Return predictability of variance differences: A fractionally cointegrated approach

Abstract

This paper examines the fractional cointegration between downside (upside) components of realized and implied variances. A positive association is found between the strength of their cofractional relation and the return predictability of their differences. That association is established via the common longmemory component of the variances that are fractionally cointegrated, which represents the volatility-of-volatility factor that determines the variance premium. Our results indicate that market fears play a critical role not only in driving the long-run equilibrium relationship between implied-realized variances but also in understanding the return predictability. A simulation study further verifies these claims. K E Y W O R D S fractional cointegration, return predictability, variance risk premium J E L C L A S S I F I C A T I O N C14; C32; C51 | INTRODUCTIONNumerous empirical studies suggest that, unlike the variance, the variance risk premium (VRP) carries nontrivial predictive power for aggregate stock market returns over quarterly to annual horizons, where the degree of return predictability is greater than that afforded by more conventional predictors, see, Bollerslev, Tauchen, and Zhou (2009), Drechsler and Yaron (2010), Bollerslev, Marrone, Xu, and Zhou (2014), and Bali and Zhou (2016), among others. Those studies also provide strong evidence that much of the predictability implicit in the VRP may be attributed to its close relation with macroeconomic uncertainty and aggregate risk aversion.The VRP is formally defined as the wedge between option-implied and realized variances. It is measured as the difference between (the square of) the CBOE VIX index and the statistical expectation of the future return variation. In Bollerslev et al. (2009), Drechsler andYaron (2010) and Bollerslev, Sizova, and Tauchen (2012), the return predictability of the VRP is investigated using a self-contained general equilibrium model. This is in the spirit of the long-run risks (LRR) model pioneered by Bansal and Yaron (2004). Specifically, Bollerslev et al. (2009) andBollerslev et al. (2012) extend the LRR model by allowing the time-varying volatility-of-volatility (vol-of-vol) within the economy to be generated by its own stochastic process. They further show that the difference between the risk-neutral and the objective expectations of return variation isolates the vol-of-vol factor, which then serves as the sole source of the true VRP. The VRP, therefore, displays good predictive power for future returns in cases where the vol-of-vol plays a more dominant role in determining variation in the equity premium. A direct link between the VRP and the vol-of-vol is also demonstrated in a purely probabilistic model introduced by Barndorff-Nielsen and Veraart (2013). U D . The SJ and IVA are, respectively, measured as the difference between RV U and RV D and the difference between IV U and IV D . Using a semiparametric approach, we demonstrate the existence of a fractionally cointegrating relation in each pair of the semivariances...

Testing for Parameter Instability and Structural Change in Persistent Predictive Regressions

“…In particular, Bandi & Perron (2006), Christensen & Nielsen (2006), Nielsen (2007), Kellard, Dunis & Sarantis (2010) and Nielsen & Frederiksen (2011) provide solid support for these hypotheses. In addition, Bollerslev, Sizova & Tauchen (2011) and Bollerslev, Osterrieder, Sizova & Tauchen (2013), Osterrieder, Ventosa-Santaularia & Vera-Valdes (2019, Li, Izzeldin & Yao (2020) confirm the findings of fractional cointegration and proceed to exploit the equilibrium relation between IV and RV to estimate (components of) the volatility risk premium and forecast asset returns.…”

“…Specifically, most studies measure the volatility risk premium as VRP t = IV t −RV t . Other studies estimate the corresponding cointegrating relation between IV t and RV t to more accurately recover the "long-run" component of the volatility risk premium; see, e.g., ), Bollerslev et al (2013), Osterrieder et al (2019 and Li et al (2020). 14 Our rejection of stability of the cointegration models for RV, thus, suggest that such measures may convey different information across predictive regimes.…”

“…As detailed in the introduction, and going back to Andersen & Bollerslev (1997), many empirical studies have documented the presence of long memory persistence in RV. Moreover, there is an extensive literature finding: (i) IV is an unbiased predictor of future RV after correcting for various methodological issues, and it subsumes the information content in historical RV, e.g., Christensen & Prabhala (1998) and Jiang & Tian (2005); (ii) IV and RV are fractionally cointegrated and their long-run relation is unbiased, e.g., Christensen & Nielsen (2006) and Bandi & Perron (2006); (iii) the fractionally cointegrating equilibrium between IV and RV carry information about the variance risk premium and forecast asset returns, e.g., Bollerslev et al (2013) and Li et al (2020).…”

“…These studies take advantage of the differing persistence among the VIX t components and use NBLS-type estimators to identify the accuracy of the most persistent component. By the same logic, , Bollerslev et al (2013), Osterrieder et al (2019 and Li et al (2020) specify fractional cointegration models between IV and RV to estimate the volatility risk premium component associated with this long-run equilibrium and use it to forecast asset returns.…”

Testing for Parameter Instability and Structural Change in Persistent Predictive Regressions

“…In particular, Bandi & Perron (2006), Christensen & Nielsen (2006), Nielsen (2007), Kellard, Dunis & Sarantis (2010) and Nielsen & Frederiksen (2011) provide solid support for these hypotheses. In addition, Bollerslev, Sizova & Tauchen (2011) and Bollerslev, Osterrieder, Sizova & Tauchen (2013), Osterrieder, Ventosa-Santaularia & Vera-Valdes (2019), Li, Izzeldin & Yao (2020) confirm the findings of fractional cointegration and proceed to exploit the equilibrium relation between IV and RV to estimate (components of) the volatility risk premium and forecast asset returns.…”