Prediction of imputed share returns

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Federal State Autonomous Educational Institution of Higher Education

National Research University

"High School of Economics"

International Institute of Economics and Finance

Graduation work

in the direction of preparation 38.03.01 "Economics"

educational program “Two-degree Program in Economics at the Higher School of Economics and the University of London”

Prediction of imputed share returns

Alexandrov Maxim Georgievich

supervisor

S.V. Gelman

Reviewer

A.E. Bulatov

Moscow 2019



Introduction

stock macroeconomic kurtosis profitability

One of the most important parameters considered when investors choose assets for their portfolios is the return. Much theoretical and empirical work has been conducted to describe and try to predict the characteristics of the returns distribution. These characteristics include the moments of the distribution, expected return, volatility, skewness and kurtosis.

As documented in theoretical and empirical literature, moments of the returns distribution contain information about the expectations of the future price of the security and, thus, may be used to predict future returns. Selected papers on this topic are discussed in the following section. The information contained in the moments implies that it is important to understand their dependency on other firm-specific and macroeconomic variables.

It is important to distinguish between option-implied and realised moments. Option-implied moments are based on option prices and are computed using some model, for example, the Black-Scholes model or the methodology proposed by Bakshi, Kapadia, Madan (2000), which is used in this paper. Unlike realised moments, option-implied moments reflect the expectations of future returns and the expectations towards risk. Option-implied moments are based on the daily data, therefore, they provide a better reflection of the market sentiment and the expectations of investors (Bliss and Panigirtzoglou, 2004).

In this paper we analyse the predictability of the option-implied risk-neutral moments of stock returns, namely, volatility, skewness and kurtosis and try to evaluate which factors significant influence these moments. We find that the returns, the liquidity measured by the bid-ask spread and the market volatility measured by the CBOE VIX Index have significant relationship with the moments. Knowledge of the relationship between the moments and other variables is important, as it may help to evaluate investors' expectations more accurately and by these means create advanced stock and options valuation methods, or use this knowledge to create trading strategies.

The rest of the paper is organised as follows. The next section discusses the relevant theoretical and empirical literature. Section 3 presents the data and the methodology used in our analysis. Section 4 contains the results, their interpretation and robustness checks. Finally, Section 5 concludes.



1. Literature review

Many researchers have considered moments of the stock returns distribution. The reason for the popularity of this topic is the amount of information the moments contain about stocks (or general assets). Clearly, the expected return is one of the main determinants of the attractiveness of a stock to investors, an important parameter of portfolio selection. Volatility is one of the most widely used measures of risk. Higher moments have been less often considered, however, there is evidence that skewness and kurtosis may be used to predict expected returns.

This section is organised as follows. Firstly, we present the research papers discussing the factors influencing volatility and the implied volatility surface of stocks and their predictability, including the implications for the pricing of the corresponding stocks and options. Secondly, we discuss the papers, which take into account higher moments of the returns distribution, specifically option-implied moments. Finally, we overview evidence of the information contained in the moments.

1.1 Research papers concerning volatility predictability

Many authors document predictability of volatility and the implied volatility surface of stock returns. Goncalves and Guidolin (2006) model the IVS as a function of moneyless and time to expiration, using VAR models to increase the quality of the modelling technique. They find that the S&P 500 index option volatility surface is highly predictable, however, these predictions can not be exploited to earn abnormal profits if transaction costs are taken into account.

Konstantinidi, Skiadopoulos and Tzagkaraki (2008) reach the similar conclusion by using several model specifications: AR(1), VAR, PCA, ARIMA and ARIFMA. They obtain point and interval forecasts which are proven to be statistically significant. However, if transaction costs are incorporated, the economic significance of the results vanishes.

Another method for modelling volatility smile or smirk is described by Zhang and Xiang (2008). They propose a quadratic function to mimic the shape of the IVS, including the slope and curvature. The approach is empirically tested using S&P index options and is shown to fit the IVS accurately. The resulting error is smaller than the minimum bid-ask spread of the corresponding options.

Finally, Bernales and Guidolin (2014) study the predictability of the S&P 500 index options using a two-stage approach: a deterministic IVS model is fitted at the first stage and VARX-type model is used for forecasting of the IVS of the stocks. The researchers document the predictability of the S&P 500 index IVS, however, the results are not economically significant after taking into account transaction costs.

1.2 Research concerning the relationship between the moments and the returns

One of the most commonly mentioned relationships between the moments of stock distribution and the returns is the “volatility feedback effect”. An increase in volatility also raises the required stock returns, lowering the price of the stock. Thus, high volatility decreases the price of the corresponding stock, leading to lower return. Campbell and Hentschel (1992) propose an asymmetric GARCH model to account for this effect. According to the model, volatility feedback effect is more pronounced during periods of greater volatility, which leads to the negative skew in case of high volatility (consequence of “bad news”).

A similar result was obtained by Ang, Hodrick, Xing, Zhang (2006). Authors document a negative price of aggregate volatility. Stocks with high sensitivities to the changes in market volatility earn lower returns, on average. Same relationship is present between the idiosyncratic volatility and the returns. These results are empirically confirmed using the US and the international data (Ang, Hodrick, Xing, Zhang, 2009). Conrad, Dittmar and Ghysels (2013) support this conclusion using the risk-neutral volatility. The same results are obtained by Christoffersen, Chang and Jacobs (2013) based on the options data for the period 1997-2007.

Amaya, Christoffersen, Jacobs and Vasquez (2015) conduct the same analysis based on the realised variance. The researchers do not find a strong relation between the volatility and next week's returns, however, after accounting for skewness the result is different. In particular, for negative skewness the relationship between volatility and returns is positive, however, if the skewness becomes positive, the link becomes negative. This reversion may be by the fact that investors accept lower returns for positive skewness. Thus, positive skewness may be an explanation to the puzzling results of Ang, Hodrick, Xing, Zhang (2006). Same explanation is appropriate for the negative relationship between the idiosyncratic volatility and returns.

Xing, Zhang, Zhao (2010) use the volatility skew to extract the information for the future returns of the underlying stock. The volatility skew is defined as the difference in the implied volatilities of OTM puts and ATM calls options. The authors prove significant predictive power of the skew: there is a positive relationship between the skew and the future returns. The result is persistent for at least six months.

Many papers consider the impact of higher moments on the expected returns. Harvey and Siddique (2000) show that systematic negative skewness requires an excess return as a premium for accepting this risk. Their model incorporates conditional skewness and shows that it is significant as an explanatory variable for returns even after accounting for size and book-to-market factors.

A similar results was obtained by Conrad, Dittmar and Ghysels (2013) using risk-neutral skewness extracted using the Bakshi, Kapadia, Madan (2000) framework. In addition, authors document a positive link between the risk-neutral kurtosis and returns. The results are still significant after controlling for the firm-specific factors. Christoffersen, Chang and Jacobs (2013) reach the same conclusion: they indicate a robust negative risk premium for the positive skewness and a weak positive relationship between the sensitivity to the changes in risk-neutral kurtosis and returns.

Amaya et. al. (2015) arrive at the same result while using realised skewness. Positive relation between kurtosis and next week's return is also found using realised kurtosis.

Doran, Diavatopoulos and Fodor (2013) show, that implied skewness and kurtosis can also be used to obtain information about future earnings announcements. They show that implied skewness and kurtosis are strongly related to the evolution returns through the announcements. The result applies to the stock and to both the put and the call options returns. Clearly, such a conclusion is a reflection of market inefficiency.

1.3 Papers concerning the predictability of higher moments

Many empirical papers are based on the framework allowing to compute the risk-neutral moments of stock returns from option prices, proposed by Bakshi, Kapadia and Madan (2000). It follows from the framework that the skewness is more negative for index returns, while the risk-neutral volatility is higher for the individual ones. In addition, the authors theoretically show that for a large class of utility functions more negative index skewness is a result of risk-aversion and fat tails of the returns distribution. The positive relationship between the risk-neutral skewness and the volatility surface slope is shown empirically: more negative risk-neutral skewness is related to a more negative IV curve slope.

Harvey and Siddique (2000) describe a negative relationship between the returns and skewness: when the past returns have been high, the skewness is expected to become more negative.

Dennis and Mayhew (2002) investigate the factors influencing risk-neutral skewness observed in prices of CBOE stock options. The computation of skew is based on the Baksi, Kapadia, Madan (2000) methodology. One of the main factors influencing the skew is proven to be market risk, as measured by beta. Skewness tends to be more negative for the stocks with high betas. This relationship imply that options can not be priced based only on non-arbitrage conditions. Further, the researchers prove volatility and corresponding index skew to make the equity skewness more negative. Other important factors are firm-specific ones, including the liquidity (measured by volume) and the firm size.

Finally, Neumann and Skeadopoulos (2013) find predictable patterns in the risk-neutral moments of the S&P 500 index options. The authors use ARIMA(X), VECM and ARFIMA models to model the RNM's. The models are evaluated based on the out-of-sample forecasting performance. The paper confirms the conclusions of Goncalves and Guidelin (2006), finding predictable patterns in the IVS of S&P index options, further, the predictability of higher RNM's is documented. Authors document autocorrelation in higher order RNM's and the cross-moments relationship. The predictability, however, is not economically significant.



2. Data description and methodology

2.1 Data

The data in this paper corresponds to a sample of 1532 US stock, trading at NYSE and NASDAQ. We select stocks according to the following criterion: the stocks were part of the Standard and Poor's 1500 index at one or more of the following dates: January 31, 1996, June 30, 2005 and September 30, 2015. We include both trading and delisted stocks to avoid any survivorship bias.

We use OptionMetrics IvyDB database to obtain the option information on the sample companies. We take the daily quotes of 1532 stocks and the corresponding Volatility Surfaces for the period from 1 January 1996 to 30 April 2016. We take on average 220 daily prices per year for each stock for the years from 1996 to 2015 and 78 daily quotes in 2016 (January - April) The Volatility Surface is calculated using the interpolation of the daily option quotes for the set of deltas ranging from 20 to 80 with increments of 5 in absolute values. The Volatility Surface provides us with the information on approximately 90% of the daily quotes of the options with maturities closest to 1 month for the period. In addition, we extract a proxy for the risk-free rate from the OptionMetrics database.

The data for the daily historical betas and the leverage ratios for the securities in the sample are obtained from the Thompson Reuters Datastream database. We selected our sample excluding the companies for which the beta values are present for less than 3 years. We also exclude the beta values calculated earlier than 1 year since the listing of the corresponding company. The betas and leverage ratios are unavailable for some periods for several companies.

We further obtain the daily quotes of VIX index for the period from the mentioned period from the finance section of yahoo.com.

The data for the daily US Treasury Yield Curve Rates for 1 months - 30 year maturities was downloaded from the US Department of The Treasury website.

The Moody's BaAA and AAAA rated corporate bonds average yields are obtained from the Moody's website for the mentioned period.

2.2 Computing risk-neutral moments

We use the methodology suggested by Bakshi, Kapadia and Madan (2003) to extract the values of Risk-Neutral Moments (RNM's) of stock returns from the data. The values of the RNM's are expressed in terms of the fair values of the following 3 contracts:

where is the continuously compounded rate of return at the moment over a period , is the price of the underlying at time , denotes the risk-free rate, denotes the risk-neutral expected value conditional on the information at time . The corresponding payoff of the contracts are:

Let the mean of the log-return over the period be:



To calculate the fair values of these 3 contracts, they can be represented as a combination of put and call options with different maturities, the underlying asset and the risk-free asset. Specifically, the linear combination of OTM calls and puts is used to determine the values of the contracts:

where and are the prices of a call and a put options with strike K and time to maturity . Finally, the expressions for risk-neutral volatility (VOL), skewness (SKEW) and kurtosis (KURT) as functions of the contracts values are:



For the implementation of the formulas (8)-(10) we need OTM option prices for the continuum of strikes. In reality, however, only finite set of option prices if available for a set of discrete strike prices. For the empirical computation of RNM's we use trapezoidal approximation, in line with Dennis and Mayhew (2002). We utilise the options with deltas ranging from 20 to 45 in absolute values, removing the options with higher deltas from the dataset. As mentioned earlier, we are using the options with maturities close to 1 month for our analysis, so we arrive at 30-days RNM's. The computed risk-neutral volatility is further annualised through the multiplication by .

As we use an approximation to overcome the unavailability of a continuum of option prices, bias may be introduced into the computed RNM's. The bias has 2 sources: discreteness of the strike prices set and the asymmetry of the domain of integration. In addition, the unavailability of the data on the deep OTM options in the Option Metrics Database harms the results. This results in the negative bias in the kurtosis calculation and positive bias in the skewness.

The computations of the risk-neutral moments were performed in R (see the script in the Appendix 1).

2.3 RNM's: a preliminary analysis

Figure 1 shows the evolution of the median risk-neutral volatility, skewness and kurtosis for the stocks in the sample for the period from 1 January 1996 to 30 April 2016. The volatility is annulized. We can observe a rise in volatility in years 2018-2019, which is an expected consequence of the financial crisis. We can observe that skewness is negative for almost the whole period, which is consistent with the existing empirical evidence, while the kurtosis is less than the 3, i.e. the kurtosis of the normal distribution. The lacking kurtosis may be explained by the bias described in the previous section.

Table 1 contains the summary statistics for the computed risk-neutral moments. As mentioned, both the median and the mean of the skewness is negative, while the same statistics for the kurtosis are less than 3. Field “Autocorrelation” contains the values of the first order autocorrelation in the RNM's series. We can see, that these values are quite high, especially for the volatility. Such phenomenon is known as “volatility clustering” effect.

Figure 1. Evolution of RNM's through the period

Graphs 1-3 show the evolution of the risk-neutral volatility, skewness and kurtosis during the period from 1 January 1996 till 30 April 2016.

Graph 1: Risk-Neutral Volatility

Graph 2: Risk-Neutral Skewness

Graph 3: Risk-Neutral Kurtosis

Prior to the panel analysis for each of the stocks in the sample we perform the Augmented Dickey-Fuller test to identify and then drop the stocks with the non-stationary RNM's. Out of the sample of 1532 companies we find that 233 companies have non-stationary Volatility series, 7 have non-stationary Skewness series and 15 have non-stationary Kurtosis series. We drop a total of 245 stocks which have at least one non-stationary RNM series from the further analysis.

Table 1. Summary statistics for the computed RNM's

	Volatility	Skewness	Kurtosis
Number of Observations	4539477	4539477	4539477
Mean	1,1551	-0,1313	2,4202
Median	1,0287	-0,1624	2,2069
Standart Deviation	0,5561	0,4897	1,2957
Autocorrelation	0,9391	0,6655	0,6670

2.4 Explanatory Variables

We obtain information on the companies-related explanatory variables and the Risk-Free Rate from OptionMetrics and Thompson Reuters Datastream, and the market-related explanatory variables (VIX, Default Spread, Term Spread) from the sources described in the section “Data”. To match the information sets from OptionMetrics and Datastream, we use the Ticker-Security ID cross-reference provided by OptionMetrics.

A. Returns

Many researchers indicated that the returns influence volatility, skewness and kurtosis. First of all, is is well known that negative news tend to increase volatility, while the positive news usually lead to lower volatility. Campbell and Hentschel (1992) describe the “volatility feedback effect”, negative asymmetric relation between volatility and the returns and stock prices. Returns can be seen as a proxy for the news, because stock prices quickly reflect the news, which is consistent with semi-strong form of market efficiency. Further, the news may influence skewness and kurtosis: the positive announcements may shift the distribution of returns in the positive direction and decrease tail risk, which results in a decrease in skewness and kurtosis.

Many authors indicate negative relationship between the volatility and the returns, same relationship between the skewness and the returns and positive link between the kurtosis and the returns.

Harvey and Siddique (2000) describe a similar relationship between the returns and skewness: when the past returns have been high, the skewness is expected to become more negative. This evidence that the returns may have explanatory power.

B. Liquidity Measure.

We use bid-ask spread as a proxy of liquidity. The bid-ask spread is calculated as a ratio of the difference between the quoted ask and bid prices and the close price. We expect a negative relationship between the liquidity and the volatility of the stock, because higher volatility leads to higher uncertainty and further sale of the stock by the traders, leading to a lower volume and, thus, higher spread.

The model by Hong and Stein (2003) predicts more negative skewness as a consequence of increased liquidity, measured by volume. The model is based on the argument that the volume is a proxy for the intensity of disagreement between the market agents, if some of them face short sale constraints.

Dennis and Mayhew (2000) find that liquidity indicated by the trading volume in line with firm size has significant explanatory power for the risk-neutral skewness.

As there is significant evidence that the liquidity influences RNM's, we include this variable in our analysis.

C. Beta

We employ historical beta as a measure of systematic risk of the stock. Historical betas are calculated by regressing the daily stock returns on the returns of the FTSE All-World index using the information for the past 60 months. The systematic risk of the stock measures its dependence on the market conditions and, thus, may affects its volatility as well as the higher RNM's.

Dennis and Mayhew (2002) show that a higher beta leads to more negative skewness of the risk-neutral density of returns. They present evidence that market risk, in line with other firm-specific factors have significant explanatory power for the skewness. To further analyse the effect of market risk on the RNM's, we include it in our analysis.

D. Leverage

We use leverage ratio to indicate the default risk of the company. The leverage ratio is calculated by dividing the sum of Long Term Debt and Short Term Debt & The Current Portion of the Long Term Debt by the sum of Total Capital and Short Term Debt & The Current Portion of the Long Term Debt, expressed in percent. Book values of debt and capital are used in calculations.

Bakshi, Kapadia and Madan (2000) show that leverage effect leads to the more negative skewness of the individual stock returns, than the skewness of the index returns. Empirically, however, they arrive at the opposite result. As the effect of the leverage is uncertain, we test in in our framework.

E. VIX

We employ the price of the CBOE VIX Index as and indicator of the market volatility. As can be seen from the graphs in the previous section, there is significant correlation between the values of volatility, skewness and kurtosis for different stocks and, thus, with market volatility.

Dennis and Mayhew (2002) present evidence that there in the periods of high market volatility, the risk-neutral skewness of the individual stocks becomes more negative. To further investigate the relation between the market volatility and the RNM's of the individual stock we include this variable.

F. Other Market-Related Factors

In addition to the variables described we employ a proxy of the risk-free rate, term spread and default spread in our analysis. These factors are used as indicators for the business cycle. A proxy for the risk-free rate is calculated through the linear interpolation of 2 yields of bonds with maturities closest to 1 month. The term spread is calculated as the difference between the 10-year and 3-months US Treasury yields. The default spread is calculated as the difference between Moody's BAAA-rated and AAAA-rated corporate bonds average yields.

Table 2 presents the summary statistics for the explanatory variables we use for our analysis. The median of the returns is zero, because our sample contains a significant number of illiquid stocks, for which the daily return is often equal to zero.

Table 2. Summary statistics for the explanatory variables

	Return	Spread	Beta	Leverage	Default Spread	Term Spread	Risk-Free Rate	VIX
Number of Observations	4539465	4539477	4492079	4476518	4539477	4519679	4539477	4539477
Mean	0,0006	0,0330	1,1631	0,3532	0,0104	0,0185	0,0234	20,7574
Median	0,0000	0,0251	1,0734	0,3220	0,0093	0,0200	0,0132	19,0300
Standart Deviation	0,0276	1,5323	0,7013	1,7078	0,0046	0,0113	0,0230	8,4935

Table 3 indicates the correlations between the explanatory variables and the RNM's we have computed. The only significant correlation we observe is the correlation between the default spread and the VIX price. This correlation, however, is not expected to affect our results significantly.

Table 3. Correlations between the dependent and the explanatory variables.

Vol, Skew, Kurt and Lev refer to the volatility, skewness, kurtosis and leverage variables, respectively

	Vol	Skew	Kurt	Return	Spread	Beta	Lev	DefautSpread	Term Spread	Risk-Free Rate	VIX
Vol	1,0000
Skew	0,3033	1,0000
Kurt	-0,0394	0,3171	1,0000
Return	-0,0133	-0,0900	-0,0283	1,0000
Spread	0,0101	0,0033	-0,0015	-0,0014	1,0000
Beta	0,3185	0,1190	-0,0534	0,0048	0,0041	1,0000
Lev	-0,0115	-0,0118	0,0031	0,0001	-0,0001	0,0179	1,0000
DefautSpread	0,2911	-0,0376	-0,0640	0,0016	0,0044	0,0604	-0,0003	1,0000
Term Spread	0,0206	-0,0716	0,0033	0,0033	0,0006	0,0337	0,0038	0,3074	1,0000
Risk-Free Rate	0,0528	0,0889	-0,0726	-0,0050	0,0014	-0,0923	-0,0061	-0,3292	-0,7932	1,0000
VIX	0,4243	0,0501	-0,1031	-0,0611	0,0079	-0,0201	-0,0029	0,6416	0,1595	0,0226	1,0000

2.5 Methodology

In order to access the effect of each factor and their significance, we separately regress Volatility, Skewness and Kurtosis on the variables listed in the previous section. As indicated in Table 3, the autocorrelations between the RNM's are small, therefore, we do not consider any interdependencies between the moments in our analysis.

The securities analysed in this paper correspond to the companies from different industries and different locations of the United States. Therefore, it is reasonable to assume that the impact of the explanatory variables on these companies and the corresponding RNM's may be different. This assumption implies that standard panel regressions with fixed or random effects can not be used, as they do not account for different sensitivities of the moments to the explanatory factors.

To illustrate the estimation problem, consider the following framework (Pesaran, 2006):

where is the dependent variable, is a vector of observed common effects, and is a vector of regressors and is an error term. is a vector of elasticities, or slopes corresponding to the regressors. Note, that observed common effects vector may also include time-invariant common effects.

In order to take into account different elasticities or so-called “heterogeneous slopes” we use the Mean Group (MG) estimator, introduced by Pesaran and Smith (1995). This estimation approach consists of two steps: firstly, group-specific regressions are estimated, where refers to the number of different groups or cross-section units. The regressions are estimated using ordinary least squares (OLS). At the second step the resulting slopes are averaged (using either arithmetic or weighted average). MG estimator allows to consistently estimate equation (1).

Another reasonable assumption for our data is the presence of cross-sectional dependence. Such cross-sectional effects may be present if some shocks influence all cross-sectional units and are not reflected in the regression equation. These shocks may include financial turbulence or the effect of unobserved macroeconomic variables. These effects refer to the presence of unobserved common factors, which are part of the error term . Let the error term be generated via the following process:

where refers to the vector of unobserved common effects and is assumed to be white noise. As the unobservable common factors are included in all error terms, the errors become correlated. Further, the unobserved common factors may also influence the regressors. Consider the following process for the regressors:

where is a vector of other factors influencing the regressors, and are the coefficients matrices corresponding to the factor vectors and and is the white noise error term. The configuration of the equation (3) causes endogeneity in the regression equation (1), leading to inconsistent estimates.

To solve both cross-sectional dependence and endogeneity problems, Pesaran (2006) introduced the Common Correlated Effects MG estimator. The estimator allows to estimate equation (1) consistently by including the cross-sectional averages of the dependent and independent variables, and in each of the group-specific regressions. The cross-sectional averages serve as a proxy for unobserved common factors, making the error of the resulting equation cross-sectionally independent.

As mentioned earlier, estimating regressions for large panels without accounting for the unobserved common factors causes omitted variable bias, leading to correlations between the error term. Chudik, Pesaran and Tosetti (2011) distinguish between two types of cross-sectional dependence. Strong cross-sectional dependence takes place if:

for some , while the errors are weakly cross-section dependent if:

Pesaran (2015) shows that CCEMG procedure allows consistent estimation even in case of weak cross-sectional dependence. Strong dependence, however, leads to inconsistency problems. To detect the strong cross-sectional dependence, Pesaran (2015) proposed the test:

where is a sample estimator of correlation coefficient of the regression residuals corresponding to the panel units and . Under the null hypothesis of weak cross-sectional dependence, the CD statistic has standard normal distribution.

As can be seen from Table 1, the risk-neutral moments we computed show high first-order autocorrelation. This implies adding a lagged dependent variable as an additional regressors may increase the prediction accuracy of the model. For this purpose, however, we need to take a slightly different estimation approach. The CCEMG estimator is consistent only for non-dynamic regression specifications. Chudik and Pesaran (2015) show that in the dynamic specifications, however, the estimator can be made consistent my adding cross-sectional lags of the dependent and independent variables. The estimation method is names Dynamic Common Correlated model. The equation to estimate becomes:

where refers to the lags of cross-sectional averages and .



3. Results

To evaluate the predictability of the risk-neutral moments and to investigate the impact of various factors on these moments we estimate the following equation for volatility:

The equation for skewness is:

Finally, the equation for kurtosis is:

In the equations (1)-(3) , and refers to the values of risk-neutral volatility, skewness and kurtosis for the stock at moment respectively.

, , correspond to the values of return, bid-ask spread and beta for the stock at moment respectively expressed in absolute values. We include these firm-specific variables with lags as they are weakly exogenous. refers to the leverage ratio of the company at moment expressed in absolute value. , , and correspond to the values of default spread, term spread, risk free rate and the price of the CBOE VIX index at moment expressed in absolute values.

Our estimation procedure is as follows. Firstly, we estimate equations (1)-(3) without including the lagged volatility, skewness and kurtosis (non-dynamic specification) respectively using the MG estimator. We further extract residuals and conduct CD test in order to investigate the presence of cross-sectional dependence. The results of the tests can be found in the Appendix #. As test for all of the three RNM's indicate the presence of the cross-sectional dependence, we need to use the CCEMG estimator. We further fit the non-dynamic specification using the CCEMG estimation procedure, i.e including the cross-sectional averages.

In the previous sections we documented the high autocorrelations present in the computed RNM's series. The moments of stock distribution, both realised and option-implied, are known to be persistent in time. This feature for volatility is known as “volatility clustering” effect.

To account for high autocorrelation, we include the lagged volatility, skewness and kurtosis respectively in the corresponding equations. We fit these models using the DCCE approach. As our data includes observations for the period from 1 January 1996 to 30 April 2016, i. e. for 5098 days, we include lags of the cross-sectional averages. The output of the CCE and DCCE models are then compared by accuracy of their predictions using the root mean square error (RMSE). As RMCE criterion indicates DCCE has greater predictive power for all of the three moments, in this section we present and discuss the output of the DCCE model. CCE results can be found in the Appendix #.

The fitting of the CCEMG and DCCEMG models requires estimating a large number of parameters. As we need to estimate regressions for the panel units separately, while each of the individual regressions includes cross-sectional lags, such calculations require significant computational power. Given the length of the time period considered (more than 20 years) and daily data, we have more that 4 500000 observations. Therefore, it is not feasible to estimate the models for the whole sample. In order to overcome this problem, we choose a subsample of 150 stocks from the total of 1293 stocks present. The sampling design is as follows: firstly, we remove the stocks for which we have less than 1000 observations. Finally, a simple random sample of 150 SecurityID's is taken from the full list of SecurityID's corresponding to the stocks we have. We include the corresponding 150 companies in our sample.

Table 4 presents the output of the DCCE model for volatility. The results indicate coefficients for several variables are significant at any reasonable confidence level. Firstly, the positive significant coefficient at the lagged volatility confirms the presence of autocorrelation documented in the previous sections. We can see that “volatility clustering effect” is present not only in the realised volatility, but in the option-implied volatility as well.

Table 4. Dynamic Common Correlated Effects model for Volatility. Volatility (-1), Return (-1), BidAskSpread (-1) and Historic Beta (-1) refer to the volatility, return, bid-ask spread and beta variables lagged one period. Leverage, DefaultSpread, TermSpread, RiskFreeRate and VIX refer to the leverage ratio, default spread, term spread risk-free rate and Vix variables. The resulting coefficients, standard errors, p-values and 95% confidence intervals are given for these variables in the table. Coefficients of the significant coefficients at 5% level are highlight in bold

	Coefficient (Standart Error)	P-Value	95% Confidence Interval
Volatility (-1)	0,7655	0,000	0,7353	0,7957
	(0,0154)
Return (-1)	-0,1932	0,000	-0,2508	-0,1356
	(0,0294)
BidAskSpread (-1)	0,2307	0,000	0,1644	0,2969
	(0,0338)
HistoricBeta (-1)	0,0213	0,162	-0,0086	0,0512
	(0,0152)
Leverage	-0,5096	0,237	-1,3544	0,3353
	(0,4311)
DefaultSpread	0,5380	0,573	-1,3343	2,4102
	(0,9552)
TermSpread	-0,2182	0,447	-0,7807	0,3442
	(0,2870)
RiskFreeRate	-0,8870	0,435	-3,1141	1,3401
	(1,1363)
VIX	0,0037	0,000	0,0029	0,0044
	(0,0004)
RMSE: 0,18

The results indicate significant negative relationship between the volatility and the return lagged one period. This relationship is consistent with the results of Ang, Hodrick, Xing and Zhang (2006 and 2009), Conrad, Dittmar and Ghysels (2013) and Christoffersen, Chang and Jacobs (2013). Low returns correspond to the stocks with higher volatility. Taking into account the volatility clustering effect, the negative link between the returns and the volatility may be explained in the context of “volatility feedback effect”, as in Campbell and Hentschel (1992). Negative news, which result in lower returns, leads to high volatility. On the contrary, positive returns corresponding to positive news decrease uncertainty, also decreasing volatility. As we include a proxy for the market volatility, the price of the VIX index, in our model equation, we can treat the result as the negative relationship between the idiosyncratic volatility and the returns, which is also consistent with the conclusion of Ang, Hodrick, Xing and Zhang (2006).

We do not include the skewness in our model equation, therefore, it is uncertain whether the results we obtained are similar to the results of Amaya, Christoffersen, Jacobs and Vasquez (2015). However, for the majority of observations in our sample skewness is negative. Having obtained the negative coefficient for the returns given negative skewness for the majority of observations, our results contradict the conclusion of Amaya, Christoffersen, Jacobs and Vasquez (2015). The reason for the different results may be the usage of the implied moments instead of the realised.

We can observe the significant positive link between the volatility and the bid-ask spread. Therefore, the result we expected has been confirmed. There is a positive relationship between liquidity and volatility. High volatility leads to a decreased trading activities with the corresponding stock, lowering the liquidity of the the security and widening the bid-ask spread.

The last significant variable is our proxy for the market volatility, the price of the CBOE VIX index. The connections between the market and the individual stock volatility is expected. This link indicated the presence of the factors affecting the volatility of many securities, i.e. causing market turbulence. We mentioned such factors, when discussing the possible cross-sectional dependency present in the moments, and the significant Vix coefficient is an additional confirmation of such dependency.

We also do not find any significant relationship between the market risk measured my beta and the volatility. The volatility as a measure of the overall risk of the security, shows no connection to the measure of market risk. Same absence of relationship applies to the leverage ratio of the corresponding firms.

The variables we used as indicators of the business cycle also show no significant influence on the volatility.

Table 5 presents results of the DCCE regression for skewness. The same set of the variables is significant at 95% confidence level. Firstly, as mentioned earlier, the significant positive coefficient for the lagged skewness confirms the autocorrelation present in the option-implied skewness.



Table 5. Dynamic Common Correlated Effects model for Skewnes. Skewness (-1), Return (-1), BidAskSpread (-1) and Historic Beta (-1) refer to the skewness, return, bid-ask spread and beta variables lagged one period. Leverage, DefaultSpread, TermSpread, RiskFreeRate and VIX refer to the leverage ratio, default spread, term spread risk-free rate and Vix variables. The resulting coefficients, standard errors, p-values and 95% confidence intervals are given for these variables in the table. Coefficients of the significant coefficients at 5% level are highlight in bold

	Coefficient (Standart Error)	P-Value	95% Confidence Interval
Skewness (-1)	0,5808	0,000	0,5522	0,6094
	(0,0146)
Return (-1)	0,3836	0,000	0,2842	0,4830
	(0,0507)
BidAskSpread (-1)	0,1179	0,023	0,0161	0,2197
	(0,0519)
HistoricBeta (-1)	-0,0046	0,752	-0,0333	0,0241
	(0,0146)
Leverage	-0,1268	0,429	-0,4410	0,1873
	(0,1603)
DefaultSpread	-0,5357	0,540	-2,2484	1,1771
	(0,8739)
TermSpread	-0,3682	0,369	-1,1708	0,4344
	(0,4095)
RiskFreeRate	-0,0690	0,973	-4,0459	3,9079
	(2,0291)
VIX	0,0066	0,000	0,0054	0,0078
	(0,0006)
RMSE: 0,32

Further, the regression indicated significant positive relationship between the lagged return and the skewness. The logic explaining such connection may be the following: when the returns are high, this decreases risk of the security and, thus, shifts the returns distribution in the more positive section. Furthermore, the abnormal returns earned by the security may attract the investors, which, in turn, increases the price of the security and leads to further positive returns. Therefore, positive past returns may shift the whole returns distribution in the positive direction, making the skewness less negative. This logic is similar to the work of Amaya et. al. (2015).

Secondly, we observe the negative influence of liquidity on the skew. Higher bid-ask spread, a consequence of decreased liquidity, increases the skewness. This result is consistent with the model of Hong and Stein (2003). Such a result is explainable: lower liquidity may be a consequence of lower trading activity, which happens if the security becomes less attractive to investors and the traders sale the stock. The sale of the security decreases its price, possibly below the fair value, and the positive returns are required in the future to return the value of the company to the fundamental value. Therefore, the distribution become more positively skewed.

Contrary to Dennis and Mayhew (2002), we do not find any significant relationship between the skewness and the market risk, measured by beta. Our different result may be explained by the different specification of the regression the authors use: they include the skewness of the S&P 500 index returns as one of the explanatory variables.

We also do not observe any relationship between the leverage ratio and the skew. This contradicts the results of Bakshi, Kapadia, Madan (2000). However, their results were ambiguous: theory implied a different relationship from the one which was observed empirically.

We document a positive relationship between the skew and market volatility, measured by VIX. This result is the most puzzling: usually the periods of high volatility correspond to lower returns, i.e the returns distribution is negatively skewed. This logic was confirmed by the results of Dennis and Mayhew (2002).

Other variables, which we used as indicators of the business cycle, do not seem to have any significant influence on the risk-neutral skewness. This result is similar to the one we obtained for volatility and for kurtosis (see the Table 6 below).

Table 6 presents the results of the DCCE model for kurtosis. Similarly to the models for the risk-neutral volatility and skewness, there is significant positive coefficient for the lagged kurtosis present, indicating high autocorrelation in the kurtosis series.

Consistent with Conrad, Dittmar and Ghysels (2013), Christoffersen, Chang and Jacobs (2013) and Amaya et. al. (2015) we find positive relationship between the returns and kurtosis. The explanation is as follows: positive kurtosis indicates higher “tail risk” of the security, i. e. shows higher probability of a large decrease in the price of the security. Such “tail risk” is compensated by higher return.

Table 6. Dynamic Common Correlated Effects model for Kurtosis. Kurtosis (-1), Return (-1), BidAskSpread (-1) and HistoricBeta (-1) refer to the kurtosis, return, bid-ask spread and beta variables lagged one period. Leverage, DefaultSpread, TermSpread, RiskFreeRate and VIX refer to the leverage ratio, default spread, term spread risk-free rate and Vix variables. The resulting coefficients, standard errors, p-values and 95% confidence intervals are given for these variables in the table. Coefficients of the significant coefficients at 5% level are highlight in bold

	Coefficient (Standart Error)	P-Value	95% Confidence Interval
Kurtosis (-1)	0,5229	0,000	0,4873	0,5584
	(0,0181)
Return (-1)	0,2849	0,002	0,1085	0,4613
	(0,0900)
BidAskSpread (-1)	-0,2721	0,021	-0,5033	-0,0410
	(0,1179)
HistoricBeta (-1)	-0,0357	0,295	-0,1024	0,0310
	(0,0340)
Leverage	-0,2863	0,307	-0,8358	0,2632
	(0,2804)
DefaultSpread	-0,3646	0,890	-5,5078	4,7787
	(2,6242)
TermSpread	-0,3499	0,676	-1,9935	1,2936
	(0,8386)
RiskFreeRate	-4,7783	0,350	-14,7924	5,2358
	(5,1093)
VIX	0,0041	0,001	0,0016	0,0065
	(0,0012)
RMSE: 0,85

There is a significant negative coefficient at the bid-ask spread variable. It indicates a positive relationship between the liquidity and the kurtosis. When the liquidity is low, the changes in the price of the security is usually small. Moreover, periods of stable, non-changing prices are usual for the illiquid securities, and the returns within these periods are zero. This provides an explanation for the lower kurtosis for less liquid securities: the presence of the periods with low returns in the absolute value decreases the probability of returns with high magnitude, leading to a decrease in kurtosis.

Finally, the last significant variable in the model for kurtosis is the VIX index price. We indicate a positive relationship between the variables. The is a quite straightforward explanation for this effect: if the volatility in the market is high, the probability of returns with high magnitude increases, increasing the kurtosis. In addition, there are common factors which increase both the market volatility and the probability of tail return: take, for example, the financial crises, when the volatility is high and many companies go bankrupt, resulting in large negative stock returns, which also increase kurtosis.



Conclusion

In this paper we investigate the relationship between the risk-neutral moments of the distribution of stock returns and the set of explanatory factors. The explanatory variables include the daily returns, bid-ask spread, beta, leverage ratio, default spread, term spread, risk-free rate and the price of the CBOE VIX Index, used as a proxy for the market volatility. We also employ lagged values of the moments as explanatory variables in the corresponding regressions. We calculate the option-implied moments using the methodology suggested by Bakshi, Kapadia and Madan (2000).

We use Chudik and Pesaran (2015) Dynamic Common Correlated Effects estimation methodology. This method allows consistent estimation of the dynamic specifications of models which are subject to heterogeneous slopes and possible cross-sectional dependence. We document the presence of these effects in our data, therefore, the usage of DCCE approach is fully justified.

Our results indicate the presence of high positive autocorrelation in the RNM's series. We observe significant positive coefficients at the lagged moments in all three regressions. Same feature of the distributional moments of stock returns were previously documented in the existing literature, this effect for the volatility is known as “volatility feedback effect”.

Consistent with the previous research on this subject, we find negative relationship between volatility and returns. This connection may be explained by the “volatility feedback effect” documented by Campbell and Hentschel (1992) or it can be understood as the negative price of volatility, as explained by Ang et. al. (2006). We also observe positive relationship between the skewness and returns and similar relationship between kurtosis and returns. Such findings are consistent with the existing literature.

We observe negative relationship between the volatility and liquidity, measured by the bid-ask spread. This is a results we expected to obtain. The connection between the skewness and liquidity is also negative. This is consistent with Hong and Stein (2003). Finally, we document a positive relationship between liquidity and kurtosis.

The last significant explanatory variable for the RNM's is the VIX. We found a positive relationship between the risk-neutral volatility and the market volatility, which is an expected result. The influence of the VIX on the skewness is also positive, contrary to Dennis and Mayhew (2000), which is a puzzling result. We also find the relationship between the market volatility and the risk-neutral kurtosis.



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