A GMM Skewness and Kurtosis Ratio Test for Higher Moment Dependence

Abstract

This article extends the variance ratio test of Lo and MacKinlay (1988) to tests of skewness and kurtosis ratios using the generalized methods of moments. In particular, overlapping observations are used in which dependencies are explicitly modeled to make the tests more powerful and have better size properties. The proposed higher-order ratio tests can be useful in risk management where risk models are estimated using daily data but multiperiod forecasts of tail risks are required for the determination of risk capital. Application of the tests finds significant higher moment dependence in the U.S. stock market returns.

Direct versus iterated multiperiod Value‐at‐Risk forecasts

“…Finally, the volatility of IBM is also symmetric with the distribution of returns being leptokurtic but symmetric. As often observed in the related literature, 10-day returns have larger skewness (in absolute value) and stronger asymmetric effects while the kurtosis is smaller than those of 1-day returns; see, among others, Ghyselset al (2016), Le (2020), Neuberger (2012), andWong (2020) for the same empirical result and Alexander et al (2021), Berd et al (2007), Colacito and Engle (2010), Engle (2004Engle ( , 2011, Fama and French (2018), Meddahi and Renault (2004), and Wong and So (2003), who explain why the presence of asymmetric volatilities increase the asymmetry of multiperiod returns even in the case of one-period innovations being symmetric. The three series of returns considered have been chosen to represent common stylized facts of daily returns often observed in real time series.…”

“…However, as far as we are concerned, only Sun et al (2009) consider the effects on VaR estimation of using overlapping returns, concluding that VaR is underestimated in this case. It could be interesting to investigate whether explicitly modeling the dependencies of overlapping returns can help recreasing these biases; see, for example, Richardson and Smith (1991), Taylor and Fang (2018), and Wong (2020) for applications to testing based on overlapping returns with corrected dependencies and Giannopoulos (2003) for an application to multistep VaR forecasting. Recently, Hedegaard and Hodrick (2016) have proposed a GMM methodology to estimate GARCH-M models using all the high-frequency data while maintaining the low frequency forecasting period.…”

“…Kole et al (2017) also conclude empirically that the RiskMetrics approach based on scaling forecasts of the volatilities constructed from daily returns in combination with the empirical distribution of standardized returns is not bad. However, there is ample evidence about the use of SRoT generating 𝑉𝑎𝑅 (10) 𝑡 forecasts with large negative biases mainly for 𝛿 = 0.01; see Colacito and Engle (2010), Danielsson and Zigrand (2006), Lönnbark (2016), McNeil andFrey (2000), Saadi and Rahman (2008), Wang et al (2011), Wang et al (2018), Wong andSo (2003), andZhou et al (2016) for negative biases. On the contrary, few authors have also found positive biases of SRoT generating over-conservative multiperiod VaR estimates; see, for example, Müller et al (1990), Jorion (2001), andWong (2020).…”

“…However, there is ample evidence about the use of SRoT generating 𝑉𝑎𝑅 (10) 𝑡 forecasts with large negative biases mainly for 𝛿 = 0.01; see Colacito and Engle (2010), Danielsson and Zigrand (2006), Lönnbark (2016), McNeil andFrey (2000), Saadi and Rahman (2008), Wang et al (2011), Wang et al (2018), Wong andSo (2003), andZhou et al (2016) for negative biases. On the contrary, few authors have also found positive biases of SRoT generating over-conservative multiperiod VaR estimates; see, for example, Müller et al (1990), Jorion (2001), andWong (2020). Wang et al (2011) carries out an analysis to disentangle the sensitivity of SRoT to different characteristics of financial returns in the context of obtaining one-period VaR using historical simulation (HS).…”

“…In particular, in this survey, we consider two popular further distributions for standardized returns, namely, the Student with 𝜈 degrees of freedom and the Skewed-generalized-t (SGT) distribution defined by Theodosiou (1998). Note that the SGT distribution with parameter 𝑘 = 2 is the same as the distribution proposed by Hansen (1994) and implemented by Anatolyev and Petukhov (2016), Bali andTheodossiou (2008), Fenunou et al (2016), Le (2020), Theodossiou (2015), Wong andSo (2003), andZhou et al (2016), among others; see Aas and Haff (2006), Nadarajah (2020), andNadarajah et al (2017) for surveys of asymmetric Student distributions often assumed in the context of financial econometrics. Fan et al (2014) deal with the distribution of the QML estimator when the maximized likelihood is non-Gaussian.…”