Characteristics of variables
The characteristics of the variables involved determine to some extent which model is a suitable representation of the data generation process (DGP). For instance, the trending properties of the variables and their seasonal fluctuations are of importance in setting up a suitable model. In the following a variable is called integrated of order d (I(d)) if stochastic trends or unit roots can be removed by differencing the variable d times (see also Chapter 29 by Bierens in this volume). In the present chapter it is assumed that all variables are at most I(1) if not otherwise stated so that, for any time series variable yit it is assumed that Ayit = yit – Vk, t-1 has no stochastic trend. Note, however, that Ayit may still have deterministic components such as a polynomial trend and a seasonal component whereas seasonal unit roots are excluded. Note also that a variable without a stochastic trend or unit root is sometimes called I(0). In other words, a variable is I(0) if its stochastic part is stationary. A set of I(1) variables is called cointegrated if a linear combination exists which is I(0). Occasionally it is convenient to consider systems with both I(1) and I(0) variables. In this case the concept of cointegration is extended by calling any linear combination which is I(0) a cointegration relation although this terminology is not in the spirit of the original definition because it can result in a linear combination of I(0) variables being called a cointegration relation. Further discussion of cointegration may also be found in Chapter 30 by Dolado, Gonzalo, and Marmol in this volume.
As mentioned earlier, we allow for deterministic polynomial trends. For these terms we assume for convenience that they are at most linear. In other words, we exclude higher order polynomial trend terms. For practical purposes this assumption is not a severe limitation.