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...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]) = for s [not equal to] t, and [PHI](L) = (1 - [[PHI].sub.1]L ... - [[PHI].sub.p][L.sup.p]), [theta](L) =...
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