Persistence in international inflation rates.

• Author: Baum, Christopher F.

1. Introduction

   An understanding of the dynamic properties of the inflation rate is essential to the ability of policy makers to keep inflation in check. Despite extensive research following the pioneering work of Nelson and Plosser (1982), disagreement remains in the literature on a key question: Does the postwar inflation rate possess a unit root? Although there is considerable evidence in support of a unit root (e.g., Barsky 1987; MacDonald and Murphy 1989; Ball and Cecchetti 1990; Wickens and Tzavalis 1992; Kim 1993), Rose (1988) provided evidence of stationarity in inflation rates. Mixed evidence has been provided by Kirchgassner and Wolters (1993). Brunner and Hess (1993) argue that the inflation rate was stationary before the 1960s but that it possesses a unit root since that time.

   A potential resolution to this debate should be of more than academic interest, as nonstationarity in the inflation process would have consequences for central banks' ratification of inflationary shocks and would affect the response of macroeconomic policy makers to external pressures. An explanation for this conflicting evidence was recently provided by modeling inflation rates as fractionally integrated processes. Using the fractional differencing model developed by Granger and Joyeux (1980), Hosking (1981), and Geweke and Porter-Hudak (1983), Baillie, Chung, and Tieslau (1996) find strong evidence of long memory in the inflation rates for the Group of Seven (G7) countries (with the exception of Japan) and those of three high-inflation countries: Argentina, Brazil, and Israel. Similar evidence of strong long-term persistence in the inflation rates of the United States, United Kingdom, Germany, France, and Italy is also provided by Hassler and Wolters (1995). Delgado and Robinson (1994) find evidence of persistent dependence in the Spanish inflation rate. The interpretation of this evidence suggests that inflation rates are mean-reverting processes, so that an inflationary shock will persist but will eventually dissipate.

   Because modeling the inflation rate as a fractionally integrated process appears to improve our understanding of inflationary dynamics, this study extends the existing long-memory evidence on inflation rates along two dimensions. First, it performs long-memory analysis on inflation rates for a number of countries not previously considered, both industrial and developing, to provide more comprehensive evidence regarding the low-frequency properties of international inflation rates and to determine whether long memory is a common feature. Second, the paper investigates the existence of long-memory properties of inflation rates on the basis of both the consumer price index (CPI), as exclusively considered in the literature, and the wholesale price index (WPI). The WPI is not as heavily influenced by the prices of nontraded goods as is the CPI, and it may therefore serve as a better indicator in tests of the international arbitrage relationship between traded goods prices and exchange rates. Measures based on the WPI have been used extensively in empirical applications, such as tests of purchasing power parity, empirical trade models, models of relative price responses, and models of the international transmission of inflation (e.g., Diebold, Husted, and Rush 1991; Fukuda, Teruyama, and Toda 1991; Rogers and Wang 1993). Therefore, we investigate and analyze the long-memory characteristics of WPI-based inflation rates, as well as their CPI-based counterparts, for both developing and industrial countries.

   Our data set consists of monthly CPI-based inflation rates for 27 countries and WPI-based inflation rates for 22 countries and covers the period 1971:1-1995:12. We estimate the fractional differencing parameter using both semiparametric (spectral regression and Gaussian semiparametric) and approximate maximum likelihood techniques. Evidence in the literature for long memory in major countries' CPI-based inflation rates is shown to generalize to both CPI- and WPI-based inflation rates for other industrial as well as developing countries. This evidence implies that policy makers may use fractionally integrated models of inflation to good advantage in modeling and forecasting the path of inflation rates. As potential sources of fractional dynamics in inflation rates, we hypothesize Granger's (1980) aggregation argument and the established presence of long memory in the growth rate of money.

   The remainder of the paper is constructed as follows. Section 2 presents the methods employed for the estimation of the fractional differencing parameter. Section 3 discusses the data and empirical results. Section 4 concludes with a summary and implications of our results.

2. Fractional Integration Estimation Methods

   The model of an autoregressive fractionally integrated moving average (ARFIMA) process of order (p, d, q), denoted by ARFIMA (p, d, q), with mean [Mu], may be written using operator notation as

   [Mathematical Expression Omitted] (1)

   where L is the backward-shift operator, [Phi](L) = 1 - [[Phi].sub.1]L - ... - [[Phi].sub.p][L.sup.p], [Theta](L) = 1 + [[Theta].sub.1]L + ... + [[Theta].sub.q][L.sub.q], and [(1 - L).sup.d] is the fractional differencing operator defined by

   [(1 - L).sup.d] = [summation of] [Gamma][(k - d)[L.sup.k] / [Gamma](-d)[Gamma](k + 1) where k = 0 to [infinity]

   with [Gamma] ([center dot]) denoting the gamma function. The parameter d is allowed to assume any real value. The arbitrary restriction of d to integer values gives rise to the standard autoregressive integrated moving average (ARIMA) model. The stochastic process y, is both stationary and invertible if all roots of [Phi](L) and [Theta](L) lie outside the unit circle and [absolute value of d] [less than] 0.5. The process is nonstationary for d [greater than or equal to] 0.5, as it possesses infinite variance (see Granger and Joyeux 1980). Assuming that d [element of] (0, 0.5) and d [+ or -] 0, Hosking (1981) showed that the correlation function, [Rho]([center dot]), of an ARFIMA process is proportional to [k.sup.2d-1] as k [approaches] [infinity]. Consequently, the autocorrelations of the ARFIMA process decay hyperbolically to zero as k [approaches] [infinity] in contrast to the faster, geometric decay of a stationary ARMA process. For d [element of] (0, 0.5), [summation of] [absolute value of [Rho](j)] where j = -n to n diverges as n [approaches] [infinity], and the ARFIMA process is said to exhibit long memory, or long-range positive dependence. The process is said to exhibit intermediate memory (antipersistence), or long-range negative dependence, for d [element of] (-0.5, 0). The process exhibits short memory for d = 0, corresponding to stationary and invertible ARMA modeling. For d [element of] [0.5, 1), the process is mean reverting, even though it is not covariance stationary, as there is no long-run impact of an innovation on future values of the process.

   The fractional differencing parameter is estimated using two semi-parametric methods, the spectral regression and Gaussian semiparametric approaches, and the frequency-domain approximate maximum likelihood method. A brief description of these estimation methods follows.

   The Spectral Regression Method

   Geweke and Porter-Hudak (1983) suggest a semiparametric procedure to obtain an estimate of the fractional differencing parameter d based on the slope of the spectral density function around the angular frequency [Xi] = 0.

   The spectral regression is defined by

   ln{I([[Xi].sub.[Lambda]])} = [[Beta].sub.0] + [[Beta].sub.1] ln {4 [sin.sup.2] ([[Xi].sub.[Lambda]]/2)} + [[Eta].sub.[Lambda]], [Lambda] = 1, ..., v, (2)

   where I([[Xi].sub.[Lambda]]) is the periodogram of the time series at the Fourier frequencies of the sample [[Xi].sub.[Lambda]] = (2[Pi][Lambda]/T), ([Lambda] = 1, ..., (T - 1)/2), T is the number of observations, and v = g(T) [much less than] T is the number of Fourier frequencies included in the spectral regression.

   Assuming that [lim.sub.T[approaches][infinity]] g(T) = [infinity], [lim.sub.T[approaches][infinity]] {g(T)/T} = 0, and [lim.sub.T[approaches][infinity]] {ln[(T).sup.2]/g(T)} = 0, the negative of the ordinary least squares (OLS) estimate of the slope...