[R] CUSUM SQUARED structural breaks approach?

[Achim Zeileis]

On Tue, 11 Jan 2005 19:33:41 +0000 Rick Ram wrote:

> Groundwork for the choice of break method in my specific application
> has already been done - otherwise I would need to rework the wheel
> (make a horribly detailed comparison of performance of break
> approaches in context of modelling post break)
> 
> If it interests you, Pesaran & Timmerman 2002 compared CUSUM Squared,
> BaiPerron and a time varying approach to detect singular previous
> breaks in reverse ordered financial time series so as to update a
> forecasting model. 

Yes, I know that paper. And if I recall correctly they are mainly
interested in modelling the time period after the last break. For this,
the reverse ordered recursive CUSUM approach works because they
essentially look back in time to see when their predictions break down.
And for their application looking for variance changes also makes sense.
The approach is surely valid and sound in this context...but it might be
possible to do something better (but I would have to look much closer at
the particular application to have an idea what might be a way to go).

> This works "fine" i.e. the plot looks correct.  The problem is how to
> appropriately normalise these to rescale them to what the CUSUM
> squared procedure expects (this looks to be a different and more
> complicated procedure than the normalisation used for the basic
> CUSUM).  I am from an IT background and am slightly illiterate in
> terms of math notation... guidance from anyone would be appreciated

I just had a brief glance at BDE75, page 154, Section 2.4. If I
haven't missed anything important on reading it very quickly, you just
need to do something like the following (a reproducible example, based
on data from strucchange, using a notation similar to BDE's):

## load GermanM1 data and model
library(strucchange)
data(GermanM1)
M1.model <- dm ~ dy2 + dR + dR1 + dp + ecm.res + season

## compute squared recursive residuals
w2 <- recresid(M1.model, data = GermanM1)^2
## compute CUSUM of squares process
sr <- ts(cumsum(c(0, w2))/sum(w2), end = end(GermanM1$dm), freq = 12)
## the border (r-k)/(T-k)
border <- ts(seq(0, 1, length = length(sr)),
             start = start(sr), freq = 12)

## nice plot
plot(sr, xaxs = "i", yaxs = "i", main = "CUSUM of Squares")
lines(border, col = grey(0.5))
lines(0.4 + border, col = grey(0.5))
lines(- 0.4 + border, col = grey(0.5))

Instead of 0.4 you would have to use the appropriate critical values
from Durbin (1969) if my reading of the paper is correct.
 
hth,
Z

> Does anyone know if this represents some commonly performed type of
> normalisation than exists in another function??
> 
> I will hunt out the 1969 paper for the critical values but prior to
> doing this I am a bit confused as to how they will
> implemented/interpreted... the CUSUM squared plot does/should run
> diagonally up from left to right and there are two straight lines that
> one would put around this from the critical values.  Hence, a
> different interpretation/implementation of confidence levels than in
> other contexts.  I realise this is not just a R thing but a problem
> with my theoretical background.
> 
> 
> Thanks for detailed reply!
> 
> Rick.
> 
> 
> > 
> > But depending on the model and hypothesis you want to test, another
> > technique than CUSUM of squares might be more appropriate and also
> > available in strucchange.
> 
> > 
> > hth,
> > Z
> > 
> > > Any help or pointers about where to look would be more than
> > > appreciated!  Hopefully I have just missed obvious something in
> > > the package...
> > >
> > > Many thanks,
> > >
> > > Rick R.
> > > 
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