Unit Root Tests

Hansen (1997) indicates that conventional tests of the null of a linear autoregression (AR) versus TAR models have nonstandard distributions, as the threshold parameter is not identified under the null of linearity (see Davies (1987)); also the sampling distribution of the threshold estimates are not standard. The model can be nonstationary within one or more regimes, though the alternation between regimes can make it overall stationary. We follow Caner and Hansen (2001) in constructing Wald tests for distinguishing between nonlinearity (threshold effects) and possible nonstationarity in real exchange rate series.⁶ We consider the following hypotheses:

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 does not have the standard χ² (chi-square) asymptotic distribution. The F-statistic is:

where ${\tilde{\sigma }}_T^2$ is the residual variance under the null hypothesis, and ${\widehat{\sigma }}_T^2({\delta }_L,{\delta }_H,k)$ is the residual variance under the alternative. Because F_T(δ_L, δ_H, k) is a decreasing function of ${\widehat{\sigma }}_T^2({\delta }_L,{\delta }_H,k)$, it follows that $F_T({\widehat{\delta }}_L,{\widehat{\delta }}_H,\widehat{k})$ is equivalent to the supremum of the pointwise test statistic F_T(δ_L, δ_H, k) over the allowable values for (δ_L, δ_H, k), that is

Thus the Wald statistic for H₀ is often called the “Sup-Wald” statistic. 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 our test statistics using bootstrap approximations to the Wald statistics.⁷ The unidentified threshold scenario, which performed better in Caner and Hansen’s Monte Carlo tests, makes use of the constrained bootstrap method,⁸ 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. In the presence of threshold effects, these threshold unit root tests have greater power than the conventional ADF unit root tests.

Estimation

We estimate equation (1) using sequential least squares (Hansen 1997). 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.⁹ 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 Δy_t−k, we estimate by ordinary least squares (OLS)¹⁰

Let ${\sigma }^2({\delta }_L,{\delta }_H,k)=T^{-1}{\Sigma }_{t=1}^T{\widehat{\epsilon }}_t{({\delta }_L,{\delta }_H,k)}^2$ be the OLS estimate of σ² for fixed (δ_L, δ_L, k). Then the least squares estimate of the threshold values is found by minimizing σ²(δ_L, δ_L, 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 $\left({\widehat{\delta }}_L,{\widehat{\delta }}_H,\widehat{k}\right).$

These estimates are used to conduct inference concerning the parameters of interest.

Model Evaluation

We evaluate the estimated models for evidence of remaining nonlinearity, based on Hamilton’s (2001) general linearity test, and their ability to replicate empirical properties of the data. If our focus is the DGP, it is natural to focus on the density describing the variable of interest. In practice, however, 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, and volatility, if our interest is uncertainty. In this paper, we focus on tests developed by Pagan (2002) and BNP (2002) that allow us to compare the performance of 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.

Suppose the analyst (policy maker) is interested in some functions of data, $\widehat{g}(y)$. Let $g(\widehat{\theta })$ be the corresponding implied population characteristic, obtained from simulated data based on the estimated model. Label the difference between these two measures as $d=\widehat{g}(y)-g(\widehat{\theta })$. Then, we can think of these tests as comparing a consistent estimator of g(y) to an efficient estimator, $g(\widehat{\theta })$, if the model is valid, enabling us to formulate the variance of d as $\mathrm{var}(d)=\mathrm{var}(\widehat{g}(y)-\mathrm{var}(g(\widehat{\theta }))$ (see Hausman (1978)). Although the variance of $\widehat{g}(y)$ is simply derived from the observed series, the analytical expression for $\mathrm{var}(g(\widehat{\theta }))$ may be difficult to obtain for complicated nonlinear specifications. Because the test statistic $T^{*}=\widehat{d}\prime {[\mathrm{var}(\widehat{g}(y)-\mathrm{var}(g(\widehat{\theta }))]}^{-1} \widehat{d}>T=\widehat{d}\prime {[\mathrm{var}(\widehat{g}(y))]}^{-1} \widehat{d}$, 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 $\mathrm{var}(\widehat{g}(y)),$ compatible with many alternative models, can be obtained using the Newey-West (1987) covariance matrix.

Empirical Characteristics

We investigate estimated the 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 economies. Summaries of the characteristics of the threshold bands and estimates of duration are shown in Tables 1 and 2 and described below.

Table 1:

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 1:

Characteristics of Threshold Bands

	δL	δH	αL	αH	κ*	%L	%H	%Cor
Advanced	0.37	0.37	0.59	0.45	5.04	0.27	0.29	0.44
G-7	0.34	0.54	0.51	0.55	5.14	0.29	0.27	0.45
Other	0.38	0.30	0.62	0.41	5.00	0.26	0.29	0.44
Developing	0.30	0.37	0.49	0.54	6.54	0.25	0.29	0.45
Asia	0.39	0.44	0.43	0.58	4.86	0.29	0.26	0.46
WH	0.23	0.38	0.49	0.43	7.60	0.24	0.33	0.43
Other	0.34	0.32	0.53	0.65	6.23	0.25	0.28	0.47
Overall	0.33	0.37	0.53	0.50	5.92	0.26	0.29	0.45

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.

View Table

Table 1:

Characteristics of Threshold Bands

	δL	δH	αL	αH	κ*	%L	%H	%Cor
Advanced	0.37	0.37	0.59	0.45	5.04	0.27	0.29	0.44
G-7	0.34	0.54	0.51	0.55	5.14	0.29	0.27	0.45
Other	0.38	0.30	0.62	0.41	5.00	0.26	0.29	0.44
Developing	0.30	0.37	0.49	0.54	6.54	0.25	0.29	0.45
Asia	0.39	0.44	0.43	0.58	4.86	0.29	0.26	0.46
WH	0.23	0.38	0.49	0.43	7.60	0.24	0.33	0.43
Other	0.34	0.32	0.53	0.65	6.23	0.25	0.28	0.47
Overall	0.33	0.37	0.53	0.50	5.92	0.26	0.29	0.45

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:

Duration and Loss Estimates

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

Table 2:

Duration and Loss Estimates

	MaxDL	MaxDH	AveDL	AveDH	CumLL	CumLH	AveLL	AveLH
Advanced	4.16	4.36	1.55	1.67	0.08	0.05	0.01	0.01
G-7	4.57	4.00	1.61	1.60	0.06	0.04	0.01	0.02
Other	4.00	4.50	1.53	1.70	0.09	0.05	0.01	0.01
Developing	4.00	4.40	1.56	1.66	0.38	0.12	0.04	0.02
Asia	4.57	3.71	1.71	1.53	0.21	0.08	0.03	0.02
WH	3.93	4.73	1.49	1.81	0.33	0.14	0.04	0.02
Other	3.77	4.38	1.55	1.54	0.52	0.13	0.05	0.02
Overall	4.07	4.38	1.56	1.66	0.25	0.09	0.03	0.02

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

View Table

Table 2:

Duration and Loss Estimates

	MaxDL	MaxDH	AveDL	AveDH	CumLL	CumLH	AveLL	AveLH
Advanced	4.16	4.36	1.55	1.67	0.08	0.05	0.01	0.01
G-7	4.57	4.00	1.61	1.60	0.06	0.04	0.01	0.02
Other	4.00	4.50	1.53	1.70	0.09	0.05	0.01	0.01
Developing	4.00	4.40	1.56	1.66	0.38	0.12	0.04	0.02
Asia	4.57	3.71	1.71	1.53	0.21	0.08	0.03	0.02
WH	3.93	4.73	1.49	1.81	0.33	0.14	0.04	0.02
Other	3.77	4.38	1.55	1.54	0.52	0.13	0.05	0.02
Overall	4.07	4.38	1.56	1.66	0.25	0.09	0.03	0.02

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

Response: The adaptive response weight parameters a_L and a_H show the quickness of response to relatively recent exchange rate variations. The response for deviations toward over-depreciation is quicker for advanced economies than that for developing countries (0.59 vs. 0.49), implying narrower, closely watched bands. In contrast, the response for over-appreciation is much quicker for developing countries (0.54 vs. 0.45). The differences are more marked in subregions. For over-depreciations, the other (non-G7) advanced economies have the fastest response (0.62), Asia the slowest (0.43); for over-appreciations, the other advanced economies have the slowest response (0.41), other developing countries the fastest (0.65). If this design of the thresholds reflects a measure of relative tolerance for these exchange rate variations, then the results suggest that Asia and other developing countries have low tolerance for over-appreciations. Further, the longer average lag for developing countries relative to advanced economies (7 vs. 5 months) suggests a more complex structure for short-term interaction between nominal exchange rates and relative prices.

On average, both advanced and developing economies display asymmetrical response to changes in the real exchange rates, with G-7 (0.55 vs. 0.51), Asia (0.58 vs. 0.43), and other developing countries (0.65 vs. 0.53) placing greater weight on recent developments relating to appreciations while predicting the tolerance margin. The opposite is true for the other advanced (0.62 vs. 0.41) and WH (0.49 vs. 0.43) economies, which react more strongly to developments relating to over-depreciations.

Duration: Maximum durations are longer for over-depreciations for G-7 and Asia and for over-appreciations in the other groups. The maximum duration for the G-7 occurs in the lower regime (4.6 months), but in the upper regime for the other advanced economies (4.5 months). Similarly, the maximum duration for Asia is in the lower regime (4.6 months), but in the upper regime for the WH countries (4.7 months) and other developing countries (4.4 months). The average durations show similar patters with the other advanced economies and WH displaying longer average durations for over-appreciations and Asia exhibiting longer average duration for over-depreciations. The G-7 and other developing countries have equal durations for both types of deviations. Given the difference in response towards depreciation and appreciation deviations of the subgroups, the evidence on duration is probably informative about the speed or effectiveness of the policy measures used to reverse deviations from forecasts. Average duration in the upper regime is greater than the average duration in the lower regime in 51 percent of developing countries, compared to 72 percent of advanced economies. Average duration varies across regions: for the other advanced economies, 83 percent record durations in the upper regime in excess of durations for the lower regime; for the Asian countries, the corresponding figure is 29 percent. As regards the distribution of observations across regimes, there is a tendency in developing countries for more observations to lie in the upper regime (29% vs. 25%), more so for WH countries (33% vs. 24%), consistent with longer average durations for and slower response to over-appreciations. For advanced economies, there is a slight tendency in G-7 for more observations in the lower regime and in other advanced economies for more observations in the upper regime.

Excess Deviation: 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 at least twice as large as that for an appreciation spell (crossing beyond the upper threshold). The overall average is 0.25 in the lower regime and 0.09 in the upper regime. For developing countries, the excess deviation for depreciations is three times higher than that for appreciations; in contrast, the factor is 1.5 for advanced economies. Further, the excess deviation for depreciations is about 4.5 times higher for developing countries relative to advanced economies; for appreciations, that relative factor is about 2.5.

For developing countries, the excess deviation per spell for depreciations (AveL_L) are about twice that for appreciations. Also, excess deviation per spell for developing countries are also larger than that for advanced economies for both depreciations and appreciations. The excess deviation per spell for depreciations in the developing countries is four times that of the advanced economies; for the group other developing countries (mostly African countries), this factor is five. Yet, developing countries have excess deviations per spell that are at most twice that of the advanced economies. Also, the excess deviation per spell in the lower regime is greater than the excess deviation per spell in the upper regime in 83 percent of developing countries (in all other developing countries group), compared to 56 percent for the advanced economies.

We use the cross-section data on duration and excess deviation to explore whether the observed asymmetry is correlated with trade openness (ratio of exports plus imports to GDP) and debt (external liabilities to GDP). PPP theory indicates that PPP deviations are corrected over time through adjustments in trade flows, suggesting that the speed of reversion may be related to trade openness. Also, Lane and Milesi-Ferretti (2001) investigate the link between the real exchange rate and net external position and find the magnitude of the “transfer-effect” varies with country characteristics like openness, size, level of development, and composition of external liabilities. Our results relate measures of real exchange rate dynamics to two country characteristics, openness and external debt. The results in Table 3 show that for the sample of 60 countries, openness is positively correlated with the average duration for over-appreciations but uncorrelated with the average duration for over-depreciations.¹³ Further, openness is negatively correlated with the excess deviation for over-depreciations but uncorrelated with the excess deviation for over-appreciations. Using the results for the 34 countries in the sample for which both debt and openness data were available (the developing countries), we obtain similar results. We find a positive correlation between average openness and average duration for over appreciations but no correlation between openness and average duration for over-depreciations; further, there is a positive correlation between the average debt-to-GDP ratio and the excess deviation per spell for over-depreciations but no correlation between the debt ratio and the excess deviation per spell for over-appreciations. For the developing countries, both the excess deviation for over-depreciation and longer average durations in the lower regime are positively correlated with the debt ratio and negatively correlated with openness. The finding 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.

Table 3:

Cross-section regression estimates

Note: 1/ Sample is restricted to countries for which both debt and openness data were available. Let subscript R depict the alternative regimes, with L corresponding to over-depreciation and H over-appreciation. AveD_t is the average duration per spell of excess deviation, AveL is the average excess deviation per spell, and CumL_s is the average excess deviation (area between the tolerance margin and the observed realizations when the band is crossed), across countries for each regime. D_L> D_H indicates average duration per spell in lower regime is greater than upper regime. Parentheses are t-statistics, based on Newey-West heteroskedasticity-consistent covariance matrix.

Table 3:

Cross-section regression estimates

	Con	G-7	OA	AS	WH	AVOP	AVDT
AveDH	1.37 (16.72)	0.08 (0.84)	0.22 (2.61)	0.044 (0.49)	0.305 (3.25)	0.24 (3.52)
AveDL	1.53 (10.14)	0.06 (0.39)	-0.009 (0.06)	0.17 (1.18)	-0.05 (0.37)	0.029 (0.35)
AveLH	0.02 (8.19)	-0.009 (2.79)	-0.011 (4.00)	-0.005 (1.97)	-0.0002 (0.05)	0.001 (0.59)
AveLH	0.049 (5.99)	-0.032 (4.24)	-0.034 (4.79)	-0.017 (2.27)	-0.005 (0.65)	-0.004 (1.01)
CumLH	0.12 (2.82)	-0.086 (2.55)	-0.074 (2.11)	-0.048 (1.23)	0.006 (0.15)	0.019 (0.70)
CumLL	0.584 (5.60)	-0.469 (4.89)	-0.455 (4.79)	-0.329 (2.93)	-0.207 (2.11)	-0.089 (2.35)
DL > DH	0.66 (3.59)	0.015 (0.05)	-0.413 (2.56)	0.138 (0.58)	-0.228 (1.24)	-0.183 (1.58)
AveDH1	1.389 (17.31)			0.039 (0.44)	0.339 (3.88)	0.284 (2.78)	-0.061 (1.36)
AveDL1	1.531 (7.42)			0.171 (1.03)	-0.061 (0.43)	-0.028 (0.36)	0.049 (0.63)
AveLH1	0.026 (7.54)			-0.006 (2.18)	0.0004 (0.13)	-0.0007 (0.46)	-0.001 (0.59)
AveLL1	0.040 (3.22)			-0.014 (1.51)	-0.002 (0.27)	-0.005 (0.84)	0.014 (2.14)
CumLH1	0.134 (2.50)			-0.052 (1.27)	0.010 (0.26)	0.018 (0.49)	-0.018 (0.84)
CumLL1	0.47 (3.83)			-0.293 (2.46)	-0.182 (1.86)	-0.097 (2.24)	0.161 (2.78)
DL > DH1	0.565 (2.08)			0.170 (0.62)	-0.280 (1.45)	-0.279 (2.32)	0.222 (2.52)

Note: 1/ Sample is restricted to countries for which both debt and openness data were available. Let subscript R depict the alternative regimes, with L corresponding to over-depreciation and H over-appreciation. AveD_t is the average duration per spell of excess deviation, AveL is the average excess deviation per spell, and CumL_s is the average excess deviation (area between the tolerance margin and the observed realizations when the band is crossed), across countries for each regime. D_L> D_H indicates average duration per spell in lower regime is greater than upper regime. Parentheses are t-statistics, based on Newey-West heteroskedasticity-consistent covariance matrix.

View Table

Table 3:

Cross-section regression estimates

	Con	G-7	OA	AS	WH	AVOP	AVDT
AveDH	1.37 (16.72)	0.08 (0.84)	0.22 (2.61)	0.044 (0.49)	0.305 (3.25)	0.24 (3.52)
AveDL	1.53 (10.14)	0.06 (0.39)	-0.009 (0.06)	0.17 (1.18)	-0.05 (0.37)	0.029 (0.35)
AveLH	0.02 (8.19)	-0.009 (2.79)	-0.011 (4.00)	-0.005 (1.97)	-0.0002 (0.05)	0.001 (0.59)
AveLH	0.049 (5.99)	-0.032 (4.24)	-0.034 (4.79)	-0.017 (2.27)	-0.005 (0.65)	-0.004 (1.01)
CumLH	0.12 (2.82)	-0.086 (2.55)	-0.074 (2.11)	-0.048 (1.23)	0.006 (0.15)	0.019 (0.70)
CumLL	0.584 (5.60)	-0.469 (4.89)	-0.455 (4.79)	-0.329 (2.93)	-0.207 (2.11)	-0.089 (2.35)
DL > DH	0.66 (3.59)	0.015 (0.05)	-0.413 (2.56)	0.138 (0.58)	-0.228 (1.24)	-0.183 (1.58)
AveDH1	1.389 (17.31)			0.039 (0.44)	0.339 (3.88)	0.284 (2.78)	-0.061 (1.36)
AveDL1	1.531 (7.42)			0.171 (1.03)	-0.061 (0.43)	-0.028 (0.36)	0.049 (0.63)
AveLH1	0.026 (7.54)			-0.006 (2.18)	0.0004 (0.13)	-0.0007 (0.46)	-0.001 (0.59)
AveLL1	0.040 (3.22)			-0.014 (1.51)	-0.002 (0.27)	-0.005 (0.84)	0.014 (2.14)
CumLH1	0.134 (2.50)			-0.052 (1.27)	0.010 (0.26)	0.018 (0.49)	-0.018 (0.84)
CumLL1	0.47 (3.83)			-0.293 (2.46)	-0.182 (1.86)	-0.097 (2.24)	0.161 (2.78)
DL > DH1	0.565 (2.08)			0.170 (0.62)	-0.280 (1.45)	-0.279 (2.32)	0.222 (2.52)

Note: 1/ Sample is restricted to countries for which both debt and openness data were available. Let subscript R depict the alternative regimes, with L corresponding to over-depreciation and H over-appreciation. AveD_t is the average duration per spell of excess deviation, AveL is the average excess deviation per spell, and CumL_s is the average excess deviation (area between the tolerance margin and the observed realizations when the band is crossed), across countries for each regime. D_L> D_H indicates average duration per spell in lower regime is greater than upper regime. Parentheses are t-statistics, based on Newey-West heteroskedasticity-consistent covariance matrix.

Parameter Estimates

Tables 4 and 5 summarize the TAR estimates and the Wald tests (see Appendix Table 1).

Table 4:

Average 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:

Average Reversion Coefficients

	Linear		Unrestricted TAR		TaRurCor
	ρLIN	ρL	ρC	ρH	ρL	ρH
Advanced	-0.023	-0.007	-0.022	0.000	-0.016	-0.015
G-7	-0.017	0.004	-0.015	-0.006	-0.019	-0.010
Other	-0.025	-0.012	-0.024	0.003	-0.014	-0.017
Developing	-0.021	-0.009	-0.011	-0.009	-0.027	-0.012
Asia	-0.012	0.005	-0.013	0.000	-0.014	-0.007
WH	-0.017	-0.001	-0.007	-0.022	-0.020	-0.008
Other	-0.030	-0.021	-0.014	-0.001	-0.039	0.017
Overall	-0.022	-0.008	-0.016	-0.005	-0.022	-0.013

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.

View Table

Table 4:

Average Reversion Coefficients

	Linear		Unrestricted TAR		TaRurCor
	ρLIN	ρL	ρC	ρH	ρL	ρH
Advanced	-0.023	-0.007	-0.022	0.000	-0.016	-0.015
G-7	-0.017	0.004	-0.015	-0.006	-0.019	-0.010
Other	-0.025	-0.012	-0.024	0.003	-0.014	-0.017
Developing	-0.021	-0.009	-0.011	-0.009	-0.027	-0.012
Asia	-0.012	0.005	-0.013	0.000	-0.014	-0.007
WH	-0.017	-0.001	-0.007	-0.022	-0.020	-0.008
Other	-0.030	-0.021	-0.014	-0.001	-0.039	0.017
Overall	-0.022	-0.008	-0.016	-0.005	-0.022	-0.013

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

Summary of Wald Tests

Table 5

Summary of Wald Tests

	Unconstructed Bootstrap				Constructed Bootstrap
	Lin vs TAR	LinUR vs TAR	TARur vs TAR	TARurCor vs TAR	Lin vs TAR	TARur vs TAR	TARurCor vs TAR
Advanced	0	0	0	0	0.20	0.08	0.28
G-7	0	0	0	0	0	0.14	0.29
Other	0	0	0	0	0.28	0.06	0.28
Developing	0	0	0	0.06	0.37	0.17	0.60
Asia	0	0	0	0.14	0.29	0.14	0.43
WH	0	0	0	0	0.27	0.07	0.53
Other	0	0	0	0.08	0.54	0.31	0.77
Overall	0	0	0	0.03	0.30	0.13	0.47

View Table

Table 5

Summary of Wald Tests

	Unconstructed Bootstrap				Constructed Bootstrap
	Lin vs TAR	LinUR vs TAR	TARur vs TAR	TARurCor vs TAR	Lin vs TAR	TARur vs TAR	TARurCor vs TAR
Advanced	0	0	0	0	0.20	0.08	0.28
G-7	0	0	0	0	0	0.14	0.29
Other	0	0	0	0	0.28	0.06	0.28
Developing	0	0	0	0.06	0.37	0.17	0.60
Asia	0	0	0	0.14	0.29	0.14	0.43
WH	0	0	0	0	0.27	0.07	0.53
Other	0	0	0	0.08	0.54	0.31	0.77
Overall	0	0	0	0.03	0.30	0.13	0.47

Tables 4 and 5 summarize the TAR estimates and the Wald tests. For the unrestricted TAR model, ρ_L > ρ_H for the other advanced economies and other developing countries, and ρ_H > ρ_L for G-7 and WH countries, but the reversion rates are faster in developing countries compared to the advanced economies. For G-7, only p_u < 0; on the other hand, for the other advanced economies, only ρ_L< 0. In the corridor regime, all reversion coefficients are negative. For the TARurCor model, |ρ_L|>| ρ_H| for all groups of developing countries, with approximate equality for G-7 and other developing economies. In the lower regime, reversion is faster for developing countries than for advanced economies. On average, WH and other developing countries revert twice as fast in the lower regime compared to G-7, other advanced economies, and Asian developing countries, which revert at similar speeds. On the other hand, Asia and WH have the slowest reversion rates in the upper regime, about one-half the speed of the other advanced economies and other developing countries. The existence of threshold effects suggest that the results for the linear model are averages across the three regimes.

We calculate Wald statistics to test for threshold effects and/or unit roots. The tests measure whether the Data Generating Process (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 5 shows the percentage of countries for which the various null hypotheses are plausible. These statistics are based on estimated bootstrap p-values, representing the percentage of Wald statistics calculated from the simulated data that exceed the Wald statistics calculated from the observed sample.

The unconstrained bootstrap results indicate an overwhelming rejection of the first three null hypotheses¹⁴. The unrestricted TAR specification outperforms the benchmark stationary ergodic linear process. It is also preferred over both the linear non-stationary I(I) specification, the p-values for which are obtained by constructing a bootstrap distribution that imposes an unidentified threshold effect, and the unit root TAR process.¹⁵ Because the unidentified threshold model was less sensitive to nuisance parameters Caner and Hansen (2001) recommend calculating p-values using the unidentified threshold bootstrap. The intermediate case, which we label as an identified threshold partial unit root process (I(I) in corridor regime combined with an otherwise stationary TAR), yields different outcomes for advanced and developing economies. While the null is still rejected against the stationary ergodic TAR for most advanced economies, 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.

Model Evaluation

Applying Hamilton’s (2001) generalized test for nonlinearity to the residuals of our estimated models indicates that both the unrestricted TAR and the TARur model explain the nonlinearity in the advanced economies. The incidence of remaining nonlinearity is about 15 percent for the advanced economies and about 33 percent for the developing countries (see Appendix Table 2). The TARurCor model shows remaining nonlinearity in about one-third of both groups of economies, suggesting that it performs as well as the TARur in developing countries but is less adequate as a characterization of the data dynamics across all economies.

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. In interpreting the reported statistics, a positive value of a statistic generally indicates a proportional under-representation of the corresponding indicator in the series implied by the model (see Appendix Table 3). 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.¹⁶,¹⁷ But 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 asymmetry effect is less impressive. An interpretation of this result is that the TAR models are more capable of explaining the mean, variance, and a specific form of persistence in the data. In about 50 percent of advanced economies, all the characteristics tested are replicated by at least one model; it is also clear that the TAR framework is less successful in replicating characteristics of the data densities of the developing countries. For the advanced economies, the linear models are also capable of replicating the characteristics of the data densities.