We use the Relevant Vector Machine, a technique of supervised learning introduced by Tipping (2001), to conduct a dynamic cointegration analysis on the time series of the price of gold and silver over the period 1971-2004. Unlike the results of traditional cointegration analysis, this study reveals that there is a dynamic long run relationship over the whole periodDynamic cointegration, relevant vector machine

Isabella Procidano

Margherita Gerolimetto

Silio Rigatti Luchini

English

Research Papers in Economics

Dynamic cointegration and Relevant Vector Machine: the
relationship between gold and silver
Margherita Gerolimetto1, Isabella Procidano2, Silio Rigatti Luchini1
1 Dipartimento di Scienze Statistiche, Universit` a di Padova.
e-mail: gerolime@stat.unipd.it, rigatti@stat.unipd.it
2 Dipartimento di Statistica, Universit` a Ca’Foscari Venezia.
e-mail: isabella@unive.it
Abstract: We use the Relevant Vector Machine, a technique of supervised learning
introduced by Tipping (2001), to conduct a dynamic cointegration analysis on the series
of the price of gold and silver over the period 1971-2004. Unlike the results of traditional
cointegrationanalysis, this study reveals that there is a dynamic long run relationship over
the whole period.
Keywords: Dynamic cointegration, Relevance Vector Machine
1. Introduction
In this paper we study the dynamic relationship between gold and silver (ﬁg.1) over the
period 1971-2004, that covers a very extensive range of economic conditions, political
change in major producers and increased sophistication in asset markets generally.
Gold and silver have been actively traded for thousands of years and remain important
and closely observed markets. They have historically been seen as close substitute for
one another and this would suggest that the two markets share the same dynamics.
However there are also economic fundamentals that may drive the prices of gold and
silver apart. While both are used extensively in industrial processes, there are signiﬁcant
differences between these uses. Therefore it seems that while they share a similar set of
drivers they each also have important unique macroeconomic drivers (see for example
Figure 1: Gold and silver time series (respectively in black and red)
Time
0 100 200 300 400
0
1
0
2
0
3
0Dooley, Isard and Taylor, 1995).
In literature there are several attempts to test for the existence and stability of the gold
silver relationship (among the others, Escribano and Granger, 1998). In this framework,
we want to give further insight to the problem. We analyse the data set of monthly
prices of gold and silver over 1971-2004 with an extremely recent approach to reveal the
existence of a dynamic cointegration relationship (Pellizzari, Pizzi and Salmasi, 2005).
The estimation of the cointegration coefﬁcient is realized with a technique of supervised
learning called Relevance Vector Machine or RVM (Tipping, 2001) that exploits a
probabilistic Bayesian learning framework to derive accurate predictions models.
In the next section we describe dynamic cointegration and the RVM. while section 3 is
devoted to the results of the application to the gold and silver data set.
2. Dynamic cointegration
The traditional concept of cointegration (Engle and Granger, 1987) assumes that the
cointegrating coefﬁcient is constant over time. If the coefﬁcient is allowed to be non
constant we have the following deﬁnition of dinamic cointegration:
Let xt a N × 1 vector, its components are dynamically cointegrated of order d,b,
CI(d,b) if i) the series in xt are all I(d) and ii) there exists a non zero vector αt such
that zt = αt xt is I(d − b), b>0.
Let us consider the case of N =2 , the (dynamic) cointegrating relation is zt =
x1t − αtx2t. Several problems can be tackled: i) to determine αt ii) to comprehend the
structure of αt iii) to develop valid estimation techniques.
If the timespan is sufﬁcientlylarge αt can be approximatedby αt ∼ =
x1t
x2t. In general, we
can assume that αt depends on several variables, αt = l(xt,x t−1,...,y t,y t−1,...,t), and
its structure can be investigated by plotting αt versus the variables that presumably affect
its dynamics. As far as the third problem is concerned, we propose the use of a Bayesian
probabilisticframework for learning, known as Relevant Vector Machine (RVM), Tipping
(2001). It is a model of identical functional form to the Support Vector Machine (SVM),
among the others Vapnik (1998), but the training takes place in a Bayesian framework and
predictive distributions of the outputs instead of point estimates are obtained.
Given a set of examples of input vectors {xn}
N
n=1 (training set) along with
corresponding targets {tn}
N
n=1, “learning” is the process of inferring a function y(x) on
which are based the predictions of t for previously unseen values of x. The RVM makes
predictions based on the function
y(x,w)=
N 
i=1
wiK(x,xi)+w0
where K(x,xi) is a kernel function, wi are the weights. One basis function is deﬁned for
each example in the training set.
The main feature of RVM (for further formal details, see Tipping, 2001) is represented
by the introduction of a prior density function over the model weights governed by a set
of hyperparameters, one associated with each weight, whose most probable values are
iteratively estimated from the data. Sparsity is achieved because in practise we ﬁnd that
the posterior distibutions of many of the weights wi are sharply peaked around zero. Forthis reason, one of the advantages in using RVM is that, with the same generalization
performance, it utilises way fewer kernel functions than an equivalent SVM.
3. Real data analysis
The data set we examine comes from the Interational Financial Statistics. The two series
are the monthly prices of gold and silver (US dollars per ounce) from the beginning of
1971to april 2004. In the ﬁrst 3 rows of table 1 we report the resultsof theADF teston the
series of gold (xt), silver(yt) and residuals of the traditionalcointegratingregressions(zt).
As we can observe, there is no evidence of linear cointegration since the null hypothesis
of unit root is always accepted.
Table 1: ADF test
Series Test ADF p-value
yt -2.5295 (lag order=7) 0.3536
xt -2.756 (lag order=7) 0.2580
zt (linear coint.) -2.6716 (lag order=7) 0.296
zt (dynamic coint.) -5.8458 (lag order=7) 0.01
After rescaling series xt and yt in (1,2) we determine
yt
xt (solid line in ﬁg.2). Since the
lattercanbethoughtasαt+ zt
xt, we employtheRVMapproachtoﬁlteroutαt. Itisassumed
a gaussian prior distribution on weigths wi, whereas for the N +1hyperparameters the
distributions are non informative Gamma. We use a gaussian kernel and set σ, the inverse
of the bandwidth parameter, equal to 5e-04.
Table 2: Relevant vectors with their time index and corresponding date
Time index 1 51 52 100 132 162 189 222 280 343 400
Date 1:71 3:75 4:75 4:79 12:81 6:84 9:86 6:89 4:94 7:99 4:04
wi 0.914 -4.627 5.101 0.6 -0.102 0.757 -0.114 0.549 0.533 0.623 0.549
The RVM algorithm ﬁnds 11 relevant vectors (tab. 2) and the αt series, obtained
ﬁltering
yt
xt, is represented with a dashed line in ﬁg.2. An ADF test is performed on the
dynamic cointegration residuals and the results are reported in the last row of table 1. As
we can see, unlike the traditional cointegration case, in this dynamic case the unit root is
rejected. This means that evidence of dynamic cointegration is found.
Interesting information is gathered by observing the time index of the relevant vectors
(tab. 2). Some of them are located where the αt series tends to exhibit a sort of level
shift, see for instance the vector with index 100 which is located just before the well
documented bubble in silver prices from roughly june 1979 to march 1980. Bearing in
mind the relevant vectors indexes, we consider, in particular, two subperiods 1:71-4:79
and 4:79-4:04. The ADF test reveals in both of them evidence of linear cointegration
(p-values, respectively, 0.06 and 0.08). The relevant vectors show also other possibilities
to split the sample, but in those cases no linear cointegration is found after 4:94.Figure 2:
yt
xt (solid line) and αt (dashed line) series
Time
a
l
f
a
0 100 200 300 400
0
.
7
0
0
.
7
5
0
.
8
0
0
.
8
5
0
.
9
0
0
.
9
5
1
.
0
0
This results show clearly how promising seems to be the use of RVM in the pioneering
context of dynamic cointegration. Indeed, RVM exhibits interesting potentiality both in
terms of accuracy when ﬁltering the series and ability in revealing where are located the
main change points of the long term dynamic.
References
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Exchange rates, country-speciﬁc shocks and gold.

Investigating the relationship between gold and silver prices.

Nonlinear cointegration and nonlinear error correction: applications of the RCC test based on ﬁrst differences of ranges.

Sparse Bayesian Learning and the Relevance Vector

Statistical learning Theory.

The evolving relationship between gold and silver 1978-2002: evidence from a dynamic cointegration analysis: A note. Discussion paper n. 55. Institute for international Integration Studies.

Dynamic cointegration and relevant vector machine: the relationship between gold and silver

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