TESTING FOR A UNIT ROOT AGAINST TRANSITIONAL AUTOREGRESSIVE MODELS

• DOI: http://doi.org/10.1111/iere.12171
• Author: Joon Y. Park,Mototsugu Shintani
• Date: 01 May 2016
• Published date: 01 May 2016

INTERNATIONAL ECONOMIC REVIEW

Vol. 57, No. 2, May 2016

TESTING FOR A UNIT ROOT AGAINST TRANSITIONAL AUTOREGRESSIVE

MODELS∗

BYJOON 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-ttest, 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.

1We 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: joon@indiana.edu.

635

C

(2016) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

636 PARK AND SHINTANI

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-ttest. Our asymp-

totics 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 Gonz´

alez

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.2Sollis

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.3Studies 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-ttest. Moreover,

the test is found to be considerably more powerful than the other existing tests when the data

are generated from stationary transitional models.4As an empirical illustration, we test for a

unit root in the target zone exchange rate model using the inf-ttest. 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

2Within 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.

3Note 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).

4The 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.

5Complete proofs of other theorems, as well as related lemmas, corollaries, and propositions, are provided in an

online technical appendix.