Discussion Paper No. 379

Simple Estimations of the Natural Rate of Interest andthe Expected Inflation Rate
Kazuhiko NAKAHIRA
Professor, Department of
Economics, Meikai University
December

2022

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

(Preliminary Working Draft: Please do not cite or quote without permission of the authors.)

Simple Estimations of the Natural Rate of Interest and the Expected Inflation Rate

Kazuhiko NAKAHIRA
Professor, Department of Economics, Meikai University
E-mail: nakahira@meikai.ac.jp

Abstract
The natural rate of interest is an important concept since it is a kind of reference for economists in that
this is a significant determinant of a variety of policy rules used to characterize monetary policy. Generally,
the natural rate of interest is defined as the real short-term interest rate consistent with the output equaling
its potential level or the one reflecting stable prices. Estimation of the natural rate of interest is requiredsince it cannot be observed directly. One of the simple ways of estimating the natural rate of interest is to
extract the trend of the real short-term interest rate. If the central bank is inclined to raise the policy interest
rate when it acknowledges an inflationary pressure or excess demand and if such a policy implementationis smoothly transmitted into adjustment processes of prices and economic activities, observed short-term
real interest rate should behave closely around the evolutionary pass of the natural rate of interest. In thissense, statistically extracted trend of real short-term interest rate can be presumed as the estimated valueof the natural rate of interest. We utilize the Hodrick-Prescott filter for our extraction. On the other hand,
we should have the estimation of expected inflation rate in order to acquire real short-term interest rate if
we follow the Fisher equation. In this paper, Kanoh (2006) procedure is adopted to estimate the expected
inflation rate. Our estimation result brings us to the conclusion that the natural rate of interest in Japan is
likely to show a substantial decline in recent 20 years.

Key Words: natural rate of interest; inflation expectations; Carlson-Parkin method; Hodrick-Prescott filter
JEL Classification Code: C53; E31; E32, E37

11. Introduction
The natural rate of interest is an important concept for economists since it is one of the references formonetary policy in that this rate is a significant determinant of a variety of policy rules used to characterizemonetary policy. Generally, the natural rate of interest is defined as the real short-term interest rateconsistent with the output equaling its potential level or the one reflecting stable prices. Estimation of thenatural real interest rate is required since it cannot be observed directly.
Because of this property, positive previous studies with regard to its theoretical analysis and empiricalestimation have been published. The most famous research of this kind is Laubach and Williams (2003).
This study uses the IS curve, the Phillips curve, and the natural rate of interest equation as the key equations
that describe the structural relationships of economy, and applies the Kalman filter technique in order toestimate some unobservable variables. Fries et al. (2016), Pescatori and Turunen (2016), Hakkio andSmith (2017), Holston et al. (2017), and Lewis and Vazquez-Grande (2017) are the other studies based on
the Laubach and Williams approach. On the other hand, Lubik and Matthes (2015) adopted the TimeVarying Parameter Vector Autoregressions approach.
Another commonly used instrument for analyzing the natural rate of interest is the so-called DSGE
(dynamic stochastic general equilibrium) model that clarifies structural interpretations of the system. Edgeet al. (2008), Justiniano and Primiceri (2010), Barsky et al. (2014), Curdia et al. (2015), Del Negro et al.
(2015, 2017), Goldby et al. (2015), Hristov (2016), Gerali and Neri (2017), Okazaki and Sudo (2018), andIiboshi et al. (2022) are the studies based on DSGE methodology.
After the remark by Summers (2013), the issue whether the natural rate is substantially been declining(and will continue to decline) has been attracted attention. We found the researches that are related to thisissue or the secular stagnation hypothesis, including Eggertsson et al. (2017), Rachel and Smith (2015),
Sajedi and Thwaites (2016), and Summers (2014).
One statistical simple method of estimating the natural rate of interest is to extract the trend componentof real short-term interest rate. If the central bank is inclined to raise the policy interest rate when it
acknowledges an inflationary pressure or excess demand and if such a policy implementation is smoothlytransmitted into adjustment processes of prices and economic activities, short-term real interest rate should
behave closely around the evolutionary pass of the natural rate of interest. In this sense, statisticallyextracted trend of real short-term interest rate can be regarded as the estimated value of the natural rate ofinterest. We utilize the Hodrick-Prescott filter for our extraction. On the other hand, we should have the2estimation of expected inflation rate in order to acquire real short-term interest rate if we follow the Fisher
equation. In our research, Kanoh (2006) procedure, a variant of the Carlson and Parkin (1975) method, isadopted to estimate the expected inflation rate.
The contribution of our paper is as follows. We utilize the Kanoh (2006) procedure, hardly ever usedin empirical research, for the empirical research when the expected inflation rate is estimated in order toacquire real short-term interest rate, and it uses the inference of the real rate of interest through the HodrickPrescott filter. Through its analysis, we found a substantial decline of Japan’s natural rate of interest in
recent years.
The rest of this paper is organized as follows. Section 2 presents the theoretical derivation of the naturalrate of interest. Section 3 summarizes the way of estimation of expected inflation rate through the CarlsonParkin (1975) and the Kanoh (2006) procedures. Section 4 sets out the empirical investigation of thenatural rate of interest by utilizing the Hodrick-Prescott filter. Lastly, Section 5 is devoted to the concluding
remarks.

2. Theoretical Derivation of Natural Rate of Interest1
The utility function is given by∞U = ∫ (𝑢(𝑐)𝑒 −𝜌𝑡 𝑑𝑡) , 𝑢 ′ (𝑐) > 0, 𝑢 ′′ (𝑐) < 0.

(1)

0

U: utility (from present to infinite future)
u: utility at each period
c: real consumption at each period of representative agent
𝜌: rete of time preferenceHousehold's Budget Constraint is as follows.
𝑘̇ = 𝜔 + (𝑟 − 𝑛)𝑘 − 𝑐.

(2)

𝑘: capital stock per capita
n: population real growth rate
𝜔: wage rate
r: real interest rate

1

The explanation in section 2 is basically based on Murata (2022) and Iwata et al. (2018).

3

By the equations (1) and (2), we have the Hamiltonian:H = 𝑢(𝑐)𝑒 −𝜌𝑡 + 𝜆[𝜔 + (𝑟 − 𝑛)𝑘 − 𝑐].

(3)

𝜆: shadow price of income
F.O.C.
𝜆 = 𝑢 ′ (𝑐)𝑒 −𝜌𝑡(4)𝜆̇ = −(𝑟 − 𝑛)𝜆(5)Transversality condition is expressed bylim [𝜆(𝑡)𝑘(𝑡)] = 0.

(6)

𝑛→∞

Taking the time derivative of (4) and plug it into (5), we get𝑢′′ (𝑐)𝑐𝑟 = −[𝑢′ (𝑐)

𝑐̇] ( ) + 𝑛 + 𝜌.

(7)𝑐

If we assume constant substitution elasticity of consumption with respect to the utility function,𝑢(𝑐) =𝑐 (1−𝜃) −1

(8)

1−𝜃

is realized, and equation (7) should be the following form:𝑐̇𝑟 = 𝜃 ( ) + 𝑛 + 𝜌.

(9)𝑐Since per capita consumption growth rate should be equal to per capita capital stock growth rate whenthe economy is on the balanced growth path, and we have:
𝑐̇

𝐾̇

𝑐

𝐾

( ) = ( ) − 𝑛.

(10)

𝐾: capital stock
Production function (homogeneous of degree one) is
𝑌 = 𝐹(𝐴𝐿, 𝐾) = 𝐴𝐿𝑓(𝑘̃).

(11)

Y: GDP
A: rate of technological progress (or Total Factor Productivity)
L: labor Input
AL: effective labor
Capital per unit of effective labor and GDP per unit of effective labor are expressed as follows.
𝐾
𝑘̃ =

(12)

𝑦̃ =

(13)

𝐴𝐿
𝑌
𝐴𝐿

Capital per unit of effective labor should be constant on the balanced growth path, and considering (10)in addition to (12), we have
𝑐̇

𝐴̇

𝑐

𝐴

( ) = ( ).

(14)

4

If we define the interest rate on the balanced growth path as the natural rate of interest, 𝑟 ∗, we have it byconsidering (10) and (14),𝐴̇𝑟∗ = 𝜃 ( ) + 𝑛

(15)

𝐴

where 𝜃 is the coefficient of relative risk aversion.
Because GDP per unit of effective labor should be constant on the balanced growth path, g: potential
growth rate (or GDP growth rate on the balanced growth path), becomes equal to the growth rate of the
Total Factor Productivity and the population growth rate. Therefore, we have𝐴̇g = + 𝑛.

(16)

𝐴

Furthermore, if we consider a dynasty with population growth, equation (15) becomes𝐴̇𝑟 ∗ = 𝜃 ( ) + 𝜌.

(17)

𝐴

In this case, the relation described below is realized as an approximation if there is no population
growth:𝑟 ∗ = 𝑔 + 𝜌.

(18)

3. Inflation Expectations and its Estimations
3.1. Survey Data for the Estimation
The purpose of our study is to estimate the natural rate of interest. However, we should have theestimation of expected inflation rate in order to obtain the real short-term interest rate by following the
Fisher equation. (In short, the Fisher equation describes that the real interest rate equals the nominal interest
rate minus the expected inflation rate.)
To examine the formation process of inflation through the estimation of inflation expectations by using
survey data enables us to implement our concrete research without any specific economic models. In
General, there are two types of survey data on inflation expectations: qualitative and quantitative. In thequalitative surveys, respondents answer in a qualitative manner to a question such as, “Do you think thatprice level (or inflation) will have gone up (or down) one year from now?” The data on inflation
expectations given by this kind of survey are presented in a qualitative statistic form indicating whether
the majority of the polled respondents expect that price level in the future will rise, remain constant, or
decline. In other words, the survey of this kind explores the general tendency of the expectation of the
future price level or inflation. On the other hand, respondents answer to the question in a quantitative
manner in the case of a quantitative survey. The problem is that it is hard to acquire an exact point forecast5

of the inflation expectation because quantitative surveys may have some defects. For instance, this kind of
direct measure has an inclination to generate some measurement and sampling errors. From this point ofview, it is worth making use of the qualitative surveys along with the special way of quantifying qualitativedata.

3.2. The Carlson-Parkin Method for Quantifying Qualitative Survey Data and The Kanoh (2006)
Methodology
A process of quantifying qualitative survey data is required to examine expectation of inflation as
mentioned in the previous section. The data given by a qualitative survey, however, often brings us tosome difficulties. For instance, the respondents only indicate whether price level (or inflation) will “rise”,“fall” or “remain unchanged” for a certain period ahead in the survey, so the data do not include a meanvalue since they are qualitative. Several techniques, such as the Carlson-Parkin method, the balancemethod, and the regression method, have been proposed to deal with these problems. The Carlson andParkin (1975) method2 is a typical probability approach for the expected inflation estimation. It postulates
that the qualitative answer in the survey follows an individual probability distribution that is statisticallyindependent of other respondents’ and normally distributed with finite mean and variance. Thus, the meanof the distribution is supposed to be reported by the respondent. The Carlson-Parkin methodology assumes
that respondents at time 𝑡 form an inflation expectation for time 𝑡 + 1 in their answers. The joint
probability distribution 𝑓(x 𝑡+1 |Ω𝑡 ) is derived by the aggregation of the individual subjective probability
distributions, where Ω𝑡 is the information set at time 𝑡 and x 𝑡+1 is the percentage change in price level inthe future for the period 𝑡 to 𝑡 + 1. This distribution is presumed to have finite first- and second-order𝑒moments, and we can express the inflation expectation for the period 𝑡 + 1 as 𝐸[x 𝑡+1 |Ω𝑡 ] = 𝜋𝑡+1.

Further, we regard that there exists an interval (−𝛿𝑡 , 𝛿𝑡 ) around 0 (𝛿𝑡 > 0) in order that the answer “nochange” will lies within this interval. With this threshold, 𝛿𝑡 , survey participants are supposed to report the
expectation of price change in the following manner:2𝑒“prices up” if 𝜋𝑡+1
> 𝛿𝑡 ,(19)𝑒“prices down” if 𝜋𝑡+1
< −𝛿𝑡 ,

(20)

The explanation of the Carlson and Parkin (1975) model given here is not exactly the same as the original procedure
but the modified one. The modified version in this section is of wide applications, and introduced by some papers,
for example, Henzel and Wollmershäuser (2006), Oral (2013), and Scheufele (2011).

6

𝑒“no change” if −𝛿𝑡 ≤ 𝜋𝑡+1
≤ 𝛿𝑡 .

(21)

The answer in the survey can be interpreted as the result of the individual probability distribution over the
possible future value of the variable in consideration and as a sampling from some aggregate distribution.
Therefore, the percentage of the responses of “prices up” (denoted by 𝑈𝑃𝑡 ) and “prices down” (denotedby 𝐷𝑂𝑊𝑁𝑡 ,) can be transformed into the associated population values:𝛿 −𝜇𝑡

𝑈𝑃𝑡 = 1 − Ф ( 𝑡

),

(22)

),

(23)

𝜎𝑡
−𝛿𝑡 −𝜇𝑡

𝐷𝑂𝑊𝑁𝑡 = Ф (𝜎𝑡where Ф is the cumulative distribution function of the standard normal distribution, and 𝜇𝑡 and 𝜎𝑡 are
respectively the mean and standard deviation of the aggregate distribution of inflation expectation. By
these two equations, we have
𝛿 −𝜇𝑡

𝑎𝑡 = Ф−1 (1 − 𝑈𝑃𝑡 ) = ( 𝑡

),

(24)

),

(25)

𝜎𝑡
−𝛿𝑡 −𝜇𝑡𝑏𝑡 = Ф−1 (𝐷𝑂𝑊𝑁𝑡 ) = (𝜎𝑡where Ф−1 is the inverse function of Ф. Further, 𝜇𝑡 and 𝜎𝑡 can be described as
𝑎 +𝑏𝑡

𝜇𝑡 = −𝛿𝑡 ( 𝑡

),

(26)

𝜎𝑡 = 2𝛿𝑡 (

),

(27)

𝑎𝑡 −𝑏𝑡1

𝑎𝑡 −𝑏𝑡if we have 𝛿𝑡 . One of the simple ways to have a plausible value of 𝛿𝑡 is to assume constant 𝛿 (i.e. 𝛿𝑡 = 𝛿
for some 𝛿) and∑𝑇𝑡=1 𝜋𝑡 = ∑𝑇𝑡=1 𝜇𝑡 ,(28)

where 𝜋𝑡 is the observed inflation rate. With these assumptions, we have∑𝑇𝜋

𝛿 = − 𝑇 𝑡=1𝑎𝑡 +𝑏𝑡 𝑡 .
∑𝑡=1(

𝑎𝑡 −𝑏𝑡(29))

Plugging this 𝛿 into (26) and (27) instead of 𝛿𝑡 , we obtain 𝜇t (expected inflation) and 𝜎𝑡 (standard
deviation).
It is often insisted that the basic (or traditional) Carlson-Parkin method have some problems. For
example, the thresholds might be asymmetric between the expectations of “prices up” and “prices down”.
In this aspect, Kanoh (2006)3 proposes the procedure that can realize two kinds of threshold by modifying
the basic model: 𝛿1 for “prices up” and 𝛿2 for “prices down.” The modification by Kanoh (2006) is asfollows.
3

Kanoh (2006) brings up several procedures for the inference of inflation expectations. The procedure applied in thispaper is one of them.

7

The respondents are supposed to indicate an expectation of change in prices as:𝑒“prices up” if 𝜋𝑡+1
> 𝛿1 ,(30)

𝑒“prices down” if 𝜋𝑡+1
< 𝛿2 ,(31)𝑒“no change” if 𝛿2 ≤ 𝜋𝑡+1
≤ 𝛿1 .

(32)

For the inferences of the mean and the variance of the expectations, the additional assumption∑𝑇𝑡=1 𝜎𝑡2 = ∑𝑇𝑡=1(𝜋𝑡 − 𝜋̅)2,(33)

where 𝜋̅ is the observed average inflation rate, is required. Then, equations (26) and (27) become𝑎 𝛿 −𝑏𝑡 𝛿1

𝜇𝑡 = ( 𝑡 2𝑎𝑡 −𝑏𝑡𝛿1 −𝛿2𝜎𝑡 = (

𝑎𝑡 −𝑏𝑡

),

(34)

),

(35)

if we assume 𝛿1𝑡 = 𝛿1 and 𝛿2𝑡 = 𝛿2 for constants 𝛿1 and 𝛿2 . After some manipulations, we obtain4
1

𝛿1 = (∑𝑇𝑡=1 𝜋𝑡 + ∑𝑇𝑡=1
𝑇1

𝛿2 = (∑𝑇𝑡=1 𝜋𝑡 + ∑𝑇𝑡=1
𝑇

𝑎𝑡

∑𝑇
̅ )2
𝑡=1(𝜋𝑡 −𝜋

𝑏𝑡

∑𝑇
̅ )2
𝑡=1(𝜋𝑡 −𝜋

𝑎𝑡 −𝑏𝑡 √∑𝑇2
1
𝑡=1(𝑎 −𝑏 )𝑡 𝑡

𝑎𝑡 −𝑏𝑡 √∑𝑇1
𝑡=1(𝑎 −𝑏 )
𝑡 𝑡

2

),

(36)

).

(37)

By the substation of 𝛿1 and 𝛿2 described in (36) and (37) into (34) and (35), we have 𝜇t (expected
inflation) and 𝜎𝑡 (standard deviation).
Kanoh (2006) procedure is one of the variants of the Carlson-Parkin methodology, but it has hardly
been used for the empirical research with a few exceptions.

3.3. Consumer Confidence Survey
The consumer confidence survey in Japan (conducted by the Economic and Social Research Instituteof the Cabinet Office)5 asks the respondents to assess the general situation and expectation about theeconomy6. It is one of the applicable data sources for empirical research with the Carlson-Parkin-type
4

Kanoh (2006) narratively shows the steps of deriving 𝛿1 and 𝛿2 without any concrete calculation processes.
However, with some manipulations applying assumptions and conditions given in his paper, we have equations (36)and (37).
5
See “http://www.esri.cao.go.jp/en/stat/shouhi/shouhi_kaisetsu-e.html” for details.
6
The noticeable points of the consumer confidence survey are as follows. (a) The survey of “price expectations a year
ahead” is conducted across three categories: “all households”, “all households except one-person households”, and“one-person households.” (b) From May 2004 to February 2007, the survey was conducted by telephone in months
other than June, September, December, and March; the survey was conducted by direct-visit and self-completion8

Table 1: Example of the Survey Result of “Price Expectations a Year Ahead”(Source: http://www.esri.cao.go.jp/en/stat/shouhi/shiken_summary_e.html)
approach. Monthly data of the survey are available from April 2004 onward. Concretely, the qualitative
data obtained from the section “price expectations a year ahead” in the survey is applied to our estimationof inflation expectations since the respondents give their expectations of future price level as “go down,”“stay the same,” “go up,” or “don’t know”. The example of the survey result is displayed in Table 1.

3.4 Estimation of Consumer’s Inflation Expectations
Monthly data of consumer price index, seasonally non-adjusted7 , change from the previous year,(excluding fresh food, whole Japan, total) spanning the period from April 2004 to September 20228 isapplied as the rate of inflation (𝜋𝑡 ) to the derivation of the expected inflation rate (𝜋𝑡𝑒 ) by the Kanoh (2006)
procedure.
The qualitative data obtained from the consumer confidence survey for “all households” are used forour estimation. The ratios of the three items in “go down” and “go up” are combined into the totals of “godown” and “go up”, respectively, for our estimation. Further, the “don’t know” answers are eliminatedquestionnaires in June, September, December, and March. (c) Since April 2013, the survey has been conducted by
mail. In addition, the number of sample households has been increased. Therefore, we are obliged to use
discontinuous survey data in our sample period.
7
The reason of adopting seasonally non-adjusted series (instead of seasonally adjusted series) of consumer price index(change from the previous year) is as follows. We also have to use the data given by the consumer confidence survey
which are not seasonally adjusted. In order to have a consistency of the format between the two kinds of data series,
seasonally non-adjusted series of consumer price index is applied.
8
The data on “consumer price index” were retrieved from the “Portal Site” of Official Statistics of Japan operated by
the Ministry of Internal Affairs and Communications, Statistics Bureau, Director-General for Policy Planning(Statistical Standards) & Statistical Research and Training Institute (in English) “http://www.estat.go.jp/SG1/estat/eStatTopPortalE.do”.

9Figure 1: Estimated Expected Inflation Rate based on Kanoh (2006) Methodology by utilizing theJapan’s “consumer confidence survey” (unit: %)

4
3
2
1
0
-1
2004

2006

2008

2010

2012

2014

2016

2018

2020

2022

from the original data set. In addition, the total of the ratios for each month often exceeds 100% due toround-off errors. To deal with this problem, the ratios are re-adjusted by proportional distribution based on
our revised total sum of the ratios.
Figure 1 reports the estimated expected inflation rates by the Kanoh (2006) procedure with ̂𝛿1 =̂2 = −1.2511. The estimated ̂−0.5025 and 𝛿
𝛿1 might not be favorable in the sense that it is inconsistent̂1 > 0, 𝛿
̂2 < 0). However, our estimated expected inflation ratewith the ideal combination of the signs (𝛿can be applied to our estimation since at least one of the two thresholds satisfies the requirement. The
estimated expected inflation rate is fluctuated through our sample period from the lowest estimated value−0.916867(%) for December 2009 to the highest 3.852244(%) for July 2022. The estimated values
are used to the inference of the natural rate of interest in Japan.

4. The Natural Rate of Interest and its Estimation
4.1. Basics of the Hodrick-Prescott Filter
To understand behavior of time series data, we often try to decompose the series into a trend and acyclical component. In this context, the “trend” means the long-term growth of the series and the “cycle”
is the deviation of the series from that trend. The Hodrick-Prescott Filter is one of the widely used10

econometric tools to remove the cyclical component of a time series from raw data. In other words, thisfilter is a kind of method to acquire a smooth estimate of the long-term trend component of a series.
Basic idea of this filter depends on the decomposition of time series. Specifically, this is usually a twosided linear filter9 for calculating the smoothed series s and y through minimizing the variance of y arounds, subject to the penalty parameter that restricts the second difference of s. In short, the Hodrick-Prescott
Filter sets s in order to minimize the value of the equation:2𝑀𝑖𝑛{𝑠𝑡} (∑𝑇𝑡=1(𝑦𝑡 − 𝑠𝑡 )2 + 𝜆 ∑𝑇−1
𝑡=2 [(𝑠𝑡+1 − 𝑠𝑡 ) − (𝑠𝑡 − 𝑠𝑡−1 )] ),where 𝒚𝒕 for t = 1,2, … , T means the logarithms of given time series variable, 𝒔𝒕 is a trend component
and 𝜆 means a penalty parameter (or smoothing parameter). The series 𝒚𝒕 consists of 𝒔𝒕, 𝒄𝒕, and 𝝐𝒕 . In
short, 𝒚𝒕 = 𝒔𝒕 + 𝒄𝒕 where 𝒄𝒕 is a cyclical component. The first term in the parenthesis describes thesummation of the squared deviations of (the logarithms of) a given time series variable and a trend
component. The second term displays the product of a smoothing parameter and the summation of thesquares of the second differences of the trend component. In other aspects, the second term penalizes
variations in the rate of growth of the trend components, that is, the larger the level of 𝜆, the higher is thepenalty. With respect to the value of this penalty parameter (or smoothing parameter), Hodrick andPrescott (1997) suggest 𝜆 = 1600 in the case of quarterly data series. On the other hand, Ravn and Uhlig(2002) insist that 𝜆 should be varied by the fourth power of the frequency observation ratio. Concretely,
for annual data, 𝜆 should be 1600/44 = 6.25, for example. Actually, 𝜆 = 100 for yearly data and 𝜆 =14400 in the case of monthly data are commonly adopted.
It is often suggested that the estimated trend by utilizing the Hodrick-Prescott Filter is relatively moresensitive than the ones by other filters when it faced with transitory shocks and short-term fluctuations at
the end of the sample period of estimation. Baxter and King (1999) suggest that this property derives
suboptimal feature of the Hodrick-Prescott Filter. Considering this problem, it is practical to eliminate theestimated values by the Hodrick-Prescott Filter for the first two years and the last two years (in the case ofyearly data) in the research.
The Hodrick-Prescott Filter is often explained from the aspect of frequency domain. With regard to
this standpoint, the penalty parameter (or smoothing parameter), 𝜆, can be associated with the cut-off
frequency of the filter, in short, the frequency at which it dampens the impact of the cyclical component

9

A one-sided version is explained in some previous studies, but this paper is concentrated on a two-sided one.

11

of the original series. Concerning this property, Nilsson and Gyomai (2011) proposes the defect of the
Hodrick-Prescott Filter. On the other hand, Hamilton (2018) points out the drawback of the HodrickPrescott Filter in that it generates series with spurious dynamic relations without the basis in the underlying
data-generation process. However, the Hodrick-Prescott Filter is broadly adopted because of its usability.

4.2. Estimation of the Natural Rate of Interest
Since the natural interest rate is not directly observable, its estimation is required. One of the statisticalways of estimating the natural rate of interest is to extract the trend component from estimated short-term
real interest rate. If the central bank is inclined to raise the policy interest rate when it acknowledges an
inflationary pressure or excess demand and if such a policy implementation is smoothly transmitted into
adjustment processes of prices and economic activities, observed short-term real interest rate should
behave closely around the evolutionary pass of the natural rate of interest. In this sense, the value that is
obtained by the extraction of the trend component from real short-term interest rate can be regarded as theestimated natural rate of interest. For the estimation of expected inflation rate in order to construct the real
short-term interest rate, the Kanoh (2006) procedure is adopted as explained in the previous section.
Concretely, we adopted the following formula by following the Fisher equation.
estimated real short-term interest rate = short-term nominal interest rate – expected inflation rate
= observed uncollateralized overnight call rate10 – expected inflation rate estimated by Kanoh (2006)
procedure
We utilize the Hodrick-Prescott filter for the extraction of the trend component from the estimated real
short-term interest rate by setting the penalty parameter (smoothing parameter), 𝜆, as 14400 in accordancewith the so-called “frequency power rule” of Ravn and Uhlig (2002). Estimation is conducted by using
monthly data spanning the period April 2004 to September 2022 to make consistent with the estimationof expected inflation.
Figure 2 displays the estimated natural rate of interest. By looking at this figure, we figure out the factthat the estimated natural rate of interest shows a substantial decline in recent 20 years although it isfluctuated through our sample period. In particular, it takes negative value from May 2012, and estimatedabout −2.3295(%) in September 2022. If this is true, the Japanese central bank is faced with the great
10

The data on “uncollateralized overnight call rate” were retrieved from “BOJ Time-Series Data Search” in thewebsite of The Bank of Japan (in English) “https://www.stat-search.boj.or.jp/index_en.html”.

12Figure 2: Estimated Natural Rate of Interest by utilizing the Hodrick-Prescott Filter (unit: %)

0.8
0.4
0.0
-0.4
-0.8
-1.2
-1.6
-2.0
-2.4
2004

2006

2008

2010

2012

2014

2016

2018

2020

2022

difficulty since it is hard to deal with such a situation by conducting a monetary policy based on the usual
framework.

5. Concluding Remarks
Our estimation result brings us to the conclusion that the natural rate of interest in Japan shows apersistent decline in recent 20 years. If the natural rate of interest is declining persistently, usual (ortraditional) framework of monetary policy cannot be pursued. In this kind of situation, monetary policy
should be conducted to guide real interest rate to negative level by raising expected inflation rate or bydecreasing nominal interest rate. One difficult problem is whether the central bank can induce real interest
rate to be negative when the natural rate is negative with the non-negative constraint of nominal rate.
Therefore, the understanding the future direction of the natural rate and the cause of its decline is theessential factor for considering effective monetary policy.

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