
 Start

1746
 Prefix

Despite
extensive research following the pioneering work of Nelson and Plosser
(1982), disagreement remains in the literature on a key question: does
the postwar inflation rate possess a unit root? Although there is
considerable evidence in support of a unit root (e.g. Barsky 1987;
MacDonald and Murphy 1989; Ball and Cecchetti 1990; Wickens and Tzavalis
1992; and
 Exact

Kim 1993),
 Suffix

Rose (1988) provided evidence of stationarity in
inflation rates. Mixed evidence has been provided by Kirchgassner and
Wolters (1993). Brunner and Hess (1993) argue that the inflation rate
was stationary before the 1960s, but that it possesses a unit root since
that time.
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 Start

2517
 Prefix

An explanation for this conflicting evidence was recently provided by
modeling inflation rates as fractionally integrated processes. Using
the fractional differencing model developed by Granger and Joyeux (1980),
 Exact

Hosking (1981), and
 Suffix

Geweke and PorterHudak (1983), Baillie, Chung, and
4
Tieslau (1996) find strong evidence of long memory in the inflation rates
for the Group of Seven (G7) countries (with the exception of Japan) and
those of three high inflation countries: Argentina, Brazil, and Israel.
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 Start

2547
 Prefix

An explanation for this conflicting evidence was recently provided by
modeling inflation rates as fractionally integrated processes. Using
the fractional differencing model developed by Granger and Joyeux (1980),
Hosking (1981), and Geweke and
 Exact

PorterHudak (1983),
 Suffix

Baillie, Chung, and
4
Tieslau (1996) find strong evidence of long memory in the inflation rates
for the Group of Seven (G7) countries (with the exception of Japan) and
those of three high inflation countries: Argentina, Brazil, and Israel.
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 Start

2981
 Prefix

Tieslau (1996) find strong evidence of long memory in the inflation rates
for the Group of Seven (G7) countries (with the exception of Japan) and
those of three high inflation countries: Argentina, Brazil, and Israel.
Similar evidence of strong longterm persistence in the inflation rates
of the U.S., U.K., Germany, France, and Italy is also provided by Hassler
and Wolters (1995). Delgado and
 Exact

Robinson (1994)
 Suffix

find evidence of persistent dependence in the Spanish inflation rate. The interpretation of
this evidence suggests that inflation rates are meanreverting processes,
so that an inflationary shock will persist, but will eventually dissipate.
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 Start

6924
 Prefix

The stochastic process is both stationary
and invertible if all roots ofandlie outside the unit
circle and. The process is nonstationary for, as
7
t
\b( )( )
0505
jjd< :d :
it possesses infinite variance, i.e. see Granger and Joyeux (1980).
26
d;: d
k
k
k
d;:j
n
(0 0 5)= 0
()
Assuming thatand,
 Exact

Hosking (1981)
 Suffix

showed that the
correlation function, , of an ARFIMA process is proportional to
as. Consequently, the autocorrelations of the ARFIMA process
decay hyperbolically to zero asin contrast to the faster,
geometric decay of a stationary ARMA process.
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 Start

8042
 Prefix

The fractional differencing parameter is estimated using two semiparametric methods, the spectral regression and Gaussian semiparametric
approaches, and the frequencydomain approximate maximum likelihood
method. A brief description of these estimation methods follows.
21
d
!1
!1
2jj
P
n
(0 0 5)( )
=
jn
!1
2d:;
(050)
d
=0
2
d:;
[0 5 1)
The Spectral Regression Method
Geweke and
 Exact

PorterHudak (1983)
 Suffix

suggest a semiparametric procedure to
d
obtain an estimate of the fractional differencing parameter based on
the slope of the spectral density function around the angular frequency
8
=0
.
The spectral regression is defined by
ln(2)
>I \f \f
< ;  ;:::; 01
2
2(1)
2
fg
ln( ) =+4 sin
2
+=1
()
Iwhereis the periodogram of the time series at the Fourier fre<
T
T
==1
()
()==0=
 ;:::;T
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 Start

8715
 Prefix

+=1
()
Iwhereis the periodogram of the time series at the Fourier fre<
T
T
==1
()
()==0=
 ;:::;T
gT<<T
gT
d
d<
d;:
quencies of the sample, is the number of
observations, and =is the number of Fourier frequencies
included in the spectral regression.
Assuming that lim, lim, and lim
, the negative of the OLS estimate of the slope coefficient in (2) provides an estimate of . Geweke and
 Exact

PorterHudak (1983)
 Suffix

prove consistency
and asymptotic normality for, while Robinson (1995a) proves consistency and asymptotic normality forin the case of Gaussian
ARMA innovations in (1).
Ooms and Hassler (1997) show that the spectral regression will contain
singularities due to prior deseasonalization of the series through standard seasonal adjustment techniques (utilizing seasonal dummy variables).
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 Start

8788
 Prefix

Assuming that lim, lim, and lim
, the negative of the OLS estimate of the slope coefficient in (2) provides an estimate of . Geweke and PorterHudak (1983) prove consistency
and asymptotic normality for, while
 Exact

Robinson (1995a)
 Suffix

proves consistency and asymptotic normality forin the case of Gaussian
ARMA innovations in (1).
Ooms and Hassler (1997) show that the spectral regression will contain
singularities due to prior deseasonalization of the series through standard seasonal adjustment techniques (utilizing seasonal dummy variables).
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 Start

9854
 Prefix

to
9
non o
2
()ln()
()
TT
gT
TT
T
gT
1
!1!1!1
0
0
(0 0 5)
2
()
I2
s
==0
; ;:::; s;s
seasonal frequencies when estimating the logperiodogram regression in
:
(2)
We refer to this method, which yields more stable and reliable estimates than those generated by the standard spectral regression approach,
as the adjusted spectral regression method.
The Gaussian Semiparametric Method
 Exact

Robinson (1995b)
 Suffix

proposes a Gaussian semiparametric estimator, GS hereH
f
f  G;H;H
dHdH
after, of the selfsimilarity parameter , which is not defined in
closed form. It is assumed that the spectral density of the time series,
denoted by, behaves as
as
()
()0
(0)(0 1)
=+
()=ln () (21)
1
ln
H
12+
!
21 2
for Gand.
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 Start

11881
 Prefix

Cheung and Diebold (1994) suggest that the
frequencydomain approximate ML estimator compares favorably, in terms of
its finitesample properties, to the much more computationally arduous
timedomain exact ML estimator proposed by
 Exact

Sowell (1992)
 Suffix

in the case that
the mean of the process is unknown.
()( )
If;\r
\r
11
3. Data and Empirical Estimates
Data
We perform the analysis on CPIbased inflation rates for 27 countries
and WPIbased inflation rates for 22 countries.
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 Start

20113
 Prefix

Care must
be exercised in estimating any regression in which two or more fractionally integrated processes appear, as they would in virtually any model
containing two or more of the series studied here. If their orders of
integration sum to greater than 0.5, ëspurious regressioní effects might
appear
 Exact

(Tsay 1995).
 Suffix

9
4. Conclusions and Implications
This paper tests for the existence of long memory, or persistence, in
international inflation rates for a number of industrial and developing
countries using semiparametric and maximum likelihood estimation methods.
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 Start

23016
 Prefix

argument put forth by Granger
(1980), which states that persistence can arise from the aggregation of
constituent processes, each of which has short memory.Granger and
Ding (1996) show that the longmemory property could also arise from
timevarying coefficient models or nonlinear models. An alternative
conjecture is that inflation inherits the longmemory property from money
growth.
 Exact

PorterHudak (1990) and
 Suffix

Barkoulas, Baum, and Caglayan (1998)
have shown that the U.S. monetary aggregates exhibit the longmemory
property, which will be transmitted to inflation, given the dependence
of longrun inflation on the growth rate of money.
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