Further long memory properties of inflationary shocks.

Richard T. Baillie (*)

Young Wook Han (+)

Tae-Go Kwon (++)

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 representation 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 para meter that is significantly different from zero and unity. (1)

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 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 investigation 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 heteroskedastic (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 thi s process are discussed, and estimation of the process is carried out by approximate 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 ARFIMA(p, d, q) model is defined as

[PHI](L)[(1 - L).sup.d]([y.sub.t] - [micro]) = [theta](L)[[epsilon].sub.t], (1)

where E([[epsilon].sub.t]) = 0, E([[epsilon].sup.2.sub.t]) = [[sigma].sup.2], and E([[epsilon].sub.t][[epsilon].sub.s]) = 0 for s [not equal to] t, and [PHI](L) = (1 - [[PHI].sub.1]L ... - [[PHI].sub.p][L.sup.p]), [theta](L) = (1 + [[theta].sub.1]L + ... + [[theta].sub.q][L.sup.q]) 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.sub.t] = [summation over(j=0, [infinity])] [[psi].sub.j][[epsilon].sub.t-j], and the infinite-order autoregressive representation is given by [y.sub.t] = [summation over(j=1, [infinity])] [[pi].sub.j][y.sub.t-j] + [[epsilon].sub.i]. For high lag j, these coefficients decay at a very slow hyperbolic rate, that is, [[psi].sub.j][approximately equal to] [c.sub.1][j.sup.d-1] and [[pi].sub.j] [approximately equal to] [c.sub.2][j.sup.-d-1]. Similarly, the autocorrelation coefficient at large lag j is [p.sub.j] [approximately equal to] [c.sub.3][j.sup.2d-1], where [c.sub.1], [c.sub.2], and [c.sub.3] are constants. For -0.5 < d < 0.5, the process is stationary and invertible, and [y.sub.t], 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 d [less than or equal to] 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 Baum, Barkoulas, and Caglayan (1999) have considered inflation to be fractionally integrated. The series of monthly inflation is defined as [y.sub.t] = 100.[DELTA]log([CPI.sub.t]), where [CPI.sub.t] is monthly CPI. The U.S. series extends from January 1947 through September 1998 (2) and 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 measures, 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 1a and b, respectively, while the autocorrelations of first-differenced inflation are graphed in Figure 1c. 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 1c 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 representation 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,

[PHI](L)[(1 - L).sup.d]([y.sub.t] - [micro]) = [theta](L)(1 - [THETA][L.sup.12])[[epsilon].sub.1], (2)

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 moving 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 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. (3) 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 component 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 generated 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 autocorrelation that is also suggestive of a form of long memory behavior. The nature of the autocorrelations 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) 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 conditional mean and the conditional variance is the ARFIMA(p, d, q)-FIGARCH(P, [delta], Q) model, given by

[PHI](L)[(l - L).sup.d]([y.sub.t] - [mirco]) = [theta](L)(1 - [THETA][L.sup.12])[[epsilon].sub.t], (3)

[[epsilon].sub.t] = [z.sub.t][[sigma].sub.t], (4)

[beta](L)[[sigma].sup.2.sub.t] = [omega] + [1 - [beta](L) - [phi](L)[(1 - L).sup.[delta]][[epsilon].sup.2.sub.t], (5)

where [PHI](L), [theta](L), and [THETA](L) are as defined earlier in Equation 1, while [beta](L) (1 - [[beta].sub.1]L ... [[beta].sub.Q][L.sup.Q]), [phi](L) = (1 + [[phi].sub.1]L + ... + [[phi].sub.p][L.sup.p]) and have all their roots outside the unit circle. Also, [E.sub.t-1][z.sub.t], = 0, [Var.sub.t-1] [z.sub.t] = 1, while [E.sub.t-1] is the expectations operator conditioned on a sigma field set of information at time t - 1, and [[sigma].sup.2.sub.t], is the conditional variance and is a positive, time-varying, and measurable function with respect to the information set, which is available at time t - 1.(4) If [[sigma].sup.2.sub.t], = [omega], a constant, the process reduces to the ARFIMA(p, d, q) model of Granger and Joyeux (1980) and Hosking (1981). Then [y.sub.t] will be covariance stationary and invertible for -0.5 < d < 0.5 and will be mean reverting for d < 1.

When [phi](L) = [theta](L) = 1, the process reduces to the FIGARCH(P, [delta], Q) conditional variance process of Baillie, Bollerslev, and Mikkelsen (1996), and the conditional variance, [[sigma].sup.2.sub.t], 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, [delta], 0) process can be expressed as [[sigma].sup.2.sub.t] =[omega]/(1 - [beta]) +[lambda][(L)[epsilon].sup.2.sub.t], where [[lambda].sub.k] = [GAMMA](k + [delta] - 1)/{[GAMMA](k)[GAMMA]([delta])}.[(1 - [beta]) - (1 - [delta])/k], and for large lags k, [[lambda].sub.k] = [(1 -[beta])/[GAMMA]([delta])].[k.sup.[delta]-1], which generates slow hyperbolic rate of decay on the impulse response weights. The process is strictly stationary and ergodic for 0 [less than or equal to] [delta] [less than or equal to] 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 [delta][not equal to] 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 conditional mean.

Assuming conditional normality, the logarithm of the likelihood can be expressed in the time domain as

L([lambda], [[epsilon].sub.1], [[epsilon].sub.2],...,[[epsilon].sub.T]) = -(T/2)In(2[pi]) - (1/2) [summation over (t=1,T)] [In([sigma].sup.2.sub.t])+ [[epsilon].sup.2.sub.1][[sigma].sup.-2.sub.t], (6)

where [lambda]' = ([micro],[[PHI].sub.1],[[PHI].sub.p],[[theta].sub.1], ..., [[theta].sub.q], [THETA], d, [omega], [phi], [delta], [beta]). 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 [[epsilon].sup.2.sub.t], t = 0, -1, -2, ..., are fixed at the sample unconditional variance. The initial observations [y.sub.0], [y.sub.-1], [y.sub.-2], ..., 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 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 [micro] either zero or known and show the MLE are [T.sup.1/2] 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 [micro] unknown. They show that the parameter estimates are asymptotically normal, with the ARMA parameter estimates again being [T.sup.1/2] consistent, while the MLE of [micro] is [T.sup.1/2 - d] consistent. For the conditional variance process, asymptotic normality and [T.sup.1/2] consistency has been derived only for the integrated generalized autoregressive conditionally heteroskedastic (IGARCH[1, 1]) model by Lee and Hansen (1994) and Lumsdaine (1996). Their proofs require [z.sub.t], in Equation 4 to be stationary and ergodic, together with three other relatively mild conditions on [z.sub.t]. While simulation evidence for FIGARCH and other complicated parametric ARCH models suggests QMLE to be consistent and asymptotically normal, a fully general theoret ical treatment is as yet unavailable. Consequently, we conjecture that with [micro] unknown, the limiting distribution of the QMLE is

[D.sub.T]([lambda] - [lambda]) [right arrow] N[0, [{[D.sup.-1.sub.T]A[([[lambda].sub.0]).sup.-1]B([[lambda].sub.0])A[([ [lambda].sub.0]).sup.-1][D.sup.-1.sub.T]}.sup.-1]], (7)

4. Estimated Models of Inflation

where A(*) and B(*) are the Hessian and outer product gradient, respectively, when evaluated at the true parameter values [[lambda].sub.0] and diag([D.sub.T]) = [[T.sup.1/2-d], [T.sup.1/2], ..., [T.sup.1/2]]. The adequacy of this estimation method was assessed by means of a detailed Monte Carlo study where ARFIMA(0, d, 1)-FIGARCH(l, [delta], 1) models were simulated for the three different parameter designs of (i) O = 0.3, d = 0.30, [beta] = 0.40, and [delta] = 0.50; (ii) 0 = 0.3, d = 0.45, [beta] = 0.65, and [delta] = 0.75; and (iii) 0 = 0, d = 0.60, [beta] = 0.70, and [delta] 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 homoskedasticity. The results of the simulati ons 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 5 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 8 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.

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 0.50. (5) For Belgium, France, Italy, Japan, the United Kingdom, and the United States, robust Wald tests can overwhelmingly reject the hypothesis that [delta] = 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 co untries, 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 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 (1980) 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.

(*.) Department of Economics and Department of Finance, Michigan State University, East Lansing, MI 48824, USA; E-mail [e-mail]; corresponding author.

(+.) 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-0071619.

Received June 2000; accepted January 2001.

(1.) These studies have either estimated the long memory parameter by semiparametric procedures or alternatively from estimating ARFIMA models.

(2.) Baillie, Chung, and Tieslau (1996) use CPI series from January 1947 through September 1990. Results from this sample and a much longer sample back to 1913 are available from the authors on request.

(3.) The application of QMLE in this study differs from that of Baillie, Chung, and Tieslau (1996), who estimated ARFIMA-GARCH(1, 1) models with conditional Student t-densities.

(4.) Some 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.

(5.) 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).

References

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[Figure 1 Omitted]

[Figure 2 omitted]

[Figure 3 Omitted]

[Figure 4 omitted]

[Figure 5 omitted]

[Figure 6 omitted]

[Figure 7 Omitted]

[Figure 8 Omitted]

Table 1

Estimated ARFIMA Models for Countries' Monthly Inflation Rates
(1 - [PHI]L)[(1 - L).sup.d] ([y.sub.t] - [micro]) = (1 + [[theta].sub.1]
+ L + [[theta].sub.2][L.sup.2](1 + [THETA][L.sup.12][[epsilon].sub.v]
[[epsilon].sub.t] are i.i.d. (0, [omega])


                 Belgium   France  Germany   Italy    Japan    Korea

[micro]           0.277    0.007    0.237    0.296    0.257    0.932
                 (0.081)  (0.170)  (0.063)  (0.194)  (0.184)  (0.446)
d                 0.290    0.409    0.211    0.407    0.440    0.418
                 (0.036)  (0.103)  (0.042)  (0.044)  (0.129)  (0.094)
[PHI]              --        --       --       --       --       --
                   --        --       --       --       --       --
[[theta].sub.1]    --        --       --       --    -0.349      --
                   --        --       --       --    (0.121)     --
[[theta].sub.2]    --        --       --       --    -0.235      --
                   --        --       --       --    (0.046)     --
[THETA]           0.169   -0.533    0.280    0.145    0.291    0.189
                 (0.036)  (0.065)  (0.034)  (0.045)  (0.038)  (0.053)
[omega]           0.100    0.108    0.089    0.152    0.449    0.582
                 (0.009)  (0.033)  (0.009)  (0.014)  (0.037)  (0.063)
ln(L)            -125.39  -150.47  -104.27  -237.83  -509.24  -396.14
Q(20)              81.08    27.23    38.55    23.87    28.39    32.52
[Q.sup.2](20)      69.17  128.358    21.22   133.33   142.83   133.91
Skewness            0.10    -0.82     0.58     0.91     0.72     0.25
Kurtosis            4.67    11.35     5.54     5.41     4.45     5.00
[W.sub.d=1]       410.88   112.73   336.40   314.44    25.87    45.46

                 United   United
                 Kingdom  States     USm

[micro]           0.004    0.368    0.288
                 (0.033)  (0.224)  (0.105)
d                 0.312    0.391    0.599
                 (0.049)  (0.063)  (0.120)
[PHI]              --     -0.134   -0.100
                   --     (0.077)  (0.125)
[[theta].sub.1]    --       --       --
                   --       --       --
[[theta].sub.2]    --       --       --
                   --       --       --
[THETA]          -0.766    0.048     --
                 (0.042)  (0.058)    --
[omega]           0.225    0.102    0.024
                 (0.033)  (0.010)  (0.003)
ln(L)            -328.86  -173.40  172.23
Q(20)              21.74    30.82   55.83
[Q.sup.2](20)      22.12   647.02  312.07
Skewness            1.18     0.18   -0.09
Kurtosis           11.32     7.45    5.20
[W.sub.d=1]       125.36    58.39   36.85

The first eight columns refer to monthly CPI inflation series, while USm
indicates the median-weighted inflation series described in the text.
In(L) is the value of the maximized Gaussian log likelihood; and QMLE
standard errors are presented in parentheses below corresponding
parameter estimates. The Q(20) and [Q.sup.2](20) are the Ljung-Box test
statistics with 20 degrees of freedom based on the standardized
residuals and squared standardized residuals, respectively. The sample
skewness and kurtosis are also based on the standardized residuals. The
series for France and United Kingdom were also seasonally differenced.
Table 2

Simulation Results of Estimating the ARFIMA(0, d, 1)-FIGARGH(1,
[delta], 1) and ARFIMA(0, d, 0)- FIGARGH(1, [delta], 1) Models

                                                      d

Model                         d        [delta]    T         Bias

ARFIMA(0 d, 1)-              0.3        0.5       500      -0.027
 FIGARGH(1, [delta], 1) (a)  0.3        0.5      1000      -0.013
                             0.45       0.75      500      -0.005
                             0.45       0.75     1000       0.002
ARFIMA(0, d,, 0)-
 FIGARGH(1, [delta], 1) (b)  0.6        0.8      1000       0.013

                                      d                   [delta]
                                       Standard
Model                        RMSE       Error    Bias       RMSE

ARFIMA(0 d, 1)-              0.093      0.089    0.030      0.223
 FIGARGH(1, [delta], 1) (a)  0.059      0.057    0.027      0.158
                             0.106      0.106    0.030      0.237
                             0.075      0.075    0.034      0.177
ARFIMA(0, d,, 0)-
 FIGARGH(1, [delta], 1) (b)  0.033      0.031    0.015      0.175

                              [delta]
                             Standard
Model                         Error

ARFIMA(0 d, 1)-               0.220
 FIGARGH(1, [delta], 1) (a)   0.156
                              0.235
                              0.173
ARFIMA(0, d,, 0)-
 FIGARGH(1, [delta], 1) (b)   0.174

(a)The table reports the averages of biases and RMSE of the QMLE of the
estimates of the d and [delta] parameters from the simulation design (i)
and (ii). The results are based on 1000 replications in all cases.

(b)The table reports the averages of biases and RMSE of the QMLE of the
estimates of the d and [delta] parameters from the simulation design
(iii). The results are based on 1000 replications in all cases.
Table 3

Estimated ARFIMA-FIGARGH Models for Countries' Monthly Inflation
Rates(1 - [PHI]L)[(1 - L).sup.d] ([y.sub.t] - [micro]) = (1 +
[[theta].sub.1]L + [[theta].sub.2][L.sup.2]) (1 + [THETA][L.sup.12])
[[epsilon].sub.t], [[epsilon].sub.t] = [Z.sub.t] [[sigma].sub.t][1 -
[beta]L][[sigma].sup.2.sub.t] = [omega] + [1 - [beta]L - [phi]L[(1 -
L).sup.[delta]]] [[epsilon].sup.2.sub.t]


                   Belgium  France   German    Italy    Japan    Korea

[micro]             0.246   -0.002    0.248    0.095    0.146    0.591
                   (0.007)  (0.034)  (0.072)  (0.141)  (0.179)  (0.390)
d                   0.281    0.347    0.226    0.362    0.412    0.352
                   (0.035)  (0.062)  (0.042)  (0.036)  (0.116)  (0.096)
[PHI]                 --       --       --       --       --       --
                      --       --       --       --       --       --
[[theta].sub.1]       --       --       --       --    -0.357      --
                      --       --       --       --    (0.113)     --
[[theta].sub.2]       --       --       --       --    -0.285      --
                      --       --       --       --    (0.042)     --
[THETA]             0.201   -0.681    0.278    0.214    0.350    0.248
                   (0.034)  (0.049)  (0.031)  (0.038)  (0.034)  (0.040)
[delta]             0.324    0.331    0.256    0.687    0.317    0.949
                   (0.135)  (0.153)  (0.147)  (0.132)  (0.101)  (0.422)
[omega]             0.011    0.001    0.004    0.001    0.033    0.025
                   (0.008)  (0.002)  (0.004)  (0.002)  (0.030)  (0.029)
[beta]              0.256    0.899    0.848    0.773    0.253    0.735
                   (0.138)  (0.280)  (0.060)  (0.070)  (0.122)  (0.336)
[phi]                 --     0.859    0.710    0.275      --       --
                      --    (0.383)  (0.113)  (0.144)     --       --
ln(L)              -106.70  -63.63   -98.32   -177.68  -471.60  -356.86
Q(20)                74.69   19.49    34.46     16.91    33.76    24.36
[Q.sup.2](20)        26.37   31.09    20.94     17.24     3.76     7.23
Skewness              0.40   -0.02     0.81      0.63     0.69     0.82
Kurtosis              4.29    6.69     5.78      4.63     4.35     5.84
[W.sub.[delta]=0]     6.85    4.70     3.02     26.51     9.03     5.05

                   United   United
                   Kingdom  States     USm

[micro]             0.031    0.518    0.269
                   (0.058)  (0.266)  (0.072)
d                   0.364    0.414    0.167
                   (0.057)  (0.08)   (0.063)
[PHI]                 --    -0.205   -0.300
                      --    (0.083)  (0.069)
[[theta].sub.1]       --       --       --
                      --       --       --
[[theta].sub.2]       --       --       --
                      --       --       --
[THETA]            -0.728    0.141      --
                   (0.035)  (0.035)     --
[delta]             0.633    0.644    0.871
                   (0.314)  (0.316)  (0.164)
[omega]             0.001    0.001    0.001
                   (0.002)  (0.001)  (0.001)
[beta]              0.893    0.811    0.670
                   (0.078)  (0.124)  (0.152)
[phi]               0.635    0.467      --
                   (0.153)  (0.123)     --
ln(L)              -284.74  -65.62   230.39
Q(20)                21.93   34.83    24.61
[Q.sup.2](20)        10.80   16.63    17.37
Skewness              0.21    0.09     0.29
Kurtosis              4.74    4.50     3.52
[W.sub.[delta]=0]     4.15    4.16    28.18

For explanation, see Table 1.