Break-points detection with atheoretical regression trees

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Break-points detection with atheoretical regression trees

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Break-points detection with atheoretical regression trees

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Marco Reale

University of Canterbury

Universidade Federal do Parana, 27th November 2006

The results presented are the outcome of joint work with:

Carmela Cappelli

and

William Rea

- A structural break is a statement about parameters in the context of a specific model.
- A structural break has occurred if at least one of the model parameters has changed value at some point (break-point).
- We consider time series data.

Their detection is important for:

- forecasting (latest update of the DGP);
- Analysis.
With regard to this point a recent debated issue is fractional integration vs structural breaks.

- Test for an a priori candidate break-point.
- Splits the sample period in two subperiods and test the equality of the parameter sets with an F statistic.
- It cannot be used for unknown dates: misinformation or bias.

- We can compute Chow statistics for all possible break-points.
- If the candidate breakpoint is known a priori, then a Chi-square statistics can be used.

- Proposed by Brown, Durbin and Evans.
- It checks the cumulative sum of the residuals.
- It tests the null of no breakpoints against one or more breakpoints.

It exploits the Quandt statistics for a priori unknown break-points.

- It finds multiple breaks at unknown times.
- Application of Fisher algorithm (1958) to find optimal exhaustive partitions.
- It requires prior indication of number of breaks.
- Applied recursively after positive indication provided by CUSUM.
- Use of AIC to decide the number of breaks.

- Eventually Fisher selects the partition with the minimum deviance.
- It is a global optimizer, but was computationally feasible only for very small n and G (even with today's computers).
- Using later results in dynamic programming Bai and Perron can use the Fisher algorithm reasonably fast for n=1000 and any G and m.
- Fisher’s algorithm is related to regression trees.

- Trees are particular kinds of directed acyclic graphs.
- In particular we consider binary trees.
- Splits to reduce heterogeneity.

Node 1 is called root.

Node 5 is called leaf.

The other nodes are called branches.

- Regression trees are sequences of hierarchical dichotomous partitions with maximum homogeneity of y projected by partitions of explanatory variables.
- y is a control or response variable.

Regression trees don't provide necessarily optimal partitions

- Any artificial strictly ascending or descending sequence as a covariate, e.g. {1,2,3,4...} would do all the optimal dichotomous partitions.
- It also works as a counter.
- It is not a theory based covariate so the name, Atheoretical regression trees ....yes it's ART.
- ART is not a global optimizer.

Trees tend to oversplit so the overgrown tree needs a pruning procedure:

- Cross Validation, is the usual procedure in regression tree, not ideal in general for time series;
- AIC (Akaike, 1973) tends to oversplit
- BIC (Schwarz, 1978) very good
All the information criteria robust for non normality, especially BIC.

- The simulations show an excellent performance.
- However ART performs better in long regimes.
- With short regimes it tends to find spurious breaks but the performance can be sensibly improved with an enhanced pruning technique (ETP).

- BP tends to find breaks any time the CUSUM rejects the null.
- It unlikely finds spurious breaks.
but

- It tends to underestimate the number of breaks.

- The Michigan-Huron lakes play a very important role in the U.S. economy and hence they are regularly monitored.
- In particular we consider the mean water level (over one year) time series from 1860 to 2000.

- We applied ART to the Campito Mountain Bristlecone Pine data which is an unbroken set of tree ring widths covering the period 3435BC to1969AD. A series of this length can be analyzed by ART in a few seconds. BPP was applied to the series and took more than 200 hours of CPU time to complete.Tree ring data are used as proxies for past climatic conditions.

…are:

- 1863-1969: Industrialization and global warming.
- 1333-1862: The Little Ice Age.
- 1018-1332: The Medieval Climate Optimum.
- 862-1017: Extreme drought in the Sierra Nevadas.

- Speed: Art has O(n(t)) while BP O(nng).
- Simplicity: it can be easily implemented or run with packages implementing regression trees.
- Feasibility: it can be used without almost any limitation on either the number of observations or the number of segments.
- Visualization: it results in a hierarchical tree diagram that allows for inputation of a priori knowledge.

... and of course you can say you're doing ART