Structural Break Detection in Time Series Models

1. 1 Structural Break Detection in Time Series Models Richard A. Davis Thomas Lee Gabriel Rodriguez-Yam Colorado State University (http://www.stat.colostate.edu/~rdavis/lectures) This research supported in part by: National Science Foundation IBM faculty award EPA funded project entitled: Space-Time Aquatic Resources Modeling and Analysis Program (STARMAP) Much of the recent interest in time series modeling has focused on data from financial markets, from communications channels, from speech recognition and from engineering applications, where the need for non-Gaussian, non-linear, and nonstationary models is clear. With faster computation and new estimation algorithms, it is now possible make significant in-roads on modeling more complex-phenomena. In this talk, we will develop estimation procedures for a class of models that can be used for analyzing a wide range of time series data that exhibit structural breaks. The novelty of the approach taken here is to combine the use of genetic algorithms with the principle of minimum description length (MDL), an idea developed by Rissanan in the 1980s, to find "optimal" models over a potentially large class of models. The underlying theme in our approach is to consider models whose parameters are piece-wise constant over different time epochs. For example, once might consider the class of autoregressive models whose coefficients are piece-wise constant. The idea behind MDL is to decompose the code length of the "data" into two pieces (see the survey paper by Lee (2001) for more details). Roughly speaking, if the code length of the "data" is the amount of memory required to store the data, then the code length of the data can be decomposed into the sum of the code length of the fitted model and the code length of the data given the fitted model, i.e., L("data")=L("fitted model") + L("data given fitted model"). Here L("fitted model") might be interpreted as the code length of the model parameters and L("data given fitted model") as the code length of the residuals from the fitted model. It follows that a more complex model is chosen provided there has been a compensating decrease in the code length of the residuals. According to the MDL principle, the "best" model is the one producing the shortest code length for the data. The genetic algorithm is then used to search for the best model. This methodology will be demonstrated in a number of applications. In addition to fitting piece-wise autoregressive models, which works well even for local stationary models that are smooth, we will also consider extensions to piece-wise nonlinear models. The latter presents some formidable challenges since a nonlinear model can often appear to have quite varied behavior over different time epochs. (This research is joint work with Thomas Lee and Gabriel Rodriguez-Yam.) Much of the recent interest in time series modeling has focused on data from financial markets, from communications channels, from speech recognition and from engineering applications, where the need for non-Gaussian, non-linear, and nonstationary models is clear. With faster computation and new estimation algorithms, it is now possible make significant in-roads on modeling more complex-phenomena. In this talk, we will develop estimation procedures for a class of models that can be used for analyzing a wide range of time series data that exhibit structural breaks. The novelty of the approach taken here is to combine the use of genetic algorithms with the principle of minimum description length (MDL), an idea developed by Rissanan in the 1980s, to find "optimal" models over a potentially large class of models. The underlying theme in our approach is to consider models whose parameters are piece-wise constant over different time epochs. For example, once might consider the class of autoregressive models whose coefficients are piece-wise constant. The idea behind MDL is to decompose the code length of the "data" into two pieces (see the survey paper by Lee (2001) for more details). Roughly speaking, if the code length of the "data" is the amount of memory required to store the data, then the code length of the data can be decomposed into the sum of the code length of the fitted model and the code length of the data given the fitted model, i.e., L("data")=L("fitted model") + L("data given fitted model"). Here L("fitted model") might be interpreted as the code length of the model parameters and L("data given fitted model") as the code length of the residuals from the fitted model. It follows that a more complex model is chosen provided there has been a compensating decrease in the code length of the residuals. According to the MDL principle, the "best" model is the one producing the shortest code length for the data. The genetic algorithm is then used to search for the best model. This methodology will be demonstrated in a number of applications. In addition to fitting piece-wise autoregressive models, which works well even for local stationary models that are smooth, we will also consider extensions to piece-wise nonlinear models. The latter presents some formidable challenges since a nonlinear model can often appear to have quite varied behavior over different time epochs. (This research is joint work with Thomas Lee and Gabriel Rodriguez-Yam.)

2. 2 Illustrative Example

3. 3 Illustrative Example

4. 4

5. 5 Introduction

6. 6 Examples

7. 7 Piecewise AR models (cont)

8. 8

9. 9 Examples (cont)

10. 10 Model Selection Using Minimum Description Length

11. 11 Model Selection Using Minimum Description Length (cont)

12. 12

13. 13 Optimization Using Genetic Algorithms

14. 14 Application to Structural Breaks�(cont)

15. 15 Implementation of Genetic Algorithm�(cont)

16. 16 Implementation of Genetic Algorithm�(cont)

17. 17 Simulation Examples-based on Ombao et al. (2001) test cases

18. 18 1. Piecewise stat (cont)

19. 19 Simulation Examples (cont)

20. 20

21. 21 3. Slowly varying AR(2) (cont)

22. 22

23. 23 2. Slowly varying AR(2) (cont)

24. 24

25. 25

26. 26

27. 27

28. 28

29. 29 Application to GARCH (cont)

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35. 35 Summary Remarks