12.5 Conclusion

13 Volatility and Density Forecasting

13.1 Role of the Loss Function

13.2 Volatility Models

13.3 Forecasts Using Realized Volatility Measures

13.4 Approaches to Density Forecasting

13.5 Interval and Quantile Forecasts

13.6 Multivariate Volatility Models

13.7 Copulas

13.8 Conclusion

14 Forecast Combinations

14.1 Optimal Forecast Combinations: Theory

14.2 Estimation of Forecast Combination Weights

14.3 Risk for Forecast Combinations

14.4 Model Combination

14.5 Density Combination

14.6 Bayesian Model Averaging

14.7 Empirical Evidence

14.8 Conclusion

III Forecast Evaluation

15 Desirable Properties of Forecasts

15.1 Informal Evaluation Methods

15.2 Loss Decomposition Methods

15.3 Efficiency Properties with Known Loss

15.4 Optimality Tests under Unknown Loss

15.5 Optimality Tests That Do Not Rely on Measuring the Outcome

15.6 Interpreting Efficiency Tests

15.7 Conclusion

16 Evaluation of Individual Forecasts

16.1 The Sampling Distribution of Average Losses

16.2 Simulating Out-of-Sample Forecasts

16.3 Conducting Inference on the Out-of-Sample Average Loss

16.4 Out-of-Sample Asymptotics for Rationality Tests

16.5 Evaluation of Aggregate versus Disaggregate Forecasts

16.6 Conclusion

17 Evaluation and Comparison of Multiple Forecasts

17.1 Forecast Encompassing Tests

17.2 Tests of Equivalent Expected Loss: The Diebold–Mariano Test

17.3 Comparing Forecasting Methods: The Giacomini–White Approach

17.4 Comparing Forecasting Performance across Nested Models

17.5 Comparing Many Forecasts

17.6 Addressing Data Mining

17.7 Identifying Superior Models

17.8 Choice of Sample Split

17.9 Relating the Methods

17.10 In-Sample versus Out-of-Sample Forecast Comparison

17.11 Conclusion

18 Evaluating Density Forecasts

18.1 Evaluation Based on Loss Functions

18.2 Evaluating Features of Distributional Forecasts

18.3 Tests Based on the Probability Integral Transform

18.4 Evaluation of Multicategory Forecasts

18.5 Evaluating Interval Forecasts

18.6 Conclusion

IV Refinements and Extensions

19 Forecasting under Model Instability

19.1 Breaks and Forecasting Performance

19.2 Limitations of In-Sample Tests for Model Instability

19.3 Models with a Single Break

19.4 Models with Multiple Breaks

19.5 Forecasts That Model the Break Process

19.6 Ad Hoc Methods for Dealing with Breaks

19.7 Model Instability and Forecast Evaluation

19.8 Conclusion

20 Trending Variables and Forecasting

20.1 Expected Loss with Trending Variables

20.2 Univariate Forecasting Models

20.3 Multivariate Forecasting Models

20.4 Forecasting with Persistent Regressors

20.5 Forecast Evaluation

20.6 Conclusion

21 Forecasting Nonstandard Data

21.1 Forecasting Count Data

21.2 Forecasting

on forecasting techniques at University of Aarhus, Denmark, and in Bertinoro, Italy, to groups of PhD students and assistant professors. Since then, we have developed the material through courses offered to participants at many institutions, including at CREATES (University of Aarhus), American University, Edhec, Bank of Italy, SoFiE (Oxford University), and Universidad del Rosario.

Our idea was to provide a unified perspective that takes both the economics and statistics of the forecasting problem seriously. The intention was to write a forecasting book that could be used by masters and Phd students as well as professionals in places such as central banks, financial institutions, and research institutes. The book can be used as a textbook. Indeed, the first section of the book provides a unified theoretical discussion of the basic approach to forecasting that is grounded in the standard statistical practice of minimizing the “risk” (expected loss) of any method. The remainder of the book can be used both as a text and as a reference to a wide range of forecasting methods. We have tried as much as possible to provide detailed descriptions of how to construct forecasts, how to evaluate such forecasts, and how to compare them across different methods. This allows the book to serve as a single source for many widely employed forecasting methods. Through empirical applications and reviews of the empirical literature, we also shed light on which methods work well in different circumstances. We use examples ranging from stock returns to macroeconomic variables and surveys of forecasters.

Nearly all researchers who are interested in developing new forecasting methods through theoretical analysis or improving their empirical performance through data analysis work within a decision-theoretic framework. For example, the provision of point forecasts is a special case of point estimation and the provision of distributional forecasts is a special case of density estimation. We use this connection as a foundation for understanding the statistical basis for forecasting analysis and gaining a better understanding of how to think about the many forecasting methods in practical use. Thus, the first premise of the book is that taking seriously the economics underlying the forecasting problem means that the forecaster’s loss function should be the starting point of the analysis.

The second premise of the book is that the joint density of the random variables that generate the observed data used to build and evaluate a forecasting model is far more complicated than we understand theoretically or empirically. As a consequence, all forecasting models are misspecified in the sense that they are approximations to the best possible forecasting model. In practice, this means choosing forecasting methods based on their risk functions (the expected loss given the data), but acknowledging that these risk functions are themselves very complicated objects that depend on the underlying (unknown) data-generating process. It is exactly the difficulties in understanding the risk functions that allow so many different forecasting approaches to be used in empirical work.

In addition to the students in our forecasting courses who have provided valuable feedback, throughout the years we have also benefitted from discussions on forecasting with many individuals. Without implying that they necessarily agree with the points of view expressed in the book, we thank our colleagues at UCSD (past and present) including Brendan Beare, Robert Engle, Clive Granger, Jim Hamilton, Ivana Komunjer, Andres Santos, Yixiao Sun, Rossen Valkanov, and Hal White. More widely in the profession we thank Frank Diebold, Peter Hansen, Andrew Patton, Hashem Pesaran, Ulrich Müller, Jim Stock, Mark Watson, and Ken West for their insights and support. We thank all of them for the inspiration they have offered over the years. This book has also benefitted more directly from the input of many friends and colleagues. In particular, we thank Peter Hansen, Kirstin Hubrich, Simone Manganelli, Andrew Patton, Davide Pettenuzzo, Barbara Rossi, and four anonymous reviewers for comments on the book. A number of PhD students provided exceptionally capable research assistance with the empirical analysis, notably Leland E. Farmer, Antonio Gargano, Rafael Burjack, Hiroaki Kaido, and Christian Constandse. Thanks also goes to Naveen Basavanhally for help with formatting the manuscript, to Alison Durham for doing an excellent job at copyediting the manuscript, and to Ali Parrington and the team at Princeton University Press for ensuring a smooth production process.

For collaboration on forecasting papers over the years we also wish to thank several former PhD students and colleagues, including Marco Aiolfi, Ayelen Banegas, Gray Calhoun, Carlos Capistran, Luis Catao, Tolga Cenesizoglu, Leland Farmer, Antonio Gargano, Veronique Genre, Dahlia Ghanem, Ben Gillen, Clive Granger, Niels Groenborg, Massimo Guidolin, Peter Reinhard Hansen, Geoff Kenny, Ivana Komunjer, Robert Kosowski, Fabian Krueger, Robert Lieli, Asger Lunde, Aidan Meyler, Andrew Patton, Bradley Paye, Thomas Pedersen, Gabriel Perez-Quiros, Hashem Pesaran, Davide Pettenuzzo, Marius Rodrigues, Steve Satchell, Larry Schmidt, Ryan Sullivan, Russ Wermers, Hal White, and Yinchu Zhu.

Last, but not least, we wish to thank our families for their understanding and inspiration during the years it took to complete the book. The book would not have been possible without their unwavering support.

I Foundations -----------------