In this section we apply the fractional integration tests with Chebyshev polynomials to examine the mean reversion of real exchange rates and the PPP theory. The absolute version of the PPP theory postulates that the price levels in two different countries should converge when measured in the same currency, so as to equalize the purchasing power of the currencies in both places. This, therefore, implies that the real exchange rate, defined as the ratio of prices in both countries, translated to a common currency using the nominal exchange rate, should converge to 1. However, it is well known within the literature that the absolute version of the PPP hypothesis may be too restrictive. Hence, a less restrictive version of PPP is the relative PPP hypothesis, which implies that prices in common currency may converge to a constant different from 1. This relative version of the PPP implies then that what is actually expected in the long run is that the real exchange rate should be reverting to a constant, which may be different from 1. The intuition behind this is related to the fact that because of the existence of trade barriers, transport costs, and different measures of price indices, there may be a gap between price levels in different countries. Hence, on average, changes in real exchange rates should be zero, according to the relative version of the PPP theory.

In view of the above comments, testing for mean reversion becomes of paramount importance when testing for the empirical validity of the PPP theory, which at the same time, can be seen as a measure of the degree of over/under-valuation of the currencies, and it is used as a base for a number of macroeconomic models, i.e. the Dornbusch model. However, real exchange rate convergence, on average, to a constant along time may not be very realistic, in particular when countries experience different levels of economic growth and productivity gains, as well as, when countries suffer from changes in economic fundamentals, which may indeed change the equilibrium value of real exchange rates. For instance, the well known dynamic Penn effect and the Balassa-Samuelson effect, may induce deterministic trends in the data (see Lothian and Taylor 2000, among others), and the existence of structural changes, may, in addition, induce changes in those trends. Hence, the importance of controlling for non-linear deterministic trends when testing for real exchange rate mean reversion.

In a recent contribution, Cushman (2008) tests for the PPP hypothesis using the Bierens (1997) unit root tests for bilateral exchange rates. He finds evidence to support that real exchange rates may in fact contain non-linear trends. However, it is not possible to test for the significance of these trends, unless the null is rejected. (See also Cuestas 2009, and Cuestas and Mourelle 2011.)

As just mentioned, our newly developed fractional integration testing procedure, taking into account Chebyshev polynomials to approximate non-linear deterministic trends, solves these problems with the flexibility of having non-integer orders of integration. Given that the residuals of the auxiliary regression are *I*(0) stationary by assumption, *t*-statistics are valid to test for the significance of the non-linear trends. This novelty solves the problem of choosing the order of the Chebyshev polynomials, which was not clearly defined by Bierens (1997). Thus, we can start from a fairly general degree of non-linearity (e.g. *m*=3) and check the *t*-values of the estimated coefficients, removing those which are found statistically insignificant. We stop with the model with all significant coefficients.

The data used in the empirical application are real effective exchange rates against each country’s 27 main trade partners, downloaded from *Eurostat* (code *ert*_*eff*_*ic*_*q*) for 40 countries, with different degrees of economic integration and development. We have used quarterly data from 1994:Q1 until 2011:Q3.

Across this section we consider the following model,

$${y}_{t}={\displaystyle \sum _{i=0}^{m}{\theta}_{i}{P}_{iT}\mathrm{(}t\mathrm{)}+{x}_{t},\text{\hspace{0.17em}}{\mathrm{(}1-L\mathrm{)}}^{d}{x}_{t}={u}_{t},}\text{\hspace{1em}(17)}$$(17)

assuming that *u*_{t} is a white noise process. The use of autoregressions for the error term *u*_{t} in (17) produced coefficients close to 0 in all cases. In fact, we also conducted an LR test to determine if the error term should be with noise or an AR(1) process and the results strongly support the white noise specification in all cases.

Although the results are not reported here, which are available upon request, we estimate *d* and the 95% confidence bands of the non-rejection values of *d* for the cases of *m*=0, 1, 2 and 3. Higher values of *m* lead to non-significant coefficients for *θ*_{i} (*i*>3) in all cases. These estimates were obtained using the Whittle function in the frequency domain and they coincide with the values of *d*_{o} that produce the lowest statistics in absolute value when using our (one-sided) testing approach with a fine grid of *d*_{o}-values (with 0.001 increments). We observe that the values of *d* are very similar across the different values for *m*, in general, observing a slight reduction in the degree of integration as we increase *m*.^{11} We also notice that most of the estimates of *d* are within the unit root interval and some of them are even significantly above 1. The only evidence of mean reversion (i.e. *d* significantly below 1) is obtained for the cases of Cyprus, Greece and Malta (for all values of *m*) and for France and Spain if *m*=2 or 3, i.e. assuming the existence of non-linearities. The results also point out that it is possible to reduce the order of integration of the variable by increasing artificially the order of the Chebyshev polynomials, *m*. This is consistent with other works that show that fractional integration and non-linearities are issues which are intimately related (Diebold and Inoue 2001; Granger and Hyung 2004; etc.). Given that, as aforementioned, inference based on *t*-statistics remain valid, this approach makes much easier the selection of the appropriate deterministic component.^{12}

The summary of the results (based only on the significant Chebyshev coefficients at the 5% level) are reported in . We see that strong evidence of non-linearities (with the two non-linear coefficients statistically significantly different from zero) is obtained for the cases of Cyprus, France, Malta, Spain, Germany, Hong-Kong and Lithuania. In the first four cases, the unit root hypothesis is rejected in favour of mean reversion, while in the remaining three cases, though the estimated values of *d* are smaller than 1, the unit root cannot be rejected. Evidence of non-linearity with significant *θ*_{2}-coefficient is observed for Austria, Greece and Slovakia, the unit root being rejected in favor of mean reversion in the case of Greece. Also, for some countries only one of the two non-linear coefficients is significant, such as China (with only *θ*_{3} being statistically significant, and an estimate of *d* of 0.979) as well as Bulgaria and Latvia (with *d*=0.827 and 1.197 respectively), and also, Belgium, Brazil and the UK (with *θ*_{2} significant but not *θ*_{3}) and the unit root being not rejected. For the remaining cases, only an intercept or a linear trend is required.

Table 5 Summary results based on the selected model for each series.

We also conducted the analysis based on weakly autocorrelated errors. We tried both seasonal and non-seasonal autoregressions and the results, not displayed, indicate that though quantitatively there are some differences when computing the results based on autocorrelated errors qualitatively the same conclusions hold, since the number of cases corresponding to “mean reversion,” “unit roots” or “explosive roots” affect exactly to the same series as in the case of white noise errors. As earlier mentioned, LR tests also support the white noise specification in all cases.

Our results pinpoint a few economic insights. We first observe that in many cases structural breaks in the form of non-linear trends are present in the data. Second, for a number of countries, for instance the Czech Republic and Hungary, a linear trend is enough to approximate the data. This implies that the Balassa-Samuelson effect might be present, which makes economic sense given the process of catching-up with Western Europe during the transition period from communism to market economies. Finally, that in all cases of mean reversion, it occurs along with structural breaks. Comparing our results to those by Cushman (2008), although the results are not directly comparable, we can say that we find evidence of mean reversion using a lower order for the Chebyshev polynomials. A similar approach as the one presented here has been recently conducted in a paper by Caporale, Carcel, and Gil-Alana (2015), examining the inflation rates in five African countries: Angola, Lesotho, Bostwana, Namibia and South Africa, and evidence of non-linearities were found in the former two countries but not in the other three.

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