of the U.S. economy. Journal of Economic Dynamics and Control,
32, 2512–2535.
Faust, J., Rogers, J. H. &Wright, J. H. (2005). News and Noise in G-7 GDP

Announcements. Journal of Money, Credit and Banking, 37, 403–419.
Gordon, R. J. (1998). Foundations of the Goldilocks economy: supply shocks andthe time-varying NAIRU. Brookings Papers on Economic Activity, 2, 297–333.
Hodrick, R. J. & Prescott, E. C. (1997). Post-War U. S. Business Cycles: An
Empirical Investigation. Journal of Money, Credit and Banking, 29, 1–16.
Kamada, K. (2005). Real-Time Estimation of the Output Gap in Japan and its
Usefulness for Inflation Forecasting and Policymaking. North American Journal of
Economics and Finance, 16, 309–332.
Kamada, K. (2009). Japan’s Equilibrium Real Interest Rate, Macroeconomy andIndustrial Structures, (K. Fukao eds.), Keio University Press. (in Japanese)
Kozicki, S. & Clark, T. (2005). Estimating equilibrium real interest rates in realtime. North American Journal of Economic and Finance, 16, 395–413.
Kuttner, K. (1994). Estimating Potential Output as a Latent Variable. Journal ofBusiness and Economic Statistics, 12, 361–368.
Laubach, T. and Williams, J. C. (2003). Measuring the Natural Rate of Interest.
Review of Economics and Statistics, 85, 1063–1070.
Mankiw, N. G., Runkle, D. E. & Shapiro, M. D. (1984). Are Preliminary
Announcements of the Money Stock Rational Forecasts? Journal of Monetary Economics,
14, 15–27.
Mankiw, N. G. & Shapiro, M. D. (1986). News or Noise? An Analysis of GNP
Revisions. NBER Working Papers, 1939.
Neiss, K. S. and Nelson, E. (2003). The real interest rate gap as an inflation
indicator. Macroeconomic Dynamics, 7, 239–262.
Oda, N. & Muranaga, J. (2003). On the Natural Rate of Interest: Its Theories andEstimates. Bank of Japan Working Paper Series, No. 03-J-5. (in Japanese)
Orphanides, A. & van Norden, S. (2002). The Uncertainty of Output-Gap Estimatesin Real Time. Review of Economics and Statistics, 84, 569–583.
Orphanides, A. & Williams, J. C. (2002). Robust monetary policy rules withunknown natural rates. Brookings Papers on Economic Activity, 2, 63–146.
Smets, F. & Wouters, R. (2003). An Estimated Dynamic Stochastic General
Equilibrium Model of the Euro Area. Journal of the European Economic Association, 1,1123–1175.
Stock, J. (1994). Deciding Between I(1) and I(0). Journal of Econometrics, 63,

105–131.
Stock, J. & Watson, M. (1998). Median Unbiased Estimation of Coefficient
Variance in a Time Varying Parameter Model. Journal of American Statistical Association,
93, 349–358.
Trehan, B. & Wu, T. (2007). Time-varying equilibrium real rates and monetary
policy analysis. Journal of Economic Dynamics and Control, 31, 1584–1609.
Wicksell, K. (1936). Interest and Prices, (tr. of 1898 edition by Kahn R. F.),
Macmillan.

Table 1: Parameter Estimatesbaseline LW
model
Estimates

t -value

Estimates

t -value

Estimates

t -value

0.660

2.508

1.025

26.7460.986

16.484

0.285

1.083

0.007

6.571

0.005

1.357

-0.114-2.253-0.006-8.665-0.015-4.214

0.688

5.516

1.305

0.571

0.109

0.595

0.198

0.146

1.026

2.078

0.047

4.590

0.008

1.006

0.039

0.857

0.006

0.883

-0.017-0.484

0.948

0.795

0.681
0.060

3.593
1.786

σy

0.948

2.472

0.362

2.880

0.413

9.063

σπ

0.619
0.846

4.990
9.940

0.491
0.233

6.526
4.461

Parametersa y ,1
a y ,2
ar ,3
bπ ,1
bπ 2
by ,3
bm
π

,4

bm

,5

π

c

Version 1

f

σ y*

σg

0.177

σz

0.886

σ r*

Version 2

0.552

MUE

λr* = 0.020

λg = 0.210

λz =0.224
Note: λg = σ g /σ y* , λz = ar ,3 (σ z /σ y ), λ * = ar ,3 (σ * /σ y ) .
r

All estimates are for 1980Q3–2010Q2.

r

Table 2: Evaluation of the uncertainty on the baseline LW model.

*

r
g
z

y− y

*

Error 1

Error 2Error 3

0.584

0.022

0.739

1.363

0.041

0.515

1.377
0.299

0.016
0.015

0.603
0.268

Note: All numbers are root square means of the revision series shown. All statistics are for
1993Q1–2010Q2.

Table 3: Evaluation of uncertainty of the modified models.

*Version 1ry − y*

Version 2*r

Error 1

Error 2Error 3

0.083

0.208

0.258

0.0002

0.002
0.069

0.258
2.602

Note: All numbers are root square means of the revision series shown. All statistics
are for 1993Q1–2010Q2.

Figure 1: Preliminary data revision and the weighted TANKAN DI.