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  Dynamics of Inflation Persistence in International Inflation Rates

Manmohan S. Kumar

International Monetary Fund

and

Tatsuyoshi Okimoto* 

Associate Professor

Faculty of Economics and IGSSS,

Yokohama National University

We are indebted to James Hamilton for his guidance and Katsumi Shimotsu for helpful

discussion. We would also like to thank Takeo Hoshi, Bruce Lehmann, Masahito

Kobayashi Laura Kodres Keiji Nagai Raghu Rajan Ken Rogoff Yixiao Sun Masahiko

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Abstract

Characteristics of inflation play a key role in policy formulation and market

analysis. Several studies have analyzed inflation persistence and reached diverging

conclusions. In this paper we investigate the dynamics of inflation persistence using

fractionally integrated processes and find that there has been a clear decline in inflation

 persistence in the U.S. over the past two decades. We also show that the presence of

fractional integration in inflation successfully explains previous diverging results. Lastly

we provide some international comparisons to examine the extent to which there has been a

commensurate decline in inflation persistence in the other G7 economies.

JFL classification: E31, C22, C14

Key words: fractional integration, rolling window estimation, long memory

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Dynamics of Inflation Persistence in International Inflation Rates

Introduction 

In recent years, there has been an active debate regarding the extent to which

dynamics of the inflation process in the world economy, particularly in the largest countries,

have changed. The discussion has been especially intense regarding the extent to which

inflation persistence has declined. This issue is important from both an analytical and a

 policy perspective, and it could be argued that changes in the degree of inflation persistence

may reflect economies’ changing resilience to shocks.

The issue of changes in the degree of inflation persistence is particularly relevant

against the background of a striking change in the realm of macro policy making globally

over the last few years. From a preoccupation with inflation over most of the post war

 period, there was a marked increase in concerns about deflation in the early part of the

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concerns abated somewhat as the effects of that crisis waned, but resurfaced in the

aftermath of the bursting of the equity price bubble in early 2000, geopolitical uncertainties

and the global slowdown.2  With the rebound in global activity since then, declining

measures of economic slack, and sharply higher oil and non-oil commodity prices, markets

and policymakers have again become preoccupied with inflation. Nonetheless, the

turnaround still leaves a key issue unresolved: the extent to which there has been a change

in the underlying characteristics of the inflation process, and its implications for policy

design and financial markets.

There have been a number of recent studies trying to empirically assess the

 persistence of inflation, especially in the United States, but they appear to reach quite

diverging conclusions. These studies can be broadly divided into two methodological types,

depending on the measure of persistence. Studies in the first category rely on order of

integration as the measure of inflation persistence, using unit root tests to classify the

inflation process as either an I(0) or I(1) process. MacDonald and Murphy (1989) found

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I(1) process during 1978 to 1992. On the other hand, Rose (1988) indicated that monthly

U.S. inflation was an I(0) process from 1947 to 1986. Mixed evidence was provided by

Brunner and Hess (1993). They concluded that the inflation rate was I(0) before the 1960’s

 but that it is characterized as I(1) since that time. Other studies include Barsky (1987), Ball

and Cecchetti (1990), Kim (1993), and Culver and Papell (1997).

A second category of studies uses AR model-based measures such as the largest

autoregressive root (LARR) and the sum of the autoregressive coefficients (SARC). For

instance, Taylor (2000) estimated the LARR and the SARC and concluded that U.S.

inflation persistence during the Volcker-Greenspan era has been substantially lower than

during the previous two decades. Similarly, Levin and Piger (2003) used the SARC, and

showed that high inflation persistence is not an inherent characteristic of industrial

economies over the period 1984-2002.3

  On the other hand, Batini (2002) found relatively

little evidence of shifts in inflation persistence for the Euro area using the SARC.

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 permanent and temporary decline in inflation persistence. To our knowledge, there are only

a few studies that systematically document the changes over time in the dynamics of

inflation persistence. Cogley and Sargent (2001, 2005) estimated a Bayesian state-space

VAR model of inflation dynamics and provided a time series of inflation persistence in the

United States. They used the spectrum at frequency zero as the measure of inflation

 persistence and indicated that there has been an unambiguous decline in inflation

 persistence. Their finding was taken to mean that indeed there had been a change in the

underlying characteristics of inflation reflecting both a change in the structural

characteristics of the economy as well possibly in the efficacy and a greater forward

looking nature of monetary policy. In complete contrast, as a comment on Cogley and

Sargent (2001), Stock (2001) estimated the LARR using rolling window estimation method.

He suggested that there is no indication of a marked decline in the persistence. Following

these studies, Pivetta and Reis (2006) estimated the LARR and the SARC using both

Bayesian and rolling window estimation methods. Their main conclusion is that inflation

 persistence in the United States has been high and approximately unchanged over the entire

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  One possible explanation for this debate and divergent results of unit root tests is

the presence of fractional integration in inflation rate,4  which is not consistent with either

an I(0) or an I(1) process. If fractional integration exists, it is possible for AR model-based

measures and unit root tests to reach diverging conclusions, as will be shown in this paper.

In fact, a number of studies find the presence of fractional integration in inflation rates. For

instance, Hassler and Wolters (1995) found that inflation is best characterized as the

fractional integral of white noise, while Baillie, Chung, and Tieslau (1996) suggested that

the ARFIMA-GARCH models best capture the dynamic properties of inflation in the major

industrial countries. Following those studies, Baum, Barkoulas, and Caglayan (1999)

estimated order of fractional integration using both CPI and WPI indices for a number of

industrial and developing countries. They conclude that inflation is best characterized as a

long memory process.5  See also Baillie, Han and Kwon (2002), Bos, Franses and Ooms

(1999, 2002).

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  The contributions of this paper are threefold. First, the paper provides new

evidence on the dynamics of inflation persistence. Specifically, we suggest using order of

fractional integration to measure inflation persistence, which can be considered as a

generalization of the first type of method that simply tries to choose between an I(0) or I(1)

representation. In addition, the method can also successfully provide an appropriate

measure of persistence related to the LAAR when there is fractional integration. As a

consequence, the method enables us to avoid possible confusion inherent in the second

methodology. We find clear evidence in the U.S. of a decline in inflation persistence since

the early 1980’s. Second, the paper reconciles the diverging results of unit root tests and

AR model-based measures reported in previous studies. Third, we summarize the dynamics

of inflation persistence for other G7 countries, to examine the extent to which there has

 been a commensurate decline in inflation persistence in other G7 countries. We find

remarkable similarities between the dynamics of inflation persistence in the U.S and other

G7 countries, with the exception of Italy.

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Section V summarizes the dynamics of inflation persistence for other G7 countries. Section

VI concludes the paper and discusses issues for further research.

II. Fractionally Integrated Processes

I. Fractionally integrated processes and measures of persistence

In this paper, we propose using order of fractional integration to assess the level of,

and changes in, inflation persistence. Fractionally integrated processes offer a

generalization of the classical dichotomy between I(0) and I(1) processes, widely used in

macroeconomic analysis, and allow us to describe long-run persistence with more

flexibility. As a consequence, we can bypass many of the difficulties inherent in alternative

approaches. Moreover, the use of fractional integration appears well justified both on

conceptual and empirical grounds as discussed in the next subsection.

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fractional difference operator d  L)1(   −   for can be defined by means of the gamma function

)(⋅Γ   as

∑∞

=   +Γ−Γ

−Γ=

⎭⎬

⎫

⎩⎨

⎧+

−−−

−+−=−

0

32

)1()(

)(

!3

)2)(1(

2

)1(1)1(

k 

k 

d 

k d 

 Ld k 

 Ld d d  Ld d dL L  

 

The parameter d is allowed to take any real value. The arbitrary restriction of d   to

integer values gives rise to the standard integrated processes. The I( d ) process is stationary

if 2/1

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slowly at a hyperbolic rate 1−d k  . Therefore, order of fractional integration is the key

determinant of the decay rates of autocorrelations and the impulse response function. We

note also that the nature of underlying I(0) process t d 

t   X  Lu )1(   −=   does not affect these

decay rates. For this reason, we view order of integration as the ideal measure of long-run

 persistence, while the persistence structure of the underlying I(0) process summarizes the

rate of the slower geometric decay.

A more fundamental attraction of using the value of d   as the measure of

 persistence is that it does not require the formulation of a specific model to estimate order

of integration. In other words, since it only requires estimating order of fractional

integration, it is not necessary to make assumptions regarding the specific model for the

underlying I(0) process of inflation process or short-run persistence structure. Moreover,

this measure will allow us to isolate persistence that is not consistent with either an I(1)

 process or an I(0) process. For these reasons, this method can be considered as a

generalization of the I(0)/I(1) dichotomy. However, one important drawback of this

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change in the degree of inflation persistence. Thus, our measure is insensitive to a change

in persistence of the underlying I(0) process and can capture only a change in long-run

 persistence. However, we also consider this as an advantage, because long-run persistence

is more important for our purpose, if it exists, which is indeed the case for the U.S. inflation

 persistence, as we will see next.

II. Evidence for fractional integration in the U.S. inflation

Our study is based on monthly CPI data from Main Economic Indicators published

 by OECD,7  with the sample period from 1960:4 to 2003:4. Confirming the results of

Hassler and Wolters (1995), Baillie, Chung, and Tieslau (1996) and so forth described in

the introduction, we find abundant evidence in our U.S. inflation data of fractional

integration. Figure 1 plots autocorrelations of US inflation, which show a clear pattern of

slow decay. In fact, autocorrelations for all lags up to the 84 months (7 years) are positive

and statistically significant. In addition, if we estimate order of fractional integration for

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asymptotic 90% confidence interval (0.32, 0.48) and a finite sample 90% confidence

interval8  (0.31, 0.53). Thus we have good initial reasons for investigating the possible

consequences of fractional integration.

In addition to this empirical evidence, there are two important conceptual

considerations weighing in favor of utilizing fractionally integrated processes to describe

and analyze inflation. These relate to issues regarding aggregation and structural change.

(i) Aggregation: Consider the case where a time series is a cross-sectional sum of a

group of individual time series, each of which follows an AR(1) model with a common

random-walk component:

⎩⎨⎧

+=

+=

−

−

,

,

1

1,

t t t 

t  jt  j j jt 

W W 

W  X  X 

ε 

 β α  

where theα  's and β  's are assumed to be drawn from independent distributions. Then under

some fairly general conditions, in particular assuming that α  's have a form of beta

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has the persistence characteristics of a fractionally integrated process with qd    −< 2 .9

Since an aggregate price index is simply a weighted sum of prices of individual goods and

services, which can be modeled as above, it seems reasonable to expect fractional

integration in an aggregate price index and hence in inflation.

(ii) Structural Change: Granger and Hyung (1999) and Diebold and Inoue (2001)

showed that processes with certain kind of structural changes in mean appear

indistinguishable from long memory processes.10

  Given the significant shocks that have

 beset the world economy over the past three decades, as well as the likelihood of structural

change occurring over this period, a measure that allows for such change is clearly

desirable. This is reinforced by the oft discussed change in monetary policy regimes in

many of the major industrial economies following the oil price shocks of the 1970s and

early 1980s.

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III. Estimation of order of fractional integration

Order of integration d   plays a central role in the definition of fractionally

integrated processes and has often been the focus of previous studies. Therefore a number

of estimators are considered. Among these estimators, semiparametric estimators appear to

 be particularly attractive because they are agnostic about the short-run dynamics of the

 procss and hence robust to misspecification of these dynamics. Unfortunately, these

estimators are still being developed. In particular, there are few semiparametric estimators

that can be used to satisfactorily assess the nonstationary case.11

  However, recently

Shimotsu and Phillips (2005) developed a new semiparametric estimator, the “Exact Local

Whittle” (ELW) estimator. They considered the fractional integration process t  X   

generated by the model

{ }  …

,2,1,0 ,11)1(   ±±=≥⋅=− t t u X  Lt t 

d 

 

where t u   is an I(0) process with mean 0 and spectral density )(λ u f    satisfying

G f u ~)(λ    for 0~λ  , and }{1 ⋅   is the indicator function. We can rewrite this expression

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  .)1()(

)(1

0

∑−

=−

+ΓΓ

+Γ=

t 

k 

k t t  uk d 

d k  X   

By defining the discrete Fourier transformation and the periodogram of a time series

nt t u 1}{ =   evaluated at frequency λ   as

2

1

)()( ,e2

1)(   λ ω λ 

π λ ω    λ  uu

n

t 

it 

t u  I un

==   ∑=

,

the (negative) Whittle likelihood of t u   up to mλ    and up to scale multiplication can be 

written

n

 j 

 f 

 I  f   j

m

 j  ju

 jum

 j

 ju

π λ 

λ 

λ λ 

2 ,

)(

)()(log

11

=+∑∑==

 

where n   is sample size, and m   is some integer less than n   and called the bandwidth.12

 

When G f u ~)(λ    for 0~λ  , the objective function is simplified to be data dependent,

.)(1

log(1

),(1

)1(

2∑=

−

−

⎥⎦

⎤⎢⎣

⎡+=

m

 j

 j X  L

d 

 jm d  I G

Gm

d GQ   λ λ   

Concentrating ),( d GQm   with respect to G , the ELW estimator is defined as

[ ])(minargˆ

21 ,

d  Rd d 

 ELW ΔΔ∈

= ,

where 1Δ   and 2Δ   are the lower and upper bounds of d   and

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Shimotsu and Phillips (2005) showed that under some conditions, in particular, for

( )21 ,ΔΔ∈d    with 2/912   ≤Δ−Δ ,

⎟ ⎠

 ⎞⎜⎝ 

⎛  ⎯→ ⎯ −

4

1,0)ˆ(  N d d m d 

 ELW ,

where m   is chosen so that it satisfies as ∞→n  

0 ,0log)(log1 221

>∀ ⎯→ ⎯ +++

γ γ 

 β 

m

n

n

mm

m.

Here β 

  represents the degree of approximation of the spectral density oft 

u ,)(λ u

 f  ,

around the origin by G . Shimotsu (2002) further developed these results so that it can

accommodate an unknown mean and a linear time trend. Hereafter, following Shimotsu

(2002), we call that estimator the “Feasible Exact Local Whittle” (FELW) estimator. He

suggested estimating unknown mean by

{ } { }   ⎟ ⎠

 ⎞⎜⎝ 

⎛ ∈≥⋅+

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Since there are no satisfactory analytical results for deciding on the appropriate value of m  

in finite sample, we choose m   based on simulation.13

  We simulate t d 

t   L X    ε −−= )1( for

t ε    Gaussian white noise with sample size14

  180 and compute the sample MSE for the

several choice of m , i.e. 8.075.07.065.06.0 ,,,, nnnnnm = . The simulation results indicate that

the sample MSE is minimized when 75.0nm =   for most of the cases.15  Therefore we

choose 75.0nm = as the bandwidth.

As a by-product of above simulation, we realized that the FELW estimator heavily

depends on the value of c   used to define )(ˆ d  . More precisely the FELW estimator has

 probability mass at the value of c , which could likely distort our empirical results. To

mitigate this problem, we use the following strategy16: If the estimate is greater than 1/2,

we reestimate d   by estimating 1−d    using the differenced series and adding back one to

13  All simulations and empirical applications are performed in Matlab. Matlab codes for

the ELW and FELW estimators are available at Shimotsu’s web site

(http://www.econ.queensu.ca/pub/faculty/shimotsu/).

14

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the estimate of 1−d  . Using simulation techniques, we confirmed that this strategy could

mitigate this problem without losing the good finite sample properties of the FELW

estimator. We refer to this estimator as the modified FELW (MFELW) estimator. The

simulation results indicate that also for the MFELW estimator the optimal choice of

 bandwidth is also 75.0nm = . Note further that this modification does not affect the

asymptotic distribution of the FELW estimator under some further conditions on d .17

 

Therefore we can construct the asymptotic confidence interval based on the asymptotic

results of the FELW estimator discussed above.

To investigate the stability of our estimates, we also compute the estimates by the

GPH estimator and the ELW estimator with estimating unknown mean by the sample

average. The GPH estimator is proposed by Geweke and Proter-Hudack (1983). The GPH

estimator is also known as the log periodogram regression estimator, since it estimates the

approximate rate of divergence of the spectrum at frequency around zero by regression of

log periodogram. More precisely, d   is estimated from the following log periodogram

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  m jn

 j d  I   j j

 j

 ju ,,1 ,2

 ,2

sin4log)(log 2 …==+⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟ ⎠

 ⎞⎜⎜⎝ 

⎛ −=

  π λ ν 

λ α λ   

where )(log  ju I    λ    is the periodogram defined above and m   is the bandwidth. The GPH

estimator is defined as the OLS estimate of d   in the log periodogram regression. Künsch

(1986), Robinson (1995), and Hurvich, Deo, and Brodsky (1998) rigorously analyzed its

asymptotic properties. Specifically, Hurvich et al. (1998) showed that under some

conditions,

⎟⎟ ⎠

 ⎞⎜⎜⎝ 

⎛  ⎯→ ⎯ −

24,0)ˆ(

2π  N d d m

d 

GPH  .

Both the GPH and the ELW estimators also have the choice of bandwidth. Based on the

simulation results we choose 8.0nm =   for the GPH estimator and75.0nm =   for the ELW

estimator. Those simulation results also indicate that the MFELW estimator has the best

finite sample properties among these three estimators.

III. Empirical Results

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constant d   against the alternative of time-varying d . The discussion below makes clear

that while the overall conclusion is unambiguous and unaffected by the sort of statistical

considerations that beset other studies in the area, there are a number of important details

that need particular attention.

For each case, we compute the three estimates discussed above: the MFELW

estimator, the ELW estimator with estimating unknown mean by sample average, and the

GPH estimator. These three estimates allow us to test the robustness of the results over a

range of priors regarding the parameters of the underlying process.

We consider first the estimates for d   provided in Table 1. The table reports

estimates and their asymptotic standard errors using three estimators for two subsample

 periods. The first estimate is for the subsample from May 1960 to April 1982; for this

 period, the MFELW estimator takes a value of 0.51. The second estimate is for the

non-overlapping subsample from May 1982 to April 2003; for this period, d   has a value

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 parameter is highly significant.18

  Results using alternative estimators (GPH and ELW)

 provide similar and equally unambiguous conclusions. The conclusion is that there has

 been a pronounced and marked decline in inflation persistence in the United States from

early 1980’s onwards compared to the two decades before.

We obtain additional insight and further support for these conclusions by estimating

the d   parameter using a 15-year rolling window.19

  The basic procedure is to estimate the

 persistence parameter d   using the first 15 years of data (specifically, from May 1960 to

April 1975). The data are then updated by 1 year increments, and the persistence parameter

is re-estimated for the updated window (that is, for the period May 1961 to April 1976).

18  We calculate asymptotic p-values based on the asymptotic theory in a conservative way.

 Namely, we compute asymptotic p-values by using upper bound of, for example,

)ˆˆ(Var  21 d d   − , viz.

)ˆ(Var )ˆ(Var )ˆ,ˆ(Cov2)ˆ(Var )ˆ(Var )ˆˆ(Var  21212121 d d d d d d d d    +≤−+=− ,

since )ˆ,ˆ(Cov 21 d d    is expected to be positive. We also compute finite sample p-values

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This procedure is repeated until the end of the sample period; i.e., the last value of d   is

 based on the period May 1988 to April 2003.

It is apparent that the fractionally integrated process implemented in this way can

yield a significant amount of information about the underlying dynamics of the inflation

 process. It can also help highlight the periods over which there would likely have been

 pronounced changes in the persistence parameters. To get a flavor of the results and

conduct formal tests, Table 3 reports results from estimates over the 15-year window for

the three nearly non-overlapping subsamples.20

  For the MFELW estimator, with the period

ending 1975, d   parameter has a value of 0.44. This increases slightly over the following

 period, so that by 1989, the parameter has a value of 0.53. However, the p-value for the null

hypothesis 210 :H d d   =   against the alternative hypothesis 211 :H d d   ≠   shown in Table 3

suggests this change is not statistically significant. This result illustrates that inflation

 persistence was high and approximately unchanged over the three-decade period before

1989. As column 3 of the Table indicates, over the following period ending in 2003, there

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hypothesis 320 :H d d   =   against the alternative hypothesis 321 :H d d   ≠   shown in Table 3

indicates, this difference between the two periods is statistically significant. Therefore, the

conclusion is that a pronounced and marked decline in inflation persistence in the United

States has begun the 1980’s. The other estimators further support the conclusion.

Tables 1 to 3 provide further support for the results by giving point estimates for

discrete periods. The methodology readily permits a generalization of the above by

allowing computation of rolling estimates over contiguous periods. We present these

estimates in Figure 2 with each graph showing three estimates of d   and the finite sample

90% confidence interval of the MFELW estimates against the end year of an estimated

sample, respectively.21  Figure 2 provides a striking illustration of the sharp decline in d  

at the beginning of the last decade. It indicates that inflation persistence increased gradually

during the first decade ending in 1985 and remained almost unchanged in the following

decade ending in 1995, then declined rapidly and remained low until the present day. Note

that Figure 2 is drawn against the end year of estimated samples. In other words, we detect

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we re-estimate the persistence parameter using other time windows, similar sharp declines

are observed, in which all of them start with samples from the early 1980’s. These results

 provide further evidence for a decline in inflation persistence around early 1980’s.

One concern regarding our empirical results is the robustness against the choice of

the bandwidth m . We choose 75.0nm =   for the MFELW estimator based on simulation.

The simulation, however, assumed that the true model is a Gaussian fractional white noise

without short-run dynamics. Therefore in the presence of additional autoregressive serial

correlation our estimates might have tremendous bias (as reported Agiakloglou, Newbod

and Wohar (1993)) for the GPH estimator.22  As a robustness check Figure 3 provides the

15-year rolling window estimates of the MFELW estimator using several bandwidth

choices. Indeed there are some differences depending on the bandwidth, but the basic

results are same. That is, we find the decline of inflation persistence around the early

1980’s. More importantly, more striking results can be found, if we use smaller bandwidths,

which are suitable for processes with short-run dynamics.

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Another concern is that the decline in inflation persistence is due to ignoring

effects of structural changes in mean. Even though most of the previous studies did not take

into account this “spurious persistence” issue, it is important to know whether the decline in

inflation persistence could be spurious or not. To investigate this, we allow for two

structural changes in mean in our model. Following Hsu (2005), we estimated the timing of

the breaks using the entire sample. The estimation result detects the structural changes

corresponding with shortly before the first oil crisis (1973:1) and shortly after the second

oil crisis (1981:9).23

  We construct a demeaned series using these structural changes and

calculate 15-year rolling window estimates. Figure 4 shows estimation results of the

MFELW estimator under two structural changes (MFELW2), along with our original

MFELW estimates. As can be seen, these estimates indeed indicate that if we include

structural changes, estimates will be smaller in most cases; however, these differences are

notably small and do not change our main finding, i.e. a decline in inflation persistence.

Our empirical results relate to many recent studies in macroeconomics which have

found growing stability in the U.S. economy. This period roughly coincides with previous

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studies’ findings on the timing of a possible structural change towards stability in the U.S.

economy. For instance, Clarida, Gali and Gertler (2000) estimated a forward-looking

monetary policy function. They showed that there were indeed substantial differences in the

 parameters of Taylor rules between pre- and post- 1979. Goodfriend (2005) argued that the

Fed adopted inflation targeting policy gradually and implicitly since the early 1980s. Also,

Kim and Nelson (1999) and McConnell and Perez-Quiros (2000) found that there was a

reduction in the volatility of output in 1984. Our empirical results suggest that there have

 been corresponding changes in inflation persistence toward stabilizing the U.S. economy.

VI. Reconciliation with previous studies

There is considerable disagreement in the previous literature regarding inflation

 persistence. Given our empirical findings, we will provide possible explanations for these

differences and try to reconcile our findings with previous studies.

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studies can be broadly divided into two methodological types, depending on persistence

measures. Studies in the first category use order of integration as the measure of inflation

 persistence. More precisely, they use unit root tests and measure inflation persistence by

classifying the inflation process as either an I(0) or I(1) process. One important and obvious

drawback of this measure is the dichotomous nature of the statistic and the researcher’s

inability to further distinguish between I(0) or I(1) process. Moreover, classification into an

I(0) or I(1) process can be far too restrictive. In other words, it is hard to capture persistence

that is not consistent with either an I(1) or an I(0) process. Indeed, as we discussed in the

 previous section, Figures 3 indicates that most of the estimates for order of integration are

significantly different from either zero or one. This may account for the fact that previous

studies provide divergent results depending on the particular sample periods and methods.

In particular, Hassler and Wolters (1994) argued that, under the presence of fractional

integration, ADF tests with large AR model have very low power to reject the I(1) null

hypothesis. As a consequence, under the presence of fractional integration the ADF test

 based on a small AR model and a large AR model can provide inconsistent conclusions.

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:H 0 inflation process is I(1), using three model specifications: AR(3), AR(6), and AR(12)

models with a constant term to capture non-zero mean. The results using the whole sample

and the range of subsamples used in the previous section are shown in Table 4 to 6. These

tables illustrate the extreme dependence on the sample periods and testing model

specifications. One interesting result in Table 4 is that if we use the AR(12) model, the US

inflation rate is classified as an I(1) process, identical to the suggestions of MacDonald and

Murphy (1989) and Evans and Wachtel (1993). On the other hand, if we use the AR(3) and

AR(6) models, the US inflation rate is described as I(0) process, in accordance to what

Rose (1988) indicated. Another disquieting result is that if we use the AR(6) and AR(12)

models and two subsamples, there is evidence of a decline in the US inflation persistence as

can be seen in Table 5; however, if we use the AR(3) model, the evidence disappears and

US inflation rate is seen to be I(0) for both subsamples. In addition, Table 6 provides

various results regarding the dynamics of inflation persistence. These inconsistent results of

ADF tests can provide one reason for the divergent results of the previous studies.

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an AR(12) model. Figure 5 shows estimates of the LARR and the SARC of AR(12) model

using a 15-year window. As can be seen, both the order of fractional integration and the

SARC declined dramatically at the beginning of the last decade, while the LARR remained

high and unchanged. Stock (2001) and Pivetta and Reis (2006) found similar results using

the LARR and concluded that inflation persistence in the U.S. was high and approximately

unchanged over the past 40 years. Therefore, if one uses only LARR as a measure of

inflation persistence, the conclusion would not be a decline in inflation persistence.

However, if one chooses the SARC or order of fractional integration as the measure, the

conclusion of declining persistence is borne out. Those results are also consistent with

findings of Taylor (2000), Cogley and Sargent (2001, 2005).24

 

V. Evidence for Other G7 Countries

In the previous sections, we found that there has indeed been a significant decline

in inflation persistence in the United States over the past two decades. It would be

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instructive to examine the extent to which there has been a commensurate decline in

inflation persistence in other G7 countries.

Each diagram in Figure 6 plots the MFELW estimates of the 15 year window for

the United States (broken line) and other G7 countries (solid line). As can be seen, there are

remarkable similarities between the dynamics of inflation persistence in the U.S. and other

G7 countries except for Italy. In particular, the similarities between the dynamics of

inflation persistence in the U.S. and France are striking. They are practically same overall

four decades. Also exhibited are parallel behaviors between the dynamics of inflation

 persistence in the United Kingdom and Germany. Their dynamics of inflation persistence

were similar to those in the U.S. for the first three decades, but their declines over the last

decade were somewhat milder than that observed in the U.S. The dynamics of inflation

 persistence in Canada moved differently for the decade ending in 1995; however, otherwise

it shifted similarly to that in the U.S. As a consequence of these observations, the degree of

inflation persistence in those five countries became closer toward the end of sample period.

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 persistence in Japan seems to be increasing, which could reflect the concern about deflation

experienced in Japan over the past five years.

Lastly, the dynamics of inflation persistence in Italy are considerably different

from those of other countries. In contrast to the decline observed in other countries,

inflation persistence in Italy has been high and approximately unchanged over the past four

decades.

VI. Concluding Remarks

There has been a significant interest in recent years in assessing the extent of a

 possible structural shift in the dynamics of the inflation process in the United States. A

number of detailed and rigorous empirical studies regarding changes in inflation persistence

have, however, reached conflicting results. Several studies suggest little or no change over

the past four decades, while others suggest a pronounced decline over the same period.

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  This paper presented a variety of evidence for the existence of long-run persistence

in the U.S. inflation. We argued that a fundamental reason for the divergence in previous

results is inappropriateness of the econometric tools given that inflation does exhibit

long-run persistence. Utilizing a more appropriate but quite general technique based on

fractional integration yields the clear conclusion that there has indeed been a significant

decline in inflation persistence in the United States over the past two decades. Moreover,

this decline was preceded by a mild increase in the persistence parameter during the late

1960s and 1970s. These conclusions are not overly sensitive to the sample period nor to the

 precise estimation procedure.

The paper also investigated the extent to which there has been a commensurate

decline in inflation persistence in other G7 countries. We found similar declines for France

and Japan, and similar but milder declines for the United Kingdom and Germany. Inflation

 persistence in Canada experienced a decline only for the last decade. We also found

inflation persistence in Italy had been high and approximately unchanged over the past four

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which a decline in the average rate of inflation and its volatility may reflect increasing

globalization and competition (both domestic and international), the role of the

informational technology revolution, increasing productivity growth, as well as improved

monetary policy design.25

  These factors are probably also important in explaining the

decline in inflation persistence. In general if the conclusions of this study are regarded as

robust, and we believe they are, an analysis of the determinants of the decline in persistence

in G7 countries would be a useful agenda for further research. In addition, it would be

instructive to examine the extent to which there has been a commensurate decline in

inflation persistence elsewhere in the other major industrial, as well as emerging market

economies, and assess the importance of the common factors that might have led to such an

outcome.

As a final contribution to econometric methods, this study opens up several possible

areas for future investigation. One of these is to consider the effect of structural changes

more carefully. This paper treats structural changes in inflation as one of the source of

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formally in the fractional integration specification is an important future work. Hidalgo and

Robinson (1996) and Bos, Franses and Ooms (1999) provide good starting points for this

 problem.

Another avenue relates to the measurement of the short-run as well as the long-run

 persistence so that we can describe the persistence structure itself more precisely. One

 possibility is the ARFIMA modeling. As discussed in Section 2, we can capture the

long-run persistence through order of fractional integration and the short-run persistence

through ARMA model. A formal modeling of time-varying variance could also be useful.

There is universal agreement on a reduction in variance and oftentimes it is misunderstood

as a reduction in persistence. By explicitly modeling time-varying variance, this confusion

can be clarified. ARFIMA-GARCH model proposed by Baillie, Chung and Tieslau (1996)

is one attractive model for those topics.

The final and perhaps the most important topic is to model the dynamics of inflation

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method could also be used in this context. Since the MCMC method provides a convenient

framework to estimating complicated models, it might be particularly useful to estimate the

time-varying order fractional integration model.

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Table 1: Estimates of order of integration (two subsamples)

Estimator 1960.5-1982.4 1982.5-2003.4 asy. p-values f.s. p-values

MFELW 0.509 0.2680.000 0.057

(asy. s.e.) (0.062) (0.063)

GPH 0.587 0.2670.000 0.022

(asy. s.e.) (0.069) (0.070)ELW 0.566 0.268

0.000 0.015(asy. s.e.) (0.062) (0.063)

Table 2: Estimates of order of integration (15-year window)

Estimator 1960.5-1975.4 1974.5-1989.4 1988.5-2003.4

MFELW 0.439 0.532 0.252

GPH 0.517 0.539 0.297

ELW 0.515 0.565 0.252

Table 3: P-values for changes in order of integration (15-year window)

Hypothesis d1=d2 d2=d3

asy. p-values f.s. p-values asy. p-values f.s. p-values

MFELW 0.514 0.437 0.050 0.029

GPH 0.848 0.850 0.033 0.077

ELW 0.725 0.663 0.028 0.011

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  44

Tables 4: Results of ADF unit root tests for US inflation (whole sample)

Sample period 1960.5-2003.4

Model AR(3) AR(6) AR(12)

 ADF statistics -4.650** -3.136* -2.573

P-values 0.000 0.025 0.099

Tables 5: Results of ADF unit root tests for US inflation (two subamples)

Sample period 1960.5-1982.4 1982.5-2003.4

Model AR(3) AR(6) AR(12) AR(3) AR(6) AR(12)

 ADF statistics -2.896* -2.059 -1.955 -6.573** -4.682** -3.182*

P-values 0.047 0.262 0.307 0.000 0.000 0.022

Tables 6: Results of ADF unit root tests for US inflation (three subsamples)

Sample period 1960.5-1975.4 1974.4-1989.5 1988.5-2003.4

Model AR(3) AR(6) AR(12) AR(3) AR(6) AR(12) AR(3) AR(6) AR(12)

 ADF statistics -2.503 -1.393 -1.654 -3.238* -2.283 -1.838 -4.551** -3.270** -2.539

P-values 0.117 0.585 0.453 0.019 0.179 0.361 0.000 0.018 0.108

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  45

Figure 1: Autocorrelations of the US CPI inflation

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 20 40 60 80 100 120

Lag

     A   u    t   o   c   o   r   r   e

     l   a    t     i   o   n

 

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  46

Figure 2: Dynamics of persistence in US inflation rate

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1975 1978 1981 1984 1987 1990 1993 1996 1999 2002

MFELW

GPH

ELW

f. s. lb of 

MFELW

f. s. ub of 

MFELW

 

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  47

Figure 3: Robustness check of the MFELW estimates

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1975 1978 1981 1984 1987 1990 1993 1996 1999 2002

m=n0.6

m=n0.65

m=n0.7

m=n0.75

m=n0.8

m=n0.6

m = n0.65

m=n0.7

m = n0.75

m = n0.8

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  48

Figure 4: Dynamics of persistence in US inflation rate under structural changes

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

1975 1978 1981 1984 1987 1990 1993 1996 1999 2002

MFELW

MFELW2

 

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  49

Figure 5: Dynamics of persistence in US inflation rate based on the LARR and SARC

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1975 1978 1981 1984 1987 1990 1993 1996 1999 2002

MFELW

LARR 

SARC

 

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  50

Figure 6: Dynamics of persistence in inflation rates for other G7 countries

Germany

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1975 1979 1983 1987 1991 1995 1999 2003

Canada

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1975 1979 1983 1987 1991 1995 1999 2003

France

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1975 1979 1983 1987 1991 1995 1999 2003

Italy

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1975 1979 1983 1987 1991 1995 1999 2003

Japan

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1975 1979 1983 1987 1991 1995 1999 2003

United Kingdom

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

1975 1979 1983 1987 1991 1995 1999 2003