New Introduction to Multiple Time Series Analysis (PDF)

15 Fitting Finite Order VAR Models to In.nite Order Processes (p.531)


15.1 Background

In the previous chapters, we have derived properties of models, estimators, forecasts, and test statistics under the assumption of a true model. We have also argued that such an assumption is virtually never ful.lled in practice. In other words, in practice, all we can hope for is a model that provides a useful approximation to the actual data generation process of a given multiple time series. In this chapter, we will, to some extent, take into account this state of a.airs and assume that an approximating rather than a true model is fitted. Speci.cally, we assume that the true data generation process is an in.nite order VAR process and, for a given sample size T, a .nite order VAR(p) is fitted to the data.

In practice, it is likely that a higher order VAR model is considered if the sample size or time series length is larger. In other words, the order p increases with the sample size T. If an order selection criterion is used in choosing the VAR order, the maximum order to be considered is likely to depend on T. This again implies that the actual order chosen depends on the sample size because it will depend on the maximum order. In summary, the actual order selected may be regarded as a function of the sample size T. In order to derive statistical properties of estimators and forecasts, we will make this assumption in the following. More precisely, we will assume that the VAR order goes to in.nity with the sample size. Under that assumption, an asymptotic theory has been developed that will be discussed in this chapter.

In Section 15.2, the assumptions for the underlying true process and for the order of the process .tted to the data are speci.ed in detail and asymptotic estimation results are provided for stable processes. In Section 15.3, the consequences for forecasting are discussed and impulse response analysis is considered