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DEPARTMENT OF ECONOMICS
Uppsala University
Master Thesis
Author: Christer Rosén
Supervisor: Lennart Berg
December 2007
Time Series Econometrics
Heteroskedasticity in Stock Return Data: Volume and Number of
Trades
versus GARCH Effects
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Abstract
The result of Lamoureux and Lastrapes and Omran and McKenzie are extended to the Swedish
stock market, and this paper examines their findings that GARCH modelling captures the serial
dependence in information flow into the market. Moreover, this paper also examines if (as a
proxy for information flow) the number of trades can challenge the volume of trade in order to
explain GARCH effects in financial time series. Using data on 25 large stocks that are traded on
The Nordic Stock Exchange, this paper finds that even though the parameter estimates of the
GARCH model becomes significantly lower for about half of the companies in this study when
volume of trade or the number of trades is used in the conditional variance of return equation, the
autocorrelation of the standardized residuals still exhibit a highly significant GARCH effect in
more than 1/3 of the companies when these two additional explanatory variables are included in
the conditional variance equation. The serial dependence in volume of trade and number of trades
does not eliminate the need for GARCH modelling of volatility.
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Contents
Abstract .................................................................................................................................2
Contents.................................................................................................................................3
1. Introduction. ................................................................................................................4
2. Background..................................................................................................................6
2.1 The information flow hypothesis ..................................................................................6
2.2 Previous studies.............................................................................................................8
2.3 This study ......................................................................................................................9
3. Market efficiency.......................................................................................................10
3.1 Theory of ARCH and GARCH models ......................................................................11
4. Data and methodology .............................................................................................13
5. Empirical results and analysis .................................................................................14
6. Conclusions ................................................................................................................16
7. References ..................................................................................................................17
Appendix A.1: Correlation between volume of trade/ number of trades
and stock return data. .................................................................................19
Appendix A.2: Correlation between volume of trade and number of trades. .....................20
Appendix B.1: Estimates of GARCH (1,1) model without volume of trade
or number of trades .....................................................................................21
Appendix B.2: Estimates of GARCH (1,1) model with volume of trade.. .........................22
Appendix B.3: Estimates of GARCH (1,1) model with number of trades. ........................23
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1. Introduction
Knowledge about volatility forecasting is very important in financial markets, and it has been
under consideration by academics and practitioners for the last two decades (Poon and Granger,
2003). Much has been written about forecasting performance of various volatility models. Good
volatility models have application in areas such as investment, security valuation, risk
management and monetary policy making. A good forecast of the volatility in the asset under
consideration over the investment holding period is a good starting point when evaluating
investment risk.
Volatility is one of the most important factors in the pricing of derivative securities. To price an
option accurate we need to know the volatility of the underlying asset from now till the option
expires.
Volatility forecasting has also taken a central roll in financial risk management; this has made
correct volatility forecasting a compulsory exercise for many financial institutions around the
world (Poon and Granger, 2003). Financial market volatility can also have a wide repercussion
on the economy as a whole, for this reason many policy makers rely on market estimates of
volatility as a barometer for the vulnerability of the financial markets and the economy. The
Chicago Board Options Exchange Volatility Index (VIX- index) measure the implied volatility of
S&P 500 index options. This VIX- index aims to measure the markets volatility over the next 30
days and is naturally valuable information to investors. In the United States, the Federal Reserve
explicitly takes into account similar volatility forecasts of stocks, bonds, currencies and
commodes when setting its monetary policy ( Nassar, 1992).
Financial time series such as stock prises can often appear to have periods with large swings
followed by periods with relatively calmer swings. This is sometime refered to as volatility
clustering in econometric literature. One hypothesis which tries to explain these auto correlation
in swings, is the information flow hypothesis. In short it states that when new information arrives
to the market, asset prices evolve. So, if information to the market varies the variance of the asset
prices will vary. Therefore, information flow can help explain volatility clustering.
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Two studies; Lamoureux and Lastrapes (1990) and Omran and McKenzie (2000) uses this
information flow hypothesis in a formal way in order to examine if the degree of information to
the market can explain the degree of volatility swings in asset prices.
The aim for this paper is to analyse if such volatility clustering described above measured by
the General autoregressive conditional heteroscedasticity (GARCH (1,1)) model can be
explained by the information flow into the Swedish stock market (volume of trade and number of
trades will be used as a proxy for information flow) for these stocks. The focus will be on
answering the question if the volume of trade and/or the number of trades is accountable for the
GARCH (volatility clustering) effects.
This paper will limit itself to the Swedish stock market and will use a data set of 25 different
large stocks traded on The Nordic Stock Exchange. The data set include; daily returns, volume of
trade and number of trades during the period from 2000-10-16 to 2006-12-08. Volume of trade is
the number of shares traded for a particular stock on a particular day, and the number of trades is
the number of realized buying and selling orders for a particular stock on a particular day.
Volume is chosen since it is the same variable used by Lamoureux and Lastrapes (1990) and
Omran and McKenzie (2000), and therefore it is a scope for comparison between these studies.
The variable number of trades is a contribution made by this paper in order to challenge the
volume of trade variable in explaining the GARCH effects in financial time series.
This paper is organized as follow: Firstly, in section two, a review of the information flow
hypothesis is presented. In addition, earlier studies on the subject are briefly discussed together
with how this study differentiates to them. Secondly, in section three, some econometric and
financial concepts are examined. Thirdly, in section four, a specification regarding the model
used in order to test the hypothesis under consideration is presented together with data and
methodology. Section five provides analysis of the empirical results. Finally, a conclusion is
presented.
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2. Background
This section includes a presentation of the information flow hypothesis, earlier studies made in
this area and also how this paper will differ from them.
A good understanding of this part will also justify the model specification used in the empirical
section.
2.1 The information flow hypothesis
The positive correlation between volume of trade and asset returns in equity markets has been
documented in literature (Karpoff, 1987). This statement might no longer be valid due to changes
in the financial market. Appendix A.1 indicates this and shows the correlation between volume
and returns and for number of trades and return data for the samples used in this text. The
information flow hypothesis discussed here is nevertheless one possible explanation regarding
the variance relationship between information and the financial market.
Because daily returns are generated by the sum of within day equilibrium returns, and because
the number of within day returns, nt, is random, daily returns are conditional to nt (Omran and
McKenzie, 2000). Further it is believed that prices evolve when new information arrives into the
market and nt is set to represent the number of information arrivals in the market on a certain day.
A possible explanation for the success of GARCH models in modelling stock returns is the
information flow hypothesis. If it is assumed that the number of information arrival and therefore
the within day equilibrium returns variable, nt, forms a serially dependent sequence, then it is
possible that GARCH is capturing the temporal dependence in this variable. To explain how
GARCH might capture the effect of time dependency in information arrivals to the market, the
following theoretical discussion is presented.
In the GARCH model the conditional variance of a time series depend upon past squared
residuals of the process.
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A possible model for daily stock returns is:
1−= ttr µ + tε (1)
tε |( ,..., 21 −− tt εε ) ~ N(0,ht) (2)
ht = 0α + 1α (L) 12
−tε + 2α (L)ht-1 (3)
Where rt represents the rate of return, 1−tµ is the mean rt conditional on past information, L is the
lag operator, and 0α is a constant. If the parameters of the lag polynominals 1α (L) and 2α (L) are
positive, then shocks to volatility persist over time. The degree of persistence is determined by
the magnitude of these parameters.
To motivate the empirical tests of this paper, let itψ denote the ith intraday equlibrium price
change in day t, which implies
tε =∑=
tn
i 1
itψ (4)
The nt is the the random variable, representing the stochastic rate at which information flows into
the market, so, equation (4) implies that daily returns are generated by a subordinated stochastic
process, in which tε is subordinated to iψ and nt is the directing process. (see Harris (1987).)
Further, if iψ is i.i.d. with mean zero and variance 2σ , and the information flow into the market
is sufficiently large, then tε |nt ~ N(0, 2σ nt). GARCH may be explained as an expression of time
dependence in the rate of evolution of intraday equilibrium returns. In order to make this point
very clear, assume that the daily number of information arrivals is serially correlated, which can
be expressed as follows:
nt = k + b(L)nt-1 + tφ (5)
Where k is a constant, b(L) is a lag polynomial of order q, and tφ is white noise. Shocks in the
information flow to the market persist according to the autoregressive structure of b(L).
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Define tΩ = E(2
tε |nt). If the information flow model is valid, then tΩ =2σ nt. Substituting the
representation of (5) into this expression for variance yields
tΩ = 2σ k + b(L) 1−Ωt + tφσ
2 (6)
Equation (6) captures the type of persistence in conditional variance that can be picked up by
estimating a GARCH model. To be precise, shocks to the information process lead to momentum
in the squared residuals of daily returns.
2.2 Previous studies
The ARCH process discovered by Engle in 1982 has been shown to provide a good fit for many
financial time series Bollerslev (1987), Lamoure and Lastrapes (1988), Baillie and Bollerslev
(1989) and Lastrapes (1989). ARCH modelling puts an autoregressive structure on conditional
variance, allowing volatility shocks to persist over time. This persistence captures the cluster
behaviour of returns over time and can explain the well-documented non-normality and non-
stability of empirical asset return distributions (Fama, 1965).
As suggested by Diebold (1986), Gallant, Hsieh, and Tauchen (1988), and Stock (1987, 1988),
GARCH might capture the time series properties (e.g. serial correlation) of the within day returns
variable. One previous study that tried to examine the validity of this explanation for daily stock
returns is that of Lamoureux and Lastrapes (1990).
The study of Lamoureux and Lastrapes (1990) used an empirical strategy to exploit that
GARCH effect in daily stock return data reflects time dependence in the information flow to the
market. The study used daily trading volume as a proxy for the information flow, and used a
sample of 20 common stocks. It was found that the GARCH effects vanished when volume was
included as an explanatory variable in the conditional variance equation. In conclusion the
Lamoureux and Lastrapes paper provides empirical support for the hypothesis that GARCH is an
expression for the daily time dependence in the rate of information arrival to the market for
individual stocks. Thus, the result found in the Lamoureux and Lastrapes (1990) paper properly
motivates the use of GARCH models to study the behaviour of asset prices.
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In a study made by Omran and McKenzie (2000), the result of Lamoureux and Lastrapes 1990
are extended to the UK stock market, and that study also finds evidence that GARCH modelling
captures the serial dependence in information flow to the market. Omran and McKenzie uses data
on 50 UK companies and found that although the parameter estimates of the GARCH (1,1) model
become insignificant when volume of trade is used in the conditional variance of return equation,
the autocorrelation of the squared residual still exhibit a highly significant GARCH effect,
something that was not examined by Lamoureux and Lastrapes.1
In conclusion the study by Omran and McKenzie find consistent result with Lamoureux and
Lastrapes 1990, that the volatility persistence, as measured by the GARCH model, become
negligible when volume of trade is introduced in the variance equation of returns. However, the
hypothesis of uncorrelated squared residuals (no GARCH effect) is rejected. There is still a
highly significant GARCH pattern in the squared standardized residuals of the model for all but
four out of 50 companies. Therefore, they conclude that GARCH effects cannot be explained
only by the serial dependence in volume of trade.
2.3 This study
As already stated briefly in the introduction, this paper contains a data set of 25 frequently traded
stocks on the Nordic Stock Exchange. The criteria that the stocks most be frequently traded is
taken from Lamoureux and Lastrapes (1990). Moreover, stocks with splits during the period of
study are excluded to eliminate possible problems from split effects on volume and number of
trades. The data set includes 1543 observations. The variable number of trades is used in order to
test if this contains a different kind of information than does the volume of trade variable. If the
number of trades are few but the volume is high means that every selling or buying order is
relatively big. Individuals that trade in this way might have access to different types of
information.
One difference between this paper and the papers by Lamoureux and Lastrapes (1990) and
Omran and McKenzie (2000) is that in these two papers the parameter estimate of the variance
1 Evidence is also found that there is a strong association in the timing of innovation outliners in returns and
volume. The result suggests that a threshold model for volume and return could prove a useful route to pursue in
future research (Omran and McKenzie, 2000).
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equation is constructed to be nonnegative. This paper does not have this restriction. The reason
for this is that the restriction is not availible in the Eviews statistical software, which is used by
this paper for estimation.
3. Market efficiency
In finance, volatility is often referred to standard deviation or variance computed from a set of
observations. In financial applications the conditional variance is more relevant. Because this
paper is concerned with time series econometrics the conditional variance is naturally used.
Market efficiency is a theory about with which precision the market prices incorporates new
information. If prices respond to all relevant new information in a rapid fashion, we say that the
market is relatively efficient.
Under the weak form of the efficient market hypothesis (EMH), stock prices are assumed to
reflect any information that may be contained in the past history of the stock price itself. Under
the weak form of EMH the yield follows a “random walk” see equation (7) below.
Rt= tC ε+ where tε ~2,0( σN ) (7)
Where Rt is the stock price at time t, C is a constant and ε t is a normal distributed error term
with expected value zero and a constant variance.
It has been found empirically that stock return distribution has “thicker tales” (leptokurtosis)
than a normal distribution. A “thicker tale” means that extreme movements are more common
than a normal distribution can explain. It has also been found that volatility in financial assets
tend to appear in cluster. Periods in which their prices show wide variations for an extended time
period followed by periods in which there is relative calm. This means that the variance is
autocorrelated in time. For equities, it is often observed that downward movements in the market
are followed by higher voltilities than upward movements of the same magnitude. The variance
in a financial asset today is dependent on yesterday’s variance in the financial asset. When asset
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prices behave in this way it is reasonable to assume that the time series variance follows an
GARCH process (Alexander, 2005).
One point to make clear is that the EMH sets no restrictions regarding volatility movements; it
can be autocorrelated without the EMH is rejected.
3.1 Theory of ARCH and GARCH models
ARCH and GARCH models are used to measure volatility in financial time series. As already
been pointed out financial time series, such as stock prices, exchange rates, inflation rates, etc.
often exhibit the phenomenon of volatility clustering. That is, periods in which their prices show
wide swings for an extended time period followed by periods in which there is relative calm
(Gujarati, 2003). Knowledge of volatility is of great importance when analysing the risk of
holding an asset or when pricing an option.
In order to model financial time series that experience volatility clustering one usually has to
take the first difference of the logarithm of the financial time series under analysis to make them
stationary and possible to extend in a meaningful way. Most financial time series are random
walks in their log level form, That is, they are nonstationary and its behaviour can only be studied
for the time period of the actual series. As a consequence, it is not possible to generalize it to
other time periods. The series used in this paper are stationary in their first difference log level
form and a formal test for this has been made but is not presented in the appendix.
In order to model “varying variance” the GARCH (1,1) can be used (Gujarati, 2003). In
developing a GARCH model two specifications must be provided, one for the conditional mean
and one for the conditional variance.
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A general GARCH(q, p) model can be written as;
ttt rr εβα ++= −1 , (8)
tε |( ,..., 21 −− tt εε ) ~ N(0,ht) (9)
∑∑=
−
=
− ++=q
j
jtj
p
i
itit
1
2
1
2
0
2 σγεαασ (10)
where (8) is the mean equation and (10) is the conditional variance equation.
The mean equation given in (8) is written as a function of an exogenous variable and an error
term. Since 2tσ is the one-period ahead forecast variable based on past information, it is called the
conditional variance.
The conditional variance equation specified in equation (10) is a function of three terms:
• The mean: (α 0).
• News about volatility from the previous period, measured as the lag of the squared
residual from the mean equation:ε 2t-1 (the ARCH term).
• Last period’s forecast variance: σ 2t-1 (the GARCH term).
The (q,p) in GARCH (q,p) refers to the presence of the order GARCH term and the order ARCH
term. An ordinary ARCH model is a special case of a GARCH specification in which there is no
lagged forecast variance in the conditional variance equation. If the sum of ARCH and GARCH
coefficients (α+γ ) is close to one, volatility shocks are quite persistent over time. Further, if
α+γ ≤ 1 the variance is stationary, if α+γ >1 the variance is explosive, and if the α≥ 0 and γ ≥ 0
the conditional variance is non-negative. Because this restriction of non-negativity is not
available in Eviews a formal test to examine if the conditional variance is stationary has been
made but not presented in the Appendix. The conditional variance series obtained after the
GARCH(1,1) modell is runned was tested by the usual ADF test. The series showed that the
series was stationary for all series and the variance is therefore not explosive.
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4. Data and methodology
The data set comprises daily returns, volume of trade and number of trades for 25 Swedish
companies during the period from 2000-10-16 to 2006-12-08. These companies were among the
biggest in Sweden during the period of the study. The data was obtained from OMX. Volume of
trade is the number of shares traded for a particular stock on a particular day. Volume of trade is
chosen since it is the same variable used by Lamoureux and Lastrapes (1990) and Omran and
McKenzie (2000), and therefore there is a scope for comparison between the studies. Moreover,
this paper adds the variable number of trades, which is the number of trades that occurred for a
particular stock on a particular day.
In the first stage of the analysis, the following model is estimated for each stock in the sample:
Mean equation:
1−= ttr µ + tε (11)
Employing three different specifications of equation (3)
Variance equations:
ht = 0α + 1α (L) 12
−tε + 2α (L)ht-1 (12)
ht = 0α + 1α (L) 12
−tε + 2α (L)ht-1+ω 1Vt (13)
ht = 0α + 1α (L) 12
−tε + 2α (L)ht-1+ω 1Tt (14)
Where rt is 100*loge(Pt/Pt-1), and Pt is the stock price at time t. Equation (11) allows for an
autoregression of order 1 in the mean of returns since most of the returns data exhibit a small but
significant first order autocorrelation (Omran and McKenzie (2000)). Equations (12), (13), and
(14) models the conditional variance of the unexpected returns,ε t, as a GARCH(1,1) process,
with the volume, Vt and number of trades, Tt, included in equation (13) and (14). In equation (12)
these two variables are set to zero.
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Following the same methodology as Lamoureux and Lastrapes (1990) and Omran and
McKenzie (2000). First, the restricted model of Equation (12) is estimated by setting the
coefficient of volume of trade and number of trades to zero, thereafter fitting a GARCH (1,1)
model to the ε t. of the mean equation. In the second stage, the unrestricted models of Equation
(13) and (14) are estimated. If volume of trade or number of trades is serially correlated, and
works as a proxy for information arrivals to the market, then it can be anticipated that ω 1 > 0 in
those two models, and the persistence in volatility as measured by the sum of 1α and 2α becomes
negligible.
The ARCH LM test is used to test the hypothesis of no GARCH effects in the residuals from the
three conditional variance models and is presented in the tables of appendix B.1, B.2 and B.3. 2
5. Empirical results and analysis
Appendix B.1 shows the result of the GARCH (1,1) model (restricted) of equation (12). This
table shows the result of estimating the GARCH (1,1) model to the data set. The GARCH model
suggests that there is volatility persistence as measured by the sum of α 1 and 2α because most
of the sums is close to 1. The table also shows the ARCH LM test at lag 10 to se whether the
standardized squared residuals (SSR) exhibit additional serial correlation. If the variance
equation is correctly specified, there should be no effect of SSR. When the variance equation is
specified as a GARCH (1,1) model the SSR do not show any significant effects for any of the 25
companies.
Appendix B.2 shows that the coefficient of volume of trade is highly significant for all
companies but three. Further, volatility persistence becomes less for only slightly more than half
of the stocks, when compared with the results reported in Appendix B.1. Moreover, when
checking the ARCH LM test in order to detect serial correlation in the SSR after fitting the
variance equation including volume of trade, there is still a highly significant serial correlation in
the SSR of the model for 11 out of the 25 companies. These results show that volatility
persistence decrease for about half of the companies when volume of trade is included in the
2 The data was also tested against the EGARCH and the result was unaffected.
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variance equation, but that the SSR shows serial correlation in 11 out of 25 companies. In
summery GARCH patterns cannot fully be explained by volume of trade.
Appendix B.3 shows that the coefficient of number of trades is significant for 18 out of 25
companies and volatility persistence becomes less for about half of all companies. Moreover, the
ARCH LM test tells that 10 out of the 25 companies experience serial correlation in the SSR after
fitting the variance equation including the number of trades as an explanatory variable. The result
from this model specification indicates that volatility persistence decrease for most companies
versus all companies when the GARCH (1,1) model was used. Further, serial correlation in the
SSR becomes present. Similar to the inference drawn from the estimates in Appendix B.2, the
GARCH structure is not fully explained by the additional variable in the conditional variance
equation.
One possible explanation of these results lies in the complex structure of equation (13) and
(14). These include past values of both conditional volatility ht-1 and volume of trade Vt or
number of trades Tt as explanatory variables. The complication arises because ht-1 is itself a
function of both Vt-1, and Tt-1. Moreover, Vt and Tt are highly correlated with its own past values,
which can lead to a multicollinearity problem between the explanatory variables used ht-1 and Vt
or ht-1 and Tt (Omran and McKenzie (2000)).
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6. Conclusions
The papers empirical results, based on data drawn from the Swedish stock market, are to some
degree different from Lamoureux and Lastrapes (1990) and Omran and McKenzie (2000). It is
possible that the difference arises because Omran and McKenzie (2000) use a restricted
parameter space, whereas no restriction was assumed for the estimations in this paper. The results
are not consistent with theirs in that the volatility persistence, as measured by the GARCH
components, become negligible for all companies under study when volume of trade is
introduced in the conditional variance equation. The result from this paper find that volatility
persistence decrease for about 50% of the companies regardless if volume of trade or number of
trades is used in the conditional variance equation. A second difference between this paper and
the Omran and McKenzie (2000) paper is that they found that serial correlation in the SSR was
present in 46 out of the 50 companies under study. The numbers for this paper are 11 out of 25
and 10 out of 25 for volume of trade and nr of trades respectively. Because of these results, this
paper concludes that GARCH effects cannot consistently be fully explained by the serial
dependence in either volume of trade nor the number of trades.
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Appendix Appendix A.1
Correlation between between returns and Volume of trade
Company Correlation Company Correlation
ASSA B -0,18 SWMA 0,24
HM B -0,01 VOLV B 0,26
NCC B 0,41 HOLM B 0,20
NDA SEK -0,06 SCA B 0,10
STE R 0,17 SAAB B 0,02
TREL B 0,13 PEAB B -0,04
VOST SDB 0,60 HOGA B 0,01
AZN -0,14 MTG B -0,09
ALIV SDB -0,14 AXFO 0,27
ERIC B -0,38 SHB B -0,02
INVE B -0,10 TIEN 0,08
NOKI SDB 0,37 OMX -0,02
SCV B 0,21
Mean correlation in absolut figures: 0,17 Minus signs: 11
Correlation between returns and Number of trades
Company Correlation Company Correlation
ASSA B -0,19 SWMA 0,68
HM B 0,18 VOLV B 0,67
NCC B 0,80 HOLM B 0,48
NDA SEK 0,33 SCA B 0,36
STE R 0,03 SAAB B 0,61
TREL B 0,55 PEAB B 0,40
VOST SDB 0,73 HOGA B 0,24
AZN -0,08 MTG B 0,45
ALIV SDB 0,25 AXFO 0,47
ERIC B 0,16 SHB B 0,09
INVE B 0,31 TIEN 0,17
NOKI SDB 0,62 OMX 0,10
SCV B 0,67
Mean correlation in absolut figures: 0,38 Minus signs: 2
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Appendix A.2 Correlation between Number of trades and Volume of trade
Company Correlation Company Correlation
ASSA B 0,80 SWMA 0,64
HM B 0,77 VOLV B 0,74
NCC B 0,57 HOLM B 0,60
NDA SEK 0,47 SCA B 0,75
STE R 0,67 SAAB B 0,25
TREL B 0,69 PEAB B 0,47
VOST SDB 0,88 HOGA B 0,42
AZN 0,87 MTG B 0,55
ALIV SDB 0,71 AXFO 0,81
ERIC B 0,58 SHB B 0,32
INVE B 0,44 TIEN 0,75
NOKI SDB 0,87 OMX 0,75
SCV B 0,47
Mean correlation: 0,63
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21
Appendix B.1
GARCH (1,1) Model
Nr Company ARCH (α1) GARCH(α2) α1+α2 ARCH LM
Test
1. ASSA B 0,027
9,000
0,971
421,900
0.998 No ARCH
2. HM B 0,011
6,710
0,986
598,280
0.997 No ARCH
3. NCC B 0,081
5,630
0,842
33,180
0.923 No ARCH
4. NDA SEK 0,112
9,610
0,880
79,370
0.992 No ARCH
5. STE R 0,047
6,340
0,943
114,850
0.990 No ARCH
6. TREL B 0,070
5,690
0,852
38,880
0,922 No ARCH
7. VOST SDB 0,123
9,770
0,807
48,500
0.930 No ARCH
8. AZN 0,035
7,510
0,951
152,750
0.986 No ARCH
9. ALIV SDB 0,150
13,180
0,833
55,180
0.983 No ARCH
10. ERIC B 0,078
12,930
0,924
147,030
1,002 No ARCH
11. INVE B 0,118
7,590
0,859
52,050
0.977 No ARCH
12. NOKI SDB 0,014
8,110
0,982
669,220
0,996 No ARCH
13. SCV B 0,107
8,140
0,837
47,950
0.944 No ARCH
14. SWMA 0,022
6,130
0,975
264,930
0.997 No ARCH
15. VOLV B 0,066
5,840
0,897
53,950
0,963 No ARCH
16. HOLM B 0,056
5,570
0,826
32,770
0.882 No ARCH
17. SCA B 0,153
7,600
0,743
26,880
0.896 No ARCH
18. SAAB B 0,078
8,130
0,910
88,390
0.988 No ARCH
19. PEAB B 0,198
7,730
0,576
13,480
0.774 No ARCH
20. HOGA B 0,056
9,410
0,931
163,980
0.987 No ARCH
21. MTG B 0,095
7,400
0,894
69,960
0.989 No ARCH
22. AXFO 0,103
8,260
0,827
43,560
0.930 No ARCH
23. SHB B 0,094
8,210
0,893
77,110
0.987 No ARCH
24. TIEN 0,055
8,630
0,926
110,870
0.981 No ARCH
25. OMX 0,099
10,910
0,900
106,600
0,999 No ARCH
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22
Appendix B.2 GARCH (1,1) Model with Volume of trade
Nr Company ARCH (α1) GARCH(α2) α1+α2 ARCH LM
Test
Volym of Trade
1. ASSA B 0,219
9,467
0,767
39,414
0.986 No ARCH 0,164
13,614
2. HM B 0,214
7,791
0,076
1,916
0.290 ARCH 1,089
20,669
3. NCC B 0,130
5,407
-0,129
-3,347
0.001 ARCH 6,253
10,966
4. NDA SEK 0,146
9,422
0,841
58,545
0.987 No ARCH 0,014
4,825
5. STE R 0,099
6,185
0,801
33,200
0.900 No ARCH 0,300
6,125
6. TREL B 0,028
1,798
-0,150
-4,276
-0.122 ARCH 5,038
13,980
7. VOST SDB 0,197
6,357
0,078
1,399
0.275 ARCH 12,617
9,921
8. AZN 0,303
7,457
0,349
9,252
0.652 ARCH 0,903
13,240
9. ALIV SDB 0,085
4,179
-0,061
-2,025
0.024 ARCH 6,219
20,081
10. ERIC B 0,115
14,914
0,883
107,758
0.998 No ARCH 0,002
5,416
11. INVE B 0,117
7,540
0,861
51,894
0.978 No ARCH -0,002
-0,211
12. NOKI SDB 0,015
0,851
-0,064
-2,446
-0.049 ARCH 1,971
25,285
13. SCV B 0,150
7,605
0,750
30,121
0.900 No ARCH 0,093
4,424
14. SWMA 0,239
7,386
0,341
6,777
0.580 ARCH 0,300
6,559
15. VOLV B 0,022
1,518
-0,218
-4,697
-0.196 ARCH 1,670
12,910
16. HOLM B 0,112
3,476
0,062
1,349
0.174 No ARCH 10,079
19,178
17. SCA B 0,259
8,374
0,579
19,498
0.838 No ARCH 0,374
8,550
18. SAAB B 0,076
8,021
0,913
89,567
0.989 No ARCH -0,092
-1,619
19. PEAB B 0,220
7,206
0,333
8,053
0.553 ARCH 13,240
10,356
20. HOGA B 0,236
7,387
0,323
6,861
0.559 ARCH 11,529
11,762
21. MTG B 0,167
9,042
0,793
44,111
0.960 No ARCH 2,900
8,361
22. AXFO 0,245
8,896
0,615
22,102
0.860 No ARCH 3,212
15,820
23. SHB B 0,089
8,012
0,898
80,148
0.987 No ARCH -0,102
-6,417
24. TIEN 0,200
16,477
0,746
46,008
0.946 No ARCH 3,399
7,503
25. OMX 0,103
10,859
0,897
102,279
1.000 No ARCH 0,071
1,194
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23
Appendix B.3 GARCH (1,1) Model with Number of trades
Nr Company ARCH (α1) GARCH(α2) α1+α2 ARCH LM
Test
Number of Trades
1. ASSA B 0,213
9,659
0,772
40,241
0.985 No ARCH 0,065
11,671
2. HM B 0,174
6,865
-0,01
-0,419
0.164 ARCH 0,298
22,650
3. NCC B 0,049
4,180
-0,412
-8,757
-0.363 ARCH 1,028
11,257
4. NDA SEK 0,112
9,404
0,881
76,912
0.993 No ARCH -0,000
-0,292
5. STE R 0,197
6,532
0,255
6,067
0.452 ARCH 0,802
14,267
6. TREL B 0,044
7,944
-0,359
-7,859
-0.315 ARCH 1,012
13,754
7. VOST SDB 0,124
8,449
0,799
42,392
0.923 No ARCH 0,016
2,376
8. AZN 0,276
6,835
0,212
5,661
0.488 ARCH 0,219
15,171
9. ALIV SDB 0,020
1,855
-0,291
-14,325
-0.271 ARCH 0,911
23,450
10. ERIC B 0,238
7,935
0,563
20,404
0.801 ARCH 0,061
11,170
11. INVE B 0,118
7,584
0,859
52,060
0.977 No ARCH -0,000
-0,061
12. NOKI SDB 0,006
0,368
-0,109
-4,136
-0.103 ARCH 0,486
25,294
13. SCV B 0,112
7,362
0,826
42,137
0.938 No ARCH 0,005
2,140
14. SWMA 0,023
5,886
0,975
219,804
0.998 No ARCH -0,000
-0,116
15. VOLV B 0,073
5,960
0,888
47,051
0.961 No ARCH 0,002
1,583
16. HOLM B 0,072
2,462
-0,028
-0,556
0.044 No ARCH 0,820
14,099
17. SCA B 0,256
8,208
0,588
19,156
0.844 No ARCH 0,049
9,602
18. SAAB B 0,078
8,154
0,912
91,515
0.990 No ARCH 0,025
2,033
19. PEAB B 0,195
6,179
0,183
5,374
0.378 ARCH 2,313
14,532
20. HOGA B 0,062
8,608
0,927
134,303
0.989 No ARCH 0,059
6,407
21. MTG B 0,092
7,092
0,900
66,061
0.992 No ARCH -0,005
-0,687
22. AXFO 0,306
8,467
0,440
11,684
0.746 ARCH 0,337
15,604
23. SHB B 0,095
8,141
0,891
75,549
0.986 No ARCH 0,033
0,956
24. TIEN 0,194
15,035
0,763
46,872
0.957 No ARCH 0,256
7,313
25. OMX 0,103
10,916
0,896
102,514
0.999 No ARCH 0,011
0,983