Econometric Forecasting

Transcription

1 Graham Elliott Econometric Forecasting Course Description We will review the theory of econometric forecasting with a view to understanding current research and methods. By econometric forecasting we mean methods that are based on past data using statistical methods to construct forecasts. The course proceeds by showing that nearly all forecasting methods used in practice fall into a decision theoretic framework, which is to say that methods are applied and evaluated using average loss as a criterion of their usefulness. This is true for nearly all the work done in econometrics and statistics, although more recent developments in other elds outside this framework will also receive attention. The rst part of the course then discusses general issues with how methods t into the framework, basic general results on optimality that arise because of this, and general approaches to estimating and formulating forecast models. Both Bayesian and Classical results will be analyzed. The main results of the theory will be presented and explained typically by example rather than by formal proofs of the statements. The second part of the course places commonly applied methods within the framework of decision theory. This provides a natural and very useful way to understand the plethora of methods and models used to forecast any particular outcome of interest. It also provides a useful structure to discuss the claims made by authors and promoters of various methods. Within this big picture, choices of methods becomes easier to understand. Throughout we will be clear on how to actually apply the various methods to data as well as understand their statistical properties. We then turn to forecast evaluation. This is a an extremely di cult statistical problem, one where over the last decade many advances have been made. We will discuss historical approaches and the recent advances, as well as the di cult issues faced by a practical forecaster in evaluating methods. As in the previous section we will also be clear on how the methods are applied in practice. Finally some special topics will be presented to enhance the breadth of applicability of the above results and methods as well as go into more detail on methods. 1

2 Background Reading. We will not follow any speci c text. However a knowledge of time series at the level of Hamilton (1994) will be helpful. Many results at this level of technicality will be presented. Course Outline. The papers listed are those we cover more closely in class as well as secondary/further readings. The plan is to give a more comprehensive list than we have time to go through. It is also likely that we will not have time to cover all the topics. A. Foundations of Forecasting Theory. 1. Introduction. We will develop forecasting as an example of decision theory hilighting the main results illustrated with simple examples. This will serve as a framework for understanding the various dimensions of special cases and the development of forecasting methods. For an overview see Elliott and Timmermann (2008). 1. Basic Decision Theory Problem Here we examine the basic decision theory problem, along with the classical (frequentist) and Bayesian approaches to making the methods feasable with data. Relationships between the methods will be reviewed. This follows classical estimation theory. 2. Loss functions. An overview of loss functions and their features, along with a discussion of the importance of correctly specifying the loss function relevant to the forecasting problem. See Granger (1969,1999) for some issues. 3. Density Forecasting A description of the basic problem of density forecasting and how it relates to the basic decision theoretic problem A dicussion of the pro s and con s of using density forecasts over point forecasts will be given. A reviews is available in Tay and Wallis (2000). B. Methods and Issues with Building Forecast Models. Typical classical approaches require methods to select the model speci cation, estimation procedure and to determine the variables to be included. In this section we mix these ideas 2

3 along with a list of popular model choices. 1. ARIMA models and extensions. Still a commonly employed approach, we review the methods brie y. Extensions to nonlinear models and other approaches such as exponential smoothing that rely only on past outcomes of the variable to be forecast are examined. The classic reference is the book by Brockwell and Davis (1991), however for our purposes Hamilton (1994) Chapters 1-5 are su cient. 2. VAR models More typically VAR models are employed. We review issues with estimation, discussing alternative estimators such as shrinkage estimators. We review recent developments involving shrinkage towards economic models. The seminal paper is Sims (1980), a comprehensive analysis is in Lutkepohl (1993). Alternative estimation strategies to OLS include Litterman (1986) and more recently Del Negro and Schorfheide (2004). 3. Dynamic Factor Models. Dynamic factor models are an alternative to VAR s, attempting to achieve a more parsimonious representation when there are many predictors without losing forecasting ability. We show how these are estimated and review recent developments. A discussion of the different approaches is in Stock and Watson (2006) and Breitung and Eickmeier (2005). Papers that introduce ideas include Forni et. al (2000) and Stock and Watson (2002). 4. Nonlinear Models Many nonlinear models have become popular, we review common approaches and discuss. Also included are nonparametric approaches such as neural networks that attempt to provide good approximations to general unknown model spec ciations. The Handbook of Forecasting chapters by Terasvirta (2006) and White (2006) provide good overviews. 5. Model Selection Methods Economic theory does not pin down a particular model, hence methods have been derived to choose between a set of models. We review these methods and place them in a uni ed context. Problems of empirical model selection are shown. A nice textbook overview is in Gourieroux and Monfort (1995) Chapter 22. 3

4 6. Model combination. Really just another estimation procedure, we examine such links as well as methods employed in practice. Both classical and Bayesian model averaging are examined. See Timmermann (2006) for a recent review. The main seminal paper is Bates and Granger (1969). Recent work includes Hansen (2007) and Raftery et. al. (1997). C. Forecast Model Evaluation The basic problem of empirical estimates of risk is reviewed (already presented in Model selection methods) and extensions are examined. 1. Estimation and inference for expected loss. Review methods to examine and do inference on empirical estimates of average loss. See West (1997, 2006). Recent work from machine learning on bounding risk is also examined. 2. Testing rationality of the forecasts Examination of inference problems, the problem of out of sample comparisons etc. are covered. The impact of misspeci ed loss functions is also examined. The notion of encompassing is also reviewed. See Chong and Hendry (1986)McCracken and West (1998), Elliott et. al. (2005). 3. Model selection methods revisited. Methods for using evaluation techniques to choose between forecast models are e ectively a sub class of model selection methods. We examine these methods with reference to the earlier discussion. See White (1997) for a recent contribution. 4. Evaluating density forecasts Review methods used to evaluate density forecasts. Discuss problems of model and parameter uncertainty and density estimation. For a review see Diebold et. al. (1998). D Special Topics 1. Kalman Filter A quick overview of the method and how it is used in forecasting. A review is available in Harvey (2006). 2. Nonstationarities - Breaks and Unit roots 4

5 Issues that arise for forecasting when there are breaks and unit roots. A particular emphasis is on what is not di erent from the above, as well as an examination of what is really feasible in such forecasting environments. See the Handbook chapter Elliott (2006) for a review. 3. Unobserved Outcomes. In a great deal of forecsting environments, the outcome is never observed (an example is volatility forecasting, although even GDP forecasting could come under this banner). We examine how the framework we have built relates to this variation. 4. Real Time Issues involved with real time forecasts and using real time data to evaluate. See the handbook chapter Croushore (2006) for a review. References A.E. Raftery, D. M., and J. Hoeting (1997): Bayesian Model Averaging for Linear Regression Models, Journal of the American Statistical Association, 92, Bates, J., and C. Granger (1969): The Combination of Forecasts, Operations Research Quarterly, 20, Breitung, J., and S. Eickmeier (2005): Dynamic Factor Models, Deutsche Bundesbank Discussion Paper 38/2005. Brockwell, P., and R. Davis (1991): Time Series: Theory and Methods. Springer-Verlag, New York, second edn. Chong, Y., and D. Hendry (1986): Econometric Evaluation of Linear Macro-Economic Models, Review of Economic Studies, 53, Croushore, D. (2006): Forecasting with Real Time Macroeconomic Data, in Handbook of Forecasting Volume 1, ed. by A. T. G. Elliott, and C. Granger, pp Elsevier, Amsterdam. 5

6 Elliott, G., and U. Mueller (2006): E cient Tests for General Persistent Time Variation in Regression Coe cients", Review of Economic Studies, 64, Elliott, G., and A. Timmermann (2005): Review of Economic Studies, 72, Estimating Loss Function Parameters, (2008): Economic Forecasting, Journal of Economic Literature, 46, Forni, M, M. H. M. L., and L. Reichlin (2000): The Generalized Dynamic Factor Model: Identi cation and Estimation, Review of Economics and Statistics, 82, F.X. Diebold, T. G., and A. Tay (1998): Evaluating Density Forecasts, International Economic Review, 39, Gourieroux, A., and A. Monfort (1995): Statistics and Econometric Models, Volume 2. Cambridge Press, Cambridge. Granger, C. (1969): Investigating Causal Relations by Econometric Models and Cross Spectral Methods, Econometrica, 37, Granger, C., and N. Hyung (1999): Occasional Structural Breaks and Long Memory, UCSD-WP Hamilton, J. (1994): Time Series Analysis. Princeton University Press, Princeton, New Jersey. Hansen, B. (2007): Least Squares Model Averaging, Econometrica, 75, Harvey, A. (2006): Forecasting with Unobserved Components Time Series Models, in Handbook of Forecasting Volume 1, ed. by A. T. G. Elliott, and C. Granger, pp Elsevier, Amsterdam. Litterman, R. (1986): Forecasting with Bayesian Autoregressions - Five Years of Experience, Journal of Business and Economic Statistics, 4, Lutkepohl, H. (1993): An Introduction to Multiple Time Series, 2nd ed. Springer Verlag, New York. 6

7 Negro, M. D., and F. Schorfheide (2004): Priors from General Equilibrium Models for VAR s, International Economic Review, 45, Sims, C. (1980): Macroeconomics and Reality, Econometrica, 48, Stock, J., and M. Watson (2002): Forecasting using Principal Components from a Large Number of Predictors, Journal of the American Statistical Association, 97, (2006): Forecasting with Many Predictors, in Handbook of Forecasting Volume 1, ed. by A. T. G. Elliott, and C. Granger, pp Elsevier, Amsterdam. Tay, A., and K. Wallis (2000): Density Forecasting: A Survey, Journal of Forecasting, 19, Terasvirta, T. (2006): Forecasting Economic Variables with Nonlinear Models, in Handbook of Forecasting Volume 1, ed. by A. T. G. Elliott, and C. Granger, pp Elsevier, Amsterdam. Timmermann, A. (2006): Forecast Combinations, in Handbook of Forecasting Volume 1, ed. by A. T. G. Elliott, and C. Granger, pp Elsevier, Amsterdam. West, K. (1997): Asymptotic Inference about Predictive Ability, Econometrica, 64, (2006): Forecast Evaluation, in Handbook of Forecasting Volume 1, ed. by A. T. G. Elliott, and C. Granger, pp Elsevier, Amsterdam. West, K., and M. McCracken (1998): Regression Based Tests of Predictive Ability, International Economic Review, 39, White, H. (1997): A Reality Check for Data Snooping, Econometrica, 68, (2006): Approximate Nonlinear Forecasting Methods, in Handbook of Forecasting Volume 1, ed. by A. T. G. Elliott, and C. Granger, pp Elsevier, Amsterdam. 7