This paper tests for the empirical fulfilment of PPP in Australia (1977-2004). Previous research focuses on the presence of structural breaks and fails to find any support for PPP (Darne and Hoarau, 2008, Henry and Olekalns, 2002). In contrast, we find that the real exchange rate is stationary once we account for a more general specification of the nonlinear deterministic components based on a Chebishev polynomials approximation

Cuestas, JC

Regis, PJ

English

Nottingham Trent Institutional Repository (IRep)

Testing for PPP in Australia: Evidence from unit root testagainst nonlinear trend stationarity alternatives Juan Carlos Cuestas Paulo José RegisNottingham Trent University University of NottinghamAbstractThis paper tests for the empirical fulfilment of PPP in Australia (1977-2004). Previousresearch focuses on the presence of structural breaks and fails to find any support for PPP(Darne and Hoarau, 2008, Henry and Olekalns, 2002). In contrast, we find that the realexchange rate is stationary once we account for a more general specification of the nonlineardeterministic components based on a Chebishev polynomials approximation.Juan Carlos Cuestas gratefully acknowledges the financial support from the CICYT and FEDER project SEJ2005-01163, theBancaja project P1.1B2005-03 and the Generalitat Valenciana Complementary Action ACOMP07/102. The authors gratefullyacknowledge Nik Masters and David Harvey for their useful comments. Juan Carlos Cuestas is a member of the INTECOresearch group. The usual disclaimer applies.Citation: Cuestas, Juan Carlos and Paulo José Regis, (2008) "Testing for PPP in Australia: Evidence from unit root test againstnonlinear trend stationarity alternatives." Economics Bulletin, Vol. 3, No. 27 pp. 1-8Submitted: February 16, 2008.  Accepted: May 16, 2008.URL: http://economicsbulletin.vanderbilt.edu/2008/volume3/EB-08C20010A.pdf1 IntroductionThe empirical fulfilment of Purchasing Power Parity (PPP) has probably been one ofthe most controversial topics in international economics in recent decades. Many authorshave contributed to the literature, using different countries, time periods and econometrictechniques. However, the results have been on many occasions contradictory1.The importance of the analysis of PPP is, at least, twofold. First, many macroeco-nomic momentary models are based on the PPP assumption. Second, the real exchangerate can be considered a measure of economic integration and external competitiveness(Wei and Parsley, 1995) and its understanding can be helpful for exchange rate policydesign.In short, the absolute version of the PPP theory establishes that the real exchangerate (RER) between two currencies must be equal to 1 and, therefore, both currenciesmust have a similar purchasing power. However, rejection of the absolute PPP hypothesisdoes not necessarily imply that national prices are exclusively explained by idiosyncraticfactors; prices may still respond to proportional instead of identical components, implyingthat the RER may be constant but different from 1. This second interpretation is therelative version of the PPP theory.It is well known within the literature that if PPP holds, it only does in the long run.This long run relation implies that shocks affecting the currencies have only effects onthe dynamics; therefore, one can only accept the PPP hypothesis when the RER is astationary process. Testing for the empirical validity of PPP is closely related to testingfor unit roots in the RER (Meese and Rogoff 1988; Mark, 1990; Ardeni and Lubian,1991; Huizinga, 1987, among others). However, it has been argued that traditional unitroot tests might suffer from power problems when the deterministic components are notproperly specified (Perron and Phillips, 1987; and West, 1988, among others). Thus, theexistence of structural changes in the series may bias the results of these unit root testsin favour of the null hypothesis.Dealing with this particular case of nonlinearity, an increasing number of authors haveapplied unit root tests with structural changes, finding in general results more favourableto the stationarity of the RER. In such a case, what we have is the so-called Quasi-PPP(Hegwood and Papell, 1998). Following this approach, a recent paper by Darne´ andHoarau (2008) analyses the empirical fulfilment of the PPP theory in Australia, applyingthe Perron and Rodr´ıguez (2003) unit root tests with structural changes. Although theyobtain a structural change in 1985, coinciding with the currency crisis suffered by theAustralian dollar, their results point to the rejection of the PPP hypothesis. Likewise,Henry and Olekalns (2002) also reject the PPP hypothesis for the Autralian RER, usingthe Zivot and Andrews (1992) and Perron (1997) unit root tests with breaks.Bearing in mind that an incorrect specification of the deterministic components maybias the results towards the integrated process hypothesis, in the present paper we applya unit root test which approximates the deterministic components with a more general1specification for structural changes, i.e. a nonlinear deterministic trend (Bierens, 1997).The existence of a stationary RER around a nonlinear trend can be understood as a timevarying or nonconstant equilibrium exchange rate. That is, variations in the fundamentalsthat determine the RER should give rise to a different RER equilibrium value. Sometimes,the nature of the changes in the fundamentals are not properly identified as a brokentrend and a more general approximation is required. This can occur when dealing withaggregate data (Granger and Tera¨svirta, 1993), since different agents do not normallytake economic or portfolio decisions at the same time.An early contribution to this approach is Lothian and Taylor (2000), who proposethat the RER deterministic trend is better approximated by a cubic function. Moreover,in recent years, the literature on the analysis of the order of integration of the RER hasstarted to pay special attention to the possibility of a stationary RER around a nonlineardeterministic trend. For instance, Sollis (2005), Assaf (2006), Cushman (2008), Camareroet al. (2008) and Cuestas (2008) find in general that the real exchange rate may be anonlinear trend stationary process instead of a unit root and therefore evidence in favourof a more relaxed version of PPP or Quasi-PPP.Summing up, there are two approaches to nonlinearities to support the Quasi-PPPhypothesis: some studies look for evidence in favour of structural changes while othersexplore the existence of a nonlinear deterministic trend. However, sometimes it is difficultto make a clear distinction between them. Other times, the evidence is more clear aboutwhat explanation seems more plausible. When sudden structural breaks are preferred,a more general nonlinear trend implies a misspecified model, where the estimation of anoverfitted model may yield to incorrect conclusions. On the other hand, both approachesmay be complementary. If the data fits better a general nonlinear trend, structural breaksmay still be useful to give an interpretation of historic events that may help in explainingthe causes of the nonlinearity.This paper aims at providing some evidence in favour of the Quasi-PPP hypothesisin Australia. First, we rely on the Darne´ and Hoarau (2008) findings for unit root testswith a sudden structural break. Second, the existence of a nonlinear deterministic trendis tested, applying Bierens (1997) unit root tests. This author proposes several tests forthe unit root hypothesis against the alternative of nonlinear trend stationarity, where thenonlinear trend is approximated by means of Chebishev polynomials. Contrary to theprevious literature, our findings point to different results: the RER in Australia appearsto be a stationary process around a nonlinear deterministic trend for the analysed period.In the next section we summarise the Bierens (1997) technique and the results ofapplying this technique. The last section concludes.22 Nonlinear unit root tests and resultsIn this section we test for the order of integration of the real exchange rate of Australia.The data correspond to the RER computed by the Reserve Bank of Australia2 fromJanuary 1977 to April 2004, the same time series used by Darne´ and Hoarau (2008).Since time series are most usually modelled by means of linear equations, unit roottests might be biased by the presence of nonlinearities in the deterministic components.It is a standard practice to introduce structural breaks and additive outliers in order toaccount for this possibility in the series of the RER. The identification of structural breaksand outliers is also informative since it provides a direct interpretation of economic events.An alternative approach is to introduce a more general approximation to the nonlineardeterministic component. The approximation would also capture the structural breakswith a smoother functional form for the transition period. It is in this direction that thispaper makes a contribution to the previous literature.To test for the order of integration of the RER, we apply the unit root tests developedby Bierens (1997). This procedure accounts for the general case of a nonlinear determin-istic trend when testing for unit roots, by extending the augmented Dickey-Fuller (ADF)test introducing orthogonal Chebishev polynomials. The ADF equation becomes∆xt = αxt−1 +p∑j=1φj∆xt−j + θTP(m)t,n + εt (1)where P(m)t,n are the Chebishev polynomials of order m. The null hypothesis is formulatedsuch that α and the lastm components of θ are not significant. In this paper we apply thetˆ(m) test, which is a t-test on the significance of the coefficient α. In addition, and in orderto check the robustness of the previous results, we also apply the Aˆ(m) = nαˆ|1−∑pi=1 φˆi| test,an alternative test for the same hypothesis. The distinction between linear or nonlineartrend stationarity depends upon the side of the rejection. Whereas right side rejection(a p-value > 0.90) implies stationarity around a nonlinear deterministic trend, left siderejection (a p-value < 0.10) does not allow us to distinguish between mean stationarityor stationarity around a deterministic trend (see Table 1).Figure 1 shows the RER series. It appears that the variable does not seems to followa linear behaviour. Instead, a nonlinear deterministic trend would probably be a betterstatistical characterisation for this variable’s deterministic component. Note that theestimated nonlinear trend adjusts smoothly the structural changes, instead of assumingsudden changes in the trend.The results are displayed in Table 2. The p-values are based on Monte Carlo sim-ulations with 5,000 replications of a Gaussian AR(p) process for ∆xt. The parametersand error variances are equal to the estimated AR(p) null model, where the lag lengthfor the ADF regression p has been selected by the Akaike Information Criterion3 (p = 0)3and the initial values have been taken from the actual series4. For comparison purposeswe have also performed the ADF test. Note that with the ADF test, it is not possibleto reject the null hypothesis of a unit root at conventional significance levels. However,the nonlinear unit root tests point to the rejection of the null hypothesis of a unit rootin favour of the alternative of stationarity around a nonlinear trend5, where m = 7.Our results are therefore different to those obtained by Darne´ and Hoarau (2008).Although these authors find a structural break in the Australian RER in 1985, a non-linear deterministic trend appears to be a better approximation for the deterministiccomponents of the RER. In addition to this, our finding does not support the Balassa-Samuelson effect, which implies that the RER should be stationary around a constanttrend. Instead, the results here reported are in favour of the Quasi-PPP interpretation.Finally, we explore whether structural breaks can explain most of the nonlinearity. Inorder to test for the importance of structural breaks as the main source of the nonlin-earity, we also perform the Bierens (1997) test over the variable previously transformedto account for a structural break in the intercept and trend6. If this structural break isimportant to explain the nonlinearity, we expect to reject the null hypothesis of a unitroot for the transformed variable with a low Chebishev polynomial order (i.e. m closeto zero). The new results7 show that the transformed variable is stationary around anonlinear trend for m equal to 5, which is very similar to our initial results of m = 7.This suggests that the sudden structural break is not able to capture the true natureof the nonlinearity. Therefore, the Chebishev polynomials appear to approximate thenonlinear deterministic component better than a single structural break.3 ConclusionPrevious literature (Darne´ and Hoarau, 2008, Henry and Olekalns, 2002) test for theempirical validity of PPP in Australia applying unit root tests with structural changes.Their results point to the rejection of the PPP hypothesis. Aimed at complementing thisanalysis, we generalise the case of structural break to a nonlinear deterministic trend.The results point to the fact that the RER is nonlinear trend stationary, accepting theQuasi-PPP hypothesis. This suggests that it is worth considering alternative forms forapproximating the deterministic trends to structural breaks in the coefficients of a linearequation when testing for the order of integration of the RER.4Notes1See Taylor (2006) for a recent survey of the literature2Available at http://www.rba.gov.au/Statistics/real exchange rate indices.xls.3The same lag length is also obtained by the Schwarz Bayesian Information Criterion.4The order of p used to obtain the statistic and the p-values are the same.5We have selected the order of m that yields more evidence against the null hypothesis.6Preliminary examination of the data reveals the most likely structural break occurs in 1985:17Available upon request to the corresponding author.ReferencesArdeni, P. G. and D. Lubian (1991): “Is there a trend reversion in purchasing powerparity?”, European Economic Review, vol. 35, pp. 1035–1055.Assaf, A. (2006): “Nonlinear trend stationarity in real exchange rates: Evidence fromnonlinear adf tests”, Annals of Economics and Finance, vol. 2, pp. 283–294.Bierens, H. J. (1997): “Testing the unit root with drift hypothesis against nonlinear trendstationarity, with an application to the U.S. price level and interest rate”, Journal ofEconometrics, vol. 81, pp. 29–64.Camarero, M., J. C. Cuestas and J. Ordo´n˜ez (2008): “Nonlinear trend stationarity ofreal exchange rates: The case of the Mediterranean countries”, International Jounralof Banking, Accounting and Finance, forthcoming.Cuestas, J. C. (2008): “Purchasing power parity in Central and Eastern European coun-tries: An analysis of unit roots and nonlinearities”, Applied Economics Letters, forth-coming.Cushman, D. (2008): “Real exchange rates may have nonlinear trends”, InternationalJournal of Finance and Economics, vol. 13, pp. 158–173.Darne´, O. and J.-F. Hoarau (2008): “The purchasing power parity in Australia: Evidencefrom unit root test with structural break”, Applied Economics Letters, vol. 15, pp. 203–206.Granger, C. W. J. and T. Tera¨svirta (1993): Modelling Nonlinear Economic Relation-ships, Oxford University Press, Oxford.Hegwood, N. D. and D. Papell (1998): “Quasi purchasing power parity”, InternationalJournal of Finance and Economics, vol. 3, pp. 279–289.Huizinga, J. (1987): “An empirical investigation of the long-run behavior of real exchangerates”, Carnegie-Rochester Conference Series on Public Policy, vol. 27, pp. 149–214.5Lothian, J. R. and M. P. Taylor (2000): “Purchasing power parity over two centuries:Stregthening the case for real exchange rate stability”, Journal of International Moneyand Finance, vol. 19, pp. 759–764.Mark, N. (1990): “Real and nominal exchange rates in the long run: An empiricalinvestigation”, Journal of International Economics, vol. 28, pp. 115–136.Meese, R. A. and K. Rogoff (1988): “Was it real? the real exchange rate-interest dif-ferential relation over the modern floating-rate period”, Journal of Finance, vol. 43,pp. 933–948.Perron, P. (1997): “Further evidence on breaking trend functions in macroeconomicvariables”, Journal of Econometrics, vol. 80, pp. 355–385.Perron, P. and G. Rodr´ıguez (2003): “GLS detrending, efficient unit root tests andstructural change”, Journal of Econometrics, vol. 115, pp. 1–27.Perron, P. and P. C. B. Phillips (1987): “Does GNP have a unit root? A reevaluation”,Economics Letters, vol. 23, pp. 139–145.Sollis, R. (2005): “Evidence on purchasing power parity from univariate models: the caseof smooth transition trend-stationarity”, Journal of Applied Econometrics, vol. 20,pp. 79–98.Taylor, M. P. (2006): “Real exchange rates and Purchasing Power Parity: Mean-reversionin economic thought”, Applied Financial Economics, vol. 16, pp. 1–17.Wei, S.-J. and D. C. Parsley (1995): “Purchasing power disparity during the float rateperiod: Exchange rate volatility, trade barriers and other culprits”, Working Paper5032, NBER.West, K. D. (1988): “Asymptotic normality when regressors have a unit root”, Econo-metrica, vol. 56, pp. 1397–1418.Zivot, E. and D. W. K. Andrews (1992): “Further evidence on the Great Crash, the oil-price shock, and the unit-root hypothesis”, Journal of Business and Economic Statis-tics, vol. 10, pp. 251–270.6Table 1: Alternative hypothesesTest Left-side rejection Right-side rejectiontˆ(m) MS, LTS or NLTS NLTSAˆ(m) MS, LTS or NLTS NLTSNote: MS= mean stationarity, LTS= linear trend stationarity, NLTS= nonlinear trend stationarity.Table 2: Unit root test resultsAˆ(m) tˆ(m) tˆADFstatistic -15.1133 -2.7758 -1.6608p-value 0.9950 0.9832 0.76007Figure 1: Australian RER and nonlinear trends8

An empirical investigation of the long-run behavior of real exchange rates”,

Asymptotic normality when regressors have a unit root”,

Does GNP have a unit root? A reevaluation”,

Evidence on purchasing power parity from univariate models: the case of smooth transition trend-stationarity”,

Further evidence on breaking trend functions in macroeconomic variables”,

Further evidence on the Great Crash, the oilprice shock, and the unit-root hypothesis”,

GLS detrending, efficient unit root tests and structural change”,

Modelling Nonlinear Economic Relationships,

Nonlinear trend stationarity of real exchange rates:

Purchasing power disparity during the float rate period: Exchange rate volatility, trade barriers and other culprits”, Working

Purchasing power parity in Central and Eastern European countries: An analysis of unit roots and nonlinearities”,

Quasi purchasing power parity”,

Real exchange rates and Purchasing Power Parity: Mean-reversion in economic thought”,

Real exchange rates may have nonlinear trends”,

Table 2: Unit root test results Aˆ(m) tˆ(m) tˆADF statistic

The purchasing power parity in Australia: Evidence from unit root test with structural break”,

Testing for PPP in Australia: evidence from unit root tests against nonlinear trend stationarity alternatives

This paper tests for the empirical fulfilment of PPP in Australia (1977-2004). Previous research focuses on the presence of structural breaks and fails to find any support for PPP (Darne and Hoarau, 2008, Henry and Olekalns, 2002). In contrast, we find that the real exchange rate is stationary once we account for a more general specification of the nonlinear deterministic components based on a Chebishev polynomials approximation.</p

Cuestas, Juan Carlos

Regis, Paulo José

Southampton (e-Prints Soton)

Testing for PPP in Australia: Evidence from unit root test against nonlinear trend stationary alternatives

Testing for PPP in Australia: evidence from unit root tests against nonlinear trend stationarity alternatives

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