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:
where yt is the dependent variable of interest,
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):
For zt = Δyt−1, Pt−1,R (Zt) = αt−1,R (Zt−1) + (1−αt−1,R))Pt−2,R (Zt−1)
Pt−1(zt) 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, Pt−1,R (zt). 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, Pt−1,H (zt), or depreciate by more than the lower forecast bound, Pt−1,L (zt). The transition variable zt = Δyt−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 δL, δH 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, ρR are the slope coefficients on yt−1; β0R are the slope coefficients on the deterministic components; and βiR are the slope coefficients on the (Δyt−1,… Δyt−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
Wald 2: Hansen’s Unidentified Threshold Scenario
Wald 3: Hansen’s Identified Threshold
Wald 4: Unit Root in Corridor Regime, Partial Unit Root
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:
where xt−1, defined as in equation 1, is a vector of exogenous variables;
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:
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.
After determining the transition function and the threshold variable, the parameters of an STR model can be estimated by nonlinear least squares (NLS). For yt = F(xt; θ) + εt, the NLS estimator is given by
If εt 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, St, 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.
where Δỹt−1 = (Δyt−1,Δyt−2,…,Δyt−(t−1)), and St is a three-state Markov chain with unknown transition probabilities Pij, given by Pij = Pr(St = j|St−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:
where ⊗ denotes element-wise matrix multiplication. To illustrate, we consider a simple two state model where states (regimes) alternate between zero and unity. Then:
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 St 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
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:
where j is an index of country i’s trade partners; Wij is the competitiveness weight put by country i on country j, Pi and Pj are consumer price indices in countries i and j; and Ri and Rj 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.
|Moments and Unit Root Tests|
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 δR are initialized through a grid search over [0,1] in steps of 0.1 increments, determining the αR, the threshold sequences, and the indicator variables (IL, IH). We use the lagged difference of the exchange rate as the transition variable and set the delay parameter to unity.19 Our choice of zt = Δyt−1 is stationary whether yt is I(1) or I(0). We also initialize St−2,R = 0, At−2,R = 0, and Ft−2,R = Δyt−2. For each triple (δL, δH, k), consisting of the lower and upper thresholds and lag k on ∆yt−k, we estimate by ordinary least squares (OLS)20
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
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
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 αL and αH 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.
Tables 4 and 6 summarize the TAR estimates and the Wald tests. For the unrestricted TAR model, ρL> ρH for developing countries, and ρH> ρL 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 ρH < 0; on the other hand, ρL < 0 and larger than ρH for WH countries. In the corridor regime, all reversion coefficients are negative. For the TARurCor model, |ρL|>|ρH| for WH countries, with approximate equality for G-7 and Asian countries. Except for the other advanced countries, for which only ρH 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.
|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|
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.
|LIN vs TAR||LINur vs TAR||TARur vs|
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
for the transition function in equation (2), with all terms evaluated at γ1= γ2 = 0, yields an auxiliary regression:
Since β3 is not dependent on c1 or c2, and all βj= 0, j = 1,…3, for γ1 = γ2 = 0, it follows that, conditional on rejecting linearity (βj ≠ 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)).
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 Δyt. The first and second blocks report p-values of F-tests for the auxiliary regression (5) with Δyt−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 ΔInyt−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.
|FLin||F3||FLin||F3||CL = CH||γL = γH||M_Lin||M_NL||V_NL/V_Lin|
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 (γL); in fact, γL> γH 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):
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 cL = cH, and asymmetry (γL = γH) 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: cL ≠ cH in 81 percent of cases, and γL ≠ γH 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 cL = cH 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.
We find qualitatively similar results to the TAR process. The G-7 and Asian countries have higher reversion coefficients in the upper regime, while other advanced and WH countries have larger reversion coefficients in the lower regime. Differences in the reversion coefficients tell only part of the story as the sum of the coefficients on the lagged dependent variables also vary significantly across regimes and countries, indicating large differences in serial correlation properties of the series. The reversion coefficients in the upper and lower regimes are also unequal, suggesting asymmetrical adjustment. We report conditional means for the three regimes, but note that they are not strictly comparable across the three classes of models estimated. This is because the predicted value from the MSIA specification is a weighted average of all three regimes, in contrast to the regime specific conditional means obtained in the TAR specifications, and the specific weights depend on the probability of being in each of the regimes at that time period.
We examine the transition probabilities pjj, j = 1,2,3, for evidence of persistence (Table 13). The probability of remaining in regime 2 at time t, given that the process was in regime 2 at time t-1 is uniformly higher across all groups, with developing countries having higher probabilities than advanced countries. It is also clear that the probability of remaining in the lower regime is uniformly less than that of remaining in the upper regime, although country specific differences exist. For almost all countries, the probability of remaining in the upper regime is greater than that of remaining in the lower regime, a fact that may be related to a fear of depreciation (Dutta and Leon (2002)). Exceptions are Germany and Spain among advanced countries and Indonesia among developing countries; for the United States and Brazil, the probabilities of remaining in the upper and lower regimes are approximately equal. For developing countries, the probability of being in the lower regime is dominated by that of being in upper regime; in contrast, two of the three major currency countries (Germany and the United States) have higher probabilities of being in the lower than the upper regime. Relative to the probabilities implied from the TAR framework (observable switching variable), the MSIA specification has higher and more variable probabilities associated with the middle regime.
D. Tests of Model Evaluation
We evaluate the performance of the models, using several measures. We consider a test of remaining nonlinearity based on Hamilton’s (2001) general linearity test (Table 16), Li’s (1995) density based non-parametric test (Table 17), and the generalized conservative test framework proposed by BNP (2002), which evaluates the properties of interest implied by the simulated models against the empirical properties of the data (Table 18). We do not control for alternative models’ sensitivity to outliers or extreme observations.
Applying Hamilton’s (2001) generalized test for nonlinearity to the residuals of our estimated models indicates that the TARur model accounts for the nonlinearity in the G-7 countries. The incidence of remaining nonlinearity is 17 percent for the other advanced countries and 39 percent for the developing countries. The TAR corridor model shows remaining nonlinearity in about one-half of the countries, suggesting that that specification is less adequate as a characterization of the data dynamics. The BSTR specification has the lowest incidence of remaining nonlinearity, with zero incidence for the G-7 countries and 12.5 percent for WH. In contrast, the MSIA performs poorly.
Li’s test is based on matching the empirical density of the observed series (estimated through a nonparametric kernel estimator) to the density implied by the simulated model, based on the estimates of each specification considered above. The reported statistics are meant to be interpreted in relative terms across the models as representations of a measure of deviations between the two densities. In terms of relative performance, the BSTR outperforms the other nonlinear models in achieving the closest match between the two densities. In about two-thirds of cases, the nonlinear models (TAR and BSTR,) outperform the linear models (LIN and LINur), conforming that our nonlinear specifications are more adequate characterizations of the data generating process than the linear models.
For BNP (2002), we consider tests for the first two moments (mean and variance), the interquartile range, (the middle 50% of the observations) and measures of asymmetry and persistence. For asymmetry and persistence, we measure how well the data simulated under the estimated models replicates the features of EGARCH-asymmetry and GARCH-persistence in the conditional variance of the empirical sample. The tests are based on the comparison of the series’ empirical density, estimated nonparametrically, and the density implied by each of the models, obtained from simulations using 1000 replications as the trimming margin. In calculating the Newey-West standard errors, 9 lags were used to account for possible serial correlation. The reported statistics should also be interpreted in relative terms across the models.
In interpreting the results, a positive value of a statistic generally indicates a proportional under-representation of the corresponding indicator in the series implied by the model. For example, the linear AR model tends to over-predict the mean relative to the linear unit root AR, and the TARur tends to over-predict the mean relative to the unrestricted TAR. For the asymmetry and persistence test, we report absolute values of the tests.
These statistics show that, in terms of relative performance, the corridor unit root model (TARurCor,) performs the least well in matching the two densities. The unrestricted (stationary) threshold model performs slightly better than the threshold model with a unit root in each regime (TARur). The performances of the linear and linear unit root models are similar, probably indicating a near unit root estimate. In contrast to the Wald tests, the BNP tests are less discriminating, because they are conservative and therefore under-reject. This suggests that when they do reject, there is an extremely strong case for rejection and any other less conservative test would also reject. More importantly, they provide critical information on the exact moment-based measure that is responsible for misspecification in the estimated model.
Although most models perform well in matching the mean and variance of the data, their ability to replicate the interquartile range and to a lesser extent asymmetry and persistence is less impressive, especially in developing countries, in about 50 percent of advanced countries, all the characteristics tested are replicated by at least one model; further, the liner models are also capable of replicating some characteristics of the data densities. A result that seems consistent across countries is that, among nonlinear models, the BSTR model comes closest to the unconditional mean and the asymmetry-based measure; among the linear models, the non-stationary model outperforms its stationary counterpart.
This paper addresses some key questions in the PPP debate: (1) Are deviations from PPP stationary? (2) Are linear specifications appropriate? (3) Is adjustment towards PPP symmetric from above and below? and (4) Which nonlinear models more adequately characterize the process generating real exchange rates? Our results indicate that the notion of a unit root in real effective exchange rates is not robust to nonlinear specifications. Second, the adjustment dynamics of real exchange rates is not symmetric and that asymmetry differs across countries. Third, a three-regime smooth transition autoregressive model with asymmetric speeds of adjustment between regimes performs best, but not across all countries. While our Markov switching model performed the least well among the models considered, we caution that the specification used can be extended in a number of directions, which could improve performance (see Hamilton and Raj (2002)).
The evidence in this paper includes a number of empirical characteristics that theory models should seek to explain. Of particular interest is the finding of asymmetrical adjustment, because different durations and frequencies of threshold crossings imply different degrees to which countries are prone to macroeconomic consequences of real exchange rate misalignments. The finding of asymmetry also suggests that transactions costs alone cannot explain the dynamics of real exchange rates;27 in contrast, asymmetry is not inconsistent with an intervention interpretation of the dynamics of real exchange rates. Asymmetry also holds on a cross-sectional basis. Using the results from identical TAR models for an expanded set of 35 countries, for which both debt and openness data were available, Leon and Najarian (2003) found a positive correlation between average openness28 and average duration for over-appreciations but no correlation between openness and average duration for over- depreciations; similarly, they found a positive correlation between the average debt to GDP ratio and the average excess deviation (as defined in this paper) for over-depreciations but no correlation between the debt ratio and the excess deviation for over-appreciations. The implication that openness may be related to duration of over-appreciation misalignments but debt ratios are related to excess deviations of over-depreciations merits further research.29 Second, the inability of the nonlinear models to explain all the characteristics of the data examined indicates the limitations of current specifications and which issues/objectives they are capable of addressing; it also points to the need to develop specifications that account, at least, for higher moments of the data. A third implication of our work, meriting further research, relates to model selection and testing. Our research suggests that formal hypothesis testing would probably be more interpretable in the context of a set of models that are capable of replicating the same characteristics of the data.
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