Note on Fully Modified Estimation for Three-Regime Threshold Cointegration Model

In this paper we consider a three-regime threshold cointegration model. The fully modified ordinary least squares (FM-OLS) regression of Phillips and Hansen [1] is used to develop new methods for estimating cointegrating coefficients. After we remove the second-order biases of parameter estimates from the three-regime threshold cointegration model, FM-OLS estimates have a limit distribution that is mixed normal for all the nonstationary coefficients.


Introduction
Since the breakthrough paper by Balke and Fomby [2], nonlinear cointegration has become one of the most important research areas in time series analysis, and a strand of econometric literature has concentrated on estimating and testing for threshold cointegration.The early works in these directions include the studies by Choi and Saikkonen [3], Hansen and Seo [4], Gaul [5], Gonzalo and Pitarakis [6] [7], Kiliç [8], Li and Lee [9], and Seo [10] among others.Numerous contributions apply the threshold cointegration concept in empirical studies.Lo and Zivot [11] and Bec and Rahbek [12] have provided comprehensive reviews of the topic.
Although there are numerous research papers in the literature, only a few of those papers focus on the estimation of multi-regime threshold cointegration.Hansen and Seo [4] developed a maximum likelihood-based estimation and testing theory for a two-regime threshold cointegration model, while Seo [10], Gonzalo and Pitarakis [6], and Li and Lee [9] constructed tests for cointegration in the presence of possible threshold effects.Gaul [5] provided a Wald test against threshold effects in a three-regime threshold vector error correction model1 .Moreover, all of the theories proposed before require the assumption of i.i.d.residuals.Only rare studies have discussed the efficiency of estimation when the error sequence is under weakly stationary, The main purpose of this paper is to develop an approach estimating for cointegrated parameters in three-regime threshold models.We use the fully modified ordinary least squares (FM-OLS) regression by Phillips and Hansen [1] to provide the optimal estimates of threshold cointegration.An asymptotic theory for FM-OLS is derived.From the asymptotic theory for three-regime threshold cointegration regression, the FM-OLS principle may be used to eliminate the possible second-order biases from weakly dependent residuals.The estimates of FM-OLS for threshold cointegration are free for possible endogeneity.
The remainder of the paper is organized as follows.Section 2 introduces the basic assumptions and asymptotic distributions of fully modified threshold cointegration estimators.Section 3 provides a conclusion, and the Appendix provides all the proofs of the theorems discussed.

Assumptions and Parameter Estimation
This study considers the following cointegrating relationship with three-regime threshold effects.
where 1t u and 2t u are scalar and p vector-valued stationary disturbances, t q is the stationary threshold variable, 1 c and 2 c are threshold parameters, ( ) For convenience, we rewrite Equation (1) in matrix format as where t Y stacks 1t y and i X , 1, 2, 3 i = stacks on the regressors of Equation ( 1) for different indicators.Asymptotic properties of FM-OLS estimates are determined by the joint error process of ( ) , Throughout this paper, we will derive the limit distributions under following assumptions.
u u is weakly stationary, ergodic and strong mixing with mixing coefficients n κ satisfying B r is a p-dimensional Brownian motion with a long-run covariance matrix given by 11 12 21 22 ′ Ω = Σ + Λ + Λ and The threshold variable t q is strictly stationary and independent of the process { } y and the serial correlation in 1t u in Equation ( 1).This treatment of the serial correlation in 1t u , like that in the study by de Jong [13] and Gonzalo and Pitarakis [7], a more general assumption in threshold cointegration regression.Assumption 2 provides some moment conditions imposed on { } 1t u and { } 2t u .Assumption 3 is a multivariate invariance principle for the partial sum process.Assumption 1 -3 are used for establishing the limit theory for partial sum process like ( ) . Assumption 4 and 5 are standard assumptions for threshold literatures.Assumption 4 excluded the possibility that that the threshold variable t q is correlated with 1t u .Assumption 5 ensures that there are enough observations in each regime.Assumption 6 rules out the possibility that the components of 2t y will themselves be cointegrated.Our estimation of threshold estimators of 1 ĉ and 2 ĉ for Equation ( 1) is based on the nonlinear least squares regression.Let The authors consider ( ) where 1, 2 i = .Once ˆi c is obtained, the 2t y can be separated by indictors.Because the regressors of Equation ( 1) are orthogonal each other.For given threshold value 1 c and 2 c , it can be estimated by using OLS procedure ( ) ( ) ( ) ( ) Although NLS procedure may be consistent, the limits of estimated parameters may contain second-order bias.Phillips and Hansen [1] propose a fully modified estimation methods which can eliminate these bias effects.Following the spirit of Phillips and Hansen [1], we suggest the FM-OLS principle for threshold cointegrating regression.From the cointegration model of Equation ( 3 Define a transformation matrix: The vector ( ) Lemma 1.Under Assumption 1 -3, for Equation (8) we have where ( ) , is a p-dimensional standard Brownian motion, ( ) ( ) The following theorem shows the limit distribution of FM-OLS estimators.Theorem 1.Under Assumption 1 -3, for Equation ( ( ) M M ϒ = .In the formulae for the bias correction terms in Equation ( 17)-( 19), 21 Ω and 21 Λ are the kernel estimates of long run covariance matrices, which can be found in Phillips [14].From the Theorem 1, the limit distribution is free of second order bias.

Conclusions
In this paper, we have established the asymptotics of fully modified ordinary least squares cointegrating estimators for three-regime threshold cointegration regression.Our results may be used under the condition that residuals with weakly dependency exist.These results improve the estimation methods available in threshold cointegration literatures.
There are a variety of potential topics for future research related to this study.First, it is possible to extend our results to more than one cointegrating relationship, although a single threshold cointegrating relationship is often used in economics and international finance.Second, testing for threshold cointegration based on the FM-OLS principle is not yet available.The authors hope to further study these issues in the future.
From the Proposition 2 in Gonzalo and Pitarakis [7], Lemma A3 in Gonzalo and Pitarakis [7] and the continuous mapping theorem give Consider fully modified estimator ˆ1, 2,3 j j ρ = . ( Assumption 1 is fairly general and covers a variety of weakly stationary posses.It also allows for the endogeneity of 2t 22Ω is nonsingular, i.e. the p-dimensional I (1) vector 2t y are not cointegrated each other.