1. Dissertation December 6 th 2001Side: 1 Introduction and Motivation for the dissertation Essays on Changing Volatility in Thinly Traded Equity Markets by Per Bjarte Solibakke

2. Dissertation December 6 th 2001Side: 2 Overview of the Dissertation’s chapters Chapter 1 Introduction Chapter 2 Unifying Theme and Core Hypotheses Chapter 3 Methodologies Chapter 4 Stylised facts on Liquidity at Oslo Stock Exchange Chapter 5 Stock return volatility in thinly traded Markets. An Empirical analysis of trading and non-trading processes for individual stocks in the Norwegian thinly traded equity market Chapter 6 Efficiently ARMA-GARCH Estimated Trading Volume Characteristics in Thinly Traded Market Chapter 9 Testing the Bivariate Conditional CAPM in Thinly Traded Markets Chapter 8 Testing the Univariate Conditional CAPM in Thinly Traded Markets Chapter 7 Non-linear Dependence and Conditional Heteroscedasticity in Stock Returns. Evidence from the Norwegian Thinly Traded Equity Market Chapter 10 Event-induced Volatility In Thinly Traded Markets Chapter 11 Calculating Abnormal Returns in Event Studies: Controlling for Non-synchronous Trading and Volatility Clustering in Thinly Traded Markets

3. Dissertation December 6 th 2001Side: 3 Introduction and Motivation The essays investigates Mean and Volatility characteristics investigating and modelling non-synchronous trading effects in univariate and bivariate return series. In formal mean and volatility model specifications, the objective is to obtain a zero mean and identical and independently distributed model residuals, implying strict white noise.

4. Dissertation December 6 th 2001Side: 4 Introduction and Motivation Hence, the dissertations main objective: Bring new insight into asset pricing processes and market dynamics with a particular emphasize to model applications in many relatively thinly traded equity markets globally. Introduction and Motivation

5. Dissertation December 6 th 2001Side: 5 Introduction and Motivation Key insight suggests that non-synchronous trading effects may have implications for: i)Random Walk tests for for example mean and variance ratios ii)Option pricing Models need revising due to changing/stochastic volatility iii)The mixture of distributions hypothesis iv)Non-linearity in asset return series rejecting the martingale and the independence hypotheses Introduction and Motivation

6. Dissertation December 6 th 2001Side: 6 Non-Synchronous Trading Effects Market Microstructure The non-synchronous trading or non-trading effect arises when time series are taken to be recorded at time intervals of one length when in fact they are recorded at time intervals of other, possibly irregular, lengths.

7. Dissertation December 6 th 2001Side: 7 Non-Synchronous Trading Effects (cont.) Market Microstructure For example, closing prices may be recorded for asset A (continuously traded) and B (thinly traded) applying the following typology: Day:12345678 A: B:Closing Prices = Closing Time for the Exchange = Ticker for a Trade

8. Dissertation December 6 th 2001Side: 8  The lagged response in B induces spurious cross- autocorrelation between stock A and B Non-Synchronous Trading effects (cont.)  Spurious own-autocorrelation in daily returns for non-traded assets (B) may occur.  Event studies may be hurt by non-synchronous trading effects due to announcements and changing information flows  Obvious implications for tests of predictability, non-linearity in asset returns and quantifying the trade-offs between risk and expected returns (beta and CAPM).

9. Dissertation December 6 th 2001Side: 9 Non-Synchronous Trading Effects (cont.) Perhaps the most natural way to view observed returns: where is the observed return, is the virtual return and is the probability of non-trading. From here it is apparent that non-trading can induce spurious serial correlation in observed returns because each contains within it the sum of past k consecutive virtual returns for every k with some positive probability

10. Dissertation December 6 th 2001Side: 10 Non-Synchronous Trading Effects (cont.) The first and second moments for becomes: where and.

11. Dissertation December 6 th 2001Side: 11 NST Effects in Mean and Volatility  Non-synchronous trading (NST) should not affect the mean  NST increases the variance for series with nonzero means  NST induces geometrically declining negative serial correlation in returns with nonzero means  NST induces geometrically declining positive serial correlation in portfolio returns that are well diversified and have common non-trading probability, yielding an AR(1) for the observed returns process.

12. Dissertation December 6 th 2001Side: 12 Non-Synchronous Trading Effects  NST induces geometrically declining cross-auto-correlation between returns of asset i and j. Asymmetric due to different non-trading probabilities.

13. Dissertation December 6 th 2001Side: 13 Stylised Facts on Liquidity for Oslo Stock Exchange (OSE) The Norwegian Equity Market quotes about 150 assets. Below we show some characteristics for continuously traded as well as thinly traded asset, portfolio and index series from 1983 to 1995. ARCH (6) : A test for conditional heteroscedasticity in returns. RESET (12,6) : A sensitivity test for mainly linearity in the mean equation. BDS (m=2,  =1): A test statistic for general non-linearity in a time series.

14. Dissertation December 6 th 2001Side: 14 Stylised Facts on Liquidity for Oslo Stock Exchange (OSE)

15. Dissertation December 6 th 2001Side: 15 Summaries for OSE  Thin trading in some assets showing increased non-normality, serial correlation and non-linearity.  Leads and lags for international indices and for daily, weekly and monthly asset return intervals.  Anomalies are found. Coefficient changes across trading volume suggesting leads and lags in time series.  Stationary series seem plausible.  Volatility seems to change and seems higher for thinly traded assets relative to continuously traded series.  Crash (Oct. 1987) seems to be more serious for continuously traded series relative to thinly traded series.

16. Dissertation December 6 th 2001Side: 16 Random walks and the weak form of the EMH RW is a stochastic process where the changes of levels are given by the addition of a random variable, , which exhibits a zero mean and a constant volatility, and where there is zero correlation between observations. Formally It is the  that has zero mean and a constant variance (  2 ). Y t has an expectation E(Y t )=  and a variance of t  2. RWs exhibit the Markov property and the martingale property.

17. Dissertation December 6 th 2001Side: 17 A time series is said to be stationary if it has a constant mean, a constant variance and a covariance which depends only on the time between lagged observations. Stationary Time-Series Clearly the index are not stationary but the index returns data may be stationary.

18. Dissertation December 6 th 2001Side: 18 Reasons for Potential Mean and Volatility Changes 1.Crisis – Great Depressions, October 1987 Crash, September 11 …. 2.Financial Leverage 3.Macroeconomic Variables: e.g. interest rate changes 4.Changes in Information Flow 5.Changes in trading Volume, Events, Stock Exchange Cross listings, popularity and index inclusions 6.Globalisation and deregulations (lower segmentations)

19. Dissertation December 6 th 2001Side: 19 Now assuming Stationary series, the empirical observations in the Norwegian indices show: 1.Return Serial correlation  Suggesting a need for conditional mean modelling (lags) 2. Squared Return Serial correlation

20. Dissertation December 6 th 2001Side: 20 Time-series Analysis: Univariate stochastic models of time-series processes Conditional Mean models suggest a return process made of past components of the time-series itself. Ex. ARIMA(2,0,2): Ex. Vector ARIMA(1,0,1) (in matrix form): An ARIMA(p,q) series {Y t } is linear if  The residual must have a mean of zero and have IID

21. Dissertation December 6 th 2001Side: 21 Time-series Analysis: Lag-structure Specification for conditional Mean The LAG structure and the values for p and q is based on the Bayes Information Criterion (BIC) (Schwarz, 1978). The criterion for ARMA(p,q) is specified for returns as:  Giving us optimal ARMA lag specifications!

22. Dissertation December 6 th 2001Side: 22 Time-series Analysis: Why non-linear models are necessary for price changes Linear AR(  ) lag models have constant conditional variances:whatever the information about the observed returns:

23. Dissertation December 6 th 2001Side: 23 Time-series Analysis: Why non-linear models are necessary for price changes

24. Dissertation December 6 th 2001Side: 24 Time-series Analysis: Why non-linear models are necessary for price changes Squared Returns show far more dependence than ordinary returns. Plausible models for the return process must have for several small lags . This constraint can be used to prove that the return process is not linear. Non-linearity is very closely associated with volatility changes and conditional heteroscedasticity.

25. Dissertation December 6 th 2001Side: 25 Serial Correlation in Ordinary and Squared Returns: Index

26. Dissertation December 6 th 2001Side: 26 Non-synchronous Trading and Non-linearity in Returns

27. Dissertation December 6 th 2001Side: 27 Time-series Analysis: Why non-linear models are necessary for price changes Non-synchronous Trading and Non-linearity in Returns Ordinary Returns from thinly traded assets show far more dependence than continuously traded asset returns. For several small lags . Squared Returns from thinly traded assets don’t show much more dependence than continuously traded asset returns. Hence, for several small lags .

28. Dissertation December 6 th 2001Side: 28 Time-series Analysis: Why non-linear models are necessary for price changes Non-synchronous Trading and Non-linearity in Returns These constraints (especially the first) can be used to prove that thinly traded return series show a different non-linearity than continuously traded asset series.  Non-linearity is associated with volatility changes and non-synchronous trading effects.

29. Dissertation December 6 th 2001Side: 29 Correlation in Ordinary & Squared Returns: Thin Trading Series

30. Dissertation December 6 th 2001Side: 30 Correlation in Ordinary & Squared Returns: Continuous Trading

31. Dissertation December 6 th 2001Side: 31 The spectral densities for NST and Serial Correlation Note, spectral densities can be used to show exactly the same results. For example for thinly and continuously traded series and value-weighted index (totx): Its variation varies among all the Fourier frequencies  No interesting shape  No obvious dependencies Most of its variation among higher Fourier frequencies  Increasing for higher Fourier Frequencies  Longer-Term return dependencies Most of its variation among lower Fourier frequencies  Decreasing for higher Fourier Frequencies  Shorter term return Dependencies

32. Dissertation December 6 th 2001Side: 32 The spectral densities for NST and Serial Correlation Squared Returns for thinly and continuously traded and value-weighted index (totx): Its variation varies among all the Fourier frequencies  No interesting shape  No obvious dependencies Most of its variation among lower Fourier frequencies  Slow Decrease for higher Fourier Frequencies  Shorter term squared-return dependencies Most of its variation among lower Fourier frequencies  Decreasing for higher Fourier Frequencies  Shorter term squared-return dependencies

33. Dissertation December 6 th 2001Side: 33 Serial Correlation in Ordinary and Squared Returns: Summary

34. Dissertation December 6 th 2001Side: 34 Serial Correlation in Ordinary and Squared Returns: Summary Q(1) Q(10) Q(20) Q(40)

35. Dissertation December 6 th 2001Side: 35 Serial Correlation in Ordinary and Squared Returns: Summary Indicate a stronger effect (Q) in the mean than in the volatility for thinly traded series. Should be visible in the RESET test statistic (Ramsey, 1969) and the BDS test statistic (Brock et al., 1988,1991).

36. Dissertation December 6 th 2001Side: 36 Box-Ljung: Q/Q 2 : Ljung and Box (1978) autocorrelation test for adjusted raw returns and squared returns. ARCH (6) : ARCH (6) is a test for conditional heteroscedasticity in returns. Low {.} indicates significant values. We employ the OLS-regression y 2 =a 0 + a 1 *y 2 t-1 +…+a 6 *y 2 t-6. T*R 2 is  2 distributed with 6 degrees of freedom. T is the number of observations, y is returns and R 2 is the explained over total variation. a 0, a 1 … a 6 are parameters. RESET (12,6) : A sensitivity test for mainly linearity in the mean equation. 12 is number of lags and 6 is the number of moments that is chosen in our implementation of the test statistic.T*R 2 is  2 distributed with 12 degrees of freedom. Finally, BDS (m=2,  =1): A test statistic for general non-linearity in a time series. The test statistic BDS =T 1/2 *[C m (  *  )- C 1 (  *  ) m ], where C is based on the correlation-integral, m is the dimension and  is the number of standard deviations. Under the null hypothesis of identically and independently distributed (i.i.d.) series, the BDS-test statistic is asymptotic normally distributed with a zero mean and with a known but complicated variance. Test statistics for the empirical observations:

37. Dissertation December 6 th 2001Side: 37 Time-series Analysis: Generalized Autoregressive Conditional Heteroscedasticity Techniques developed in the 1980s by Engle (1982), Bollerslev (1986) and Nelson (1991) have given us the econometric tools for making predictions of future time- varying volatility. (G)ARCH model specifications have been developed to take account of the non-stationary of the variance. Moreover, financial markets show a need for improved forecasts of the volatility of financial time series, the advent of financial options and the generally greater volatility of financial markets in the last 20 or so years.

38. Dissertation December 6 th 2001Side: 38 Time-series Analysis: Generalized Autoregressive Conditional Heteroscedasticity The unconditional variance of a random variable is given as We are interested in the conditional mean, m t, and the conditional variance h 2 t. The conditional mean is the expected value of a random variable when the expectation is influenced (conditional) by knowledge of other random variables. m t is the expected value of y t, conditioned on the set of information, F, available in the previous time period. The information set F could contain any sources of information.

39. Dissertation December 6 th 2001Side: 39 Time-series Analysis: Generalized Autoregressive Conditional Heteroscedasticity The difference between y t and the mean is  t. We can derive the conditional variance, h 2 t, as a function of past squared residuals of the conditional mean equation (one one lag): and generalised ARCH becomes (only one lag): Similarly, the conditional variance is the variance of a random variable conditioned by knowledge of other random variables.

40. Dissertation December 6 th 2001Side: 40 Time-series Analysis: Lag-structure Specification for conditional Volatility The criterion reward good fits as represented by small (ln    and uses the term (m+n)T -1 lnT to penalise good fits that is got by means of excessively rich parameterisations. The criterion for GARCH(m,n) is specified for the squared residuals (  2 ) from the ARMA(p,q) residuals determined above as:  Optimal GARCH lag specification!

41. Dissertation December 6 th 2001Side: 41 Consequences of non-linearity in returns 1.Random Walk tests require changes  Mean and Variance Ratios for individual assets 2.Option Valuation deserves more consideration  Changing Volatility Models  Stochastic Volatility Models 3.The distribution of returns is a mixture distribution  Non-Normal conditional return distributions

42. Dissertation December 6 th 2001Side: 42 Dissertation paper: A Random Walk/Brownian Motion application. Chapter 5. Stock return Volatility in thinly Traded Markets. An Empirical analysis of trading and non-trading processes for individual stocks in the Norwegian thinly traded market(AFE, 2000, 10, pp.299-310)

43. Dissertation December 6 th 2001Side: 43  Models of Changing Volatility across NST and CT series  Remedy non-linearity found in raw and adjusted asset, portfolio and Index return series. 2. Option Valuation deserves more consideration

44. Dissertation December 6 th 2001Side: 44 Dissertation papers for ARMA-GARCH lag specifications: Chapter 6. Efficiently Estimated ARMA-GARCH Trading Volume Characteristics in Thinly Traded Equity Markets (AFE, 2001, 11, pp.539-556) Chapter 7. Non-linear Dependence and Conditional Heteroscedasticity. Evidence from the Norwegian Thinly Traded Equity Market (EJF, 2001, pp. 1 –30) Univariate Specifications Multivariate/Bivariate Specifications Chapter 8. Testing the Univariate conditional CAPM in thinly Traded Markets (AFE, 2001, 11, in press) Chapter 9. Testing the Bivariate conditional CAPM in thinly Traded Markets (Forskpub Molde 2001) Chapter 10. Event Induced Volatility(Forskpub, Molde 2001) Chapter 11. Calculating Abnormal Returns In Event studies: Controlling for Non- Synchronous Trading and Volatility Clustering in Thinly Traded Markets (Managerial Finance, 2002, in press)

45. Dissertation December 6 th 2001Side: 45  Models of Stochastic Volatility  Remedy non-linearity found in raw and adjusted asset, portfolio and Index return series. 2. Option Valuation deserves more consideration

46. Dissertation December 6 th 2001Side: 46 A Stochastic Volatility with Diagnostics for Thinly Traded Equity Markets (JMFM, 2001, 11, pp.539-556) Newly Published Papers for SNP/EMM specifications: