Autoregressive Conditional Skewness and Kurtosis on Russian and US Markets

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Bachelor's work

Autoregressive Conditional Skewness and Kurtosis on Russian and US Markets

Аннотация

bevel kurtosis stock market

В данной работе исследуется динамическая модель для 3 и 4 моментов распределения доходностей - скоса и куртозиса. Имеются две основных задачи: сравнить результаты оценки данной модели для двух рынков, американского и российского, и для разных активов, что позволит сделать вывод об относительном уровне риска. Вторая задача заключается в том, чтобы проверить, коррелирует ли размер дивидендных выплат российских акций и величины их скоса и куртозиса. Это позволит оценить, какие акции более привлекательны для инвесторов: те, что платят относительно высокие дивиденды, или те, которые платят относительно низкие, либо вообще не платят.

Установлено, что относительный риск российского и американского рынков акций согласно данной модели примерно одинаков. Существенная разница проявляется скорее между группами активов на одном рынке, чем между двумя рынками. Также было выяснено, что на российском рынке доходность портфеля, состоящего из высоко дивидендных акций, имеет более тонкие хвосты, но при этом более выраженный отрицательный скос, чем портфель, состоящий из акций, платящих низкие дивиденды, либо не платящих их вовсе. С точки зрения инвестора низкодивидендный портфель более привлекателен, чем высокодивидендный, если не брать в расчет сам размер дивидендов в качестве потенциально дополнительного фактора полезности.

Abstract

This paper studies the dynamic model of skewness and kurtosis known as autoregressive conditional density model. There are two major goals: firstly, to compare the results of estimation of this model for various assets on two markets, US and Russian which will allow to distinguish assets and markets in terms of risk. The second goal is to look whether there is a correlation between dividend payments, skewness and kurtosis on Russian stock market. This will allow to tell, which stocks are more desirable for investors: those that pay relatively high dividends, and those which pay little or do not pay at all.

It was found that Russian and US markets possess relatively the same level of riskiness. The difference appears rather across various assets on the same market, than across markets. It was also shown that on Russian market the returns of high dividend portfolio have lighter tails than returns of low dividend one, but more pronounced negative skewness. In terms of investor's choice, the low dividend portfolio turned out to be more attractive, however, only if we do not take into account the potential additional utility brought by dividend payments.

Introduction

Modelling financial data is a non-trivial task. There are several reasons for that. Firstly, it is difficult to find the probability distribution which will adequately fit the data. The common knowledge is just that the tails of the distributions of financial returns are much heavier than that of the normal distribution, making the latter not a valid choice. However, there are still numerous distributions with tails heavier than Gaussian which can potentially suit our needs. These include Student's t, Pareto, Levy Stable, Entropy and many more. Choosing the best one is not easy: financial data is often significantly influenced by rare but extreme events which can potentially change the shape of the whole distribution and make the previous conclusions invalid.

Another issue comes when we consider modelling the moments of higher order. There exists well-developed literature on models of ARCH-GARCH type, which are used for modelling volatility, each of them having an advantage over another, so it is not simple to identify the best one. These models can vary in two ways: firstly, the functional form itself may be different. Secondly, we may assume different distributions for the error term. Moreover, we should not forget higher order moments as well. Skewness (the 3^rd moment) and kurtosis (the 4^th moment), although may not be so easily interpretable as volatility, but play very important role as well. Skewness indicates us where probability mass of the density function is located- in the left (in this case skewness will be negative) or in the right (then skewness will be positive) tail of distribution. This is important as negative skewness (or asymmetry) indicates that there are potentially more negative than positive outcomes for a specific asset, which undermines its attractiveness. Kurtosis is also an important indicator, as it measures the “tails' heaviness” of the distribution: higher kurtosis would mean relatively more frequent extreme events which increases the risk of holding a specific asset or portfolio.

Higher order moments play a huge role in certain areas of finance, specifically, in risk management and asset pricing. So, how to measure skewness and kurtosis? Of course one can simply use standard formulas for sample analogues and be satisfied with that. But such calculations are not robust enough: they neglect the overall shape of distribution and are also prone to outliers. Moreover, it might also come in handy to be able to predict future values of skewness and kurtosis which can then be used in forecasting and risk management problems, which cannot be done with sample statistics. One natural way to do so is to apply dynamic modelling to 3^rd and 4^th moments, just as it is used for volatility. This possibility opens for a researcher a large class of so-called Autoregressive Conditional Density (ARCD) models. Such modelling may allow capturing the moments which simple sample calculations was not able to detect. Obviously, such model requires the usage of probability distribution other than normal one. The distribution that may capture both heavy tails and asymmetry. The skewed (or generalized) Student's t distribution is an ideal candidate for this role and thus will be used in the study.

This paper will apply the ARCD model to returns of various financial assets of two countries: the US and Russia. The assets analyzed will be market indices, stock prices and exchange rates. The overall goal is to see, how the financial assets differ from each other in terms of the skewness and kurtosis they possess, whether there is a significant difference across the markets and across the classes of assets. After such analysis it will be possible to distinguish between Russian and American assets more rigorously in terms of their riskiness, as heavier tailed underlying distributions would generally imply higher volatility and more frequent extreme events, while negative skewness would mean higher concentration of adverse outcomes.

Apart from predicting skewness and kurtosis based on their past values, one might want to identify some external, exogeneous factors that affect these parameters. Why would it be potentially useful? By making such analysis an investor would be able to identify potential future values of skewness and kurtosis based on some publicly available data. These factors may come from the nature of the asset analyzed, its specific features which other assets do not possess. Unfortunately, this branch of literature is not developed well: there are very few papers which try to find factors that are correlated with skewness and kurtosis. Most of the literature concentrates on how the higher order parameters influence some variables (such as price), rather than what parameters affect skewness and kurtosis themselves. In this thesis the potential influence of one of such factors on these parameters will be examined. In this work I would like to propose the dividend yield of the stock as one of such factors, and see whether there is indeed a correlation between it and skewness with kurtosis. Moreover, I would also like to elaborate on whether higher dividend yield is necessarily a desirable thing for an investor. This will be done with the help of a simple model which calculates an expected utility of an investor holding a portfolio of stocks. These expected utilities will be calculated for two portfolios: the one which consists of dividend paying stocks and the one consisting of stocks which do not pay dividends at all, or pay them very little.

The paper will proceed as follows: in the next chapter the literature review will be given which will outline the major developments in the area of financial econometrics which deals with applications of non-Gaussian distributions and autoregressive models to modelling the higher order moments. Next, the data will be presented. Then, the tests for normality of distribution of returns will be performed and the distribution used for the further analysis will be presented. The fourth chapter starts with presentation and fitting of the model when both skewness and kurtosis parameters are constant on various assets from Russian and US markets. Then, one of the parameters is made dynamic and the estimation results along with the model performance are compared with the first model. Next, the second parameter is made dynamic, the model is fitted and the results are compared with two previous models. Finally, in the fifth chapter, the method which aim is to tell, whether the skewness and kurtosis parameters are correlated with publicly known financial data (dividend yields) will be presented and applied to the new set of data: portfolios consisting of Russian stocks. This chapter finishes with the description and application of the simple model which aim is to tell, which of the portfolios is more desirable from investors' point of view. The paper ends up with conclusion, where all the findings will be summarized.

1. Literature review

In this literature review, at first, the view on how non-Gaussian distributions have been applied to modelling the returns of financial assets will be given, then a brief history of volatility modelling will be mentioned. Further, some of the most influential papers which deal with modelling of higher moments shall be described and, finally, the papers which use skewness and kurtosis for answering the questions in finance theory related to portfolio choice and asset pricing will be addressed.

The fact that financial returns exhibit heavy tails was known for quite a long time. The usage of heavy-tailed distributions in economical applications may be traced back to the works of V. Pareto at the beginning of 20^th century. He discovered that income follows a power law distribution of the form . This distribution has polynomial tail and is noticeable for having infinite moments of order less than . Among the first researchers who applied the heavy-tailed distribution to the analysis of financial indicators was Benoit Mandelbrot. In his work dating back to 1963 he used the class of stable distributions to describe the behaviour of various financial assets, including stocks and commodities prices. He regarded such distributions as better descriptors of the behavior of financial assets than the normal one.

Numerous papers on the applications of non-Gaussian distributions were written since then. At first, the Student's t distribution was used. Compared to standard Gaussian, it can capture some of the tail-heaviness which financial returns tend to experience. However, in its basic version it is unable to describe the asymmetry often seen on financial markets: financial data often has more observations in one of the tails. Hansen (1994) proposed the asymmetric t distribution which includes the shape parameter in its equation. If >0, the random variable which follows such distribution is skewed to the right, and if it is less than 0, skewed to the left. When =0, the density reduces to standard t distribution. Some papers also used the above-mentioned Levэ-stable distribution. In particular, Mantegna & Stanley (1995) fitted it to the price differences of S&P 500 index with scale parameter of . They found out that this distribution remarkably well describes the dynamics of index right up to magnitude of approximately 6. After that, there is a substantial deviation of empirical data from theoretical, with former demonstrating exponential, rather than power law behavior.

There have also been quite exotic choices of modelling distributions. One of the popular classes of probability densities used in literature are the so-called generalized hyperbolic distributions. Their density function is defined as follows: , where X is normally distributed with zero mean and unit variance, whereas V has a Generalized inverse Gaussian distribution. https://en.wikipedia.org/wiki/Generalised_hyperbolic_distribution. Special case of this distribution, called Normal-inverse Gaussian distribution was used by Barndorff-Nielsen in 1997 paper to describe the financial returns. This distribution uses inverse Gaussian distribution as a mixing density. Its interesting feature is that two tails behave differently, meaning they decay at different rates. But they are both “semi-heavy”, declining at the rate slower than usual normal distribution, but faster than the stable distributions other than Gaussian (such as Levэ). However, some researchers pointed out that NIG distribution cannot handle substantial skewness which is sometimes present in financial returns and developed new density functions which can handle this issue. For instance, Aas and Haff (2006) introduced the Generalized Hyperbolic Skew Student's t-distribution. As it follows from its name, this density is from the same class of distributions as the NIG one. It has a substantial difference, however: the two tails decay at significantly different speed, one behaving exponentially and another polynomially. This feature helps describe the asymmetry appearing in financial returns: the negative tail, for example, is often heavier than the positive one. The GHS Student's distribution was fitted to log-returns of Norwegian stocks, international bonds, EUR/NOK exchange rate and European 5-year interest rate. Overall fit turned out to be better than from NIG distribution. Moreover, for the very skewed data this distribution is also superior to the skewed Student's t-distribution.

Apart from modelling the direct distribution of returns, there is another important area in financial econometrics which deals with modelling of time-varying higher moments. The first person to propose such kind of setting was Engle who in 1982 introduced the Autoregressive Conditional Heteroscedasticity model. In this framework variance is no longer constant in time and is modelled according to the following equation: , where is conditionally normally distributed with mean 0 and variance . The main idea of using such framework is to capture the volatility clustering effect: “large changes tend to be followed by large changes, of either sign, while small changes tend to be followed by small changes”. Mandelbrot (1963). This means that periods with high volatility are usually followed by high volatility, and periods with low volatility- by low volatility. Mathematically it can be depicted as for some , while still the correlation between simple returns is 0. Bollerslev (1986) introduced the generalization of an ARCH process- GARCH, which in its most general form (GARCH(p,q)) can be written as . The transition from ARCH(p) to GARCH(p,q) is similar to the transition from AR(p) to ARMA(p,q). And indeed, it can be shown that ARCH(p) is AR(p) for u², while GARCH(p,q) is ARMA(max{p,q},q) for u².

After the seminal papers by Engle and Bollerslev, the literature on autoregressive conditional heteroscedasticity has sprung. Modelling ARCH-type models with underlying distribution other than normal has become one of the directions in which this sphere has been developing. Both ARCH and GARCH models in their original specifications have a significant drawback: they assume the normality of errors' distribution. It is well-known that it is an unrealistic assumption and some heavier-tailed distributions need to be used. The attempts to apply some non-Gaussian distribution to GARCH framework goes back to Bollerslev who in his paper dated back to 1987 allowed for errors to be conditionally t-distributed. In this paper the GARCH(1,1)-t model was fitted to the exchange rates of British Pound and German Mark to US Dollar and also to some US stock indices, including S&P 500 and others. Estimates showed a reasonably well fit, which turned out to be better than in case of standard GARCH(1,1) model. Moreover, the theoretical level of kurtosis, the fourth moment, implied by the used specification are pretty close to its estimated value which also indicates the validity of such model. The question is, if there is some distribution which provides even better fit?

Usage of various distributions in GARCH framework has been rapidly developing since. Recent researches often apply it to option pricing, with the first paper of such kind by Duan (1995), who used the simple GARCH(1,1) model with normal distribution. Other papers include, for instance, the one by Menn & Rachev (2005) who developed the GARCH model, where innovations follow the stable distribution, which possess some of the attractive features of the Gaussian distribution, but also allowing for much heavier tails. Kim et al. (2010) used two different types of stable distribution - the classical tempered stable and the rapidly decreasing tempered stable and compared it to the model with normally distributed innovations. Both papers have found out that stable distributions unsurprisingly can describe excess skewness and kurtosis of financial data much better than the normal one.

Modelling of higher order moments started to develop more recently, but has become pretty deep and recognizable branch of literature. There are two broad models which are used for this purpose. They try to answer the same questions, but slightly in a different manner. The first models the dynamics of higher moments directly, by allowing them to vary over time and then estimating the equations. Such models are called ARCM models. One of the examples of the implementation of such setting is presented in Jondeau and Rockinger (2001). They applied Entropy distribution to model the returns, GARCH(1,1) for variance and simple autoregressive framework for skewness and kurtosis which are dependent on past values of residuals:

Here is the authorized domain, region where skewness and kurtosis exist. Also, specifications for skewness and kurtosis involving standardized residuals were estimated:

It was found out that in both cases the coefficients b₁ and c₁ are statistically insignificant. Probable explanation of this phenomenon is that both skewness and kurtosis are largely determined by rare extreme events and because of the rare occurrence of these, statistical tests may lack power and produce insignificant estimates.

The main disadvantage of ARCM models is that this approach may be very computationally intensive. Another approach started by Hansen (1994) is called the Autoregressive Conditional Density model. Unlike ARCM models, in ARCD ones the time dynamics is imposed not on higher moments directly, but on the shape parameters of the distribution, and skewness with kurtosis are derived from these time-varying parameters. The motivation for such extension is that complete description of conditional distribution is required for the efficient estimation of the equations for conditional mean and variance. Moreover, the accuracy of the chosen predictive distribution depends on the knowledge of the correct conditional distribution for the normalized error. Finally, the specification of the full conditional model may turn out to be quite useful in some asset pricing models, especially in the context of option pricing, since the options prices depend on the conditional distributions of prices of underlying assets and are some complicated functions of these conditional distributions. So, Hansen applied ARCD model to the interest and exchange rates assuming for the above-mentioned Skewed t-distribution and compared the goodness-of-fit to the cases with standard and asymmetric t-distribution with moments constant in time and to the model with time-varying moments and standard distribution. The “law of motion” for the shape parameter is a quadratic function: , where are the residuals. The improvement in the model fit was not very noticeable when only one of the tools, either asymmetric distribution or ARCD structure was applied. But when both of these were used, the overall quality of the model improved significantly, with LR statistic having the p-value of 0.02, hence indicating that the data has both the asymmetry and autoregressive conditional density structure. In subsequent papers the ARCD model was further developed, fitted with other functional forms and some other density functions.

First of all, the paper by Harvey & Siddique (1999) should be mentioned. It was one of the first in this field of study, along with Hansen's one. The authors again used skewed Student's distribution, but now the cubic equation was chosen for conditional skewness parameter:

with being the residuals. The model was fitted together with asymmetric GARCH specification (GARCH-M) to daily, weekly and monthly S&P500, DAX30 and Nikkei 225 indices returns for the period of 1969-1997 for US, 1975-1997 for German and 1980-1997 for Japan index. It was found out that both autoregressive skewness and asymmetric variance are important in describing these returns.

Among the papers which use other probability density functions, for instance, Yan's (2005) paper can be mentioned. It used two heavy-tailed distributions, namely Pearson's Type IV and Johnson's SU in the ARCD setting applied to S&P500 daily returns spanned from January 1990 to June 2000. Both these distributions have two shape parameters, to be estimated, and ARCD structure is imposed on them. Positive or negative skewness corresponds to being positive or negative; increase in holding other parameter fixed reduces kurtosis. Both of these, however, do not correspond to exactly 3^rd and 4^th moment, which need to be estimated separately. Author used the following specifications for these shape parameters:

Here z_t are standardized residuals and . It should be noted that if c₁ and c₂ are insignificant while c₃ is, the results of the regression might be spurious.

Since the model reduces to which is not distinguishable from constant model when converges to stationary level,. Author fitted several models using both the Pearson's IV and Johnson's SU distributions and adopting various specifications for skewness and kurtosis: in some of them the former was time-varying and in others-the latter, and compared them to the model where both higher moments are made time-varying. The results show that the specification with both moments being time-varying is the best choice. Also it is worth noting that results of fitting these two distributions are quite similar, as two of these have comparable degree of tail thickness. Pearson's distribution, however, is slightly heavier tailed and, therefore, is more applicable to the cases when data has more extreme events.

Among the more recent works, the paper by Feunou et al. (2016) can be mentioned. In this article authors focus on the evaluation of parametric conditional skewness models. The part of this paper which is of most interest to me focuses on evaluating the quality of parametric models for skewness and kurtosis in autoregressive conditional density setting. Authors take the data for S&P500 index returns and use three different distributions to model the dynamics of density parameters: the skewed Student-t, the skewed generalized error and the binormal. They estimate 8 different models for conditional parameters using the logistic transformation similar to Hansen. Overall, authors found out that the model which allowed for the leverage effect in conditional skewness and asymmetric GARCH dynamics for both skewness and kurtosis in the conditional distributions performed the best. The preferred specification according to the article is thus:

Here and are skewness and kurtosis parameters, respectively. They are estimated as the free values and then transformed to obtain the true parameters in order to bound the skewness between -1 and 1 and kurtosis in such a way for density to exist. The mappings are the following:

for skewed Student-t distribution, for generalized error, with the same transformation for skewness parameter as before, and for the standardized binormal distribution. Among three applied distributions, the generalized error one performed the best. Authors explain this by pointing out that this distribution allows for a richer dynamics in conditional skewness and kurtosis.

As we can see, there is a broad set of literature consisting of quite technical papers that deal with the estimation procedure of ARCD model. But how one can apply their knowledge of skewness and kurtosis parameters' dynamics to more concrete problems in finance? One of the possible applications is the theory of portfolio optimization. The usual problem is to maximize the expected return of a portfolio subject to unit variance, but one can also include maximization of skewness into this problem. Indeed, higher skewness of the portfolio would imply more right-tailed distribution and hence higher frequency of positive returns, which is desirable for investors. One of the papers which solves this problem is the one by Lai (1991). The optimization problem is the following:

s.t.

Here r is the return on risky investment, - risk free rate of return, X- vector of portfolio weights, and V-variance-covariance matrix. Depending on investor's preferences, different solutions may be attained. Such optimization may be extended to include not only skewness, but also kurtosis and even higher moments. In theory, one may apply dynamic models of higher moments to such portfolio optimization problems in order to obtain solutions which vary for different time periods.

Another possible area where dynamic skewness and kurtosis models may be useful is asset pricing. One might try to include skewness and kurtosis as factors which influence the assets' expected returns. Chang, Christoffersen and Jacobs (2013) extended the classical CAPM model by including the factors of deviations in market volatility, market skewness and market kurtosis from their conditional expectations into time series regressions:

Here . The first slope coefficient comes from the standard CAPM model and represents sensitivity to a market risk premium, while all other regressors are new, and betas at them measure asset's exposures to market volatility, skewness and kurtosis risks. The estimation results show that all betas are significant, even after controlling for other factors, such as size, book-to-market, momentum and others. It was found out that stocks with high sensitivities to innovations in implied market volatility and skewness exhibit lower returns on average, whereas those with high sensitivities to innovations in implied market kurtosis exhibit higher returns on average.

What can be said about the literature concerning the potential exogeneous factors that influence higher order moments? As it was mentioned in the introduction, the literature on this subject is not very well developed. In fact, the only paper I managed to find out is the one by Aggarwal & Rao (1990). It argues that variance, skewness and kurtosis of equity returns are inversely related to the extent of institutional ownership. To find out whether this hypothesis is true, authors formed quintiles based on the extend of institutional ownership, which was measured by the proportion of outstanding stock held by institutions and the number of institutions holding the stock. Then, sample standard deviation, skewness and kurtosis was computed for each stock in each quintile. It was found out that there is indeed a negative relationship between the values of these parameters and degree of institutional ownership in the stock.

As we can see, the literature concerning the modelling of time-varying skewness and kurtosis is pretty well established. Most of the existing literature, however, deals with fitting the models to some specific type of data (for example stock or exchange rate returns) for one specific market and do not perform the analysis to multiple financial indicators. The relative novelty of this paper is that it takes two different markets and several different assets on them, fits the models and compares the results of estimation with the respect to the nature of asset and market from which it originates. Moreover, to my concern, there are no (or maybe very few) papers which have studied the correlation of dividend payments with the distribution of portfolio returns. This study will try address this problem as well.

2. Data description

In this part of the study the data which will be analyzed in this paper will be presented, along with some motivation of why exactly it was chosen.

One of the aims of this work is to make an analysis for a wide range of financial assets, and to do so for two different economies: the “developed”, and the “developing” ones. For this purpose two countries were chosen: the USA, which has the largest economy in the world and which is considered to be a developed one, and Russian Federation, which has one of the largest developing economies in the world. The first major goal of this paper is to examine, whether there are significant differences in skewness and risk across the economies of these two countries.

The financial assets which were analyzed in the first part of the study (chapter 4) consist of returns on market indices, stock returns, and returns on the exchange rates. Starting with indices, the one which was taken for the US market is S&P500. It measures the stock performance of 500 largest companies that are listed on stock exchanges in the United States. It is also often considered as one of the best measures of the aggregate performance of the US market, so it is crucial for my analysis to understand the situation on the market as a whole. The data was taken starting from the beginning of 1980 and ending with the end of 2019. The reason for choosing such data span is that it is large enough to cover both periods of economic growth and economic downturn. Indeed, it covers such events as the “Black Monday” of 1987, which was the day of the largest one-day percentage decline in stock market history, recession of early 1990's, growing and bursting of the dot-com bubble (2001), the world's financial crisis of 2007-2008, which is considered the most severe economic crisis since the Great Depression of 1930's. The data span ends with the end of 2019, as there was not enough data for 2020 for some of the assets I was going to analyze. The main analogue of S&P500 on the Russian market is the RTS Index. It is a capitalization-weighted index of 50 Russian stocks (of the companies considered to be the largest ones on the Russian market) traded on the Moscow Exchange and calculated in the US dollars. It was introduced on September, 1 in 1995, hence in this paper it will be analyzed for a period from 01.09.1995 to 31.12.2019. This period covers several financial crises on the Russian market as well, including the crisis of 1998, when the Russian rubble was devalued and Russian government defaulted on its debt; the crisis of 2007-2008 which stroke Russia slightly later than the US and Europe, but which has lasted at least until the beginning of 2010; and the crisis which started in the end of 2014 and which lasted until approximately the end of 2016 - beginning of 2017 (some people consider it has been lasting until now). The main reasons for this final crisis were declining oil prices and economic sanctions imposed on Russia. These two factors together caused the loss of confidence of investors in the Russian economy and subsequent sharp decline in the value of Russian ruble. As we can see, both for American and Russian economy these time periods were rich on different events which resulted in wide range of their values, which would thus result in pretty accurate description of their true distribution. The data for both indices was taken from Yahoo Finance https://finance.yahoo.com..

Proceeding to the stock returns, this paper will analyze stock of the companies from two different industries: technological and resource extraction ones. It is natural to believe that technological companies are less stable and, as consequence, have wider range of possible realizations of their share prices and returns. This is one of the hypothesis that will be tested in this paper: whether technological companies are likely to have more extreme fluctuations in their shares' returns than natural resources ones. Again two countries will be taken: Russia and US. Technological companies taken are Yandex from Russia and Google from the US, whereas natural resources ones are Gazprom and Chevron. Data for Chevron starts from 1980, for Google, Yandex and Gazprom- from the moment of their IPO's in 2004, 2011 and 2006 respectively. All this data was again obtained from Yahoo Finance.

And the final group of assets that will be analyzed is the exchange rates. The reason why I decided to analyze them is that investments in exchange rates are often considered less risky than investments in equity, as exchange rates are considered less volatile. I wanted to test this hypothesis. Moreover, I wanted to see whether there is some difference in the behavior of exchange rates taken with both reserve currencies and with one reserve and the other which is not. Hence, I took USD/CHF as the former and USD/RUB as the latter exchange rates. The data for former spans from 1988 and for the latter- from 1998. The data for both these exchange rates was obtained from the website investing.com https://www.investing.com..

For the second part of the study (chapter 5), where effect of dividend yield on higher-order moments is examined, only the data for the Russian stock market was used. The reason is that Russian stocks possess quite broad set of possible values of dividend yield: from around 1% to almost 15% per year. Such wide range of possible values for dividend yield implies a rich overall picture and will make the analysis more robust. Stocks on American market, on the contrary, in recent years provide relatively low dividend yields, and if we analyzed them, the obtained range of dividend yields might have been not wide enough. The dividend yields on S&P 500, for instance, have seldom risen higher than 3% during the past 30 years https://www.investopedia.com/articles/markets/071616/history-sp-500-dividend-yield.asp..

Hence, the data for a larger number of Russian companies needed to be obtained. Luckily, all the data for closing prices of the newly selected Russian stocks could still be obtained from the Yahoo Finance, as it contains the data for all Russian stocks traded on Moscow Stock Exchange. The data was downloaded for 22 different companies for 4 years period: from the start of 2016 till the end of 2019. There was still a question of obtaining the dividend yields for each of the companies, however. Some part of the required data was obtained from the website place.moex.com which is one of the official websites of Moscow Stock Exchange created specifically for investors-beginners https://place.moex.com.. This website contains information on the most popular stocks traded on MSE: their current price, dynamics and, which interests us the most, the dividend yields over the past several years.

Unfortunately, not all of the stocks analyzed in this paper are available for investigation on the MSE website. For the remaining ones the following procedure has been applied: the website dohod.ru https://www.dohod.ru/ik/analytics/dividend. contains information on the total dividends paid by various Russian companies each year; the figure for each year has been taken from there and then divided by the average price of the stock over the year, the obtained figure is the average dividend yield during that particular year. As for the choice of the stocks, the sample represents shares of the largest public Russian companies, both paying and not paying any dividends, starting from 2016. These companies conduct different kind of businesses, from trading of natural resources (such as Gazprom, Lukoil, Nornickel and others) to mobile and other technologies (Yandex, MTS, Rostelecom) and banking (Sberbank, VTB). Thus, I expect the sample to be representative of the whole market.

We now turn to the discussion of the distributional issues related to our data.

3. The distribution

In this chapter theoretical background on the distribution which will be used in this paper shall be given and several tests for normality will be described and implemented.

3.1 Testing for (non)normality

In order to implement some non-standard heavy-tailed distribution, we should first answer the question whether the normal distribution provides a non-adequate representation of the data. If it does fit the data well, there is no need in using the more complex densities.

There are quite a lot of statistical tests which examine the data for non-normality. In this work two of them will be used: Jarque-Bera and Anderson-Darling. The first of them is quite intuitive: it uses the fact that for the standard normal distribution skewness (which is the third standardized moment) is equal to 0, together with the excess kurtosis (which is defined as fourth standardized moment minus 3, which is the value of kurtosis for standard normal distribution). The test statistic for the test is calculated as , where is sample skewness, - sample excess kurtosis and n is the sample size. Under the null hypothesis of normality the test statistic follows the chi-squared distribution with 2 degrees of freedom.

The second test is not so intuitive. It uses the following test statistic:

Here F() is the cdf of the theoretical distribution, in our case normal. The obtained test statistic is then compared against the critical values of the normal distribution.

The reason for choosing these two tests is that they both have pretty high statistical power. For instance, Anderson-Darling test came out as one of the winners according to 2011 study Razali et al. (2011) “Power comparisons of Shapiro-Wilk, Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests”.: it outperformed Kolmogorov-Smirnov and Lilliefors tests ceding only slightly to Shapiro-Wilk test. Moreover, although Jarque-Bera test is quite intuitive and generally works well, it has several disadvantages, such as low power for distributions with short tails. Although this case generally does not apply to financial data, for higher confidence two tests will be performed.

The table below summarizes the results of these two tests performed for the data. The 1^st column contains the names of the assets, two others - the p-values of the respective tests

Table 1. Normality tests p-values

As it was expected, both tests reject the null hypothesis about normality at any reasonable significance level, which is actually not surprising. This suggests using some other distribution, and one of the potential candidates will be described in the next section.

3.2 Choice of the density function

As it was mentioned before, there are numerous distributions allowing for fat tails which were used in numerous different papers when modelling financial data. This paper will use skewed (or generalized) Student's t-distribution: it can allow for very heavy tails, together with the asymmetry which is crucial for modelling financial returns.

Density of the generalized t-distribution (GST) has the following formula:

Here and Г() is the gamma function. The parameter controls skewness: if , GST reduces to standard t-distribution. If , the distribution is skewed to the right, while if , it is skewed to the left. is the degrees of freedom parameter. The smaller is its value, the heavier the tails will be. That is why this parameter may be seen being in charge of kurtosis or riskiness of an asset (and sometimes, further in this paper, it will be called kurtosis parameter). Both these parameters have restrictions: skewness should be in range between -1 and 1, while should be higher than 2 in order for density to exist- same, as with the standard t-distribution. Another restriction typically imposed on parameter is that it should be less than 30. The reason is that when Student's distribution has degrees of freedom higher than 30, it becomes virtually undistinguishable from the normal distribution. In this paper this density function will be fitted to the real data and estimates of the coefficients and computed and used as skewness and kurtosis parameters.

Here one question needs to be clarified before we continue: why to bother with such complicated method and not to just use more standard methods of skewness and kurtosis calculation? After all, one can just download all the data into Excel and use Excel functions to compute the values of these parameters. To start with, sample skewness and kurtosis are extremely sensitive to outliers, they will jump from one value to another when new data will be coming. The parameters described above use information about the whole distribution of returns and are, therefore, more accurate and stable representations of the general picture. Moreover, which is even more important, such estimators for skewness and kurtosis allow for dynamic modelling, which makes it possible to forecast the future values of higher moments. This can be particularly useful for asset pricing, portfolio choice and risk management problems.

The next chapter will describe the method for estimating skewness and kurtosis parameters based on the density function. This method will be applied to the data in three possible specifications: constant skewness and kurtosis, dynamic skewness but constant kurtosis and dynamic skewness and kurtosis. The obtained estimates will then be discussed.

4. Skewness and kurtosis on Russian and US markets

We now turn to the first major part of this paper, namely the modelling of skewness and kurtosis. But prior to that, few words should be said about the volatility specification. There exist numerous models for variance. This paper will use probably the most famous one: classical GARCH (1,1). There are two reasons for that. Firstly, it is done for simplicity: volatility modelling is not the main concern of this paper, so there is no need to overcomplicate and use more advanced models. Secondly, despite its simplicity, GARCH (1,1) actually performs pretty decent, often not giving in to more complicated models, and sometimes even outperforming them. In particular, GARCH (1,1) is not outperformed in exchange rates analysis, but it is usually inferior to models accommodating leverage effects in analysis of stock returns. Hansen, Lunde (2005) “A forecast comparison of volatility models: does anything beat a GARCH(1,1)?”. In this paper, however, leverage effect will be taken into account via the use of dynamic modelling for skewness and kurtosis, so GARCH (1,1) should be enough.

4.1 Constant parameters model

Now let us turn to the discussion of the model for skewness and kurtosis. In this section we will look at the model for log-returns where these parameters are held constant and only volatility is changing, while the next section will discuss the models where either skewness or kurtosis or both of them are time-varying. For simplicity, we do not assume any ARMA dynamics in the model for the mean, which is a relatively realistic assumption for a data studied over long periods of time as in our case. The model now therefore looks like this:

,

Here GST means generalized (skewed) Student's t-distribution. Such model is estimated almost like the standard GARCH - using the maximum likelihood method, just the two new parameters are added to the likelihood function and the new likelihood function is then maximized with respect to the whole set of parameters (This model will be called Model 1.

The table below summarizes the results of estimation of this model on the empirical data and provides estimates of the parameters and , the p-values for and log-likelihood of the model. The p-values for the parameters are not provided, as they are equal to 0 in all cases.

Table 2. Skewness and kurtosis parameters in constant parameters model

It can be seen that skewness parameter is generally negative, implying skewness to the left. It is, however, significant only for three assets: S&P500 and Chevron returns and USD/RUB exchange rate with latter actually having significant positive asymmetry. Assets also tend to “group” by the degrees of freedom parameter. The group with the highest values of this parameter consists of natural resources companies - Gazprom and Chevron, the one with the lowest values - of technological companies, namely Yandex and Google. Market indices are in the middle between these two groups. These results are quite expected: it was already mentioned that lower values of parameter imply heavier tails, that is higher probabilities of extreme outcomes. It is known that technological companies are very volatile due to the nature of business, that is why we obtained relatively low values for degrees of freedom parameter for them. Natural resources companies on the contrary are pretty stable, hence this parameter has relatively high values. Market indices are composed of numerous companies with different natures of businesses and are, therefore, situated somewhere in the middle. It is also worth mentioning that significant values for the degrees of freedom parameter in all cases implies that our choice of the density function being of the family of the Student's distribution is reasonable enough. For assets with insignificant asymmetry coefficient, the standard t-distribution would also be a suitable choice.

It is also worth noting that based on the estimated values of the parameter the hypothesis about Russian assets being more risky than American seems to be not correct: for all studied assets the value of this parameter is not lower, but is actually higher for Russian assets than for the US ones. The only exception is USD/RUB exchange rate, which returns have the lowest estimated value of DF parameter in the studied samples. The estimated value of it is 3.663. This implies that USD/RUB exchange rate returns possess quite heavy tails and may be the riskiest asset out of all studied in this paper. This exchange rate, however, actually has significantly positive estimated skewness and low value of the degrees of freedom parameter might indicate that the significant mass of the distribution may be situated in the positive right tail of the distribution, which is actually good for investors. Such dynamics might be explained that Russian ruble has experienced several pretty serious devaluations: in 1998, in 2008-2009 and in 2014-2015.

4.2 Model with dynamic parameters

In this part of the chapter the dynamic models of skewness and kurtosis will be fitted, and their performance in comparison to the model with constant parameters will be examined. At first, we will look at the model where only skewness parameter is dynamic, and next the model where both skewness and degrees of freedom parameters are allowed to be time-varying.

In the first case the complete specification looks like this:

,

(1)

This model will be called Model 2. Notice that equation (1) specifies the functional form for the dynamic asymmetry coefficient, where are standardized residuals. Such equation allows for both ARCH type dynamics (by including the lagged term ) and asymmetry or leverage effects which will be captured by coefficients and . Coefficient accounts for the impact of negative news on the skewness parameter, while shows the impact of positive news. This specification of dynamic skewness comes from Feunou et al. (2016) and is basically a slightly modified version of the model used by Jondeau and Rockinger (2003) Now, apart the GARCH (1,1) coefficients the likelihood function also includes the coefficients for the dynamic skewness and is again maximized with respect to the whole set of coefficients.

We would also like to study the performance of such model relative to the model with constant parameters in order to see whether it is worth to make such complications. This will be done with the help of the likelihood ratio test. For its implementation two log-likelihood functions are needed: the one for restricted version of the model, which in our case is the model with constant parameters (the restrictions are ), and the one for unrestricted version. The null hypothesis is that restricted and unrestricted models are equal in performance, hence, we can go on with restricted version. The test statistic is then formed: . Under the null hypothesis it has a chi-square distribution with q degrees of freedom, where q is the number of restrictions (in this case 3).