Abstract
In this paper we analyze the validity of the purchasing power parity (PPP) in a nonlinear framework using data for 18 bilateral US dollar exchange rates. Following Enders and Ludlow (2002), we use unit root and cointegration tests that do not assume a specific nonlinear adjustment. We find evidence of non-linear mean reversion in deviations from the PPP equilibrium in 11 out of 18 currencies. Additionally, to disentangle the respective contribution of exchange rate and prices to the adjustment toward the long run equilibrium, we estimate a Vector Error Correction Model. According to our empirical analysis, there exists a nonlinear mechanism to correct for deviation from the PPP equilibrium that comes mainly from the exchange rates. This is consistent with theoretical arguments on international goods markets under transaction costs as well as with an emerging strand of empirical literature. These results highlight the importance of neglecting the possibility of nonlinearity in the debate about the PPP and provide empirical evidence that supports the scenario of the PPP hypothesis as a reality.
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Notes
Enders and Loudlow (2002) have already provided an empirical evidence of the relative PPP, but it is applied only to the logarithm of the real French franc/DM exchange rate.
Kapetanios et al. (2003), Enders and Siklos (2001) or Bec et al. (2008) propose testing procedures to detect the presence of nonstationarity against nonlinear stationarity when the nonlinear dynamics process leading the adjustment to the deviations of PPP is a function of the size of deviations that can be modelized by some kind of threshold model. The Enders and Ludlow’s tests we use in this paper will still detect nonlinearity in such a situation.
There also exists the absolute version of PPP that posits equality between the price level in one country and the exchange rate adjusted price level in the other. Strictly speaking, the absolute PPP hypothesis states that the exchange rate between the currencies of two countries should equal the ratio of the price levels of the two countries. The measures of consumer prices published by national statistical agencies are typically reported as indexes relative to a base year (say, 1995 = 100). Thus, they only measure the rate of change of the price level from the base year, not its absolute level. Therefore, it does not make sense testing this version of PPP when working with consumer index prices as it is done in this paper.
The US dollar exchange rate for participating currencies in the single European currency after January 1999 has been calculated using the official conversion rates announced on December 31, 1998 and the Euro/Dollar exchange rate. Greece joins the European Economic Community in April 1981. This leads the Bank of Greece to adjust the trade-weighted system adopted following the collapse of the Bretton-Woods agreement placing a greater weight on European currencies and smaller weight on the dollar. Greece is joining the euro group as the 12th member on January 1, 2001. The Netherlands fixes its currency to the German Mark until January 1981. After the country’s public finances went bankrupt in 1982, Mexico began a profound transformation. The foreign debt crisis of 1981-82 had a severe and irreversible effect on public finances and political thinking. Until that date, Mexico’s economy was heavily subsidized and protected from competition.
Before 1991 the consumer prices refers only to West Germany.
DF-GLS test has substantially improved power when an unknown mean or trend is present. As Elliott et al. (1996) prove, the modified test works well in small samples.
All individual series (nominal exchange rates and prices) have also been tested for the presence of unit root using the ADF and DF-GLS tests. Consistent with the literature, the null hypothesis of a single unit root cannot be rejected in any case. To save space, these results are not reported here, but are available upon request.
Some researchers, such as Cheung et al. (1998), and Koedijk et al. (1998), have found that the stochastic processes of some of the real exchange rates cannot be adequately modeled without the inclusion of a linear deterministic time trend. The linear deterministic time trend is generally interpreted as representing systematic differences in productivity growth between tradable and non-tradable goods in the two countries. On the other hand, other researchers, for example, Papell and Theodoridis (1998), and Amara and Papell (2006) consider a linear time trend in the real exchange rate as inconsistent with long-run PPP.
Perron and Rodriguez (2001) analyze residual based tests for cointegration. Among other cases, they consider the standard ADF test, derive their asymptotic distribution assuming a general quasi-differencing parameter and tabulate its critical values. Their simulations reveal an important power gain from using GLS detrended data, especially if the quasi-difference parameter is set as suggested by Elliott et al. (1996).
When the model is linear in parameters, it is not able to account for a number of important properties of time series data, such as the clustering of large and small errors or thick-tailed distributions. Models with asymmetric adjustment are able to obtain good forecasts for heteroskedastic time series. As Ludlow and Enders (2000) show, the Fourier approximation captures the conditional volatility present in the NYSE Transportation Index.
Let the function α(t) have a Fourier expansion: \( \alpha \left( t \right) = a_0 + \sum\limits_{k = 1}^{\infty } {\left[ {ak\;\sin \left( {{{2\pi kt} \mathord{\left/ {\vphantom {{2\pi kt} T}} \right. } T}} \right) + bk\;\cos \left( {{{2\pi kt} \mathord{\left/ {\vphantom {{2\pi kt} T}} \right. } T}} \right)} \right]}, \) and define F s (t) to be the sum of the Fourier coefficients: \( F_s \left( t \right) = \sum\limits_{k = 1}^s {\left[ {a_k \sin \left( {{{2\,\pi \,k\,t} \mathord{\left/ {\vphantom {{2\,\pi \,k\,t} T}} \right. } T}} \right) + b_k \cos \left( {{{2\,\pi \,k\,t} \mathord{\left/ {\vphantom {{2\,\pi \,k\,t} T}} \right. } T}} \right)} \right]}, \) then for any arbitrary positive number h, there exists a number N such that: \( \left| {\alpha (t) - F_s \left( t \right)} \right|\kern1.5pt<\kern1.5pth \) for all \( s \ge N \).
Half the sample size is a natural upper bound arising from the standard deterministic harmonic decomposition.
For time series analysis the frequency domain often provides valuable information on the dynamics of the series. The frequency domain properties of an economic time series may provide a useful complement to its time domain properties. The frequency approach is useful to identify the length of business cycles and seasonal patters that may be important. There may also be unknown cycles at other frequencies that should be identified. The method allows quantitative definition of the cycle, and extraction of long, medium or short term components, according to the researcher’s wish. The frequency information gives quality insight to the function shape where rapid changes imply high frequencies and gradual changes imply low frequencies.
We can represent \( \alpha \left( t \right) = a_0 + a_1 \;{ \sin }\left( {{{2\,\pi \,k\,t} \mathord{\left/ {\vphantom {{2\,\pi \,k\,t} T}} \right. } T}} \right) + b_1 \;{ \cos }\left( {{{2\,\pi \,k\,t} \mathord{\left/ {\vphantom {{2\,\pi \,k\,t} T}} \right. } T}} \right) \) as \( \alpha (t) = a_0 + r{ \cos }\left( {{{2\,\pi \,k\,t} \mathord{\left/ {\vphantom {{2\,\pi \,k\,t} T}} \right. } T} + d} \right), \) where \( r = \sqrt {a_1^2 + b_1^2 } \) and \( d = \arcsin \left( {{{a_1 } \mathord{\left/ {\vphantom {{a_1 } r}} \right. } r}} \right) \). If a 0 + r > 1, and since \( { \cos }\left( {{{2\,\pi \,k\,t} \mathord{\left/ {\vphantom {{2\,\pi \,k\,t} {T + d}}} \right. } {T + d}}} \right) \) can equal unity, there will be k periods when α(t) exceed the unity, there being explosive periods. Oscillations appear when α(t) < 0, that being the case when a 0 < r and \( { \cos }\left( {{{2\,\pi \,k\,t} \mathord{\left/ {\vphantom {{2\,\pi \,k\,t} {T + d}}} \right. } {T + d}}} \right) = - 1 \).
Additionally, we apply the test on the demeaned real exchange rate. To save space, we only present the demeaned and detrended case, but results are available upon request.
We use differenced data to allow for comparison with the Engle and Granger (1987) test for the linear case.
However, if the F_trig test correctly indicates r > 0, rejecting the null of de c* test against the alternative hypothesis c < 0, is sufficient to guarantee reversion.
We also analyze the demeaned unrestricted PPP. The results are almost the same that in the detrended case. For this reason, to conserve space, these results are not reported here, but are available upon request.
The estimated models appear to pass all the standard diagnostic tests. To save space, we do not include these results but are available upon request.
The explanatory variables in the long run relationship are highly correlated and this may affect the signs and values of the estimated parameters.
Our previous results on the stability of deviations from PPP only allow to understand the α i1, i = 1,2,3 coefficients as adjustment coefficient to past disequilibria for those countries with evidence in favor of the PPP. For the other countries, these coefficients capture responses to changes in a nonstationary relation of the variables.
We approximate the distribution of these significance test statistics by a normal standard distribution.
To test the existence of short-term causality, we start from the estimated VECM in expression (11) and carry out a joint significance test of the lags of the causal variable in the equation of the caused one.
References
Amara J, Papell D (2006) Testing for purchasing power parity using stationary covariates. Appl Financ Econ 16(1–2):29–39. doi:10.1080/09603100500389374
Baum CF, Barkoulas JT, Caglayan M (2001) Non-linear adjustment to purchasing power parity in the Post-Bretton Woods era. J Int Money Finance 20:379–399
Bec F, Guay A, Guerre E (2008) Adaptive consistent unit root tests based on autoregressive threshold model. J Econom 142(1):94–133. doi:10.1016/j.jeconom.2007.05.011
Cheung YW, Lai KS, Kon S (1998) Parity reversion in real exchange rates during the post-Bretton Woods period. J Int Money Finance 18:751–768
Cheung YW, Lai KS, Bergman M (2004) Dissecting the PPP puzzle: the unconventional roles of nominal exchange rate and price adjustments. J Int Econ 64:135–150. doi:10.1016/S0022-1996(03) 00076-X
Corbae D, Ouliaris S (1988) Cointegration and tests of purchasing power parity. Rev Econ Stat 70:508–521. doi:10.2307/1926790
Davidson R, MacKinnon J (1993) Estimation and inference in econometrics. Oxford University Press, Oxford
Diebold F, Husted S, Rush M (1991) Real exchange rate under the gold standard. J Polit Econ 99:1151–1158. doi:10.1086/261799
Dumas B (1992) Dynamic equilibrium and the real exchange rate in a spatially separated world. Rev Financ Stud 5:153–180. doi:10.1093/rfs/5.2.153
Edison HJ, Fisher E (1991) A long-run view of the European monetary system. J Int Money Finance 10:53–70. doi:10.1016/0261-5606(91)90026-G
Elliott G, Rothenberg TJ, Stock JH (1996) Efficient tests for an autoregressive unit root. Econometrica 64:813–836. doi:10.2307/2171846
Enders W, Dibooglu S (2001) Long-run purchasing power parity with asymmetric adjustment. South Econ J 68:433–445. doi:10.2307/1061603
Enders W, Siklos PL (2001) Cointegration and Threshold Adjustment. J Bus Econ Stat 19:166–176
Enders W, Loudlow J (2002) “Non-linear decay: Tests for an attractor using a Fourier approximation.” Working Paper 01-02-02, University of Alabama. http://www.cba.ua.edu/efl/working_papers/pdf/WP01-02-02.pdf
Enders W, Hoover GA (2003) The effect of robust growth on poverty: a nonlinear analysis. Appl Econ 35(9):1063–1071. doi:10.1080/0003684032000080871
Engel C, Morley J (2001) The adjustment of prices and the adjustment of the exchange rate. NBER
Engel CM, Hendrickson MK, Rogers JH (1997) “Intra-National, Intra-Continental, and Intra-Planetary PPP” Board of Governors of the Federal Reserve System International Finance, Discussion Paper 589. doi:10.2139/ssrn.76928
Engle R, Granger C (1987) Co-integration and error correction: representation, estimation, and testing. Econometrica 55:251–276. doi:10.2307/1913236
Fernández JL, Peruga R (1999) “Un Contraste ADF Secuencial para la Detección de Cambios en la Tendencia Estocástica”, WP 5/99. Universidad Europea, Madrid
Fernández JL, Robles MD (2008) Time-series model forecasts and structural breaks: evidence from Spanish pre-EMU interest rates. Appl Econ 40(13):1707–1721. doi:10.1080/00036840600895640
Fernández-Serrano JL, Sosvilla S (2003) Modelling the linkages between US and Latin American stock markets. Appl Econ 35(12):1423–1434. doi:10.1080/0003684032000100409
Frankel J, Rose A (1996) A panel project on purchasing power parity: mean reversion within and between countries. J Int Econ 40:209–224. doi:10.1016/0022-1996(95)01396-2
Granger C (1986) Developments in the study of cointegrated variables. Oxf Bull Econ Stat 48:213–228
Gregory AW, Hansen BE (1996) Residual based tests for cointegration in models with regime shifts. J Econom 70:99–126. doi:10.1016/0304-4076(69)41685-7
Grilli V, Kaminsky G (1991) Nominal exchange rate regimes and the real exchange rate: evidence for the United States and Great Britain, 1885–1986. J Monet Econ 27:191–212. doi:10.1016/0304-3932(91)90041-L
Juvenal L, Taylor MP (2007) The law of one price: Nonlinearities in sectoral real exchange rate dynamics. University of Warwick
Kapetanios G, Shin A, Snell Y (2003) Testing for a unit root in the nonlinear STAR framework. J Econom 112:359–379. doi:10.1016/S0304-4076(02)00202-6
Kapetanios G, Snell A, Shin Y (2006) Testing for cointegration in nonlinear smooth transition error correction models. Econom Theory 22:279–303. doi:10.1017/S0266466606060129
Koedijk K, Schotman P, Van Dijk M (1998) The re-emergence of PPP in the 1900’s. J Int Money Finance 17:51–61. doi:10.1016/S0261-5606(97)98051-7
Liu PC, Maddala GS (1996) “Do panel data cross-country regressions rescue purchasing power parity (PPP) theory?” Working Paper, Department of Economics, Ohio State University
Lothian J (1997) Multi-country evidence on the behaviour of purchasing power parity. J Int Money Finance 16:19–35. doi:10.1016/S0261-5606(96)00047-2
Ludlow J, Enders W (2000) Estimating non linear ARMA models using fourier coefficients. Int J Forecast 70:261–290
MacKinnon JG (1996) Numerical distribution functions for unit root and cointegration tests. J Appl Econ 11:601–618. doi:10.1002/(SICI)1099-1255(199611)11:6<601::AID-JAE417>3.0.CO;2-T
Meese R, Rogoff K (1988) Was it real? The exchange rate interest differential relation over the modern floating exchange rate period. J Finance 43:933–948. doi:10.2307/2328144
Michael P, Nobay A, Peel D (1997) Transaction costs and non-linear adjustments in real exchange rates: an empirical investigation. J Polit Econ 105:862–879. doi:10.1086/262096
O’Connell PGJ (1998) The overvaluation of purchasing power parity. J Int Econ 44:1–19. doi:10.1016/S0022-1996(97)00017-2
Oh KY (1996) Purchasing power parity and unit root tests using panel data. J Int Money Finance 15:405–418. doi:10.1016/0261-5606(96)00012-5
Papell DH (1997) Searching for stationarity: purchasing power parity under the current float. J Int Econ 43:313–332. doi:10.1016/S0022-1996(96)01467-5
Papell DH, Theodoridis H (1998) Increasing evidence of purchasing power parity over the current float. J Int Money Finance 17:41–50. doi:10.1016/S0261-5606(97)00050-8
Pedroni P (1997). “Panel cointegration: asymptotic and finite sample properties of pooled time series tests with an application to the PPP hypothesis (new results).” Working paper. Department of Economics, Indiana University
Perron P, Rodriguez G (2001) Residual based test for cointegration with GLS detrended data. University of Montreal, PhD dissertation
Robles MD, Nieto ML, Fernández A (2004) Nonlinear Intraday dynamics in Eurostoxx50 index markets. Stud Nonlinear Dyn Econom 8(4), art. 3
Rothman P (1998) Forecasting asymmetric unemployment rates. Rev Econ Stat 80:164–168. doi:10.1162/003465398557276
Sercu P, Uppal R, Van Hull C (1995) The exchange rate in the presence of transaction costs: implications for tests of purchasing power parity. J Finance 50:1309–1319. doi:10.2307/2329354
Taylor A (1996) “International capital mobility in history: Purchasing-power parity in the long run.” Working Paper 5742. National Bureau of Economic Research, Cambridge, MA
Taylor A, Taylor M (2004) The purchasing power parity debate. J Econ Perspect 18:135–158. doi:10.1257/0895330042632744
Uppal R (1993) A general equilibrium model of international portfolio choice. J Finance 48:529–553. doi:10.2307/2328911
Wu Y (1996) Are real exchange rates non-stationary? Evidence from a panel data test. J Money Credit Bank 28:54–63. doi:10.2307/2077966
Acknowledgement
We are grateful to A. Novales and two anonymous referees for helpful comments. The paper has benefited of comments from seminar participants at the George Washington University and from participants in the Unit root and Testing Cointegration conference. We acknowledge the financial support of the Ministerio de Ciencia y Tecnología, Spain, through the Project SEJ2006-14354/ECON and the Comunidad de Madrid through the Project 940063.
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Jiménez-Martin, J.A., Robles-Fernandez, M.D. PPP: Delusion or Reality? Evidence from a Nonlinear Analysis. Open Econ Rev 21, 679–704 (2010). https://doi.org/10.1007/s11079-009-9113-0
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DOI: https://doi.org/10.1007/s11079-009-9113-0
Keywords
- Unit-root test
- Cointegration test
- Fourier approximation
- Nonlinear VECM
- Exchange rates
- Purchasing power parity