Long Memory Inflationary Dynamics: The Case of Brazil
It has been argued by several authors that the inflationary dynamics in Brazil follow a unit root process, thus displaying some inertia. Indeed, Cati, et al. (Journal of Applied Econometrics, 1999) have found that the inflationary dynamics in Brazil are nearly fully inertial. We estimate the fractional differencing parameter using an ARFIMA specification for the inflation rate in that country and our results suggest that the inflationary dynamics are better modeled by a long memory process than by a unit root mechanism, thus implying that there is no inertia in inflation, contrary to what has been found by other researchers. We also found that the estimates of the fractional parameter are invariant to first differencing.
Studies in Nonlinear Dynamics & Econometrics Volume 7, Issue 3 2003 Article 3 Long Memory Inflationary Dynamics: The Case of Brazil Valderio A. Reisen Francisco Cribari-Neto Mark J. Jensen University Federal do Espirito Santo, University Federal de Pernambuco, Brigham Young University, jensenm@byu.edu Copyright c 2003 by The Berkeley Electronic Press. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher, bepress, which has been given certain exclusive rights by the author. Studies in Nonlinear Dynamics & Econometrics is produced by The Berkeley Electronic Press (bepress). http://www.bepress.com/snde Long Memory Inflationary Dynamics: The Case of Brazil Valderio A. Reisen, Francisco Cribari-Neto, and Mark J. Jensen Abstract It has been argued by several authors that the inflationary dynamics in Brazil follow a unit root process, thus displaying some inertia. Indeed, Cati, et al. (Journal of Applied Econometrics, 1999) have found that the inflationary dynamics in Brazil are nearly fully inertial. We estimate the fractional differencing parameter using an ARFIMA specification for the inflation rate in that country and our results suggest that the inflationary dynamics are better modeled by a long memory process than by a unit root mechanism, thus implying that there is no inertia in inflation, contrary to what has been found by other researchers. We also found that the estimates of the fractional parameter are invariant to first differencing. 1. Introduction It is common to find time series that exhibit long memory dynamics in a wide number of fields, such as astronomy, hydrology, computer science, economics, among others. The ARFIMA (autoregressive fractionally integrated moving average) process has been widely used to model time series with such a property; see Beran (1994) for a review. When d, the integration parameter in the ARFIMA specification, belongs to the interval ( 0.5, 0.5), the process is stationary, invertible, and may display either the persistence or the antipersistency property. The most interesting case corresponds to d > 0 (long memory or persistence). Here, the autocorrelations are not summable and the spectral density tends to infinity as frequencies approach zero. Many estimators of the parameter d based on parametric and semiparametric estimation have been proposed in the literature. Among the semiparametric estimation approaches, there are those of Geweke and Porter Hudak (1983), Reisen (1994), and Robinson (1995). They all make use of the regression equation of the log of the spectral density of the process. Robinson (1994) and Lobato and Robinson (1996) develop an alternative semiparametric approach based on averages of the periodogram over a band of frequencies around the origin. The parametric estimators of the degree of fractional differencing are usually obtained using maximum likelihood and pseudo-maximum likelihood methods by assuming that the form of the spectral density is known (see Dahlhaus, 1989; Fox and Taqqu, 1986; Sowell, 1992; and Taqqu, et al., 1995). Recent simulation studies comparing different estimation procedures for long memory processes are Bisaglia and Guegan (1998), and Hurvich and Deo (1999). They focus on the estimation of the parameter d. In the ARFIMA(p,d,q) model, however, all parameters, including the autoregressive and moving average ones, have to be estimated. These parameters can be simultaneously estimated using the parametric approach. In the case of semiparametric methods, the parameters are estimated in two steps: at the outset one estimates d; the autoregressive and moving average parameters are estimated in a second step. Smith, et al. (1997) and Reisen, et al. (2001) investigated the bias in both the estimate of the fractional integration parameter d and in the the short-run autoregressive (AR) and moving average (MA) parameter estimates in ARFIMA models using parametric and semiparametric estimation methods. Recent interest has been devoted to the situation where d > 0.5. Here, the process is still long memory but becomes non-stationary. Special interest lies with the case where 0.5 < d < 1.0 (level reversion); see Hurvich and Ray (1995) and Lopes, et al. (2003). Reisen et al.: Long Memory Inflationary Dynamics: The Case of Brazil 1 Published by The Berkeley Electronic Press, 2003 The performance of traditional unit root tests under long memory has also been investigated. As shown in Diebold and Rudebush (1991) and Hassler and Wolters (1994), Dickey Fuller type tests have low power against fractional alternatives. Consequently, such tests may often suggest the need to take (integer) differences of the data when this operation is not appropriate. A discussion of the misspecification of stationary ARFIMA processes as nonstationary ARIMA models can be found in Crato and Taylor (1996). The consistency of the KPSS stationarity test against fractionally integrated alternatives, i.e., the ARFIMA process, is shown by Lee and Amsler (1997). The authors show how the KPSS statistic can be used to distinguish between the following: (a) short memory, i.e., d = 0.0; (b) stationary long memory, i.e., 0.0 < d < 0.5; (c) nonstationary long memory, i.e., 0.5 d < 1 and (d) unit root (d = 1). The results suggest that one can consistently distinguish (a) and (b) from (c) and (d), but cannot distinguish between (c) and (d). The goal of this paper is to investigate whether the inflationary dynamics in Brazil display long memory properties. Brazil has often been cited as a country where inflation displays inertia . That means that in the absence of further shocks, inflation tends to reproduce itself from one period to the next (see, e.g., Cati, et al., 1999; Durevall, 1998; and Novaes, 1993). Full inertia corresponds to a random walk in inflation, where innovations are fully persistent in the sense that a one-percent shock to inflation today changes one s long-run inflation forecast by exactly one percent. When inflation follows an ARIMA(p,1,q) process, it displays some inertia, which can be small (close to zero) or large (greater than one). When d < 1, however, inflation displays long memory (i.e., it takes a long time for shocks to die out) but no inflation inertia (i.e., shocks do die out).1 A number of unorthodox stabilization plans have been put in effect between the mid 1980s and the early 1990s based on the diagnosis that the Brazilian inflationary dynamics were mainly inertial. Our results suggest that such a diagnosis was incorrect. 2. The ARFIMA(p,d,q) model The discrete time ARFIMA(p, d, q) model is given by (B)(1 B)dXt = (B) t, d IR, t WN(0, 2 ), (2.1) 1See Baillie, et al. (1996), Hassler and Wolter (1995), and Baum et al. (1999) for empirical evidence of long memory behavior in the inflation data of G7, developing, and high inflation rate countries. 2 Studies in Nonlinear Dynamics & Econometrics Vol. 7 [2003], No. 3, Article 3 http://www.bepress.com/snde/vol7/iss3/art3 where t = 1, 2, . . . , n, B is the lag operator, (B) = 1 1B pBp and (B) = 1 1B qBq are polynomials of orders p and q, respectively, with all roots outside of the unit circle, and WN stands for white noise. When d ( 0.5, 0.5) the ARFIMA process is stationary and invertible, and its spectral density, f(w), is given by f(w) = fu(w){2 sin(w/2)} 2d, w [ , ], (2.2) where fu( ) is the spectral density of an ARMA(p, q) process. As shown by Jensen (1999), the variance of the ARFIMA process wavelet coefficients equals 2m = 2m+1 2 m+1 2 m f(w) dw, (2.3) where m is the wavelet coefficients scaling parameter. See Beran (1994), Hosking (1981) and Reisen (1994) for detailed accounts of the ARFIMA process. We consider five alternative estimators for the parameter d. Three of them are obtained using semiparametric procedures based on regression equations constructed from the logarithm of expression (2.2); the fourth estimator is an approximate, parametric, maximum likelihood estimator (MLE) in the frequency domain proposed by Fox and Taqqu (1986); and lastly the fifth is an approximate, parametric, maximum likelihood estimator in the wavelet domain proposed by Jensen (1999). These estimators are briefly described below. The first estimator, hereafter denoted by dp, was proposed by Geweke and Porter Hudak (1983) who used the periodogram function I(w) as an estimate of the spectral density function in expression (2.2). The number of observations in the regression equation (bandwidth) is a function of the sample size n, that is, g(n) = n , with 0 < < 1. The second estimator, hereafter denoted by dsp, was introduced by Reisen (1994). This regression estimator is obtained by replacing the spectral density function in expression (2.2) by the smoothed periodogram function based on the Parzen lag window. In this estimation method, g(n) is chosen as above and the truncation point in the Parzen lag window is m = n , 0 < < 1. The appropriate choices for and were investigated by Geweke and Porter Hudak (1983) and Reisen (1994), respectively. The third estimator we consider is the one proposed by Robinson (1995). It is hereafter denoted by drb. This estimator is a modified form of the logperiodogram one which regresses {ln I(wi)} on ln {2 sin(wi/2)}2, for i = , + 1, , g(n), where is the lower truncation point, with diverging to infinity more slowly than g(n). Reisen et al.: Long Memory Inflationary Dynamics: The Case of Brazil 3 Published by The Berkeley Electronic Press, 2003 The fourth estimator is a parametric estimator suggested by Fox and Taqqu (1986), and will be hereafter denoted by dW. The estimator dW is based on the periodogram and involves the function Q( ) = I(w) fX(w, )dw, (2.4) where fX(w, ) is the known spectral density function at frequency w and denotes the vector of unknown parameters. The Whittle estimator is the value of which minimizes the function Q( ). For the ARFIMA(p,d,q) process, the vector contains the parameter d and also all of the unknown autoregressive and moving average parameters. For more details, see Beran (1994), Dahlhaus (1989), and Fox and Taqqu (1986). In practice, the estimator dW is obtained by using the discrete form of Q( ), as in Dahlhaus (1989, p. 1753), that is, Ln( ) = 1 2n n 1 j=1 ln fX(wj, ) + I(wj) fX(wj, ) . (2.5) The fifth estimator, constructed by Jensen (1999), is an alternative, approximate maximum likelihood estimator to Fox and Taqqu (1986). Based on the wavelet representation of the ARFIMA process likelihood function, the wavelet MLE of d, which we will denote as da, is found by maximizing the ARFIMA process wavelet coefficients likelihood function Lw( ) = 1 2 [lo g2 n] m=1 n 2m log m( ) + w mwm 2m ( ) , (2.6) where w m = (wm,1, . . . ,wm,n/2m), is the vector of wavelet coefficients, wm,j corresponding to scale m, and translation j, and [ ] is the integer operator. If the number of observations n is not a power two the data is zero padded for estimation of the wavelet MLE. The asymptotic properties of some of these estimators under nonstationarity have recently been derived; see, e.g., Velasco (1999a,b). Monte Carlo results on the finite-sample performance of these (and other) estimators under nonstationarity are presented in Lopes, et al. (2003). Their results suggest that the estimators perform well even when 0.5 d < 1. 3. Data and results The data consist of the inflation index known as IGP DI ( Indice Geral de Pre cos, Disponibilidade Interna ) computed by the Getulio Vargas Founda- 4 Studies in Nonlinear Dynamics & Econometrics Vol. 7 [2003], No. 3, Article 3 http://www.bepress.com/snde/vol7/iss3/art3 0 2 4 6 8 0.0 0.2 0.4 0.6 0.8 1.0 lag Figure 1: Sample autocorrelation function for Brazil s inflation rate where the major ticks on the x-axis represent the lag argument in years. tion, and ranges from February 1944 through February 2000 (673 observations). Figure 1 displays the sample autocorrelation of the series function up to lag 100. A visual examination of the correlogram suggests that the data are nonstationary and possibly characterized by long memory behavior since the sample autocorrelations decay very slowly. The semiparametric estimation results of d for different bandwidths, g(n), are presented in Table 1. The truncation point in the smoothed periodogram was set equal to 0.9 (see Reisen, 1994, for further details), and the lower truncation in the Robinson method was taken to be 2. The results in Table 1 show that, regardless of the value of g(n), the estimates strongly indicate nonstationary long memory behavior with 0.5 < d < 1.0 (level reversion). Also, when all frequencies are used (g(n) = 336), the estimated values of d are very similar to the one obtained with the Whittle method (see the case ARFIMA(0, dw,0) in Table 2). The standard errors (s) are also close to the their asymptotic values ( d. ). The best fitted ARFIMA(p,d,q) models by the parametric and nonparametric estimators are reported in Table 2, along with the parametric Whittle and wavelet estimates of the ARFIMA(0,d,0) model. Other than the Reisen et al.: Long Memory Inflationary Dynamics: The Case of Brazil 5 Published by The Berkeley Electronic Press, 2003 bandwidth smoothed periodogram periodogram Robinson g(n) = 25 dsp = 0.705 dsp = 0.065 s dsp = 0.096 dp = 0.753 dp = 0.157 s dp = 0.143 drb = 0.769 drb = 0.102 s drb = 0.179 g(n) = 95 dsp = 0.615 dsp = 0.029 s dsp = 0.039 dp = 0.595 dp = 0.072 s dp = 0.053 drb = 0.585 drb = 0.052 s drb = 0.058 g(n) = 336 dsp = 0.832 dsp = 0.017 s dsp = 0.022 dp = 0.841 dp = 0.040 s dp = 0.033 drb = 0.856 drb = 0.027 s drb = 0.039 Table 1: Semiparametric estimation results where s d is the estimated standard error and d is the asymptotic standard error. The denotes the result obtained using all possible frequencies, i.e., g(n) = [n 2 ], where [.] denotes the greatest integer function. ARFIMA results where p = q = 0, the results reported for the parametric estimators are of the ARFIMA(p,d,q) model whose Akaike Information Criterion (AIC) was the smallest from the array of models where p, q = 0, 1, 2, 3. For the semiparametric estimators, when a model with autoregressive and/or moving average components is estimated, we first filter the inflation series with d (g(n) = n0.5 = 25, see Reisen, 1994) and use the filtered series, (1 L) dXt, in the estimation of the ARFIMA(p, d, q) model s short-memory parameters. Lastly, the best semiparametrically estimated model is chosen as the model with the smallest AIC from among the ARFIMA models with p, q = 0, 1, 2, 3. The semiparametric estimation of d and the parametric estimation of d jointly with AR and/or MA parameters were performed using FORTRAN code linked to IMSL libraries. The estimation of AR and/or MA parameters for the filtered series obtained from semiparametric estimates of d was carried out using S-PLUS. 6 Studies in Nonlinear Dynamics & Econometrics Vol. 7 [2003], No. 3, Article 3 http://www.bepress.com/snde/vol7/iss3/art3 method model AR and/or MA AIC/p-values * estimates periodogram ARFIMA(3,d,0) 1 = 0.2021 1790.2; 0.51, 0.14 dp = 0.753 2 = 0.0517 3 = 0.1627 smoothed ARFIMA(3,d,0) 1 = 0.2447 1788.0; 0.65, 0.19 periodogram dsp = 0.705 2 = 0.0619 3 = 0.1553 Robinson ARFIMA(3,d,0) 1 = 0.1882 1791.0; 0.45, 0.12 drb = 0.769 2 = 0.0481 3 = 0.165 Whittle ARFIMA(0,d,0) 1827.2; 0.0, 0.0 dw = 0.867 dw = 0.049 Whittle ARFIMA(1,d,0) = 0.4658 1796.3; 0.005, 0.04 dw = 0.4874 dw = 0.049 wavelet MLE ARFIMA(0,d,0) 3723.7; 0.0, 0.0 da = 0.7942 wavelet MLE ARFIMA(2,d,1) 1 = 0.8256 3412.6; 0.0, 0.0 da = 0.9333 2 = 0.4665 = 0.6818 unit root ARFIMA(1,1,0) = 0.0012 1834.0; 0.0, 0.0 unit root ARFIMA(3,1,2) 1 = 0.2673 1789.0; 0.37, 0.14 2 = 0.5156 3 = 0.1689 1 = 0.3194 2 = 0.5631 Table 2: Estimation and identification results. The p-values are for the modified Box Pierce 2 statistic with 4 and 32 degrees of freedom respectively. Reisen et al.: Long Memory Inflationary Dynamics: The Case of Brazil 7 Published by The Berkeley Electronic Press, 2003 The reported estimates in Table 2 reveal that the value of d range from 0.49 through 0.94. The smallest estimate (0.49) is obtained from the ARFIMA(1,d,0) model using Whittle s method, and the largest one is obtained with the wavelet estimate of the ARFIMA(2,d,1) model. The smallest AIC value corresponds to the ARFIMA(3,d,0) where estimation was carried out by smoothed periodogram. In all cases, the estimates are below one, i.e., below the unit root case. In short, the results of Table 2 suggest that the series is nonstationary and displays long memory. However, the order of integration is less than one, thus implying that there is no inflation inertia in the Brazilian inflationary process. These findings are similar to what Baillie et al. (1996) found for Brazilian inflationary process using an ARFIMA model with conditionally heteroskedastic innovations. A remark on the Whittle estimated ARFIMA(0,d,0) model versus the ARFIMA(1,d,0) model is in order. As is well know, in the Whittle estimation the short and long memory parameters are estimated simultaneously in the frequency domain with the parameters entering the objective function through the power spectrum found in (2.2). In comparison with the Whittle estimate of the integration parameters when no short-memory parameters are present ( dw = 0.8967), the long memory behavior becomes diluted into a smaller valued integration parameter, dw = 0.4874, and into the autoregressive parameter, = 0.4658, when short memory parameters are included. Our estimates suggest that the Whittle estimator of d may be sensitive to the presence of short memory parameters. Our findings with the wavelet estimator reveal that including short memory parameters in a long memory model does not present a problem for the wavelet estimator. In both the ARFIMA(0,d,0) and ARFIMA(2,d,1) model, the wavelet estimator of the integration parameter are slightly less than one, with da = 0.7942 when no short memory parameters are found in the model, and da = 0.9333 when the model is a ARFIMA(2,d,1). Since in both models the wavelet estimates of d are near a unit root, we estimate the ARIMA(2,1,1) model: (1 0.891B + 0.092B2)(1 B)Xt = (1 0.946B) t. Other than the autoregressive parameter estimate for the second lag, the short memory parameter estimates are similar to the those estimated with the wavelet estimator; with the first-order autoregressive parameter estimates being most alike, and the signs of the short memory parameters estimates being the same. 8 Studies in Nonlinear Dynamics & Econometrics Vol. 7 [2003], No. 3, Article 3 http://www.bepress.com/snde/vol7/iss3/art3 bandwidth smoothed periodogram periodogram Robinson g(n) = 25 dsp = 0.290 dp = 0.250 drb = 0.248 Table 3: Estimation results based on first differences. This invariance to first differencing was also found to hold for the semiparametric estimates of d when g(n) = 95 and g(n) = [n 2 ]. Hurvich and Ray (1995) investigated the invariance property of the Geweke and Porter Hudak (1983) estimator. They showed that this estimator, in general, is not invariant to first differencing. However, our estimates of the first order differenced series indicate that such invariance does hold for Brazilian inflation. Overall, the semiparametric estimates have proved invariant to first differencing, i.e., the estimates of d based on the original data are quite close to one plus the estimates of d based on the differenced data (see Table 3). This was also observed for the Whittle and wavelet estimates in the case where there are no AR and/or MA parameters. The Whittle and wavelet estimates were, respectively, dW = 0.1269 and da = 0.1238. The invariance to first differencing is taken as further evidence that d < 1. The next step is to investigate the forecasting performance of the different estimated models, in order to identify which one has the best forecasting ability. The forecasting experiment consists of estimating the model s parameters with the information found in the first 620 observation (February, 1944 to September, 1995) and predicting out-of-sample 1, 5, 10, and 20 months into the future. We choose not to iteratively update the parameter estimates with an ever increasing information set because when we re-estimated the models with each additional monthly observation added to the original 620 observations we find that the parameter estimates are only different at the fourth order of precision from the estimates found with the first 620 observations. The h-step ahead forecasts of inflation are obtained using a truncated version of the infinite AR representation of the ARFIMA process as in Brockwell and Davis (1991, p. 533) and Reisen and Lopes (1999). The latter analyzes forecasting issues in long memory processes, including the effect of the truncation point in the forecast expression on the generated forecasts. Based on their results, we chose the truncation point where the estimate of the last AR coefficient in the infinite AR representation of the ARFIMA process is smaller than 0.0001. [See also the discussion in Brodsky and Reisen et al.: Long Memory Inflationary Dynamics: The Case of Brazil 9 Published by The Berkeley Electronic Press, 2003 Hurvich (1998).] The forecasts are evaluated on three measures; the mean squared forecast error (MSE): MSE = 1 n h m + 1 n h k=m em(k + h)2, the mean absolute percentage error (MAPE): MAPE = 100 n h m + 1 n h k=m em(k + h) Xk+h , and the mean forecasting error (MFE): MFE = 1 n h m + 1 n h k=m em(k + h), where em(k + h) = Xk+h Xm(k + h) is the out-of-sample forecast error of the k + h observation given information up to time period m = 620 k. Table 4 summarizes each estimator and model s forecast performance.2 The best performing models are the ARFIMA(0,d,0) model estimated with the Whittle and wavelet methods. In the immediate short run (h = 1, 5) the wavelet estimated ARFIMA(0,0.794,0) model produces the best forecasts as gauged by all reported measures. For the intermediate (h = 10) and long-run (h = 20) it is the forecasting measures of the Whittle estimated ARFIMA(0,0.867,0) model that are smallest. These two models delivered the best out-of-sample forecasts, although they did not display the smallest AIC values (see Table 2). It is noteworthy that the three measures of accuracy of forecasts considered (MSE, MAPE and MFE) point to the same conclusion, namely: for moderately short forecasting horizons (h = 1, 5), the best forecasts are obtained from the ARFIMA(0,d,0) model, where the fractional integration parameter is estimated by wavelet methods; on the other hand, for moderately long forecasting horizons (h = 10, 20), it is best to employ forecasts from the same class of models, but with the Whittle estimate of d. The results show, indeed, that the choice of estimation method for the fractional integration parameter has a sizeable effect on the quality of the forecasts of the ARFIMA(0,d,0) model. For instance, for h = 10 (forecasts ten months 2We do not report the MSE, MAPE, nor the MFE for the Robinson estimator because this estimator s forecast measures are nearly indistinguishable from those compute from the periodogram estimator. 10 Studies in Nonlinear Dynamics & Econometrics Vol. 7 [2003], No. 3, Article 3 http://www.bepress.com/snde/vol7/iss3/art3 h method/model 1 5 10 20 MSE periodogram; ARFIMA(3,0.753,0) 0.070 2.587 7.036 17.840 MSE smoothed periodogram; ARFIMA(3,0.705,0) 0.035 7.720 18.594 37.840 MSE Whittle; ARFIMA(0,0.867,0) 0.379 0.687 0.517 1.994 MSE Whittle; ARFIMA(1,0.487,0) 0.194 10.272 26.426 51.139 MSE wavelet; ARFIMA(0,0.794,0) 0.007 0.645 2.468 8.613 MSE wavelet; ARFIMA(2,0.933,1) 34.910 55.650 44.148 35.503 MSE unit root; ARFIMA(3,1,2) 3.690 50.310 106.800 162.410 MSE unit root; random walk 1.688 4.283 4.430 3.713 MAPE periodogram; ARFIMA(3,0.753,0) 4.780 26.610 48.673 75.955 MAPE smoothed periodogram ARFIMA(3,0.705,0) 3.380 47.290 80.743 113.400 MAPE Whittle; ARFIMA(0,0.867,0) 11.080 14.377 12.799 23.635 MAPE Whittle; ARFIMA(1,0.487,0) 7.930 57.154 97.410 132.969 MAPE wavelet; ARFIMA(0,0.794,0) 1.480 11.857 27.204 50.356 MAPE wavelet; ARFIMA(2,0.933,1) 106.400 154.687 138.178 115.806 MAPE unit root; ARFIMA(3,1,2) 34.58 134.474 202.500 244.370 MAPE unit root; random walk 23.39 42.920 44.043 38.656 MFE periodogram; ARFIMA(3,0.753,0) 0.265 1.077 2.221 3.781 MFE smoothed periodogram ARFIMA(3,0.705,0) -0.188 2.253 3.833 5.687 MFE Whittle; ARFIMA(0,0.867,0) 0.616 0.596 0.075 0.868 MFE Whittle; ARFIMA(1,0.487,0) 0.441 2.717 4.614 6.656 MFE wavelet; ARFIMA(0,0.794,0) 0.082 0.374 1.206 2.509 MFE wavelet; ARFIMA(2,0.933,1) 5.909 7.328 6.502 5.631 MFE unit root; ARFIMA(3,1,2) 1.920 6.358 9.562 12.161 MFE unit root; random walk 1.299 1.979 2.029 1.844 Table 4: Measures of h-step ahead out of sample forecasting errors. Reisen et al.: Long Memory Inflationary Dynamics: The Case of Brazil 11 Published by The Berkeley Electronic Press, 2003 ahead) and ARFIMA(0,d,0), the MSE and MAPE measures when d is estimated using the Whittle approach are 79% and 53% smaller than the same accuracy measures when d is estimated using the wavelets approach. It is also interesting to compare the long memory model s forecast to the unit root model (last two rows in each grouping of forecasting measures of Table 4; the final row corresponds to the fully inertial case). Both long memory models are superior to the unit root models at a forecast horizon of five months. When h = 10, 20 the strength of the long-memory model forecast is only found in the Whittle estimated model, with the margin of improvement decreasing with the length of the time horizon. We conclude from the results in Table 4 that the forecasts of the inflationary dynamics can be greatly improved by taking into account inflation s long memory behavior. It could be argued that the long memory, no unit root behavior described above was the result of bias due to the presence of potential inliers in the data. These correspond to observations for which the inflation level was artificially low due to the implementation of shock plans that were destined to fail a few months later, when inflation would resume its natural path; see Cati, et al. (1999) for details. In order to check whether that is the case, we have estimated the fractional integration parameter d for a truncated series that ranges from February 1944 through December 1985 (503 observations). This period is known to not contain inliers since the first shock plan (the so-called Cruzado Plan ) took place in February 1986. The estimates of d corresponding to the estimation methods/models described in Table 2 were now dp = 0.934, dsp = 0.860, dw = 0.576, 0.586 (p = 0 and p = 1, respectively) and drb = 0.583. Also, da = 0.543, 0.658 (p = 0 and p = 1, respectively). These values also suggest long memory behavior, as did the estimates in Table 2. It is noteworthy that the difference between the Whittle estimates for p = 0 and p = 1 is now quite small since they are both approximately equal to 0.6. The Robinson estimate of d is also around 0.6, thus being smaller than the figure obtained for the complete series (see Table 2). If the argument of bias induced from the presence of possible inliers were valid, one would expect to obtain estimates of d much closer to one for the truncated series and long memory behavior for the complete series. This is not what we observe. Overall, the results seem to be insensitive to the inclusion in the sample of the period characterized by shock plans and large fluctuations.3 3Using G7 inflation rate data, Bos et al. (1999) also find their long memory estimates to be robust to level shifts. Modeling these level shifts with a ARFIMA model with mean 12 Studies in Nonlinear Dynamics & Econometrics Vol. 7 [2003], No. 3, Article 3 http://www.bepress.com/snde/vol7/iss3/art3 4. Concluding remarks Brazil has often been cited as a country where the inflationary process is mainly driven by inflation inertia, and as a consequence traditional monetary and fiscal policies would fail to curb inflation. Several papers have identified some inertia in the inflationary dynamics in that country. In particular, Cati, et al. (1999) find that inflation in Brazil is almost entirely driven by inertia. A number of stabilization plans have been put into effect by Brazilian policymakers between the mid 1980s and the early 1990s, based on the diagnosis that inflation followed an inertial dynamics. Our results suggest, however, that there is no inflation inertia in this country. Instead, the inflationary dynamics display long memory. In short, it takes a long time for shocks to inflation to die out, but they eventually do die out. That is, shocks to inflation are not (totally or partially) persistent. Colophon Vald erio A. Reisen, Dept. de Estat stica, CCE PMEA,CT, Univ. Federal do Esp rito Santo, Vit oria, ES 29060-900, Brazil, email: valderio@cce.ufes.br, Francisco Cribari Neto Dept. de Estat stica, CCEN, Univ. Federal de Pernambuco, Recife, PE 50740-540, Brazil, email: cribari@ufpe.br, and Mark J. Jensen, Dept. of Economics, Brigham Young University, 130 Faculty Office Building, P.O. Box 22363, Provo, Utah 84602-2363, U.S.A., email: jensen@haar.byu.edu The first two authors gratefully acknowledge partial financial support from CNPq/Brazil. The third author gratefully acknowledges the financial support from a research grant provided by Brigham Young University s College of Family, Home and Social Sciences. 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