1.1.4 Part IV

1.1.4 Part IV

The fourth part of the book covers a variety of topics that are specific to forecasting. Chapter 19 discusses predictions under model instability. This chapter builds on the earlier observation that all forecasting models are simplified representations of a much more complex and evolving data-generating process. A key source of model misspecification is the constant-parameter assumption made by many prediction models. Empirical evidence suggests that simple ARMA models are in fact misspecified for many macroeconomic variables. The chapter first discusses how model instability can be monitored before moving over to discuss prediction approaches that specifically incorporate time-varying parameters, including random walk or mean-reverting parameters and regime switching parameters.

The previous chapters deal with cases where the forecast horizon is relatively short. Chapter 20 directly attacks the case where the forecast horizon can be long. Oftentimes a policy maker or budget office is interested in 5 or 10-year forecasts of revenue or expenditures. Interest may also lie in forecasts of the average growth rate over some period. From an estimation perspective, whether the forecast horizon is short or long is measured relative to the length of the data sample. We discuss these issues in chapter 20.

Real-time forecasting methods emphasize the need to ensure that all information and all methods used to construct a forecast would have been available in real time. This consideration becomes particularly relevant in so-called pseudo out-of-sample forecasts that simulate a sequence of historical forecasts. Many macroeconomic time series are subject to revisions that become available only after the date of the forecast. Since the selection of a forecasting model and estimation of its parameters may depend on the conditioning information set, which vintage of data is used can sometimes make a material difference. Similar issues related to data availability are addressed by a relatively new field known as nowcasting which uses filtering and updating algorithms to account for the jagged-edge nature of data, i.e., the fact that data are released at different frequencies and on different dates. These issues are covered in chapter 21.

This chapter also covers models for predicting data that take the format of either counts, and so are restricted to being an integer number, or durations, i.e., the length of the time intervals between certain events. The nature of the dependent variable gives rise to specific forecasting models, such as Poisson models, that are different from the models covered in the previous chapters of the book. Count models have gained widespread popularity in the context of analysis of credit events such as bankruptcies or credit card default, while duration analysis is used to predict unemployment spells and times between trades in financial markets.

1.2 TECHNICAL NOTES

Throughout the book we follow standard statistical methods which view the data as realizations of underlying random variables. Objective functions and other functions of interest are then also functions of random variables. Further, we assume that all functions are measurable, including functions that arise from maximizations of functions over parameters. We are rarely explicit about these assumptions, though this is seldom an issue for the functions examined in the book.

The decision-theoretic approach relies on the existence of risk or expected loss. For loss functions that are bounded, this is usually not problematic, but many popular loss functions are not bounded. For example, mean squared error loss and mean absolute error loss are the most popular loss functions in practice, and neither is bounded. It is fairly standard in the forecasting literature to simply assume that the expected loss exists, and further assume that the asymptotic limit of expected loss is the expected value of the limiting random variable that measures the loss. Throughout the book we follow this practice without giving conditions. Forecasting practice in some instances does seem to enforce “boundedness” of a sort on forecast losses; for example, in evaluating nonlinear models with mean squared error loss, often extreme forecasts that could lead to very large losses are removed and so the loss is in effect bounded.

Throughout the book we tend not to present results as fully worked theorems but instead give the main conditions under which the results hold. Original papers with the full set of conditions are cited. The reasons for this approach are twofold. First, often there are many overlapping sets of conditions that would result in lengthy expositions on often very straightforward methods if we were to include all the details of a result. Second, many of the conditions are highly technical in nature and often difficult or impossible to verify.