# 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

ADF* = 1infT {ADF(t0)}.

The null hypothesis of no cointegration and no structural break is rejected if the statistic ADF* is smaller than the corresponding critical value. In this case the structural break will be located at time t* where the inf of the ADF test is obtained. The work of Gregory and Hansen is opening an extensive research on analyzing the stability of the parameters of multivariate possibly cointegrated systems models like the VECM in (30.16). Further work in this direction can be found in Hansen and Johansen (1993), Quintos (1994), Juhl (1997), and Arranz and Escribano (2000).

4 Concluding Remarks

The considerable gap in the past between the economic theorist, who had much to say about equilibrium but relatively less to say about dynamics and the econometrician whose models concentrated on the short-run dynamics disregarding the long-run equilibrium, has been bridged by the concept of cointegration. In addition to allowing the data to determine the short-run dynamics, cointegration suggest that models can be significantly improved by including long-run equilibrium conditions as suggested by economic theory. The generic existence of such long-run relationships, in turn, should be tested using the techniques discussed in this chapter to reduce the risk of finding spurious conclusions.

The literature on cointegration has greatly enhanced the existing methods of dynamic econometric modeling of economic time series and should be considered nowadays as a very valuable part of the practitioner’s toolkit.

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