Asymmetric Adjustment and Nonlinear Dynamics in Real Exchange Rates

Gene L. Leon; Serineh Najarian

doi:10.5089/9781451857696.001.A001

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Asymmetric Adjustment and Nonlinear Dynamics in Real Exchange Rates

Author: Mr. Gene L. Leon and Serineh Najarian¹

¹ 0000000404811396https://isni.org/isni/0000000404811396International Monetary Fund

Contributor Notes

Author’s E-Mail Address: hleon@imf.org; serineh.najarian@hertford.oxford.ac.uk

Publication Date:: 01 Jul 2003

eISBN:: 9781451857696

Language:: English

Keywords:: WP; exchange rate; standard deviation; positive correlation; nominal exchange rate; corridor regime; countries duration; numbers in symmetry

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This paper examines whether deviations from PPP are stationary in the presence of nonlinearity, and whether the adjustment toward PPP is symmetric from above and below. Using alternative nonlinear models, our results support mean reversion and asymmetric adjustment dynamics. We find differences in magnitudes, frequencies, and durations of the deviations of exchange rates from fixed and time-varying thresholds, both between over-appreciations and over-depreciations and between developed and developing countries. In particular, the average cumulative sum of deviations during periods when exchange rates are below forecasts is twice that of the sum during periods of over-appreciation, and is larger for developing than for advanced countries.

Abstract

I. Introduction

Purchasing power parity (PPP) states that national price levels should be equal when expressed in a common currency. Therefore, variations in the real exchange rate (RER), defined as the nominal exchange rate adjusted for relative national price levels, represent deviations from PPP. While an exact PPP relationship is not expected to hold at every period, researchers have been concerned about the almost universal finding that deviations from PPP appear to persist for very long periods (that is, have unit roots). Sarno and Taylor (2002) list three reasons why we should care if the real exchange rate has a unit root. First, the degree of persistence can be used to infer the principal impulses driving real exchange rate movements, high persistence indicating principally supply-side shocks. Second, nonstationarity raises questions about a large part of open economy macroeconomic theory that assumes PPP. Third, policy based on estimates of PPP exchange rates may be flawed if the real exchange rate contains a unit root. Research on PPP has therefore focused on the credibility of the unit root finding and on why deviations from PPP exist.

One explanation of the unit root finding relates to the low power of unit root tests. Consequently, a number of researchers have sought to increase the power of unit root tests by increasing the span of the data (Lothian and Taylor (1996), Cheung and Lai (1998)), and by using panel unit root tests (Frankel and Rose (1996), Taylor and Sarno (1998)). Another explanation, and that which we examine in this paper, is that standard unit root tests arc likely to be biased and have low power in rejecting the null of a unit root because real exchange rates follow a nonlinear adjustment process (Yilmaz (2001); Bergman and Hansson (2000); De Grauwe and Vansteenkiste (2001); Kilian and Taylor (2001); Michael, Nobay, and Peel (1997); Taylor (2001)).

A. Nonlinear Adjustment and Asymmetry

Nonlinear exchange rate adjustment may arise from transaction costs in international arbitrage (Sercu, Uppal, and Van Hulle (1995); Obstfeld and Taylor (1997); Coleman (1995); O’Connell and Wei (2002)).² Deviations from PPP are assumed not corrected if they are small relative to the costs of trading.³ Proportional or “iceberg” costs create a band (thresholds) for the real rate, within which the marginal cost of arbitrage exceeds the marginal benefit. Dixit (1989) and Krugman (1989) argue that thresholds may also arise because of sunk costs of international arbitrage and the tendency for traders to wait for sufficiently large arbitrage opportunities before entering the market. Another explanation is government intervention (Dutta and Leon (2002)). Governments care about large and persistent deviations because real exchange rates are likely to affect net exports, as well the cost of servicing debt denominated in foreign currency. In fact, Calvo, Reinhart, and Veigh (1995) concluded that the RER is perhaps the most popular real target in developing countries.

These models suggest that the exchange rate can be modeled as a regime-switching process, with a band of inaction. Thus, the exchange rate will at least revert to a range. Two issues arise: the choice of the switching function governing the regime change and the symmetry of rates of reversion on either side of the band of inaction. In some models the jump to mean or range reversion is sudden (Obstfeld and Taylor (1997)), while in others it is smooth (Michael and others (1997)). Dumas (1992), and Teräsvirta (1994) argue that time aggregation and nonsynchronous trading favor smooth transition between regimes. It can also be argued that the averaging implicit in the compilation of the real exchange rate index would suggest a smooth rather than discontinuous adjustment process, given that the underlying goods traded have different arbitrage costs. For the second issue, the accepted view is that the transactions-cost model requires symmetry of thresholds and adjustment parameters (Lo and Zivot (2001)). For example, Michael, Nobay, and Peel (1994) argue that because adjustment to PPP deviations must be the same for positive and negative deviations from equilibrium, it is appropriate to specify a symmetric threshold autoregressive (TAR) model with the same autoregressive parameters in the outer regimes. Similarly, Taylor, Peel, and Sarno (2001) propose a nonlinear model that implies random behavior near equilibrium but mean-reverting behavior for large departures from fundamentals. They estimate an exponential threshold autoregressive (ESTAR) model that implies symmetric adjustment of the exchange rate above and below equilibrium.⁴ On the other hand, Dutta and Leon (2002) argue that countries may choose to defend depreciations more or less vigorously than appreciations, thereby generating asymmetric adjustment behavior.

We address these issues by estimating and evaluating three classes of regime switching models for a range of advanced and developing economies.⁵ The first model is a time-varying threshold autoregressive model (TVTAR), which allows asymmetrical adjustment when real exchange rates deviate from forecasts. The estimated model allows us to calculate the magnitudes, frequencies, and durations of these deviations from forecasts, both for depreciations and appreciations. The second specification is an adaptation of Silverstov’s (2000) bi-parameter smooth transition regression (BSTR), which allows for asymmetric adjustment between the middle and outer regimes. The third specification is a Markov switching model (MSM), where the change in the regimes in exchange rates dynamics is governed by an unobservable Markov chain. Thus, our design compares smooth versus sudden switching in regimes, includes fixed and time-varying thresholds, and allows for asymmetry in adjustment.

B. Issues in Testing for Unit Roots

If the true model is nonlinear, then estimates from a linear model will average the potentially reverting data outside of the band with the nonstationary nonreverting data within the band, leading to biases, especially if bias due to nonlinearity interacts with bias due to temporal aggregation (Taylor (2001)). Therefore, the effect of nonlinearity needs to be considered in tests for nonstationarity. Goering and Pipenger (1993 and 2000) argue that the presence of threshold nonlinearities reduces the power of standard unit root and cointegration tests; Michael and others (1997) argue that cointegration or unit root tests may be biased when the linear alternative neglects nonlinearity of the smooth transition autoregressive (STAR) type. Nelson, Piger, and Zivot (2001) show that standard augmented Dickey-Fuller (ADF) tests have low power against stable but occasionally integrated alternatives. In fact, these nonlinear models may be globally stationary even if they have a unit root in the middle regime.

Testing for unit roots when the data generating process (DGP) is nonlinear poses two problems. First, which nonlinear model should be used? Most researchers consider one process and few comparisons exist (see Carrasco (2001) and Taylor and van Dijk (2002)). Yet, the failure to confirm a regime shift may be due to misspecification of the alternative. Second, how should we test for nonstationarity in the presence of nonlinearity? We address these issues by estimating alternative nonlinear specifications and employing recent developments in the joint analysis of nonstationarity and nonlinearity, proposed by Balke and Fomby (1997) in the context of threshold cointegration and subsequently developed by Berben and van Dijk (1999), Caner and Hansen (2001), Kapetanios and Shin (2002), and Lo and Zivot (2001) in the context of TAR models, and by Kapetanios, Shin, and Snell (2003) when the alternative is a stationary ESTAR process.

C. Summary of Results

Our research contributes to the literature in three related ways. First, we introduce the TVTAR and provide evidence on nonstationarity in the presence of nonlinearity. For the TVTAR models, we follow Caner and Hansen (2001), who allow for both effects simultaneously, in computing Wald tests for unit roots (nonstationarity) when the threshold nonlinearity is either present or absent.⁶ Second, we focus on potential asymmetries in the short-run dynamics of real exchange rates by allowing the parameters of the models to be estimated unrestrictedly. In particular, we estimate BSTR models that allow different adjustment speeds from the lower-to-middle and middle-to-upper regimes, providing direct evidence on asymmetrical adjustment. Third, we implement tests that allow comparisons of alternative specifications. We follow Breunig, Najarian, and Pagan (2002) (BNP) who develop tests to compare the implied densities of the estimated models with that of the data. We complement the BNP tests with Hamilton’s (2001) flexible parametric nonlinearity test and Li’s (1996) test of density equivalence.⁷

We estimate the models for 26 countries, using monthly data on real effective exchange rates. Our sample includes all G-7 countries, a selection of advanced countries, and some emerging market countries from Asia and Latin America.⁸ Our results provide support for both stationary regime-switching processes and asymmetric adjustment. For the threshold models, the Wald tests show that the unrestricted TVTAR outperforms both the linear specifications (stationary as well as nonstationary) and the identified threshold nonstationary model (unit root with threshold effects). We find support in some developing countries for the threshold model with a unit root in the corridor regime. For the smooth transition models, we find reversion to the mean in almost every case when the nonlinear component is included. As regards asymmetry, we calculate the speed of response to deviations from forecasts and duration of time spent outside threshold bands to gauge the potential impact of real exchange rate misalignments. For the TAR models, we find that while advanced countries respond faster than developing to over-appreciations and over-depreciations, Asian and G-7 (other advanced and Western Hemisphere) countries in our sample respond more strongly to developments relating to over-appreciations (over-depreciations). We find asymmetric speeds of adjustment between regimes in the smooth transition models in more than half of the countries. In general, durations are longer for over–appreciations. For the threshold model, durations are longer for over–depreciations in the G-7 and Asian countries, but for over–appreciations in the other advanced countries and countries in Western Hemisphere (WH). The excess deviation measure of these over–depreciations is uniformly about twice that for over–appreciations and larger for developing countries than for advanced countries. We evaluate all the models estimated for their ability to replicate five characteristics of the densities of the data. We find that the nonlinear specifications better explain the first two moments and the asymmetry and persistence characteristics to a lesser extent, but do less well, especially for developing countries, in replicating the observed interquartile range. In general, the BSTR specification, which captures best the characteristics of interest, adequately characterizes the nonlinearity in the observed data and provides a realistic insight into the short-run dynamics.

Some potential implications of our results relate to the effects of real exchange rate misalignment. Countries with longer durations of misalignment, larger deviations from threshold bands, or higher excess deviations could have a higher probability of experiencing hysteresis effects. These probabilities appear higher for over–depreciations than for over–appreciations and more so for developing countries than for advanced economies. Consequently, an argument can be made for interventionist policies aimed at reducing the variability and length of duration of misalignments outside a desired range.

The rest of this paper is organized as follows. Section II discusses the nonlinear frameworks used in estimating the real exchange rate dynamics. In Section III we present the results. A brief summary follows in Section IV.

II. Nonlinear Frameworks

Nonlinear modeling of economic variables assumes that different states of the world or regimes exist and that the dynamic behavior of economic variables depends on the regime occurring at a point in time. Therefore certain properties of the time series, such as means and autocorrelations, vary with each regime. We consider nonlinear models that are characterized as piecewise linear processes, such that the process is linear in each regime. Each model is distinguished by a different stochastic process governing the change of regime. Our models are intentionally eclectic and nonnested to provide a measure of robustness to the results. Our generic functional form is:

y_{t} = {\tilde{π}}_{1}^{'} {\tilde{Z}}_{t} + {\tilde{π}}_{2}^{'} {\tilde{Z}}_{t} Φ (v_{t}; ψ) + ξ_{t}

where y_t is the dependent variable of interest, ${\tilde{Z}}_{t}$ is a vector of lagged dependent variables, ${\tilde{π}}_{j}$ are the parameter vectors, Φ is the regime-switching function, v_t is the transition variable, ψ is the threshold vector, and ξ_t ~ i.i.d.(0, σ²). Thus, each model reduces to a linear process under the null hypothesis ${\tilde{π}}_{2} = 0$ . We consider two classes of regime-switching models. The first class assumes that the regimes are determined by an observable variable. We examine a threshold model, with a discrete jump at a threshold value, and a smooth transition model, with a continuous function determining the weight assigned to the regimes. In both models, the switching function is dependent on the value of the transition variable relative to the threshold value. In the second class, the regimes are not observed but are inferred from an unobservable stochastic process. We examine the Markov switching model, with changes driven by an unobservable exogenous Markov chain, S_t.

In all three models, testing is problematic because of nuisance parameters in the transition function, which are identified only under the alternative (Davies (1987), Hansen (1996)). In the threshold and smooth transition models, the nuisance parameters are the parameters of the transition function (values of the thresholds and delay factor of the transition variable), while in the Markov switching model, the nuisance parameters are the transition probabilities.

A. Threshold Autoregressions (TAR)

In the TAR model, introduced and popularized by Tong (1978) and Tong and Lim (1980), the parameters of the process generating the data depend on the value of the regime-switching variable. The series can then be categorized into states consistent with the threshold variable reaching the threshold values separating the regimes. In the context of real exchange rates, the TAR model allows for a band within which no adjustment to the deviations from PPP takes place. This implies that within the band, deviations from PPP may exhibit unit root behavior, but the adjustment process is reverting or stationary in the outer bands. Because the bands of inaction may vary over time, due to changes in relative transactions costs, other market frictions, and/or policy intervention, Leon and Najarian (2002) introduce and estimate the following time-varying TAR (TVTAR):

\begin{array}{l} \begin{array}{l} Δ y_{t} = θ_{L}^{'} x_{t - 1} I_{t, L} + θ_{H}^{'} x_{t - 1} I_{t, H} + θ_{C}^{'} x_{t - 1} + ε_{t} & (1) \\ x_{t - 1} = (1, y_{t - 1}, Δ y_{t - 1}, \dots, Δ y_{t - k}), θ_{R}^{'} = (β_{0 R}, ρ_{R}, β_{1 R}, \dots, β_{K R}), R = L, C, H, and \\ I_{t, L} = {\begin{array}{l} 1 if z_{t} < 0 \land | z_{t} | > | p_{t - 1, L} (z_{t}) | \\ 0 otherwise \end{array} \\ I_{t, H} = {\begin{array}{l} 1 if z_{t} > 0 \land | z_{t} | > | p_{t - 1, H} (z_{t}) | \\ 0 otherwise \end{array} \end{array} \end{array}

For z_t = Δy_t−1, P_t−1,R (Z_t) = α_t−1,R (Z_t−1) + (1−α_t−1,R))P_t−2,R (Z_t−1)

\begin{array}{l} α_{t - 1, R} = | \frac{S_{t - 1, R}}{A_{t - 1, R}} |, with \\ S_{t - 1, R} = δ_{R} de v_{t - 1, R} + (1 - δ_{R}) S_{t - 2, R} \\ A_{t - 1, R} = δ_{R} | de v_{t - 1, R} | + (1 - δ_{R}) A_{t - 2, R}, and \\ de v_{t - 1, R} = z_{t - 1} - P_{t - 2, R} (z_{t - 1}) \end{array}

P_t−1(z_t) is the expected forecast value of the transition variable, based on exponential smoothing with adaptive response (time varying) weights for the exponential rate of decay. Thus, the 3-regime TVTAR divides the regression according to whether the absolute value of the percentage change in the real exchange rate exceeds the upper and lower forecast bounds, P_t−1,R (z_t). The corridor regime occurs when the change in the real exchange rate during one month does not appreciate by more than the upper forecast bound, P_t−1,H (z_t), or depreciate by more than the lower forecast bound, P_t−1,L (z_t). The transition variable z_t = Δy_t−d is assumed to be known, stationary, and have a continuous distribution; however, the delay factor d, the lag length k, and the threshold values are unknown. Each δ_L, δ_H depends on a functional of the sample. I(A) denotes the indicator function for the event A, such that I(A) = 1 if A is true and I(A) = 0 otherwise. In interpreting the coefficients, R is an index for the alternative regimes, ρ_R are the slope coefficients on y_t−1; β_0R are the slope coefficients on the deterministic components; and β_iR are the slope coefficients on the (Δy_t−1,… Δy_t−k) in the alternative regimes. The model can be nonstationary within one or more regimes, though the alternation between regimes can make it overall stationary.

Unit Root Tests

Following Caner and Hansen (2001), Leon and Najarian (2002) compute the following Wald statistics for distinguishing between nonlinearity (threshold effects) and possible nonstationarity (unit roots) in real exchange rate series:⁹

Wald 1: Linear Stationary-ergodic AR versus Unrestricted TAR

\begin{array}{l} H_{0} : θ_{L} = θ_{H} = 0, ρ_{C} < 0 \\ H_{A} : θ_{L} \neq 0, θ_{H} \neq 0 \end{array}

Wald 2: Hansen’s Unidentified Threshold Scenario

\begin{array}{l} H_{0} : θ_{L} = θ_{H} = 0, ρ_{C} = 0 \\ H_{A} : Unrestricted 3 - regime TAR \end{array}

Wald 3: Hansen’s Identified Threshold

\begin{array}{l} H_{0} : θ_{L} \neq 0, θ_{H} \neq 0, ρ_{L} = ρ_{H} = ρ_{C} = 0 \\ H_{A} : θ_{L} \neq 0, θ_{H} \neq 0, ρ_{L} < 0, ρ_{H} < 0, ρ_{C} < 0 (unrestricted 3 - regime TAR) \end{array}

Wald 4: Unit Root in Corridor Regime, Partial Unit Root

\begin{array}{l} H_{0} : θ_{L} \neq 0, θ_{H} \neq 0, ρ_{L} < 0, ρ_{H} < 0, ρ_{C} = 0 \\ H_{A} : Unrestriscted 3 - regime TAR \end{array}

The test is an F-statistic calculated as the ratio of residual variance of the linear model (null) to that of the TAR model (alternative); however, the F-statistic docs not have the standard χ² (chi-square) asymptotic distribution. Given the dependence of the critical values on the particular null and alternative, as well as the presence of nuisance (unidentified under the null) parameters, we calculate the critical values for the test statistics using bootstrap approximations to the asymptotic distributions of the Wald statistics.¹⁰ The unidentified threshold scenario, which performed better in Caner and Hansen’s (2001) Monte Carlo tests, makes use of the constrained bootstrap method,¹¹ and the identified threshold bootstrap is conducted through a simulation from a unit root TAR. The Wald 1 is a test for the existence of a threshold; Wald 2 tests for a unit root when there is no threshold effect; Wald 3 tests for a unit root in the presence of threshold effects; and Wald 4 tests for a (partial) unit root only in the corridor regime.

B. Smooth Transition Regressions (STR)

In contrast to the TAR model, where the switch between regimes occurs abruptly at a specific value of the threshold variable, smooth transition regression models allow a more gradual transition between regimes. STR models, introduced by Chan and Tong (1986) and popularized by Granger and Terasvirta (1993), are a more general class of state-dependent nonlinear time series models capable of accounting for deterministic changes in parameters over time, in conjunction with regime switching behavior (see survey in van Dijk and others (2002)). The STR model can be viewed as a weighted average of two linear models, with weights determined by the value of a transition function, typically defined as either a logistic or an exponential function.¹²

The STR model of order r is:

\begin{matrix} Δ y_{t} = θ_{L}^{'} x_{t - 1} + θ_{2}^{'} x_{t - 1} F (z_{t}^{d}; γ, c) + μ_{t} & (2) \end{matrix}

where x_t−1, defined as in equation 1, is a vector of exogenous variables; $z_{t}^{d}$ is the transition variable and may include a linear combination of several variables; F is the transition function determining the weights of the regimes and is bounded between 0 and 1; γ measures the speed of transition from one regime to the next; and c is the location variable (threshold) for the transition function. As γ becomes very large, the change of $F (z_{t}^{d}; γ, c)$ from 0 to 1 becomes almost instantaneous at $z_{t}^{d} = c$ , and the transition function approaches the indicator function $I [z_{t}^{d} > c]$ . The conventional STAR model is a special case of the smooth transition model when $z_{t}^{d} = Δ y_{t - d}$ .

A natural counterpart to the multiple regime TAR model is the multiple regime smooth transition autoregressive (MSTAR) model, which has multiple transition functions, each with its own location and slope parameters. Silverstovs (2000) argues that the greater flexibility of the MSTAR model may also be a drawback in the case of a 3-regime model with two identical outer regimes and with asymmetric speed of transition between regimes. He proposes the bi-parameter smooth transition regression (BSTR) model, with the following transition function:

\begin{matrix} F_{t} (γ_{1}, c_{t}, γ_{2}, c_{2,}; z_{t}^{d}) = \frac{\exp [- γ_{1} (z_{t}^{d} - c_{t})] + \exp [γ_{2} (z_{t}^{d} - c_{2})]}{1 + \exp [- γ_{1} (z_{t}^{d} - c_{1})] + \exp [γ_{2} (z_{t}^{d} - c_{2})]}, γ_{1}, γ_{2} > 0, c_{1} < c_{2} & (3) \end{matrix}

where γ₁ and γ₂ determine the speed of transition at their corresponding transition locations. In particular, the slopes of the transition functions at the two threshold parameters are different, thus allowing the transition speed from the lower-outer to middle regime and from the middle to higher-outer regime to be asymmetric. With four parameters, the BSTR(ρ) offers a large variety of shapes, with the magnitude of each slope parameter determining the steepness of the slope of the transition function.¹³ Smooth transition models are arguably more appropriate in modeling foreign exchange markets than threshold autoregressive or Markov regime-switching models because of the large number of investors, different investment horizons, and varying learning speeds, which suggest smooth rather than discrete adjustment.

Estimation

After determining the transition function and the threshold variable, the parameters of an STR model can be estimated by nonlinear least squares (NLS). For y_t = F(x_t; θ) + ε_t, the NLS estimator is given by $\hat{θ} = \arg \min_{θ} \sum_{t = 1}^{T} {(y_{t} - F (x_{t}; θ))}^{2} = \arg \min_{θ} \sum_{t = 1}^{T} ε_{t}^{2}$ .

If ε_t is normal, NLS is equivalent to maximum likelihood (MLE). Otherwise, NLS can be interpreted as a Quasi-maximum Likelihood Estimator (QMLE). Potsher and Prucha (1997) demonstrate that NLS is consistent and asymptotically normal under appropriate regularity conditions.

C. Markov Switching Models (MSM)

In Markov switching models, the parameters of the process generating the dependent variable depend on the unobservable regime variable, S_t, which indicates the probability of being in a particular state of the world.¹⁴ The process generating a change in regime depends on an exogenous unobservable Markov chain. Here we model real exchange rate appreciations and depreciations as switching regimes of the stochastic process underlying the data generating process. Thus appreciations and depreciations are associated with different conditional distributions of the change in the real exchange rate. The parameters of each regime are estimated unrestrictedly.

We consider

\begin{matrix} \begin{array}{l} Δ y_{t} = \sum_{k = 1}^{m} E [Δ y_{t} | S_{t} = R; Δ {\tilde{y}}_{t - 1}] \Pr (S_{t} = R | Δ {\tilde{y}}_{t - 1}) \\ = \sum_{R = 1}^{m} (α_{R} + ρ_{R} y_{t - 1} + \sum_{i = 1}^{k} β_{i R} Δ y_{t - i}) \Pr (S_{t} = R | Δ {\tilde{y}}_{t - 1}) + σ v_{t}, v_{t} \sim N (0, 1) & (8) \end{array} \end{matrix}

where Δỹ_t−1 = (Δy_t−1,Δy_t−2,…,Δy_t−(t−1)), and S_t is a three-state Markov chain with unknown transition probabilities P_ij, given by P_ij = Pr(S_t = j|S_t−1 =i). Thus, the conditional density is weighted by the predicted probability of being in a specific regime at time t, given the information set. The sequence of predicted probabilities, which indicate the likelihood of the variable being in a particular state in each time period, is:

\Pr (S_{t} | Δ {\tilde{y}}_{t - 1}) = \frac{p^{T} [\Pr (Δ y_{t - 1} | Δ {\tilde{y}}_{t - 2}, S_{t - 1}) \otimes \Pr (S_{t - 1} | Δ {\tilde{y}}_{t - 2}]}{{\Pr {(Δ y_{t - 1} | Δ {\tilde{y}}_{t - 2}, S_{t - 1})}^{T} \Pr (S_{t - 1} | Δ {\tilde{y}}_{t - 2})}}

where ⊗ denotes element-wise matrix multiplication. To illustrate, we consider a simple two state model where states (regimes) alternate between zero and unity. Then:

\begin{matrix} Δ y_{t} = θ_{1}^{'} x_{t - 1} (1 - S_{t}) + θ_{2}^{'} x_{t - 1} S_{t} + ε_{t} & (9) \end{matrix}

The null hypothesis of linearity can generally be formulated in terms of restrictions on θ₁ or θ₂, leaving the transition probabilities unidentified. This well-documented identification problem poses a challenge for conventional specification and evaluation tests.

The parameters of the model are estimated by maximum likelihood, with normality assumed to ensure consistency. Because S_t is not observed, inference about the states is carried out using an Expectation Maximization (EM) algorithm, with smoothed probabilities of the unobserved states replacing the conditional regime probabilities in the likelihood function.¹⁵ Critical values for the test statistics are generated by simulation methods.

D. Model Evaluation

Despite the recent proliferation in the use of nonlinear models, the relative merits of alternative classes of models still remain a nontrivial problem because alternative specifications are not nested and the use of standard asymptotic theory is often highly questionable. Most specification tests for nonlinear models tend to be based on time series analysis of standardized residuals. Breunig, Najarian, and Pagan (2002) (BNP) argue that because formal procedures such as Likelihood ratio tests of hypotheses may be difficult to interpret in nonlinear models, given their sensitivity to particular observations, it is necessary to complement these procedures with informal methods of evaluation. For example, if the act of simulating a model demonstrates that there is a fundamental flaw with it, this raises doubts about the validity of the maximum likelihood theory used in constructing a formal test (sec Breunig and Pagan (2001) and Pagan (2001)). BNP (2002) develop tests based on simulations of models that allow the discovery of population characteristics that can be compared with the corresponding sample equivalents. These tests allow us to compare the performance of the competing nonlinear models without a priori assumptions that either model is the true DGP. This is particularly important because most times the researcher does not know which model may have generated the hypothesized shift in regime.

If our focus is the DGP, it is natural to focus on the density describing the variable of interest. Because the density is generally unknown, we have to estimate it, preferably with an estimator that does not already assume that the null hypothesis is correct. One way of doing this is to use a nonparametric estimator – that is free from all parametric assumptions regarding the moments of the distribution – which will converge to the true density whether or not the parametric model is correctly specified. We can compare this density with that implied by the estimated model. Clearly, the density implied by the estimated model will converge to the true density only if the model is correctly specified. A measure of the distance between the two density estimates provides a natural statistic to test the null hypothesis of correct parametric specification. Ait-Sahalia (1996) uses this notion to compare a nonparametric density estimate with a parametric density estimate from the estimated parametric model. In contrast, we report results for a test of closeness between two unknown density functions, due to Li (1996), which compares an empirical density (nonparametric kernel) to a nonparametric density based on simulated data from the estimated models.

In practice, researchers tend to focus on some characteristics of the density, depending on the objectives of the modeling exercise. For example, these may include the conditional mean (if the objective is prediction of a point estimate), volatility (if our interest is uncertainty), skewness (if interest is in the relative balance of upside and downside risk), and asymmetry (if we are interested in comovements across markets during periods of crises). So, suppose the analyst (policy maker) is interested in some functions of data, ĝ(y). Let $g (\hat{θ})$ be the corresponding implied population characteristic, obtained from simulated data based on the estimated model. Label the difference between these two measures as $d = \hat{g} (y) - g (\hat{θ})$ . Then, we can think of these tests as comparing a consistent estimator of g(y) to an efficient estimator, $g (\hat{θ})$ , if the model is valid, enabling us to formulate the variance of d as $var (d) = var (\hat{g} (y) - var (g (\hat{θ}))$ (see Hausman (1978)). Although the variance of ĝ(y) is simply derived from the observed series, the analytical expression for $var (g (\hat{θ}))$ may be difficult to obtain for complicated nonlinear specifications. Because the test statistic $T^{*} = \hat{d}' {[var (\hat{g} (y) - var (g (\hat{θ}))]}^{- 1} \hat{d} > T = \hat{d}' {[var (\hat{g} (y))]}^{- 1} \hat{d}$ , Pagan (2002) suggests using the conservative test T. A rejection based on T (compared to χ² (1)) would imply an even stronger rejection than if based on T^*. A robust estimator of var(ĝ(y)), compatible with many alternative models, can be obtained using the Newey-West (1987) covariance matrix.

III. Estimation and Results

We examine real effective exchange rates for 26 countries, 13 of which are industrial countries.¹⁶^,¹⁷ All data are taken from the International Financial Statistics (IFS) database of the International Monetary Fund (IMF). The real effective exchange rate (REER), based on consumer prices, measures movements in the nominal exchange rate adjusted for differentials between the domestic price index and trade-weighted foreign price indices. The IMF’s CPI-based REER indicator (year 1995=100) of country i is:

e_{t} = {\prod_{j \neq i} (\frac{P_{i} R_{i}}{P_{j} R_{j}})}^{W i j}

where j is an index of country i’s trade partners; W_ij is the competitiveness weight put by country i on country j, P_i and P_j are consumer price indices in countries i and j; and R_i and R_j represent the nominal exchange rates of countries i and j’s currencies in US dollars. An increase (appreciation) in a country’s index indicates a decline in international competitiveness.

A preliminary evaluation of the data shows that real exchange rates in the developing countries in our sample are more volatile (have higher standard deviations) than those of the advanced countries. Their distributions are also more skewed. Non-normality is common across all regions.¹⁸ We calculate both the ADF and Ng and Perron (2001) unit root tests, given the significant moving average coefficients found in estimated ARMA (1,1) models. We find that, except for Brazil and Costa Rica (using ADF), we cannot reject the unit root hypothesis (see Table 1). As indicated earlier, these conventional tests, which do not account for nonlinearity, may be misleading; however, our initial unit root results are consistent with the existing literature. In what follows we estimate nonlinear models and re-evaluate the evidence for the unit root hypothesis.

Table 1:

Descriptive Statistics

Note: SD is the standard deviation, KT is kurtosis, and J-B the Jarque-Bera normality test. The Ng and Perron (2001) tests reported are modified forms of the Phillips and Perron Za and Zt statistics, the Bhargava (1986) R1 statistic, and the Elliot, Rothenberg, and Stock (1997) Point Optimal statistic. The 5 % critical values are -8.10 for Mza, -1.98 for MZt, 0.23 for MSB, 3.17 for MPT, and -2.87 for ADF.

A. TAR Estimates

We estimate equation 1 using sequential least squares (Hansen 1997), for the period 1981:03 to 2001:12, with Ox Professional 3.0. Our δ_R are initialized through a grid search over [0,1] in steps of 0.1 increments, determining the α_R, the threshold sequences, and the indicator variables (I_L, I_H). We use the lagged difference of the exchange rate as the transition variable and set the delay parameter to unity.¹⁹ Our choice of z_t = Δy_t−1 is stationary whether y_t is I(1) or I(0). We also initialize S_t−2,R = 0, A_t−2,R = 0, and F_t−2,R = Δy_t−2. For each triple (δ_L, δ_H, k), consisting of the lower and upper thresholds and lag k on ∆y_t−k, we estimate by ordinary least squares (OLS)²⁰

Δ y_{t} = {\hat{θ}}_{L}^{'} (δ_{L}, δ_{H}, k) x_{t - 1} I_{t, L} + {\hat{θ}}_{H}^{'} (δ_{L}, δ_{H}, k) x_{t - 1} I_{t, H} + θ_{C}^{'} (δ_{L}, δ_{H}, k) x_{t - 1} + ε_{t} (δ_{L}, δ_{H}, k)

Let $σ^{2} (δ_{L}, δ_{H}, k) = T^{- 1} \sum_{t = 1}^{T} {\hat{ε}}_{t} {(δ_{L}, δ_{H}, k)}^{2}$ be the OLS estimate of σ² for fixed δ_L, δ_H,k. Then the least squares estimate of the threshold values is found by minimizing σ² (δ_L, δ_H,k)

({\hat{δ}}_{L}, {\hat{δ}}_{H}, \hat{k}) = \arg \min_{(δ_{L}, δ_{H}, k) \in Λ_{L} Λ_{H} Λ_{K}} {\hat{σ}}^{2} (δ_{L}, δ_{H}, k)

The parameters of the model can be estimated consistently as long as the true threshold values lie in the interior of the grid space and each regime has sufficient data points to produce reliable estimates of the autoregressive parameters. The least square estimates of the other parameters and residuals are found by substitution of the point estimates $({\hat{δ}}_{L}, {\hat{δ}}_{H}, \hat{k})$ .

Empirical Characteristics

We investigate estimated lag lengths, speed of response to deviations from forecasts, time spent outside threshold bounds, and a measure of deviations between actual changes and forecast thresholds during periods outside of thresholds. We present results for groupings of advanced and developing countries. Summaries of the characteristics of the threshold bands and estimates of duration are shown in Tables 2 and 3 and described below.²¹

Table 2:

Characteristics of Threshold Bands

Note: Let subscript R depict the alternative regimes, with L corresponding to over-depreciation, H to over-appreciation, and Cor to the corridor. The columns report the parameters from the forecast measure that characterizes the time-varying bands (δ_R and α_R), the optimal lag-length (κ*), and the percentage of times the series spends in each of the intervention regimes.

Table 2:

Characteristics of Threshold Bands

		δ_L	δ_H	α_L	α_H	κ*	%L	%H	%Cor
Advanced		0.38	0.52	0.56	0.53	5.00	0.27	0.28	0.45
	G-7	0.34	0.54	0.51	0.55	5.14	0.29	0.27	0.45
	Other	0.43	0.48	0.62	0.51	4.83	0.26	0.29	0.46
Developing		0.31	0.37	0.50	0.47	6.77	0.26	0.30	0.45
	Asia	0.42	0.44	0.45	0.63	6.00	0.28	0.26	0.47
	WH	0.24	0.33	0.53	0.38	7.25	0.24	0.33	0.43
Overall		0.35	0.44	0.53	0.50	5.88	0.26	0.29	0.45

Table 2:

Characteristics of Threshold Bands

		δ_L	δ_H	α_L	α_H	κ*	%L	%H	%Cor
Advanced		0.38	0.52	0.56	0.53	5.00	0.27	0.28	0.45
	G-7	0.34	0.54	0.51	0.55	5.14	0.29	0.27	0.45
	Other	0.43	0.48	0.62	0.51	4.83	0.26	0.29	0.46
Developing		0.31	0.37	0.50	0.47	6.77	0.26	0.30	0.45
	Asia	0.42	0.44	0.45	0.63	6.00	0.28	0.26	0.47
	WH	0.24	0.33	0.53	0.38	7.25	0.24	0.33	0.43
Overall		0.35	0.44	0.53	0.50	5.88	0.26	0.29	0.45

Table 3:

Duration and Loss Estimates

Note: Let subscript R depict the alternative regimes, with L corresponding to over-depreciation, and H to over-appreciation. MaxD_R shows average maximum duration of excess deviations on each side of the band (number of periods); AveD_R is the average duration per spell of excess deviation, across countries for each regime; CumL_R is the cumulative excess deviation (area between the tolerance margin and the observed realizations when the band is crossed); and AveL_R is the average excess deviation, across countries for each regime.

Table 3:

Duration and Loss Estimates

		MaxD_L	MaxD_H	AveD_L	AveD_H	CumL_L	CumL_H	AveL_L	AveL_H
Advanced		4.23	4.08	1.59	1.62	0.10	0.05	0.02	0.01
	G-7	4.57	4.00	1.61	1.60	0.06	0.04	0.01	0.02
	Other	3.83	4.17	1.57	1.64	0.13	0.06	0.02	0.01
Developing		4.15	4.46	1.53	1.72	0.32	0.15	0.04	0.02
	Asia	4.60	3.60	1.65	1.51	0.26	0.09	0.03	0.02
	WH	3.88	5.00	1.46	1.86	0.35	0.18	0.04	0.03
Overall		4.19	4.27	1.56	1.67	0.21	0.10	0.03	0.02

Table 3:

Duration and Loss Estimates

		MaxD_L	MaxD_H	AveD_L	AveD_H	CumL_L	CumL_H	AveL_L	AveL_H
Advanced		4.23	4.08	1.59	1.62	0.10	0.05	0.02	0.01
	G-7	4.57	4.00	1.61	1.60	0.06	0.04	0.01	0.02
	Other	3.83	4.17	1.57	1.64	0.13	0.06	0.02	0.01
Developing		4.15	4.46	1.53	1.72	0.32	0.15	0.04	0.02
	Asia	4.60	3.60	1.65	1.51	0.26	0.09	0.03	0.02
	WH	3.88	5.00	1.46	1.86	0.35	0.18	0.04	0.03
Overall		4.19	4.27	1.56	1.67	0.21	0.10	0.03	0.02

Lag Length: On average, the specifications for developing countries are characterized by longer lags of exchange rate changes. The average lag for advanced countries is 5 compared to 6.8 for developing countries. For countries in Western Hemisphere (WH), the average lag is as high as 7.3. This suggests a more complex structure for short-term interaction between nominal exchange rates and relative prices; it also highlights the importance of correct lag length in tests of unit roots because omission of short-run dynamics could affect tests based on the long-term impact matrix (see the Π matrix in the Johansen test).

Response: The adaptive response weight parameters α_L and α_H show the quickness of response to relatively recent exchange rate variations. Advanced countries respond faster than developing countries to both over-depreciations (0.56 vs. 0.50) and over-appreciations (0.53 vs. 0.47), implying narrower and probably closely watched bands. The differences are more marked in subregions. For over-depreciations, the other (non G-7) advanced countries have the fastest response (0.62), Asia the slowest (0.45); for over-appreciations, the countries of WH have the slowest response (0.38), Asia the fastest (0.63). If this design of the thresholds reflects a measure of relative tolerance for these exchange rate variations, then the results suggest that G-7 and Asian countries exercise greater caution against over-appreciations.

Asymmetry of response: On average, both advanced and developing countries display asymmetrical response to changes in the real exchange rates, with G-7 (0.55 vs. 0.51) and Asia (0.63 vs. 0.45) placing greater weight on recent developments relating to appreciations while predicting the tolerancc margin. The opposite is true for the other advanced (0.51 vs. 0.62) and WH (0.38 vs. 0.53) countries, which react more strongly to developments relating to over-depreciations.

Maximum durations of spells: These are somewhat longer for over-appreciations in WH and other advanced countries but longer for over-depreciations for G-7 and Asian countries. As in the other statistics, the subgroups reveal differences. The maximum duration for the G-7 occurs in the lower regime (4.6 months), but in the upper regime for the other advanced countries (4.2 months). Similarly, the maximum duration for Asia is in the lower regime (4.6 months), but in the upper regime for the WH countries (5 months).

Average duration of spells: In general, the average duration of periods between threshold crossings is somewhat higher for appreciations than for depreciations. The G-7 countries have equal durations for both types of deviations while Asian countries having higher durations for over-depreciations. The WH countries have the largest difference in average duration. Given the difference in response towards depreciation and appreciation deviations of the subgroups, the evidence of duration is probably informative about the speed or effectiveness of the policy measures used to reverse deviations from forecasts.

Asymmetry in duration of deviations: Average durations in the lower regime exceeds that in the upper regime in 38 percent of both other advanced countries and developing countries, but these percentages mask inter-regional differences. Specifically, average duration in the lower regime is greater than the average duration in the upper regime in 57 percent of G-7 and 80 percent of Asian countries, compared to 17 percent of other advanced countries and 13 percent for WH countries.

Frequency of thresholds being crossed: For developing countries, there is a tendency for more observations to lie in the upper regime (30% vs. 26%), more so for WH countries; however, with longer average durations for over-appreciations, the lower regime is characterized with a higher frequency of threshold crossings. The advanced economies experience similar frequency and duration of deviations on both sides of the bands, though slightly less pronounced. The observation that the developing countries sampled seem to watch their depreciation thresholds more closely is consistent with their recording more deviations in the upper regime and having a higher frequency of crossings in the lower regime.

Cumulative excess deviation per spell: If we define the cumulated difference between the actual exchange rate change and the expected change for the duration of a crossing as an excess deviation measure, we find that, for all groups, the excess deviation for a depreciation spell (crossing beyond the lower threshold) is twice as large as that for an appreciation spell (crossing beyond the upper threshold). The overall average is 0.21 in the lower regime and 0.10 in the upper regime. For the Asia countries, the excess deviation per depreciation spell is three times higher than that per appreciation spell; in contrast, the factor is 1.5 for G-7 countries. Further, the excess deviation per depreciation and appreciation spells is about three times higher for developing countries relative to advanced countries.

Average excess deviation per spelt We calculate the average excess deviation per spell and find that, for both the advanced and developing countries, average excess deviation for depreciations are about twice that for appreciations. Also, average excess deviations per spells of depreciation and appreciation for developing countries is about twice that for advanced countries. But there are differences among sub-groupings. For the advanced countries, the average excess deviation per appreciation spell is twice that of a depreciation spell in the G-7; in contrast, the average excess deviation per depreciation spell is twice that of an appreciation spell in the other (non G-7) advanced countries. The average excess deviation for depreciations in the developing countries is four times that of the G-7 countries; on the other hand, the average excess deviation for appreciations in the developing countries is the same as that for the G-7 countries. We compare average excess deviation per spell in the upper and lower regimes and find that the average excess deviation per spell in the lower regime is greater than the average excess deviation per spell in the upper regime in all of the developing countries, compared to 57 percent of G-7 and 83 percent of other advanced countries.

Parameter Estimates

Tables 4 and 6 summarize the TAR estimates and the Wald tests. For the unrestricted TAR model, ρ_L > ρ_H for developing countries, and ρ_H > ρ_L for advanced countries, consistent with faster reversion in developing countries for over-depreciations and faster reversion in advanced countries for over-appreciations. For G-7 and Asian countries, only ρ_H < 0; on the other hand, ρ_L < 0 and larger than ρ_H for WH countries. In the corridor regime, all reversion coefficients are negative. For the TARurCor model, |ρ_L|>|ρ_H| for WH countries, with approximate equality for G-7 and Asian countries. Except for the other advanced countries, for which only ρ_H is negative, the reversion coefficients are negative and larger for depreciations relative to appreciations and for developing countries relative to advanced countries. As suggested by Caner and Hansen (2001), our tests are likely to be more powerful for WH, given the size of the threshold effects.

Table 4:

Summary Reversion Coefficients

Note: Subscripts depict the alternative regimes, with L corresponding to over-depreciation, H to over-appreciation, and C to the corridor. LIN refers to the linear model.

Table 4:

Summary Reversion Coefficients

		Linear	Unrestricted TAR			TARurCor
		ρ_LIN	ρ_L	ρ_C	ρ_H	ρ_L	ρ_H
Advanced		-0.0213	0.0000	-0.0218	0.0030	-0.0076	-0.0140
	G-7	-0.017	0.0042	-0.0153	-0.006	-0.019	-0.010
	Other	-0.0257	-0.0049	-0.0294	0.0140	0.0059	-0.0191
Developing		-0.0302	-0.0185	-0.0101	-0.0140	-0.0391	-0.0212
	Asia	-0.0104	0.0080	-0.0097	-0.0075	-0.0167	-0.0119
	WH	-0.0426	-0.0351	-0.0104	-0.0180	-0.0531	-0.0270
Overall		-0.0257	-0.0093	-0.0160	-0.0055	-0.0234	-0.0176

Note: Subscripts depict the alternative regimes, with L corresponding to over-depreciation, H to over-appreciation, and C to the corridor. LIN refers to the linear model.

Table 4:

Summary Reversion Coefficients

		Linear	Unrestricted TAR			TARurCor
		ρ_LIN	ρ_L	ρ_C	ρ_H	ρ_L	ρ_H
Advanced		-0.0213	0.0000	-0.0218	0.0030	-0.0076	-0.0140
	G-7	-0.017	0.0042	-0.0153	-0.006	-0.019	-0.010
	Other	-0.0257	-0.0049	-0.0294	0.0140	0.0059	-0.0191
Developing		-0.0302	-0.0185	-0.0101	-0.0140	-0.0391	-0.0212
	Asia	-0.0104	0.0080	-0.0097	-0.0075	-0.0167	-0.0119
	WH	-0.0426	-0.0351	-0.0104	-0.0180	-0.0531	-0.0270
Overall		-0.0257	-0.0093	-0.0160	-0.0055	-0.0234	-0.0176

Note: Subscripts depict the alternative regimes, with L corresponding to over-depreciation, H to over-appreciation, and C to the corridor. LIN refers to the linear model.

Table 5:

TAR Parameters of Interest

Note: Subscripts depict the alternative regimes, with L corresponding to over-depreciation, H to over-appreciation, and C to the corridor. LIN refers to the linear model

Table 5:

TAR Parameters of Interest

			Linear		TAR						TARUR			TARURCOR
			Cond’l Mean	Reversion	Cond’l Mean			Reversion			Cond’l Mean			Cond’l Mean			Reversion
			μ	ρ	μ L	μ C	μ U	ρ L	ρ C	ρ U	μ L	μ C	μ U	μ L	μ C	μ U	ρ L	ρ u
Advanced
	*G-7*
Canada			-0.0008	-0.0049	-0.0039	0.0003	0.0005	-0.0462	0.0178	-0.0326	-0.0039	0.0003	0.0005	-0.0146	0.0043	0.0122	-0.0138	-0.0027
France			-0.0007	-0.0252	-0.0032	0.0003	0.0011	-0.0384	-0.0008	-0.0604	-0.0032	0.0003	0.0011	-0.0086	0.0024	0.0083	-0.0298	-0.0147
Germany			-0.0007	-0.0266	-0.0038	0.0007	0.0042	0.0161	-0.0426	0.0387	-0.0038	0.0007	0.0042	-0.0095	0.0034	0.0112	0.0004	-0.0064
Italy			0.0001	-0.0165	-0.0047	0.0017	0.0045	0.0150	-0.0248	0.0198	-0.0047	0.0017	0.0045	-0.0123	0.0043	0.0105	-0.0245	-0.0307
Japan			0.0014	-0.0144	-0.0062	0.0044	0.0125	0.0163	-0.0098	-0.0029	-0.0062	0.0044	0.0125	-0.0241	0.0120	0.0301	-0.0147	0.0078
United Kingdom			0.0006	-0.0249	-0.0062	0.0035	0.0075	0.0233	-0.0211	-0.0224	-0.0062	0.0035	0.0075	-0.0174	0.0085	0.0185	-0.0202	-0.0206
United States			0.0010	-0.0097	-0.0052	0.0036	0.0098	0.0430	-0.0260	0.0156	-0.0052	0.0036	0.0098	0.0165	0.0084	-0.0176	-0.0315	0.0007
	*Other*
Australia			-0.0012	-0.0130	-0.0096	0.0016	0.0042	0.0169	-0.0401	0.0494	-0.0096	0.0016	0.0042	-0.0292	0.0083	0.0196	0.0063	-0.0266
Belgium			-0.0007	-0.0250	-0.0039	0.0010	0.0027	-0.0187	-0.0219	0.0193	-0.0039	0.0010	0.0027	-0.0071	0.0028	0.0079	0.0073	0.0035
Israel			0.0009	-0.0410	-0.0005	0.0013	0.0059	-0.1095	-0.0227	0.0162	-0.0005	0.0013	0.0059	-0.0240	0.0076	0.0179	-0.0250	-0.0144
Korea			-0.0007	-0.0298	-0.0150	0.0042	0.0086	0.0380	-0.0313	-0.0047	-0.0150	0.0042	0.0086	-0.0233	0.0070	0.0186	0.0760	-0.0644
New Zeuland			-0.0005	-0.0296	-0.0064	0.0010	0.0041	0.0433	-0.0433	0.0158	-0.0064	0.0010	0.0041	-0.0252	0.0059	0.0173	0.0191	-0.0138
Spain			-0.0004	-0.0157	-0.0053	0.0053	0.0050	0.0007	-0.0122	-0.0121	-0.0053	0.0013	0.0050	-0.0117	0.0034	0.0105	-0.0335	0.0011
*Developing*
	*Asia*
		India	-0.0030	-0.0031	-0.0093	-0.0006	0.0036	0.0028	-0.0018	-0.0034	-0.0093	-0.0006	0.0036	-0.0225	0.0044	0.0173	0.0112	-0.0074
		Indonesia	-0.0043	-0.0106	-0.0182	0.0020	0.0069	0.0340	0.0010	-0.0502	-0.0182	0.0020	-0.0069	0.0454	0.0147	0.0389	-0.0254	-0.0357
		Malaysia	-0.0014	-0.0072	-0.0058	0.0002	0.0047	-0.0182	-0.0040	-0.0069	-0.0058	0.0002	0.0047	-0.0182	0.0054	0.0138	0.0035	-0.0070
		Philippines	-0.0012	-0.0201	-0.0152	0.0025	0.0087	-0.0152	-0.0282	0.0300	-0.0152	0.0025	0.0087	-0.0350	0.0079	0.0220	-0.0251	-0.0064
		Thailand	-0.0011	-0.0111	-0.0042	0.0003	0.0110	0.0364	-0.0155	-0.0066	-0.0042	0.0003	0.0110	-0.0218	0.0080	0.0191	-0.0479	-0.0030
	WH
		Argentina	-0.0003	-0.0213	-0.0063	0.0018	0.0073	0.1009	-0.0567	0.0833	-0.0063	0.0018	0.0073	-0.0614	0.0225	0.0356	0.0519	-0.0692
		Brazil	-0.0012	-0.0410	-0.0132	0.0027	0.0102	-0.0765	-0.0145	-0.0218	-0.0132	0.0027	0.0102	-0.0472	0.0136	0.0325	-0.0081	-0.0517
		Chile	-0.0024	-0.0146	-0.0113	0.0008	0.0078	-0.0643	0.0023	-0.0178	-0.0113	0.0008	0.0078	-0.0304	0.0073	0.0231	-0.0540	0.0013
		Colombia	-0.0014	-0.0060	-0.0084	0.0013	0.0072	-0.0010	-0.0066	0.0012	-0.0084	0.0013	0.0072	-0.0236	0.0069	0.0190	-0.0010	-0.0117
		Costa Rica	0.0003	-0.1757	-0.0102	0.0039	0.0075	-0.0732	0.0322	-0.1213	-0.0102	0.0039	0.0075	-0.0249	0.0089	0.0137	-0.1021	-0.0472
		Mexico	0.0005	-0.0439	-0.0146	0.0032	0.0118	-0.0697	-0.0377	0.0022	-0.0146	0.0032	0.0118	-0.0656	0.0124	0.0233	-0.2031	-0.0191
		Paraguay	0.0027	-0.0195	0.0002	-0.0037	0.0021	-0.0252	-0.0115	-0.0265	0.0002	-0.0037	0.0021	-0.0455	0.0117	0.0291	-0.0632	0.0168
		Uruguay	0.0002	-0.0184	-0.0120	0.0039	0.0111	-0.0715	0.0092	-0.0438	-0.0120	0.0039	0.0111	-0.0334	0.0106	0.0259	-0.0454	-0.0350

Note: Subscripts depict the alternative regimes, with L corresponding to over-depreciation, H to over-appreciation, and C to the corridor. LIN refers to the linear model

Table 5:

TAR Parameters of Interest

			Linear		TAR						TARUR			TARURCOR
			Cond’l Mean	Reversion	Cond’l Mean			Reversion			Cond’l Mean			Cond’l Mean			Reversion
			μ	ρ	μ L	μ C	μ U	ρ L	ρ C	ρ U	μ L	μ C	μ U	μ L	μ C	μ U	ρ L	ρ u
Advanced
	*G-7*
Canada			-0.0008	-0.0049	-0.0039	0.0003	0.0005	-0.0462	0.0178	-0.0326	-0.0039	0.0003	0.0005	-0.0146	0.0043	0.0122	-0.0138	-0.0027
France			-0.0007	-0.0252	-0.0032	0.0003	0.0011	-0.0384	-0.0008	-0.0604	-0.0032	0.0003	0.0011	-0.0086	0.0024	0.0083	-0.0298	-0.0147
Germany			-0.0007	-0.0266	-0.0038	0.0007	0.0042	0.0161	-0.0426	0.0387	-0.0038	0.0007	0.0042	-0.0095	0.0034	0.0112	0.0004	-0.0064
Italy			0.0001	-0.0165	-0.0047	0.0017	0.0045	0.0150	-0.0248	0.0198	-0.0047	0.0017	0.0045	-0.0123	0.0043	0.0105	-0.0245	-0.0307
Japan			0.0014	-0.0144	-0.0062	0.0044	0.0125	0.0163	-0.0098	-0.0029	-0.0062	0.0044	0.0125	-0.0241	0.0120	0.0301	-0.0147	0.0078
United Kingdom			0.0006	-0.0249	-0.0062	0.0035	0.0075	0.0233	-0.0211	-0.0224	-0.0062	0.0035	0.0075	-0.0174	0.0085	0.0185	-0.0202	-0.0206
United States			0.0010	-0.0097	-0.0052	0.0036	0.0098	0.0430	-0.0260	0.0156	-0.0052	0.0036	0.0098	0.0165	0.0084	-0.0176	-0.0315	0.0007
	*Other*
Australia			-0.0012	-0.0130	-0.0096	0.0016	0.0042	0.0169	-0.0401	0.0494	-0.0096	0.0016	0.0042	-0.0292	0.0083	0.0196	0.0063	-0.0266
Belgium			-0.0007	-0.0250	-0.0039	0.0010	0.0027	-0.0187	-0.0219	0.0193	-0.0039	0.0010	0.0027	-0.0071	0.0028	0.0079	0.0073	0.0035
Israel			0.0009	-0.0410	-0.0005	0.0013	0.0059	-0.1095	-0.0227	0.0162	-0.0005	0.0013	0.0059	-0.0240	0.0076	0.0179	-0.0250	-0.0144
Korea			-0.0007	-0.0298	-0.0150	0.0042	0.0086	0.0380	-0.0313	-0.0047	-0.0150	0.0042	0.0086	-0.0233	0.0070	0.0186	0.0760	-0.0644
New Zeuland			-0.0005	-0.0296	-0.0064	0.0010	0.0041	0.0433	-0.0433	0.0158	-0.0064	0.0010	0.0041	-0.0252	0.0059	0.0173	0.0191	-0.0138
Spain			-0.0004	-0.0157	-0.0053	0.0053	0.0050	0.0007	-0.0122	-0.0121	-0.0053	0.0013	0.0050	-0.0117	0.0034	0.0105	-0.0335	0.0011
*Developing*
	*Asia*
		India	-0.0030	-0.0031	-0.0093	-0.0006	0.0036	0.0028	-0.0018	-0.0034	-0.0093	-0.0006	0.0036	-0.0225	0.0044	0.0173	0.0112	-0.0074
		Indonesia	-0.0043	-0.0106	-0.0182	0.0020	0.0069	0.0340	0.0010	-0.0502	-0.0182	0.0020	-0.0069	0.0454	0.0147	0.0389	-0.0254	-0.0357
		Malaysia	-0.0014	-0.0072	-0.0058	0.0002	0.0047	-0.0182	-0.0040	-0.0069	-0.0058	0.0002	0.0047	-0.0182	0.0054	0.0138	0.0035	-0.0070
		Philippines	-0.0012	-0.0201	-0.0152	0.0025	0.0087	-0.0152	-0.0282	0.0300	-0.0152	0.0025	0.0087	-0.0350	0.0079	0.0220	-0.0251	-0.0064
		Thailand	-0.0011	-0.0111	-0.0042	0.0003	0.0110	0.0364	-0.0155	-0.0066	-0.0042	0.0003	0.0110	-0.0218	0.0080	0.0191	-0.0479	-0.0030
	WH
		Argentina	-0.0003	-0.0213	-0.0063	0.0018	0.0073	0.1009	-0.0567	0.0833	-0.0063	0.0018	0.0073	-0.0614	0.0225	0.0356	0.0519	-0.0692
		Brazil	-0.0012	-0.0410	-0.0132	0.0027	0.0102	-0.0765	-0.0145	-0.0218	-0.0132	0.0027	0.0102	-0.0472	0.0136	0.0325	-0.0081	-0.0517
		Chile	-0.0024	-0.0146	-0.0113	0.0008	0.0078	-0.0643	0.0023	-0.0178	-0.0113	0.0008	0.0078	-0.0304	0.0073	0.0231	-0.0540	0.0013
		Colombia	-0.0014	-0.0060	-0.0084	0.0013	0.0072	-0.0010	-0.0066	0.0012	-0.0084	0.0013	0.0072	-0.0236	0.0069	0.0190	-0.0010	-0.0117
		Costa Rica	0.0003	-0.1757	-0.0102	0.0039	0.0075	-0.0732	0.0322	-0.1213	-0.0102	0.0039	0.0075	-0.0249	0.0089	0.0137	-0.1021	-0.0472
		Mexico	0.0005	-0.0439	-0.0146	0.0032	0.0118	-0.0697	-0.0377	0.0022	-0.0146	0.0032	0.0118	-0.0656	0.0124	0.0233	-0.2031	-0.0191
		Paraguay	0.0027	-0.0195	0.0002	-0.0037	0.0021	-0.0252	-0.0115	-0.0265	0.0002	-0.0037	0.0021	-0.0455	0.0117	0.0291	-0.0632	0.0168
		Uruguay	0.0002	-0.0184	-0.0120	0.0039	0.0111	-0.0715	0.0092	-0.0438	-0.0120	0.0039	0.0111	-0.0334	0.0106	0.0259	-0.0454	-0.0350

Note: Subscripts depict the alternative regimes, with L corresponding to over-depreciation, H to over-appreciation, and C to the corridor. LIN refers to the linear model

Table 6:

Summary of Wald Tests

Note: Numbers are percentages of do not reject, based on Wald tests

Table 6:

Summary of Wald Tests

		TARur vs. TAR	TARurCor vs. TAR
Advanced		0.08	0.23
	G-7	0.14	0.29
	Other	0.00	0.17
Developing		0.15	0.69
	Asia	0.20	0.60
	WH	0.13	0.75
Overall		0.12	0.46

Note: Numbers are percentages of do not reject, based on Wald tests

Table 6:

Summary of Wald Tests

		TARur vs. TAR	TARurCor vs. TAR
Advanced		0.08	0.23
	G-7	0.14	0.29
	Other	0.00	0.17
Developing		0.15	0.69
	Asia	0.20	0.60
	WH	0.13	0.75
Overall		0.12	0.46

Note: Numbers are percentages of do not reject, based on Wald tests

On the basis of the estimated TAR models, we calculate Wald statistics to test for threshold effects and/or unit roots. The tests measure whether the DGP under the null produces a residual variance that is significantly larger than the residual variance obtained from the fit of the alternative hypothesis, in our case the unrestricted TAR specification. Table 6 shows the percentage of countries for which the various null hypotheses are plausible (see Table 7 for details). These statistics arc based on estimated unconstrained bootstrap ρ-values, representing the percentage of Wald statistics calculated from the simulated data that exceed the Wald statistics calculated from the observed sample.

Table 7:

Wald Tests

Note: W1 is a test for the existence of a threshold; W2 tests for a unit root when there is no threshold effect; W3 tests for a unit root in the presence of threshold effects; and W4 tests for a (partial) unit root only in the corridor regime. Unc indicates rejection (0) of null based on the unconstrained bootstrap critical values. We report absolute values but, in a few cases, we obtained small and negative statistics for W3 and W4, arising from the small sample adjustment to the variance under the null and the alternative.

The results indicate an overwhelming rejection of the first three null hypotheses. The unrestricted TAR specification outperforms the benchmark stationary ergodic linear process. It is also preferred over both the linear non-stationary I(1) specification, the ρ-values for which are obtained by constructing a bootstrap distribution that imposes an unidentified threshold effect, and the unit root TAR process.²² Because the unidentified threshold model was less sensitive to nuisance parameters Caner and Hansen (2001) recommend calculating ρ-values using the unidentified threshold bootstrap. The intermediate case, which we label as an identified threshold partial unit root process (I(1) in corridor regime combined with an otherwise stationary TAR), yields different outcomes for advanced and developing countries. While the null is still rejected against the stationary crgodic TAR for most advanced countries, the developing countries do not reject the partial unit root TAR as their preferred specification. Thus, the partial unit root model could characterize the data dynamics for these countries.

B. STR Estimates

Testing for Linearity

The first step in estimating an STR model is to test for linearity against STR-type nonlinearity, which implies testing the null hypothesis $H_{0} : {θ^{'}}_{2} = 0$ in equation (2). Under the null hypothesis, the parameters γ and c arc not identified. The solution advocated by Luukkonen and others (1988) and adopted by Terasvirta (1994) is to replace the transition function by a suitable Taylor series approximation. We propose considering a third-order Taylor expansion of the transition function for the BSTR model.²³ Substituting

\begin{matrix} T_{3} = F_{t}^{*} (z_{t}^{d}) + \sum_{j} γ_{i} \frac{\partial F_{t}^{*} (z_{t}^{d})}{\partial γ_{i}} + \frac{1}{21} \sum_{i} \sum_{j} γ_{i} γ_{j} \frac{\partial^{2} F_{t}^{*} (z_{t}^{d})}{\partial γ_{i} \partial γ_{j}} + \frac{1}{31} \sum_{i} \sum_{j} \sum_{k} γ_{i} γ_{t} γ_{k} \frac{\partial^{3} F_{t}^{*} (z_{t}^{d})}{\partial γ_{i} \partial γ_{j} \partial γ_{k}} + R_{3} & (4) \end{matrix}

for the transition function in equation (2), with all terms evaluated at γ₁= γ₂ = 0, yields an auxiliary regression:

\begin{matrix} Δ y_{t} = β_{0} x_{t - 1} + β_{1} x_{t - 1} z_{t}^{d} + β_{2} x_{t - 1} {(z_{t}^{d})}^{2} + β_{3} x_{t - 1} {(z_{t}^{d})}^{3} + η_{t} & (5) \end{matrix}

where:

\begin{matrix} \begin{array}{l} η_{t} = θ^{'} x_{t} R_{3} + ε_{t} and \\ β_{0} = \underset{I}{\underset{︸}{ϕ^{'} + \frac{1}{9} θ^{'} (c_{1} γ_{1} - c_{2} γ_{2})}} + \underset{I I}{\underset{︸}{θ^{'} (\frac{1}{54} c_{1}^{2} γ_{1}^{2} + \frac{2}{27} c_{1} c_{2} γ_{1} γ_{2} + \frac{1}{54} c_{2}^{2} γ_{2}^{2})}} + \underset{I I I}{\underset{︸}{\frac{1}{162} θ^{'} (c_{2}^{3} γ_{2}^{3} - c_{1}^{3} γ_{1}^{3})}} \\ β_{1} = \frac{1}{9} θ^{'} \underset{I}{\underset{︸}{(- γ_{1} + γ_{2})}} + \underset{I I}{\underset{︸}{\frac{1}{27} θ^{'} (- c_{1}^{2} γ_{1}^{2} - 2 c_{1} γ_{1} γ_{2} - c_{2} γ_{1} γ_{2} - c_{2} γ_{2}^{2})}} + \underset{I I I}{\underset{︸}{\frac{1}{54} θ^{'} (c_{1}^{2} γ_{1}^{3} - c_{2}^{2} γ_{2}^{3})}} \\ β_{2} = \underset{I I}{\underset{︸}{\frac{1}{54} θ^{'} γ_{1}^{2} + θ^{'} (\frac{2}{27} γ_{1} γ_{2} + \frac{1}{54} γ_{2}^{2})}} + \underset{I I I}{\underset{︸}{\frac{1}{54} θ^{'} (c_{2} γ_{2}^{3} - c_{2} γ_{1}^{3})}} \\ β_{3} = \underset{I I I}{\underset{︸}{\frac{1}{162} θ^{'} (γ_{1}^{3} - γ_{2}^{3})}} & (6) \end{array} \end{matrix}

Since β₃ is not dependent on c₁ or c₂, and all β_j= 0, j = 1,…3, for γ₁ = γ₂ = 0, it follows that, conditional on rejecting linearity (β_j ≠ 0, j = 1,…3), a do not reject of the hypothesis β₃ = 0 indicates γ₁ = γ₂ and suggests a symmetric three-regime STR model. If the hypothesis of symmetry is not rejected, tests exist for choosing among logistic and exponential smooth transition models (see Terasvirta (1999), Escribano and Jorda (1999)).

Parameter Estimates

Table 10 includes results of linearity test against smooth transition alternatives. In executing the linearity tests, the lag length p was chosen based on Akaike information criterion (AIC) applied to a linear AR for Δy_t. The first and second blocks report p-values of F-tests for the auxiliary regression (5) with Δy_t−1 and time as the transition variables, respectively.²⁴

With the exception of the United States and India, the linearity test results provide uniformly strong evidence against linearity in favor of STR-form nonlinearity for a number of transition variables considered. The tests show stronger rejection of linearity (across potential transition variables) for developing countries relative to advanced countries; linearity is rejected against smooth transition time variation only in developing countries. For the BSTR alternative, the hypothesis of symmetry in regime transition, (F₃): β₃ = 0 in equation 5, is rejected for almost one-half of the countries, less so for the G-7 countries. Using the lag length chosen by AIC for the corresponding linear AR specifications, and with the choice of ΔIny_t−1 as transition variable (linearity test result), the appropriate BSTR models are estimated and their results reported in Tables 8 and 9.²⁵ Following Teräsvirta (1998), the transition parameter was standardized through division by its sample variance and the initial value of γ, the adjustment speed parameter, was fixed at 1 for the estimation algorithm.

Table 8:

BSTR Summary Coefficients

Note: c_L,c_H are threshold values, γ_L, γ_H are speeds of regime transition, and reversion shows the percentage of times the difference between the conditional mean from the nonlinear model (M_NonLin) and the unconditional mean (M_{∆ H}) is less than the corresponding difference for the linear model (M_Lin).

Table 9:

BSTR Parameters of Interest

Note: c_L, c_H are threshold values, γ_l, γ_H are speeds of regime transition,M_Lm and M_VmLu are conditional means from the linear and nonlinear models, M_Δv is the unconditional mean. The parameters determine the transition function of the BSTR model

Δ y_{t} = θ_{1}^{'} x_{t - 1} + θ_{2}^{'} x_{t - 1} F (z_{t}^{d}; γ, c) + μ_{t}

, where

F_{t} (γ_{L}, c_{L}, γ_{H}, c_{H},; z_{t}^{d}) = \frac{\exp [- γ_{L} (z_{t}^{d} - c_{l})] + \exp [γ_{H} (z_{t}^{d} - c_{H})]}{1 + \exp [- γ_{l} (z_{t}^{d} - c_{l})] + \exp [γ_{H} (z_{t}^{d} - c_{H})]}, γ_{L}, γ_{H} > 0, c_{L} < c_{H}

Table 9:

BSTR Parameters of Interest

	Parameters				Cond’l Mean		Mean	Duration
	C_L	C_H	γ_L	γ_H	M_Lin	M_NonLin	M_sy	d_L	d_c	d_H
*Advanced*
*G-7*
Canada	-0.006	0.010	0.333	2.468	-0.0121	-0.0007	-0.0006	1.58	1.96	1.49
France	-0.010	0.006	0.160	1.298	0.0029	-0.0007	-0.0007	1.28	3.84	1.34
Germany	-0.010	0.004	1.053	0.682	-0.0012	-0.0007	-0.0007	1.76	1.88	1.78
Italy	-0.011	0.001	0.847	0.369	0.0401	0.0002	0.0000	1.47	2.35	2.66
Japan	-0.010	0.019	1.123	1.148	0.0072	0.0016	0.0017	1.63	2.00	1.50
United Kingdom	-0.013	0.003	0.726	0.417	0.0056	0.0007	0.0010	1.57	2.27	1.73
United States
*Other*
Australia	-0.017	0.013	1.131	0.425	0.0096	-0.0012	-0.0011	1.75	2.24	1.50
Belgium	-0.013	0.002	0.536	0.744	0.0098	-0.0008	-0.0008	1.57	3.85	1.77
Israel	-0.003	0.028	1.275	0.222	-0.0001	-0.0234	0.0009	2.11	2.80	1.00
Korea	-0.017	0.011	0.775	0.702	0.2011	-0.0014	-0.0008	1.42	2.72	1.77
New Zealand	-0.018	0.005	0.689	0.788	-0.0007	-0.0002	-0.0004	1.45	2.23	1.96
Spain	-0.012	0.001	0.824	0.578	0.0036	-0.0004	0.0000	1.65	2.42	2.00
*Developing*
*Asia*
India	0.271	0.830	1.179	0.918	0.0125	-0.0034	-0.0029	1.69	2.67	1.17
Indonesia	-0.018	-0.011	0.685	0.146	-0.0126	-0.0040	-0.0035	1.44	1.13	3.92
Malaysia	-0.010	0.008	0.725	0.418	0.0419	-0.0020	-0.0014	1.63	2.65	1.56
Philippines	-0.017	-0.001	0.738	0.497	-0.0194	-0.0015	-0.0010	1.61	1.28	3.08
Thailand	-0.042	-0.015	0.488	0.531	-0.0143	-0.0006	-0.0009	1.29	1.29	9.88
WH
Argentina	0.008	-0.023	1.778	0.323	-0.0053	-0.0001	0.0003	3.16	1.00	8.50
Brazil	-0.022	0.016	0.608	0.825	-0.0473	-0.0011	-0.0006	1.57	2.57	1.69
Chile	-0.011	0.017	1.107	0.529	0.0361	-0.0026	-0.0018	1.69	2.38	1.55
Colombia	-0.024	0.006	0.624	1.437	-0.0062	-0.0014	-0.0014	1.42	2.37	2.27
Costa Rica	-0.010	0.009	0.907	1.016	-0.1565	-0.0009	-0.0012	1.59	2.67	2.03
Mexico	-0.009	0.012	1.162	0.836	0.1669	-0.0020	0.0008	1.75	2.13	2.02
Paraguay	-0.018	0.011	1.018	0.923	-0.0872	-0.0027	-0.0028	1.53	2.56	1.86
Uruguay	-0.013	0.009	1.463	0.654	0.2983	-0.0032	0.0004	1.30	2.00	1.88

Δ y_{t} = θ_{1}^{'} x_{t - 1} + θ_{2}^{'} x_{t - 1} F (z_{t}^{d}; γ, c) + μ_{t}

, where

F_{t} (γ_{L}, c_{L}, γ_{H}, c_{H},; z_{t}^{d}) = \frac{\exp [- γ_{L} (z_{t}^{d} - c_{l})] + \exp [γ_{H} (z_{t}^{d} - c_{H})]}{1 + \exp [- γ_{l} (z_{t}^{d} - c_{l})] + \exp [γ_{H} (z_{t}^{d} - c_{H})]}, γ_{L}, γ_{H} > 0, c_{L} < c_{H}

Table 9:

BSTR Parameters of Interest

	Parameters				Cond’l Mean		Mean	Duration
	C_L	C_H	γ_L	γ_H	M_Lin	M_NonLin	M_sy	d_L	d_c	d_H
*Advanced*
*G-7*
Canada	-0.006	0.010	0.333	2.468	-0.0121	-0.0007	-0.0006	1.58	1.96	1.49
France	-0.010	0.006	0.160	1.298	0.0029	-0.0007	-0.0007	1.28	3.84	1.34
Germany	-0.010	0.004	1.053	0.682	-0.0012	-0.0007	-0.0007	1.76	1.88	1.78
Italy	-0.011	0.001	0.847	0.369	0.0401	0.0002	0.0000	1.47	2.35	2.66
Japan	-0.010	0.019	1.123	1.148	0.0072	0.0016	0.0017	1.63	2.00	1.50
United Kingdom	-0.013	0.003	0.726	0.417	0.0056	0.0007	0.0010	1.57	2.27	1.73
United States
*Other*
Australia	-0.017	0.013	1.131	0.425	0.0096	-0.0012	-0.0011	1.75	2.24	1.50
Belgium	-0.013	0.002	0.536	0.744	0.0098	-0.0008	-0.0008	1.57	3.85	1.77
Israel	-0.003	0.028	1.275	0.222	-0.0001	-0.0234	0.0009	2.11	2.80	1.00
Korea	-0.017	0.011	0.775	0.702	0.2011	-0.0014	-0.0008	1.42	2.72	1.77
New Zealand	-0.018	0.005	0.689	0.788	-0.0007	-0.0002	-0.0004	1.45	2.23	1.96
Spain	-0.012	0.001	0.824	0.578	0.0036	-0.0004	0.0000	1.65	2.42	2.00
*Developing*
*Asia*
India	0.271	0.830	1.179	0.918	0.0125	-0.0034	-0.0029	1.69	2.67	1.17
Indonesia	-0.018	-0.011	0.685	0.146	-0.0126	-0.0040	-0.0035	1.44	1.13	3.92
Malaysia	-0.010	0.008	0.725	0.418	0.0419	-0.0020	-0.0014	1.63	2.65	1.56
Philippines	-0.017	-0.001	0.738	0.497	-0.0194	-0.0015	-0.0010	1.61	1.28	3.08
Thailand	-0.042	-0.015	0.488	0.531	-0.0143	-0.0006	-0.0009	1.29	1.29	9.88
WH
Argentina	0.008	-0.023	1.778	0.323	-0.0053	-0.0001	0.0003	3.16	1.00	8.50
Brazil	-0.022	0.016	0.608	0.825	-0.0473	-0.0011	-0.0006	1.57	2.57	1.69
Chile	-0.011	0.017	1.107	0.529	0.0361	-0.0026	-0.0018	1.69	2.38	1.55
Colombia	-0.024	0.006	0.624	1.437	-0.0062	-0.0014	-0.0014	1.42	2.37	2.27
Costa Rica	-0.010	0.009	0.907	1.016	-0.1565	-0.0009	-0.0012	1.59	2.67	2.03
Mexico	-0.009	0.012	1.162	0.836	0.1669	-0.0020	0.0008	1.75	2.13	2.02
Paraguay	-0.018	0.011	1.018	0.923	-0.0872	-0.0027	-0.0028	1.53	2.56	1.86
Uruguay	-0.013	0.009	1.463	0.654	0.2983	-0.0032	0.0004	1.30	2.00	1.88

Δ y_{t} = θ_{1}^{'} x_{t - 1} + θ_{2}^{'} x_{t - 1} F (z_{t}^{d}; γ, c) + μ_{t}

, where

F_{t} (γ_{L}, c_{L}, γ_{H}, c_{H},; z_{t}^{d}) = \frac{\exp [- γ_{L} (z_{t}^{d} - c_{l})] + \exp [γ_{H} (z_{t}^{d} - c_{H})]}{1 + \exp [- γ_{l} (z_{t}^{d} - c_{l})] + \exp [γ_{H} (z_{t}^{d} - c_{H})]}, γ_{L}, γ_{H} > 0, c_{L} < c_{H}

Table 10:

Linearity, Symmetry, and Encompassing Tests

Note: For the linearity test, (Δy_t-1 and Time as transition variables) we report:

(F_{L m}) H_{j}^{L m} : β_{1} = β_{2} = β_{3} = 0

and (F₃) H,: β₃ = 0. The encompassing tests are calculated by estimating by NLLS an MNM form equation containing all of the explanatory variables for both models under consideration and then testing the restrictions necessary to obtain each model through F-tests. Subscripts Lin and NL refer to linear and nonlinear, respectively. V is variance. Numbers in symmetry and encompassing columns are p-values.

Table 10:

Linearity, Symmetry, and Encompassing Tests

	Linearitv				Symmetry		Encompassing
	ΔY_t-1		Time
	F_Lin	F₃	F_Lin	F₃	C_L = C_H	γ_L = γ_H	M_Lin	M_NL	V_NL/V_Lin
*Advanced*
*G-7*
Canada	0.018	0.161	0.102	0.419	0.017	0.000	0.000	0.818	0.940
France	0.035	0.096	0.251	0.105	0.165	0.000	0.000	1.000	0.956
Germany	0.082	0.575	0.420	0.228	0.000	0.349	0.000	1.000	0.964
Italy	0.000	0.713	0.431	0.247	0.000	0.001	0.000	0.956	0.842
Japan	0.000	0.014	0.185	0.955	0.000	0.909	0.000	0.136	0.858
United	0.003	0.105	0.222	0.426	0.000	0.011	0.000	0.713	0.862
Kingdom
United Slates	0.685	0.257	0.111	0.439
*Other*
Australia	0.099	0.501	0.078	0.563	0.000	0.064	0.026	1.000	0.962
Belgium	0.000	0.000	0.666	0.677	0.001	0.221	0.000	0.777	0.918
Israel	0.003	0.077	0.461	0.681	0.000	0.008	0.000	0.637	0.937
Korea	0.000	0.000	0.419	0.909	0.000	0.709	0.038	1.000	0.856
New Zealand	0.001	0.051	0.521	0.379	0.016	0.695	0.000	1.000	0.952
Spain	0.000	0.000	0.440	0.689	0.000	0.001	0.000	1.000	0.895
*Developing*
*Asia*
India	0.779	0.695	0.068	0.410	0.000	0.291	0.000	0.419	0.957
Indonesia	0.000	0.000	0.077	0.436	0.677	0.000	0.000	0.153	0.659
Malaysia	0.000	0.000	0.138	0.504	0.000	0.008	0.000	1.000	0.691
Philippines	0.000	0.013	0.015	0.032	0.169	0.091	0.000	1.000	0.894
Thailand	0.000	0.000	0.050	0.521	0.181	0.656	0.000	1.000	0.850
WH
Argentina	0.000	0.000	0.009	0.011	0.149	0.038	0.000	0.991	0.789
Brazil	0.000	0.006	0.002	0.087	0.001	0.429	0.019	0.370	0.887
Chile	0.000	0.021	0.000	0.000	0.000	0.005	0.000	1.000	0.794
Colombia	0.055	0.698	0.056	0.086	0.005	0.001	0.001	1.000	0.951
Costa Rica	0.000	0.000	0.000	0.000	0.016	0.544	0.000	0.883	0.706
Mexico	0.000	0.000	0.000	0.034	0.001	0.206	0.000	0.834	0.734
Paraguay	0.038	0.213	0.055	0.177	0.023	0.771	0.000	0.711	0.951
Uruguay	0.000	0.000	0.029	0.002	0.000	0.047	0.009	0.719	0.755

Note: For the linearity test, (Δy_t-1 and Time as transition variables) we report:

(F_{L m}) H_{j}^{L m} : β_{1} = β_{2} = β_{3} = 0

Table 10:

Linearity, Symmetry, and Encompassing Tests

	Linearitv				Symmetry		Encompassing
	ΔY_t-1		Time
	F_Lin	F₃	F_Lin	F₃	C_L = C_H	γ_L = γ_H	M_Lin	M_NL	V_NL/V_Lin
*Advanced*
*G-7*
Canada	0.018	0.161	0.102	0.419	0.017	0.000	0.000	0.818	0.940
France	0.035	0.096	0.251	0.105	0.165	0.000	0.000	1.000	0.956
Germany	0.082	0.575	0.420	0.228	0.000	0.349	0.000	1.000	0.964
Italy	0.000	0.713	0.431	0.247	0.000	0.001	0.000	0.956	0.842
Japan	0.000	0.014	0.185	0.955	0.000	0.909	0.000	0.136	0.858
United	0.003	0.105	0.222	0.426	0.000	0.011	0.000	0.713	0.862
Kingdom
United Slates	0.685	0.257	0.111	0.439
*Other*
Australia	0.099	0.501	0.078	0.563	0.000	0.064	0.026	1.000	0.962
Belgium	0.000	0.000	0.666	0.677	0.001	0.221	0.000	0.777	0.918
Israel	0.003	0.077	0.461	0.681	0.000	0.008	0.000	0.637	0.937
Korea	0.000	0.000	0.419	0.909	0.000	0.709	0.038	1.000	0.856
New Zealand	0.001	0.051	0.521	0.379	0.016	0.695	0.000	1.000	0.952
Spain	0.000	0.000	0.440	0.689	0.000	0.001	0.000	1.000	0.895
*Developing*
*Asia*
India	0.779	0.695	0.068	0.410	0.000	0.291	0.000	0.419	0.957
Indonesia	0.000	0.000	0.077	0.436	0.677	0.000	0.000	0.153	0.659
Malaysia	0.000	0.000	0.138	0.504	0.000	0.008	0.000	1.000	0.691
Philippines	0.000	0.013	0.015	0.032	0.169	0.091	0.000	1.000	0.894
Thailand	0.000	0.000	0.050	0.521	0.181	0.656	0.000	1.000	0.850
WH
Argentina	0.000	0.000	0.009	0.011	0.149	0.038	0.000	0.991	0.789
Brazil	0.000	0.006	0.002	0.087	0.001	0.429	0.019	0.370	0.887
Chile	0.000	0.021	0.000	0.000	0.000	0.005	0.000	1.000	0.794
Colombia	0.055	0.698	0.056	0.086	0.005	0.001	0.001	1.000	0.951
Costa Rica	0.000	0.000	0.000	0.000	0.016	0.544	0.000	0.883	0.706
Mexico	0.000	0.000	0.000	0.034	0.001	0.206	0.000	0.834	0.734
Paraguay	0.038	0.213	0.055	0.177	0.023	0.771	0.000	0.711	0.951
Uruguay	0.000	0.000	0.029	0.002	0.000	0.047	0.009	0.719	0.755

Note: For the linearity test, (Δy_t-1 and Time as transition variables) we report:

(F_{L m}) H_{j}^{L m} : β_{1} = β_{2} = β_{3} = 0

Tables 8 and 9 show that the threshold range is wider in developing countries and the speed of adjustment is greater at the lower threshold (γ_L); in fact, γ_L > γ_H in 62 percent of countries. Comparing conditional and unconditional means, we find that in 96 percent of cases the addition of the nonlinear component to the model indicates reversion to the mean. The duration estimates indicate a higher probability of being in the upper regime; exceptions among advanced countries are Australia, Canada, Germany, and Japan. As an interpretational example, we reproduce below (equation 10) the BSTR result for Canada. The lower and upper thresholds are at -0.6 and 1 percent, respectively, indicating a higher threshold tolerance for appreciations. The reversion coefficient, which is significantly different from zero, interacts with the transition function, indicating different reversion speeds, depending on the value of the transition function. The speeds of adjustment are 0.33 from the lower to the middle regime and 2.47 between the middle and upper regimes, indicating a quicker move between the corridor and appreciation regimes than between the depreciation and corridor regimes.

Estimated parsimonious BSTR model for Canada (p=2):

\begin{matrix} \begin{array}{l} Δ y_{t} = - \underset{(21.3)}{0.10} - \underset{(2.77)}{0.80} Δ y_{t - 2} + {\underset{(4.04)}{0.20} - \underset{(3.73)}{0.04} y_{t - 1} + \underset{(2.98)}{0.32} Δ y_{t - 1} - \underset{(3.06)}{1.46} Δ y_{t - 2}} F (Δ y_{t - 1}) \\ F (Δ y_{t - 1}) = \frac{\exp [- (0.33 / σ_{Δ y}) (Δ y_{t - 1} - (0.006))] + \exp [- (2.47 / σ_{Δ y}) (Δ y_{t - 1} - 0.01)]}{1 + \exp [- (0.33 / σ_{Δ y}) (Δ y_{t - 1} - (0.006))] + \exp [- (2.47 / σ_{Δ y}) (Δ y_{t - 1} - 0.01)]} & (10) \end{array} \end{matrix}

Table 10 also presents results on symmetry and encompassing. The third block reports tests for the feasibility of regime reduction (from three to two regimes), that is c_L = c_H, and asymmetry (γ_L = γ_H) ²⁶ The fourth block reports encompassing tests of the linear model relative to the nonlinear model. The final column reports the ratio of the variance of the STR residuals to variance of the linear residuals. We find ample evidence consistent with 3-regime switching regressions and asymmetric adjustment speeds between regimes: c_L ≠ c_H in 81 percent of cases, and γ_L ≠ γ_H in 58 percent of countries. Among G-7 countries, we cannot reject symmetry for the two major currency countries, Germany and Japan. Further, for these two countries durations in each regime are approximately equal, probably reflecting the market microstructure of these advanced economies. The results show that c_L = c_H in France (among advanced countries) and in Indonesia, Philippines, Thailand, and Argentina (among developing countries). Only Thailand does not reject both symmetry and adequacy of two-regimes.

The results of the encompassing tests, based on the minimal nesting model (MNMJ framework, indicate that the linear model does not encompass the nonlinear alternative while the nonlinear BSTR models encompass the corresponding linear models for all countries. Although the rich parameterization in an MNM framework is believed to endanger the convergence properties in tests of parsimonious encompassing (the BEGS algorithm may either not converge or converge to a local minimum), we did not encounter any convergence problems; in fact, the smooth convergence found suggests that the parameter estimates are very close to their optimal values. In terms of variance reduction, the largest improvements occur for the developing countries.

C. MSM Estimates

Table 11 shows that an initial test of linearity versus non-linearity of a Markov switching form rejects the linear specification. The results for a Markov switching intercept and autoregressive (MSIA) specification are reported in Tables 12, 13, and 14.

Table 11:

Linearity vs. MSM Nonlinearity

Note: UKD is United Kingdom and USA is United States.

Table 11:

Linearity vs. MSM Nonlinearity

G-7	Canada	France	Germany	Italy	Japan	UKD	USA
F-test	33.58	62.14	22.82	189.4	73.19	71.74	37.61
p-value	0.002	0.000	0.004	0.000	0.000	0.000	0.000
Other	Australia	Belgium	Israel	Korea	New Zealand	Spain
F-test	86.13	117.4	78.6	259.7	74.42	124.1
p-value	0.000	0.000	0.000	0.000	0.000	0.000
Developing
Asia	India	Indonesia	Malaysia	Philippines	Thailand
F-test	121.7	506.6	246.5	185.7	331.4
p-value	0.000	0.000	0.000	0.000	0.000
WH	Argentina	Brazil	Chile	Colombia	Casta Rica	Mexico	Paraguay	Uruguay
F-test	347.5	259.3	219.7	50.16	514.0	437.8	147.2	364.3
p-value	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000

Note: UKD is United Kingdom and USA is United States.

Table 11:

Linearity vs. MSM Nonlinearity

G-7	Canada	France	Germany	Italy	Japan	UKD	USA
F-test	33.58	62.14	22.82	189.4	73.19	71.74	37.61
p-value	0.002	0.000	0.004	0.000	0.000	0.000	0.000
Other	Australia	Belgium	Israel	Korea	New Zealand	Spain
F-test	86.13	117.4	78.6	259.7	74.42	124.1
p-value	0.000	0.000	0.000	0.000	0.000	0.000
Developing
Asia	India	Indonesia	Malaysia	Philippines	Thailand
F-test	121.7	506.6	246.5	185.7	331.4
p-value	0.000	0.000	0.000	0.000	0.000
WH	Argentina	Brazil	Chile	Colombia	Casta Rica	Mexico	Paraguay	Uruguay
F-test	347.5	259.3	219.7	50.16	514.0	437.8	147.2	364.3
p-value	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000

Note: UKD is United Kingdom and USA is United States.

Table 12:

MSIA Summary Coefficients

Note: ρ_i are coefficients of y_t−1 in

Δ y_{t} = \sum_{R = 1}^{m} (α_{R} + ρ_{R} y_{t - 1} + \sum_{i = 1}^{k} β_{i R} Δ y_{t - i}) \Pr (S_{t} = R | Δ {\tilde{y}}_{t - 1}) + σ v_{t}

(equation 8); R_i are conditional means.

Table 13:

MSIA Probabilities and Duration Estimates

Note: Let j = 1,2,3 denote the alternative regimes. Then, p_jj indicates the probability of being in regime j given that we were in regime j the previous period; Pr_Mj is the unconditional probability of being in regime j; d_j is the average duration in regime j.

		Moments and Unit Root Tests
		Mean	SD	Skew	KT	J-B	MZa	MZt	MSB	MPT	ADF	MA	MA_t
*Advanced*
	*G-7*
Canada		4.49	0.11	-0.12	1.60	22.07	-0.87	-0.41	0.47	15.06	-1.09	0.26	4.18
France		4.59	0.04	-0.06	3.31	1.23	-1.79	-0.61	0.34	9.75	-2.25	0.31	5.09
Germany		4.61	0.05	0.29	2.58	5.66	-6.83	-1.82	0.27	3.70	-2.23	0.29	4.69
Italy		4.46	0.08	0.51	2.58	13.17	-2.98	-1.22	0.41	8.20	-1.83	0.52	9.41
Japan		4.67	0.20	-0.50	2.14	18.95	-2.06	-0.93	0.45	11.07	-1.84	0.39	6.49
United Kingdom		4.59	0.09	-0.10	1.83	15.46	-2.05	-1.01	0.49	11.96	-1.73	0.36	5.95
United States		4.70	0.12	0.64	2.36	22.32	-3.25	-1.13	0.35	7.41	-1.95	0.39	6.61
	*Other*
Australia		4.55	0.13	0.31	1.98	15.69	-0.79	-0.38	0.48	15.80	-1.73	0.37	6.22
Belgium		4.59	0.05	0.44	2.93	8.56	-5.77	-1.60	0.28	4.55	-2.82	0.30	4.80
Israel		4.62	0.07	0.43	2.34	13.10	-3.83	-1.17	0.30	6.56	-2.12	0.27	4.25
Korea		4.53	0.12	-1.07	4.25	67.05	-5.66	-1.59	0.28	4.62	-2.30	0.60	11.42
New Zealand		4.55	0.09	0.09	2.52	2.82	-7.61	-1.86	0.24	3.56	-1.99	0.43	7.40
Spain		4.45	0.09	0.26	2.89	3.09	-4.08	-1.43	0.35	6.01	-1.72	0.37	6.14
Developing
	*Asia*
India		4.65	0.38	0.31	1.39	32.79	0.60	0.96	1.59	152.66	-1.62	0.13	2.05
Indonesia		4.72	0.44	0.02	2.65	1.38	0.10	0.07	0.70	31.95	-1.37	0.16	2.55
Malaysia		4.71	0.18	0.34	2.08	14.52	-0.92	-0.49	0.54	17.55	-1.28	0.25	3.94
Philippines		4.79	0.16	0.56	2.22	20.53	-0.83	-0.44	0.53	17.51	-1.93	0.22	3.46
Thailand		4.65	0.16	0.13	2.62	2.29	-0.04	-0.02	0.68	29.09	-1.43	0.27	4.21
	WH
Argentina		4.87	0.37	-0.63	2.44	20.67	-2.62	-1.14	0.44	9.37	-1.96	0.03	0.41
Brazil		4.33	0.19	-0.03	1.78	16.25	-10.4	-2.17	0.21	2.78	-2.30	0.29	4.53
Chile		4.88	0.25	1.09	3.24	52.79	0.12	0.12	0.97	55.03	-2.47	0.11	1.66
Colombia		4.98	0.25	0.46	2.04	19.48	-0.33	-0.28	0.83	37.88	-1.97	0.45	7.80
Costa Rica		4.65	0.13	0.64	9.45	474.7	-2.21	-0.82	0.37	9.50	-4.61	0.46	7.72
Mexico		4.68	0.20	-0.30	2.45	7.14	-4.37	-1.47	0.34	5.62	-2.67	0.30	4.86
Paraguay		4.89	0.24	0.76	2.44	28.50	0.77	0.98	1.28	104.93	-1.84	0.12	1.87
Uruguay		4.96	0.23	-0.05	1.50	24.85	-2.60	-1.14	0.44	9.44	-1.78	0.04	0.65

		Moments and Unit Root Tests
		Mean	SD	Skew	KT	J-B	MZa	MZt	MSB	MPT	ADF	MA	MA_t
*Advanced*
	*G-7*
Canada		4.49	0.11	-0.12	1.60	22.07	-0.87	-0.41	0.47	15.06	-1.09	0.26	4.18
France		4.59	0.04	-0.06	3.31	1.23	-1.79	-0.61	0.34	9.75	-2.25	0.31	5.09
Germany		4.61	0.05	0.29	2.58	5.66	-6.83	-1.82	0.27	3.70	-2.23	0.29	4.69
Italy		4.46	0.08	0.51	2.58	13.17	-2.98	-1.22	0.41	8.20	-1.83	0.52	9.41
Japan		4.67	0.20	-0.50	2.14	18.95	-2.06	-0.93	0.45	11.07	-1.84	0.39	6.49
United Kingdom		4.59	0.09	-0.10	1.83	15.46	-2.05	-1.01	0.49	11.96	-1.73	0.36	5.95
United States		4.70	0.12	0.64	2.36	22.32	-3.25	-1.13	0.35	7.41	-1.95	0.39	6.61
	*Other*
Australia		4.55	0.13	0.31	1.98	15.69	-0.79	-0.38	0.48	15.80	-1.73	0.37	6.22
Belgium		4.59	0.05	0.44	2.93	8.56	-5.77	-1.60	0.28	4.55	-2.82	0.30	4.80
Israel		4.62	0.07	0.43	2.34	13.10	-3.83	-1.17	0.30	6.56	-2.12	0.27	4.25
Korea		4.53	0.12	-1.07	4.25	67.05	-5.66	-1.59	0.28	4.62	-2.30	0.60	11.42
New Zealand		4.55	0.09	0.09	2.52	2.82	-7.61	-1.86	0.24	3.56	-1.99	0.43	7.40
Spain		4.45	0.09	0.26	2.89	3.09	-4.08	-1.43	0.35	6.01	-1.72	0.37	6.14
Developing
	*Asia*
India		4.65	0.38	0.31	1.39	32.79	0.60	0.96	1.59	152.66	-1.62	0.13	2.05
Indonesia		4.72	0.44	0.02	2.65	1.38	0.10	0.07	0.70	31.95	-1.37	0.16	2.55
Malaysia		4.71	0.18	0.34	2.08	14.52	-0.92	-0.49	0.54	17.55	-1.28	0.25	3.94
Philippines		4.79	0.16	0.56	2.22	20.53	-0.83	-0.44	0.53	17.51	-1.93	0.22	3.46
Thailand		4.65	0.16	0.13	2.62	2.29	-0.04	-0.02	0.68	29.09	-1.43	0.27	4.21
	WH
Argentina		4.87	0.37	-0.63	2.44	20.67	-2.62	-1.14	0.44	9.37	-1.96	0.03	0.41
Brazil		4.33	0.19	-0.03	1.78	16.25	-10.4	-2.17	0.21	2.78	-2.30	0.29	4.53
Chile		4.88	0.25	1.09	3.24	52.79	0.12	0.12	0.97	55.03	-2.47	0.11	1.66
Colombia		4.98	0.25	0.46	2.04	19.48	-0.33	-0.28	0.83	37.88	-1.97	0.45	7.80
Costa Rica		4.65	0.13	0.64	9.45	474.7	-2.21	-0.82	0.37	9.50	-4.61	0.46	7.72
Mexico		4.68	0.20	-0.30	2.45	7.14	-4.37	-1.47	0.34	5.62	-2.67	0.30	4.86
Paraguay		4.89	0.24	0.76	2.44	28.50	0.77	0.98	1.28	104.93	-1.84	0.12	1.87
Uruguay		4.96	0.23	-0.05	1.50	24.85	-2.60	-1.14	0.44	9.44	-1.78	0.04	0.65

		ρ₁	ρ₂	ρ₃	R₄	R₂	R₃
Advanced		-0.115	-0.004	-0.099	-0.020	-0.003	0.009
	G-7	-0.005	0.001	-0.064	-0.012	0.000	0.013
	Other	-0.028	-0.034	0.009	-0.028	-0.005	0.003
Developing		-0.037	-0.010	-0.037	-0.080	-0.007	0.008
	Asia	-0.090	-0.004	-0.131	-0.053	-0.016	-0.007
	WH	-0.084	-0.009	-0.063	-0.099	-0.001	0.020
Overall		-0.049	-0.011	-0.059	-0.048	-0.005	0.009

		ρ₁	ρ₂	ρ₃	R₄	R₂	R₃
Advanced		-0.115	-0.004	-0.099	-0.020	-0.003	0.009
	G-7	-0.005	0.001	-0.064	-0.012	0.000	0.013
	Other	-0.028	-0.034	0.009	-0.028	-0.005	0.003
Developing		-0.037	-0.010	-0.037	-0.080	-0.007	0.008
	Asia	-0.090	-0.004	-0.131	-0.053	-0.016	-0.007
	WH	-0.084	-0.009	-0.063	-0.099	-0.001	0.020
Overall		-0.049	-0.011	-0.059	-0.048	-0.005	0.009

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Series

About

Resources

Contributor Notes

Abstract

I. Introduction

A. Nonlinear Adjustment and Asymmetry

B. Issues in Testing for Unit Roots

C. Summary of Results

II. Nonlinear Frameworks

A. Threshold Autoregressions (TAR)

Unit Root Tests

B. Smooth Transition Regressions (STR)

Estimation

C. Markov Switching Models (MSM)

D. Model Evaluation

III. Estimation and Results

A. TAR Estimates

Empirical Characteristics

Parameter Estimates

B. STR Estimates

Testing for Linearity

Parameter Estimates

C. MSM Estimates