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

From: Simone Giannerini <sgiannerini_at_gmail.com>
Date: Fri, 14 Sep 2007 16:48:57 +0200

Dear Paul,

there is no mention to the pacf in the multivariate setting in the book you suggested.
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
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> 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/
______________________________________________________

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