Estimation And Testing For Threshold Autoregression,Asymmetric Unit Root And Threshold Cointegration

Economic research shows that many economy variables display nonlinear dynamic adjustment mechanism; it is beyond doubt that this has brought forward a new challenge to classical time series analysis methodology. If still employing a linear model to describe these economy variables'dynamic features, then it is improper. In recent years, the threshold autoregressive (TAR) method has already become one of the main research fields about nonlinear time series analysis. TAR principle is based on the piecewise linearity approximation over the whole state space, that is to say, it splits the whole state space to several subspaces, and employs different linear autoregressive model to be an approximation to each subspace. Therefore, TAR model describes the nonlinear dynamic mechanism for a time series data with a different linear model based on each subspace (we refer to this subspace as one of those TAR regimes). At present, from taking a look on the literature related to TAR method, TAR theory is studied mainly from two different directions: single-equation TAR model and multi-equation TAR model. First of all, this paper carries out a set of Monte Carlo simulations to study Chan'method (1993) for single-equation TAR model. Simulation results indicate that the method has high test power, but has serious test size distortions in the meantime. Second, the purpose of this part lays emphasis on studying Gonzalo and Pitarakis method (2005, GP) and Hansen method (1996) under multi-equation system. GP method premises the hypothesis that random error follows an independently and identically distribution, and the asymptotic distribution of the test statistic is dependent on variance-covariance matrix (in relation to data generation process). Then, critical values cannot be tabulated. According to this, on the one hand the paper applies Hansen's (1996) Bootstrap method to improve GP method; On the other hand extends Hansen's method from single-equation to multi-equation system, and uses Seemingly Unrelated Regressions (SUR) to estimate unknown parameters in the model and variance-covariance matrix of the parameter estimators by generalized LS estimation. Through simulation shows that GP has larger test size distortions than that of SUR method, but has higher test power. The reason lies that is asymptotically effective only under large sample size by estimating the unknown parameters based on feasible generalized LS with SUR method, however, test power is low under small sample size.Just as classical time series analysis, how to make a distinction between globally stationary TAR process and unit root process is one of the hot research fields in nonlinear time series analysis. This is known as a so-called asymmetric unit root test. This paper studies it mainly from three aspects: The first is through setting up a new"Z statistics"to reveal that the traditional ADF and PP methods are not suitable to the context of TAR. The research indicates there are some major factors affecting the test power about conventional ADF and PP method, these factors include asymmetric degree and mean-reversion time of the data process. The second makes a systematic study to two-regime asymmetric unit root tests method. Consequently, in order to increase the test power and decrease the test size distortions, the paper proposes the residual-based block Bootstrap method(RBB) suggested by Paparoditis&Politis(2003) and Myunghwan Seo (2005) to improve EG(1998) method. Through simulation based on the RBB method under various conditional heteroskedasticity, simulation evidence finds that EG method and BVD method have serious test size distortions, but EG-Bootstrap method and CH method have relatively low test size distortions, and EG-Bootstrap method has lowest test size distortions among them. Therefore, the method applied RBB can reduce the test size distortions, and improve whose test power, too. But CH's test method possesses a serious drawback: the test statistic value has no way to be obtained on certain conditions. The third sets forth three-regime asymmetric unit root test. This part lays stress on a special three-regime threshold autoregressive model, and this special three-regime globally stationary process follows the random walk in the corridor regime. And because KS's three-regime TAR model specification is consistent with our three-regime TAR, this paper explains KS method at some length and proposes a procedure to improve KS method. KS method aims to obtain the asymptotic distribution of the statistic through applying a bounded threshold intervals and increase test power, but brings about a fateful consequence: the statistic has serious test size distortions. This paper suggests that RWB bootstrap method can improve finite sample performance of KS method; simulation evidence indicates that the RWB bootstrap method really can decrease test size distortions to a certain extent, but it still has large test size distortions. At the same time, the RWB method hasn't an obvious power advantage over the original KS method, and has lower test power than that of KS method even in most of the cases. The reason lies that detrending towards the data process is problematic because of biased OLS estimator, and RWB isn't suitable to the TAR model because it is constructed based on linear autoregressive model.According to the two-step cointegration test procedure of Engle and Granger (1987), the asymmetric unit root test can be applied to test for threshold cointegration directly based on residual in principle. The threshold cointegration test methodology has been developed mainly from two directions: the first is to test whether or not it is threshold cointegration based on residuals of long-run relation among economic variables; the second is to test for threshold cointegration through a threshold error correction model (TECM). Since the original EG method has low test power in asymmetric unit root test, the paper goes a further step to modify the EG method through RWB bootstrap and fixed regressor bootstrap (FRB, Hansen, 2000). A set of simulation shows: the test power of FRB method is higher than that of RWB bootstrap method, and the power of FRB is more than that of RWB by 20% or so. In addition, the two of them all have higher power than ADF method even though the cointegrating relation is linear. This part studies Myunghwan Seo's (2006) threshold cointegration test through Monte Carlo simulation systematically. Unlike the EG method, M.Seo method constructs a new SupWald statistic to develop a cointegration test based on the threshold ECM model. In order to increase test power and decrease test size distortions, the paper proposes RWB bootstrap and FRB bootstrap to improve M.Seo method. Simulation results show that RWB and FRB have higher power than that of M.Seo's residual-based bootstrap method, and the power of FRB is highest among them. Whether the disturbance term is homoskedasticity or heteroskedasticity, FRB is better than RWB or M.Seo's residual-based bootstrap method.