Generalized long memory and mean reversion of the real exchange rate

Abstract

Definitive evidence regarding a rapid mean reversion of the real exchange rate is not present when using standard linear methodology, including unit root tests and fractional integration. To consider the robustness of these results, we use an encompassing model, the Gegenbauer AutoRegressive Moving Average (GARMA) model, which nests as special cases the existing linear methods. The GARMA model accommodates a complete notion of persistence and allows shocks to dissipate slowly in a cyclical manner. We find evidence supporting a weak version of purchasing power parity, where equilibrium errors are long memory with strongly persistent cycles. However, this new form of cyclical mean reversion is likely too slow to be economically meaningful. The inability to find a strong equilibrium attractor process, using a very general encompassing linear methodology provides support for the recent models that allow for a nonlinear attraction process and for shifting real exchange rate equilibria.

Notes

¹ Other studies supporting the finding of nonlinear dynamics in the real exchange rate include Arghyrou et al. (2006), Enders and Dibooglu (2001), Enders and Chumrusphonlert (2004), Leon and Najarian (2005) and Taylor and Taylor (2004). Additionally, Taylor et al. (2001) and Smallwood (2008) present evidence that half lives to deviations from PPP are substantially reduced when using threshold nonlinear models relative to linear models, including long-memory fractional models.

² All data considered in this study are monthly data from the IFS CD-ROM 2002.

³ Smallwood and Norrbin (2006) and Ramachandran and Beaumont (2001) employ the same definition in the context of cointegration using the GARMA model.

⁴ Smallwood and Norrbin (2007) propose an algorithm that exploits the different convergence rates of the model parameters and also provide additional technical details about the CSS estimator. We do not include the full details here, but refer the interested reader to Chung 1996a, b and Smallwood and Norrbin (2007).

⁵ The GARMA model possesses a spectral density function that has a pole at the Gegenbauer frequency, v = cos⁻¹(u). As further corroborative evidence of the estimates reported here, it is interesting to note the periodogram for the UK real exchange rate is maximized at 0.0179π, corresponding to u = 0.9984.

⁶ Weak evidence of PPP for Canada and the United States is common in the literature. Murray and Papell (2002) show the half life of deviations from PPP for this pair is three times longer than for any other country pair.

⁷ For Canada, Japan, Germany and the United Kingdom, the estimated cointegrating vectors using the Johansen algorithm are [1.00, 4.89, −3.36]′, [1.00, 5.32, −1.70]′, [1.00, 8.57, 0.89]′ and [1.00, −0.04, 0.45]′. Coefficients displaying unexpected sizes and signs are consistent with the findings of Smallwood and Norrbin (2004) who document the large biases in the Johansen procedure when the equilibrium relationship is a near unit root GARMA variable. Another potential explanation for the odd cointegrating vectors is the potential existence of measurement errors in the price levels as suggested by, for example, Taylor (1988).