INTERNATIONAL ECONOMIC REVIEW Vol. 57, No. 2, May 2016

TESTING FOR A UNIT ROOT AGAINST TRANSITIONAL AUTOREGRESSIVE MODELS∗ BY JOON Y. PARK AND MOTOTSUGU SHINTANI1 Indiana University, U.S.A., and Sungkyunkwan University, Korea; University of Tokyo, Japan, and Vanderbilt University, U.S.A. This article develops a novel test for a unit root in general transitional autoregressive models, which is based on the infimum of t-ratios for the coefficient of a parametrized transition function. Our test allows for very flexible specifications of the transition function and short-run dynamics and is significantly more powerful than all the other existing tests. Moreover, we develop a large sample theory general enough to deal with randomly drifting parameter spaces, which is essential to properly test for a unit root against stationary transitional models. An empirical application of our test to the exchange rate data is also provided.

1.

INTRODUCTION

In many economic models, the economic agents face some types of costs that prevent an instantaneous adjustment of variables toward their long-run equilibrium levels. As a result of comparing the cost and benefit by the agents, the speed of adjustment naturally depends on the size of deviation from the equilibrium. Empirically, such adjustment dynamics can be conveniently described by a stationary transitional autoregressive (AR) model that allows transition from one regime with a faster adjustment to the other regime with a slower adjustment. This class of the model, however, is known to be difficult to be discriminated from the unit root model, that is, the model with no long-run equilibrium. In particular, the poor power performance of the standard unit root test against transitional AR models has been reported by many studies, including Balke and Fomby (1997), Taylor (2001), and Taylor et al. (2001). In this article, we develop a novel test for the unit root model against the alternative of a variety of transitional AR models. Our test, called the inf-t test, is based on the infimum of t-ratios for the coefficient on the cross-product of a lagged variable and the transition function taken over all possible values of the parameter that is identified only under the alternative. Our framework is very general and accommodates virtually all potentially interesting models with the threshold, discrete and smooth transition functions, including as special cases all the models considered previously in the literature such as the threshold autoregressive (TAR) models, exponential smooth transition autoregressive (ESTAR) models, and logistic smooth transition autoregressive (LSTAR) models. Moreover, we only require very mild assumptions on the short-run dynamics, allowing the underlying time series to be generated as a general linear process driven by the martingale difference innovations with conditional heteroskedasticities. ∗ Manuscript

received September 2009; revised August 2014. We are very grateful to Frank Schorfheide and three anonymous referees for many helpful comments. We also thank Yoosoon Chang, Robert de Jong, Walt Enders, Emmanuel Guerre, Lutz Kilian, Vadim Marmer, and Ingmar Prucha, and seminar and conference participants at Kyoto University, Rice University, University of British Columbia, University of Maryland, University of Michigan, the 9th World Congress of the Econometric Society, and the 17th SNDE Conference for their helpful comments and useful discussions. Earlier versions of this article have been circulated since 2005. Park gratefully acknowledges the financial support from the National Research Foundation of Korea grant funded by the Korean Government (NRF-2012S1A5A2A01020692). Shintani gratefully acknowledges the financial supports by the JSPS KAKENHI Grant No. 26285049 and by the NSF Grant No. SES-0524868. Please address correspondence to: Joon Y. Park, Department of Economics, Indiana University, 100 S. Woodlawn, Bloomington, IN 47405-7104. E-mail: [email protected]. 1

635  C

(2016) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social and Economic Research Association

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An arbitrary lag delay is also permitted in the transition function. For such a large class of transitional AR models, we fully establish the large sample theory of our inf-t test. Our asymptotics pose new technical problems, since they require the weak convergence of a sequence of random functions involving unit root processes and random parameter spaces with unbounded support. It is therefore necessary to appropriately normalize the parameter spaces, so that they have well-defined distributional limits under the unit root hypothesis. Fortunately, this is generally possible for a wide variety of transitional AR models. The test for a unit root in transitional AR models has recently been investigated by many authors. They are all, however, restricted to some special class of models and do not provide ´ appropriate statistical theory for general transitional AR models. For example, both Gonzalez and Gonzalo (1998) and Caner and Hansen (2001) look at the properties of the unit root test against the TAR model. However, the former focuses on the case of a known threshold value, and the latter only considers the case of a stationary transition variable. Enders and Granger (1998) propose a unit root test in the TAR framework with unknown threshold value and a possibly nonstationary transition variable, but they do not provide any theoretical results.2 Sollis et al. (2002) extend the approach of Enders and Granger (1998) to the case of the (second-order) LSTAR model but again without asymptotic theory. Kapetanios et al. (2003) consider the unit root test against the ESTAR model, but their methodology is based on the Taylor approximation of the transition function, which is not as effective as our approach in the article.3 Studies on the three-regime TAR models by Bec et al. (2004), Kapetanios and Shin (2006), and Bec et al. (2008) are more closely related to our approach in the article. However, as we discuss later in detail, their results rely on some restrictive assumptions, which may not be valid in general. For them, we provide a more adequate asymptotic theory to properly deal with randomly drifting parameter spaces. Through simulations, we show the good finite sample performance of the inf-t test. Moreover, the test is found to be considerably more powerful than the other existing tests when the data are generated from stationary transitional models.4 As an empirical illustration, we test for a unit root in the target zone exchange rate model using the inf-t test. Our empirical results imply that, for some economic and financial time series, the stationary transitional autoregressions can be much more plausible alternatives to the random walk models than the usual stationary linear autoregressions. The rest of the article is organized as follows. Section 2 provides some motivating examples. It introduces two prototypical models and the hypotheses to be tested, which are followed by some important discussions on the normalization and parameter space. The assumptions on the transition functions and the preliminary asymptotic results are given in Section 3. In Section 4, the asymptotic null distribution of the test statistic is obtained and its nuisance parameter dependency is analyzed. The test consistency is also established. Section 5 provides the results of Monte Carlo experiments designed to evaluate the finite sample performance of the proposed test. An empirical application to the target zone model is also included. Section 6 concludes the article. Some useful lemmas and mathematical proofs of main theorems are provided in the Appendix.5 2 Within the TAR framework, some theoretical results are obtained by Berben and van Dijk (1999). Recent studies by de Jong et al. (2007) and Seo (2008) also consider the TAR model with a more general serial correlation structure. 3 Note that both the linearity test and the unit root test against the transitional AR model are subject to the Davies problem since threshold and transition parameters are not identified under the null. Using Taylor approximation has been successful in the linearity test under the stationary framework. See van Dijk et al. (2002). 4 The recent simulation study by Choi and Moh (2007) also clearly demonstrates that our test has desirable power properties against a large class of nonlinear stationary alternatives. 5 Complete proofs of other theorems, as well as related lemmas, corollaries, and propositions, are provided in an online technical appendix.

TRANSITIONAL AUTOREGRESSIVE MODELS

2.

637

MOTIVATING EXAMPLES: EQ-TAR AND ESTAR MODELS

We first look at two prototypical versions of the transitional AR model for the purpose of illustrating the basic strategy employed in the article. Let us consider a model that involves the transition between two regimes with two first-order AR parameters ρ1 and ρ2 and is given by (1)

yt = ρ1 yt−1 π(yt−d , θ) + ρ2 yt−1 {1 − π(yt−d , θ)} + εt ,

where π(yt−d , θ) ∈ [0, 1] is a known transition function using a lagged dependent variable, yt−d , as a transition variable, d is a known positive integer lag delay parameter, θ is an unknown parameter of transition function, and εt is an i.i.d. white noise error term with variance σ2 . With a U-shaped transition function, the AR parameters ρ1 and ρ2 , respectively, represent the speed of adjustment in the outer regime and in the inner regime. In what follows, we tentatively set 0 < ρ1 < 1 and ρ2 = 1 to reflect a commonly used assumption in the empirical literature that there is no adjustment in the inner regime.6 In the simple trade cost model with fixed trade cost μ, price deviations between two locations are corrected by arbitrage only if they exceeds μ. In such a case, we can consider a three-regime TAR process with a two-sided abrupt transition function defined as (2)

π(yt−d , θ) = 1{|yt−d | ≥ μ},

where θ = μ > 0. Balke and Fomby (1997) referred to the model as the Equilibrium-TAR (EQTAR) model (see Figure 1a). Throughout this article, we classify the threshold value μ as a type of “location” parameter. In addition to the application to the law of one price deviations, the model is useful in describing the monetary policy intervention, including exchange rate target zones, and spread between the Fed Funds rate and discount rate (Balke and Fomby, 1997, p. 628). The ESTAR model is a convenient parsimonious specification to introduce a smooth transition, rather than an abrupt transition, based on an exponential transition function defined as

(3)

π(yt−d , θ) = 1 − e−κ

2 2 yt−d

,

where θ = κ > 0 (see Figure 1b). A single parameter κ controls the degree of smoothness in the transition, and we classify it as a type of “scale” parameter in this article. This model has often been employed in the analysis of the aggregate real exchange rate dynamics including the studies by Michael et al. (1997), Taylor et al. (2001), and Kilian and Taylor (2003) among others. Kapetanios et al. (2003) recommend using a unit root test based on Taylor-series expansion of the ESTAR model specification. For both EQ-TAR model and ESTAR model, (1) with the restriction ρ2 = 1 can be rewritten as (4)

yt = λyt−1 π(yt−d , θ) + εt ,

where λ = ρ1 − 1. When −1 < λ < 0, model (4) with a given value of θ = μ or θ = κ represents a transitional AR model with a state-dependent speed of adjustment. However, if λ = 0, there will be only a single regime with a unit root that represents no adjustment toward a long-run equilibrium. For this reason, in model (4), it is of interest to test the null hypothesis (5)

H0 : λ = 0

6 Assumptions of the U-shaped transition function with its range [0, 1] and a partial unit root in one regime are not necessary in our subsequent theoretical development.

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1.0

(a) EQ-TAR model

0.8 0.6 0.4 0.2 0.0 1.0

(b) ESTAR model

0.8 0.6 0.4 0.2 0.0 1.0

(c) EQ-LSTAR model

0.8 0.6 0.4 0.2 0.0

FIGURE 1 TRANSITION FUNCTIONS

against the alternative hypothesis H1 : λ < 0.7 To test the hypothesis of our interest, we rely on the extremum type test statistics regarding the parameter θ, which is not identified under the null hypothesis of a unit root. For a given choice of the parameter value for θ, the model can be estimated by running the regression of the form (6)

yt = λwt (θ) + εt ,

where wt (θ) = yt−1 π(yt−d , θ). The test statistic can be constructed by computing the infimum of the t statistic of the least squares estimator  λn (θ) for each possible value of the parameter θ given by (7)

 λn (θ) , θ∈ n s( λn (θ))

T n = inf T n (θ) = inf θ∈ n

where n is the parameter space of θ and s( λn (θ)) is the standard error of  λn (θ). In the following sections, we show that the limiting distribution of this inf-t statistic is free from nuisance parameters and depends only on the choice of transition function π(yt−d , θ) along with the parameter space n . Regarding the choice of n , we follow the usual convention in practice and set the parameter space to be randomly given by the functions of the data. In case of the location parameter μ, it is customary to set the parameter space to be given by some fixed percentiles in the range of 7 The alternative hypothesis H : λ < 0 is not necessarily equivalent to the stationarity of (y ). In general, we need t 1 extra conditions to ensure that (yt ) is stationary. See, for example, Chan et al. (1985).

TRANSITIONAL AUTOREGRESSIVE MODELS

639

(y1 , . . . , yn ) or (|y1 |, . . . , |yn |). For example, as suggested by Caner and Hansen (2001), we can set the parameter space as (8)

n = [Qn (15), Qn (85)],

where Qn (15) and Qn (85) are, respectively, the 15th and 85th percentiles of (y1 , . . . , yn ) or (|y1 |, . . . , |yn |). Whether we use (y1 , . . . , yn ) or (|y1 |, . . . , |yn |) depends on the type of location it is usually searched over some fixed interval parameter.8 In case of the scale parameter κ,  ¯ 2 /n)1/2 of the data (y1 , . . . , yn ). For normalized by the sample standard deviation ( nt=1 (yt − y) example, similar to the choice of van Dijk et al. (2002) for their study of stationary smooth transition AR models, we can search κ over the interval given by (9)

n = [10−2 Pn , 102 Pn ],

 where Pn = ( nt=1 (yt − y) ¯ 2 /n)−1/2 . In the subsequent development of our theory, we mainly consider the parameter space for μ and κ given by (8) and (9). The other specifications are, of course, also possible, and we may easily modify our results to accommodate them. When (yt ) is stationary, both (8) and (9) yield a well-defined sequence ( n ) of parameter spaces contained for all large n in a compact set. This is not so, however, if (yt ) has a unit root. The sequence ( n ) of parameter spaces for the location parameter expands √ at the rate of √ n, whereas for the scale parameter it shrinks toward the origin at the rate of n. Therefore, the usual assumption on the compactness of the parameter space is fatally inadequate for the development of asymptotic theory. In the limit, the assumption simply cannot be met for the location parameter μ since its parameter space becomes unbounded, and it yields degenerate asymptotics with respect to the scale parameter κ that has the parameter space with empty interior. To clarify the role of random parameter space in the limit null distribution of our test statistic, we introduce the normalized parameters (θn ) and the corresponding normalized parameter space ( nn ). In case of the EQ-TAR model, arguments in the transition function can be rewritten as       yt−d  μ yt−d = π √ , θn , (10) π(yt−d , θ) = 1{|yt−d | ≥ μ} = 1  √  ≥ √ n n n √ where θn = μn = μ/ n. The corresponding normalized parameter space is given by nn = n −1/2 [Qn (15), Qn (85)]. Similarly, in case of the ESTAR model, arguments in the transition function can be rewritten as      2 2

√ 2 yt−d 2 yt−d (11) π(yt−d , θ) = 1 − exp −κ yt−d = 1 − exp −( nκ) √ = π √ , θn , n n √ where θn = κn = nκ. The corresponding normalized parameter space is given by nn = n 1/2 [10−2 Pn , 102 Pn ]. Note that our normalization of parameter and parameter space here has no bearing on the actual implementation of our testing procedure because searching θ over

n and searching θn over nn are indifferent in the computation of inf-t statistic (7). For the location parameter, μ lying in an interval given by certain percentiles of (y1 , . . . , yn ) or is equivalent to an√interval given by the corresponding per(|y1 |, . . . , |yn |) √ √ to μn belonging √ centiles of (y1 / n, . . . , yn / n) or (|y1 | n, . . . , |yn | n). For the scale parameter, requiring κ to n y) ¯ 2 /n)1/2 amounts to assuming κn to be in the be in a fixed interval normalized n by ( t=12(yt 2−1/2 ¯ /n ) . Therefore, our normalization of parameter same interval normalized by ( t=1 (yt − y) 8 In case of the EQ-TAR model under consideration, an obvious choice would be (|y |, . . . , |y |) instead of n 1 (y1 , . . . , yn ).

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and parameter space here is just an instrument to obtain proper asymptotics. Under the null hypothesis of a unit root, we have nn →d , where is a well-defined random interval that is compact almost surely.9 Using these results on the parameter space, the limiting distribution of the inf-t statistic (7) for the case of EQ-TAR model can be derived as 

 W(r)1 W(r) ≥ μ dW(r)  , 

2  1   0 W(r)1 W(r) ≥ μ dr

1 (12)

0

inf

μ∈[Q(15),Q(85)]

where Q is the quantile function for a standard Brownian motion W or |W| over r ∈ [0, 1], that

1

1 is, 0 1{W(r) ≤ Q(100s)} dr = s or 0 1{|W(r)| ≤ Q(100s)} dr = s for s ∈ [0, 1]. Similarly, for the case of the ESTAR model, the distribution can be expressed as

1 (13)

inf

κ∈[10−2 P,102 P]

  2 dW(r) 0 W(r) 1 − exp −κW(r)  ,

1

  2 2 dr 0 W(r) 1 − exp −κW(r)

1

1 where P = ( 0 (W(r) − 0 W(s)ds)2 dr)−1/2 . For both cases, the distribution does not depend on unknown parameters, and the critical values can be tabulated. At the 5% significance level, the critical values of the EQ-TAR-based test and ESTAR-based test are −2.83 and −2.28, respectively. We reject the unit root in favor of transitional AR models if the test statistic (7) takes values below these critical values.

3.

THE MODEL AND ASSUMPTIONS

3.1. The General Model. Let us now consider a broader class of transitional AR models, which will be covered in our analysis. To test the null hypothesis H0 : λ = 0 against an alternative hypothesis H1 : λ < 0, we can generalize the regression model (6) introduced in the previous section and consider yt = λwt (θ) +

(14)

p 

αi yt−i + εt ,

i=1

where the first regressor is given by wt (θ) = yt−1 π(yt−d , θ) and additional regressors yt−i ’s are now included to allow for the possible serial correlation under the null hypothesis of a unit root that is analogous to the augmented Dickey-Fuller (ADF) regression. Note that, under the null hypothesis, we have yt = ut , with its error term following the linear AR model given by α(L)ut = εt ,

(15) where α(z) = 1 − If we define

(16)

p k=1

αk zk .

An (θ) =

n  t=1

wt (θ)yt −

n  t=1

wt (θ)x t

 n  t=1

−1 xt x t

n 

xt yt ,

t=1

9 This random interval, in general, depends on the unknown parameter σ 2 . However, we can easily make this random interval free of nuisance parameter by further renormalizing the parameters and parameter space using the variance σ 2 so that we may effectively set σ2 = 1 without loss of generality.

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(17)

Bn (θ) =

n  t=1

w2t (θ)

−

n 

wt (θ)x t

t=1

 n 

−1 xt x t

t=1

n 

xt wt (θ),

t=1

where xt = (yt−1 , . . . , yt−p ) , then, we can express the least squares estimator of λ and its standard error as λˆ n (θ) =  article, we An (θ)/Bn (θ) and s(λˆ n (θ)) = σˆ n (θ)/ Bn (θ), respectively. Here and elsewhere in the  denote the regression residual by εˆ t and the error variance estimate by σˆ n2 (θ) = (1/n) nt=1 εˆ 2t (θ). The inf-t test is defined as in (7), that is, the infimum of the t-ratio (18)

T n (θ) =

λˆ n (θ) , s(λˆ n (θ))

taken over all values of θ ∈ n .10 The null hypothesis of λ = 0, namely, a unit root, is rejected in favor of the transitional AR model if the statistic is smaller than the critical value. Naturally, the parameter θ may be estimated by θˆ n = argmin T n (θ), θ∈ n

and we may also define the inf-t test as T n = T n (θˆ n ). The estimator θˆ n is in general not identical ˜ ˜ ˜ to the nonlinear 2  least squares estimator θn , say, of θ. As is well known, θn is given by θn =  ˆ argmax T n (θ) θ ∈ n . In contrast, we may define θn by 

θˆ n = argmax T n2 (θ) λˆ n (θ) < 0, θ ∈ n , as long as the set of θ ∈ n for which λˆ n (θ) < 0 is nonempty. Under the alternative of stationarity and with suitable regularity conditions, it is well expected that both θˆ n and θ˜ n are consistent. The former, however, will be more efficient than the latter, if the true value of λ is negative. We will not further study the properties of these estimators in the article, since our main purpose is to test for the unit root hypothesis. We have already introduced two popular transitional AR models based on U-shaped transition functions, namely, the EQ-TAR and ESTAR models in the previous section. Another popular class of models relies on S-shaped transition functions, such as the logistic function given by (19)

π(yt−d , θ) = [1 + e−κyt−d ]−1 ,

where θ = κ > 0. Combining (1) with (19) yields the LSTAR model. Unlike the EQ-TAR and ESTAR models, however, the LSTAR model typically considers two stationary regimes by imposing restrictions 0 < ρ1 < 1 and 0 < ρ2 < 1. In particular, the partial unit root assumption

More generally, we may consider tests based on the statistics of the form F (T n (θ))G(dθ) for some function F on R and measure on n . Their asymptotics can be easily obtained from our theory developed subsequently. For the Lagrange multiplier, Wald, and likelihood ratio tests in the standard stationary models, Andrews and Ploberger (1994) indeed show that the optimal tests are of an average exponential form. However, their result is not applicable for our nonstationary model. Here, we consider the inf-t test, since it is more directly comparable to the sup-Wald test that has been used extensively in practical applications. 10

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ρ2 = 1 is usually not imposed in the LSTAR model. In this case, model (1) cannot be expressed in the form of (4). To deal with the models without the partial unit root assumption ρ2 = 1, we may rewrite (1) as (20)



yt = λ γyt−1 π(yt−d , θ) + (1 − γ)yt−1 [1 − π(yt−d , θ)] + εt ,

where λ = ρ1 + ρ2 − 2 and γ = (ρ1 − 1)/(ρ1 + ρ2 − 2) with the restriction 0 < γ < 1. The test for the null hypothesis H0 : λ = 0 against the alternative hypothesis H1 : λ < 0 may then still be based on regression (6) with its first regressor replaced by (21)

wt (γ, κ) = γyt−1 [1 + e−κyt−d ]−1 + (1 − γ)yt−1 [1 + eκyt−d ]−1 .

Obviously, the inf-t statistic (7) should now be computed by searching not only for θ = κ over

n given by (9) but also for γ over [0, 1]. Note that, unlike θ, the parameter space for γ is independent of the data. The ESTAR model does not nest the EQ-TAR model as the limiting case of κ → ∞. Note that the two-sided abrupt transition function 1{|x| ≥ μ} can be written as the sum of 1{x ≤ −μ} and 1{x ≥ μ} and that two indicator functions can be viewed as the limits of two logistic transition functions, [1 + eκ(x+μ) ]−1 and [1 + e−κ(x−μ) ]−1 , as the scale parameter κ(> 0) tends to infinity. Model (4) with a U-shaped transition function, (22)

π(yt−d , θ) = [1 + eκ(yt−d +μ) ]−1 + [1 + e−κ(yt−d −μ) ]−1 ,

where θ = (μ, κ) , thus nests the EQ-TAR model as a special case (see Figure 1c). Because of its relationship to the EQ-TAR model, this model may be called the Equilibrium-LSTAR (EQ-LSTAR) model. A sup-type unit root test against the EQ-LSTAR model has recently been proposed by Bec et al. (2010).11 In the computation of inf-t statistic (7), μ is searched over the parameter space (8) using (y1 , . . . , yn ), and κ is searched over the parameter space (9). More generally, we may allow for regimes by introducing multiple transition functions (πi )  and additional weighting parameters (γi ) where i=1 γi = 1. Additional location parameters (νi ) can also be introduced to represent possibly regime-dependent and nonzero intercepts in mean-reverting regimes. This additional location parameter differs from μ since it appears outside of the transition functions but the same parameter space (8) can be employed. Our testing procedure relies on the inf-t statistic (7) using regression (6) or (14) with wt (θ) replaced by wt (γ, ν, θ) = i=1 γi (yt−1 − νi )πi (yt−di , θ), where γ = (γ1 , . . . , γ ) and ν = (ν1 , . . . , ν ). Using this general representation, our procedure allows for a version of three-regime TAR model where points of attraction correspond to threshold values in two outer regimes by setting

= 2, γ1 = γ2 = 1, ν1 = −μ, ν2 = μ, π1 = 1{yt−d ≤ −μ}, and π2 = 1{yt−d ≥ μ} or (23)

wt (μ) = (yt−1 + μ)1{yt−d ≤ −μ} + (yt−1 − μ)1{yt−d ≥ μ}.

This model is referred to as the Band-TAR (B-TAR) model by Balke and Fomby (1997) and is employed by Obstfeld and Taylor (1997) and Taylor (2001) in their analysis of price adjustment across intranational and international cities. Note that the B-TAR model differs from the EQTAR model since the latter process tends toward zero in two outer regimes instead of threshold values. By setting ν1 = −ν and ν2 = ν, or (24)

wt (ν, μ) = (yt−1 + ν)1{yt−d ≤ −μ} + (yt−1 − ν)1{yt−d ≥ μ},

11 An alternative approach to allow smooth transition in the EQ-TAR model is to consider the second-order LSTAR model with transition function π(x, θ) = [1 + exp{−κ2 (x − μ)(x + μ)}]−1 , where θ = (μ, κ) and μ > 0. See van Dijk et al. (2002).

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TABLE 1 TRANSITIONAL AUTOREGRESSIVE MODELS

Acronym

Equation

(a) Threshold autoregressive (TAR) models EQ-TAR (2) B-TAR (23) D-TAR (24) TAR (26)

π(x) or w(x, y)

Parameter

1{|x| ≥ μ} (y + μ)1{x ≤ −μ} + (y − μ)1{x ≥ μ} (y + ν)1{x ≤ −μ} + (y − ν)1{x ≥ μ} γy1{x ≤ μ} + (1 − γ)y[1 − 1{x ≤ μ}]

μ>0 μ>0 ν, μ > 0 μ, 0 < γ < 1

(b) Exponential smooth transition autoregressive (ESTAR) models 2 2 ESTAR (3) 1 − e−κ x 2 2 ESTAR-M (25) (y − ν)[1 − e−κ (x−ν) ]

κ κ, ν

(c) Logistic smooth transition autoregressive (LSTAR) models EQ-LSTAR (22) [1 + eκ(x+μ) ]−1 + [1 + e−κ(x−μ) ]−1 LSTAR (21) γy[1 + e−κx ]−1 + (1 − γ)y[1 + eκx ]−1

μ > 0, κ > 0 κ > 0, 0 < γ < 1

the model nests both the EQ-TAR and B-TAR models.12 In this article, we call this general three-regime TAR model the Double-TAR (D-TAR) model to emphasize the presence of two abrupt transition functions. We can also consider a possibly nonzero mean ESTAR model by 2 2 setting = 1, γ1 = 1, ν1 = ν, and π1 = 1 − e−κ (yt−d −ν) or (25)

wt (ν, κ) = (yt−1 − ν)[1 − e−κ

2

(yt−d −ν)2

],

which we call the ESTAR-M model. Finally, we may accommodate the standard two-regime TAR model considered by Enders and Granger (1998) in their test for a unit root by setting = 2, γ1 = γ, γ2 = 1 − γ, ν1 = ν2 = 0, π1 = 1{yt−d ≤ μ}, and π2 = 1 − 1{yt−d ≤ μ} or (26)

wt (γ, μ) = γyt−1 1{yt−d ≤ μ} + (1 − γ)yt−1 [1 − 1{yt−d ≤ μ}] .

We will refer to this model simply as the TAR model. The relationships among the transitional AR models provided in Table 1 can be summarized as follows. The D-TAR model specification includes the EQ-TAR and B-TAR models as special cases. Similarly, the EQ-LSTAR model nests the EQ-TAR model whereas the ESTARM model nests the ESTAR model. Our methodology and theory will be developed mainly using the simple model (6) with wt (θ) = yt−1 π(yt−d , θ), which contains the EQ-TAR, ESTAR, and EQ-LSTAR models. As we willshow later, however, they can be readily extended to the general model with wt (γ, ν, θ) = i=1 γi (yt−1 − νi )πi (yt−di , θ), which contains the B-TAR, D-TAR, TAR, ESTAR-M, and LSTAR models. 3.2. Assumptions on Transition Functions. Let us now define the class of transition functions π(yt−d , θ) allowed in our procedure. Here, yt−d is the transition variable with known lag delay d ≥ 1, θ is an m-dimensional parameter, and π is a real-valued function on R × Rm . To obtain a well-defined limit null distribution of our test, we need to introduce the normalized parameters (θn ) and require the corresponding normalized parameter space ( nn ) to have a well-defined distributional limit.

12 Another type of the three-regime TAR model, which is referred to as the returning drift TAR model by Balke and Fomby (1997), imposes the unit root in outer regimes with nonzero drift terms. Such a model may be also considered if we employ wt (ν, μ) = −ν1{x ≤ −μ} + ν1{x ≥ μ}.

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ASSUMPTION 1. We let θn ∈ nn be the normalized parameter defined as (27)

  yt−d π(yt−d , θ) = π √ , θn , n

and assume under the null hypothesis of a unit root that nn →d with an a.s. compact random subset of Rm ,13 jointly with all other weak convergences introduced later. The limit parameter space is a random subset of Rm given by the distributional limit of the random sequence of parameter spaces ( nn ) for the normalized parameter (θn ). Though the limit random parameter set is required to be compact a.s., it is typically not contained in any fixed compact subset of Rm . In virtually all transitional models of potential interests, the transition function π is given as   π(x, θ) =  a(x, μ), b(κx) , where a is homogeneous of degree zero, and μ and κ are, respectively, the location and scale parameters with θ = (μ, κ) . In this case, we may write        x x x = π √ , θn π(x, θ) =  a √ , μn , b κn √ n n n √ √ by defining μn = μ/ n and κn = nκ with θn = (μn , κn ) . Under the conventional choice of the parameter space such as (8) and (9), Assumption 1 is therefore met under the null hypothesis of a unit root for a wide class of transitional AR models, including all models considered in the article. In the previous literature, none of the limit null distributions is obtained under appropriate assumptions on the parameter space. Kapetanios and Shin (2006) derive the limit null distribution of the sup-Wald test in a three-regime TAR model under the assumption of the compact parameter space, that is, the condition comparable to that in Assumption 1 imposed on the unnormalized parameter. Therefore, it is not applicable if the usual convention is followed to set the parameter space. On the other hand, Bec et al. (2004) allow the location parameter space to expand in investigating the sup-Wald, LM, and LR tests for their three-regime TAR model. However, their derivations of the limit null distributions rely critically on the stochastic equicontinuity result established in Bec et al. (2008, Lemma 3), which is valid only for a nonrandom and compact parameter space. The usual parameter space employed in the literature remains to be random in the limit and has support on the entire Rm under the null hypothesis, even if appropriately normalized. The limit theory of the three-regime LSTAR model in Bec et al. (2010) is also based on Bec et al. (2008) for the stochastic equicontinuity and has a similar problem.14 We now introduce the precise conditions for the transition function. To present the required conditions for π, it will be convenient to introduce some preliminary regularity conditions. These we will provide in Definitions 1 and 2 below.

13 To define the weak convergence

nn →d more precisely, we should introduce a topology in the space of subsets of Rm . Here, we use the topology induced by the Hausdorff metric for the space of compact subsets of Rm . In most cases, nn can be written as an m-product of intervals of the form [an , bn ], where (an ) and (bn ) converge weakly to some functionals of Brownian motion, and therefore, we have nn →d in the topology we use here. 14 Bec et al. (2010) simply refer for the proof of their stochastic equicontinuity to Bec et al. (2008), which does not explicitly deal with the three-regime LSTAR model. However, we believe that the required extension is possible as long as the normalized parameter space has a nonrandom limit.

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645

DEFINITION 1. A transformation υ on R is said to be regular if and only if, for any compact subset K of R given, there exists a δ-sequence of continuous

functions (υδ ) and (υδ ) such that υδ (x) ≤ υ(y) ≤ υδ (x) for all |x − y| < δ on K, and such that K (υδ − υδ )(x) dx → 0 as δ → 0. The regularity condition in Definition 1 is satisfied by a large class of functions including all continuous and piecewise continuous functions as noted by Park and Phillips (2001), who introduced it earlier in their study of nonlinear regressions with integrated time series.15 Here and elsewhere in the article, · denotes the usual Euclidean norm if applied to a vector. DEFINITION 2. A functional  on R × Rm is said to be regular if and only if (a) for all θ0 ∈ Rm , (·, θ0 ) is regular, and (b) for each θ0 ∈ Rm given, there exists δ-sequence of regular functions (δ ) and (δ ) such that δ (x) ≤ (x, θ) ≤ δ (x) for all θ − θ0 < δ, and such that δ (x) − δ (x) → 0 as δ → 0 a.e. x ∈ R. The regularity conditions in Definition 2 are comparable to those in Park and Phillips (2001). Our conditions are, however, weaker than theirs, especially in that ours allow for the functionals that are discontinuous in θ ∈ Rm . This is necessary to accommodate the indicator type transition functions in our model. Now we are ready to introduce the conditions required for π. ASSUMPTION 2. We assume that π is given by either (a) a simple function in x ∈ R over a union of intervals defined by θi ’s, θ = (θi ) ∈ Rm , or (b) a bounded function on R × Rm such that π(·, θ0 ) is regular for all θ0 ∈ Rm , and |π(x, θ1 ) − π(x, θ2 )| ≤ π(x) θ ¯ 1 − θ2 for all θ1 , θ2 ∈ Rm with some regular function π¯ such that π(x) ¯ ≤ ec|x| for some c > 0 as |x| → ∞. The conditions in Assumption 2 are not stringent and satisfied for virtually all transition functions that can possibly be used in practical applications. Clearly, all indicator functions such as 1{x ≤ μ} and 1{|x| ≥ μ} satisfy part (a) of Assumption 2 Moreover, part (b) of Assumption 2 holds widely for π that is differentiable with respect to θ. For instance, it is satisfied if ˙ θ)| ≤ π(x) ¯ for a regular function π¯ increasing at a polynomial rate, where π(x, ˙ θ) = supθ∈Rm |π(x, (∂/∂θ)π(x, θ). LEMMA 1. Under Assumption 2, π is regular. Assumption 2 is therefore sufficient to ensure that the transition function π is regular as a functional on R × Rm . 3.3. Assumptions on Errors. For the subsequent development of our asymptotic theories, we make the following assumptions for the innovations (εt ) generating the errors (ut ) in (15). ASSUMPTION 3. (εt , Ft ) is a martingale difference sequence, with some filtration (Ft ), such that (a) E ε2t = σ2 for all t ∈ Z, (b) supt∈Z E|εt |r < ∞ for some r ≥ 4, and  (c) sup1≤i≤n n1 it=1 [E(ε2t |Ft−1 ) − σ2 ] → p 0. 15 They also require that the regular function be continuous at infinity. This condition, however, is unnecessary and not invoked here.

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Assumption 3 is quite general and allows for the conditional heteroskedasticity in the innovation sequence (εt ). In the unit root literature, it is routinely assumed that the fourth conditional moment of the innovation sequence is bounded. See, for example, Stock (1994) and Park and Phillips (2001). We do not impose this uniform boundedness of the conditional fourth moment, which is not satisfied, for instance, by the usual ARCH processes. Our assumption here holds for a wide class of ARCH-type processes.  As is well expected, the usual (infeasible) variance estimator σn2 = (1/n) nt=1 ε2t is consistent under Assumption 3, as we show in the following lemma: LEMMA 2. Under Assumption 3, we have σn2 → p σ2 . The result in Lemma 2 is essential for the consistent estimation of the error variance. Moreover, conditions (a) and (b) of Assumption 3 are sufficient to ensure that (εt ) satisfies an invariance principle, as shown in, for example, Hall and Heyde (1980, Theorem 2, p. 99). A properly constructed partial sum process of (εt ) would therefore weakly converge to a Brownian motion. For our subsequent theory, however, it would be more convenient to use a more direct method of embedding a distributionally equivalent copy of the partial sum process of (εt ) into a Brownian motion. We achieve this using the so-called Skorokhod embedding, which we state below as a lemma. LEMMA 3. Under Assumption 3, there exists a probability space supporting a Brownian motion U with variance σ2 and a time change (i.e., a nondecreasing sequence of stopping times) τi such that i τ  1  i εt =d U √ n n t=1

for i = 1, . . . , n, and that (28)

   τi i   sup  −  → p 0. n 1≤i≤n n

The reader is referred to Hall and Heyde (1980, Appendix) for the Skorokhod embedding introduced in Lemma 2, and to Park and Phillips (1999, 2001) for its application to the analysis of nonlinear models with integrated processes. In what follows, we set τ0 ≡ 0. ASSUMPTION 4. α(z) has roots outside the unit circle. Then it follows that i i  1 1  ut = √ εt + o p (1) √ n t=1 nα(1) t=1

uniformly in 1 ≤ i ≤ n, as shown in, for example, Phillips and Solo (1992). Consequently, if we define a Brownian motion V by V (r) =

1 U(r), α(1)

TRANSITIONAL AUTOREGRESSIVE MODELS

647

that is, V is a Brownian motion with variance ω2 , where ω2 =

(29)

σ2 , α(1)2

then we have i τ  1  i + o p (1) ut =d V √ n n t=1

uniformly in 1 ≤ i ≤ n. In what follows, we denote by Vn the process defined as Vn (r) =

n  i=1

V

τ

 τ τ   τn  τi  i−1 n 1 +V 1 r≥ ≤r< n n n n n

i−1

with the convention τ0 = 0 a.s. Clearly, Vn →d V on [0, 1], due in particular to our result in (28). We now establish a functional central limit theory for the nonlinear transformation of integrated time series that can be applied to a class of transition functions introduced in this section. The main results of this section are given below. In what follows, we let 

r

Mn (r, θ) =

 π(Vn (s), θ) dU(s)

M(r, θ) =

0

r

π(V (s), θ) dU(s)

0

be the stochastic processes defined on (r, θ) ∈ [0, 1] × Rm . THEOREM 1. Under Assumptions 1, 2, 3, and 4, we have Mn →d M jointly with Vn →d V . PROPOSITION 1. M has a modification which is almost surely continuous. The functional central limit theory established in Theorem 1 is crucial in developing our subsequent asymptotic theories.16 Due to Proposition 1, we may assume without loss of generality that the limit process M has a.s. continuous sample paths on [0, 1] × Rm , by taking such a modification if necessary. This convention will be made throughout the article. As a consequence, we may interpret Mn →d M in Theorem 1 as the weak convergence in the space C([0, 1] × Rm ) endowed with the usual topology of uniform convergence on compact subsets, where C(D) denotes the set of continuous functions defined on D. The reader is referred to, for example, Revuz and Yor (1994, p. 487) for more details on the topology we use here.17 It is important to note that the space for the parameter θ in the processes Mn and M is not restricted to be a compact space. This is crucial for our asymptotic theories, which allow for the parameter space that remains to be random in the limit. 16 Note that (M ) and M are defined on the unbounded support [0, 1] × Rm , and we have M → M on any compact n n d subset of [0, 1] × Rm , that is, [0, 1] × [−K, K] for any K > 0. 17 More precisely, we need the product topology of uniform convergence on compact subsets here. Since the topology of uniform convergence on compact subsets coincides with the conventional uniform topology over any compact set, we may just endow C([0, 1]) with the uniform topology.

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4.

ASYMPTOTIC THEORY

4.1. Main Results. To obtain the limit null distribution of our test statistic T n , we first show LEMMA 4. Let Assumptions 1, 2, 3, and 4 hold. Then, under the null hypothesis H0 : λ = 0, we have   n  1     (30) yt−1 π(yt−d , θ)ut−i  → p 0 sup  3/2   n θ∈ n t=1

for i = 1, . . . , p . The result in Lemma 4 establishes the asymptotic orthogonality of the leading transition term and the lagged differences. This is not surprising and indeed well expected from the regression theory for the unit root models. Note that the orthogonality in Lemma 4 applies uniformly in θ ∈ n . The inclusion of the lagged differences would thus have no effect on the testing for the transition term over all possible values of the transition parameter θ ∈ n . It simply washes away the serial correlation in the innovations. The limit distribution of our inf-t statistic in (7) may now be easily deduced from Theorem 1 and Lemma 4. Throughout the article, W signifies the standard Brownian motion. THEOREM 2. Let Assumptions 1, 2, 3, and 4 hold, and let T n be defined as in (7). Under the null hypothesis H0 : λ = 0, we have T n = inf T n (θ) →d T = inf T (θ), θ∈ n

θ∈

where T (θ) is a stochastic process defined by  (31)

1

T (θ) =  0 

(ωW(r), θ) dW(r) 1/2

1

2 (ωW(r), θ) dr

0

with (x, θ) = xπ(x, θ). The distribution of T , in general, depends upon the nuisance parameter ω, which is the long-run variance of (ut ) introduced in (29), as well as the transition function π and the limit parameter space . The dependency of the distribution of T on ω, however, is very simple to deal with, and we may easily get rid of it for most of the transition functions used in practical applications. This is shown in the following corollary. ∗ ∗ θ) = π(x, not depend upon COROLLARY 1. Suppose ∗ ∗that π(ωx,

θ (ω, θ)) for some θ , which does ∗ ∗ x ∈ R. If we let = θ |θ = θ (ω, θ), θ ∈ , then we have T = infθ∗ ∈ ∗ T ∗ (θ∗ ), where



(32)

1

(W(r), θ∗ ) dW(r) T (θ ) =  1/2  1 2 ∗  (W(r), θ ) dr ∗

∗

0

0

with  defined as in Theorem 2.

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TABLE 2 ASYMPTOTIC CRITICAL VALUES OF INF-T TESTS

Probability of a Smaller Value Test Statistic

1%

5%

10%

25%

50%

75%

90%

95%

99%

(a) TAR model-based test inf-tEQ-TAR −3.33 inf-tB-TAR −3.23 inf-tD-TAR −3.84 inf-tTAR −3.47

−2.83 −2.71 −3.24 −2.94

−2.55 −2.44 −2.95 −2.69

−2.12 −2.00 −2.46 −2.26

−1.65 −1.54 −1.94 −1.82

−1.16 −1.07 −1.43 −1.41

−0.44 −0.32 −0.85 −1.01

0.07 0.25 −0.38 −0.64

0.94 1.14 0.55 0.10

(b) ESTAR model-based test −2.86 inf-tESTAR −3.78 inf-tESTAR-M

−2.28 −3.29

−1.98 −3.02

−1.52 −2.59

−1.02 −2.16

−0.39 −1.75

0.31 −1.43

0.74 −1.25

1.51 −0.94

(c) LSTAR model-based test inf-tEQ-LSTAR −3.30 inf-tLSTAR −3.19

−2.76 −2.64

−2.47 −2.33

−2.03 −1.86

−1.55 −1.38

−1.07 −0.97

−0.33 −0.57

0.17 −0.02

1.02 0.96

NOTES: Based on discrete approximation to the Brownian motion by partial sums of standard normal random variable with 1000 steps and 10,000 replications.

If the conditions in Corollary 1 are met, the asymptotic critical values of our inf-t test based on T n depend only upon the transition function π and the limit parameter space . Especially, the dependency of the distribution of T on the long-run variance of (ut ) disappears. The conditions hold virtually all the transition functions used in practical applications. For the models in Table 1 with parameters θ = μ (EQ-TAR model), θ = κ (ESTAR model), and θ = (μ, κ) (EQLSTAR model), we may easily see that they are satisfied for θ∗ = μ∗ , θ∗ = κ∗ , and θ∗ = (μ∗ , κ∗ ), respectively, with μ∗ = μ/ω and κ∗ = ωκ. The actual implementation of our test is fairly simple, due in particular to the result in Corollary 1. Let W be the standard Brownian motion given by V = ωW. Also, let QV be

1 the quantile function for V or |V | over r ∈ [0, 1], that is, 0 1{V (r) ≤ QV (100s)} dr = s or

1

1

1 2 −1/2 . Then we 0 1{|V (r)| ≤ QV (100s)} dr = s for s ∈ [0, 1], and PV = ( 0 (V (r) − 0 V (s)ds) dr) may easily deduce that [QV (15), QV (85)] = ω[Q(15), Q(85)] and [10−2 PV , 102 PV ] = (1/ω)[10−2 P, 102 P]. Therefore, if we denote by μ∗ and κ∗ the redefined location and scale parameters that are given by μ∗ = μ/ω and κ∗ = ωκ, their parameter spaces are given by [Q(15), Q(85)] and [10−2 P, 102 P], respectively. Consequently, the limit null distribution of the inf-t test depends only upon the transition function and the limit parameter space. The critical values for the inf-t tests with the parameter spaces in (8) and (9) are tabulated in Table 2 for EQ-TAR, ESTAR, and EQ-LSTAR models in Table 1. It may be worth emphasizing once again that they are invariant with respect to the long-run variance of (ut ), but dependent upon the limit parameter space in a very critical manner. We now establish the consistency of our inf-t test, which is given in the following proposition. PROPOSITION 2. Suppose that (yt ) is a stationary process with finite second moment, and that there exists a sequence θ¯ n ∈ n such that plimn→∞ λˆ n (θ¯ n ) < 0 and the conditional variance of (yt−1 π(yt−d , θ¯ n )) given (yt−1 , . . . , yt−p ) is nonzero. Moreover, let σˆ n (θ) be bounded away from zero a.s. uniformly in θ ∈ n and all large n. Then we have T n → p −∞.

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The conditions in Proposition 2 are of course satisfied if (yt ) is indeed a stationary process generated by the transitional AR model (4) with the error sequence given by (15). In this case, we have in particular λˆ n (θ0 ) → p λ0 < 0, where we denote by θ0 and λ0 the true values of θ and λ, respectively. Proposition 2 shows that the inf-t test is also consistent against other stationary models as long as for all large n there are parameter values θ¯ n ∈ n to fit them with transitional ˆ θ¯ n ) < 0. AR models (4) with some λ( 4.2. Models with Multiple Transition Functions. Now we consider the general model, which includes the rest of models in Table 1 as special cases. The null hypothesis H0 : λ = 0 can be tested in this model based on regression (14) with wt (θ) replaced by (33)

wt (γ, ν, θ) =



γi (yt−1 − νi )πi (yt−d , θi ).

i=1

 We let γ = (γ1 , . . . , γ ) , ν = (ν1 , . . . , ν ) , and θ = (θ1 , . . . , θ ) , and set i=1 γi = 1 for identification and γi ≥ 0 for all i = 1, . . . , . Also, we denote by n , n , and n the sequences of random parameter spaces given, respectively, for γ, ν, and θ as functions of (y1 , . . . , yn ). Under the null hypothesis of a unit root, we follow (27) and write the transition term (33) as

(34)

wt (γn , νn , θn ) =

 i=1

 γni

   yt−1 yt−d √ − νni πi √ , θni , n n

where (γn , νn , θn ) ∈ (nn , nn , nn ) with γn = (γn1 , . . . , γn ) , νn = (νn1 , . . . , νn ) , and θn =



, . . . , θn ) . We assume (θn1 ASSUMPTION 5. (nn , nn , nn ) →d (, , ), where (, , ) is a compact subset of R + × R × Rm a.s.

Assumption 5 is entirely analogous to Assumption 1 with the same notation used for the distributional convergence of the sequence of random parameter spaces. The motivation for the definition of the transition term in (34) is precisely the same as the one given Note √ for (27). √ that the normalized parameters (νni ) and (γni ) in (34) are given by νni = νi / n and γni = nγi in terms of the unnormalized original parameters (νi ) and (γi ), for i = 1, . . . , . ASSUMPTION 6. πi satisfies the conditions in Assumption 2 for i = 1, . . . , . Like the parameter θ, the parameters γ and ν are identified only under the alternative hypothesis. Therefore, the infimum of the t-ratio should now be taken with respect to γ and ν as well as θ. If we let An (γ, ν, θ) and Bn (γ, ν, θ) be defined similarly as in (16) and (17) with wt (θ) replaced by wt (γ, ν, θ), then the t-ratio is given for each value of (γ, ν, θ) by T n (γ, ν, θ) =

λˆ n (γ, ν, θ) , s(λˆ n (γ, ν, θ))

 where λˆ n (γ, ν, θ) = An (γ, ν, θ)/Bn (γ, ν, θ) and s(λˆ n (γ, ν, θ)) = σˆ n (θ)/ Bn (γ, ν, θ) with the usual error variance estimate σˆ n2 (θ) introduced earlier. The inf-t test may then be defined as (35) in place of (7).

Tn =

inf

(γ,ν,θ)∈n ×n × n

T n (γ, ν, θ)

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TRANSITIONAL AUTOREGRESSIVE MODELS

Unless there are reasons to specify otherwise, it seems natural to also set the parameter spaces for the parameters νi s and γi s to be exactly as those for the parameters μi s and κi s, respectively, in (8) and (9). Note that νi s and γi s also designate the location and scale of the transition functions. We use these parameter spaces in our simulations and empirical applications. THEOREM 3. Let Assumptions 3, 4, 5, and 6 hold, and let T n be defined as in (31). Under the null hypothesis H0 : λ = 0, we have Tn =

inf

(γ,ν,θ)∈n ×n × n

T n (γ, ν, θ) →d T =

inf

(γ,ν,θ)∈××

T (γ, ν, θ)

as n → ∞, where T (γ, ν, θ) is a stochastic process defined similarly as in (31) with (ωW(r), θ) replaced by (ωW(r), γ, ν, θ) for (x, γ, ν, θ) = i=1 γi (x − νi )π(x, θi ). The results in Theorem 3 are comparable to those in Theorem 2. The distribution of T given in Theorem 3 depends upon the nuisance parameter ω, just as that of T in Theorem 2, as well as the transition function π and the limit parameter space  ×  × . The dependency on ω of the limit distribution of T , however, can easily be dealt with similarly as in Corollary 1. The following corollary extends our results in Corollary 1. COROLLARY 2. Suppose that π(ωx, θ) = π(x, θ∗ (ω, θ)) θ∗ , which does not de ∗for some ∗ ∗ ∗ ∗ ∗ ∗ pend upon x ∈ R. If we let = θ |θ = θ (ω, θ), θ ∈ ,  = {ν |ν = ν/ω, ν ∈ }, and ∗ = {γ ∗ |γ ∗ = ωγ, γ ∈ }, then we have T = inf(γ ∗ ,ν∗ ,θ∗ )∈∗ ×∗ × ∗ T ∗ (γ ∗ , ν∗ , θ∗ ), where T ∗ (γ ∗ , ν∗ , θ∗ ) is a stochastic process defined similarly as in (32) with (W(r), θ∗ ) replaced by (W(r), γ ∗ , ν∗ , θ∗ ) for  defined as in Theorem 3. Corollary 2 implies that the critical values of our inf-t test are only dependent upon the transition function and the limit parameter space for a wide class of transitional AR models. This is exactly identical to what is implied by Corollary 1. Table 2 tabulates the critical values for all the remaining models listed in Table 1. Finally, the consistency of the inf-t test can be established for the general models exactly as in Proposition 2. It is, however, trivial, and we do not state it as a separate proposition to save the space. 4.3. Models with Errors of Unknown Form. It is possible to further accommodate a broader class of transitional AR models in many other directions. In particular, we may show that our tests are valid for the models with (ut ) driven by a general linear process. Now we specify (ut ) as (36)

ut = ϕ(L)εt =

∞ 

ϕi εt−i ,

i=1

where we assume the following. ASSUMPTION 7. Let ϕ(z) = 0 for all |z| ≤ 1, and ϕk = O(k−s−δ ) for some s ≥ 2 and δ > 0. With the coefficient summability condition in Assumption 7, condition (a) of Assumption 3 implies that (ut ) is weakly stationary. Moreover, condition (b) of Assumption 3 guarantees that the r th moment of (ut ) exists and is uniformly bounded for all t ∈ Z, due to the MarcinkiewiczZygmund inequality in, for example, Stout (1974, Theorem 3.3.6). Due to the so-called Beveridge-Nelson decomposition, we may write (ut ) given in (36) as ut = ϕ(1)εt + (u˜ t−1 − u˜ t ) ,

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where u˜ t =

∞ 

ϕ˜ i εt−i , ϕ˜ i =

∞ 

ϕj .

j =i+1

i=0

 ˜ i | < ∞ as shown in Phillips and Solo (1992), and therefore Under Assumption 7, we have ∞ i=0 |ϕ (u˜ t ) is well defined both in a.s. and an Lr sense (see Brockwell and Davis, 1991). Moreover, we have i i 1  1  ut = ϕ(1) √ εt + o p (1) √ n t=1 n t=1

uniformly in 1 ≤ i ≤ n. Therefore, the asymptotics in Theorem 1 hold also in this case with a new Brownian motion V (r) = ϕ(1)U(r) with a variance ω2 = σ2 ϕ(1)2 .  i Under Assumptions 3 and 7, we may let α(L)ut = εt with α(z) = 1 − ∞ i=1 αi z , and approximate (ut ) in r th mean by a finite order AR process ut = α1 ut−1 + · · · + α p ut−p + εt,p with εt,p = εt +

∞ 

αi ut−i .

i=p +1

As is well known (see,  e.g., Brillinger, 1975), the condition in Assumption 7 implies that αk = −(s−1) ). Given the existence of the r th moment of (ut ) O(k−s−δ ), and we have ∞ i=p +1 |αi | = o(p implied by Assumptions 3 and 5, we therefore have ⎛ E|ε p,t − εt |r ≤ E|ut |r ⎝

∞ 

⎞r |αi |⎠ = o(p −r(s−1) ).

i=p +1

The approximation error thus becomes small as p gets large. It is now well expected that our tests are valid for the models driven by general linear processes of unknown order under suitable conditions, if we let the order of fitted autoregression increase as the sample size gets large. This is what we will show below.18 We start by introducing assumptions on the order of fitted autoregression and the transition function. For the order of fitted autoregression, we write p = p n to make it explicit that p is a function of the sample size n and assume the following. ASSUMPTION 8. Let p n → ∞ and p n = o(n 1/2 ) as n → ∞. The condition in Assumption 8 is very mild and the same as the one used in Chang and Park (2002) to derive the asymptotics for the ADF unit root t-test. We now consider the model with possibly multiple transition functions satisfying the conditions in the following. ASSUMPTION 9. πi (·, θ) is differentiable for all θ ∈ Rm with derivatives bounded uniformly in θ on any compact subset K of Rm , for i = 1, . . . , . 18 We may also estimate ω2 nonparametrically to allow for the serial correlation of unknown form. Such an alternative approach is pursued by de Jong et al. (2007) in their unit root test against the TAR model.

TRANSITIONAL AUTOREGRESSIVE MODELS

653

The conditions in Assumption 9 are stronger and less general than those introduced originally in Assumption 2. In particular, they do not hold for the class of TAR models. However, they are satisfied for all smooth transition AR models including ESTAR and LSTAR models. LEMMA 5. Let Assumptions 1, 3, 6, 7, and 9 hold. Then, under the null hypothesis H0 : λ = 0, we have  n  1     yt−1 π j (yt−d , θ)ut−i  = Op (1) sup    n θ∈ n t=1

for i ≥ 1 and j = 1, . . . , . The result in Lemma 5 ensures the asymptotic orthogonality between the transition term and all of the lagged differences, the number of which increases as the sample size gets large. It is comparable to the result in Lemma 4. However, the obtained bound for the cross product of the transition function and the differenced lag terms is tighter in Lemma 5, compared with the one given in Lemma 4. This is to allow for the number of the lagged differences to increase. The tighter bound obtained in Lemma 5 requires the differentiability of the transition function introduced in Assumption 9. THEOREM 4. Under Assumptions 3, 5, 6, 7, 8, and 9, Theorem 3 holds. Under appropriate conditions, our previous results on the inf-t tests are therefore applicable for the models driven by general linear processes of unknown order.

5.

SIMULATION AND EMPIRICAL EXAMPLE

5.1. Finite Sample Performance. We conduct simulation experiments to investigate the finite sample performance of the inf-t tests. Since our theory is robust in the presence of the ARCHtype innovations, we first investigate the finite sample √ size property of the inf-t tests using model (6) with λ = 0 and the ARCH(1) error ut = ht εt where (εt ) is i.i.d. N(0, 1) and ht = (1 − α) + αε2t−1 with α ∈ {0.0, 0.5}. We consider inf-t tests based on all transitional AR models listed in Table 1. The parameter spaces are set according to (8) and (9), respectively, for the location and scale parameters.19 Rejection frequencies of the unit root hypothesis are computed using the asymptotic critical values provided in Table 2. All the results reported in this section are based on 10,000 replications and for the nominal 1%, 5%, and 10% tests.20 Table 3 reports the actual rejection frequencies of the inf-t tests for the samples of the sizes n = 100, 200, and 400. The results are summarized as follows: First, in absence of the ARCH effect (α = 0.0), all the inf-t tests seem to have reasonable size properties. Second, when the ARCH effect is present (α = 0.5), the inf-t tests based on ESTAR-M model over reject the null hypothesis somewhat more than other inf-t tests, whereas only the inf-t test based on the EQTAR model tends to under reject the null. The size distortions for all tests, however, become smaller as sample size increases. Next, we evaluate the finite sample powers of the inf-t tests in comparison to other tests.21 To incorporate the differences in size, we focus on the size-adjusted powers. For this purpose, the size-corrected critical values of all tests for the sample sizes of n = 100 and 200 are first computed 19 The parameter space for ν is the same as that for μ, and the parameter space for γ is [0,1]. When a nonnegative restriction is imposed on the location parameter, absolute transformation is applied to the parameter space. The infimum of the t statistic is searched over 10,000 grid points for all the specification. 20 For both size and power evaluations, the initial values are set at y = 0. 0 21 Shintani (2013) examines the finite sample power of the inf-t tests based on the ESTAR model and its variant. His simulation results include a direct comparison of the inf-t test and the test by Kapetanios et al. (2003), which is also based on the same ESTAR specification.

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PARK AND SHINTANI

TABLE 3 SIZE OF INF-T TESTS

inf-tEQ-TAR α 0.0

0.5

inf-tB-TAR

0.0

0.5

inf-t TAR

n

10%

5%

1%

10%

5%

1%

10%

5%

1%

10%

5%

1%

100 200 400 100 200 400

7.9 8.7 10.0 8.6 8.6 9.3

3.7 4.3 4.7 4.6 4.4 4.7

0.9 0.9 0.9 1.4 1.1 1.1

8.3 9.2 9.7 12.5 11.6 11.3

4.3 4.4 5.0 7.8 6.9 6.4

0.8 0.9 0.9 2.8 2.3 1.9

9.1 9.5 9.9 12.3 11.4 11.0

4.6 4.9 5.1 7.6 6.3 6.2

1.0 0.9 0.9 2.5 1.9 1.6

8.7 8.6 9.1 10.8 10.3 9.9

4.8 4.6 4.8 6.9 5.9 5.3

1.0 1.0 0.9 2.2 1.5 1.4

inf-tESTAR α

inf-tD-TAR

inf-tESTAR-M

inf-tEQ-LSTAR

inf-tLSTAR

n

10%

5%

1%

10%

5%

1%

10%

5%

1%

10%

5%

1%

100 200 400 100 200 400

10.0 10.1 10.2 12.3 11.2 10.7

5.2 5.3 5.3 7.3 6.1 5.9

1.2 1.1 1.2 2.4 1.7 1.5

10.7 10.2 10.1 15.4 13.1 11.9

5.7 5.3 5.1 9.6 7.5 6.7

1.5 1.1 1.3 4.0 2.8 2.2

8.9 9.4 10.1 10.0 9.6 10.1

4.2 4.4 4.9 5.3 5.0 5.2

0.9 0.9 0.9 1.6 1.2 1.2

8.7 8.8 9.0 12.4 11.5 11.5

4.5 4.3 4.3 7.8 6.9 6.3

1.0 1.0 1.0 3.1 2.3 2.0

NOTES: Empirical rejection rate of 10%, 5%, and 1% level tests using asymptotic critical values when data are generated from unit root processes with i.i.d. error (α = 0.0) and ARCH(1) error (α = 0.5). Results are based on 10,000 replications.

by generating the data from yt = εt where (εt ) is i.i.d. N(0, 1). The rejection frequencies based on such critical values are then computed by generating the data from transitional AR models described below. The power performance of the inf-t tests is also compared to that of other competing nonlinear unit root tests as well as the conventional Dickey-Fuller test (hereafter DF test). Table 4 reports the rejection frequencies of the inf-t test based on the D-TAR specification in Table 1, when the data are generated from the three-regime TAR model given by (37)

yt = (ρ1 − 1) [(yt−1 + ν)1{yt−1 ≤ −μ} + (yt−1 − ν)1{yt−1 ≥ μ}] + εt

for the sample sizes of n = 100 and 200. The range of parameter values for power comparisons is ρ1 ∈ {0.7, 0.9, 0.95}, μ ∈ {1, 3, 5}, and ν = 0. Within this range of parameter values, the model becomes most persistent when ρ1 = 0.95, which corresponds to the AR parameter for two outer regimes. In addition, when the threshold parameter μ increases, it will be more difficult to reject the unit root hypothesis since more data are likely to be observed in the inner regime with a unit root. Bec et al. (2008) used essentially the same model in the power evaluation of their Sup sup-Wald test for a unit root (WB in their notation, hereafter the BGG test). They found that the BGG test outperforms the DF test when the value of μ is very large. For this reason, the results from both the BGG test and the DF test with an intercept are also included in Table 4. In almost all cases, the inf-t test outperforms the BGG and DF tests. When ρ1 = 0.7, the powers of all tests quickly approach 100% as the sample size increases. In this sense, the advantage of the inf-t test is more evident in the near unit root cases such as ρ1 = 0.95. The BGG tests performs better than the DF test in the case of a faster mean reversion (ρ1 = 0.7 and 0.9) combined with a largest threshold value (μ = 5). In contrast, the inf-t test works well even if the threshold value is relatively small (μ = 1). Table 5 reports the results for the inf-t test based on the TAR specification in Table 1 when the data are generated from the two-regime TAR model given by (38)

yt = ρ1 yt−1 1{yt−1 ≤ μ} + ρ2 yt−1 1{yt−1 > μ} + εt

655

TRANSITIONAL AUTOREGRESSIVE MODELS

TABLE 4 SIZE-ADJUSTED POWER OF INF-T TESTS AGAINST THREE-REGIME TAR MODEL

inf-tD-TAR

BGG

DF

ρ1

μ

n

10%

5%

1%

10%

5%

1%

10%

5%

1%

0.7

1

100 200 100 200 100 200 100 200 100 200 100 200 100 200 100 200 100 200

99.9 100.0 98.0 100.0 70.4 99.4 50.5 93.9 48.9 92.6 37.9 79.1 22.9 47.7 22.8 48.8 19.6 43.8

99.4 100.0 92.7 100.0 56.9 96.6 32.2 82.2 31.4 80.7 23.6 62.2 12.3 29.1 12.6 30.0 10.2 26.6

90.2 100.0 67.7 99.9 33.2 81.9 8.8 45.4 9.0 44.4 6.9 31.8 2.5 7.9 2.6 8.6 2.1 8.5

85.6 100.0 77.0 99.5 67.7 96.2 22.3 55.1 24.4 48.4 25.7 48.1 13.2 21.8 14.1 21.7 15.2 23.5

71.0 100.0 63.4 98.1 55.4 92.0 12.5 35.8 14.6 32.1 16.0 33.6 6.8 11.7 7.7 11.9 8.6 13.9

36.6 98.9 36.4 87.5 33.8 77.4 3.1 11.0 4.5 11.9 5.2 13.8 1.6 2.8 1.8 2.8 2.2 4.1

100.0 100.0 78.8 100.0 29.2 75.8 48.9 94.9 28.6 78.7 21.7 33.8 21.6 47.7 18.5 36.2 17.3 24.3

99.4 100.0 50.3 100.0 18.7 45.0 29.3 83.7 15.8 52.6 13.1 19.4 11.3 29.3 9.6 19.6 9.4 13.9

88.5 100.0 13.9 91.2 7.2 11.6 7.5 41.5 4.3 12.9 4.0 6.2 2.2 7.0 2.2 4.2 2.3 3.9

3 5 0.9

1 3 5

0.95

1 3 5

NOTES: Empirical rejection rate of 10%, 5%, and 1% level tests using size-adjusted critical values when data are Sup generated from three-regime TAR (D-TAR) processes. Results are based on 10,000 replications. BGG is WB test statistic by Bec et al. (2008) and DF is the Dicky-Fuller t test statistic with an intercept. TABLE 5 SIZE-ADJUSTED POWER OF INF-T TESTS AGAINST TWO-REGIME TAR MODEL

inf-tTAR

EG

DF

ρ1

ρ2

μ

n

10%

5%

1%

10%

5%

1%

10%

5%

1%

0.7

0.85

−1.0

100 200 100 200 100 200 100 200 100 200 100 200 100 200 100 200 100 200 100 200 100 200 100 200

98.7 100.0 98.6 100.0 99.0 100.0 65.2 90.3 63.8 89.4 72.1 93.9 46.2 84.4 46.6 84.4 47.3 85.0 28.6 46.8 28.3 46.2 29.9 48.9

94.5 100.0 94.4 100.0 95.5 100.0 51.0 83.9 49.1 82.2 59.0 89.0 28.2 68.9 28.6 68.9 29.3 69.9 16.2 32.6 16.0 32.2 17.6 34.8

73.1 99.9 73.4 99.9 77.2 100.0 26.3 61.3 25.1 58.6 34.7 71.5 8.7 30.2 8.8 30.3 9.3 31.6 4.5 11.0 4.5 10.5 5.2 12.2

93.8 100.0 94.4 100.0 94.9 100.0 36.3 70.5 37.0 72.3 40.6 73.4 25.9 65.8 25.9 66.3 26.3 66.8 15.9 25.0 15.7 25.0 16.5 25.9

83.3 100.0 84.6 100.0 85.9 100.0 24.4 53.8 24.6 55.5 29.0 58.0 14.6 45.4 14.6 45.7 15.0 46.4 8.6 14.5 8.4 14.4 9.1 15.4

46.2 99.3 48.0 99.4 51.3 99.5 7.7 24.4 7.4 25.4 10.9 30.1 3.3 14.6 3.3 14.8 3.4 15.4 1.8 3.8 1.8 3.5 2.0 4.2

95.3 100.0 95.9 100.0 96.1 100.0 38.3 71.0 39.3 72.7 41.5 72.7 30.4 72.0 30.7 72.5 31.1 72.7 18.3 27.8 18.1 27.9 18.9 28.4

86.8 100.0 87.7 100.0 88.3 100.0 25.3 53.7 25.9 55.7 28.9 56.3 17.5 50.8 17.5 51.4 18.1 51.7 9.9 16.5 9.7 16.5 10.3 17.2

51.2 99.5 53.4 99.5 56.1 99.5 8.5 23.8 8.5 24.8 11.5 27.1 4.3 17.2 4.3 17.4 4.5 17.9 2.5 4.4 2.4 4.2 2.6 4.9

0.0 1.0 0.97

−1.0 0.0 1.0

0.9

0.95

−1.0 0.0 1.0

0.99

−1.0 0.0 1.0

NOTES: Empirical rejection rate of 10%, 5%, and 1% level tests using size-adjusted critical values when data are generated from two-regime TAR processes. Results are based on 10,000 replications. EG is the μ test statistic by Enders and Granger (1998) and DF is the Dicky-Fuller t test statistic with an intercept.

656

PARK AND SHINTANI

for the sample sizes of n = 100 and 200. The range of parameter values is (ρ1 , ρ2 ) ∈ {(0.7, 0.85), (0.7, 0.97), (0.9, 0.95), (0.9, 0.99)} and μ ∈ {−1, 0, 1}. Since Enders and Granger (1998) considered a similar two-regime stationary TAR model as an alternative hypothesis, Table 5 includes the results of their F test (μ in their notation, hereafter the EG test) for comparison, along with the DF test with an intercept. The AR parameter ρ1 in the first regime with yt−1 ≤ μ is set at a value less than the AR parameter ρ2 in the second regime with yt−1 > μ. Note that less frequent rejection of the unit root hypothesis can be expected for the case of larger ρ1 (ρ2 ) when ρ2 (ρ1 ) is fixed. Consistent with this prediction, Table 5 shows that power of all three tests declines as ρ1 or ρ2 increases. Although the evidence is somewhat more ambiguous, the power tends to be higher for μ = 1 since more data are expected to be observed in the first regime (less persistent regime) for a larger value of the threshold μ. The table clearly shows the dominance of the inf-t test over the EG and DF tests in terms of the power. For the least persistent case of ρ1 = 0.7 combined with ρ2 = 0.85, powers of all tests are near 100% when n = 200. For the same specification, however, the difference between the power of the inf-t test and those of other tests is obvious when n = 100. For a more persistent case of ρ1 = 0.9 combined with either ρ2 = 0.95 or ρ2 = 0.99, the power of the inf-t test using the 5% significance level is often twice as large as that of other two tests when n = 100. Finally, note that as long as assumptions in Proposition 1 are satisfied, our test is expected to have a power not only against the transitional AR model on which the test is based, but also against a larger class of nonlinear stationary alternatives. In a recent simulation study, Choi and Moh (2007) examine the power of various unit root tests, including our inf-t test, with an extensive list of empirically relevant nonlinear models. They find that the inf-t test has an advantage over other nonlinear unit root tests because it has a good power against a variety of stationary nonlinear processes. 5.2. Target Zone Model. As an empirical example, we apply inf-t tests to the target zone exchange rate model. During the 1980s and 1990s, the exchange rates within the European Monetary System (EMS) were subject to intervention by central banks based on the bands set at ±2.25% around the central parity in most of the member countries. As pointed out by Balke and Fomby (1997, p. 628), the target zone systems may well be characterized by the three-regime TAR model with a partial unit root in the inner regime, since the exchange rates are allowed to fluctuate freely within the band. Although the official band is set at ±2.25% around the central parity, it is likely that intervention begins before the rate actually hits ±2.25% points. Therefore, it seems reasonable to consider unknown threshold values in the transitional AR models. For this reason, we employ inf-t tests based on the D-TAR, ESTAR-M, and EQ-LSTAR models, respectively, specified as yt = λ [(yt−1 + ν)1{yt−1 ≤ −μ} + (yt−1 − ν)1{yt−1 ≥ μ}] +

p 

αi yt−i + εt ,

i=1

  2 2 yt = λ(yt−1 − ν) 1 − e−κ (yt−1 −ν) + αi yt−i + εt , p

i=1

yt = λ[yt−1 [1 + eκ(yt−1 +μ) ]−1 + [1 + e−κ(yt−1 −μ) ]−1 ] +

p 

αi yt−i + εt ,

i=1

where λ = ρ1 − 1. We use the daily French franc/Deutschemark exchange rates measured in log deviations from the central parity. The sample period is from 1979/3/13 to 1993/7/30 (n = 3, 645), during which the central rate was realigned six times. The data are plotted in Figure 2. The results of the three inf-t tests are reported in the panel A of Table 6. In addition to inf-t tests, we also consider Kapetanios et al.’s (2003, KSS) test, the BGG test and the ADF test.

657

TRANSITIONAL AUTOREGRESSIVE MODELS

0.030

0.020

0.010

0.000

-0.010

-0.020

-0.030 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993

NOTE: Third realignment is shown with a dotted line. FIGURE 2 FRENCH FRANC/DEUTSCHEMARK EXCHANGE RATE: DEVIATION FROM CENTRAL PARITY

TABLE 6 UNIT ROOT TEST RESULTS FOR FRENCH FRANC/DEUTSCHEMARK EXCHANGE RATE

Test Statistics inf-tD-TAR

inf-tESTAR-M

inf-tEQ-LSTAR

KSS

BGG

DF

Sample A: 1979/3/13-1993/7/30 (n = 3645) p =0 −7.51* −8.39* p = 5† −6.84* −7.78*

−7.48* −6.82*

−5.91* −5.26*

89.37* 76.42*

−5.21* −4.68*

Sample B: 1979/3/13-1982/6/11 (n = 819) p = 0† −4.13* −4.19* p =5 −4.05* −4.08*

−4.51* −4.47*

−4.19* −4.09*

11.24 11.17

−2.41 −2.26

NOTES: Rejection of the unit root hypothesis at the 1% significance level is shown by *. The lag length p that minimizes AIC for the augmented Dickey-Fuller regression with an intercept is indicated by †. KSS is the tNL,μ test statistic by Sup Kapetanios et al. (2003), BGG is the WB test statistic by Bec et al. (2008), and DF is the Dicky-Fuller t test statistic with an intercept.

For the lag order p we report both p = 0 and p = 5, where the latter corresponds to the lag selected by AIC applied to the linear AR model. For all six tests, the unit root hypothesis is rejected at the 1% significance level. Thus, it appears that rejection of a unit root is strong and unambiguous. This result, however, is not very surprising given the fact that the series is by definition confined within a band as well as the availability of the series with a large sample size. In fact, the conclusion can differ drastically from the above-mentioned if we consider subsamples of a shorter sample period. To demonstrate such a possibility, we consider a subsample based on the series only up to the time of the third realignment (in 1982) in the total of six realignments. The results from the additional tests are provided in the panel B of Table 6.22 For this subsample (n = 819), all three inf-t tests and the KSS test still reject the unit root. However, the BGG and DF tests fail to reject the unit root hypothesis. This difference in the subsample analysis is consistent with the simulation evidence of the better finite sample power properties of the inf-t tests than the other tests. 22

For the subsample, p = 0 is suggested by AIC

658

PARK AND SHINTANI

6.

CONCLUSION

In this article, we consider the test of a unit root against the transitional AR models. Our framework is truly general, allowing for a wide range of AR models with discrete and smooth transition dynamics. The models that we study in the article include as special cases all the transitional AR models considered previously in the literature. Moreover, we only impose very mild conditions on the innovation sequence, which are minimal in the sense that they are also required for the validity of standard unit root tests in the linear AR models. Under this very general and flexible specification, we investigate the inf-t test that can be used to effectively discriminate the unit root model from the stationary transitional AR models. Such a test is motivated by the fact that the parameters in the transitional AR models are not identified under the null hypothesis of a unit root. The full asymptotic theory for the inf-t test is developed in the article. In particular, we show that the test has well-defined limit distribution that is free of any nuisance parameters and depends only on the transition function and the limit parameter space. The critical values of the transitional AR models that are used most frequently in practical applications are tabulated in the article. The small sample performance of the inf-t test is very encouraging. The test has a reasonably good size property with sample sizes typically available in practice. It also has a rather satisfactory power property in finite samples. Although the power performance somewhat depends on the range of parameters when the sample size is small, the inf-t test generally has significantly better power than all other existing unit root tests. The power gain is often very substantial, for some of the empirically relevant cases. Finally, our empirical application shows that the transitional AR models can be potentially much more promising than the linear AR models in explaining the data-generating process of some economic and financial time series. Most of all, it seems apparent that the ubiquitous unit root in many economic and financial time series is likely to disappear, as we allow for the nonlinear transition dynamics. APPENDIX

Here we only provide some selected lemmas and the proofs of main theorems. Complete proofs of other theorems, as well as related lemmas, corollaries, and propositions, are provided in an online technical appendix. Useful Lemmas. LEMMA A1. (a) If π is regular on R × Rm , then so are |π| and π2 . (b) If π is regular on R × Rm and υ is regular on R, then the functional υπ, defined by (υπ)(x, θ) = υ(x)π(x, θ), is regular on R × Rm . LEMMA A2. Let Assumption 3 holds. If  is regular on R × Rm , then 

1 0

 (Vn (r), θ) dr →a.s

1

(V (r), θ) dr

0

uniformly in θ ∈ K, as n → ∞, where K is any compact subset of Rm . Proofs of Theorems 1 and 2. Proof of Theorem 1. We prove the convergence of finite-dimensional distributions and the tightness. The former is trivial. We will establish the latter below. The tightness for the transition function π satisfying part (a) of Assumption 2 is established in Lemma 3.3 of Park and Whang (2005). Therefore, we only consider the transition function π satisfying part (b) of Assumption

659

TRANSITIONAL AUTOREGRESSIVE MODELS

2. For the required tightness, it suffices to show that there exist strictly positive constants a, b, c such that E[ sup |Mn (r, θ1 ) − Mn (r, θ2 )|a ] ≤ c θ1 − θ2 m+b

(A.1)

0≤r≤1

for all θ1 , θ2 ∈ Rm . This is due to Kolmogorov’s criterion in Theorem 1.8 of Chapter XIII in Revuz and Yor (1994). Strictly, the exact form of the criterion presented there applies only to stochastic processes with scalar parameters. It is, however, straightforward to extend their criterion to make it applicable for stochastic processes with vector parameters. For instance, we may readily define the Wiener space for stochastic processes having vector parameters in parallel to the one for stochastic processes having scalar parameters that they use and note that all their results continue to apply for stochastic processes with vector parameters in the newly defined Wiener space. Moreover, our condition in (A.1) is slightly different from theirs, since we use Exercise 2.10, instead of Theorem 2.1, in Chapter I of Revuz and Yor (1994). Finally, the sup norm for Rm in their condition is replaced by the usual Euclidean norm in ours. Clearly, this would not affect any of the subsequent results, since the two norms are equivalent to each other in Rm . Note that (A.1) implies that there exists a modification of Mn (r, θ) that is continuous in both r ∈ [0, 1] and θ ∈ Rm due to Exercise 2.10 in Chapter I of Revuz and Yor (1994). To prove (A.1), we set a, b so that a>m

(A.2)

b=a−m>0

and

and let θ1 and θ2 be chosen arbitrarily in Rm . Moreover, we note that 

r

Mn (r, θ1 ) − Mn (r, θ2 ) =

[π(Vn (s), θ1 ) − π(Vn (s), θ2 )] dU(s)

0

is a martingale with parameter r ∈ [0, 1] for any fixed θ1 , θ2 ∈ Rm . Therefore, we may deduce from the Burkholder-Davis-Gundy inequality (see, e.g., Revuz and Yor, 1994, p. 153) that 

 (A.3) E

sup |Mn (r, θ1 ) − Mn (r, θ2 )|

a

 ≤ cE

1

a/2 |π(Vn (r), θ1 ) − π(Vn (r), θ2 )| dr 2

0

0≤r≤1

for some constant c > 0. Due to (A.2) and (A.3), we now only need to show that 

1

E

(A.4)

a/2 |π(Vn (r), θ1 ) − π(Vn (r), θ2 )| dr 2

≤ c θ1 − θ2 a

0

for some constant c > 0 to establish (A.1). To establish (A.4), we note that 

1

 |π(Vn (r), θ1 ) − π(Vn (r), θ2 )|2 dr ≤ θ1 − θ2 2

0

1

π¯ 2 (Vn (r))dr,

0

from which it follows that 

1

(A.5) E

a/2 |π(Vn (r), θ1 ) − π(Vn (r), θ2 )| dr 2

 ≤ θ1 − θ2 E

1

a

0

 ≤ θ1 − θ2

a

a/2 π¯ (Vn (r))dr 2

0



sup E π¯ (Vn (r)) . a

0≤r≤1

660

PARK AND SHINTANI

We may assume without loss of generality that π¯ is monotone increasing around the origin, so that we have   π(V ¯ n (r)) ≤ π¯ sup |Vn (r)| ≤ π(V ¯ max + 1), 0≤r≤1

for all large n and all r ∈ [0, 1], where Vmax = sup0≤r≤1 |V (r)|. Consequently, we have sup E π¯ a (Vn (r)) ≤ E π¯ a (Vmax + 1) < ∞,

(A.6)

0≤r≤1

since π(x) ¯ ≤ ec|x| for some c > 0 as |x| → ∞. Now we may easily deduce (A.4) from (A.5) and (A.6), and the proof is complete.  Proof of Theorem 2. Throughout the proof, we assume that An (θ) and Bn (θ) are redefined in terms of the normalized parameter θn , which for notational brevity we will denote simply by θ. Moreover, we may assume d = 1, without loss of generality. This is because     n n 1 yt−1 yt−d yt−d 1 yt−d ut−i √ π √ , θ = ut−i √ π √ , θ n n n n n n t=1

t=1

+

    n yt−1 yt−d yt−d 1 π √ ,θ , ut−i √ − √ n n n n t=1

and, due to Cauchy-Schwarz inequality and the boundedness of π, we have  n     1  yt−d yt−1 yt−d   π √ ,θ  ut−i √ − √  n  n n n t=1 c ≤√ n



n 1 2 ut−i n

1/2 

t=1

n 1 (yt−1 − yt−d )2 n

1/2

t=1

uniformly in θ ∈ Rm , for some constant c > 0. Let   yt−1 yt−1 wnt (θ) = √ π √ , θ . n n Under the null hypothesis H0 : λ = 0, An and Bn in (16) and (17) reduce to n An (θ) 1  = √ wnt (θ)εt − n n t=1 n 1 2 Bn (θ) = wnt (θ) − n2 n t=1





n 1 wnt (θ)x t n t=1

n 1 wnt (θ)x t n



t=1



n 1

xt xt n t=1

n 1

xt xt n t=1

−1 

−1 

 n 1  xt εt , √ n t=1

 n 1 xt wnt (θ) . n t=1

Furthermore, we have σˆ n2 (θ)

n 1 2 1 = εt − n n t=1



n 1 

εt xt √ n t=1



n 1

xt xt n t=1

−1 

n 1  xt εt √ n t=1



661

TRANSITIONAL AUTOREGRESSIVE MODELS

⎡  ⎤2  n −1  n n n 1 

1

1 1⎣ 1  − wnt (θ)εt − √ εt xt xt xt xt wnt (θ) ⎦ √ n n n n t=1 n t=1 t=1 t=1 ⎡

n 1 2 ·⎣ wnt (θ) − n t=1



n 1 wnt (θ)x t n



t=1

n 1

xt xt n

−1 

t=1

⎤−1 n 1 xt wnt (θ) ⎦ n t=1

under the null hypothesis H0 : λ = 0. Note that  (A.7)

n 1

xt xt n

−1

t=1

n 1  , √ xt εt = Op (1). n t=1

Moreover, we have n n 1  1 wnt (θ)εt = Op (1) and xt wnt (θ) = o p (1) √ n n t=1 t=1

(A.8)

uniformly in θ ∈ nn , due, respectively, to Theorem 1 and Lemma 4. Finally, as shown in Park and Phillips (2001), we have as n → ∞ (A.9)

 1  1 n 1 2 wnt (θ) =d Vn (r)2 π2 (Vn (r), θ) dr + o p (1) →a.s. V (r)2 π2 (V (r), θ) dr n 0 0 t=1

uniformly in θ ∈ nn , due to Lemmas A1 and A2. Under the null hypothesis H0 : λ = 0, it follows from (A.7)–(A.9) that (A.10)

n 1  An (θ) =√ wnt (θ)εt + o p (1), n n t=1

(A.11)

n Bn (θ) 1 2 = wnt (θ) + o p (1), n2 n t=1

uniformly in θ ∈ nn , and that (A.12)

σˆ n2 =

n 1 2 εt + o p (1) n t=1

uniformly in θ ∈ nn . However, due to Kurtz and Protter (1991), we have from Theorem 1 and Proposition 1 that (A.13)

 1  1 n 1  wnt (θ)εt =d Vn (r) Mn (dr, θ) + o p (1) →d V (r) M(dr, θ). √ n t=1 0 0

The stated result may now be easily deduced from (A.10), (A.11), (A.12), and Assumption 1, using (A.9), (A.13), and Lemma 2. Here we may apply the continuous mapping theorem, since T (θ) is continuous a.s. in θ and the operation of taking infimum of a continuous functional T

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PARK AND SHINTANI

on Rm over a subset of Rm is continuous with respect to the standard Hausdorff metric in the space of compact subsets of Rm . Note that Mn (dr, θ) = π(Vn (r), θ)dU(r) and

M(dr, θ) = π(V (r), θ)dU(r),

from which it follows immediately that  0

1



1

Vn (r)Mn (dr, θ) =



1

Vn (r)π(Vn (r), θ)dU(r),

0



1

V (r)M(dr, θ) =

0

V (r)π(V (r), θ)dU(r),

0

and that, in case π(x, θ) = 1, our weak convergence in (A.13) reduces to  1 n 1 yt−1 εt →d V (r)dU(r), n 0 t=1

which is well known in the unit root asymptotics.



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