The Japanese Economic Review

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The Journal of the Japanese Economic Association

The Japanese Economic Review Vol. 64, No. 1, March 2013

doi: 10.1111/jere.12005

THE INF-T TEST FOR A UNIT ROOT AGAINST ASYMMETRIC EXPONENTIAL SMOOTH TRANSITION AUTOREGRESSIVE MODELS* By MOTOTSUGU SHINTANI† †Vanderbilt University This paper proposes a new test for a unit root against an alternative of asymmetric exponential smooth transition autoregressive models, by extending the infimum test developed by Park and Shintani. Simulation results suggest that the test performs reasonably well in finite samples. The proposed test is also applied to real exchange rates to examine their asymmetric and nonlinear mean-reverting properties. JEL Classification Numbers: C12, C22, F31.

1

Introduction

Among a wide class of nonlinear time-series models, the exponential smooth transition autoregressive (ESTAR) model has been used extensively in economic applications when some form of friction prevents instantaneous adjustment of price and quantity. For example, in the presence of transaction costs, price deviations between two locations are corrected by arbitrage only if deviations are sufficiently large. The ESTAR model, which was originally introduced by Haggan and Ozaki (1981) and later generalized by Granger and Teräsvirta (1993), allows smooth transitions between the slowly adjusting inner regime with small price deviations and the quickly adjusting outer regime with large price deviations. By estimating this model, the possibility of a nonlinear adjustment of aggregate real exchange rates is emphasized by Michael et al. (1997), Sarantis (1999), Taylor et al. (2001) and Kilian and Taylor (2003), among others. A failure of rejection of a unit root hypothesis in real exchange rates implies a violation of long-run purchasing power parity. In considering a linear adjustment, it is reasonable to employ linear unit root tests such as the Dickey–Fuller test to examine the meanreverting properties. However, in considering a nonlinear adjustment in the ESTAR framework, the mean-reverting properties are better examined by unit root tests designed to have power against ESTAR models. In the development of unit root tests against nonlinear models, two main approaches have been applied in the literature to circumvent the problem caused by the lack of identification of parameters under the null hypothesis. The first type of approach involves using a Taylor series expansion of the nonlinear model around the parameter value under the null. The test by Kapetanios et al. (2003) falls in the group of the first approach. The second type of approach is to construct the test based on the extremum over the parameter space of the nonlinear model. For example, the inf-t test

* The author thanks an anonymous referee, Tomoyoshi Yabu and seminar participants at Hitotsubashi University for helpful comments and suggestions. Financial support from the National Science Foundation Grant SES-1030164 is gratefully acknowledged.

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proposed by Park and Shintani (2005) utilizes this second approach.1 In general, simulation studies suggest that the two approaches complement each other (e.g. Choi and Moh, 2007). In some applications, price adjustment can be not only nonlinear but asymmetric as well. For example, the downward rigidity of nominal wages as a result of a high adjustment cost of wage cuts is a widely observed empirical regularity. Using an asymmetric version of the ESTAR model, Anderson (1997) investigates the nonlinear adjustment process of 3-month and 6-month US Treasury Bill rates. Sollis et al. (2002) also estimate an asymmetric STAR model of real exchange rates, and claim that the adjustment is faster when the currency is overvalued against the US dollar. A unit root test against an asymmetric ESTAR model based on the first approach is proposed by Sollis (2009), which can be viewed as an extended version of the Kapetanios et al. (2003) test. However, an equivalent unit root test against an asymmetric ESTAR model based on the second approach has not been examined.2 In this paper, we extend the inf-t test of Park and Shintani (2005) to allow for an asymmetric ESTAR model and evaluate its performance in comparison to the first approach. We then apply the proposed test to revisit the nonlinear mean reversion of real exchange rate series. The rest of the paper is organized as follows. In Section 2, a brief review on unit root tests and the ESTAR model is provided, followed by an introduction of the new test statistic. In Section 3, the simulation results are explained. An empirical application to real exchange rates is given in Section 4. Concluding remarks are made in Section 5.

2

Test statistics 2.1

Asymmetric exponential transition function

Suppose a smooth transition autoregressive model given by

yt = ρ1 yt −1π ( zt − d , θ ) + ρ2 yt −1{1 − π ( zt − d , θ )} + ut , where p (zt-d, q) ∈ [0, 1] is a continuous U-shaped transition function of a transition variable zt-d and ut is a white noise error term. The functional form of p(zt-d, q) is known but the parameter q is unknown. With a U-shaped transition function, the autoregressive (AR) parameters r1 and r2, respectively, represent the speed of adjustment in the outer regime (with a large zt-d in absolute value) and in the inner regime (with a small zt-d in absolute value). In the presence of transaction costs in the adjustment, an assumption of 0 < r1 < r2 ⱕ 1 is often imposed. To investigate the asymmetric role of transaction costs in asset prices, Anderson (1997) proposes the following asymmetric exponential transition function:

π ( zt − d , θ ) = 1 − exp{−κ 2 zt2− d × h( zt − d , γ )}

(1)

1

Other studies on unit root tests based on the second approach include Caner and Hansen (2001), Bec et al. (2008) and Bec et al. (2010).

2

Sollis et al. (2002) employ a unit root test combined with an estimated asymmetric STAR model but not in the form of an extremum of the t-statistic.

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M. Shintani: The Inf-t Test For a Unit Root

FIGURE 1.

Transition functions of asymmetric exponential smooth transition autoregressive models

with

h( zt − d , γ ) = 0.5 + [1 + exp(γ zt − d )]−1, where q = (k, g), k ⱖ 0 and g ∈ R. The parameter g in the logistic transition function h(zt-d, g) controls the degree of asymmetry.3 When g > 0, the adjustment is faster when zt-d < 0. In contrast, when g < 0, the adjustment is faster when zt-d > 0. When g = 0, the transition function becomes symmetric around zt-d = 0. The role of the parameter k is the same as the dispersion parameter in the symmetric ESTAR model and controls the degree of curvature. When k = 0 (k → •), the model reduces to a linear autoregressive model with an AR parameter r2 (r1) so that neither r1 (r2) nor g can be identified. Figure 1 shows the typical shape of Anderson’s (1997) asymmetric exponential transition function of the transition variable zt-d when g takes non-negative values. The figure contains a symmetric transition case with g = 0 where the coefficient on zt2− d in Equation (1) is k2 = 0.1 for both negative and positive regions of zt-d. When g increases and becomes as large as 10, the transition function is nearly a combination of two exponential transition functions where the coefficient on zt2− d is given by 0.15 (= 1.5 ¥ k2) when zt-d < 0 and 0.05 (= 0.5 ¥ k2) when zt-d > 0. For moderate values of g, the curve will be in the middle of the two extremes because the coefficient on zt2− d changes smoothly between 0.05 and 0.15. 3

There are many other possibilities for introducing asymmetric U-shaped transition functions. For example, Shintani et al. (2013) modify the three-regime logistic smooth transition autoregressive model used in Bec et al. (2010) and consider the asymmetric adjustment by allowing the parameter difference between two logistic functions.

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2.2 Unit root tests against symmetric exponential smooth transition autoregressive models The most commonly used test for a unit root is based on the augmented Dickey–Fuller (ADF) regression of the form: k

Δyt = λ yt −1 + ∑ φ j Δyt − j + ε t

(2)

j =1

for t = 1, . . ., n, where et is a white noise error term. Under the null hypothesis of a unit root, H0: l = 0, the t test statistic of the least square estimator λˆ has the following non-standard limiting distribution:

λˆ τ= ⇒ s(λˆ )

1

∫ W (r )dW (r ) ∫ W (r ) dr 0

1

(3)

2

0

as n → •, where s(λˆ ) is the standard error of λˆ , ⇒ denotes weak convergence and W(r) is the standard Brownian motion on r ∈ [0,1]. When the constant term is included in the ADF regression (Equation (2)), the t test statistic, tm, follows a demeaned variant of the limiting distribution, with W(r) in Equation (3) replaced by the demeaned Brownian motion Wm(r). Similarly, when a linear time trend is included, the limiting distribution of the t test statistic involves the detrended Brownian motion Wt(r). In general, however, the ADF test and other linear unit root tests may have low power against ESTAR alternatives. For this reason, several studies recommend incorporating the ESTAR structure into the unit root test based either on the Taylor series approximation approach or the extremum test statistic approach. Kapetanios et al. (2003) apply the first approach and propose the unit root test against a symmetric ESTAR model, where the transition function is given by imposing g = 0 on Equation (1), or

π ( zt − d , κ ) = 1 − exp{−κ 2 zt2− d }. For the purpose of comparison with Kapetanios et al. (2003), in what follows, we consider the case of zt-d = yt-d with d = 1 along with the partial unit root assumption.4 With a stationary root in the outer regime -1 < r1 = r < 1 and a unit root in the inner regime r2 = 1, the ESTAR model can be written as

Δyt = λ yt −1[1 − exp{−κ 2 yt2−1}] + ut ,

(4)

where l = r - 1, which nests the unit root case with l = 0. In this setting, testing for a unit root against the ESTAR model is equivalent to testing H0: k = 0 against H1: k ⫽ 0 and l < 0 but l is not identified under the null hypothesis. As in the case of a linearity test of 4

Geometric erogodicity of such a model is shown by Kapetanios et al. (2003).

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M. Shintani: The Inf-t Test For a Unit Root

Teräsvirta (1994) in an ESTAR model, Kapetanios et al. (2003) rely on a first-order Taylor series approximation of the transition function around kyt-1 = 0, or

π ( yt −1, κ ) ≈ κ 2 yt2−1, and propose running an auxiliary regression of the form

Δyt = δ yt3−1 + ε t ,

(5)

or its augmented version k

Δyt = δ yt3−1 + ∑ φ j Δyt − j + ε t .

(6)

j =1

Their unit root test is a one-sided test for H0: d = 0 against H1: d < 0 using the least square estimator δˆ , and the t-statistic follows the non-standard limiting distribution

t NL

{

3 1 1 W (1) − ∫ W ( r ) 2 dr ˆ δ 2 0 = ⇒ 4 1 ˆ s(δ ) ∫ W (r )6 dr

}

(7)

0

under the null hypothesis of a unit root. When a nonzero long-run level of yt is allowed, zt-d = yt-d in the transition function can be replaced by zt-d = yt-d - m. To consider such a case, Kapetanios et al. (2003) replace yt in the auxiliary regression with yt − y , where y is the sample mean of yt, and show that the limiting distribution is given by Equation (7) with W(r) replaced by Wm(r). The demeaned version of the Kapetanios et al. (2003) test (KSS test) statistic will be denoted by tNL,m. Similarly, the detrended version of the KSS test statistic has the limiting distribution that involves Wt(r). Instead of using a Taylor series approximation, Park and Shintani (2005) rely on the extremum-type test statistics over the region for the parameter q, which is not identified under the null hypothesis of a unit root. For a given choice of the parameter value for q, the model can be estimated by running the regression k

Δyt = λ (θ )wt (θ ) + ∑ φ j Δyt − j + ε t ,

(8)

j =1

where the first regressor is given by

wt (θ ) = yt −1π ( yt −1, θ ) = yt −1[1 − exp{−κ 2 yt2−1}] so that q = k. The test statistic for H0: l = 0 against H1: l < 0 can be constructed by computing the infimum of the t-statistic of the least squares estimator λˆ (θ ) for each possible value of the parameter q ∈ Qn, with its limiting distribution given by 7 © 2013 Japanese Economic Association

The Japanese Economic Review TABLE 1 Asymptotic critical values of inf-t tests Symmetric ESTAR model

Asymmetric ESTAR model

Significance level

inf-tE

inf-tE,m

inf-tE,t

inf-tAE

inf-tAE,m

inf-tAE,t

1% 5% 10%

-2.85 -2.27 -1.98

-3.71 -3.13 -2.84

-4.18 -3.64 -3.35

-3.04 -2.42 -2.12

-3.81 -3.25 -2.97

-4.30 -3.79 -3.50

Notes: Based on discrete approximation to the Brownian motion by partial sums of a standard normal random variable with 1,000 steps and 10,000 replications. ESTAR, exponential smooth transition autoregressive.

1

(W ( r )[1 − exp{−κ *2W ( r ) 2 }])dW ( r ) λˆ (θ ) ∫ 0 , inf t E = inf ⇒ inf 1 ˆ (θ )) θ *∈Θ* θ ∈Θ n s( λ 2 2 2 1 − exp − κ * ( W ( r )[ { W ( r ) }]) dr ∫

(9)

0

where s(λˆ (θ )) is the standard error of λˆ (θ ) and q* = k*. Note that the inf-t-statistic is free from nuisance parameters but depends on the choice of Qn. Similar to Park and Shintani (2005), we consider k ∈ [10-1,10] ¥ Pn, where −1 2 Pn = (∑ yt2−1 n) so that the limiting parameter space is given by Q* = Qk = [10-1,10] ¥ P,

(

)

1

−1 2

where P = ∫ W ( r ) 2 dr . As for the KSS test, a demeaned version of the inf-t test, 0 denoted by inf-tE,m, can be considered by replacing yt in Equation (8) with yt − y , and the limiting distribution of the test statistic is given by Equation (9), with W(r) replaced by Wm(r).5 Similarly, a detrended version of the inf-t test, denoted by inf-tE,t, asymptotically follows Equation (9), with W(r) replaced by Wt(r). Asymptotic critical values of inf-tE, inf-tE,m and inf-tE,t are provided in Table 1.6 2.3 Unit root tests against asymmetric exponential smooth transition autoregressive models Let us now consider the unit root test against asymmetric ESTAR models. Using the asymmetric transition function (Equation (1)), the basic regression model (Equation (4)) can be rewritten as

Δyt = λ yt −1[1 − exp{−κ 2 yt2−1 × (0.5 + [1 + exp(γ yt −1 )]−1 )}] + ut ,

(10)

where l = r - 1. Using a similar but not identical type of asymmetric ESTAR model, Sollis (2009) utilizes a Taylor series approximation and proposes including an additional regressor yt4−1 in Equation (5) so that an extended auxiliary regression is given by

5

Park and Shintani (2005) also recommend using an alternative regressor, wt (θ ) = ( yt −1 − μ )π ( yt −1, θ ) = ( yt −1 − μ )[1 − exp{−κ 2 ( yt −1 − μ ) 2}], and use the inf-t statistic computed over the parameter q = (k, m).

6

Asymptotic critical values are computed by generating artificial random walk series with n = 1,000 in 10,000 replications.

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M. Shintani: The Inf-t Test For a Unit Root

Δyt = δ1 yt3−1 + δ 2 yt4−1 + ε t ,

(11)

or Equation (6) is now extended as k

Δyt = δ1 yt3−1 + δ 2 yt4−1 + ∑ φ j Δyt − j + ε t .

(12)

j =1

The two-sided F test for H0: d1 = d2 = 0 based on the least square estimators δˆ1 and δˆ 2 is used as a unit root test. The limiting distribution of the F test statistic, denoted as FAE, under the null hypothesis of a unit root is given by

FAE ⇒ h ′Q−1h 2,

(13)

where

h ′ = ⎡ ∫ W ( r )3 dW ( r ) ⎣ 0 1

1

∫ W (r)

4

0

dW ( r ) ⎤ ⎦

and

⎡ W ( r )6 dr ∫0 Q=⎢ 1 ⎢ 7 ⎢⎣ ∫0 W ( r ) dr

∫ W (r ) dr ⎤⎥ .

1

1

7

0

1

⎥

∫ W (r ) dr ⎥⎦ 8

0

The demeaned version of the test statistic, denoted as FAE,m, can be introduced by replacing yt in Equations (11) and (12) with yt − y . Its limiting distribution is given by Equation (13), with W(r) replaced by Wm(r). Similarly, the detrended version of the test statistic can be obtained by using detrended yt, and its limiting distribution can be obtained by using Wt(r). Let us now construct an inf-t-type test statistic instead of using a Taylor series approximation. The first regressor in Equation (8) is now replaced by

wt (θ ) = yt −1π ( yt −1, θ ) = yt −1[1 − exp{−κ 2 yt2−1 × (0.5 + [1 + exp(γ yt −1 )]−1 )}] so that q = (k, g). For the parameter space of g, we consider g ∈ [-10,10] ¥ Pn, where −1 2 Pn = (∑ yt2−1 n) . The inf-t test statistic is obtained as an infimum of the t-statistic of the least squares estimator λˆ (θ ) for each possible value of the parameter q = (k, g) ∈ Qn. The asymptotic properties of the statistic can be derived by noticing that the asymmetric exponential transition function satisfies the conditions used in Park and Shintani (2005). The limiting distribution is given by

λˆ (θ ) ˆ (θ )) θ ∈Θ n s( λ

inf t AE = inf

1

⇒ inf

θ *∈Θ *

∫ (W (r )[1 − exp{−κ * W (r ) × (0.5 + [1 + exp(γ *W (r ))] )}])dW (r ) , ∫ (W (r )[1 − exp{−κ * W (r ) × (0.5 + [1 + exp(γ *W (r ))] )}]) dr 2

−1

2

(14)

0

1

2

2

−1

2

0

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where q* = (k*, g*). The limiting parameter space under the null hypothesis of a unit root is given by Q* = Qk ¥ Qg, where Qk = [10-1, 10] ¥ P and Qg = [10, 10] ¥ P. We denote the demeaned version of the same test by inf-tAE,m and point out that its limiting distribution is also given by Equaion (14), with W(r) replaced by Wm(r). Similarly, a detrended version of the inf-t test, denoted by inf-tAE,t, asymptotically follows Equation (14) with W(r) replaced by Wt(r). Asymptotic critical values of inf-tAE, inf-tAE,m and inf-tAE,t are provided in Table 1 along with those of inf-tE, inf-tE,m and inf-tE,t.

3

Monte Carlo simulation 3.1

Size of the test

We conduct simulation experiments to investigate the finite sample performance of the inf-t tests designed for symmetric and asymmetric ESTAR models. We first investigate the finite sample size property of the inf-t tests. In particular, we evaluate rejection frequencies of the nominal 5% tests using samples of sizes n = 100 and 200 generated from

yt = yt −1 + ut with possibly serially correlated errors ut = fut-1 + et, where et ~ i.i.d. N(0, 1) for f = {0, 0.5}.7 Following the simulation design employed by Kapetanios et al. (2003), we omit intercepts and time trends in the data-generating processes and use the correctly specified regression with one lag augmentation in the presence of serial correlation. In addition, because the theoretical result of Park and Shintani (2005) allows for autoregressive conditional heteroskedasticity (ARCH)-type errors, we also consider the case of ARCH(1) errors ut = ht ε t, where et ~ i.i.d. N(0, 1) and ht = (1 − α ) + αε t2−1 with a = 0.5. All the results are based on 10,000 replications. Table 2 reports the actual rejection frequencies of the inf-t tests based on the asymptotic critical values provided in Table 1. The top and middle panels show the results for the i.i.d. error when f = 0 and for theAR(1) error when f = 0.5, respectively. For the i.i.d. and AR(1) errors, inf-t tests seem to have reasonable size properties. The bottom panel shows the result when the data is generated using ARCH(1) errors with a = 0.5. In the presence of ARCH effects, the inf-t tests based on the nominal level of 5% over-reject the null. The degree of over-rejection is larger for the demeaned and detrended tests. The size distortions, however, become smaller as the sample size increases. 3.2

Power of the test

Next, we evaluate the finite sample powers of the inf-t tests against asymmetric ESTAR models. Their power performance is examined in comparison to the ADF test and KSS-type tests. Here, only results for demeaned tests are reported because they are more relevant in practice. To incorporate the different size distortion among tests under consideration, we focus on the size-adjusted powers. For this purpose, the size-corrected critical values of all tests are first computed for n = 100 and 200 by setting l = 0 in Equation (10) along with 7

For initial values, u0 is drawn from N(0, (1-f2)-1) and y0 = 0.

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M. Shintani: The Inf-t Test For a Unit Root TABLE 2 Size of inf-t tests Symmetric ESTAR model

(A) i.i.d. error n = 100 n = 200 (B) AR(1) error (f = 0.5) n = 100 n = 200 (C) ARCH(1) error (a = 0.5) n = 100 n = 200

Asymmetric ESTAR model

inf-tE

inf-tE,m

inf-tE,t

inf-tAE

inf-tAE,m

inf-tAE,t

5.3 5.4

5.4 5.1

6.0 4.8

5.0 4.8

5.3 4.9

5.7 4.9

5.5 5.5

5.4 5.3

6.2 5.1

5.4 5.3

5.5 5.4

6.0 5.1

6.9 6.1

8.4 6.5

10.5 8.1

7.3 6.9

9.3 7.8

11.6 9.0

Notes: Empirical rejection rates of 5% level tests using asymptotic critical values when data are generated from a unit root process with i.i.d., AR(1) and ARCH(1) errors. Results are based on 10,000 replications. AR, autoregressive; ARCH, autoregressive conditional heteroskedasticity; ESTAR, exponential smooth transition autoregressive.

ut = et, where et ~ i.i.d. N(0, 1). The rejection frequencies based on such critical values are then computed by setting l < 0 in Equation (10). Table 3 reports the results for the two inf-t tests (inf-tE,m and inf-tAE,m), the ADF test (tm) and two KSS-type tests (tNL,m and FAE,m) when the data is generated from the asymmetric ESTAR model (Equation (10)) with a range of parameter values l ∈ {-0.1, -0.2}, g ∈ {0.5, 1.0, 10.0} and k2 ∈ {0.1, 1.0}. Note that we also include inf-tE,m and tNL,m for comparison even though they are originally designed for symmetric models. The upper panel of the table shows the result for k2 = 0.1. When n = 100, the power of the ADF test is the lowest among all five tests. Furthermore, for almost all cases, two inf-t tests, inf-tE,m and inf-tAE,m, outperform two KSS-type tests, tNL,m and FAE,m. The difference in power between the two types of ESTAR-based tests, however, is relatively small compared to the difference between the ESTAR-based tests and the ADF test. Note that the degree of asymmetry increases as the logistic parameter g increases from 0.5 to 10.0. Incorporating asymmetry in the inf-t test seems to result in some advantage over the symmetric ESTARbased inf-t test when the degree of asymmetry increases. However, for the l = -0.2 case, the advantage of the asymmetric ESTAR-based inf-t test becomes ambiguous when n = 200 because the power of both inf-t tests becomes sufficiently large.8 In the same case, the power of the ADF test becomes larger than that of KSS-type tests but not as large as that of the inf-t tests. The lower panel of the table shows the result for k2 = 1.0. In general, the powers of all tests are expected to become larger when k2 increases because more observations will be available to estimate the mean reversion parameter in the outer regime. In fact, when k2 becomes as large as 1.0, it takes the value in the inner regime much less often so that the data generated from ESTAR models match well with those generated from the linear autoregressive model in small samples. For this reason, Kapetanios et al. (2003) report that 8

While not reported, we also conducted simulations for the symmetric case with g = 0 and for the cases of negative g. The results for g = 0 were very close to the case of g = 0.5. The results for negative g are confirmed to be almost the same as the cases of positive g of the same absolute values.

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The Japanese Economic Review TABLE 3 Size-adjusted power of unit root tests against the asymmetric ESTAR model inf-t test

(A) k2 = 0.1 n = 100

g

inf-tE,m

inf-tAE,m

tm

tNL,m

FAE,m

-0.1

0.5 1.0 10.0 0.5 1.0 10.0 0.5 1.0 10.0 0.5 1.0 10.0

17.0 17.4 17.9 43.0 44.2 46.4 56.3 56.8 57.6 98.5 98.7 98.8

17.4 17.8 18.3 44.9 46.4 48.6 56.4 57.1 58.1 98.3 98.4 98.6

15.8 15.9 16.4 35.5 35.9 38.4 52.7 53.1 54.5 97.6 97.7 98.1

17.5 17.5 17.8 42.9 44.0 45.0 53.9 54.2 54.4 92.7 92.9 92.5

16.9 17.0 17.4 42.7 43.8 46.0 53.1 53.8 54.2 94.5 94.9 95.3

0.5 1.0 10.0 0.5 1.0 10.0 0.5 1.0 10.0 0.5 1.0 10.0

27.3 27.3 27.4 78.8 78.8 78.9 77.5 77.5 77.4 100.0 100.0 100.0

26.9 27.0 27.1 76.3 76.3 76.4 74.2 74.2 74.3 99.9 99.9 99.9

28.8 28.9 28.9 83.2 83.2 83.5 83.4 83.4 83.4 100.0 100.0 100.0

24.2 24.2 24.2 60.3 60.4 60.3 60.4 60.4 60.6 94.9 94.8 94.8

23.8 23.7 23.8 61.6 61.6 61.5 60.8 60.7 60.7 96.6 96.6 96.6

-0.1 -0.2

(B) k2 = 1.0 n = 100

-0.1 -0.2

n = 200

KSS-type test

l

-0.2 n = 200

ADF test

-0.1 -0.2

Notes: Empirical rejection rate of 5% level tests using size-adjusted critical values. Results are based on 10,000 replications. inf-tE,m and inf-tAE,m are demeaned inf-t tests against the symmetric exponential smooth transition autoregressive (ESTAR) model and asymmetric ESTAR model, respectively. tm is the augmented Dickey–Fuller (ADF) test with a constant. tNL,m is the demeaned test by Kapetanios et al. (2003) (KSS) and FAE,m is the demeaned test of Sollis (2009).

the ADF test always outperforms their test when the data is generated from the symmetric ESTAR model with k2 = 1.0. Our simulation results confirm that the same is true when the data is generated from the asymmetric ESTAR model.9 Even in the case when the ADF test is dominant with k2 = 1.0, the inf-t tests clearly outperform KSS-type tests. In this sense, the usefulness of the inf-t test remains to some extent, even when k2 is large. In summary, all ESTAR-based tests, namely, inf-t tests and KSS-type tests, perform better than the ADF test when k2 takes small values, while the ADF test performs better when k2 becomes large. Judging from the simulation results, it seems fair to say that inf-t tests are at least as useful as the KSS-type test. In terms of the introduction of the asymmetric structure, the advantage of inf-tAE,m over inf-tE,m is not always clear as inf-tE,m performs reasonably well even if the model is misspecified. This is consistent with the 9

When k2 = 1.0, inf-tE,m uniformly outperforms inf-tAE,m even if the former test is based on the misspecified model. Recall that when k is large, the ADF test based on a parsimonious (but wrong) linear specification is often more powerful than other tests that involve a greater number of parameters. For the same reason, the increased number of parameters from inf-tE,m to inf-tAE,m may have caused some loss in power.

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simulation study of Choi and Moh (2007), who find that the inf-t test based on the symmetric ESTAR model is a powerful test against a wide class of nonlinear processes.

4

An application to real exchange rates

The ESTAR model has been used in many studies to describe a continuous nonlinear adjustment in the aggregate real exchange rates, including Michael et al. (1997), Sarantis (1999), Taylor et al. (2001) and Kilian and Taylor (2003). At the same time, Sollis et al. (2002) and Sollis (2009) argue that asymmetric nonlinear mean reversion is an important feature of data on real exchange rates. For example, the speed of adjustment may be different in two directions if the government is more likely to conduct foreign exchange market intervention in response to appreciation of the domestic currency rather than depreciation.10 In this section, the inf-t test for a unit root is applied to real exchange rate series for the purpose of incorporating the possibility of such an asymmetric adjustment. The data set consists of real effective exchange rates for 61 countries constructed by the Bank for International Settlements (BIS). It is a monthly series and the sample period is from January 1994 to May 2012 (n = 221). Unlike the bilateral rates, effective rates reflect all the trading partners. Related studies using real effective exchange rates include Sarantis (1999), who estimates the ESTAR model, and Bahmani-Oskooee et al. (2007), who employ the KSS test for a unit root against the symmetric ESTAR model. However, none of the previous studies on real effective exchange rates apply unit root tests allowing for the possibility of asymmetric ESTAR models. Below, five test statistics, inf-tE,m, inf-tAE,m, tm, tNL,m and FAE,m, are used to test for a unit root.11 The results are reported in Table 4. In many cases, the null hypothesis of a unit root is not rejected based on the ADF tests at the 5% significance level. In contrast, the unit root hypothesis is rejected more often against the ESTAR model using inf-t tests and KSS-type tests. In particular, the number of countries that reject the null increases from 6 when tm is used to 17 when inf-tE,m and tNL,m are used and to 16 when inf-tAE,m and FAE,m are used. In some cases, the unit root hypothesis can be rejected only if the possibility of an asymmetric structure is taken into consideration. For example, the unit root hypothesis is rejected for Malaysia only when asymmetric ESTAR-based tests, inf-tAE,m and FAE,m, are used. At the 10% level of significance, the null is also rejected for the Netherlands and New Zealand by asymmetric ESTAR-based tests but not by other tests. Overall, the results suggest that the ESTAR models with and without asymmetric structure are a convincing alternative to the linear models in describing the real exchange rates.

5

Conclusion

Economic theory suggests that the adjustment process of some economic variables is not only nonlinear but asymmetric as well. Such an asymmetric nonlinear dynamic structure 10

In the case of the intervention in the yen/dollar market, Ito and Yabu (2007) provide empirical evidence that Japanese monetary authorities are much more tolerant toward yen depreciation than yen appreciation.

11

For the purpose of comparison, all tests are based on a common lag order selected by AIC applied to Equation (2).

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The Japanese Economic Review TABLE 4 Unit root tests for real effective exchange rates inf-t test

ADF test

KSS-type test

Country

inf-tE,m

inf-tAE,m

tm

tNL,m

FAE,m

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61

-1.93 -2.21 -1.60 -2.13 -2.29 -2.21 -4.31** -1.64 -2.72 -2.68 -1.88 -2.05 -1.71 -3.24** -1.77 -2.73 -5.10** -2.15 -2.27 -2.04 -1.76 -2.43 -1.77 -2.26 -2.59 -2.57 -5.03** -1.75 -1.54 -4.65** -2.47 -3.41** -3.44** -4.46** -2.29 -2.05 -2.41 -3.39** -2.50 -2.35 -3.87** -2.79 -1.88 -2.68 -2.64 -3.75** -2.44 -2.18 -0.91 -0.70 -3.48** -3.14** -2.10 -2.55 -3.57** -7.15** -3.15** -1.72 -2.45 -1.65 -3.15**

-2.20 -2.33 -2.18 -2.36 -2.59 -2.21 -4.53** -1.65 -2.72 -2.87 -2.35 -2.16 -2.11 -3.31** -2.00 -2.80 -5.51** -2.65 -2.45 -2.06 -1.98 -2.50 -1.77 -2.58 -2.70 -2.90 -5.07** -1.84 -1.67 -4.76** -2.47 -3.66** -3.47** -4.73** -2.34 -3.49** -2.44 -3.46** -3.00* -3.16* -4.36** -2.84 -1.92 -2.69 -2.64 -3.96** -2.44 -2.41 -1.09 -1.41 -3.50** -3.14* -2.10 -2.56 -3.65** -8.32** -3.18* -2.24 -2.68 -1.75 -3.41**

-4.64** -1.38 -1.37 -1.65 -1.86 -1.55 -1.64 -0.81 -2.67* -2.78* -1.18 -2.23 -1.76 -1.77 -1.28 -2.53 -5.61** -1.89 -2.06 -1.38 -1.26 -1.98 -0.53 -1.24 -1.28 -2.53 -2.93** -1.37 -1.37 -2.31 -2.32 -2.37 -2.84* -4.47** -1.71 -1.97 -2.01 -2.15 -2.09 -1.87 -3.33** -2.40 -1.71 -2.57 -1.77 -1.55 -1.69 -1.63 -0.30 -0.46 -3.65** -2.64* -1.28 -1.89 -1.65 -2.68* -2.23 -1.67 -1.12 -1.36 -1.90

-15.62** -2.24 -1.58 -2.19 -2.31 -2.23 -4.21** -1.68 -2.28 -2.70* -1.86 -1.94 -1.80 -3.24** -1.69 -2.77* -7.72** -2.17 -2.06 -2.02 -1.73 -2.33 -1.79 -2.27 -2.62 -2.59 -5.09** -1.76 -1.55 -4.71** -2.48 -3.29** -2.98** -3.57** -2.30 -1.89 -2.25 -3.45** -2.46 -2.36 -3.89** -2.78* -1.57 -2.02 -2.58 -3.90** -2.36 -2.22 -0.92 -0.69 -3.10** -3.15** -2.16 -2.58 -3.65** -7.33** -2.92* -1.59 -2.48 -1.65 -3.14**

125.92** 2.85 2.56 2.58 3.14 2.50 13.53** 1.40 2.58 3.72 2.53 2.63 2.45 5.47** 1.45 3.92 30.84** 2.64 3.10 2.08 1.65 2.92 1.60 3.47 3.56 4.43* 14.23** 1.71 1.39 11.75** 3.08 5.40** 4.61* 8.71** 2.68 18.67** 2.55 6.70** 4.20* 4.90* 9.45** 4.02 1.46 2.72 3.32 12.73** 2.79 3.11 0.56 1.26 6.54** 4.94* 2.31 3.35 7.89** 38.42** 4.33* 2.65 3.07 1.53 5.81**

Algeria Argentina Australia Austria Belgium Brazil Bulgaria Canada Chile China Chinese Taipei Colombia Croatia Cyprus Czech Republic Denmark Estonia Euro area Finland France Germany Greece Hong Kong Hungary Iceland India Indonesia Ireland Israel Italy Japan Korea Latvia Lithuania Luxembourg Malaysia Malta Mexico Netherlands New Zealand Norway Peru Philippines Poland Portugal Romania Russia Saudi Arabia Singapore Slovakia Slovenia South Africa Spain Sweden Switzerland Thailand Turkey United Arab Emirates UK USA Venezuela

Notes: ** and * denote rejection of the unit root hypothesis at the 5 and 10% level of significance, respectively. Critical values at the 5 and 10% level are -1.95 and -1.61 for tm, -2.22 and -1.92 for tNL,m, and 4.971 and 4.173 for FAE,m (n = 200 case in Sollis, 2009), respectively. See Table 1 for the critical values for inf-tE,m and inf-tAE,m.

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M. Shintani: The Inf-t Test For a Unit Root

can conveniently be estimated by the asymmetric variant of ESTAR models. This paper proposes a new test for a unit root against an alternative of the asymmetric ESTAR model by extending the inf-t test originally developed by Park and Shintani (2005) based on the symmetric ESTAR model. This class of test, constructed from the extremum over the space of a parameter that is unidentified under the null, can be a reasonable alternative to the other class of test based on Taylor series expansion of the nonlinear model around the parameter value under the null. This view is supported by the simulation results which compare the finite sample performance of inf-t tests and tests proposed by Kapetanios et al. (2003) and Sollis (2009). The usefulness of the proposed test is also confirmed in an empirical application to real exchange rates. Final version accepted 15 November 2012.

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