Re: [Rd] partial correlation function for multivariate time series

From: Paul Gilbert <pgilbert_at_bank-banque-canada.ca>
Date: Fri, 14 Sep 2007 11:50:35 -0400

Simone Giannerini wrote:
> Dear Paul,
>
> there is no mention to the pacf in the multivariate setting in the book
> you suggested.

My apologies, I should have looked more carefully and realized the pacf discussion in Granger and Newbold is all univariate.

> As I mentioned in private I suspect that pacf() in the multivariate case
> computes the
> partial autoregression matrix (in the terminology of Reinsel) rather
> than the partial autocorrelation matrix

It looks like pacf() uses the calculation in stats:::ar.yw.default , so roughly sounds consistent with what you say. I do think parts of this code pre-date Reisel's book, so inconsistency with his book would probably be imperfect foresight.

BTW, I think bug reports with suggested fixes are usually more useful. And, at the very least, it seems the references in the documentation could be improved.

Paul Gilbert

> as the two coincide in the univariate case but are different in the
> multivariate case as stated explicitly
> in Reisel (Sec. 3.3).
> This would explain the coefficients well above 1 (in modulus) in the
> example I have given.
> To support my statement you can fit A VAR model to the data and compare
> the coefficients with the results
> from pacf():
>
> set.seed(10)
> x <- rnorm(1000,sd=10000)
> y <- rnorm(1000,sd=1)
>
> library(vars);
> mod1 <- VAR(ts(cbind(x,y)),p=4,type="none"); # fit a VAR (OLS)
>
> ## Have a look at estimated coefficients
>

>>  noquote(formatC(mod1$varresult$x$coefficients,format="f"))

> ## Edited by me, compare with the first column of the results below from
> the pacf (pacf(....)$acf[,1,])
> x.l1 x.l2 x.l3 x.l4
> 0.047 0.013 0.004 0.014
> y.l1 y.l2 y.l3 y.l4
> 374.117 72.758 -526.452 126.610
>>  noquote(formatC(mod1$varresult$y$coefficients,format="f"))

> ## Edited by me, compare with the second column of the results below
> from the pacf (pacf(....)$acf[,2,])
> x.l1 x.l2 x.l3 x.l4
> 0.000 -0.000 0.000 0.000
> y.l1 y.l2 y.l3 y.l4
> -0.046 -0.025 -0.033 -0.020
>
> pacf(ts(cbind(x,y)),plot=FALSE,lag.max=4)
>
> Partial autocorrelations of series 'ts(cbind(x, y))', by lag
>
> , , x
>
> x y
> 0.047 ( 1) 0.000 (-1)
> 0.011 ( 2) 0.000 (-2)
> 0.005 ( 3) 0.000 (-3)
> 0.013 ( 4) 0.000 (-4)
>
> , , y
>
> x y
> 374.052 ( 1) -0.045 ( 1)
> 66.717 ( 2) -0.024 ( 2)
> -535.810 ( 3) -0.031 ( 3)
> 120.802 ( 4) -0.020 ( 4)
>
> As you can see the coefficients fairly agree.
> I might file a bug report in some days unless someone will prove me
> wrong before.
>
> Regards,
>
> Simone
>
>
>
> On 11/09/2007, *Paul Gilbert* <pgilbert_at_bank-banque-canada.ca
> <mailto:pgilbert_at_bank-banque-canada.ca>> wrote:
>
> I think the reference for pacf is
>
> @BOOK{GraNew77,
> author = {Granger, C. W. J. and Newbold, Paul},
> title = {Forecasting Economic Time Series},
> publisher = {Academic Press},
> year = 1977
> }
>
> It certainly would not be Reisel's book, as parts of the code predate
> that by many years.
>
> Paul Gilbert
>
> Simone Giannerini wrote:
> > Dear all,
> >
> > I found the following behaviour with pacf() in the multivariate
> case,
> >
> > set.seed(10)
> > x <- rnorm(1000,sd=10000)
> > y <- rnorm(1000,sd=1)
> > pacf(ts(cbind(x,y)),plot=FALSE,lag.max=10)
> >
> > Partial autocorrelations of series 'cbind(x, y)', by lag
> >
> > , , x
> >
> > x y
> > 0.047 ( 1) 0.000 ( -1)
> > 0.011 ( 2) 0.000 ( -2)
> > 0.005 ( 3) 0.000 ( -3)
> > 0.013 ( 4) 0.000 ( -4)
> > 0.050 ( 5) 0.000 ( -5)
> > 0.034 ( 6) 0.000 ( -6)
> > 0.026 ( 7) 0.000 ( -7)
> > -0.029 ( 8) 0.000 ( -8)
> > -0.010 ( 9) 0.000 ( -9)
> > -0.013 ( 10) 0.000 (-10)
> >
> > , , y
> >
> > x y
> > 374.052 ( 1) -0.045 ( 1)
> > 66.717 ( 2) -0.024 ( 2)
> > -535.810 ( 3) -0.031 ( 3)
> > 120.802 ( 4) -0.020 ( 4)
> > 142.824 ( 5) 0.004 ( 5)
> > -211.711 ( 6) -0.010 ( 6)
> > 201.856 ( 7) 0.058 ( 7)
> > 286.079 ( 8) -0.035 ( 8)
> > -134.057 ( 9) 0.032 ( 9)
> > -18.200 ( 10) 0.019 ( 10)
> >
> > Since there are multiple ways of defining the pacf for
> multivariate time
> > series
> > (see e.g. G.C. Reinsel, Elements of multivariate time series
> analysis, II
> > edition, Springer, section 3.3) and given that
> > in ?pacf there is no reference to articles or books regarding its
> > computation
> > I do not know whether this behaviour is expected or it is a bug
> instead.
> > In the first case could you provide a reference for it? In the
> second case I
> > might file a bug report.
> > Thank you for the great work you are doing for the scientific
> community.
> >
> > kind regards,
> >
> > Simone Giannerini
> >
> > WINDOWS
> >
> > platform i386-pc-mingw32
> > arch i386
> > os mingw32
> > system i386, mingw32
> > status
> > major 2
> > minor 5.1
> > year 2007
> > month 06
> > day 27
> > svn rev 42083
> > language R
> > version.string R version 2.5.1 (2007-06-27)
> >
> > LINUX
> >
> >
> >>R.Version()
> >
> > $platform
> > [1] "x86_64-unknown-linux-gnu"
> >
> > $arch
> > [1] "x86_64"
> >
> > $os
> > [1] "linux-gnu"
> >
> > $system
> > [1] "x86_64, linux-gnu"
> >
> > $status
> > [1] ""
> >
> > $major
> > [1] "2"
> >
> > $minor
> > [1] "5.1"
> >
> > $year
> > [1] "2007"
> >
> > $month
> > [1] "06"
> >
> > $day
> > [1] "27"
> >
> > $`svn rev`
> > [1] "42083"
> >
> > $language
> > [1] "R"
> >
> > $version.string
> > [1] "R version 2.5.1 (2007-06-27)"
> ====================================================================================
>
> La version française suit le texte anglais.
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> --
> ______________________________________________________
>
> Simone Giannerini
> Dipartimento di Scienze Statistiche "Paolo Fortunati"
> Universita' di Bologna
> Via delle belle arti 41 - 40126 Bologna, ITALY
> Tel: +39 051 2098262 Fax: +39 051 232153
> http://www2.stat.unibo.it/giannerini/
> ______________________________________________________

La version française suit le texte anglais.


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