Note on Fully Modified Estimation for Three-Regime Threshold Cointegration Model ()
Keywords
Fully Modified Regression, Nonlinearity, Threshold Cointegration
1. 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 stationaryThe 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.
2. Assumptions and Parameter Estimation
This study considers the following cointegrating relationship with three-regime threshold effects.
(1)
(2)
where 
and 
are scalar and p vector-valued stationary disturbances, 
is the stationary threshold variable, 
and 
are threshold parameters, 
is indicator function. For convenience, we rewrite Equation (1) in matrix format as
(3)
where 
stacks 
and
, 
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.
Assumption 1. The sequence 
is weakly stationary, ergodic and strong mixing with mixing coefficients 
satisfying
for some
.
Assumption 2. 
and 
for some
.
Assumption 3.
where 
is a p-dimensional Brownian motion with a long-run covariance matrix given by
with 
and 
where 
and
Assumption 4. The threshold variable 
is strictly stationary and independent of the process
, and has a distribution function 
that is continuous and strictly increasing.
Assumption 5. The threshold parameter 
and 
are such that 
and 
a closed and bounded subset of the sample space of the threshold variable.
Assumption 6. 
is nonsingular, i.e. the p-dimensional I (1) vector 
are not cointegrated each other.
Assumption 1 is fairly general and covers a variety of weakly stationary posses. It also allows for the endogeneity of 
and the serial correlation in 
in Equation (1). This treatment of the serial correlation in
, 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 
and
. 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 
is correlated with
. Assumption 5 ensures that there are enough observations in each regime. Assumption 6 rules out the possibility that the components of 
will themselves be cointegrated.
Our estimation of threshold estimators of 
and 
for Equation (1) is based on the nonlinear least squares regression. Let
The authors consider
(4)
where
. Once 
is obtained, the 
can be separated by indictors. Because the regressors of Equation (1) are orthogonal each other. For given threshold value 
and
, it can be estimated by using OLS procedure
(5)
(6)
(7)
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), the residuals 
are weakly dependent. 
may be correlated with
. It will cause efficiency loss.
Consider 
and
. Redefine
(8)
Define a transformation matrix:
(9)
The vector 
can be written as
(10)
Lemma 1. Under Assumption 1 - 3, for Equation (8) we have
(11)
(12)
(13)
where 
denotes a Brownian sheet, 
, 
, is a p-dimensional standard Brownian motion, 
and
.
Lemma 2. Under Assumption 1 - 3, for Equation (8) we have
(14)
(15)
(16)
The following theorem shows the limit distribution of FM-OLS estimators.
Theorem 1. Under Assumption 1 - 3, for Equation (8) we have
(17)
(18)
(19)
where
.
In the formulae for the bias correction terms in Equation (17)-(19), 
and 
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.
3. 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.
Acknowledgements
The author thanks financial support from Ministry of Science and Technology of Taiwan under grant NSC 95- 2415-H-305-010 and Shu-Yu Lin and Yi-Ting Tsai for excellent research assistants.
Mathematical Appendix
Proof of Lemma 1
Let 
and
. The asymptotic distribution follows directly from Phillips [14] and Gonzalo and Pitarakis [6]
Proof of Lemma 2
From the Proposition 2 in Gonzalo and Pitarakis [7], Lemma A3 in Gonzalo and Pitarakis [7] and the continuous mapping theorem give
Proof of Theorem 1
Consider fully modified estimator
.
For denominator,
by Lemma 2 For numerator, by virtue of Lemma 1
According to the continuous mapping theorem and functional central limit theorem, it follows that
where
. 
NOTES
1Hansen and Seo and Gonzalo and Pitarakis consider MLE methods, but they need normality for residuals.