Unit roots and purchasing power parity: another kick at the can

Abstract

Lopez et al. (2005) demonstrated that single-equation unit-root tests cannot provide conclusive evidence of whether real exchange rates are stationary because inference depends critically on the lag-lengths used to construct the test statistics, a result reinforced by a recent work by Sweeney (2006). The purpose of this article is to revisit the issue, first demonstrating the necessary conditions under which this approach of testing for Purchasing Power Parity (PPP) is appropriate.

Acknowledgements

I thank Mark Jensen, Claude Lopez, Laura Mayoral, Morten Nielsen, Alan Taylor and Dimitrios Vougas for sharing their data and computer code, and Daniel L. Thornton for helpful comments. Any errors or omissions are my own.

Notes

¹ This analysis will focus on single-equation tests rather than panel-based tests of unit roots and stationarity in real exchange rates. Breuer et al. (2001), Taylor (2003) and Sweeney (2006) provide evidence using panel methods.

² With appropriate recognition of post hoc ergo prompter hoc.

³ LMP report test values (lag-lengths) of −2.62 (1), −3.55 (1) and −2.95 (1) for Austria, France and Germany, respectively, with results for Austria significant at the 10% level, France at the 1% level and Germany at the 5% level. Increasing the maximum lag-length to 12 leads to test values (lag-lengths) of −1.29 (9), −1.89 (10), and −2.91 (12) for Austria, France and Germany, respectively. Using MacKinnon's (1991) response surface estimates, the latter statistic has a probability value of just under 5% (4.75%). Lag selection was based on variable reduction, using the 10% level of significance.

⁴ Simulations are based on 50 000 replications. Calculations are based on an autoregressive coefficient of 0.80 to generate the data. The first 30 data points are discarded in creating the series.

⁵ To examine the power of the test, I simulated the distributions of the test statistics, both allowing for additive outliers and not; see Otero and Smith (2005) and Darne (2004). The tests are most powerful for a ‘hyperparameter’ equals 2 while the test sizes were best for a much higher setting of 8. The closer the process was to a unit root, the worse were the sizes of the tests. These results are also presented in Table 4. Extensions to endogenously chosen structural change along the lines of Harvey and Mills (2004) is the subject of ongoing work.

⁶ Sollis (2005) examines a similar question but assumes PPP requires the smooth transitions to start and end at the same values.

⁷ Examining the sensitivity of this test of lag selection is the subject of ongoing work.

⁸ See Baillie (1996), Phillips and Xiao (1998) and Sephton (2002), for examples. Tests for threshold unit roots are also popular; see Bec et al. (2004).

⁹ See Geweke and Porter-Hudak (1983), Andrews and Guggenberger (2003), Jensen (2004) and Stoev and Taqqu (2005), and the references therein for examples.

¹⁰ See, for example, Dolado et al. (2005).