# Nonlinear error correction models

When discussing the role of the cointegrating relationship zt in (30.3) and (30.3′), we motivated the EC model as the disequilibrium mechanism that leads to the particular equilibrium. However, as a function of an I(0) process is generally also I(0), an alternative more general VECM model has zt-1 in (30.3) and (30.3′) replaced by g(zt-1) where g(z) is a function such that g(0) = 0 and E[g(z)] exists. The function g(z) is such that it can be estimated nonparametrically or by assum­ing a particular parametric form. For example, one can include z+ = max{0,zt} and z~ = min{0, zt} separately into the model or large and small values of z according to some prespecified threshold in order to deal with possible sign or size asym­metries in the dynamic adjustment. Further examples can be found in Granger and Terasvirta (1993). The theory of nonlinear cointegration models is still fairly incomplete, but nice applications can be found in Gonzalez and Gonzalo (1998) and Balke and Fomby (1997).

3.1 Structural breaks in cointegrated systems

The parameters in the cointegrating regression model (30.5) may not be constant through time. Gregory and Hansen (1995) developed a test for cointegration allowing for a structural break in the intercept as well as in the slope of model (30.5). The new regression model now looks like

yu = a 1 + a 2D(t0) + P1 y2t + p2 y2tD(t0) + zt, (30.27)

where D(t0) is a dummy variable such that D(t0) = 0 if 0 < t < t0 and D(t0) = 1 if t0 < t < T. The test for cointegration is conducted by testing for unit roots (for instance, with an ADF test) on the residuals flt for each t0. Gregory and Hansen propose and tabulate the critical values of the test statistic