Testing PPP hypothesis under temporary structural breaks and asymmetric dynamic adjustments

ABSTRACT

We test the empirical validity of the PPP proposition under temporary structural breaks and dynamic nonlinear adjustments. Although several testing procedures have recently been proposed in the existing literature to investigate stochastic properties of the series under gradual breaks and nonlinear adjustments, none of these tests are compatible with the PPP proposition. Therefore, we propose new testing procedures that restrict the break to be temporary while simultaneously allowing for asymmetric dynamic nonlinear adjustment towards equilibrium. Using these newly proposed tests, we test stationarity of real exchange rate of 24 OECD countries vis-à-vis USA, and find support in favour of PPP proposition in majority of the countries.

KEYWORDS:

• PPP
• Temporary Gradual Break
• Nonlinear Symmetric and Asymmetric Dynamic Adjustments

JEL CLASSIFICATION:

• F31
• F40
• C12
• C22

Acknowledgements

We are grateful to the editor and an anonymous reviewer of this journal for their insightful comments and suggestions that improved quality and presentation of the article significantly. The usual disclaimer applies.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ See Officer (1976), Rogoff (1996), Taylor, Peel and Sarno (2001), Taylor (2003) and Carvalho and Julio (2012) for an extensive survey of the literature.

² Interaction of heterogeneous traders in goods and/or foreign exchange markets, diversity in agents’ beliefs, central banks’ intervention into the foreign exchange markets, market imperfections and frictions may also give cause to nonlinear adjustment towards equilibrium. See also Taylor (2003) and Hasanov (2014).

³ See also Omay and Yıldırım (2014), who also used a logistic function to model gradual break in the series.

⁴ For a thorough discussion of smooth transition regression models see Teräsvirta (1994) and van Dijk (1999).

⁵ As the logistic transition function $F_t\left({\theta }_2,u_{t-1}\right)$ given in Eq. (7)(7) $F_t\left({\theta }_2,u_{t-1}\right)={\left[1+\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\theta }_2{u}_{t-1}\right)\right]}^{-1},{\theta }_2>0,$(7) is bounded between zero and one, it follows that the deviations from equilibrium $u_t$ is geometrically ergodic, and hence, asymptotically stationary. When ${\theta }_1=0$, the exponential function given in Eq. (6)(6) $G_t\left({\theta }_1,u_{t-1}\right)=1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\theta }_1\left(u_{t-1}^2\right)\right),{\theta }_1>0$(6) will also be equal to zero, $G_t\left({\theta }_1,u_{t-1}\right)=0$, and hence, Eq. (5)(5) $\mathrm{\Delta }{u}_t=G_t\left({\theta }_1,u_{t-1}\right)\left\{F_t\left({\theta }_2,u_{t-1}\right){\rho }_1\text{break}+\left(1-F_t\left({\theta }_2,u_{t-1}\right)\right){\rho }_2\right\}{u}_{t-1}+{ϵ}_t$(5) will collapse to $\mathrm{\Delta }{u}_t={ϵ}_t$, implying that deviations from the equilibrium follow a unit root process. On the other hand, if ${\theta }_1>0$, $u_t$ will follow a nonlinear but stationary process provided that $-2<\left[F_t{\rho }_1+\left(1-F_t\right){\rho }_2\right]<0$, which is assumed to hold. See Kapetanios, Shin, and Snell (2003) and Omay, Emirmahmutoglu and Hasanov (2018).

⁶ Here, we assume that the serially correlated errors are linear. Alternatively, as Sollis (2009) argues, if higher order dynamics are nonlinear, then the augmentation terms can be interpreted as a first-order approximation.

⁷ See also Kapetanios, Shin and Snell (2003), Sollis (2004), Hasanov (2014), Omay, Emirmahmutoglu and Hasanov (2018).

⁸ For comparison of performance of alternative estimation techniques see Omay and Emirmahmutoglu (2017).

⁹ Although asymptotic distribution of the test statistics cannot be derived analytically, we generated density functions of the tests by stochastic simulations. Simulated density functions of the test statistics are presented in the Appendix A.

¹⁰ According to Omay and Emirmahmutoglu (2017), the SQP algorithm performs best among considered alternatives in terms of accuracy of the estimated nonlinear parameters. The second best performer is the Genetic algorithm, which considerably reduces computation time. However, both the SQP and Genetic algorithms produce the same critical values. Therefore, in this article, we used the Genetic algorithm to produce the critical values. We used extensive grid search to find the initial values for the Genetic algorithm.

¹¹ In particular, we use the same specification of the series given in Eqs. (3(3) $y_t={\alpha }_1+{\beta }_{1}t+{\alpha }_2{S}_t\left(\gamma ,\tau \right)+{\beta }_{2}t{S}_t\left(\gamma ,\tau \right)+u_t,$(3) –7(7) $F_t\left({\theta }_2,u_{t-1}\right)={\left[1+\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\theta }_2{u}_{t-1}\right)\right]}^{-1},{\theta }_2>0,$(7) ) with the only difference being in that, instead of the transition function $S_t\left(\gamma ,\tau \right)=1-exp\left[-\gamma {\left(t-\tau T\right)}^2\right]$, $\gamma >0$, given in (2.2), we use $S_t\left(\gamma ,\tau \right)=1-exp\left[-{\gamma }^2{\left(t-\tau T\right)}^2\right]$. Using this specification restriction adds further nonlinearity into transition function. As nonlinear optimization algorithms are sensitive to specification of the nonlinear structure, such a specification restriction ultimately alters the estimated deterministic component, and hence, critical values of the tests also change slightly. For this purpose, we also simulated the critical values of the proposed tests using this specification restriction. The critical values of the tests under this specification restriction are presented in Table 1B in Appendix.

¹² We also repeated the simulation exercises by varying other parameters as well. However, the simulation results were both qualitatively and quantitatively similar. In order to preserve space, we do not report results of these simulation exercises, which are available from authors upon request.

¹³ See Appendix B.

¹⁴ We use the same data and time span as Corakcı, Emirmahmutoğlu and Omay (2017) to facilitate comparison with their results.

¹⁵ We thank an anonymous reviewer for this suggestion.