# Forecasting Economic. Time Series

The construction and interpretation of economic forecasts is one of the most publicly visible activities of professional economists. Over the past two decades, increased computer power has made increasingly sophisticated forecasting methods routinely available and the role of economic forecasting has expanded. Economic forecasts now enter into many aspects of economic life, including business planning, state and local budgeting, financial management, financial engineering, and monetary and fiscal policy. Yet, with this widening scope comes greater opportunities for the production of poor forecasts and the misinterpretation of good forecasts. Responsible production and interpretation of economic forecasts requires a clear understanding of the associated econometric tools, their limits, and an awareness of common pitfalls in their application.

This chapter provides an introduction to the main methods used for forecasting economic time series. The field of economic forecasting is large, and, because of space limitations, this chapter covers only the most salient topics. The focus here will be on point forecasts, that is, forecasts of future values of the time series. It is assumed that the historical series is relatively "clean," in the sense of having no omitted observations, being observed at a consistent sampling frequency (e. g. monthly), and either having no seasonal component or having been seasonally adjusted. It is assumed that the forecaster has quadratic (i. e. mean squared error) loss. Finally, it is assumed that the time series is sufficiently long, relative to the forecast horizon, that the history of the time series will be informative for making the forecast and for estimating parametric models.

This chapter has four substantive sections. Section 2 provides a theoretical framework for considering some of the tradeoffs in the construction of economic forecasts and for the comparison of forecasting methods. Section 3 provides a

glimpse at some of the relevant empirical features of macroeconomic time series data. Section 4 discusses univariate forecasts, that is, forecasts of a series made using only past values of that series. Section 5 provides an overview of multivariate forecasting, in which forecasts are made using historical information on multiple time series.

There are many interesting and important aspects of economic forecasting that are not covered in this chapter. In some applications, it is of interest to estimate the entire distribution of future values of the variable of interest, conditional on current information, or certain functions of that conditional distribution. An example that arises in macroeconomics is predicting the probability of a recession, an event often modeled as two consecutive declines in real gross domestic product. Other functions of conditional distributions arise in finance; for examples, see Diebold, Gunther, and Tay (1998).

In some cases, time varying conditional densities might be adequately summarized by time varying conditional first and second moments, that is, by modeling conditional heteroskedasticity. For example, conditional estimates of future second moments of the returns on an asset can be used to price options written on that asset. Although there are various frameworks for estimating conditional heteroskedasticity, the premier tool for modeling conditional heteroskedasticity is Engle’s (1982) so-called autoregressive conditional heteroskedasticity (ARCH) framework and variants, as discussed in Bollerslev, Engle, and Nelson (1994).

Another topic not explored in this chapter is nonquadratic loss. Quadratic loss is a natural starting point for many forecasting problems, both because of its tractability and because, in many applications, it is plausible that loss is symmetric and that the marginal cost of a forecast error increases linearly with its magnitude. However, in some circumstances other loss functions are appropriate. For example, loss might be asymmetric (would you rather be held responsible for a surprise government surplus or deficit?); see Granger and Newbold (1986, ch. 4.2) and West, Edison, and Cho (1993) for examples. Handling nonquadratic loss can be computationally challenging. The classic paper in this literature is Granger (1969), and a recent contribution is Christoffersen and Diebold (1997).

Another important set of problems encountered in practice but not addressed here involve data irregularities, such as missing or irregularly spaced observations. Methods for handling these irregularities tend to be model-dependent. Within univariate linear models and low-dimensional multivariate linear models, these are typically well handled using state space representations and the Kalman filter, as is detailed by Harvey (1989). A somewhat different set of issues arise with series that have large seasonal components. Issues of seasonal adjustment and handling seasonal data are discussed in Chapter 31 in this volume by Ghysels, Osborn, and Rodrigues.

Different issues also arise if the forecast horizon is long relative to the sample size (say, at least one-fifth the sample size) and the data exhibit strong serial correlation. Then the long-run forecast is dominated by estimates of the long-run correlation structure. Inference about the long-run correlation structure is typically nonstandard and, in some formulations, is related to the presence of large, possibly unit autoregressive roots and (in the multivariate setting) to possible cointegration

among the series. Unit roots and cointegration are respectively discussed in this volume in Chapter 29 by Bierens and Chapter 30 by Dolado, Gonzalo, and Marmol. The construction of point forecasts and forecast intervals at long horizons entails considerable difficulties because of the sensitivity to the long-run dependence parameters, and methods for handling this are examined in Stock (1996).

A final area not addressed here is the combination of competing forecasts. When a variable is forecasted by two different methods that draw on different information sets and neither model is true, typically a combination of the two forecasts is theoretically preferred to either individual forecast (Bates and Granger, 1969). For an introduction to this literature, see Granger (1989), Diebold and Lopez (1995), and Chan, Stock, and Watson (1998).

This chapter makes use of concepts and methods associated with unit autoregressive roots, cointegration, vector autoregressions (VARs), and structural breaks. These are all topics of separate chapters in this volume, and the reader is referred to those chapters for background details.

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