Break-points detection with atheoretical regression trees

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Break-points detection with atheoretical
regression trees

• Marco Reale
• University of Canterbury
• Universidade Federal do Parana, 27th November 2006

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Acknowledgements

• The results presented are the outcome of joint
  work with
• Carmela Cappelli
• and
• William Rea

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Structural Breaks

• 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.

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Relevance

• 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.

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Milestones Chow 1960

• 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.

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Milestones Quandt 1960

• 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.

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Milestones CUSUM 1974

• 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.

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Milestones Andrews 1993

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

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Bai and Perron 1998, 2003

• 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.

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Fishers algorithm
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Examples with G2,3 and m1
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Example with G3 and m2
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Bai, Perron and Fisher

• 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 n1000 and any G and m.
• Fishers algorithm is related to regression trees.

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Trees (1)

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

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Trees (2)
Node 1 is called root. Node 5 is called leaf. The
other nodes are called branches.
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Regression Trees (1)

• 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.

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Regression trees (2)
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Regression trees optimality

• Regression trees don't provide necessarily
  optimal partitions

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Atheoretical Regression Trees

• 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.

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Pruning the tree

• 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.

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Single break simulations
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Noisy square simulations
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CUSUM on noisy square
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ART on noisy square
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Some comments

• 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).

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Bai and Perron on noisy square
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Some comments

• 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.

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Application to Michigan-Huron

• 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.

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Michigan-Huron (2)
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Michigan-Huron (3)
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Michigan-Huron (4)
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Campito Mountain

• 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.

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Campito Mountain (2)
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Campito Mountain (3)
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The four most recent periods

• 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.

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Niceties of ART

• 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.

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and

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

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Dedicated to Paulo