# The random walk hypothesis

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## Test for heteroscedasticity

When using the Ordinary Least Squares (OLS) regression model, it is usually assumed that coefficients are fixed, and the disturbances are homoscedastic, which indicates that the variance is constant over time. Breusch and Pagan (1979) argue that this should be questioned. When the requirements regarding homoscedasticity are not met, the efficiency in using a general linear model is weak and it may lead to substantial errors in the estimates. Adjusting for heteroscedasticity by introducing random coefficient variation allows for the potential for the dependent variable to take on different variances at each observation.
The heteroscedasticity can be tested by computing a Breusch-Pagan test where the null hypothesis states that homoscedasticity exists, while the alternative hypothesis indicates the existence of heteroscedasticity. Hence, rejecting the null hypothesis suggests heteroscedasticity, in other words, that there are non-constant volatility in the selected sample. When heteroscedasticity is present, linear regression is not the appropriate statistical model to use, hence a conditional heteroscedasticity test might be appropriate (Breusch & Pagan, 1979).

### GARCH (1/1)

Generalised Autoregressive Conditional Heteroscedasticity or the GARCH model was first introduced by Bollerslev (1986) and it was developed to examine volatility in returns. The main difference between GARCH and a simple linear regression is that the GARCH model allows for the variances of errors to be time dependent. The output of a GARCH model estimates the predictability of future stock returns and determines the nature of volatility, in other words if volatility is persistent or not.
A GARCH model consists of two components; a mean equation and a variance equation. The mean equation is stated in equation (2) and the variance equation is specified in equation 3:
where Ht is the conditional variance, ω is the constant term. βht-1 is the GARCH term which represents the influence of new shocks to volatility. The ARCH term is denoted as αε2t-1, which measures how intensely the volatility reacts to market shocks. Furthermore, α and β represents the market volatility and these are the variables to be estimated. A large error coefficient α implies that volatility reacts strongly to movements in the market. A large value of the β coefficient means that shocks to conditional variance take long time to fade out, in other words this measures the persistence of volatility (Bollerslev, 1986).
Significant values of α and β imply that historical stock market volatility affects current volatility and that volatility is highly persistent which in turn implies market inefficiency. The main drawback of the GARCH model is that it does not capture the asymmetry effect (leverage effect). The GARCH model tests two different hypotheses, one for the mean equation and one for the variance equation. The hypothesis for the mean equation is the same as that one of the OLS regression, while for the variance equation it is different. For an efficient market, α and β should be equal to zero. Hence, by the acceptance of the null hypothesis of the variance equation, α and β should not be statistically different from zero:
H0: ??=0
H0: ??=0
therefore, the alternative hypothesis suggests that α and β are not equal to zero.
The TGARCH or Threshold GARCH-model is an extension of the GARCH which was developed by Zakoin (1994). The TGARCH differs from the GARCH model in the sense that this model also accounts for the asymmetry effect, which is the tendency for bad news to have a greater impact on volatility compared to good news. The TGARCH contain one additional term that accounts for the asymmetry effect. The mean equation is specified in equation 2 and the variance equation is stated in equation 4
where Yt-1It-1 captures the asymmetry effect with gamma being the parameter to be estimated by the model. The hypothesis of the mean equations is similar to the one of the OLS regression. The variance equation of the TGARCH tests if γ=0, hence the null hypothesis is stated as: H0: γ =0
gamma equal to zero (γ=0) states that good and bad news have the same impact on stock market volatility. If gamma is different from zero, this implies that good and bad news have a different impact on stock market volatility. More specifically, if γ>0, this means that bad news increases stock market volatility and if γ<0, this implies that good news tend to increase volatility more than bad news (Zakoian, 1994).

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1 Introduction
1.1 Background
1.2 Problem
1.3 Purpose
1.4 Hypothesis
1.5 Delimitations.
1.6 Abbreviations.
2 Theoretical background
2.1 Efficient markets.
2.1.1 Samuelson´s dictum
2.1.2 The random walk hypothesis
2.2 Market Anomalies
2.3 Behavioural finance
3 Literature review
3.1 Summary of previous findings
3.2 Review of pre-2000 studies
3.3 Review of post-2000 studies .
4 Method
4.1.1 Method Summary
4.1.2 Scientific Philosoph
4.2 Statistical Theory.
4.2.1 Computation of Returns.
4.2.2 Volatility
4.2.3 Comparing mean using T-stat
4.2.4 Model of the day-of-the-week effect
4.2.5 Test for heteroscedasticity.
4.3 Quality Criteria.
5 Empirical Results
5.1 Descriptive Statistics
5.2 T-test for mean
5.2.1 Sample 1
5.2.2 Sample 2
5.2.3 Sample 3 .
5.2.4 Full Period
5.3 Ordinary least squares regression.
5.3.1 Sample 1
5.3.2 Sample 2 .
5.4 Test for Heteroscedasticity
5.5 ARCH/GARCH/TGARCH .
6 Analysis
6.1 Non-existing day-of-the-week effect
6.2 Evidence on Leverage Effect.
6.3 Evidence on Volatility Clustering
7 Discussion
8 Conclusion
9 References .
10 Appendix

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Stock market anomalies: The day-of-the-week-effect An empirical study on the Swedish stock market: A GARCH model analysis