The analysis of time series data is a vast enterprise. With this fact in mind, the previous chapters introduced the core concepts and analytic tools that form a foundational understanding of time series analysis. This chapter presents four more advanced topics: fractional integration, heterogeneity, forecasting, and estimating and modeling with unknown structural breaks. Although by no means an exhaustive list, the topics presented in this chapter represent concerns of the contemporary literature: they extend some of the previously discussed concepts, provide additional means of evaluating time series models, and are a means through which time series analysis can inform policy.

Fractional integration is an extension of the preceding discussion of unit roots and of tests for unit roots. The first few chapters assumed that our time series data was stationary, but it was subsequently presented that this may not necessarily be the case; as a result, tests for unit roots or an integrated series were presented in detail in Chapter 5. However, as intuition may suggest, it may not always be the case in practice that every series can be appropriately characterized as either stationary or integrated, as shocks may enter the series, persist for a nontrivial amount of time, and eventually dissipate. In such a case, the series is neither stationary nor integrated, because the shocks do not rapidly exit the series, nor do they persist indefinitely.