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Further Long Memory Properties of Inflationary Shocks Richard T

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					Further Long Memory Properties of Inflationary Shocks

         Richard T. Baillie; Young Wook Han; Tae-Go Kwon

         Southern Economic Journal, Vol. 68, No. 3. (Jan., 2002), pp. 496-510.

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Southern Econovnic Journal 2002, 68(3), 496-510




Further Long Memory Properties of
Inflationary Shocks
Richard T. Baillie," Young Wook Han,? and Tae-Go Kwoni:


     Several previous studies have found fractionally integrated, or long memory behavior, in the
     conditional mean of inflation. This paper notes that extremely similar phenomena are also
     apparent in the squared and absolute values of residuals from fractionally filtered inflation series.
     Hence, the inflation process appears to have a dual long memory feature in both its first and
     its second conditional moments. We suggest a parametric model of long memory in both the
     conditional mean and the conditional variance. Some Monte Carlo evidence is presented that
     supports estimation of the model by approximate maximum likelihood methods. We then report
     estimated models for the inflation series for several different industrialized countries, including
     the United States. For nearly all of the countries in our study, there is strong evidence of
     statistically significant long memory parameters in both the conditional mean and the variance.
     We note some of the implications for modeling inflation.




1. Introduction

      Many previous studies have considered the properties of the univariate time-series repre-
sentation of monthly inflation. A central issue in much of this research has been the degree of
persistence of the shocks and is related to the controversy concerning the possible existence of
a unit root in inflation. In particular, Nelson and Schwert (1977), Barsky (1987), Ball and
Cecchetti (1990), and Brunner and Hess (1993) have argued that U.S. inflation contains a unit
root so that shocks to inflation are completely persistent. Alternatively, Hassler and Wolters
(1995); Baillie, Chung, and Tieslau (1996); and Baum, Barkoulas, and Caglayan (1999) have
found evidence that inflation is fractionally integrated. The fractionally integrated model implies
that the autocorrelations and impulse response weights of inflation exhibit very slow hyperbolic
decay. The previously mentioned articles provide quite consistent evidence across countries and
time periods that inflation is fractionally integrated with a differencing parameter that is signif-
icantly different from zero and unity.'
      The contribution of this paper is to note that very similar long memory properties are also
present in the second moment of inflation. In particular, the squared and absolute values of


       * Department of Economics and Department of Finance, Michigan State University, East Lansing. MI 48824, USA;
E-mail baillie@msu.edu: corresponding author.
       t Department of Economics and Finance, City University of Hong Kong. Tat Chee Avenue, Kowloon, Hong Kong.
People's Republic of China.
       $ Industrial Bank of Korea. 50 Ulchiro 2-ga, Chung-gu, 100-78, Seoul, Korea.
       The authors gratefully acknowledge the helpful comments of the editor, David Papell. and also those of two
anonymous referees. The first and second authors are also grateful for financial support from the National Science
Foundation Grant DMS-007 16 19.
       Received June 2000; accepted January 2001.
I These studies have either estimated the long memory parameter by semiparametric procedures or alternatively from

  estimating ARFIMA models.
                                                                 Long Memory of Inflation          497


inflation residuals, from applying a fractional filter to the conditional mean, also possess long
memory. An implication of this finding is that the conditional variance of inflation can probably
be modeled as a long memory autoregressive conditional heteroskedastic (ARCH) process.
Hence, inflation has the rather curious and hitherto undetected property of persistence in both
its first and its second conditional moments.
      The plan of the rest of this paper is as follows. Section 2 briefly summarizes the standard
autoregressive fractionally integrated moving average (ARFIMA) model, which has the long
memory property in the mean. The model is estimated for the consumer price index (CPI)
inflation series of eight different countries, including the United States, and also for a new
median-weighted CPI series. These results support previous findings of long memory, and in-
vestigation of the residuals of the model provides evidence suggestive of similar long memory
behavior in the squared and absolute standardized residuals. Section 3 introduces a model that
is sufficiently flexible to handle the type of long memory behavior encountered in inflation;
namely, a hybrid ARFIMA-fractionally integrated generalized autoregressive conditional het-
eroskedastic (ARFIMA-FIGARCH) model, which generates the long memory property in both
the first and the second conditional moments of the inflation process. Some of the theoretical
properties of this process are discussed, and estimation of the process is carried out by approx-
imate maximum likelihood estimation (MLE) assuming a Gaussian density and subsequent
inference based on quasi-maximum-likelihood estimation (QMLE). This section also includes
results of the small sample properties of the estimation and inference from a relatively detailed
Monte Carlo study. Section 4 then reports estimates of ARFIMA-FIGARCH models for the
eight separate countries CPI inflation series and also for an alternative measure of inflation that
has recently been proposed that is based on the U.S. median-weighted CPI inflation. The hybrid
long memory model is generally found to be the most appropriate representation for the inflation
series. The estimated model implies the eventual mean reversion of both the conditional mean
and the conditional variance following the impact of shocks.



2. Conditional Mean of Inflation

   Following Granger (1980), Granger and Joyeux (1980), and Hosking (1981), the ARFI-
MA@, d, q) model is defined as



where E(E,) = 0, E(e2,) = u2, and E(E,E,) = 0 for s i and @(L) = (1 - @,L . . . - @p),
                                                        t,
0(L) = (1 + 0,L + . . . + 0,Lq) and have all their roots outside the unit circle. The Wold
decomposition, or infinite-order moving-average representation of this process, is given by y,
- C,=,,
-       Vje,,, and the infinite-order autoregressive representation is given by y, = Z,=,,, n,yr,
+ E,. For high lag j, these coefficients decay at a very slow hyperbolic rate, that is, V, .= c c , j d '
and n, .= cj-"-l. Similarly, the autocorrelation coefficient at large lag j is p .= cj2"-I, where
                                                                                ,
c,, c,, and c, are constants. For -0.5 < d < 0.5, the process is stationary and invertible, and
y, is said to be fractionally integrated of order d, or I(d). Hence, the parameter d represents the
degree of "long memory" behavior for the series. For 0.5 5 d < 1, the process does not have
a finite variance, but for d < 1, the impulse response weights are finite, which implies that
shocks to the level of the series are mean reverting.
      Previous studies by Hassler and Wolters (1995); Baillie, Chung, and Tieslau (1996); and
498        Baillie, Han, and Kwon

Baum, Barkoulas, and Caglayan (1999) have considered inflation to be fractionally integrated.
The series of monthly inflation is defined as y, = lOO.Alog(CPI,), where CPI, is monthly CPI.
The U.S. series extends from January 1947 through September 199g2and for the other countries
from February 1957 through either September 1998 or October 1998, while the series for Korea
was from February 1970 through October 1998.
      While it is traditional to measure inflation as the differenced logarithm of the CPI series,
some other definitions have also been proposed. In particular, Bryan, Cecchetti, and Wiggins
(1997) and Bryan and Cecchetti (1999) have argued that since disaggregate price data generally
possess high kurtosis, the standard method of taking arithmetic averages of prices may not be
the most appropriate procedure for measuring inflation. They suggest several alternative mea-
sures, including the trimmed mean and also the median of the first-differenced logarithm of the
CPI series. In order to widen this study to include a potentially interesting new measure of
inflation, this paper also considers U.S. median-weighted inflation, which extends from January
1967 through September 1998.
      The U.S. CPI inflation series and the first 120 autocorrelations of the levels of U.S. inflation
are plotted in Figure l a and b, respectively, while the autocorrelations of first-differenced infla-
tion are graphed in Figure lc. The autocorrelations of the inflation series possess the very slow
decay associated with fractionally integrated processes. Furthermore, the autocorrelations of the
differenced U.S. inflation series in Figure l c display some negative values at low lags, which
is strongly suggestive of overdifferencing. Very similar plots for the other countries are omitted
for reasons of brevity but are available from the authors on request.
      The results from estimating ARFIMA models for the different countries are given in Table
1. For the United States, an ARFIMA(1, d, 0) model was found to provide an adequate repre-
sentation of the inflation series, with the estimate of d being 0.39 and a robust asymptotic
standard error of .06. The other countries' inflation series all exhibited considerable seasonality.
Accordingly, the following seasonal ARFIMA model,



was estimated. Interestingly, for two countries (France and the United Kingdom), the seasonality
was quite strong, and the most parsimonious model was found to also seasonally difference the
inflation series before estimating the model in Equation 2. This transformation seemed necessary
in order to produce a more parsimonious ARFIMA model, while the application of tests for the
presence of a seasonal unit root was inconclusive, probably because of the power of the test
statistics. Without the use of seasonal differencing, a higher-order seasonal autoregressive mov-
ing average (ARMA) structure was required. These models contained more parameters than the
seasonally differenced ones and are not reported for reasons of conserving space. The use of
the seasonal differencing operator did not significantly change the estimated value of the long
memory parameter for any of the countries. Again, full results are available on request.
      The point estimates of the long memory parameter in the conditional mean for the regular
inflation series were found to be in the range of 0.21 through 0.44. The model for the new U.S.
median-weighted inflation series was quite similar to the regular U.S. CPI inflation series, albeit
with a slightly higher estimated long memory parameter of 0.59 and rather less kurtosis in its
standardized residuals. Inference in the estimated models is based on QMLE, so that robust


'	 aillie. Chung, and Tieslau (1996) use CPI series from January
 B                                                            1947 through September 1990. Results from this sample
 and a much longer sample back to 1913 are available from the authors on request.
                                                                                                                                                                 Long Memory o f Infition                    499




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Figure 1, (a) Graph of the U.S. CPI inflation. The vertical axis shows monthly U.S. CPI inflation, that is, lOO.Alog(CP1,). The
series is from January 1947 through September 1998. (b) Autocorrelations of U.S. CPI inflation. The vertical axis represents
the first 120 lags of the autocorrelations of monthly U.S. CPI inflation. (c) Autucorrelations of differenced U.S. CPI inflation.
The vertical axis represents the first 120 lags of the autucorrelations of the first differences of monthly U.S. CPI inflation.
500   Baillie, Han, and Kwon


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                                      [ b l Autocorrelations o f squared res~duals




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Figure 2. Autocorrelations of transformations of the residuals of U.S. CPI inflation. The vertical axis represents the first
120 lags of the autocorrelations of the residuals. squared residuals, and absolute values of residuals from an ARFIMA(1.
0.39. 0)-seasonal ARMA(0, I ) model.



asymptotic standard errors for the estimated parameters appear beneath corresponding MLEs.
Robust Wald tests indicated overwhelming rejection of both the d = 0 and d = 1 hypotheses.'
The QMLE procedure is described for the more general ARFIMA-FIGARCH model in section
3. While the estimated models in Table 1 appear to adequately describe the dynamics in the
conditional mean, the residuals clearly possess ARCH effects.
      An indication of the rather unusual properties of inflation can be heuristically observed
from the autocorrelations of the residuals from the previously estimated ARFIMA models. One
interpretation of the residuals is that they are formed from a filter containing a fractional com-
ponent plus short memory components. The first 120 autocorrelation coefficients of the U.S.
inflation residuals are plotted in Figure 2 and are consistent with the hypothesis of being gen-
erated by an uncorrelated process. However, Figure 2 also plots the autocorrelations of the
squared and absolute values of the residuals, or fractionally filtered series. Interestingly, the
autocorrelations of the squared and absolute residuals display extremely persistent autocorre-
lation that is also suggestive of a form of long memory behavior. The nature of the autocor-
relations of the squared and absolute residuals is extremely similar to that of many observed
financial market return series. This fact was originally noted by Ding, Granger, and Engle (1993)


'	 he application of QMLE in this study differs from that of Baillie. Chung, and Tieslau (1996), who estimated ARFIMA-
 T
 GARCH(1, 1) models with conditional Student t-densities.
502       Baillie, Hail, and Kwon




Figure 3. Finite sample densities of the QMLE of d. Simulation design I for ARFIMA(0, d, 1)-FIGARCH(1, 6, I ) , with
d = 0.3 and 6 = 0.5. The figure graphs the kernel estimates of the simulated small sample densities of the QMLE for
d from the data-generating process of an ARFIMA(0, d, I)-FIGARCH(1. 6. 1) model with d = 0.3, H = 0.3, P = 0.3.
and 6 = 0.5. The solid line is for a sample size of T = 1000, and the broken line is for T = 500.




for equity returns and by Dacorogna et al. (1993) for exchange rates. Some of these stylized
facts are consistent with the FIGARCH model of Baillie, Bollerslev, and Mikkelsen (1996).




3. The ARFIMA-FIGARCH Model


     A model that is capable of representing the dual long memory features in both the con-
ditional mean and the conditional variance is the ARFIMAb, d, q)-FIGARCH(P,S, Q) model,
given by

                    @(L)(l    -   L)"().,   -   k ) = 0(L)(1   -   OL'2)€t,                                     (3)




where @(L), 0(L), and O(L) are as defined earlier in Equation 1, while P(L) = (1 - P,L . . .
PC&"), +(L) = (I + + , L + . . . + +,LP) and have all their roots outside the unit circle. Also,
El+,:, = 0, Var,-,z, = I, while El-, is the expectations operator conditioned on a sigma field
set of information at time t - 1, and u2, is the conditional variance and is a positive, time-
varying, and measurable function with respect to the information set, which is available at time
                                                                        Long Memory of Zn$ation               503

Table 2. Simulation Results of Estimating the ARFIMA(0, d, 1)-FIGARGH(1, 6, 1) and AR-
FIMA(0, d, 0)-FIGARGH(1, 6, 1) Models
                                                                    d                              6
                                                                             Stan-                         Stan-
                                                                             dard                          dard
Model                          d       6        T        Bias      RMSE      Error      Bias     RMSE      Error

ARFIMA(0, d, 1)-             0.3     0.5       500     -0.027      0.093     0.089     0.030     0.223    0.220
 FIGARGH(1, 6, 1)"           0.3     0.5      1000     -0.013      0.059     0.057     0.027     0.158    0.156
                             0.45    0.75      500     -0.005      0.106     0.106     0.030     0.237    0.235
                             0.45    0.75     1000       0.002     0.075     0.075     0.034     0.177    0.173
ARFIMA(0, d,, 0)-
 FIGARGH(1, 6, l ) b         0.6     0.8      1000       0.013     0.033 0.031         0.015 0.175        0.174
 The table reports the averages of biases and RMSE of the QMLE of the estimates of the d and 6 parameters from the 

 simulation design (i) and (ii). The results are based on 1000 replications in all cases. 

 The table reports the averages of biases and RMSE of the QMLE of the estimates of the d and 6 parameters from the 

 simulation design (iii). The results are based on 1000 replications in all cases. 



t -     If a2,= w, a constant, the process reduces to the ARFIMAb, d, q) model of Granger
and Joyeux (1980) and Hosking (1981). Then y, will be covariance stationary and invertible for
-0.5 < d < 0.5 and will be mean reverting for d < 1.
      When +(L) = 0(L) = 1, the process reduces to the FIGARCH(P, 6. Q) conditional variance
process of Baillie, Bollerslev, and Mikkelsen (1996), and the conditional variance, a2,, has a
slow hyperbolic rate of decay in terms of lagged squared innovations. The associated impulse
response weights also exhibit quite persistent hyperbolic decay. As an illustration, the
FIGARCH(1, 6, 0) process can be expressed as a2,= w/(l - P) + A(L)e2,,where A, = T(k +
                                                                                          which
S - l)l{T(k)T(S)].[(l - P) - (1 - S)lk], and for large lags k, A, = [ ( l - (3)lT(S)l.k6-1,
generates slow hyperbolic rate of decay on the impulse response weights. The process is strictly
stationary and ergodic for 0 5 S 5 1, and shocks will have no permanent effect. As noted
previously, the pure ARFIMA(p, d, q)-homoskedastic process will have a finite variance for
-0.5 < d < 0.5. However, the ARFIMA-FIGARCH process will have an infinite unconditional
variance for all d given a 6 # 0. This fact is discussed in the context of the pure FIGARCH
model by Baillie, Bollerslev, and Mikkelsen (1996); the presence of the FIGARCH volatility
process imposes an undefined unconditional variance independent of the dynamics in the con-
ditional mean.
      Assuming conditional normality, the logarithm of the likelihood can be expressed in the
time domain as



where A' = ( k , @,, @,, 0,, . . . , 0,, O, d, o,+, 6, p). The QMLE of the parameters are obtained
by an analogous methodology to that described by Baillie, Bollerslev, and Mikkelsen (1996),
where the likelihood function is maximized conditional on initial conditions and the presample
values of e2,, t = 0, - 1, -2, . . . , are fixed at the sample unconditional variance. The initial
observations yo, y- ,, y-,, . . . , are also assumed fixed, in which case minimizing the conditional
sum of squares function will be asymptotically equivalent to MLE. This procedure is known as

"ome previous researchers such as Ling and Li (1997) and Teyssiere (1997) have also considered the possibility of
 nonstandard behavior occurring in both means and variances of some time series.
504       Baillie, Hun, and Kwon




Figure 4. Finite sample densities of the QMLE of 6. Simulation design I for ARFIMA(0. d, I)-FIGARCH(1, 6, I), with
d = 0.3 and 6 = 0.5. The figure graphs the kernel estimates of the simulated small sample densities of the QMLE for 6
from the data-generating process of an ARFIMA(0, d, 1)-FIGARCH(1. 6, 1) model with d = 0.3. H = 0.3. and 6 = 0.5.
The solid line is for a sample size of T = 1000, and the broken line is for T = 500.




minimizing the conditional sum of squares (CSS) function and is widely used in similar models
(e.g., see Baillie, Chung, and Tieslau 1996, among others). The consistency and asymptotic
normality of the QMLE has been established only for specific special cases of the ARFIMA
and/or FIGARCH model. Li and McLeod (1986) consider the ARFIMA(p, d, q)-homoskedastic
model with p. either zero or known and show the MLE are T k consistent and asymptotically
normal. Dahlhaus (1988, 1989) and Moehring (1990) have extended the proof to the case of
the ARFIMA(p, d, q)-homoskedastic model with p. unknown. They show that the parameter
estimates are asymptotically normal, with the ARMA parameter estimates again being Th        con-
sistent, while the MLE of  p. is T*-d consistent. For the conditional variance process, asymptotic
normality and T %consistency has been derived only for the integrated generalized autoregressive
conditionally heteroskedastic (IGARCH[l, 11) model by Lee and Hansen (1994) and Lumsdaine
(1996). Their proofs require z, in Equation 4 to be stationary and ergodic, together with three
other relatively mild conditions on z,. While simulation evidence for FIGARCH and other
complicated parametric ARCH models suggests QMLE to be consistent and asymptotically
normal, a fully general theoretical treatment is as yet unavailable. Consequently, we conjecture
that with p unknown, the limiting distribution of the QMLE is
                                                                         Long Memory of InJlation               505




Figure 5. Finite sample densities of the QMLE of d. Simulation design I1 for ARFIMA(0, d. 1)-FIGARCH(1, 6, l), with
d = 0.45 and 6 = 0.75. The figure graphs the kernel estimates of the simulated small sample densities of the QMLE for
d from the data-generating process of an ARFIMA(0, d, I)-FIGARCH(1, 6, 1) model with d = 0.45. 8 = 0.3, and 6 =
0.75. The solid line is for a sample size of T = 1000, and the broken line is for T = 500.




where A(.) and B(.) are the Hessian and outer product gradient, respectively, when evaluated
at the true parameter values A, and diag(D,) = [TM-< . . . , TI>].The adequacy of this
                                                              TI>,
estimation method was assessed by means of a detailed Monte Carlo study where ARFIMA(0,
d, 1)-FIGARCH(1, 6, 1) models were simulated for the three different parameter designs of (i)
0 = 0.3, d = 0.30, P = 0.40, and 6 = 0.50; (ii) 0 = 0.3. d = 0.45, P = 0.65, and 6 = 0.75;
and (iii) 0 = 0, d = 0.60, P = 0.70, and 6 = 0.80. The sample sizes of T = 500 and T =
1000 were investigated for the different designs for 1000 replications in all cases. Design (iii)
implies an undefined unconditional variance even in the presence of conditional homoskedas-
ticity. The results of the simulations for all three designs are summarized in Table 2, which
gives the average biases and root mean squared errors (RMSE). The distributions of the QMLE
of d and 6 for design (i) are shown in Figures 3 and 4, respectively, and for design (ii) they
are graphed in Figures 5 and 6, respectively. The results for other designs, including (iii), are
very similar and are omitted for reasons of saving space. The overall quality of the application
of the QMLE is generally very satisfactory with relatively small parameter estimate biases for
d and 6 in either design. Corresponding results for other parameter estimation biases are quite
similar and are not reported in the interest of conserving space but are available from the authors
on request. Table 2 also gives details of the within-replication RMSE for each parameter estimate
compared with the mean standard error computed from the QMLE. The use of the asymptotic
t-test also appears satisfactory for all three designs.
506       Baillie, Hun, and Kwon




Figure 6. Finite sample densities of the QMLE of 6. Simulation design I1 for ARFIMA(0. d, 1)-FIGARCH(1, 6 , 1). with
d = 0.45 and 6 = 0.75. The figure graphs the kernel estimates of the simulated small sample densities of the QMLE for
d from the data-generating process of an ARFIMA(0. d, 1)-FIGARCH(1, 6. 1) model with d = 0.45. 8 = 0.3. and 6 =
0.75. The solid line is for a sample size of T = 1000. and the broken line is for T = 500.



4. Estimated Models of Inflation

      Given all the preceding, some hybrid ARFIMA-FIGARCH models were estimated for the
monthly U.S. inflation series. Details of the most appropriate models are given in Table 3. The
estimated value of the long memory parameter in the conditional mean is generally similar to
that of the simpler ARFIMA with homoskedasticity model and is significantly different from
zero or one. As for Table 1, the estimated long memory conditional mean parameter, d, lies in
the range of 0.23 to 0.42, while the U.S. median-weighted inflation series has an estimated d
of 0.61 but is fewer than two robust standard errors away from 0 . 5 0 . V o r Belgium, France,
Italy, Japan, the United Kingdom, and the United States, robust Wald tests can overwhelmingly
reject the hypothesis that 6 = 0, indicating strong evidence of long memory in the conditional
variance as well as the conditional mean. For Germany, the robust Wald statistic is 3.02, and
the hypothesis of stable GARCH(1, 1) cannot be rejected at the 0.05 level. For the other coun-
tries, the FIGARCH model is the preferable parameterization.
      The implied impulse responses for both the conditional mean and the conditional variance
of the United States are given in Figures 7 and 8, respectively. Again, extremely similar results


'There is evidence that a model with 0.5 < d < 1 can still be efficiently estimated by QMLE or alternatively estimated
 on the differenced series (see Smith, Sowell, and Zin 1993: Baillie. Chung. and Tieslau 1996: and part of section 4 of
 Baillie 1996 for a discussion of related issues).
508       Baillie, Hun, and Kwon




      0      1C       21
                       5        7 P
                               2L        40        50       60        '0      80       33       100
                                                                                            Iags
Figure 7. Cumulatl\e Response We~ghts Cond~bonalMean of U S CPI Inflat~on
                                    for




are also available for the other countries but are omitted for reasons of space. In general, the
various diagnostic statistics all indicate the appropriateness of modeling long memory in both
the first two conditional moments for the eight inflation series.



5. Conclusion

     This paper has noted that monthly CPI inflation for eight different industrialized countries
appears to have long memory behavior in both its first and its second conditional moments.
This is the only economic variable that we are aware of that has this property. We suggest a
parametric ARFIMA-FIGARCH model to represent the dual long memory phenomenon, and a
detailed simulation study reveals that the QMLE procedure works well for inferential purposes
in this new model.
     An interesting issue for future research concerns the reasons for the finding of long memory
in data series and whether extensions of the aggregation arguments in Granger ( 1 980) and Ding
and Granger (1996) can account for this phenomenon in inflation. In particular, since the CPI
series are aggregates of two-digit industry classifications, an interesting area for future research
concerns the behavior of different levels of aggregation of the contemporaneous price series.
                                                                           Long Memory of Injution                 509




Figure 8. Cumulative Response Weights for Conditional Variance of U.S. CPI Inflation


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        Further Long Memory Properties of Inflationary Shocks
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[Footnotes]

    4
     On Fractionally Integrated Autoregressive Moving-Average Time Series Models With
    Conditional Heteroscedasticity
    Shiqing Ling; W. K. Li
    Journal of the American Statistical Association, Vol. 92, No. 439. (Sep., 1997), pp. 1184-1194.
    Stable URL:
    http://links.jstor.org/sici?sici=0162-1459%28199709%2992%3A439%3C1184%3AOFIAMT%3E2.0.CO%3B2-Z



References

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    Richard T. Baillie; Ching-Fan Chung; Margie A. Tieslau
    Journal of Applied Econometrics, Vol. 11, No. 1. (Jan. - Feb., 1996), pp. 23-40.
    Stable URL:
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    Persistence in International Inflation Rates
    Christopher F. Baum; John T. Barkoulas; Mustafa Caglayan
    Southern Economic Journal, Vol. 65, No. 4. (Apr., 1999), pp. 900-913.
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    Inflation and the Distribution of Price Changes
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    Small Sample Effects in Time Series Analysis: A New Asymptotic Theory and a New Estimate
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    Efficient Parameter Estimation for Self-Similar Processes
    Rainer Dahlhaus
    The Annals of Statistics, Vol. 17, No. 4. (Dec., 1989), pp. 1749-1766.
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    Long Memory in Inflation Rates: International Evidence
    Uwe Hassler; Jürgen Wolters
    Journal of Business & Economic Statistics, Vol. 13, No. 1. (Jan., 1995), pp. 37-45.
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                        LINKED CITATIONS
                                  - Page 3 of 3 -



    On Fractionally Integrated Autoregressive Moving-Average Time Series Models With
    Conditional Heteroscedasticity
    Shiqing Ling; W. K. Li
    Journal of the American Statistical Association, Vol. 92, No. 439. (Sep., 1997), pp. 1184-1194.
    Stable URL:
    http://links.jstor.org/sici?sici=0162-1459%28199709%2992%3A439%3C1184%3AOFIAMT%3E2.0.CO%3B2-Z


    Short-Term Interest Rates as Predictors of Inflation: On Testing the Hypothesis that the Real
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    Charles R. Nelson; G. William Schwert
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