Conclusions and Extensions
Since the publication of Sims’ (1980) critique of classical econometric modeling, VAR processes have become standard tools for macroeconometric analyses. A brief introduction to these models, their estimation, specification, and analysis has been provided. Special attention has been given to cointegrated systems. Forecasting, causality, impulse response, and policy analysis are discussed as possible uses of VAR models. In some of the discussion exogenous variables and deterministic terms are explicitly allowed for and, hence, the model class is generalized slightly relative to standard pure VAR processes.
There are now different software packages that support VAR analyses. For example, PcFiml (see Doornik and Hendry, 1997) and EVIEWS may be used. Furthermore, packages programmed in GAUSS exist which simplify a VAR analysis (see, e. g. Haase et al., 1992).
In practice, further model generalizations are often useful. For instance, to obtain a more parsimonious parameterization allowing for MA terms as well and, hence, considering the class of vector autoregressive moving average processes may be desirable (see Hannan and Deistler, 1988; Lutkepohl and Poskitt, 1996). Generalizations of the concept of cointegration may be found in Chapter 30 by Dolado, Gonzalo, and Marmol in this volume. Especially for financial time series modeling the conditional second moments is sometimes of primary interest. Multivariate ARCH type models that can be used for this purpose are, for instance, discussed by Engle and Kroner (1995). Generally, nonlinearities of unknown functional form may be treated nonparametrically, semiparametrically, or semi-nonparametrically. A large body of literature is currently developing on these issues.
* I thank Jorg Breitung and Moses Salau for helpful comments on an earlier draft of this chapter and the Deutsche Forschungsgemeinschaft, SFB 373, as well as the European Commission under the Training and Mobility of Researchers Programme (contract No. ERBFMRXCT980213) for financial support. An extended version of this chapter with more explanations, examples and references is available from the internet at http://sfb. wiwi. hu-berlin. de in subdirectory bub/papers/sfb373.
 The estimator 0 is defined, uniquely in a neighborhood in 0, by the k equations RT(0)r(0) = 0;
 for any root-n consistent 0, a consistent estimate of var(plim n1/2(0 – 0O)) is given by the inverse of n-1RT(0)R(0). Formally,