Time Series Econometrics Heteroskedasticity in Stock Return Data: Volume and Number of Trades versus GARCH Effects

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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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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, n_t, is random, daily returns are conditional to n_t (Omran and McKenzie, 2000). Further it is believed that prices evolve when new information arrives into the market and n_t 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, n_t, 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

= _t

rt µ +ε_t (1) εt|(ε_t₋₁,ε_t₋₂,...) ~ N(0,h_t) (2) ht = α₀+α₁(L) 2 1

−

ε t +α₂(L)ht-1 (3)

Where r_t represents the rate of return, µ_t₋₁ is the mean r_t conditional on past information, L is the lag operator, and α₀ is a constant. If the parameters of the lag polynominals α₁(L) and α₂(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 ψ_it denote the ith intraday equlibrium price change in day t, which implies

εt=

∑

= nt

i 1

ψit (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 ε_t is subordinated to ψ_i and n_tis the directing process. (see Harris (1987).) Further, if ψ_i is i.i.d. with mean zero and variance σ², and the information flow into the market is sufficiently large, then ε_t|nt ~ N(0, σ²nt). 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 + φ_t (5)

Where k is a constant, b(L) is a lag polynomial of order q, and φ_t 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 Ω = E(_t ε_t²|n_t). If the information flow model is valid, then Ω =_t σ²n_t. Substituting the representation of (5) into this expression for variance yields

Ω = t σ²k + b(L)Ω_t₋₁ + σ²φ_t (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

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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=C+ε_t where ε_t~N(0,σ²) (7)

Where Rt is the stock price at time t, C is a constant and εt 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;

r_t =α +r_t₋₁β +ε_t, (8)

ε_t|(ε_t₋₁,ε_t₋₂,...) ~ N(0,ht) (9)

∑ ∑

=

−

=

− + +

=

q

j

j t j p

i

i t i t

1 2 1

2 0

2 α α ε γ σ

σ (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 σ_t² 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: (α₀).

• News about volatility from the previous period, measured as the lag of the squared residual from the mean equation:ε²t-1 (the ARCH term).

• Last period’s forecast variance: σ²t-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

= _t

rt µ +ε_t (11)

Employing three different specifications of equation (3)

Variance equations:

h_t= α₀+α₁(L) 2 1

−

ε t +α₂(L)h_t-1 (12)

ht = α₀+α₁(L) 2 1

−

ε t +α₂(L)ht-1+ω1Vt (13)

ht = α₀+α₁(L) 2 1

−

ε t +α₂(L)ht-1+ω1Tt (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,εt, as a GARCH(1,1) process, with the volume, V_t and number of trades, T_t,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 εt. 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 ω1 > 0 in those two models, and the persistence in volatility as measured by the sum of α₁and α₂ 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 α₂ 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 V_t-1, and T_t-1. Moreover, V_tand T_t 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 h_t-1 and T_t (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

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

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Appendix A.2

Correlation between Number of trades and Volume of trade

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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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

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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 0,763 0.957 No ARCH 0,256