What is cointegration test

In the current macroeconomic econometric analysis, the cointegration method proposed by Granger (1987) has become one of the most important tools to analyze the quantitative relationship between non-stationary economic variables, and the linear adjustment mechanism between economic variables is portrayed through the linear error correction model (ECM), which is called the linear cointegration method. In recent years, with the development of economic theory, especially in the economic analysis of transaction costs and policy responses, the traditional linear cointegration analysis is no longer an appropriate analysis method, in view of which Balk and Fomby (1997) put forward the so-called Threshold Cointegration (Threshold Cointegraion) method, which depicts the nonlinear adjustment mechanism between economic variables. For example, in the process of stock trading, due to transaction costs, trading policy and other factors will lead to asymmetric adjustment of stock prices; the country's monetary policy due to institutional reasons will also have asymmetric adjustment behavior of the inflation rate. Therefore threshold cointegration methodology is one of the most powerful tools for analyzing such economic problems. Threshold cointegration is a crucial development of the concept of cointegration proposed by Granger (1987) to describe the long-run relationship between economic variables. It is well known that cointegration means that if there is a long-run cointegration relationship between economic variables and the regularized cointegration vector is (1, -β′), the long-run equilibrium relationship between them can be expressed as:

Where: the β parameter is the vector of cointegration coefficients between the variables, γ is the threshold variable, d is the transformation variable, d is the lagged parameter, then this cointegration is called threshold cointegration. If the covariance error term is a data generating mechanism shaped as in Eq. (2), it is called Two-Regime threshold cointegration; if it is an error generating mechanism shaped as in Eq. (3), it is called Three-Regime threshold cointegration. In previous studies, most of the studies on the threshold cointegration represented by Eqs. (2) and (3) have focused on the case where all the four parameters, ρ, q, θ, and λ, are less than 1, and fewer studies have been conducted on the other cases (Enders and Granger (1998)[3]). In this paper, we mainly study the following cases, namely:

Then the threshold cointegration expressed by equations (2) and (3) is the so-called partial cointegration (Partial Cointegration). For Partial Cointegration test, caner and Hansen (2001) proposed a statistic and Gouveia and Rodrigues (2004) applied this statistic to the threshold cointegration test, but they did not study the test potential of this statistic. And in our previous study, we found that this statistic has a low potential in testing threshold cointegration.