key: cord-0076720-493b74u6
authors: Theodossiou, Panayiotis; Ellina, Polina; Savva, Christos S.
title: Stochastic properties and pricing of bitcoin using a GJR-GARCH model with conditional skewness and kurtosis components
date: 2022-04-06
journal: Rev Quant Finan Acc
DOI: 10.1007/s11156-022-01055-x
sha: cc862c2e68b4f2acf69f0b4a4320066ee6410bd0
doc_id: 76720
cord_uid: 493b74u6

Using a flexible statistical framework that accounts for time-varying skewness and leptokurtosis, we examine the stochastic behavior of Bitcoin in comparison to five major currencies. The empirical findings reveal that the distribution of all series is leptokurtic. Once the effect of skewness-kurtosis is considered, the true price of risk is obtained, with implications on policymakers’ and investors’ strategies.

Cryptocurrencies have undergone an extraordinary development, especially if we consider that they are based on a decentralized system (Nakamoto 2008; Yermack 2015) . They are used for business and private transactions, as well as speculation and hedging. Unlike traditional currency transactions guaranteed by a third party such as a bank, cryptocurrency transactions are secured by their strong underlying cryptography. Cryptocurrencies are also popular for reasons other than prudent economic behavior. They are often used in illegal activities such as money laundering, tax evasion, terrorist organization financing, and ransom payments (Vigna 2019; Popper 2019; Justin and Demos 2021; Osipovich 2021) .

The conditional parameters of the ST-GJR GARCH model are estimated via the maximum likelihood method using a sample likelihood specification based on the flexible skewed generalized error distribution (SGED). The findings reveal higher-order moment dependencies and important skewness-kurtosis pricing effects. The understanding of the stochastic dynamics and pricing of Bitcoin is important to portfolio managers, forex traders, speculators, policymakers, as well as businesses, since Bitcoin can be accepted as a form of payment (Akhtaruzzaman et al. 2020) .

The rest of this paper is organized as follows: Sect. 2 presents the ST-GJR GARCH model, equations for the transition probabilities and risk measures for upside and downside markets, and a risk-neutral pricing equation for Bitcoin. Section 3 discusses the model's estimation and major empirical findings. The paper ends with a summary and conclusions in Sect. 4.

Volatility clustering and negative asymmetric volatility have been documented in financial, currency, and commodity prices, including Bitcoin (Corbet et al. 2018) . Volatility clustering refers to the tendency of large price shocks to set off large price shocks of smaller magnitude but random sign. On the other hand, negative asymmetric volatility refers to the tendency of large drops in the price of financial assets to be followed by larger and more persistent volatility shocks, compared to equivalent increases in the price of financial assets. In general, volatility clustering and asymmetric volatility trigger skewness and excess kurtosis in the unconditional distribution of financial returns (Theodossiou 2015) .

Under the GJR GARCH model, the conditional variance of returns in market i, based on the information set I t-1, , is modeled as a function of past innovations, their squared values, and past conditional variances. That is, where i,t = r i,t − i,t is an innovation or an error term. The indicator variable

takes the value of one for negative values of ε i,t and zero otherwise.

The coefficient a N,i measures the impact of past negative errors on current volatility. A positive value is indicative of a larger conditional variance during negative markets and vice versa (asymmetric volatility). The coefficients a i and β i measure the persistence of volatility shocks over time (volatility clustering). Note that for a N,i = 0, GJR GARCH yields the standard GARCH model of Bollerslev (1986) .

The conditional mode and conditional mean of returns are specified as linear functions of past returns and their contemporaneous conditional standard deviations. That is, and (1)

where m 0,i is an intercept, b i an autoregressive coefficient and c i and δ i,t are the in-mean and time-varying skewness coefficients, respectively. The sum ξ i,t = c i + δ i,t , interpreted as the total price of risk, measures the impact of risk on expected returns. The coefficients c i and δ i,t are respectively the pure and the skewness prices of risk (Theodossiou and Savva, 2016) . The analytical equation for the skewness price of risk is where and where λ i,t and k i,t are respectively the conditional asymmetry and conditional shape parameters of the distribution of returns r i,t and Γ(·) is the gamma function. These equations are obtained from Theodossiou and Savva (2021).

Theoretical models postulate a positive relationship between risk and expected returns, as per Markowitz (1952) and Sharpe (1964) . In the intertemporal CAPM of Merton (1973) , stochastic factors can influence equilibrium risk premia in financial markets. These factors trigger fluctuations in the risk-return tradeoff and, as such, are a source of skewness and kurtosis when returns are computed over discrete time intervals. Because investors are constantly hedging against such fluctuations, higher moments are likely to be priced. Ignoring the impact of higher moments on the distribution of equilibrium returns has been the source of contradictory and inconclusive empirical findings on the risk and expected return tradeoff of financial assets (Theodossiou and Savva 2016) .

Using the analytical framework of Eq. (3), Savva and Theodossiou (2018) investigated the risk and expected returns relationship across a wide range of international financial markets. They concluded that the contradictory findings in the literature regarding risk and return were the result of skewness. That is, negative skewness often switched the total price of risk from positive to negative.

Standardized deviations of returns from their conditional mode are used as proxies for market shocks; see also Feunou et al. (2013) for a similar definition. They are computed using the equation where m i,t and σ i,t are respectively the conditional mode, and conditional standard deviation of returns r i,t , in market i. The variable based on negative market shocks is used as a proxy for downside markets. On the other hand, based on positive market shocks is used as a proxy for upside markets. The conditional asymmetry parameter of the distribution of returns is specified as where is a time-varying asymmetry index. Note that for h i,t = 0, λ i,t = 0. Moreover, h i,t has a positive monotonic impact on λ i,t or the skewness price of risk δ i,t , i.e., i,t h i,t > 0 and

The coefficient γ N,i measures the marginal impact of downside market shocks on the asymmetry index h i,t , the asymmetry parameter λ i,t , and the skewness price of risk δ i,t . A positive value indicates that past downside market shocks, on average, have a positive impact on current values of the three parameters and vice versa. Similarly, the coefficient γ P,i measures the marginal impact of past upside market shocks on the three parameters. A positive value implies that past upside market shocks, on average, have a positive impact on the three parameters. The coefficient γ h,i measures the persistence of past market shocks. The intercept γ 0,i is an autonomous component.

The time-varying behavior of the conditional shape parameter k i,t is modeled like in Mazur and Pipień (2018) . That is,

where is a time-varying shape index and k L and k U are pre-set lower and upper bounds for the time-varying shape parameter k i,t . Similarly, the relationship between the g i,t and k i,t is monotonic and positive. Moreover, for For excessively volatile assets (highly leptokurtic), k L is set to a value of less than one. In this paper, the lower and upper bounds of the shape parameter are set to k L = 0.4 and k U = 1.6, respectively. This range can accommodate Pearson's moment coefficient of kurtosis KU values between 3.5 and 73.2. 3 Such a wide range of values can accommodate the empirical distributions of the series of returns investigated, as well as many other financial assets. For example, for k L = 0.4, k U = 1.6 and λ i,t = 0, the KU values associated with the index values of g i,t = − ∞, 0, ∞ are:

The Pearson's moment coefficient of kurtosis, being the fourth moment of standardized returns, z i,t = (r i,tμ i,t )/σ i,t , can be written equivalently as thus Therefore, KU measures the dispersion of volatility shocks. Larger KU values imply greater dispersion. Note that there is an inverse relationship between the shape parameter k and KU. Lower values of k correspond to fat tails and peakness around the mode.

The returns for each series are modeled using the centered SGED of Theodossiou (2015) . That is,

where μ i,t , σ i,t , δ i,t , θ i,t , λ i,t , and k i,t are as defined previously. Because of its flexibility, the SGED has been employed in the literature for the measurement of risk, pricing of options, and modeling of the time-series behavior of, amongst others, returns of stock indices, currencies, as well as oil and precious metals. We chose the SGED instead of the skewed generalized t, often used in empirical work, because it enables the development of pricing equations for Bitcoin. Note that the SGED gives for k i,t = 1 the skewed Laplace or double exponential distribution, and for k i,t = 2 the skewed normal distribution used in Feunou et al. (2013) and Roon and Karehnke (2017) .

The conditional probabilities for downside and upside markets are computed using the equations 4

and For investors with long positions in assets, downside returns are synonymous to downside risk, computed using the equation On the other hand, returns above their conditional mode can be viewed as measures of upside uncertainty, computed using the equation

The equations for the conditional Pearson's moment coefficients of skewness and kurtosis implied by the SGED are respectively

and where j = 1, 2, 3, 4 and t = 1, 2, …, T.

The expected price of asset i at time t is where and z i,t is a standardized return for asset i. The pdf for z i,t is obtained from that of Eq. (11) by setting μ i,t = 0 and σ i,t = 1.

In the absence of arbitrage opportunities, prices will be martingale processes. Therefore, their expected value at time t discounted by the period's required rate of return.

where r f,i,t is the conditional risk-free rate of interest in the country of asset i and ρ i,t the conditional risk premium for asset i in period t-1 to t, gives

This section discusses the estimation of the model and presents the empirical findings.

The data, obtained from DATASTREAM, includes daily US dollar prices for Bitcoin (BTC), the euro (EUR), the Japanese yen (JPY), the Canadian dollar (CAN), the British pound (GBP) and the Chinese yuan (RMB). It covers the period January 3, 2012 to September 17, 2021. Continuously compounded daily returns are computed using the equation where X i,t is the US dollar price of asset i at time t and i = BTC, EUR, JPY, CAN, GBP, RMB. To simplify the discussion of the results, henceforth, the subscript i is dropped from the coefficients and time-varying parameters. Table 1 reports the mean, standard deviation, minimum, median, maximum, Pearson's moment coefficients of skewness and kurtosis and the Bera-Jarque test statistic for normality of returns for each of the series. BTC's mean return is positive and large, reflecting its steep upward price trend over time. The mean returns for the other series are negative and close to zero. The standard deviation of returns for BTC is 5.3697 and is between ten to twenty-five times larger than those of the other series.

The Pearson's moment coefficients of skewness and kurtosis are respectively computed using the equations SK = m 3 m 3∕2 2

the j th moment around the sample mean and j = 1, 2, 3, 4. Negative skewness is present in the return series of BTC, GBP, and RMB. Low positive skewness is present in the return series of the remaining three currencies. Leptokurtosis is present in all six-return series. It is noted that the KU value for BTC of 22.701 is between three to four times larger than those of EUR, JPY, and CAN. The KU value for GBP of 24.3391 is quite large because of a few outliers in the data. A large KU value of 13.2278 is also present in RMB returns. The Bera-Jarque test statistics reject the null hypothesis of normality in all cases, indicating the presence of asymmetry and/or leptokurtosis in the data. It is noted that these results do not fully reflect the true extent of skewness and kurtosis in the data. These will be computed more accurately using the SGED estimated parameters for each return series.

Maximum likelihood estimates (MLE) for the coefficients of the unconditional and conditional mean, variance, asymmetry, and shape parameters of the distribution of returns are obtained via the Berndt et al. (1974) optimization procedure of the sample log-likelihood below:

where f y (θ | r t I t -1 ) is the SGED sample likelihood function for the returns of each series, given by Eq. (11), and θ is a column vector of coefficients for the conditional mean, variance, asymmetry, and shape parameters specified previously. The estimated values for the skewness price of risk, denoted by δ t , are obtained via the substitution of the MLE estimates for k t and λ t into Eq. (4). In the case of the unconditional distribution, θ includes the parameters μ, σ, λ and k. Robust standard errors for the MLE estimates denoted by ̃ are obtained from the equation As shown in Bollerslev and Wooldridge (1992) and Engle and Gonzalez-Rivera (1991) , the above estimators are more appropriate in the case of misspecified sample likelihood functions. Table 2 presents the MLE estimates of the parameters of the unconditional distributions of returns of the six series. The estimated means and standard deviations are similar to those in Table 1 . The asymmetry parameter λ of the distribution of returns is positive (right skewed) and statistically significant for BTC and statistically insignificant for the remaining series. The estimated values of the shape parameter k for BTC and RMB are lower than one, indicating excessive amounts of kurtosis. For the remaining series, it ranges from 1.0139 (JPY) to 1.2329 (CAN). These values deviate significantly from k = 2, which is the value of the shape parameter for the normal distribution.

The Pearson's moment coefficients of kurtosis KU for BTC and RMB, computed using Eq. (17), are respectively 10.9342 and 13.0491. For the other series, they range between 4.5992 and 5.8844. Thus, the resulting variances of volatility shocks for BTC and RMB, computed using the equation var(z 2 ) = KU -1, are about three times larger than those of EUR, JPY, CAN, and GBP. Note that for the normal distribution, var(z 2 ) = 2.

The resulting SGED probability curves of the returns of the six series, constructed using the estimated values of the parameters from Table 2 , along with their respective histograms

(non-parametric) and normal probability curves are presented in Fig. 1 . Interestingly, the SGED appears to provide an excellent fit to the empirical distribution of the data. This is attributed to its flexibility in accommodating asymmetry and fat-peaked tails. The normal probability curves, especially for BTC and RMB, deviate significantly from their respective histograms. The visual superiority of the SGED is statistically confirmed by the loglikelihood ratio test statistics for normality (LR-Normal), also presented in Table 2 . These statistics reject the null hypothesis of normality for the six series. Table 3 presents the test statistics for the presence of non-linearities, such as asymmetric volatility, volatility clustering, and higher-order moment dependencies often present in the distribution of financial returns. Panel A presents the test statistics for asymmetric volatility and volatility clustering. These statistics are computed using Equations (A1) and (A2) in the Appendix. With the exception of BTC and GBP, asymmetric volatility is absent in the remaining return series. The latter is attributed to the double-sided property of exchange rates (Theodossiou, 1994) . As expected, the statistics for past squared shocks are highly significant, confirming the presence of strong volatility clustering.

Panel B presents the test statistics for the impact of past shocks and their squared values on skewness shocks, measured by z 3 , and, indirectly, by the asymmetry parameter λ. These are computed using Equations (A3) and (A4) in the Appendix. The results suggest that past shocks have a significant impact on current skewness and thus the asymmetry parameter λ. The test statistics for past squared shocks are generally insignificant, suggesting a weak relationship with the asymmetry parameter. Estimates are based on the maximum likelihood estimation method. Parentheses include robust standard errors. SK = Ez 3 and KU = Ez 4 are the Pearson's moment coefficients of skewness and kurtosis computed using Eqs. (17) and (18), respectively. LogL is the sample log-likelihood value. LR-Normal = -2 (LogL -Log-Normal) is the sample log-likelihood ratio test statistic for normality of returns; it follows chi-square with 2 d.f. Its 5% and 1% critical values are respectively 5.99 and 9.21 ** and * indicate significance at the 1% and 5% level, respectively Panel C presents the test statistics for the impact of past shocks and their squared values on kurtosis shocks, measured by z 4 , and, indirectly, by the shape parameter of the distribution k. These are computed using Equations (A5) and (A6) in the Appendix. Interestingly, almost all statistics are significant, suggesting a strong impact of past shocks on the shape of the distribution.

In summary, the results confirm the presence of strong volatility, skewness, and kurtosis clustering. These preliminary findings motivate the formulations of the mean, variance, asymmetry, and shape parameters as time-varying. Table 4 presents the results of the estimated coefficients of the conditional equations for the variance, mean, asymmetry, and shape parameters of the six series, given respectively by Eqs. (1), (3), (9), and (10).

As for the estimated parameters of the conditional variance presented in Panel A of Table 4 , α N is positive and significant for CAN and GBP; positive and insignificant for EUR and JPY; and negative and insignificant for BTC and RMB. Although insignificant, the latter result is in line with previous findings regarding the presence of negative asymmetric volatility for cryptocurrencies (Cheikh et al. 2020 ). This can often be attributed to uninformed investors who do not want to lose out on the potentially rising price of Bitcoin (Baur and Dimpfl 2018), and consequently, on a profitable investment. This behavior is commonly known as 'fear of missing out', or FOMO. Furthermore, the findings of the asymmetric volatility on foreign exchange rates reflect the two-sided effect of these assets (Theodossiou, 1994) . The coefficient a is positive and significant for all series. The parameter β is positive and significant for all cases, indicating that volatility is persistent over time. Moreover, their sum is close to 0.98, indicating the presence of strong volatility clustering.

Regarding the conditional mode equation and specifically the risk-return tradeoff (Panel B), the pure price of risk c is positive and statistically significant for BTC and GBP; negative and statistically significant for RMB; and insignificant for the remaining series. With the exception of EUR, the estimated values of the autoregressive coefficient b are negative and statistically significant, indicating that, to some extent, returns are predictable by their own lag values.

Panel C reports the estimated coefficients of the conditional asymmetry index h t . The estimated values for γ Ν are negative and statistically significant for all series except EUR. These findings imply that past downside market shocks have a negative impact on the asymmetry index h t and asymmetry parameter λ t . The estimated values for γ P are positive and statistically significant for BTC only, suggesting that past upside market shocks have a positive impact on the asymmetry index and parameter. Furthermore, this coefficient is greater (in absolute values) for all cases compared to the downside asymmetry coefficient, except in the case of EUR. The persistence of past upside and downside market shocks (γ h ) is significant for EUR only, while the estimated parameter γ 0 is insignificant in all cases.

Panel D reports the estimated coefficients of the shape index g t . The constant d 0 is negative and significant for BTC and RMB; positive and significant for EUR; and positive but insignificant for the remaining series. The coefficient d N is positive and significant only for BTC and negative and significant for RMB, while d P is positive and significant only for BTC, indicating that past downside and upside market shocks impact The coefficients a N , a and β are respectively measures of volatility asymmetry, clustering, and persistence. m t = m 0 + b r t−1 + c t is the conditional mode (most probable value of the distribution) of returns and c the pure price of risk. The conditional mean of returns is t = m t + t t , where δ t is the time-varying skewness price of risk and ξ t = c + δ t is the total price of risk. Standardized excess to mode returns u t = r t − m t t , are used as proxies for market shocks,u − t = | | u t | | for u t < 0, and u − t = 0 for u t ≥ 0 is used as a proxy for downside markets and u + t = 0 for u t < 0, and u + t = u t for u t ≥ 0 for upside markets. h t and g t are respectively asymmetry and shape indices. k L = 0.4 and k U = 1.6 are pre-set lower and upper bounds for the shape parameter k. The parameters γ 0 , γ N , γ P , and γ h determine the size and time-varying behavior of the asymmetry parameter λ t . The coefficients d 0 , d N , d P , and d h control the shape of the distribution to the left and right of the conditional mode. Parentheses include standard errors of the estimates. SK and KU are the sample means of the time-varying Pearson's moment coefficients of skewness and kurtosis computed using Eqs. (17) and (18), respectively. The estimated parameters are statistically insignificant unless otherwise noted ** and * are statistically significant at 1% and 5%. The sample spans the period January 3, 2012 to September 17, 2021 the shape of the distribution of returns. The parameter d g is statistically significant for BTC, EUR, JPY, and CAN.

Furthermore, Panel E presents the simple arithmetic means of the conditional asymmetry and shape parameters with their standard errors. The means of the conditional asymmetry parameters, denoted by , are positive for BTC and JPY and negative for EUR, CAN, GBP, and RMB. Except for RMB, these means are statistically significant at the 1% level. The fact that these means are close to zero indicate that the time-varying distributions of returns revert over time to a symmetric shape. The average estimated coefficients of the skewness price of risk are positive and significant for BTC and JPY, and negative and significant for EUR, CAN, and GBP. Overall, these findings highlight the importance of skewness for the risk-return relationship, as outlined by León et al. (2005) , Theodossiou and Savva (2016) , and Savva and Theodossiou (2018) . If ignored, the impact of the risk on mean returns could be underestimated or overestimated. The simple arithmetic means of the total price of risk, denoted by , is positive and significant for BTC and GBP, and negative and significant for RMB. For the remaining series, these are statistically insignificant.

The arithmetic means of the shape parameters of the six series of returns, denoted by k, range between 0.7787 (RMB) and 1.4460 (CAN), suggesting that the distributions of returns are characterized by excessive volatility. RMB and BTC appear to be the most volatile of the six assets under consideration. The simple arithmetic means of the time-varying Pearson's moment coefficient of kurtosis, denoted by KU , indicate significant deviations of the empirical distributions of returns from the normal distribution, which equal to three. Table 5 presents various quantile values for the conditional means, conditional standard deviations, conditional asymmetry parameters, and conditional shape parameters of the six series. The conditional means of BTC range between −35.242% and 22.41%. The conditional means of the five currencies fall between −5.5% and 0.643%. The median daily returns are 0.271% for BTC, −0.006 for EUR, −0.011 for JPY, −0.007 for CAN, −0.005 for GBP, and −0.002 for RMB. Clearly, the return series for BTC reflect its extraordinary price growth over the sample period.

The conditional standard deviations (volatility) of daily returns range between 2.40% and 20.473% for BTC, and between 0.085% and 1.22% for the other five currencies. The median volatility values are 4.117% for BTC, 0.445% for EUR, 0.463% for JPY, 0.427% for CAN, 0.492% for GBP, and 0.145% for RMB. BTC prices are relatively more volatile (about ten times more). In this respect, BTC prices are extremely risky for investors, but offer higher compensation.

The conditional asymmetry parameters range between −0.329 and 0.567 for BTC, −0.147 and 0.048 for EUR, −0.524 and 0.338 for JPY, −0.412 and 0.157 for CAN, −0.317 and 0.208 for GBP, and −0.566 and 0.131 for RMB. Finally, the conditional shape parameter of all series ranges between 0.401 and 1.600. These results imply leptokurtic shapes for their empirical distributions. The median values of the shape parameters are 0.885 for BTC, 1.381 for EUR, 1.223 for JPY, 1.484 for CAN, 1.475 for GBP, and 0.801 for RMB. Table 6 reports the paired contemporaneous correlations of standardized returns of the six series. These correlations range between 0.0079 (BTC vs EUR) and 0.5483 (EUR vs GBP). Interestingly, the paired correlations between BTC and the five currencies are the lowest. The largest correlation is between BTC and CAN (0.047). These results imply that BTC can be used for hedging purposes by investors and businesses alike (Briere et al. 2015) . Table 7 presents the downside and upside mean probabilities computed using Eqs. (12) and (13). In the case of EUR, CAN, GBP, and RMB, the upside probability is lower than The quantile values are based on the series of conditional parameters obtained from the MLE estimation of the ST-GJR GARCH model. μ t is the conditional mean, σ t , the conditional standard deviation, λ t , the conditional asymmetry parameter, and k t , the conditional shape parameter Table 6 Correlation matrix of standardized residuals

The correlations are based on standardized residuals, computed using z t = (r tμ t ) / σ t ,, where μ t and σ t are respectively the conditional mean and conditional standard deviations of returns the downside, indicating that there is a higher probability of a negative, rather than a positive, shock to occur (i.e., a negative probability distribution). Overall, the main findings suggest that the exchange rates are well captured by GARCH models because of the significant first-order positive serial correlation and heteroscedasticity. The SGED estimated parameter, k, indicates that the distribution of all series is leptokurtic (especially for Bitcoin; see also Takaishi, 2018) . Moreover, the results suggest that ignoring skewness and kurtosis effects from the risk and return relationship may lead to misleading inferences.

To test the robustness of the results across time, the model is re-estimated using pre-COVID-19 data from the period January 1, 2012 to January 29, 2020 (1836 observations). The results are similar to a large extent with the findings of the full sample. They suggest that all series are leptokurtic with significant asymmetry and kurtosis parameters, indicating that the use of a flexible statistical framework that accounts for time-varying skewness and leptokurtosis is necessary, irrespective of the timespan of the data sample. A full set of results and analysis is available from the authors upon request.

This paper expands the literature on the investigation of and comparison between the stochastic properties of equilibrium returns of Bitcoin and five major currencies. The investigation extends beyond the conditional mean and conditional variance of the distribution of returns using a stochastic dynamic framework that is based on the flexible SGED.

Our empirical findings suggest that the higher-order dependencies in the series under study affect their risk-return tradeoff. Once the effect of skewness-kurtosis is considered, the true price of risk is obtained.

Furthermore, the dynamic behavior and pricing characteristics of Bitcoin and other currencies can be used in hedging and risk management by speculators and portfolio managers, as well as by regulators and policymakers. Specifically, the estimated equations for computing the conditional mean, variance, asymmetry, and kurtosis parameters provide a way to compute the required rate of return for the purpose of forecasting future prices.

More broadly, these findings have serious implications for theoretical and empirical research on asset pricing, as well as for practitioners interested in asset pricing in global markets. 

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Let z t = r t − , for t = 1, 2, …, T, be a series of standardized returns. Under the null hypothesis that z t 's are normal and identically and independently distributed (i.i.d.) across time, the statistics below, scaled to a zero mean, can be used to test for the presence of asymmetric volatility, volatility clustering, and other higher-order moment dependencies in the data. That is, where j = 1, 2, … and T is the sample size.Note that test statistic nt nd i = nd i √ var nd i , for i = 1, 2, …, 6, is asymptotically normal.

For any normal i.i.d. standardized return, the sth moment is given by.All odd moments are equal to zero, i.e., Ez s = 0 for s = 1, 3, 5, … The statistics given by Equations (A1) to (A6) can be written concisely as and where l ≡ Ez q t−j z r t = Ez q t−j Ez r t , i = 1, 2, …, 6, q = 1, 2, and r = 2, 3, 4. Note that for q = 2, r = 2,Ez 2 t−j z 2 t = 1 and q = 2, r = 4,Ez 2 t−j z 4 t = 3 . For other values of q and r, Ez q t−j z r t = 0. The first term of var nd i is Substitution of the values j and Ez s t into the above equation gives the variances of the six statistics.