key: cord-0939100-egmzupmz
authors: Shum, Wai Yan
title: Modelling conditional skewness: Heterogeneous beliefs, short sale restrictions and market declines
date: 2020-01-31
journal: The North American Journal of Economics and Finance
DOI: 10.1016/j.najef.2019.101070
sha: 5d19d9dfc3b5ba09243665cfaa3f652b06fff3ea
doc_id: 939100
cord_uid: egmzupmz

Abstract This paper tests the relationship among heterogeneous beliefs, short sale restrictions and time-varying conditional skewness under different market conditions. The results show that heterogeneous beliefs and short sale restrictions have negative impacts on conditional skewness during periods of market decline but have negative, positive or no impacts during periods of market growth. This evidence reconciles conflicting evidence in recent empirical studies on the relationship among heterogeneous beliefs, short sale restrictions and conditional skewness.

allows the other skewness factors such as the volatility feedback effect, the stochastic bubble effect and the Hong and Stein's "hidden bad news" effect to bring negativity to the overall level of market skewness. However, the theory does predict that the relative skewness conditional on the degree of price convexity remains positive even with the presence of the other skewness factors. Similar to the empirical studies in Charoenrook and Daouk (2004) , Chang et al. (2007) , Blau and Pinegar (2009) , and Hueng and McDonald (2005) mentioned above, empirical tests conducted by Xu (2007) find evidence against the Hong and Stein (2003) theory. In sum, they find that stock prices react more to good news than to bad news, short sale constrains increase skewness and heterogeneous beliefs increase skewness.

None of the above empirical analyses of skewness include the behavior of skewness in different market conditions. We argue that the relationship among short sale restrictions, heterogeneous beliefs, and conditional skewness behaves differently during periods of market decline and periods of market growth. We believe that when the market is on the rise and a sustained increase in the market price is expected, a sudden price fall will not trigger a bearish investor who believes in his/her own negative signal to enter the market since it is profitable for other rational buyers to buy now and sell later, not only because shares would gain in dividends but also because rational buyers expect that prices will rise more quickly than discounts (Diba & Grossman, 1988) .

To elaborate on our point, we take a closer look at the original Hong and Stein (2003) theory which assumes stock price depends on the terminal value of the dividend paid and then we modified their theory by allowing stock gains in dividends as well as its expected price when sold.

The original Hong and Stein (2003) theory assumes that a market with one stock to trade involves trader A, trader B, and a group of risk-neutral rational arbitrageurs. Traders A and B are subject to short sale constraints, but the arbitrageurs are not. The traders and the arbitrageurs will trade for four periods: time 0, 1, 2, and 3. Time 0 is the starting time of the trade, where no one receives any information. At time 1, trader B observes S B , and at time 2, trader A observes S A . The signals are equally informative for the arbitrageurs, and the stock will pay a terminal dividend of D at time 3 where = +

and is a normally distributed shock with zero mean and unit variance. The market clearing price of the stock at time 2 is the terminal value of the dividend paid,

. To obtain the equilibrium price, let us assume that trader A, trader B, the arbitrageurs, and the auctioneer are sitting in a room to "trade". The auctioneer starts by announcing a "high" trail price p, which is known to be higher than anybody's highest possible valuation. The initial "high" trail price creates negative excess demand. The auctioneer continues to lower the price when the excess demand remains negative. Conversely, once he/she reaches a point where the excess demand is positive, he/she raises the price. The process continues until the auctioneer reaches a price where the excess demand is zero and the market is cleared.

During the entire trading process, the arbitrageurs know that the average of signals S A and S B is the best estimate of the price. However, the arbitrageurs may not always see both signals. To further examine this, we look at the trading process at different time periods. First, at time 1, trader B obtains a bad signal S B , so their valuation of the stock is well below that of A. Because of short sale constraints, trader B will not enter the market, and the only trade will be between trader A and the arbitrageurs. The arbitrageurs can deduce that B's signal is well below A's but cannot determine the exact amount. Therefore, the market price at time 1 impounds A's prior information but cannot reflect B's time 1 signal. Next, at time 2, two possible scenarios arise. If A receives a good signal at time 2, then B's signal remains hidden. If, however, A gets a bad signal at time 2, then it is possible that B's previously hidden time 1 signal may be revealed at time 2. For example, if B steps in and is willing to buy after the price drops by 5%, the arbitrageurs learn that B's time 1 signal was not very negative. If, however, B does not step in even after the price drops by 20%, the arbitrageurs must conclude that B's time 1 signal was more negative than their prior expectation. In other words, trader B's sideline behavior in the face of A's selling is additional bad news for the arbitrageurs above and beyond the bad news implicit in A's selling.

The original model described above captures two elements of a market crash. First, the price movement may be totally out of proportion to the news arriving in the market (i.e., the signal received by A). Second, at time 2, when A gets a positive signal, it is revealed in the price, but the same is not true for the bad signal S B received at time 1. In contrast, when A gets a negative signal, not only is the bad signal S A received at time 2 revealed but the bad signal S B received at time 1 may also be revealed. This result implies that the largest price movements will be declines. Apart from capturing market crashes, the original model also captures the skewness of return distributions: hidden bad news at time 1 translates into overpricing and thus positive asymmetry, while a piece of related bad news at time 2 generates negative asymmetry with a larger magnitude.

The model can be modified if we assume the stock gains in dividends as well as its expected price when sold. The discounted future price remains positive if an agent can rationally plan to sell the overvalued asset to another agent at some finite future date (O'Connell & Zeldes, 1988; Tirole, 1982 

where is the terminal value of a rational bubble process that arises when the expected future price is growing faster than the discount factor. A few good examples of these rational bubbles are provided in Blanchard and Watson (1982) and Diba and Grossman (1988) . We continue assuming that traders A and B only believe in their signals, but we also assume that the arbitrageurs expect that they can sell the stock at time 3 to another agent with a markup b added onto the fundamental price of the stock. Therefore, the terminal value of the bubble process can be written as:

where is a random variable (or combination of random variables) generated by a stochastic process that satisfies = E ( ) 0. This process is similar to the process in Diba and Grossman (1988) . Under the assumptions of perfect information, rational expectation, risk neutrality, constant marginal utility, and constant discount rates, Diba and Grossman (1988) show that a negative rational bubble ends at a negative price. Given free disposal, a negative bubble is not possible since investors cannot rationally expect the value of the stock to become negative in a finite time. Therefore, bubbles can only be positive, i.e., 0. With the assumption of positive bubbles, trader B's signal S B remains hidden as long as b is high enough. When b is high, before the trial price p called out by the auctioneer ever reaches S B , it reaches the arbitrageurs' conditional estimate of the stock value, and the market is cleared. This is true for both time 1 and time 2. Therefore, when trader A gets a negative signal at time 2, trader B will not enter the market, since the arbitrageurs will clear the market with an overvalued price. Their negative signals will be revealed if and only if the bubble bursts, i.e., = 0, or when b is small enough.

In other words, the Hong and Stein (2003) theory of rational bubbles predicts negative asymmetries if and only if the bubble bursts and the market enters a downward adjustment period. On the other hand, when the bubble is forming, traders enter the market not only because shares would gain in dividends but also because rational buyers expect that prices will rise more quickly than discounts (Diba & Grossman, 1988) . This implies that rational traders can plan to sell an overvalued asset to another agent at some finite future date (O'Connell & Zeldes, 1988; Tirole, 1982) . It follows that a sudden price fall (or a technical downward adjustment) will not trigger traders with bad private signals about the terminal dividend to enter the market since it is profitable for other rational buyers to buy now and sell later to clear the market even at an overvalued price. As a result, the modified theory predicts less negative (i.e., more positive) asymmetries or no asymmetries during the bubble phase since the bubble process prevents traders with bad private signals to enter the market.

Therefore, the theory proposed by Hong and Stein (2003) , together with the bubble process, predicts more pronounced negative skewness during market declines than expansions, and thus, negative asymmetries are more positively related to the degree of heterogeneous beliefs and short sale restrictions during periods of general market declines.

The aim of this paper is therefore to examine the effects of short sale restrictions, heterogeneous beliefs, and market direction on time-varying conditional skewness. In particular, we would like to see how the relationship among heterogeneous beliefs, short sale restrictions, and conditional skewness changes under different market conditions. In the next section, we present a skew normal generalized autoregressive conditional heteroskedasticity (SN-GARCH) model that helps us test this idea.

The SN-GARCH model extends the GARCH model to allow for time-varying conditional skewness by assuming that conditional returns follow a skew normal distribution (Azzalini, 1985) . The class of skew normal distributions extends the normal distribution by including an additional shape parameter to regulate skewness and allows a continuous transition from normality to non-normality. There are other alternatives such as the inverse hyperbolic sine function proposed by Johnson (1949) , the stable Paretian distributions proposed by Mandelbrot (1963) , the skew-t distribution proposed by Hansen (1994) , the skewed generalized error distribution proposed by Hansen, McDonald, and Newey (2010) , and the centered parametrized skew-t distribution proposed by Azzalini (2013) . All these distributions allow us to model both skewness and kurtosis. Since our paper focuses on skewness but not kurtosis and the parametric tests can still perform well with data that are fat-tailed if the sample size is large enough, we have chosen the skew normal distribution since the skew normal distribution is mathematically tractable even under a multivariate setting, and no transformation due to parameter constraints imposed by the distribution is necessary (Arellano-Valle & Azzalini, 2008; Azzalini & Dalla_Valle, 1996) . Thus, the skew normal distribution is a natural choice for modeling skewness when the data under analysis exhibits unimodal empirical distribution with some skewness present. First, we look at the properties of the skew normal distribution, and then we present our SN-GARCH model.

To define the skew normal random variables, let X and Y~N(0,1). For any z, α ∈ R, the random variable

with the density function where < < z is referred to as a standard skew normal variable and where ϕ(·) and Φ(·) are the standard normal density and distribution function, respectively. The class of standard skew normal variables is denoted by SN(α), with α representing the skewness coefficient of the distribution. When α is zero, the skew normal distribution reduces to a simple standard normal distribution. The distribution is left-skewed if α is negative and right-skewed if α is positive. The moment generating function of the random variable Z above is given by

where δ = α/(1 + α 2 ) 1/2 . From the moment generating function, we can obtain the first two moments of the skew normal random variables

where b = (2/π) 1/2 . Note that the second moment of the skew normal variate is the same as the second moment of the normal variate; in fact, both have the same ( ) 1 2 distribution when squared. Indeed, all even moments of a skew normal variable coincide with the corresponding even moments of the normal. It follows from the first two moments presented above that the variance of the skew normal distribution is given by

In practice, one can apply a linear transformation to generate an entire family of skew normal distributions, as follows:

where the random variable V is a skew normal variate with the location parameter ξ and the scale parameter ω. We denote this distribution as SN(ξ,ω 2 ,α). When α = 0, we obtain the normal distribution, N(ξ,ω 2 ). The expected mean, variance, and skewness of the skew normal variable V are

and

We can thus apply the usual transformation to have a more general skew normal density function 

In this section, we introduce the two variations on the SN-GARCH model we used in the empirical section: the Glosten-Jagannathan-Runkle GARCH (GJR-GARCH) model (Glosten et al., 1993) and the quadratic GARCH (Q-GARCH) model (Sentana, 1995) . To model conditional skewness, we consider the specification of returns in which

where r t is the daily return at day t, μ is the conditional mean, and u t is the unexpected part of returns, which is generally referred to as "news" in the markets. According to the GJR-GARCH model, the conditional variance follows an asymmetric GARCH process as follows:

where I + = 1 if u t−1 > 0 and I -= 0 if u t−1 ≤ 0. We expect the coefficient on the asymmetric term to be negative so that bad news has a larger impact than good news on the conditional variance of the return.

In the Q-GARCH model, the variance equation is specified as follows:

Since arbitrage forces unexpected returns to have zero mean, we assume the innovation term ε t to be a sequence of independent, identically distributed random variates from the general skew normal distribution, SN(κ,τ 2 ,α). Similar to Liseo and Loperfido (2006) , the location parameter is

and the scale parameter is

such that the innovation term ε t has zero mean, as required by the arbitrage-free condition, and the scale parameter t 2 is retained to be the conditional variance of the model. Unlike Liseo and Loperfido (2006) , we allow the shape parameter α, and thus, the skewness in the Q-GARCH model to be time varying 

Note that when b 3 > 0, good news has a positive impact and bad news has a negative impact on the skewness of the returns. The use of ε t−1 instead of u t−1 in the skewness equation is to prevent conditional variance from having an effect on skewness. S t represents all the relevant skewness factors and can be defined as follows:

The model has three main skewness factors. DTO t , a detrended turnover, which is a proxy for heterogeneous beliefs; DSI t , a detrended short interest, which is a proxy for short sale restrictions. (For examples of these uses of proxies, see Miller (1977) , Epps and Epps (1976) , Figlewski (1981) , Jones and Lamont (2002) , and others.) The third skewness factor is the market direction indicator, t I , which is equal to one during periods of general market declines and zero otherwise. We expect the signs of the coefficients on "DTO t t I " and "DSI t t I " to be negative since the Hong and Stein (2003) theory, together with the bubble process, predicts that negative asymmetries are more positively related to the degree of heterogeneous beliefs and short sale restriction during periods of general market decline. The other variables control any interaction effects among the three variables. We refer to the model that uses the GJR-GARCH conditional variance as the SN-GJR-GARCH model, and we refer to the model that uses the Q-GARCH conditional variance as the SN-Q-GARCH model. An estimation has been performed using the "optimize" routine in Mata; the code is available from the author upon request.

The stocks analyzed in this paper consist of all the components of the Hang Seng Index (HSI). Our data, including short interest, trading volume, and total shares outstanding of individual stocks that constitute the HSI over the period Jan 4, 1999 to May 31, 2011, were purchased from the Hong Kong Stock Exchange website (www.hkex.com.hk). The daily return series for individual stocks is calculated as r i,t = ln(p i,t ) − ln(p i,t−1 ), where the daily closing prices for individual stocks, p i,t , are obtained from the Reuters EcoWin Pro database. Market return, turnover, and short interest are capitalization weighted and consist of all HSI components. Historical changes to the list of HSI constituents can be downloaded from the HSI website (www.hsi.com.hk). The turnover and short interest series are measured in the number of shares traded per day.

Following the methodology used by Chen et al. (2001) , the normal level of heterogeneous beliefs and short sale restrictions in this paper are approximated by a centered moving average of market turnover and short interest over a 120-day window where the centered moving average with the i-day window is defined as =

. Both the turnover and short interest series are 

North American Journal of Economics and Finance 51 (2020) 101070 detrended by first taking their natural log and then subtracting the moving average trends from the logged series. The degree of heterogeneous beliefs and short sale restrictions are the highest (lowest) when the level of turnover or short interest has the highest positive (negative) deviation from the normal levels. We use a centered instead of a backward moving average since we are interested in detrending and not forecasting. Fig. 1 plots the market return, whereas Figs. 2 and 3 plot the detrended turnover and detrended short interest series, respectively. We define four different market direction indicators, namely, the crisis indicator ℑ crisis, t , the yearly market direction indicator ℑ year, t , the quarterly market direction indicator ℑ qtr, t , and the weekly market direction indicator ℑ week,t . We first define the crisis indicator. There were two major financial crises in Hong Fig. 4 shows the HSI series along with the start and end dates of the crises. Since the purpose of the crisis indicator is to indicate general market declines, we pick the starting and ending dates by using ex post data.

The crisis indicator is somewhat arbitrary; therefore, we also look at other market direction indicators. The yearly market direction indicator ℑ year,t is equal to 1 at day t for the year y − 1 if the yearly price difference calculated as P y − P y−1 is negative, where the variable y records the number of years from the start of January 4, 1999 to May 31, 2011 and P y is the last observation of the daily price series in year y. The quarterly market direction indicator ℑ qtr,t is equal to 1 at day t for the quarter q − 1 if the quarterly price difference calculated as P q − P q−1 is negative, where the variable q records the number of quarters from the start of January 4, 1999 to May 31, 2011, and P q is the last observation of the daily price series at quarter q. The weekly market direction indicator ℑ week,t is equal to 1 at day t for the week w if the weekly price difference calculated as P w − P w−1 is negative, where the variable w records the number of weeks from the start of January 4, 1999 to May 31, 2011 and P w is the last observation of the daily price series at week w.

Figs. 5-7 show the yearly, quarterly, and weekly price difference indicators, respectively.

Most financial data are characterized by time-varying and predictable skewness. For example, Hansen (1994), Harvey and 

where r t the daily log return at time t, r is the sample mean and s is the sample standard deviation. The second measure is the quantile-based measure developed by Hinkley (1975) , which is defined as 

1 , for (0, 1] is the unconditional quantiles of the log continuously compounded n-period return = = + r r t n j n t j , 0 1 . We set = 0.75 as Yule and Kendall (1968) showed that when = RA r 0.75, ( ) t n , is very robust to outliers. From Fig. 8 , we can see that all the rolling unconditional skewness estimates show the variation in skewness month by month for the HSI index. From the unconditional estimates alone, however, we have no concrete evidence for instability of unconditional skewness over time. However, it has been shown that failure to detect unconditional skewness in the sample would not rule out the possibility of time-varying conditional skewness over the estimation period (Ghysels, Plazzi, & Valkanov, 2011) . The same can also be applied to the stability of conditional and unconditional skewness; even if unconditional skewness proves to be stable, we cannot rule out the possibility of the presence of unstable conditional skewness. To see whether there is any instability in conditional skewness, we apply the robust conditional quantile-based skewness estimate proposed by Ghysels et al. (2011 Ghysels et al. ( , 2016 

is a conditional variable representing daily conditioning information with lag of d days and the weight is . We have also followed the convention to set = 0.75 (Yule & Kendall, 1968 ) and denote the robust conditional quantile-based skewness estimate as SK75. Fig. 9 displays the robust conditional skewness measure and compares it with our crisis periods. We note that the robust conditional skewness measure is unstable over time and is more negative during the two crisis periods. This gives us the first piece of evidence for our modified theory, which predicts less negative (i.e., more positive) asymmetries during the bubble phase. Standard errors in parentheses *p < 0.10, **p < 0.05, ***p < 0.01.

North American Journal of Economics and Finance 51 (2020) 101070 Table 1 shows the estimation results and various specification tests for the SN-GJR-GARCH and SN-Q-GARCH models presented in the previous section.

First, we look at the variance equation in Table 1 . In the variance equations of both models, all the signs are consistent with our expectations. Coefficients on the asymmetric terms in the variance equations are negative and strongly significant at the 1% level for various specifications, implying that negative shocks tend to cause higher volatility than positive shocks. This result is consistent with the studies of Nelson (1990) , Engle and Ng (1993) , Glosten et al. (1993) , and other empirical asymmetric GARCH studies.

Second, in the skewness equations of both models in Table 1 , the coefficients are generally significant at the 5% level, indicating that conditional skewness is time-varying, which is consistent with the findings of Harvey and Siddique (1999) , Jondeau and Rockinger (2003) , Brooks et al. (2005) , Leon et al. (2005) , and Lanne and Saikkonen (2007) . The coefficients on t 1 in the skewness equations of both models are positive, implying that good news has a positive impact and bad news has a negative impact on the skewness of the returns. This shows that our asymmetric terms in the variance equations and skewness equations are consistent with each other. The asymmetric terms in the variance equations tell us that the distribution of the returns is negatively skewed for the estimation period, whereas the asymmetric terms in the skewness equations tell us that today's returns are more negatively skewed when the market receives bad news on the previous day.

Third, in the skewness factors equation of Table 1 , the coefficients on DTO t Crisis t , I and DSI t Crisis t , I are negative and strongly significant for various specifications in both models, indicating that both turnover and market short interest have a statistically significant power in predicting conditional skewness during the two crisis periods. Specifically, during the two crisis periods, negative conditional skewness is more pronounced when people disagree about the market more or the short sale constraints are increasingly binding. However, heterogeneous beliefs and short sale restrictions do not seem important to the determination of market conditional skewness during market growth since the coefficients on DTO t and DSI t in our baseline specifications, specification (1) for the SN-GJR-GARCH model and specification (4) for the SN-Q-GARCH model, respectively, are not statistically significant. In other words, our results indicate that the short sale constraints and differences in opinions contributed to negative asymmetries during the two crisis periods but did not do so during the periods of market growth. These results confirmed our prediction, since the Hong and Stein (2003) theory of rational bubbles predicts negative asymmetries if and only if the bubble bursts and the market enters a downward adjustment period. Note that in specifications (3) and (6), when we omit the crisis indicator and its related terms, heterogeneous Fourth, we decompose the skewness equation in our baseline specifications, i.e., specification (1) for the SN-GJR-GARCH model and specification (4) for the SN-Q-GARCH model in Table 1 , into two parts:

where part I is defined as

and part II is defined as

The first part of the skewness equation s t viewed conditional skewness as analogous to heteroskedasticity and modeled conditional skewness as a function of lagged skewness and shocks. Under the specification s t is stable over time and thus, the instability of the conditional skewness is brought to us by S t . Fig. 10 plot the conditional skewness decomposition for the SN-GJR-GARCH and SN-Q-GARCH model. It shows us how the skewness factors bring instability to conditional skewness. First, the skewness factors contribute more negative skewness during the crisis periods. Second, conditional skewness clearly does not mean reverting during the crisis periods.

Last, we compare the robust conditional skewness measure proposed by Ghysels et al. (2011 Ghysels et al. ( , 2016 in Fig. 9 with the conditional skewness estimated by the SN-GJR-GARCH and SN-Q-GARCH model (i.e., specification (1) and (4) in Table 1 respectively). The comparison is shown in Fig. 11 . From the figure, we can see that both measures predict less negative (i.e., more positive) asymmetries during the bubble phase and, thus, give us additional evidence to support our modified theory. Table 2 shows the estimation results of the SN-GJR-GARCH and SN-Q-GARCH models with different market direction indicators. As mentioned in the previous section, the crisis indicator is somewhat arbitrary; therefore, as a robustness check, we also test the yearly, quarterly, and weekly market direction indicators. As shown in Figs. 4 and 5, the yearly market indicator spots similar turning points in the market, whereas the quarterly market indicator spots more frequent changes in the market direction than the crisis indicator. The crisis and yearly indicators help us test how conditional skewness responds to the general market decline for a relatively long period of time. The quarterly market indicator, which can be seen in Fig. 6 , helps us test how conditional skewness responds to more frequent changes in the market direction, and the weekly market direction indicator, which can be seen in Fig. 7 , helps us test how conditional skewness responds to short-term changes in the market direction. The estimation results are presented in Table 2 . All coefficients on "DTO t I and "DSI t I" are negative and strongly significant when the yearly and quarterly market indicators are used. However, we find that market turnover has only mild statistical power in forecasting conditional skewness, while market short interest is not important in determining the skewness of market return even during periods of general market declines when the weekly market direction indicator is used. This result may imply that the degree of heterogeneous beliefs and short sale restrictions in the market has an effect on conditional skewness when the downward trend in the market persists for more than a week. It is also possible that the weekly market direction indicator that we have used to obtain the above results may be a noisy proxy for indicating real changes in the market direction.

Finally, in Table 3 , we test the effect of past returns on conditional skewness by including cumulative returns in our model. Both Hong and Stein (2003) and Hueng and McDonald (2005) found that past returns as far back as 36 months are negatively related to conditional skewness. Therefore, following Hueng and McDonald (2005) and Hong and Stein (2003) , we define cumulative returns as

, where R t is the simple return between times + t 1 andt . For t < 750, we set τ as the number Standard errors in parentheses. *p < 0.10, **p < 0.05, ***p < 0.01.

of maximum possible days that we can use. We drop the first five observations such that our shortest cumulation period is five days. Fig. 12 plots cumulative returns (RET t ) against time. For each specification, we estimate the SN-GJR-GARCH and SN-Q-GARCH models with RET t as additional skewness factors measured in percentage. Table 3 presents the estimation results. We find that past returns have no predictive power on conditional skewness when skewness factors are included. However, when we omit the skewness factors, the coefficients on the cumulative return are negative and strongly significant. This result is similar to the results shown in Hong and Stein (2003) and Hueng and McDonald (2005) .

In this paper, we have analyzed the relationship among short sale restrictions, heterogeneous beliefs, and conditional skewness by using a skew normal generalized autoregressive conditional heteroskedasticity model. Unlike previous studies, our paper considers the relationship under different market conditions. We show that negative conditional skewness is more pronounced when people disagree more about the market or the short sale constraints are increasingly binding during the general market declines, but the Standard errors in parentheses *p < 0.10, **p < 0.05, ***p < 0.01.

North American Journal of Economics and Finance 51 (2020) 101070 effect is undetermined during periods of market growth. We demonstrated the importance of market conditions on conditional skewness and reconciled conflicting evidence in recent empirical studies on the relationship among heterogeneous beliefs, short sale restrictions, and conditional skewness by combining Hong and Stein (2003) theory with bubble processes. 

The centred parametrization for the multivariate skew-normal distribution

A class of distributions which includes the normal ones

The skew-normal & related families

The multivariate skew-normal distribution

Studies in stock price volatility changes. Proceedings of the 1976 business meeting of the business and economics statistics section

Bubbles, rational expectations and financial markets

Expected idiosyncratic skewness

Autoregressive conditional kurtosis

Dispersion and volatility in stock returns: An empirical investigation

Short-sales constraints and price discovery: Evidence from the Hong Kong market

Conditional skewness of aggregate market returns

Forecasting crashes: Trading volume, past returns, and volatility

The stochastic behavior of common stock variances: Value, leverage, and interest rate effects

A study of market-wide short-selling restrictions. SSRN Working Paper 687562 Social Science Research Network

Measuring and testing the impact of news on volatility

The stochastic dependence of security price changes and transaction volumes: Implications for the mixture-of-distributions hypothesis

The informational effects of restrictions on short sales: Some empirical evidence

Is the leverage effect a leverage effect? SSRN Working Paper 256109 Social Science Research Network

Conditional skewness of stock market returns in developed and emerging markets and its economic fundamentals, Swiss Finance Institute Research Paper Series 11-06

Why invest in emerging markets? The role of conditional return asymmetry

On the relation between the expected value and the volatility of the nominal excess return on stocks

Autoregressive conditional density estimation

Instrumental Variables Estimation with Flexible Distributions

Speculative investor behavior in a stock market with heterogeneous expectations

Autoregressive conditional skewness

On power transformations to symmetry

Differences of opinion, short-sales constraints, and market crashes

Forecasting asymmetries in aggregate stock market returns: Evidence from conditional skewness

Systems of frequency curves generated by methods of translation

Conditional volatility, skewness, and kurtosis: Existence, persistence, and comovements

Short-sale constraints and stock returns

Modeling conditional skewness in stock returns

Autoregressive conditional volatility, skewness and kurtosis. The Quarterly Review of Economics and Finance

Default Bayesian analysis of the skew-normal distribution

Risk, uncertainty, and divergence of opinion

Arch models as diffusion approximations

Rational Ponzi games

The variability of the market factor of the

Business cycle asymmetries in stock returns: Evidence from higher order moments and conditional densities

Risk, inflation, and the stock market

Business cycles, financial crises and stock volatility. Carnegie-Rochester conference series on public policy

Why does stock market volatility change over time?

Quadratic arch models. The Review of Economic Studies

On the possibility of speculation under rational expectations

Price convexity and skewness

An introduction to the theory of statistics

The author would like to thank Professor Peter Moffatt (University of East Anglia, UK) for his insightful suggestions. This research was financially supported by the University of East Anglia through the Economics Research and Teaching Studentship.

Supplementary data to this article can be found online at https://doi.org/10.1016/j.najef.2019.101070.