Median-unbiased estimation in DF-GLS regressions and the PPP puzzle

Abstract

Using median-unbiased estimation based on Augmented Dickey–Fuller (ADF) regressions, recent research has questioned the validity of Rogoff's ‘remarkable consensus’ of 3–5 year half-lives of deviations from Purchasing Power Parity (PPP). The confidence intervals of these half-life estimates, however, are extremely wide, with lower bounds of about 1 year and upper bounds of infinity. We extend median-unbiased estimation to the Dickey–Fuller Generalized Least Square (DF-GLS) regression of Elliott et al. (1996). We find that combining median-unbiased estimation with this regression has the potential to tighten confidence intervals for the half-lives. Using long-horizon real exchange rate data, we find that the typical lower bound of the confidence intervals for median-unbiased half-lives is just below 3 years. Thus, while previous confidence intervals for median-unbiased half-lives are consistent with virtually anything, our tighter confidence intervals are inconsistent with economic models with nominal rigidities as candidates for explaining the observed behaviour of real exchange rates and move us away from solving the PPP puzzle.

Keywords:

• DF-GLS
• median unbiased
• PPP
• half-life

JEL Classification:

• C22
• F30

Acknowledgements

We thank Lutz Kilian, Barbara Rossi, seminar participants at Duke University, Texas Econometrics Camp 8, the Southern Economic Association meetings, Econometrics Brown bag seminar at the Banque de France and two anonymous referees for helpful comments and discussions. Papell thanks the National Science Foundation for financial support.

Notes

¹ Engel (2000) raises the question of whether these rejections are caused by size distortions.

² See Kilian (1998, 1999), Hansen (1999), and Inoue and Kilian (2002) for further discussion of bootstrapping autoregressive processes with unit roots or near unit roots.

³ See Kilian and Zha (2002) for a Bayesian perspective.

⁴ Taylor (2001) and Imbs et al. (2005) investigate time aggregation and sectoral heterogeneity bias, respectively, although their results remain controversial. We treat the Consumer Price Index (CPI) based real exchange rate as the object of interest, and thus dealing with these potential sources of bias is beyond the scope of this article.

⁵ Panel methods have been used extensively to test for unit roots in post-1973 real exchange rates. Murray and Papell (2005), Choi et al. (2006) and Lopez (2008) examine real exchange rate persistence with panel methods. Elliott and Pesavento (2006) use univariate unit root tests with stationary covariates to investigate PPP in post-1973 data. These covariate augmented unit root tests have not yet been extended to calculate unbiased half-life estimates.

⁶ Murray and Papell (2002) also analyse six long-horizon (1900–1996) real exchange rates, but the set of counties is nonoverlapping with the series used here, and they are constructed with Wholesale Price Indexes (WPIs) rather than CPIs as in Taylor (2002).

⁷ The regression with only a constant and a lagged dependent variable is Case 2 in Andrews (1993). Cases 1 and 3 have no deterministic regressors, and a constant and time trend respectively. Since we are interested in the strict interpretation of PPP, for our purposes Case 2 is appropriate.

⁸ We note that the αs in Equations 2 and 3 are in general not the same, but we use the same notation for convenience.

⁹ Again, since we are interested in the strict interpretation of PPP, we do not allow for deterministic time trends, although doing so is straightforward.

¹⁰ See Andrews and Chen (1994) and Murray and Papell (2002) for further details concerning the computation of approximately median-unbiased estimators.

¹¹ While our subsequent empirical application reports 95% confidence intervals, we report 90% confidence intervals in Table 1 in order to directly compare our estimator with Andrews’ estimator, for which he does not report 95% confidence intervals.

¹² Using a different methodology, Rossi (2005) only reports confidence intervals for half-lives.

¹³ The gain in precision leads to a smaller variance, hence narrower confidence intervals.

¹⁴ We note that if the series exhibits mild structural change, where the size of the break is small relative to the innovation SD, then it is possible to reject the unit root null.

¹⁵ Although we do not report them, we have constructed confidence intervals for α, and thus the half-life, based on the OLS estimates. We used the delta-method, as well as both a parametric and a nonparametric bootstrap. In every case, these confidence intervals are shifted to the left of those reported in Table 3, and the coverage probabilities are much less than 95%.

¹⁶ The GS lag selection starts with a maximum lag, typically eight in annual data, and does a sequence of hypothesis tests to determine the significance of the coefficient on the longest lagged first difference term. The procedure stops once a significant coefficient is found. See Hall (1994) and Ng and Perron (1995) for further discussion.

¹⁷ We do not report these results here, except for the US/UK real exchange rate, where the selected lags are equal.