Bootstrap testing for the null of no cointegration in a threshold vector error correction model
Introduction
Threshold cointegration, introduced by Balke and Fomby (1997), generalizes standard linear cointegration to allow adjustment toward long-run equilibrium to be nonlinear and/or discontinuous. Such nonlinear short-run dynamics are predicted by many economic phenomena, such as policy interventions and the presence of transaction costs or any other transaction barriers. Lo and Zivot (2001) provide an extensive review of the growing literature regarding the application of threshold cointegration. Despite significant applied interest, econometric theory has not been developed satisfactorily.
Two testing issues arise: one is testing for the presence of cointegration, that is, the presence of long-run equilibrium, and the other is testing for the linearity of short-run dynamics. Commonly adopted in the literature is the two-step approach proposed by Balke and Fomby (1997), in which the linear no cointegration null hypothesis is first examined against the linear cointegration alternative, and then the linear cointegration null hypothesis is tested against the threshold cointegration alternative. For example, investigating the linearity of the term structure of interest rates, Hansen and Seo (2002) apply the ADF test to the interest rate spread and then apply the SupLM test they developed for a two-regime threshold vector error correction model (TVECM). However, this approach can be quite misleading because the standard cointegration tests can suffer from substantial power loss when the alternative is threshold cointegration, as demonstrated by Pippenger and Goering (2000), Taylor (2001), and the simulation study in this paper. Therefore, a new test is required to examine the linear no cointegration null hypothesis in a TVECM or in a threshold autoregression (TAR), either of which allows both linear and threshold cointegration alternative.1 Although Enders and Granger (1998) and Enders and Siklos (2001) propose such tests in TAR's, they do not provide a formal distribution theory. No such test is developed in a TVECM.
This paper develops a cointegration test in a TVECM with a prespecified cointegrating vector, in which the linear no cointegration null hypothesis is examined. Economic models often imply simple and known cointegrating vectors, and the use of a prespecified cointegrating vector is common in the literature as in Lo and Zivot (2001) and Hansen and Seo (2002). In addition, the power of the cointegration test improves significantly by using prespecified cointegrating vectors. (See Horvath and Watson (1995) for examples and more discussion.) Unlike the cointegrating vector, however, few economic theories predict the threshold parameter.
The testing is nonstandard since the threshold parameter is not identified under the null hypothesis. The inference problem when a nuisance parameter is not identified under the null hypothesis has been studied by Davies (1987), Andrews and Ploberger (1994), and Hansen (1996), among others. This paper employs the sup-Wald type statistic (hereafter supW) following Davies (1987) and derives its asymptotic distribution based on a newly developed asymptotic theory. This development contributes to the literature by extending the nonlinear nonstationary asymptotics of Park and Phillips (2001) to a uniform convergence over a class of functions.
This paper proposes a residual-based bootstrap to approximate the distribution of the statistic supW, and establish its consistency. In other words, the distribution of the bootstrapped supW is shown to converge to the same asymptotic distribution of supW by establishing an invariance principle for a bootstrap partial sum process. The bootstrap with nonstationary data has become popular recently, and this paper is the first that shows the consistency of such a bootstrap in a nonlinear model. For more information concerning the bootstraps and their consistency in standard unit root testing, see, for example, Park (2002) and Paparoditis and Politis (2003).
The finite sample properties are investigated and compared to those of the conventional cointegration tests, such as the ADF test and the Wald test of Horvath and Watson (1995). We find that the bootstrap of supW approximates the finite sample distribution of supW reasonably well. Furthermore, it exhibits much higher empirical power against threshold cointegration alternatives than the conventional tests. The discrepancy in power becomes larger as the sample size increases. Therefore, the use of supW is advisable when threshold cointegration is under consideration.
This paper is organized as follows. Section 2 introduces the model and our supW test statistic, and develops an asymptotic distribution of the statistic. We describe the bootstrap procedure and establish its asymptotic validity in Section 3. Finite sample performances of the bootstrap are examined in Section 4. Section 5 illustrates the usefulness of the proposed testing strategy by an application of the law of one price hypothesis. Proofs of all the theorems are presented in the appendix.
Section snippets
Testing the linear no cointegration null in a TVECM
We consider a Band-TVECMwhere and is a qth-order polynomial in the lag operator defined as . The error correction term is defined as for a known cointegrating vector . The threshold parameter satisfying takes values on a compact set . The model (1) allows for the no-adjustment region in the middle , which arises due to the presence of transaction barriers or policy interventions.
Residual-based bootstrap
In this section, motivated by the asymptotic pivotalness of the supW statistic, we propose a residual-based bootstrap approximation to the distribution of supW and show that the bootstrap is first-order consistent. Refinement of finite sample performance is investigated by Monte Carlo simulations in Section 4.
Since the time-series is nonstationary, we do not resample the data directly. Instead, we make use of the assumption that is independent and identically distributed. Since it is
Monte Carlo experiment
The finite sample performance of the supW statistic is examined and compared to the performances of the ADF test and the Wald test by Horvath and Watson (1995) (hereafter HW). The number of simulations is 1000, and that of bootstrap replications is 200. We examine samples of sizes 100 and 250. The statistic supW is computed based on the two-regime TVECM when the sample size is 100, and from the band TVECM when it is 250.
If we adopt the latter with a small sample size, the number of observations
Empirical illustration
We illustrate the use of our test strategy using the price indexes of used car markets from 29 different locations in the US, and investigate the Law of One Price (LOP) hypothesis. The hypothesis has been frequently examined in the literature, and Lo and Zivot (2001), among others, study the hypothesis and the presence of threshold effect in the adjustment toward the long-run equilibrium for many categories of goods. We use the same data set as they do, namely the US Bureau of Labor Statistics
Concluding remarks
We have developed the supW test examining the linear no cointegration null hypothesis in the Band TVECM, a residual-based bootstrap for the test, and required asymptotic results to justify the test and the bootstrap. Furthermore, we have demonstrated the advantage of this test over the conventional cointegration tests within the context of the commonly used two-step approach.
Two issues have not been studied enough in the literature nor in this paper. First is how to test the linear no
Acknowledgements
I would like to thank Bruce E. Hansen for his excellent advice, encouragement and insightful comments. I gratefully acknowledge helpful comments from an anonymous associate editor and two referees.
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