On the persistence of prices in Mexico: a fractional integration approach

ABSTRACT

A relevant yet often overlooked characteristic of the inflation rate is its mean-reverting property. If a series has this feature, shocks eventually dissipate, whereas if it does not, they have a permanent effect on the series. The usual I(1) versus I(0) dichotomy in time-series econometrics goes only so far towards disentangling this issue. By employing a methodology that estimates the persistence of inflation by allowing (i) fractional integration and (ii) persistence and level shifts in the series, we aim to define whether it is stationary and/or mean reverting and, if so, during which periods. The results of our analysis for the period 1987–2015 are threefold: firstly, inflation in the eighties and nineties should be seen as a highly persistent yet mean-reverting process (not a random walk); secondly, inflation remained mean reverting, though became a short-memory (less persistent) process around the date of the implementation of the inflation-targeting framework of 2001; thirdly, during the later phase, the level of inflation also decreased and is now within the inflation target range set by Banco de México, namely 3 per cent with an interval of ±1 percentage point.

KEYWORDS:

• Mean reversion
• fractional integration
• inflation
• persistence

JEL CLASSIFICATION:

• C12
• C22
• E31
• E58

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ See Ramos-Francia and Torres-García (2005) and Ortiz Martínez (2009) for a comprehensive analysis of inflation-targeting regime adoption after the 1995 crisis.

² Such an estimate of the persistence, however, is based upon the estimation of an AR(p), an intrinsically short-memory specification that de facto rules out the possibility of long-range dependence. In other words, once the unit root hypothesis has been discarded, they assume that the inflation series are short-memory ones and discard possible long-range dependencies.

³ Allowing for changes in persistence is important given the previous findings; see, for example, Capistrán and Ramos-Francia (2009), Chiquiar, Noriega, and Ramos-Francia (2010), Caporale and Paxton (2011), among others.

⁴ More recently, Perron and Qu (2010) made a strong argument about the danger of confusing long memory and short memory with level shifts and proposed an LM test of long memory (null) against short memory with shifts in the mean. Qu (2011) proposed an alternative test with the same null hypothesis of stationary long memory against the alternative hypothesis that the process is affected by regime change or a smoothly varying trend. Kruse (2015) further extends Qu’s proposal to allow for the I(d) process to be contaminated with level and trend shifts.

⁵ There are several studies that consider long memory/fractional integration a more appropriate methodology to study the dynamics of inflation, inter alia: Hassler and Wolters (1995), Gadea and Mayoral (2006) Gil-Alana (2008), Gil-Alana and Moreno (2012), and Gil-Alana and Pestana Barros (2009). Other studies consider the possibility that inflation behaves as a nonlinear stationary process; see, Nobay, Paya, and Peel (2010) and Zhou (2013) among others, all of whom employ a different approach to model inflation: The nonlinear exponential smooth autoregressive model (ESTAR).

⁶ When $0.5\le d<1$ the process is no longer covariance-stationary. In this case, it is a process whose mean does not depend on t, but its variance does. In other words, ‘`the process will still return to its equilibrium in the long run but will also possess an infinite variance’. Tkacz (2001, p.20). We would like to thank an anonymous referee for pointing out to us the importance of the distinction between (weak) nonstationarity and mean reversion.

⁷ For instance, the sample period we use is almost 17 years longer than that of Chiquiar, Noriega, and Ramos-Francia (2010). Our sample period starts 8 years earlier than theirs and ends almost 9 years later. This allows us to better understand the effects of the 2008 global crisis on price dynamics in Mexico.

⁸ These measures of inflation are calculated as $\left[1200\ast ln\left(P_t/P_{t-1}\right)\right]$ and $\left[400\ast ln\left(P_t/P_{t-1}\right)\right]$ for the monthly and quarterly data, respectively; $P_t$ is the corresponding price index at time t.

⁹ Appendix B shows three alternate estimates of the persistence parameter as a robustness test of the main results. The alternative estimators are those proposed by Geweke and Porter-Hudak (1983), Robinson (1995), and Jensen (1999). Results are very similar to those reported in Table 1. We would like to thank an anonymous referee for suggesting this robustness exercise.

¹⁰ Chiquiar, Noriega, and Ramos-Francia (2010) conclude that inflation in Mexico behaved as a nonstationary I(1) process from 1995 to 2000 and switched to a stationary I(0) process around the end of 2000 or at the beginning of 2001; Noriega, Capistrán, and Ramos-Francia (2013) suggest that inflation behaved as a nonstationary I(1) process from 1981:12 to 2002:08, and as a stationary I(0) process from 1961:04 to 1981:11 and from 2002:09 to 2008:04.

¹¹ When $0<d<1$, standard unit-root tests, which use $d=1$ under the null, may fail to detect mean reversion in the series and conclude that the series contains a unit root. Fractionally integrated processes are mean reverting, even when they are (weakly) nonstationary ($0.5<d<1$). This can be understood as a process whose failure to satisfy the weak stationarity definition is due to its second moment depending on time, but not its first moment. In other words, when $0.5<d<1$, although the process is no longer covariance stationary, the mean reversion property is maintained. This is in sharp contrast with I(1) process where neither stationarity nor mean reversion are maintained: see, inter alia, Gil-Alana (2000, 2008), Tkacz (2001), and Caporale and Gil-Alana (2002).

¹² This result also coincides with Chiquiar, Noriega, and Ramos-Francia (2010), who estimated that the change in persistence – from nonstationary to stationary – of headline inflation occurred before the observed change in core inflation (i.e. December 2000 compared to April 2001).

¹³ As can be seen from Table 1, the other three measures of inflation display a similar behaviour, with decreasing means during the stationary period. At the end of the sample, we estimate that the level of inflation for all series is within Banxico’s target range.

¹⁴ We found a fourth estimator: Haslett and Raftery (1989). Space-Time Modelling with Long-Memory Dependence: Assessing Ireland’s Wind Power Resource. Journal of the Royal Statistical Society, Series C (Applied Statistics), 38:1, 1–50. Unfortunately, this estimator is only valid for stationary processes ($d<0.5$).