The process followed by PPP data. On the properties of linearity tests

Abstract

Recent research has reported the lack of correct size in stationarity test for PPP deviations within a linear framework. However, theoretically well motivated non-linear models, such as the ESTAR, appear to parsimoniously fit the PPP data and provide an explanation for the PPP ‘puzzle’. Employing Monte Carlo experiments the size and power of the non-linear tests are analysed against a variety of nonstationary hypotheses. Aslo the ESTAR model is fitted to data from high inflation economies. The results provide further support for ESTAR specification.

Acknowledgment

Ivan Paya acknowledges financial support from the IVIE.

Notes

¹ Estimates of the non-linear models have been reported employing data sampled at different levels of aggregation. For example, Taylor et al. (2001), report ESTAR models employing monthly data, Kilian and Taylor (2003) quarterly data whilst Micheal et al. (1997), and Paya and Peel (2003) report results employing annual data. Paya and Peel (2004) show that the empirical results obtained at different degrees of temporal aggregation are consistent with the non-linear data generating process (DGP) obtained at the highest frequency observable (monthly).

² Rogoff drew attention to the fact that in linear models it is difficult to reconcile high short-term volatility of real exchange rates with extremely slow convergence to PPP. The average reported half-life of PPP deviations based on linear models is around 3–5 years, seemingly far too long to be explained by nominal rigidities. An important property of the non-linear models is that their impulse response functions show that whilst the speed of adjustment for small shocks around equilibrium can be extremely slow, larger shocks mean-revert much faster than the ‘glacial rates’ obtained in the linear estimates (see e.g. Taylor et al., 2001, and Paya et al., 2003).

³ Taylor et al. (2001) and Pippenger and Goering (1993) demonstrate that the standard Dickey–Fuller unit root tests have low power against data simulated from an ESTAR model.

⁴ For both high-inflation economies and also for bilateral real exchange rates against the US dollar under the post-war float.

⁵ Byers and Peel (2000) show that data that is generated from an ESTAR process can appear to exhibit the fractional property (as conjectured by Acosta and Granger, 1995) thus potentially providing an explanation of the results reported employing the fractional processes.

⁶ Caner and Kilian (2001) examine the size distortions of stationarity tests within the PPP framework. However, their study assumes linear stationary real exchange rates.

⁷ These sample sizes would correspond to available data at monthly or quarterly frequency for the post-Bretton Woods period. The actual sample size in each replication were of 1360 and 1120 data points, respectively. Then the first 1000 data points are discarded.

⁸ The ESTAR process is of the form

⁹ The χ² version of the test yielded similar results.

¹⁰ See Terasvirta (1994) when d is unknown.

¹¹ A maximum number of 6 lags and the minimum lag length is one is used initially.

¹² Empirical marginal significance levels of the estimated parameter γ has to be obtained through Monte Carlo simulation as it is not defined under the null. In particular, the model is assumed to follow a unit root linear autoregressive process and then a non-linear ESTAR specification (Equation 5) is estimated, computing the appropriate confidence interval of significance for γ. See Paya and Peel (2004) for a detailed description of the procedure.

¹³ The results reported correspond to the restricted cases where β ₁ = 1, β ₂ = 0 in the ‘monthly’ case of sample of 360, and β ₁ + β ₂ = 1 in the ‘quarterly’ case of samples of 120. The results of the unconstrained cases are quantitatively similar and available upon request from the authors.

¹⁴ See Kapetanios et al. (2003) for a new developed test for non-linearity when processes are non-linear STAR models. Their tests has better power than the tests used in this paper against the random walk hypothesis. However, it can only be used to discriminate between unit root and globally mean reverting STAR processes.

¹⁵ The model Q_t  =  (1 + δ _t )Q_{t −1} + v was also estimated, where . The results were quantitatively similar to the case above and available upon request from the authors.

¹⁶ Taylor and Van Dijk (2002) demonstrate that the stochastic unit root tests of McCabe and Tremayne (1995) appear to display little or no power against threshold unit root processes.

¹⁷ Except in the case of a stochastic unit root null hypothesis.

¹⁸ Data kindly provided by Michael Bleaney. Bleaney et al. (1999) reported that data for Brazil was a pure unit root process. A parsimonious ESTAR process is fitted to that data set. On the other hand, they report a stochastic unit root process for Chile but a parsimonious ESTAR process to data cannot be fitted for the country.

¹⁹ Panel B in Table 3 provides the half-lives of shocks to the non-linear ESTAR model using the Generalized Impulse Response Function (GIRF) introduced by Koop et al. (1996) that successfully confronts the challenges that arise in defining impulse responses for non-linear models. Taylor et al. (2001) and Paya et al. (2003) report similar estimates of non-linear impulse response functions for bilateral and effective real exchanges at monthly frequencies for post-Bretton Woods period. Panel B also reports the non-linearity tests. However, the results of previous section show that those tests do not have the appropriate size and power to discriminate between stochastic unit root process and globally mean reverting non-linear ESTAR.