Modeling Conditional Skewness in Stock Returns

Abstract

In this paper, we propose a new GARCH-in-Mean (GARCH-M) model allowing for conditional skewness. The model is based on the so-called z distribution capable of modeling skewness and kurtosis of the size typically encountered in stock return series. The need to allow for skewness can also be readily tested. The model is consistent with the volatility feedback effect in that conditional skewness is dependent on conditional variance. Compared to previously presented GARCH models allowing for conditional skewness, the model is analytically tractable, parsimonious and facilitates straightforward interpretation.Our empirical results indicate the presence of conditional skewness in the monthly postwar US stock returns. Small positive news is also found to have a smaller impact on conditional variance than no news at all. Moreover, the symmetric GARCH-M model not allowing for conditional skewness is found to systematically overpredict conditional variance and average excess returns.     

Firm size and volatility analysis in the Spanish stock market

Using Spanish stock market data, this paper examines volatility spillovers between large and small firms and their impact on expected returns. By using a conditional capital asset pricing model (CAPM) with an asymmetric multivariate GARCH-M covariance structure, it is shown that there exist bidirectional volatility spillovers between both types of companies, especially after bad news. After estimating the model, a positive and significant price of risk is obtained. This result is consistent with the volatility feedback effect, one of the most popular explanations of the asymmetric volatility phenomenon, and explains why risk premiums are much more sensitive to negative return shocks coming from the whole market or other related markets.     

The empirical relationship between risk and return: evidence from the UK stock market

Previous studies reach no consensus on the relationship between risk and return using data from one market. We argue that the world market factor should not be ignored in assessing the risk-return relationship in a partially integrated market. Applying a bivariate generalized autoregressive conditional heteroscedasticity in mean (GARCH-M) model to the weekly stock index returns from the UK and the world market, we document a significant positive relationship between stock returns and the variance of returns in the UK stock market after controlling for the covariance of the UK and the world market return. In contrast, conventional univariate GARCH-M models typically fail to detect this relationship. Nonnested hypothesis tests supplemented with other commonly used model selection criteria unambiguously demonstrate that our bivariate GARCH-M model is more likely to be the true model for UK stock market returns than univariate GARCH-M models. Our results have implications for empirical assessments of the risk-return relationship, expected return estimation, and international diversification.     

Sensitivity of the bank stock returns distribution to changes in the level and volatility of interest rate: A GARCH-M model

The objective of this paper is to employ the generalized autoregressive conditionally heteroskedastic in the mean (GARCH-M) methodology to investigate the effect of interest rate and its volatility on the bank stock return generation process. This framework discards the restrictive assumptions of linearity, independence, and constant conditional variance in modeling bank stock returns. The model presented here allows for shifts in the volatility equation in response to the changes in monetary policy regime in 1979 and 1982 to be estimated. ARCH, GARCH, and volatility feed back effects are found to be significant. Interest rate and interest rate volatility are found to directly impact the first and the second moments of the bank stock returns distribution, respectively. The latter also affects the risk premia indirectly. The degree of persistence in shocks is substantial for all the three bank portfolios and sensitive to the nature of the bank portfolio and the prevailing monetary policy regime.     

Time-varying risk The case of the American computer industry

A bivariate GARCH-in-mean model for individual stock returns and the market portfolio is designed to model volatility and to test the conditional Capital Asset Pricing Model versus the conditional Residual Risk Model. We find that a univariate model of volatility for individual stock returns is misspecified. A joint modelling of the market return and the individual stock return shows that a major force driving the conditional variances of individual stocks is the history contained in the market return variance. We find that a conditional residual risk model, where the variance of the individual stock return is used to explain expected returns, is preferred to a conditional CAPM. We propose a partial ordering of securities according to their market risk using first and second order dominance criteria.     

Stock market volatility and equity returns: Evidence from a two-state Markov-switching model with regressors

This paper proposes a two-state Markov-switching model for stock market returns in which the state-dependent expected returns, their variance and associated regime-switching dynamics are allowed to respond to market information. More specifically, we apply this model to examine the explanatory and predictive power of price range and trading volume for return volatility. Our findings indicate that a negative relation between equity market returns and volatility prevails even after having controlled for the time-varying determinants of conditional volatility within each regime. We also find an asymmetry in the effect of price range on intra- and inter-regime return volatility. While price range has a stronger effect in the high volatility state, it appears to significantly affect only the transition probabilities when the stock market is in the low volatility state but not in the high volatility state. Finally, we provide evidence consistent with the ‘rebound’ model of asset returns proposed by Samuelson (1991), suggesting that long-horizon investors are expected to invest more in risky assets than short-horizon investors.     

Modeling Time-Varying Volatility and Expected Returns: Evidence from the GCC and MENA Regions

The aim of this study is to investigate empirically the underlying nexus of stock market returns and volatility in the Gulf Cooperation Council (GCC) countries and Middle East and North Africa (MENA) region by using the GARCH-M model. We find that volatility is time-varying in all countries, which indicates substantial variation in the degree of risk across time. However, we do not find empirical support that this time-varying volatility significantly explains expected returns, except in the case of Kuwait, United Arab Emirates, and the MENA region portfolio. Our findings show that stock return volatility is negatively correlated with stock returns in these three markets under the assumption of investor risk aversion. This lends some support to the hypothesis of a volatility-driven negative relationship in the literature. The policy implications of our results are discussed.     

Regime switching in the relationship between equity returns and short-term interest rates in the UK

This paper examines the relationship between UK equity returns and short-term interest rates using a two regime Markov-Switching EGARCH model. The results suggest one high-return, low variance regime within which the conditional variance of equity returns responds persistently but symmetrically to equity return innovations. In the other, low-mean, high variance, regime equity volatility responds asymmetrically and without persistence to shocks to equity returns. There is evidence of a regime dependent relationship between shorter maturity interest rate differentials and equity return volatility. Furthermore, there is evidence that events in the money markets influence the probability of transition across regimes.     

How Does Information Quality Affect Stock Returns?

Using a simple dynamic asset pricing model, this paper investigates the relationship between the precision of public information about economic growth and stock market returns. After fully characterizing expected returns and conditional volatility, I show that (i) higher precision of signals tends to increase the risk premium, (ii) when signals are imprecise the equity premium is bounded above independently of investors' risk aversion, (iii) return volatility is U-shaped with respect to investors' risk aversion, and (iv) the relationship between conditional expected returns and conditional variance is ambiguous.     

The cross-sectional relation between conditional heteroskedasticity,the implied volatility smile,and the variance risk premium

This paper estimates how the shape of the implied volatility smile and the size of the variance risk premium relate to parameters of GARCH-type time-series models measuring how conditional volatility responds to return shocks. Markets in which return shocks lead to large increases in conditional volatility tend to have larger variance risk premia than markets in which the impact on conditional volatility is slight. Markets in which negative (positive) return shocks lead to larger increases in future volatility than positive (negative) return shocks tend to have downward (upward) sloping implied volatility smiles. Also, differences in how volatility responds to return shocks as measured by GARCH-type models explain much, but not all, of the variations in excess kurtosis and multi-period skewness across different markets.     

ON THE RATIONALITY OF COMMON STOCK RETURN VOLATILITY

This paper derives the relationship between the population unconditional variance of common stock returns and the variance of expected returns conditional on a well-specified information set. As a consequence, a lower bound is obtained for the variance of common stock returns. The sample counterpart of this bound is then empirically tested against the sample variance of returns. The paper's main conclusion can be stated as follows: the observed volatility of real (inflation-adjusted) common stock returns is not “irrationally” large. The paper admits of this conclusion because the point estimate of the lower-bound variance derived in this model is actually larger than the point estimate of common stock return volatility. However, since these point estimates are found to have a statistically insignificant difference, equality of the two variances cannot be ruled out. Hence, “rationality” of common stock returns—as implied by a utility-based valuation conditional on a specified information set—cannot be rejected.     

MEAN AND VOLATILITY SPILLOVERS ACROSS MAJOR NATIONAL STOCK MARKETS: FURTHER EMPIRICAL EVIDENCE

This paper provides additional insight into the nature and degree of interdependence of stock markets of the United States, Japan, the United Kingdom, Canada, and Germany, and it reports the extent to which volatility in these markets influences expected returns. The analysis uses the multivariate GARCH-M model. Although they are considered weak, statistically significant mean spillovers radiate from stock markets of the U.S. to the U.K., Canada, and Germany, and then from the stock markets of Japan to Germany. No relation is found between conditional market volatility and expected returns. Strong time-varying conditional volatility exists in the return series of all markets. The own-volatility spillovers in the U.K. and Canadian markets are insignificant, supporting the view that conditional volatility of returns in these markets is “imported” from abroad, specifically from the U.S. Significant volatility spillovers radiate from the U.S. stock market to all four stock markets, from the U.K. stock market to the Canadian stock market, and from the German stock market to the Japanese stock market. The results are robust and no changes occur in the correlation structure of returns over time.     

Asymmetric risk premium in value and growth stocks

There are two competing explanations for the existence of a value premium, a rational market risk explanation, whereby value stocks are inherently more risky than growth stocks, and a market over-reaction hypothesis, where agents overstate future returns on growth stock. Using asymmetric GARCH-M models this paper tests the predictions of the two hypotheses. Specifically, examining whether returns exhibit a positive (negative) risk premium resulting from a negative (positive) shock and the relative size of any premium. The results of the paper suggest that following a shock, volatility and expected future volatility are heightened, leading to a rise in required rates of return which depresses current prices. Further, these effects are heightened for value stock over growth stock and for negative shocks over positive shocks. Thus, in support of the rational risk interpretation, with a volatility feedback explanation for predictive volatility asymmetry.     

EXCESS RETURNS AND RISK AT THE LONG END OF THE TREASURY MARKET: AN EGARCH-M APPROACH

In this paper we model weekly excess returns of ten-year Treasury notes and long-term Treasury bonds from 1968 through 1993 using an exponential generalized autoregressive conditional heteroskedasticity in mean (EGARCH-M) approach. The results indicate the presence of conditional heteroskedasticity and a strong tendency for the ex-ante volatility of excess returns to increase more following negative excess return innovations compared with positive innovations of equal magnitude. In addition, increases in ex-ante volatility are associated in some subperiods with rising excess returns on longer-term instruments, although the slope of the yield curve and lagged excess returns generally remain significant predictors of excess returns.     

News Arrival, Jump Dynamics, and Volatility Components for Individual Stock Returns

This paper models components of the return distribution, which are assumed to be directed by a latent news process. The conditional variance of returns is a combination of jumps and smoothly changing components. A heterogeneous Poisson process with a time‐varying conditional intensity parameter governs the likelihood of jumps. Unlike typical jump models with stochastic volatility, previous realizations of both jump and normal innovations can feed back asymmetrically into expected volatility. This model improves forecasts of volatility, particularly after large changes in stock returns. We provide empirical evidence of the impact and feedback effects of jump versus normal return innovations, leverage effects, and the time‐series dynamics of jump clustering.     

Asymmetric Price and Volatility Adjustments in Emerging Asian Stock Markets

This paper tests the hypothesis that stock returns in emerging stock markets adjust asymmetrically to past information. The evidence suggests that both the conditional mean and the conditional variance respond asymmetrically to past information. In agreement with studies dealing with developed stock markets, the conditional variance is an asymmetrical function of past innovations, rising proportionately more during market declines. More importantly, the conditional mean is also an asymmetrical function of past returns. Specifically, positive past returns are more persistent than negative past returns of an equal magnitude. This behaviour is consistent with an asymmetric partial adjustment price model where news suggesting overpricing (negative returns) are incorporated faster into current prices than news suggesting underpricing (positive returns). Furthermore, the asymmetric adjustment of prices to past information could be partially responsible for the asymmetries in the conditional variance if the degree of adjustment and the level of volatility are positively related.     

Modeling the time varying volatility of housing returns: Further evidence from the U.S. metropolitan condominium markets

This study extends the literature on modeling the volatility of housing returns to the case of condominium returns for five major U.S. metropolitan areas (Boston, Chicago, Los Angeles, New York, and San Francisco). Through the estimation of ARMA models for the respective condominium returns, we find volatility clustering of the residuals. The results from an ARMA‐TGARCH‐M model reveal the absence of asymmetry in the conditional variance. Dummy variables associated with the housing market collapse unique to each metropolitan area were statistically insignificant in the conditional variance equation, but negative and statistically significant in the mean equation. Condominium markets in Los Angeles and San Francisco exhibit the greatest persistence to volatility shocks.     

Good News,Bad News,Volatility, and Betas

We investigate the conditional covariances of stock returns using bivariate exponential ARCH (EGARCH) models. These models allow market volatility, portfolio-specific volatility, and beta to respond asymmetrically to positive and negative market and portfolio returns, i.e., “leverage” effects. Using monthly data, we find strong evidence of conditional heteroskedasticity in both market and non-market components of returns, and weaker evidence of time-varying conditional betas. Surprisingly while leverage effects appear strong in the market component of volatility, they are absent in conditional betas and weak and/or inconsistent in nonmarket sources of risk.     

There is a risk-return trade-off after all

This paper studies the intertemporal relation between the conditional mean and the conditional variance of the aggregate stock market return. We introduce a new estimator that forecasts monthly variance with past daily squared returns, the mixed data sampling (or MIDAS) approach. Using MIDAS, we find a significantly positive relation between risk and return in the stock market. This finding is robust in subsamples, to asymmetric specifications of the variance process and to controlling for variables associated with the business cycle. We compare the MIDAS results with tests of the intertemporal capital asset pricing model based on alternative conditional variance specifications and explain the conflicting results in the literature. Finally, we offer new insights about the dynamics of conditional variance.