In this paper the long memory and non-linear properties of share prices in the UK’s Stock Exchange and AIM are explored. The results suggest that the most commonly traded shares exhibit long memory thus raising interesting issues about the validity of normal assumptions of market efficiencies.Efficient Markets; Long Memory; Nonlinear Models

Bond, Derek

Dyson, Kenneth

English

Research Papers in Economics

Long memory and non-linearity in Stock Markets
Derek Bond∗ and Kenneth A. Dyson
School of Business, Retail and Financial Services,
University of Ulster at Coleraine,
BT52 1SA, United Kingdom
*corresponding author: d.bond@ulster.ac.uk
September 20, 2006
Abstract
In this paper the long memory and non-linear properties of share prices
in the UK’s Stock Exchange and AIM are explored. The results suggest
that the most commonly traded shares exhibit long memory thus rais-
ing interesting issues about the validity of normal assumptions of market
eﬃciencies.
11 Introduction
Gil-Alana (2006) highlights the issue of whether the time series behaviour of
stock market indexes can be described by a fractional representation. Such a
possibility could be due to long memory or nonlinearities and has serious conse-
quences for the eﬃcient market hypothesis in stock markets. In this paper the
issues are examined further using the time-series of individual share prices for
one hundred companies from the London Main(FTSE) and Alternative Invest-
ment(AIM) markets. A simple fractional augmented Dickey-Fuller (FADF)
test is used to test for long memory utilising a range of consistent estimates
of the fractional diﬀerencing parameter d. The possibility of non-linearity is
investigated by applying both random ﬁeld based tests and traditional tests for
structural breaks.
2 Background
Weak form market eﬃciency posits that the current share price fully reﬂects
the information implicit in the share price history of a company. This implies
that the share price time series, say {Pit}T
t=0 for share i, does not exhibit serial
correlation, or ’memory’. Recent literature puts forward the case for a generic
random walk type model, albeit one with less stringent assumptions than those
imposed in earlier literature on market eﬃciency. Traditionally the analysis
has been whether {Pit}T
t=0 is I(1) or I(0). In the I(0) case Pit is covariance
stationary and displays the property of short-memory. In the I(1) case Pit is
non-stationary and non-mean reverting. Fractional integration relaxes the as-
sumption that the diﬀerencing parameter d is integer. A time-series process
with 0 < d < 1 is deﬁned as a long-memory process. If 0 < d < 0.5 the series
Pit is covariance stationary, but the autocorrelation function is more persistent,
with a longer decay period, than when d = 0. If 0.5 ≤ d < 1 Pit is not covari-
ance stationary, but is still mean reverting, with a persistent, slowly decaying
autocorrelation function. Thus, if Pit is I(d) with 0 < d < 1 it exhibits memory
2and is thus not consistent with weak-form eﬃciency.
While studies focusing on the non-stationarity of time series have dominated
the literature, increasingly attention is focusing on the issue of whether series
that appear integrated of order > 0 are actually non-linear but stationary series.
Diebold & Inoue (2001) showed that long memory and structural change are
closely related in the I(d) case, but warned:
‘.. that the temptation to jump to the conclusion of “structural
change producing spurious inferences of long memory” should be
resisted.’
(Diebold & Inoue 2001, p.157)
3 Methodology
To test whether share prices exhibit long memory and/or non-linearity, a va-
riety of techniques are applied to the time-series of the top 25 and bottom 25
(by market capitalisation) companies in both the FTSE and the AIM from the
1st January 1998 to 31st December 2004, inclusive. The selection allows for the
investigation of two hypotheses. The ﬁrst is that there are diﬀerences in how
information is processed between listed markets. The expectation being that
AIM listed companies will be subject to less scrutiny by market participants
than their FTSE counterparts and more prone to information ineﬃciencies.
Therefore their share prices are more likely to exhibit long memory. The second
hypothesis is the relative size hypothesis. Larger companies should be subject
to greater scrutiny by market participants and therefore be less prone to infor-
mation ineﬃciencies. The tendency towards long memory should thus be more
prominent in smaller companies.
Firstly, long memory is investigated using Dolado, Gonzalo & Mayoral (2002)
FADF test and consists of estimating the equation:
∆Pt = θ∆dPt−1 +
p X
i=1
ςi∆Pt−i + t (1)
3The test of H0 : θ = 0 is a test of the hypothesis that d = 1. Using various
consistent estimates of d (Shimotsu (2006) exact Whiittle estimators1, semi-
parametric estimator of Geweke & Porter-Hudak (1983) and two parametric
estimators, the exact maximum likelihood and nonlinear least squares, available
in the ARFIMA package of Oxmetrics (Doornik & Ooms 1999) the test statistic
is asymptotically normal under H0.
To investigate the possibility of non-linearity in the series both traditional
parametric tests for structural breaks and random ﬁeld based inference are used.
In both cases the possibility of the share prices following a non-linear AR(1)
process or some form of transition models was investigated. Using the suite of
tests proposed by Bai & Perron (2003) the models
Pt = α0 +
Pp
i=1 αiδit + t and
Pt = β0Pt−1 +
Pp
i=1 βiPt−1δit + t

 
 
δit =

 
 
0 (t < t∗
i)
1 (t ≥ t∗
i)
p < 6
(2)
were investigated.2 The random ﬁeld tests for non-linearity are based on two
special cases of the model introduced in Hamilton (2001).
Pt = α + βPt−1 + λm(g  P) + t and
Pt = α + βPt−1 + ψt + λm(g  x) + t
(3)
where xt = (Pt,t), and  denotes element by element multiplication. In (3)
m(.) are random ﬁelds and g is a (k×1) vector k = 1 or 2. The parameter g can
be seen as a measure of the impact of either P or t on the non-linearity. The
scalar parameter λ as a measure of the degree of non-linearity. A simple test
for non-linearity is to test H0 : λ = 0. Hamilton’s approach was to assume a
gaussian random ﬁeld, meaning that the ﬁeld is deﬁned by its ﬁrst two moments,
and deriving a simple χ2 test of the null hypothesis derived from a generalised
linear regression model interpretation. In Dahl & Gonzalez-Rivera (2003) the
basic test was extended by the introduction of three tests that took account,
to various degrees, the problems of nuisance parameters. It was shown in Dahl
4& Gonzalez-Rivera (2003) that the four test were generally more powerful than
other tests of non-linearity and fairly insensitive to model misspeciﬁcation.3
4 Results
Tables 1 and 2 summarise the principal ﬁndings of the study. A series is classiﬁed
as either fractional or I(1) when over 85% of the FADF tests indicate this
otherwise the series has been classiﬁed as indeterminate. In all fractional cases
d was non-stationary.
For the FTSE 64% of the largest companies exhibit strong signs of fraction-
ality whilst none of the smallest companies do. Conversely, 32% of the largest
companies and 92% of the smallest companies exhibit signs of weak form ef-
ﬁciency. The remaining 4% and 8% being indeterminate. These result seem
contrary to the relative size hypothesis. The results of the non-linearity tests
are generally inconclusive. The one exception is the single large company where
the FADF was inconclusive - here there is a strong indication of some form
of non-linearity. Similar, less strong, results follow for the 2 small companies
where the FADF is inconclusive. Amongst the smaller companies there is more
evidence that in some cases the non-stationarity could be due to non-linearity.
For the AIM the results are signiﬁcantly diﬀerent, 8% of the largest com-
panies and 24% of the smallest companies exhibit signs of fractionality. The
majority of large companies, 76%, are weak form eﬃcient whilst only 44% of
the smallest companies are. There are many more companies, 16% and 32%
of the largest amd smallest respectively, being indeterminate. These result
support the relative size hypothesis but when compared with the FTSE are
contrary to the relative markets eﬃciency hypothesis. Again the results of
the non-linearity tests are generally inconclusive. The increased probability of
non-linearity amongst smaller companies together with the larger number of
inconclusive fractionality results might be explained by thin trading.
55 Discussion and Conclusion
In this paper the work of Gil-Alana (2006) has been extended by investigating
the time-series characteristics of individual share price series using a simple
FADF test and a suite on non-linearity tests. Data from the top and bottom
25 ﬁrms in the FTSE 100 index and the AIM were investigated. The results of
the study appear to cast doubt on whether the larger companies in the FTSE
are weak form eﬃcient. Thus someone using some form of trading rule with
these companies, operating in ‘real-time’, may be capable of proﬁting from these
ineﬃciencies. This is an interesting ﬁnding, as theory posits that stock price
markets are information eﬃcient. The concept of market eﬃciency is built upon
the foundation that markets are at least weak form eﬃcient. If this does not
hold, then neither do the stricter deﬁnitions. If anything, the results provide
evidence that information eﬃciency is more prevalent the smaller the company,
and given the caveat re thin trading, the smaller the market.
The possibility of ineﬃciencies in the FTSE needs to be investigated fur-
ther. For example, a possible explanation for the ineﬃciency in the FTSE may
be the use from the late 1980’s onwards of automated trading systems. Such
systems may create self-perpetuating patterns, thereby creating long-memory,
from which traders may proﬁt.
Given the inconclusive nature of the Hamilton-Dahl and Bai-Perron tests
the issue of whether the non-stationarity in share prices is in part due to non-
linearity needs to be investigated further. This could be by trying to model the
possible non-linearity through random ﬁeld or STR models. Or it could be by
examining the behaviour of the consistent semi-parametric estimators of d in
line with Diebold & Inoue (2001).
6References
Bai, J. & P. Perron (2003), ‘Computation and analysis of multiple structural
change model’, Journal of Applied Econometrics 18, 1–22.
Dahl, C. M. & G. Gonzalez-Rivera (2003), ‘Testing for neglected nonlinearity
in regression models based on the theory of random ﬁelds’, Journal of
Econometrics 114, 141–164.
Diebold, F. X. & A. Inoue (2001), ‘Long memory and regime switching’, Journal
of Econometrics 105, 131–159.
Dolado, J. J., J. Gonzalo & L. Mayoral (2002), ‘A fractional dickey-fuller test
for unit roots’, Econometrica 70, 1963–2006.
Doornik, J.A. & M Ooms (1999), ‘A package for estimating,forecasting and
simulating arﬁma models: Arﬁma package 1.0 for ox’. Discussion Paper,
Nuﬃeld College, Oxford.
Geweke, J. & S. Porter-Hudak (1983), ‘The estimation ansd application of long-
memory time series models’, Journal of Time Series Analysis 4, 221–237.
Gil-Alana, L.A. (2006), ‘Fractional integration in daily stock market indexes’,
Review of Financial Economics 15, 28–48.
Hamilton, J. D. (2001), ‘A parametric approach to ﬂexible nonlinear inference’,
Econometrica 69, 537–573.
Shimotsu, K. (2006), Exact local whittle estimation of fractional integration
with unknown mean and time trend, Working Papers 1061, Queen’s Uni-
versity, Department of Economics.
7Table 1: Stationarity Results
Market 25 Largest Companies 25 Smallest Companies
Fractional I(1) Indeterminate Fractional I(1) Indeterminate
FTSE
Number
Percent
16
64%
8
32%
1
4%
0
0%
23
92%
2
8%
AIM
Number
Percent
2
8%
19
76%
4
16%
6
24%
11
44%
8
32%
Table 2: Percentage of linearity tests signiﬁcant at 5% levels
Hamilton-Dahl Bai-Perron
model model
Pt−1 Pt−1,t β α
FTSE Largest % % % %
Fractional
(n=16)
11 13 6 38
I(1)
(n=8)
31 38 0 63
Indeterminate
(n=1)
75 75 0 100
FTSE Smallest
Fractional
(n=0)
. . . .
I(1)
(n=23)
8 33 30 30
Indeterminate
(n=2)
38 38 0 50
AIM Largest % % % %
Fractional
(n=2)
50 50 0 50
I(1)
(n=19)
8 14 16 11
Indeterminate
(n=4)
0 0 0 25
AIM Smallest
Fractional
(n=6)
16 46 0 50
I(1)
(n=11)
7 25 27 45
Indeterminate
(n=8)
50 32 25 63
Note: Each variable has 4 tests applied for each method/model
8Notes
1using the MATLAB code supplied by the authors.
2the Gauss code provided by the authors was used.
3The Gauss code supplied by Hamilton (http://weber.ucsd.edu/∼jhamilto) was used.
9

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