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

2

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.

3

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

4

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.

5

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.

6

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.

7

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).

8

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.

9

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).

10

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

11

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.

12

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.

13

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.

14

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.

15

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)).

16

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.

17

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18

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19

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

20

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

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

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

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