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# Break-points detection with atheoretical regression trees - PowerPoint PPT Presentation

Marco Reale University of Canterbury Universidade Federal do Parana, 27 th November 2006. Break-points detection with atheoretical regression trees. Acknowledgements. The results presented are the outcome of joint work with: Carmela Cappelli and William Rea. Structural Breaks.

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University of Canterbury

Universidade Federal do Parana, 27th November 2006

Break-points detection with atheoretical regression trees

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.

Examples with G=2,3 and m=1

Example with G=3 and m=2

• 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