Testing for Nonlinearity in Exchange Rate Dynamics

Purchasing Power Parity (PPP) embodies the idea that, when expressed in the same currency units, price levels should be equal across nations (Cassel, 1918). Variations in the real exchange rate can be thought of as deviations from PPP. A long literature has attempted to reconcile the high short-term volatility in real exchange rates with the slow rate at which convergence to PPP seems to occur. This has become known as the PPP puzzle (Rogoff, 1996).

The (log) real exchange rate can be expressed as qt = st −pt +pt∗, where st is the (log)

nominal exchange rate,ptis the logarithm of the domestic price level, andpt∗is the logarithm of

the foreign price level. This formulation allows one to interpret the real exchange rate as a measure of deviation from Purchasing Power Parity. Taylor et al. (2001) and others note that studies of the effect of transaction costs on PPP suggest that exchange rate adjustments resemble a non-linear process in which the rate appears to be a unit root process within a band and a stationary process outside of that band. They model real exchange rate dynamics with a model that allows a smooth transition at the boundary of the band. In particular, they examine the STAR model (Granger and Ter¨asvirta, 1993) qt−µ= p X j=1 βj(qt−j−µ) + h_Xp j=1 β_j∗(qt−j−µ) i Φ(γ;qt−d−µ) +εt

where{qt}is assumed stationary and ergodic with εt ∼ iid(0, σ2)and the exponential transition

function

Φ(γ;qt−d−µ) = 1−exp(−γ2(qt−d−µ)2).

Alongside the exponential transition function, the model is referred to as the ESTAR model. Sim- ilar models, including the Logistic (LSTAR) model with transition function

Φ(γ;qt−d−µ) = h

1−exp(−γ(qt−d−µ)) i−1

have been used as specification tests for the estimated models. van Dijk et al. (2002) provide an extensive review of smooth transition models.

Two potential issues appear in this modeling exercise. First the unknown value of d must be selected. Second, given the non-linearity of the chosen model, parameter identification failure may result under some situations, and in particular parameter identification failure occurs under the null hypothesis when testing no omitted non-linearity. For the first point, Taylor et al. (2001) provide economic intuition in favor of smaller values of the parameter d, namely that we should not expected a long lag between a shock and the adjustment response from the exchange rate. The second issue is handled less satisfactorily, as the modeling procedure is based on a linearization of the non-linear model about the point of identification failure. This method addresses issues that arise from identification failure, but as recent research indicates, this may provide a poor approximation to the desired model (Kilic, 2016).

Further, Hill (2008) notes that the traditional method involving a truncated Taylor approxima- tion simply “directs power toward low order polynomials” and is therefore not truly a test against smooth transition alternatives. He draws attention to the fact that treatingd as a parameter to be estimated yields a non-standard limiting distribution, a fact that was ignored in the early literature. Importantly, he notes that a test that only considers a finite number of conditions (e.g. a small support ford or a finite-order polynomial approximation) can give rise to inconsistency. Francq, Horvath, and Zako¨ıan (2010) also examine non-standard tests that result due to the presence of nui- sance parameters when testing for linearity against smooth transition autoregressive alternatives.

Taylor et al. (2001) follow a sequential modeling procedure similar to those suggested in Granger and Ter¨asvirta (1993), Terasvirta (1994), and Eitrheim and Ter¨asvirta (1996). Kilic (2016) follows the specification procedure in Ter¨asvirta (2004) and utilizes the diagnostic tests suggested by Eitrheim and Ter¨asvirta (1996) for the first differenced model

∆qt = h β₀∗+ p X i=1 β_i∗∆qt−i i Φ(γ,∆qt−d) +ut.

This class of models fits into the class of additive nonlinear models yt = p X j=1 βjgj(Xj,t, πj) +Zt0ζ+εt.

In particular, there is no need for different models for each d, as the model corresponding to a particulardis just a restriction on a larger model:

yt = s X j=1 ˜ βjyt−j+ r X d=1 s X j=1 ˜ β_j,d∗ yt−jΦ(γ;yt−d) +εt =Z_t0ζ+ p X j=1 βjgj(Xj,t, πj) +εt

where p = rs, Zt = (yt−1, . . . , yt−s), and gj(Xj,t, πj) ≡ yt−jΦ(γ;yt−d). Letting r → ∞, we

can then form a test of no nonlinearity via the null hypothesis thatβ = 0 or a test of no omitted nonlinearity with the null hypothesis that a subset ofβis the zero vector.

Typically in this literature,γ = 0drives identification failure inβ. This parameterization may lead to issues with inference under the framework presented here, sinceβ = 0would also induce the identification failure ofγ. We are not aware of any study of such ‘double identification failure.’ For this setup, we will require that eitherγ >0orβ >0so that only a single point of identification failure exists.